THEOEETICAL ELEMENTS ELECTRICAL ENGINEERING EPS PROTEUS STEINMETZ, A.M., PttD. THIRD EDITION THOROUGHLY REVISED AND GREATLY CORRECTED The paper in this volume is brittle or the Inner margins are extremely narrow. We have bound or rebound the volume utilizing the best GENERAL BOOKBINDING Co,, CHESTER LAND, OHIO COPYRIGHT, 1909, BY THE MoGRAW-HILL BOOK COMPANY NEW YORK PREFACE. THE first part of the following volume originated from a series of University lectures which I once promised to deliver. This part can, to a certain extent, be considered as an intro- duction to my work on "Theory and Calculation of Alternating Current Phenomena/' leading up very gradually from the ordinary sine wave representation of the alternating current to the graphical representation by polar coordinates, from there to rectangular components of polar vectors, and ultimately to the symbolic representation by the complex quantity. The present work is, however, broader in its scope, in so far as it comprises the fundamental principles not only of alternating, but also of direct currents. The second part is a series of monographs of the more impor- tant electrical apparatus, alternating as well as direct current. It is, in a certain respect, supplementary to "Alternating Current Phenomena." While in the latter work I have presented the general principles of alternating current phenomena, in the present volume I intended to give a specific discussion of the particular features of individual apparatus. In consequence thereof, this part of the book is somewhat less theoretical, and more descriptive, my intention being to present the most impor- tant electrical apparatus in all their characteristic features as regard to external and internal structure, action under normal and abnormal conditions, individually and in connection with other apparatus, etc. I have restricted the work to those apparatus which experi- ence has shown as of practical importance, and give only those theories and methods which an extended experience in the design and operation has shown as of practical utility. I con- sider this the more desirable as, especially of late years, electri- iii iv PREFACE. cal literature has been haunted by so many theories (for instance of the induction machine) which are incorrect, or too compli- cated for use, or valueless in practical application. In the class last mentioned are most of the graphical methods, which, while they may give an approximate insight in the inter-relation of phenomena, fail entirely in engineering practice owing to the great difference in the magnitudes of the vectors in the same diagram, and to the synthetic method of graphical representa- tion, which generally require one to start with the quantity which the diagram is intended to determine. I originally intended to add a chapter on Rectifying Apparatus, as arc light machines a-nd alternating current rectifiers, but had to postpone this, due to the incomplete state of the theory of these apparatus. The same notation has been used as in the Third Edition of "Alternating Current Phenomena/ 7 that is, vector quantities denoted by dotted capitals. The same classification and nomen- clature have been used as given by the report of the Standardiz- ing Committee of the American Institute of Electrical Engineers. PREFACE TO THE THIRD EDITION. NEARLY eight years have elapsed since the appearance of the second edition, during which time the book has been reprinted without change, and a revision, therefore, became greatly desired. It was gratifying, however, to find that none of the contents of the former edition had to be dropped as superseded or anti- quated. However, very much new material had to be added. During these eight years the electrical industry has progressed at least as rapidly as in any previous period, and apparatus and phenomena which at the time of the second edition were of theoretical interest only, or of no interest at all, have now assumed great industrial importance, as for instance the single- phase commutator motor, the control of commutation by commutating poles, etc. Besides rewriting and enlarging numerous paragraphs through- out the text, a number of new sections and chapters have been added, on alternating-current railway motors, on the control of commutation by commutating poles ("interpoles"), on converter heating and output, on converters with variable ratio of conversion ("split-polo converters")* on three-wire generators and converters, short-circuit currents of alternators, stability and regulation of induction motors, induction generators, etc. In conformity with the arrangement used in my other books, the paragraphs of the text have been numbered for easier reference and convenience. When reading the book, or using it as text-book, it is recom- mended : After reading the first or general part of the present volume, to read through the first 17 chapters of " Theory and Calculation of Alternating Current Phenomena," omitting, however, the mathematical investigations as far as not absolutely required vi PREFACE TO THE THIRD EDITION. for the understanding of the text, and then to take up the study of the second part of the present volume, which deals with special apparatus. When reading this second part, it is recom- mended to parallel its study with the reading of the chapter of "Alternating Current Phenomena" which deals with the same subject in a different manner. In this way a clear insight into the nature and behavior of apparatus will be imparted. Where time is limited, a large part of the mathematical dis- cussion may be skipped and in that way a general review of the material gained. Great thanks are due to the technical staff of the McGraw- Hill Book Company, which has spared no effort to produce the third edition in as perfect and systematic a manner as possible, and to the numerous engineers who have greatly assisted me by pointing out typographical and other errors in the previous edition. CHARLES PROTEUS STEINMETZ. SCHENECTADY, September, 1909. CONTENTS. PART I. GENERAL SURVEY. PAGE 1. Magnetism and Electric Current. 1 2. Magnetism and E.M.F. 9 3. Generation of E.M.F. 12 4. Power and Effective Values. 16 5. Self-Inductance and Mutual Inductance. 21 6. Self-Inductance of Continuous-Current Circuits. 25 7. Inductance in Alternating-Current Circuits. 32 8. Power in Alternating-Current Circuits. 41 9. Polar Coordinates. 43 10. Hysteresis and Effective Resistance. 53 11. Capacity and Condensers. 59 12. Impedance of Transmission Lines. 62 13. Alternating-Current Transformer. 73 14. Rectangular Coordinates. 83 15. Load Characteristic of Transmission Line. ' 91 16. Phase Control of Transmission Lines, 96 17. Impedance and Admittance. 105 18. Equivalent Sine Waves. 114 PART II. SPECIAL APPARATUS. INTRODUCTION. 120 A. SYNCHRONOUS MACHINES. I. General. 125 II. Electromotive Forces. 127 III. Armature Reaction. 129 IV. Self-Inductance. 132 V. Synchronous Reactance. 136 VI. Characteristic Curves of Alternating-Current Generator. 138 viii CONTENTS. PAGE SYNCHRONOUS MACHINES (continued). VII. Synchronous Motor. 141 VIII. Characteristic Curves of Synchronous Motor. 143 IX. Magnetic Characteristic or Saturation Curve. 146 X. Efficiency and Losses. 149 XI. Unbalancing of Polyphase Synchronous Machines. 150 XII. Starting of Synchronous Motors. 151 XIII. Parallel Operation. 152 XIV. Division of Load in Parallel Operation. 154 XV. Fluctuating Cross-Currents in Parallel Operation. 155 XVI. High Frequency Cross-Currents between Synchronous Machines. 159 XVII. Short-Circuit Currents of Alternators. 160 B. DIRECT-CURRENT COMMUTATING MACHINES. I. General. 166 II. Armature Winding. 168 III. Generated Electromotive Forces. 178 IV. Distribution of Magnetic Flux. 179 V. Effect of Saturation on Magnetic Distribution. 183 VI. Effect of Commutating Poles. 185 VII. Effect of Slots on Magnetic Flux. 190 VIII. Armature Eeaction. 192 IX. Saturation Curves. 194 X. Compounding. 196 XI. Characteristic Curves. 197 XII. Efficiency and Losses. 198 XIII. Commutation. 198 XIV. Types of Commutating Machines. 206 A. Generators. Separately excited and Magneto, Shunt, Series, Compound. 208 B, Motors. Shunt, Series, Compound. 215 C. ALTERNATING CURRENT COMMUTATING MACHINES. I. General. 219 II. Power Factor. 220 III. Field Winding and Compensation Winding. 226 IV. Types of Varying Speed Single-Phase Commutator Motors. 230 V. Commutation. 236 VL Motor Characteristics. 250 VII. Efficiency and Losses. 259 VIII. Discussion of Motor Types. 260 IX. Other Commutator Motors. 266 CONTENTS. ix PAGE D. SYNCHRONOUS CONVERTERS. I. General. 270 II. Ratio of E.M.Fs. and of Currents. 271 III. Variation of the Ratio of E.M.Fs. 277 IV. Armature Current and Heating. 279 V. Armature Reaction. 292 VI. Reactive Currents and Compounding. 297 VII. Variable Ratio Converters (Split-Pole Converters). 299 VIII. Starting. 328 IX. Inverted Converters. 330 X. Frequency. 332 XL Double-Current Generators. 333 XII. Conclusion. 335 XIII. Direct-Current Converter. 337 XIV. Three-Wire Generator and Converter. 345 E. INDUCTION MACHINES. I. General. 352 II. Polyphase Induction Motor. 1. Introduction. 356 2. Calculation. 357 3. Load and Speed Curves. 363 4. Effect of Armature Resistance and Starting. 368 III. Single-phase Induction Motor. 1. Introduction. 372 2. Load and Speed Curves. 376 3. Starting Devices of Single-phase Motors. 380 4. Acceleration with Starting Device. 385 IV. Regulation and Stability. 1. Load and Stability. 387 2. Voltage Regulation and Output. 392 V. Induction Generator. 1. Introduction. 407 2. Constant Speed Induction or Asynchronous Gen- erator. 409 3. Power Factor of Induction Generator. 410 VI. Induction Booster. 417 VII. Phase Converter. 418 VIII. Frequency Converter or General Alternating-Current Transformer. 421 IX. Concatenation of Induction Motors. 423 X. Synchronizing Induction Motors. 428 XI. Self-exciting Induction Machines. 435 PAET L GENERAL THEORY. i. MAGNETISM AND ELECTRIC CURRENT. i. A magnet pole attracting (or repelling) another magnet pole of equal strength at unit distance with unit force* is called a unit magnet pole. The space surrounding a magnet pole is called a magnetic field of force , or magnetic field. The magnetic field at unit distance from a unit magnet pole is called a unit magnetic field, and is represented by one line of magnetic force (or shortly "one line") per sq. cm., and from a unit magnet pole thus issue a total of 4 K lines of magnetic force. The total number of lines of force issuing from a magnet pole is called its magnetic flux. The magnetic flux $ of a magnet pole of strength m is, <> = 4 *m. At the distance l r from a magnet pole of strength m, and therefore of flux $ = 4 nm, assuming a uniform distribution in all directions, the magnetic field has the intensity, $ m 47T/ 3 I 2 Tt 7iLf 6j since the $ lines issuing from the pole distribute over the area of a sphere of radius l r , that is the area 4 xlf. A magnetic field of intensity 3C exerts upon a magnet pole of strength m the force, Thus two magnet poles of strengths m l and m v and distance lr from each other, exert upon each other the force, * That is, at one centimeter distance with such force as to give to the mass of one gram the acceleration of one centimeter per second, 1 2 ELEMENTS OF ELECTRICAL ENGINEERING. 2. Electric currents produce magnetic fields also; that is, the space surrounding the conductor carrying an electric current is a magnetic field, which appears and disappears and varies with the current producing it, and is indeed an essential part of the phenomenon called an electric current. Thus an electric current represents a magnetomotive force (m.m.f.). The magnetic field of a straight conductor, whose return conductor is so far distant as not to affect the field, consists of lines of force surrounding the conductor in concentric circles. The intensity of this magnetic field is directly proportional to the current strength and inversely proportional to the dis- tance from the conductor. Since the lines of force of the magnetic field produced by an electric current return into themselves, the magnetic field is a magnetic circuit. Since an electric current, at least a steady current, can exist only in a closed circuit, electri- city flows in an electric circuit. The magnetic circuit produced by an electric current surrounds the electric circuit through which the electricity flows, and inversely. That is, the elec- tric circuit and the magnetic circuit are interlinked with each other. Unit current in an electric circuit is the current which produces in a magnetic circuit of unit length the field intensity 4 TT, that is, produces as many lines of force per square centimeter as issue from a unit magnet pole. In unit distance from an electric conductor carrying unit current, that is in a magnetic circuit of length 2 n, the field intensity is = 2, and in the distance 2 the field intensity 2 7T is unity; that is, unit current is the current which, in a straight conductor, whose return conductor is so far distant as not to affect its magnetic field, produces unit field intensity in distance 2 from the conductor. One tenth of unit current is the practical unit, called one ampere. 3. One ampere in an electric circuit or turn, that is, one ampere-turn, thus produces in a magnetic circuit of unit length the field intensity, 0.4 TT, and in a magnetic circuit of length MAGNETISM AND ELECTRIC CURRENT. 3 04" I the field intensity '--'-, and & ampere-turns produce In a 6 magnetic circuit of length I the field intensity : JC = ~~~~ lines of force per sq. cm. 6 regardless whether the 5 ampere-turns are due to ^ amperes F in a single turn, or one ampere in SF turns, or - amperes in n turns. n &, that is, the product of amperes and turns, is called magneto- motive force (m.m.f.). The m.m.f. per unit length of magnetic circuit, or ratio: v m.m.f. eft- == . length of magnetic circuit is called the magnetizing force. Hence, m.m.f. is expressed in ampere-turns; magnetizing force in ampere-turns per centimeter (or in practice frequently ampere-turns per inch), field intensity in lines of magnetic force per square centimeter. At the distance l r from the conductor of a loop or circuit of 2F ampere-turns, whose return conductor is so far distant as not to affect the field, assuming the m.m.f. = SF, since the length of the magnetic circuit = 2 nl Tj we obtain as the magnetizing force, and as the field intensity, 4. The magnetic field of an electric circuit consisting of two parallel conductors (or any number of conductors, in a poly- phase system), as the two wires of a transmission line, can be considered as the superposition of the separate fields of the conductors (consisting of concentric circles). Thus, if there are / amperes in a circuit consisting of two parallel conductors (conductor and return conductor), at the distance Z t from the 4 ELEMENTS OF ELECTRICAL ENGINEERING. first and 1 2 from the second conductor, the respective field intensities are, ^ 9 r _!irl_. j - y and 027 and the resultant field intensity, if T = angle between the direc- ions of the two fields, 3C - \/X 2 + 3C 3 2 + 2 CK^OCj cos T, = ~ vV + Z 2 2 +2^ 2 cosr. 6^2 In the plane of the conductors, where the two fields are in the same, or opposite direction, the resultant field intensity is,, where the plus sign applies to the space between, the minus sign the space outside of the conductors. The resultant field of a circuit of parallel conductors con- sists of excentric circles, interlinked with the conductors, and crowded together in the space between the conductors. The magnetic field in the interior of a spiral (solenoid, helix, coil) carrying an electric current, consists of straight lines. 5. If a conductor is coiled in a spiral of I centimenter axial length of spiral, and N turns, thus n = y turns per centimeter I length of spiral, and / = current, in amperes, in the conductor, the m.m.f. of the spiral is $ =IN, and the magnetizing force in the middle of the spiral, assuming the latter of very great length, thus the field intensity in the middle of the spiral or solenoid, X = 0.4 7i3C = 0.4 nnL MAGNETISM AND ELECTRIC CURRENT. 5 Strictly this is true only in the middle part of a spiral of such length that the m.m.f. consumed by the external or magnetic return circuit of the spiral is negligible compared with the m.m.f. consumed by the magnetic circuit in the interior of the spiral, or in an endless spiral, that is a spiral whose axis curves back into itself, as a spiral whose axis is curved in a circle. Magnetomotive force & applies to the total magnetic circuit, or part of the magnetic circuit. It is measured in ampere- turns. Magnetizing force <3C is the m.m.f. per unit length of mag- netic circuit. It is measured in ampere-turns per centi- meter. Field intensity 3 is the number of lines of force per square centimeter. If I = length of the magnetic circuit or a part of the magnetic circuit; fjn = 0.4 TT^C 3C = 0.4 7T = 1.257 3C 3C = 0.796 3G. 6. The preceding applies only to magnetic fields in air or other unmagnetic materials. If the medium in which the magnetic field is established is a " magnetic material/ 7 the number of lines of force per square centimeter is different and usually many times greater. (Slightly less in diamagnetic materials.) The ratio of the number of lines of force in a medium, to the number of lines of force which the same magnetizing force would produce in air (or rather in a vacuum), is called the permeability or magnetic conductivity /* of the medium. The number of lines of force per square centimeter in a mag- netic medium is called the magnetic induction (B. The number of lines of force produced by the same magnetizing force in air is called the field intensity 3C. 6 ELEMENTS OF ELECTRICAL ENGINEERING. In air, magnetic induction (B and field intensity X are equal. As a rule, the magnetizing force in a magnetic circuit is changed by the introduction of a magnetic material; due to the change of distribution of the magnetic flux. The permeability of air = 1 and is constant. The permeability of iron and other magnetic materials varies with the magnetizing force between a little above 1 and values as high as 6000 in soft iron. The magnetizing force 3C in a medium of permeability /JL produces the field intensity 3C = 0.4 7t3& and the magnetic induction (B = 0.4 ^/z3C. EXAMPLES. 7. (1.) A pull of 2 grams at 4 cm. radius is required to hold a horizontal bar magnet 12 cm. in length, pivoted at its center, in a position at right angles to the magnetic meridian. What is the intensity of the poles of the magnet, and the number of lines of magnetic force issuing from each pole, if the horizontal intensity of the terrestrial magnetic field X = 0.2, and the acceleration of gravity = 980? The distance between the poles of the bar magnet may be assumed as five-sixths of its length. Let m = intensity of magnet pole. l r = 5 is the radius on which the terrestrial magnetism acts. Thus 2 m3l r = 2 m = torque exerted by the terrestrial mag- netism. 2 grams weight = 2 X 980 = 1960 units of force. These at 4 cm. radius give the torque 4 X 1960 7840 g cm. Hence 2 m = 7840. m 3920 is the strength of each magnet pole and $ = 4 nm = 49,000, the number of lines of force issuing from each pole. 8. (2.) A conductor carrying 100 amperes runs in the direc- tion of the magnetic meridian. What position will a compass needle assume, when held below the conductor at a distance of 50 cm., if the intensity of the terrestrial magnetic field is 0.2? The intensity of the magnetic field of 100 amperes 50 cm. from the conductor, is 3C = ~^~- = 0.2 X ~ = 0.4, the direc- IT 50 MAGNETISM AND ELECTRIC CURRENT. 7 tion is at right angles to the conductor; that is at right angles to the terrestrial magnetic field. If r = angle between compass needle and the north pole of the magnetic meridian, I = length of needle, m = intensity of its magnet pole, the torque of the terrestrial magnetism is JCml sin T = 0.2 ml sin r, the torque of the current is ^ 1 0.2 Iml cos r A 7 3Cm cos r = = 0.4 mi cos r. L T In equilibrium, 0.2 wZ sin T = 0.4 #iZ cos r, or tan T = 2, r = 63.4. 9. (3.) What is the total magnetic flux per I = 1000 m. length, passing between the conductors of a long distance transmission, Fig. 1. Diagram of Transmission Line for Inductance Calculation. line carrying / amperes of current, if Id = 0.82 cm. is the diameter of the conductors (No. B. & S.), k = 45 cm. the spacing or distance between them? At distance l r from the center of one of the conductors (Fig. 1), the length of the magnetic circuit surrounding this conductor is 2 nlr, the m.m.f., / ampere turns; thus the magnetizing force 3C = ~- } and the field intensity 3C = 0.4 nK = -~ 9 and the 2 nlr Lr flux in the zone dl r is d4> = -^ -, and the total flux from the Lr surface of the conductor to the next conductor is, 0.2 Ildlr = IT 0.2 II ["log e lr 1 = 0.2 II bg e ~ - 8 ELEMENTS OF ELECTRICAL ENGINEERING. The same flux is produced by the return conductor in the same direction, thus the total flux passing between the trans- mission wires is, or per 1000 m. = 10 5 cm. length, on 2 $ =0.4 X 10 5 / log e ~ - 0.4 X 10 5 X 4.70 / = 0.188 X 10 6 /, U.Q-i or 0.188 / megalines or millions of lines per line of 1000 m. of which 0.094 / megalines surround each of the two con- ductors. 10. (4.) In an alternator each pole has to carry 6.4 millions of lines, or 6.4 megalines magnetic flux. How many ampere- turns per pole are required to produce this flux, if the magnetic circuit in the armature of laminated iron has the cross section of 930 sq. cm. and the length of 15 em., the air-gap between stationary field poles and revolving armature is 0.95 cm. in length and 1200 sq. cm. in section, the field-pole is 26.3 cm. in length and 1075 sq. cm. in section, and is of laminated iron, and the outside return circuit or yoke has a length per pole of 20 cm. and 2250 sq. cm. section, and is of cast iron? The magnetic densities are; (B 1 = 6880 in the armature, (B 2 = 5340 in the air-gap, (B 3 = 5950 in the field-pole, and (E 4 = 2850 in the yoke. The permeability of sheet iron is ^ = 2550 at ^ = 6880, /. 3 =2380 at & 3 =5950. The permeability of cast iron is // 4 =280 at (B 4 =2S50. Thus the field intensity = ~ is, 3e x = 2.7, X 2 = 5340, X 3 - 2.5, OC 4 = 10.2. The magnetizing force \^\ is, X 1 = 2.15, 5C 2 = 3C 3 =1.99, 3t 4 =8.13 ampere-turns per cm. Thus the m.m.f. (fr = 3CZ) is, SF X = 32, F 2 - 4040, CF 3 = 52, F 4 = 163, or the total m.m.f. per pole is SF = 3^+ CF 2 + ^3+ $F 4 = 4290 ampere-turns. The permeability fi of magnetic materials varies with the density = 25 megalines? During each revolution the copper cylinder cuts 25 megalines. It makes 50 rev. per sec. Thus it cuts 50 X 25 X 10 8 = 12.5 X 10 s Pig. 3. Unipolar Generator. lines of magnetic flux per second. Hence the generated e.m.f. is E = 12.5 volts. Such a machine is called a "unipolar/' or more properly a "nonpolar" or an " acyclic 77 generator. 14. (2.) The field spools of the 20-pole alternator in Section 1 } Example 4 ; are wound each with 616 turns of wire No. 7 (B. & S.), 0.106 sq. cm. in cross section and 160 cm. mean length of turn. The 20 spools are connected in series. How many amperes and how many volts are required for the excitation of this alternator field, if the resistivity of copper is 1.8 X 10"" 6 ohms per cm. 3 * Since 616 turns on each field spool are used, and 4280 ampere- 4280 turns required, the current is -~-r = 6.95 amperes. * Cm. 3 refers to a cube whose side is one centimeter, and should not be confused with cu. cm. 12 ELEMENTS OF ELECTRICAL ENGINEERING. The resistance of 20 spools of 616 turns of 160 cm. length, 0.106 sq. cm. section, and 1.8 X 10"" 6 resistivity is, 20 X 616 X 160 X 1.8 X IP"" 6 = ^ ^ 0.106 and the e.m.f. required 6.95 X 33.2 = 230 volts. 3. GEITORATION OF E.M.F, 15. A closed conductor, convolution or turn, revolving in a magnetic field, passes during each revolution through two positions of maximum inclosure of lines of magnetic force A in Fig. 4, and two postions of zero inclosure of lines of mag- netic force B in Fig. 4. Fig. 4. Generation of e.m.f. Thus it cuts during each revolution four times the lines of force inclosed in the position of maximum inclosure. If $ = the maximum number of lines of flux inclosed by the conductor, /= the frequency in revolutions per second or cycles, and n = number of convolutions or turns of the con- ductor, the lines of force cut per second by the conductor, and thus the average generated e.m.f. is, E = 4 fn$ absolute units, 10" 8 volts. If /is given in hundreds of cycles, <& in megalines, E = 4n$ volts. If a coil revolves with uniform velocity through a uniform magnetic field, the magnetism inclosed by the coil at any instant is, < COS r GENERATION OF E.M.F. 13 where $ = the maximum magnetism inclosed by the coil and r = angle between coil and its position of maximum inclosure of magnetism. $ The e.m.f. generated in the coil, p-^ which varies with the rate of cut- ting or change of <> cos T, is thus, E sin Q I in megalines, /in hundreds of cycles per second). This is t"he formula of the direct-current generator. EXAMPLES. 17. (1.) A circular wire coil of 200 turns and 40 cm. mean diameter is revolved around a vertical axis. What is the horizontal intensity of the magnetic field of the earth, if at a speed of 900 revolutions per minute the average e.m.f. generated in the coil is 0.028 volts? The mean area of the coil is - = 1255 sq. cm., thus the GENERATION OF E.M.F. 15 terrestrial flux inclosed is 1255 X, and at 900 revolutions per minute or 15 revolutions per second, this flux is cut 4X 15 = 60 times per second by each turn, or 200 X 60 = 12,000 times by the coil. Thus the total number of lines of magnetic force cut by the conductor per second is 12,000 X 1255 X == 0.151 X 10 s X, and the average generated e.m.f. is 0.151 5C volts. Since this is = 0.028 volts, X = 0.1S6. 18. (2.) In a 550-volt direct-current machine of 8 poles and drum armature, running at 500 rev. per min., the average volt- age per commutator segment shall not exceed 11, each armature coil shall contain one turn only, and the number of commutator segments per pole shall be divisible by 3, so as to use the machine as three-phase converter. What is the magnetic flux per field- pole? 550 volts at 11 volts per commutator segment gives. 50, or as next integer divisible by 3, n = 51 segments or turns per pole. 8 poles give 4 cycles per revolution, 500 revs, per min. gives 500/60 8.33 revs, per sec. Thus the frequency is, /= 4 X 8.33 = 33.3 cycles per sec. The generated e.m.f. is E = 550 volts, thus by the formula of direct-current generator, or, 550 = 4 X 0.333 X 51 *, = 8.1 megalines per pole. 19. (3.) What is the e.m.f. generated in a single turn of a 20- pole alternator running at 200 rev. per min., through a mag- netic field of 6.4 megalines per pole? The frequency is /= 2 ^ X = 33,3 cycles. 2t X oU e = .E sin r, E =2xfn$, *=6.4, =1, /= 0.333. Thus, $ = 2 ic X 0.333 X 6.4 = 13.4 volts maximum, or e = 13.4 sin 0. 16 ELEMENTS OF ELECTRICAL ENGINEERING 4. POWER AND EFFECTIVE VALUES. 20. The power of the continuous e.m.f. ; E producing con- tinuous current / is P = EL The e.m.f. consumed by resistance r is E l = Ir, thus the power consumed by resistance r is P = Pr. Either E l = E 9 then the total power in the circuit is con- sumed by the resistance, or E^ E } then only a part of the power is consumed by the resistance, the remainder by some counter e.m.f., E E v If an alternating current i = 7 sin 6 passes through a resist- ance r, the power consumed by the resistance is, fr = 7 2 r sin 2 6 = (1 - cos 2 6), 2i thus varies with twice the frequency of the current, between zero and 7 2 r. The average power consumed by resistance r is, since avg. (cos) = 0. Thus the alternating current i = J sin 9 consumes in a resist- ance r the same effect as a continuous current of intensity The value / = Jb is called the effective value of the alter- v2 nating current i = 7 sin 6; since it gives the same effect. JF Analogously E = t is the effective value of the alternating v2 e.m.f., e E sin #. Since E Q =2 nfn$, it follows that jE? = V2 Ttfn = 4.44 /n$ ; is the effective alternating e.m.f., generated in a coil of n turns rotating at a frequency of / (in hundreds of cycles per second) through a magnetic field of $ megalines of force. This is the formula of the alternating-current generator. POWER AND EFFECTIVE VALUES. 21. The formula of the direct-current generator, holds even if the e.m.fs. generated in the individual turns are not sine waves, since it is the average generated e.m.f. The formula of the alternating current generator, does not hold if the waves are not sine waves, since the ratios of average to maximum and of maximum to effective e.m.f. are changed. If the variation of magnetic flux is not sinusoidal, the effective generated alternating e.m.f. is, # = 7 V2 xfn$. 7 is called the form factor of the wave, and depends upon its shape, that is the distribution of the magnetic flux in the mag- netic field. Frequently form factor is defined as the ratio of the effec- tive to the average value. This definition is undesirable since it gives for the sine wave, which is always considered the standard wave, a value differing from one. EXAMPLES. 22. (1.) In a star connected 20-pole three-phase machine, revolving at 33.3 cycles or 200 rev. per min., the magnetic flux per pole is 6.4 megalines. The armature contains one slot per pole and phase, and each slot contains 36 conductors. All these conductors are connected in series. What is the effective e.m.f. per circuit, and what the effective e.m.f. between the terminals of the machine? Twenty slots of 36 conductors give 720 conductors, or 360 turns in series. Thus the effective e.m.f. is, t = V2 Ti = 4.44 X 0.333 X 360 X 6.4 = 3400 volts per circuit. The e.m.f. between the terminals of a star connected three- phase machine is the resultant of the e.m.fs. of the two phases, 18 ELEMENTS OF ELECTRICAL ENGINEERING. which differ by 60' degrees, and is thus 2 sin 60 = \/3 times that of one phase, thus, E = E l V3 = 5900 volts effective. 23. (2.) The conductor of the machine has a section of 22 sq. cm. and a mean length of 240 cm. per turn. At a resis- tivity (resistance per unit section and unit length) of copper of p 1.8 X 10", what is the e.m.f. consumed in the machine by the resistance, and what the power consumed at 450 kw. output? 450 kw. output is 150,000 watts per phase or circuit, thus ,, , 7 150,000 ,, ,. the current / = = 44.2 amperes effective. O "iUU The resistance of 360 turns of 240 cm. length, 0.22 sq. cm. section and 1.8 X 10" 6 resistivity, is 360 X 240 X 1.8 X 10~ 6 A ^ , ... r = - - = O./l ohms per circuit. 44.2 amp. X 0.71 ohms gives 31.5 volts per circuit and (44.2) 2 X 0.71 - 1400 watts per circuit, or a total of 3 X 1400= 4200 watts loss. 24. (3.) What is the self-inductance per wire of a three- phase line of 14 miles length consisting of three wires No. (Id =0.82 cm.), 45 cm. apart, transmitting the output of this 450 kw. 5900-volt three-phase machine? 450 kw. at 5900 volts gives 44.2 amp. per line. 44.2 amp. effective gives 44.2 \/2 = 62.5 amp. maximum. 14 miles = 22,400 m. The magnetic flux produced by / amperes in 1000 m. of a transmission line of 2 wires 45 cm. apart and 0.82 cm. diameter was found in paragraph 1, example 3, as 2 $ - 0.188 X 10 /, or * = 0.094 X 10 6 / for each wire. Thus at 22,300 m. and 62.5 amp. maximum, the flux per wire is $ = 22.3 X 62.5 X 0.094 X 10 6 = 131 megalines. Hence the generated e.m.f., effective value, at 33.3 cycles is, = 4.44 X 0.333 X 131 = 193 volts per line; POWER AXD EFFECTIVE VALUES. 19 the maximum value is, E Q - E X VT= 273 volts per line; and the instantaneous value, e = E sin (0 - = 273 sin (0 - or, snce = xt = have, = 273 sin 210 (/ 25. (4.) What is the form factor (a) of the e.m.f. gene- rated in a single conductor of a direct-current machine hav- ing 80 per cent pole arc and negligible spread of the mag- netic flux at the pole corners, and (6) what is the form fac- tor of the voltage between two collector rings connected to diametrical points of the arm- ature of such a machine? (a.) In a conductor during the motion from position A, shown in Fig. 6, to position B, no e.m.f. is generated; Fig ' 6 ' of Bipolar Generator. from position B to C a constant e.m.f. e is generated, from C to E again no e.m.f., from E to F a constant e.m.f. e, I r 1 \ r I i 1 F 1 A B C /r E F L 3 Fig. 7. E.m.f. of a Single Conductor, Direct-Current Machine 80 per cent Pole Arc. and from F to A again zero e.m.f. The e.m.f. wave thus is as shown in Fig. 7. The average e.m.f. is e l = 0.8 e] 20 ELEMENTS OF ELECTRICAL ENGINEERING. hence, with this average e.m.f., if it were a sine wave, the maxi- mum e.m.f. would be and the effective e.m.f. would be __ e^ __ 0.4 xe 63 "" V2 " \/2 The actual square of the e.m.f. is er for 80 per cent and zero for 20 per cent of the period, and the average or mean square thus is 0.8 e\ and therefore the actual effective value, The form factor y, or the ratio of the actual effective value e 4 to the effective value e 3 of a sine wave of the same mean value and thus the same 'magnetic flux, then is ~ = 4 = 1.006; that is, practically unity. (6.) While the collector leads a, b move from the position F 9 C } as shown in Fig. 6, to 5, E, constant voltage E exists between them, the conductors which leave the field at C being replaced by the conductors entering 'the field at 5. During the motion of the leads a, b from B, E to C, F, the voltage steadily decreases, Pig. 8. E.m.f. between two Collector Rings connected to Diametrical Points of the Armature of a Bipolar Machine having 80 per cent Pole Arc. reverses, and rises again, to E } as the conductors entering the field at E have an e.m.f. opposite to that of the conductors leaving at C. Thus the voltage wave is, as shown by Fig. 8, triangular, with the top cut off for 20 per cent of the half wave. XELF-IXDVCTAXCE AND MUTUAL IXDUCTAXCE. 21 Then the average e.m.f. is e, = 0.2 E + 2 X ^ - 0.6 E. The maximum value of a sine wave of this average value is and the effective value corresponding thereto is = 2 0.3 r.E 3 V2~ V2 The actual voltage square is E 2 for 20 per cent of the time, and rising on a parabolic curve from to E 2 during 40 per cent of the time, as shown in dotted lines in Fig. 8. The area of a parabolic curve is width times one-third of height, or 3 ' hence, the mean square of voltage is and the actual effective voltage is ..- hence, the form factor is or, 2.5 per cent higher than with a sine wave. 5. SELF-INDUCTANCE AND MUTUAL INDUCTANCE. 26. The number of interlinkages of an electric circuit with the lines of magnetic force of the flux produced by unit current in the circuit is called the inductance of the circuit. The number of interlinkages of an electric circuit with the lines of magnetic force of the flux produced by unit current in a second electric circuit is called the mutual inductance of the second upon the first circuit. It is equal to the mutual indue- 22 ELEMENTS OF ELECTRICAL ENGINEERING tance of the first upon the second circuit, as will be seen, and thus is called the mutual inductance between the two circuits. The number of interlinkages of an electric circuit with the lines of magnetic flux produced by unit current in this circuit and not interlinked with a second circuit is called the self- inductance of the circuit. If i = current in a circuit of n turns, $ flux produced thereby and interlinked with the circuit, n$ is the total number of interlinkages, and L = ~- the inductance of the circuit. % If <3> is proportional to the current i and the number of turns n, = , and L = the inductance. (R ' E for t = 0, i Q = - = c. Substituting this value, the current is i== E e ~ ( -^ } r and the generated e.m.f. is Substituting i Q = , the current is r+n i = i e L ' and the generated e.m.f. is / - \ At t = 0, that is, the generated e.m.f. is increased over the previously impressed e.m.f. in the same ratio as the resistance is increased. When r 1 = 0, that is, when in withdrawing the impressed e.m.f. E the circuit is short-circuited, E - T J- _rL . i = L = I Q B L the current, and !l _ !l e^^ E e L = ijre L the generated e.m.f. * In this case, at t == 0, e 1 = E, that is, the e.m.f. does not rise. In the case r t = oo ; that is, if in withdrawing the e.m.f. E, the circuit is broken, we have t = and e l = oo ; that is, the e.m.f. rises infinitely. The greater r v the higher is the generated e.m.f. e v the faster, however, do e l and i decrease. If r l = r, we have at t = 0, CONTINUOUS-CURRENT CIRCUITS. 29 and e n - ijr = E; that is, if the external resistance r t equals the internal resistance r, at the moment of withdrawal of the e.m.f. E the terminal voltage is E. The effect of the e.m.f. of inductance in stopping the current at the time t is ...o r-fr. f ie l = v (r + r,} * ~ L ; thus the total energy of the generated e.m.f. W = _ that is, the energy stored as magnetism in a circuit of current i Q and inductance L is which is independent both of the resistance r of the circuit and the resistance r 1 inserted in breaking the circuit. This energy has to be expended in stopping the current. EXAMPLES- 32. (1.) In the alternator field in Section 1, Example 4, Sec- tion 2, Example 2, and Section 5, Example 1, how long a time after impressing the required e.m.f. E= 230 volts will it take for the field to reach (a) J strength, (6) T \ strength? (2.) If 500 volts are impressed upon the field of this alternator, and a non-inductive resistance inserted in series so as to give the required exciting current of 6.95 amperes, how long after impressing the e.m.f. E = 500 volts will it take for the field to reach (a) J strength, (6) & strength, (c) and what is the resist- ance required in the rheostat? (3.) If 500 volts are impressed upon the field of this alter- nator without insertion of resistance, how long will it take for the field to reach full strength? (4.) With full field strength what is the energy stored as magnetism? 30 ELEMENTS OF ELECTRICAL ENGINEERING. (1.) The resistance of the alternator field is 33.2 ohms (Section 2, Example 2), the inductance 112 h. (Section 5, Example 1), the impressed e.m.f. is E = 230, the final value of current = rr = 6.95 amperes. Thus the current at time t, is r I = \ = G.95 (1 - (a.) i strength t = -^-, hence (1 - e-"- 89 ") = 0.5. ,J -' = 0.5, - 0.296 1 log e = log 0.5, t = ~^. g . ' 5 , and . log 2.34 seconds. (6.) A strength: i = 0.9 i , hence (1 - e- - 296 *) = 0.9, and t = 7.8 seconds (2.) To get i = 6.95 amperes, with E = 500 volts, a resist- 500 ance r = ~ - = 72 ohms, and thus a rheostat having a resist- 6.95 ance of 72 33.2 = 38.8 ohms is required. We then have = 6.95 (1 - e"- 643 0. i (a.) i = ^, after t = 1.08 seconds. ^ (6.) i = 0.9 i , after t = 3.6 seconds. (3.) Impressing E = 500 volts upon a circuit of r = 33.2, L = 112, gives . EL -st\ 2, _ H e L ) r v 7 = 15.1 (1 - -' 2mt ). i = 6.95, or full field strength, gives 6.95 = 15.1 (1 - -- 296 0. and t = 2.08 seconds. CONTINUOUS-CURRENT CIRCUITS. 31 (4.) The stored energy is ^' 2 L 6.95 2 X 112 070A , , i i ^j_ _ _ 2720 watt-seconds or joules 2i 2i = 2000 foot-pounds. (1 joule = 0.736 foot-pounds.) Thus in case (3), where the field reaches full strength in 2.08 2000 seconds ; the average power input is j~ = 960 foot-pounds J.OS per second, = 1.75 hp. In breaking the field circuit of this alternator, 2000 foot- pounds of energy have to be dissipated in the spark, etc. 33. (5.) A coil of resistance r = 0.002 ohm and inductance L = 0.005 milhenry, carrying current 7=90 amperes, is short- circuited. (a.) What is the equation of the current after short cir- cuit? (5.) In what time has the current decreased to 0.1, its initial value? (a.) i = 7e"i = 90s- 400 '. (6.) i = 0.1 7, -%-* that is, the effective value of the counter e.m.f. of inductance equals the reactance, x, times the effective value of the current, I, and lags 90 time degrees, or a quarter period, behind the current. 35. By the counter e.m.f. of inductance, e/ = x/ cos 6, which is generated by the change in flux due to the passage of the current i = J sin d through the circuit of reactance x, an equal but opposite e.m.f. e 2 = z/ cos 6 is consumed, and thus has to be impressed upon the circuit. This e.m.f. is called the e.m.f. consumed by inductance. It is 90 time degrees, or a quarter period, ahead of the current, and shown in Fig. 10 as a drawn line e T Thus we have to distinguish between counter e.m.f. of induc- tance 90 time degrees lagging, and e.m.f. consumed by inductance 90 time degrees leading. INDUCT AXCE IX ALTERXATIXG-CTRREXT CIRCUITS. 35 These e.m.fs. stand in the same relation as action and reaction in mechanics. They are shown in Fig. 10 as e./ and as . , r I 1.8 X 10- 6 X 2.23 X 10 6 (a.) Resistance per line, r = p = -~~ A u.52o == 7.60 ohms. 2 Z Reactance per line, x = 4 nfllog,- X 10~ 9 == 4 K X 33.3 X id 2.23 X 10 6 X log 110 X 10- 9 - 4.35 ohms. The impedance per line, z \/r 2 + x 2 = 8.76 ohms. Thus if / = 44 amperes per line, the e.m.f. consumed by resistance is E l = rl = 334 volts, the e.m.f. consumed by reactance is E 2 = xl = 192 volts, and the e.m.f. consumed by impedance is E 3 = zl = 385 volts. INDUCTANCE IN ALTERNATING-CURRENT CIRCUITS. 39 (&.) 5500 volts between lines at receiving circuit give 5500 V3 3170 volts between line and neutral or zero point (Fig. 12), or per line, corresponding to a maximum voltage of 3170 V2 = 4500 volts. 44 amperes effective per line gives a maximum value of 44 \/2 = 62 amperes. Denoting the current by i = G2 sin 6, the voltage per line at the receiving end with non-inductive load is e = 4500 sin 0. The e.m.f. consumed by resistance, in phase with the current, of effective value 334, and maximum value 334 \/2 - 472, is e l = 472 sin 0. The e.m.f. consumed by reactance, 90 time degrees ahead of the current, Fig . 12> Voltage D ia ^ ram for of effective value 192, and maximum a Three-Phase Circuit, value 192 V2 = 272, is e 2 = 272 cos d. Thus the total voltage required per line at the generator end of the line is e Q = e + BI + e 2 = (4500 + 472) sin + 272 cos 6 = 4972 sin d + 272 cos 6. . 272 Denoting 4972 tan , we have sin # = tan fl fl 272 Vl + tan 2 tf n 4980 cos<9 n 4972 + tan 2 d n 4980 Hence, e Q = 4980 (sin 9 cos # + cos 6 sin ) - 4980 sin (9 + 9 ). Thus d is the lag of the current behind the e.m.f. at the generator end of the line, = 3.2 time degrees, and 4980 the 40 ELEMENTS OF ELECTRICAL ENGINEERING. 4980 maximum voltage per line at the generator end; thus E Q = == = 3520, the effective voltage per line, and 3520 V3 = 6100, the effective voltage between the lines at the generator. (c.) If the current i = 62 sin 6 lags in time 45 degrees behind the e.m.f. at the receiving end of the line, this e.m.f. is expressed by e = 4500 sin (6 + 45) - 3170 (sin + cos ff) ; that is, it leads the current by 45 time degrees, or is zero at =* 45 time degrees. The e.m.f. consumed by resistance and by reactance being the same as in (6), the generator voltage per line is e o = e + &i + e 2 = 3642 sin + 3442 cos 0. Denoting ^ = tan , we have e = 5011 sin (0 + ). Thus , the time angle of lag of the current behind the gen- erator e.m.f., is 43 degrees, and 5011 the maximum voltage; hence 3550 the effective voltage per line, and 3550 \/3 6160 the effective voltage between lines at the generator. (d.) If the current { = 62 sin leads the e.m.f. by 45 time degrees, the e.m.f. at the receiving end is e - 4500 sin (0 - 45) = 3170 (sin 8 - cos 0). Thus at the generator end e o *= e + e i + e 2 * 3642 sin 6 - 2898 cos 0. Denoting r- = tan , it is OUttA e, -4654 sin (0 - ). Thus , the time angle of lead at the generator, is 39 degrees, and 4654 the maximum voltage; hence 3290 the effective voltage per line and 5710 the effective voltage between lines at the generator. POWER IX ALTERNATING-CURRENT CIRCUITS. 41 8. POWER IN ALTERNATING-CURRENT CIRCUITS. 39. The power consumed by alternating current i = J sin 6, of effective value / = ^, in a circuit of resistance r and reac- V2 tance x = 2 ;r/L, is P = ei, where e / sin (0 + ) is the impressed e.m.f., consisting of the components e 1 = r/ sin 0, the e.m.f. consumed by resistance and e 2 = xI Q cos 5. the e.m.f. consumed by reactance. z = Vr + x 2 is the impedance and tan = - the time-phase angle of the circuit; thus the power is p = 2/ 2 sin 6 sin (6 + ) = ^-(cos0 - cos (20 + )) = zP (cos - cos (2 + )). Since the average cos (20 + ) = zero, the average power is p = Z P cos that is, the power in the circuit is that consumed by the resistance, and independent of the reactance. Reactance or self-inductance consumes no power, and the e.m.f. of self-inductance is a wattless e.m.f., while the e.m.f. of resistance is a power e.m.f. The wattless e.m.f. is in quadrature, the power e.m.f. in phase with the current. In general, if = angle of time-phase displacement between the resultant e.m.f. and the resultant current of the circuit, / = current, E = impressed e.m.f., consisting of two com- ponents, one, E t = E cos 0, in phase with the current, the other, E 2 === E sin 0, in quadrature with the current, the power in the circuit is IE l = IE cos 0, and the e.m.f. in phase with the current E t = E cos is a power e.m.f., the e.m.f. in quadrature with the current E = E sin a wattless or reactive e.m.f. 42 ELEMENTS OF ELECTRICAL ENGINEERING. 40. Thus we have to distinguish power e.m.f. and wattless or reactive e.m.f., or power component of e.m.f., in phase with the current and wattless or reactive component of e.m.f., in quadra- ture with the current. Any e.m.f. can be considered as consisting of two components, one, the power component, e v in phase with the current, and the other, the reactive component, e 2 , in quadrature with the current. The sum of instantaneous values of the two com- ponents is the total e.m.f. e == e l + e r If E, E v E 2 are the respective effective values, we have E = VE* + E*, since E 1 = E cos 6, E 2 = E sin 0, where d = time-phase angle between current and e.m.f. Analogously, a current / due to an impressed e.m.f. E with a time-phase angle 6 can be considered as consisting of two component currents, 7 X = 7 cos 9j the power component of the current, and 7 2 == 7 sin 0, the wattless or reactive component of the current. The sum of instantaneous values of the power and reactive components of the current equals the instantaneous value of the total current, i l + i a = i, while their effective values have the relation 7 = v / 7 1 3 +7 2 2 . Thus an alternating current can be resolved in two com- ponents, the power component, in phase with the e.m.f., and the wattless or reactive component, in quadrature with the e.m.f. An alternating e.m.f. can be resolved in two components, the power component, in phase with the current and the watt- less or reactive component, in quadrature with the current. The power in the circuit is the current times the e.m.f. times the cosine of the time phase angle, or is the power component of the current times the total e.m.f., or the power component of the e.m.f. times the total current. POLAR COORDINATES. 43 EXAMPLES. 41. (1.) What is the power received over the transmission line in Section 7, Example 2, the power lost in the line, the power put into the line, and the efficiency of transmission with non- inductive load, with 45-time-degree lagging load and 45-clegree leading load? The power received per line with non-inductive load is P = El = 3170 X 44 = 139 kw. With a load of 45 time degrees phase displacement, P = El cos 45 = 98 kw. The power lost per line P l = PR - 44 2 X 7.6 = 14.7 kw. Thus the input into the line P = P + P l = 151.7 kw. at non-inductive load, and = 111.7 kw. at load of 45 degrees phase displacement. The efficiency with non-inductive load is = 90.3 per cent, P 151.7 and with a load of 45 degrees phase displacement is p~ = 1 - jjjy ^ 86 - 8 P er cent - The total output is 3 P = 411 kw. and 291 kw., respectively. The total input 3 P = 451.1 kw. and 335.1 kw., respectively. 9. POLAR COORDINATES. 42. In polar coordinates, alternating waves are represented, with the instantaneous values as radius vectors, and the time as an angle, counting left-handed or counter clockwise, and one revolution or 360 degrees representing one complete period. The sine wave of alternating current i = I sin 9 is repre- sented by a circle (Fig. 13) with the vertical axis as diameter, equal in length 0/ to the maximum value J , and shown as heavy drawn circle. The e.m.f. consumed by inductance, e 2 = x/ cos 0, is repre- sented by a circle with diameter OE 2 in horizontal direction to the right, and equal in length to the maximum value, xI Q . Analogously, the counter e.m.f. of self-inductance E 2 r is repre- 44 ELEMENTS OF ELECTRICAL ENGINEERING. sented by a circle OE^in Fig. 13; the e.m.f. consumed by resistance r by circle OE l of a diameter ^= r/ , and the counter e.m.f. of resistance $/ by circle OE^. The counter e.m.f. of impedance is represented by circle OE' of a diameter equal in length to E', and lagging 180 - # behind the diameter of the current circle. This circle passes through the points $/ and EJ, since at the moment 6 = 180 degrees, Fig. 13. Sine waves represented in Polar Coordinates. 6/ 0, and thus the counter e.m.f. of impedance equals the counter e.m.f. of reactance ef = e 2 ', and at 6 = 270 degrees, e a ' = 0, and the counter e.m.f. of impedance equals the counter e.m.f. of resistance e r e/. The e.m.f. consumed by impedance, or the impressed e.m.f., is represented by circle OE having a diameter equal in length to E, and leading the diameter of the current circle by the angle . This circle passes through the points E 1 and E r An alternating wave is determined by the length and direction of the diameter of its polar circle. The length is the maximum value or intensity of the wave, the direction the phase of the maximum value, generally called the phase of the wave. POLAR COORDINATES. 45 43- Usually alternating waves are represented in polar co- ordinates by mere vectors, the diameters of their polar circles, and the circles omitted, as in Fig. 14. Fig. 14. Vector Diagram. Fig. 15. Vector Diagram of two e.m.fs. Acting in the same Circuit. Two e.m.fs., ^ and e v acting in the same circuit, give a result- ant e.m.f. e equal to the sum of their instantaneous values. In polar coordinates e l and e 2 are represented in intensity and in phase by two vectors, OE 1 and OE^ Fig. 15. The instantane- ous values in any direction OX are the projections Oe v Oe 2 of OE 1 and OE 2 upon this direction. 46 ELEMENTS OF ELECTRICAL ENGINEERING. Since the sum of the projections of the sides of a parallelo- gram is equal to the projection of the diagonal, thejsum of the projections Oe^ and Oe 2 equals thejprojection Oe of OE, the diago- nal of the parallelogram with OE t and OE 2 as sides, and OE is thus the diameter of the circle of resultant e.m.f . ; that is, in polar coordinates alternating sine waves of e.m.f., current, etc., are combined and resolved by the parallelogram or polygon of sine waves. Since the effective values are proportional to the maximum values, the former are generally used as the length of vector of the alternating wave. In this case the instantaneous values are given by a circle with \/2 times the vector as diameter. 44. As phase of the first quantity considered, as in the above instance the current, any direction can be chosen. The further quantities are determined thereby in direction or phase. In polar coordinates, as phase of the current, etc., is here and in the following understood the time or the angle of its vector, that is, the time of its maximum value, and a current of phase zero would thus be denoted analytically by i = 7 cos 6. The zero vector OA is generally chosen for the most frequently used quantity or reference quantity, as for the current, if a number of e.m.fs. are considered in a circuit of the same cur- rent, or for the e.m.f., if a number of currents are produced by the same e.m.f., or for the generated e.m.f. in apparatus such as transformers and induction motors, synchronous apparatus, etc. With the current as zero vector, all horizontal components of e.m.f. are power components, all vertical components are reactive components. With the e.m.f. as zero vector, all horizpntal components of current are power components, all vertical components of current are reactive components. By measurement from the polar diagram numerical values can hardly ever be derived with sufficient accuracy, since the magnitudes of the different quantities used in the same diagram are usually by far too different, and the polar diagram is there- fore useful only as basis for trigonometrical or other calculation, and to give an insight into the mutual relation of the different quantities, and even then great care has to be taken to distin- POLAR COORDINATES. 47 guish between the two equal but opposite vectors, counter e.ni.f. and e.m.f. consumed by the counter e.m.f., as explained before. 45. In the polar coordinates described in the preceding, and used throughout this book, the angle represents the time, and is counted positive in left-handed or counter-clockwise rotation, with the instantaneous values of the periodic function as radii, so that the periodic function is represented by a closed curve, and one revolution or 360 degrees as one period. This "time diagram" is the polar coordinate system universally used In other sciences to represent periodic phenomena, as the cosmic motions in astronomy, and even the choice of counter-clockwise as positive rotation is retained from the custom of astronomy, the rotation of the earth being such. In the time diagram, the sine w r ave is given by a circle, and this circle of instantaneous values of the sine wave is represented, in size or position, by its diameter. That is, the position of this diameter denotes the time, t, or angle, = 2 nft, at which the sine wave reaches its maximum value, and the length of this diameter denotes the intensity of the maximum value. If then, in polar coordinate representation, Fig. 16, OE denotes an e.m.f., 01 a current, this means that the maximum value of e.m.f. equals OE, and is reached at the time, t v represented by angle AOE = O l = 2 nft r The current in this diagram then has a maximum value equal to 01, and this maximum value is reached at the time, t v represented by angle AOI = 2 = 2 xft y If then angle 6 2 > 6 V this means that the current reaches its maximum value later than the e.m.f., that is, the current in Fig. 16 lags behind the e.m.f., by the angle EOI = # 3 - O l = 2 nf(t 2 ^), or by the time 3 t r Frequently in electrical engineering another system of polar coordinates is used, the so-called "crank diagram." In this, sine waves of alternating currents and e.m.fs. are represented as projections of a revolving vector upon the horizontal. That is, a vector, equal in length to the maximum of the alternating wave, revolves at uniform speed so as to make a complete revolution per period, and the projections of this revolving vector upon the horizontal then denote the instantaneous values of the wave. 48 ELEMENTS OF ELECTRICAL ENGINEERING. Obviously, by the crank diagram only sine waves can be represented, while the time diagram permits the representation of any wave shape, and therefore is preferable. Let, for instance, 01 represent in length the maximum value of current, i = / cos (0 2 ). Assume, then, a vector, OL to Fig. 16. Representation of Current and e.m.f. by Polar Coordinates. revolve, left-handed or in positive direction, so that it makes a complete revolution during each cycle or period. If then at a certain moment of time, this vector stands in position 01 1 (Fig. 17), the projection, OA lt of 01 i on OA represents the instan- taneous value of the current at this moment. At a later moment 01 has moved farther, to 07 2 , and the projection, OA 2J of 07 2 on OA is the instantaneous value. The diagram thus shows the instantaneous condition of the sine waves. Each sine wave reaches the maximum at the moment when its revolving vector, 01 j passes the horizontal. If Fig. JL8 represents the crank diagram of an e.m.f., OE, and a current, 07, and if angle APE > AOI, this means that the e.m.f., OE, is ahead of the current, 07, passes during the revolution the zero line or line of maximum intensity, OA, earlier than the current, or leads; that is, the current lags behind the e.m.f. The same Fig. 18 considered as polar diagram would mean that the current leads the e.m.f.; that is, the maximum value of current, 07, occurs at a smaller angle, A 07, that is, at an earlier time., than the maximum value of the e.m.f,, OE. POLAR COORDINATES. 46. In the crank diagram, the first quantity therefore can be put in any position. For instance, the current, 07, in Fig. IS, could be drawn in position 07, Fig. 19. The e.m.f. then being ITig. 17, Crank Diagram showing instantaneous values. Fig. 18. Crank Diagram of an e.m.f. and Current. ahead of the current by angle EOI = 6 would come into the position OE, Fig. 19. A polar diagram, Fig. 16, with the current, 0/ ? lagging behind the e.m.f., OE, by the angle, 0, thus considered as crank diagram would represent the current leading the e.m.f. by the angle, 6 f and a crank diagram, Fig, 18 or 19, with the current lagging behind the e.m.f. by the angle, 6, would as polar diagram repre- sent a current leading the e.m.f. by the angle, 6. The main difference in appearance between the crank diagram and the polar diagram therefore is that, with the same direction of rotation, lag in the one diagram is represented in the same manner as lead in the other diagram, and inversely. Or, a 50 ELEMENTS OF ELECTRICAL ENGINEERING. representation by the crank diagram looks like a representation by the polar diagram, with reversed direction of rotation, and vice versa. Or, the one diagram is the image of the other and can be transformed into it by reversing right and left, or top and bottom. Therefore the crank diagram, Fig. 19, is the image of the polar diagram, Fig. 16. Fig. 19. Crank Diagram. Since the time diagram, in which the position of the vector represents its phase, that is, the moment of its maximum value, is used in all other sciences, and also is preferable in electrical engineering, it will be exclusively used in the following, the positive direction being represented as counter-clockwise. EXAMPLES. 47. In a three-phase long-distance transmission line, the volt- age between lines at the receiving end shall be 5000 at no load, 5500 at full load of 44 amperes power component, and propor- tional at intermediary values of the power component of the current; that is, the voltage at the receiving end shall increase proportional to the load. At three-quarters load the current shall be in phase with the e.m.f. at the receiving end. The generator excitation however and thus the (nominal) generated e.m.f. of the generator shall be maintained constant at all loads, and the voltage regulation effected by producing lagging or leading currents with a synchronous motor in the receiving cir- cuit. The line has a resistance r l = 7.6 ohms and a reactance x t = 4.35 ohms per wire, the generator is star connected, the resistance per circuit being r 2 = 0.71, and the (synchronous) reactance is 3, = 25 ohms. What must be the wattless or POLAR COORDINATES. 51 reactive component of the current, and therefore the total cur- rent and its phase relation at no load, one-quarter load, one- half load, three-quarters load, and full load, and what will be the terminal voltage of the generator under these conditions? The total resistance of the line and generator is r = r l + r 2 = 8.31 ohms; the total reactance, x = x l + -r 2 = 29.35 ohms. Fig. 20. Polar Diagram of e.m.f. aud Current in Transmission Line. Current Leading. Let, in the polar diagram, Fig. 20 or 21, OE = E represent the voltage at the receiving end of the line, 01 1 = 7 t the power component of the current corresponding to the load, in phase with OE } and 0/ 2 = 7 2 the reactive component of the current Fig. 21. Polar Diagram of e.m.f. and Current in Transmission Line. Current Lagging. in quadrature with OE y shown leading in Fig. 20, lagging in Fig. 21. _ We then have total current 7 = 07. Thus the e.m.f. consumed by resistance, OE = r7, is in phase with 7, the e.m.f. consumed by reactance, OE 2 = #7, is 90 degrees 52 ELEMENTS OF ELECTRICAL ENGINEERING. ahead of /, and their resultant is OF 37 the e.m.f. consumed by impedance. OEs combine^ with OE, the receiver voltage, gives the gener- ator voltage OE Q . Resolving all e.m.fs. and currents into components in phase and in quadrature with the received voltage E, we have PHASE QUADRATURE COMPONENT. COMPONENT. Current / t I 2 E.m.f. at receiving end of line, E E E.m.f. consumed by resistance, E l = rl t rI 2 E.m.f. consumed by reactance, E 2 = xI 2 xl^ Thus total e.m.f. or generator voltage, E Q = E + E l + E 2 = E + rl t + xI 2 r/ 2 - xl : Herein the reactive lagging component of current is assumed as positive, the leading as negative. The generator e.m.f. thus consists of two components, which give the resultant value E, = V(E + rl, + xI 2 r + (rl t - xI. substituting numerical values, this becomes E Q = V(E + 8.31 1, + 29.35 7 2 ) 2 + (8.31 / 2 - 29.35 /J 2 ; at three-quarters load, 5375 E = =~ = 3090 volts per circuit, ^=33, J 2 = 0, thus E Q = V(3090 4- 8.31 X 33) 2 + (29.35 X 33) 2 = 3520 volts per line or 3520 X V3 = 6100 volts between lines as (nominal) generated e.m.f. of generator. Substituting these values, we have 3520 = V(JS + 8.31 /, + 29.35 / 2 ) 2 + (8.31 /, - 29.35 J,) 2 . The voltage between the lines at the receiving end shall be: No i } | FULL LOAD. LOAD. LOAD. LOAD. LOAD. Voltage between lines, _ 5000 5125 5250 5375 5500 Thus, voltage per line, W3, #-= 2880 2950 3020 3090 3160 HYSTERESIS AND EFFECTIVE RESISTANCE 53 The power components of current per line, I, = 11 22 33 44 Herefrom we get by substituting in the above equation Reactive component of Lo ^ D . Lo * AD . Lo i D . Lo | D . j current, 7 2 - 21.6 16.2 9.2 -9.7 hence, the total current, / = VI* + / 2 2 = 21.6 19.6 23.9 33.0 45.05 and the power factor, L cos g o 56.0 92.0 100.0 97.7 the lag of the current, 0= 90 61 23 -11.5 the generator terminal voltage per line is + 7.6 1, + 4.35 / 2 ) 2 + (7.6 / 2 - 4.35 / t ) thus: No i i I LOAD. LOAD. LOAD. LOAD. LOAD Per line, E^ = 2980 3106 3228 3344 3463 Between lines, E' V$ = 5200 5400 5600 5800 6000 Therefore at constant excitation the generator voltage rises with the load, and is approximately proportional thereto. 10. HYSTERESIS AND EFFECTIVE RESISTANCE. 48. If an alternating current 01 = /, in Fig. 22, exists in a circuit of reactance x 2 TtfL and of negligible resistance, the magnetic flux produced by the current, 0$ = $, is in phase with the current, and the e.m.f. generated by this flux, or counter e.m.f. of self-inductance, OE'" = E f " = xl, lags 90 degrees behind the current. The e.m.f. consumed by self-inductance or impressed e.m.f. OE" = E" = xl is thus 90 degrees ahead of the current. Inversely, if the e.m.f. OE" = E" is impressed upon a circuit of reactance x = 2 nfL and of negligible resistance, the current i ~c\n 01 = I = lags 90 degrees behind the impressed e.m.f. x 54 ELEMENTS OF ELECTRICAL ENGINEERING. This current is called the exciting or magnetizing current of the magnetic circuit, and is wattless. If the magnetic circuit contains iron or other magnetic material, energy is consumed in the magnetic circuit by a frictional resist- ance of the material against a change of |" magnetism, which is called molecular mag- I netic friction. If the alternating current is the only avail- able source of energy in the magnetic circuit, the expenditure of energy by molecular magnetic friction appears as a lag of the magnetism behind the m.m.f. of the current, that is, as magnetic hysteresis, and can be measured thereby. Magnetic hysteresis is, however, a dis- tinctly different phenomenon from molecular magnetic friction, and can be more or less eliminated, as for instance by mechanical vibration, or can be increased, without changing the molecular magnetic friction. E/' 49. In consequence of magnetic hys- Fig. 22. Phase Reia- teresis, if an alternating e.m.f. OE" = E" is tions of Magnetizing impressed upon a circuit of negligible resist- n f d ance, the exciting current, or current pro- "" x ' * ducing the magnetism, in this circuit is not a wattless current, or current of 90 degrees lag, as in Fig. 22, but Jags less than 90 degrees, by an angle 90 a, as shown by 01 = I in Fig. 23. _ Since the magnetism 0$ = is in quadrature with the e.m.f. E" due to it, angle a is the phase difference bet ween the magnetism and the m.m.f., or the lead of the m.m.f., that is, the exciting current, before the magnetism. It is called the angle ofhysteretic lead. In this case the exciting current 01 = / can be resolved in two components, the magnetizing current 01 2 = 7 2 , in phase with the magnetism 0$ = , that is, in quadrature with the e.m.f. OE /f = W r , and thus wattless, and the magnetic power component of the current QY the hysteresis current 01 1 = I v in phase with the e.m.f. OE" = E", or in quadrature with the magnetism 0$ = $. HYSTERESIS AND EFFECTIVE RESISTANCE. 55 Magnetizing current and hysteresis current are the two com- ponents of the exciting current. If the circuit contains^ besides the reactance x = 2 ~/L, a resistance r, the e.m.f. OE" = E" in the preceding Figs. 22 and 23 is not the impressed e.m.f., but the e.m.f. consumed by self- inductance or reactance, and has to be combined, Figs, 24 and 25, with the e.m.f. consumed__by the resistance, QE' = E' = Ir, to get the impressed e.m.f. OE = E. Due to the hysteretic lead a, the lag of the current is less in Figs. 23 and 25, a circuit expending energy in molecular mag- netic friction, than in Figs. 22 and 24, a hysteresisless circuit. Fig. 23. Angle of Hysteretic Lead. Fig. 24. Effective Resistance on Phase Relation of Impressed e.m.f in a Hysteresisless Cir- cuit. As seen in Fig. 25, ir a circuit whose ohmic resistance is not negligible, the hysteresis current and the magnetizing current are not in phase and in quadrature respectively with the im- pressed e.m.f., but with the counter e.m.f. of inductance or e.m.f. consumed by inductance. Obviously the magnetizing current is not quite wattless, since energy is consumed by this current in the ohmic resistance of the circuit. . Resolving, in Fig. 26, the impressed e.m.f. OE = E into two components, OW l J B 1 in phase, and OE 2 - E 2 in quadrature with the current 01 = /, the power component of the e.m.f., 56 ELEMENTS OF ELECTRICAL ENGINEERING. OE l = E v is greater than W Ir, and the reactive component OE 2 = E 2 is less than W = /x. Fig. 25. Effective Resistance on Phase Relation of Im- pressed e.m.f. in a Circuit having Hysteresis. Fig. 26. Impressed e.m.f. Resolved into Components in Phase and in Quadrature with the Exciting Current. The value r' ___ power e.m.f. . total current is called the effective resist- ance t and the value x f wattless e.m.f. . is called the ap- / total current parent or effective reactance of the circuit. 50. Due to the loss of energy by hysteresis (eddy currents, etc.), the effective resistance differs from, and is greater than, the ohmic resistance, and the apparent reactance is less than the true or inductive reactance. The loss of energy by molecular magnetic friction per cubic centimeter and cycle of magnetism is approximately W=* ^(B 1 - 6 , where (B the magnetic flux density, in lines per sq. cm. W energy, in absolute units or ergs per cycle (= 10~~ 7 watt-seconds or joules), and TJ is called the coef- ficient of hysteresis. In soft annealed sheet iron or sheet steel, >? varies from 0.75 X 10" 3 to 2.5 X 10~ 3 , and can in average, for good material, be assumed as 2.00 X 10' 3 . HYSTERESIS AND EFFECTIVE RESISTANCE. 57 The loss of power in the volume, I r , at flux density (B and frequency /, is thus p = r/j lagging current, and 6 = a non-inductive receiver circuit. IMPEDANCE OP TRANSMISSION LINES. 63 The capacity of the transmission line shall be considered as negligible. Assuming the phase of the current 07 = 7 as zero in the polar diagram, Fig. 28, the e.m.f. E is represented by vector OE, ahead of 01 by angle 0. The e.m.f. consumed by resistance r is OE 1 = E l = 7r in phase with the current, and the e.m.f. consumed by reactance x is OE 2 = E 2 = Ix, 90 time degrees ahead of the current; thus the total e.m.f. con- sumed by the line, or e.m.f. consumed by impedance, is the resultant OE% of OE { and OE 2) and is E 3 = Iz. Combining OE Z and OE gives , the e.m.f. impressed upon the line. Fig. 28. Polar Diagram of Cur- rent and e.m.fs. in a Trans- mission Line Assuming Zero Capacity. Denoting tan t = - the time angle of lag of the line impe- dance, it is, trigonometrically, Since we have and OE* - OE* + BE, 2 - 2 EE Q cos OEE Q . EE - OE, Iz, OEE, - 180 - 6 1 + 0, = E 2 + /) ^^ n 4 EIz sin 3 L ~ > 2t and the drop of voltage in the line, / 7 7/7 TJ1 4 / / Tr I //w\2 jG/ """"" Cj s==; \/ I/-/ "T" LZ) 57. That is, the voltage E Q required at the sending end of a line of resistance r and reactance or, delivering current 7 at voltage E, and the voltage drop in the line, do not depend upon current and line constants only, but depend also upon the angle of time-phase displacement of the current delivered over the line. 64 ELEMENTS OF ELECTRICAL ENGINEERING. If 8 = 0, that is, non-inductive receiving circuit, (E that is, less than ^ + Iz, and thus the line drop is less than Iz. If Q = iy E is a maximum, = E + Iz, and the line drop is the impedance voltage. With decreasing 0, E Q decreases, and becomes E', that is, no drop of voltage takes place in the line at a certain negative Fig. 29. Locus of the Generator and Receiver e.m.fs. in a Transmission Line with Varying Load Phase Angle. value of 6 which depends not only on z and O t but on E and /. Beyond this value of 0, E Q becomes smaller than E; that is, a rise of voltage takes place in the line, due to its reactance. This can be seen best graphically; Choosing the current vector 01 as the horizontal axis, for the sameje.m.f. E received, but different phase angles 0, all vectors OE lie on a circle e with as center. Fig. 29. Vector OE B is constant for a given line and given current I. IMPEDANCE OF TRANSMISSION LINES. 65 Since EJE^ = OE = constant, E Iks on a circle e with E 3 as center and OE = E an radius. To const ruct_ the diagram for angle 7 OE is drawn at the angle with 01, and M" parallel to OE 3 . The distance EE Q between the two circles on vector OE Q is the drop of voltage (or rise of voltage) in the line. Fig. 30. Locus of the Generator and Receiver e.m.fs. in a Transmission Line with. Varying Load Phase Angle. As seen in Fig. 30, J5? is maximum in the direction OE 3 as that is for = 6 Q , and is less for greater as well, 0$ ", as smaller angles 0. It is = E in the direction OE Q "', in which case 9 < 0, and minimum in the direction OE Q IV . The values of E corresponding to the generator voltages /, EQ", EJ", E are shown by the points W E" E rn S IV 66 ELEMENTS OF ELECTRICAL ENGINEERING. respectively. The voltages E Q " and E correspond to a watt- less receiver circuit E" and E JV . For non-inductive receiver circuit ~OEv the generator voltage is OE*. 58. That is, in an inductive transmission line the drop of voltage is maximum and equal to Iz if the phase angle of the receiving circuit equals the phase angle Q of the line. The drop of voltage in the line decreases with increasing difference between the phase angles of line and receiving circuit. It be- comes zero if the phase angle of the receiving circuit reaches a certain negative value (leading current). In this case no drop of voltage takes place in the line. If the current in the receiving circuit leads more than this value a rise of voltage takes place in the line. Thus by varying phase angle of the receiving circuit the drop of voltage in a transmission line with cur- rent 7 can be made anything between Iz and a certain negative value. Or inversely the same drop of voltage can be produced for different values of the current / by varying the phase angle. Thus, if means are provided to vary the phase angle of the receiving circuit, by producing lagging and leading currents at will (as can be done by synchronous motors or converters) the voltage at the receiving circuit can be maintained constant within a certain range irrespective of the load and generator voltage. In Fig. 31 let OE = E } the receiving_voltage; 7, the power component of the line current; thus OE^ = E^ = Iz, the e.m.f. consumed by the power component of the current in the impe- dance. This e.m.f. consists of the e.m.f. consumed by resist- ance "OE^ and the e.m.f. consumed by reactance OE 2 . Reactive components of the current are represented in the diagram in the direction OA when lagging and OB when leading. The e.m.f. consumed by these reactive components of the current in the impedance is thus in the direction gen- erates e.m.fs. E l and E i in secondary and in primary circuit, "P which are to each other as the ratio of turns, thus ^= - - * a Let E = secondary terminal voltage, 7 t = Secondary current, 1 = lag of current 7j behind terminal voltage E (where O l < denotes leading current). Denoting then in Fig. 35 by a vector (IB = E the secondary terminal voltage, 01 i = / x is the secondary current lagging by the angle EOI - d r The e.m.f. consumed by the secondary resistance r t is OE^ = EI = I l r l in phase with 7 r The e.m.f. consumed by the secondary reactance x l is OE" = E" = I 1 x v 90 time degrees ahead of 7 r Thus the e.m.f. con- sumed by the secondary impedance z l = Vr t 2 -I- x t 2 is the resultant of OF/ and MS?, or OF/ 77 - E,"' =_!&. OE,"' combined with the terminal voltage OE = E gives the secondary e.m.f. OE l == E r Proportional thereto by the ratio of turns and in phase there- with is the e.m.f. generated in the primary OEi = Ei where *-. 74 ELEMENTS OF ELECTRICAL ENGINEERING. To generate e.m.f. E^ and E^ the magnetic flux 0 = $ is required, 90 time degrees ahead of OE l and OE^ To produce flux < the m.m.f. of $ ampere-turns is required, as determined from the dimensions of the magnetic circuit, and thus the primary current 7 00 , represented by vector 07 00 , leading 0$ by the angle a. Since the total m.m.f. of the transformer is given by the primary exciting current 7 00 , there must be a component of Pig. 35. Vector Diagram of e.m.s. and Currents in a Transformer. primary current 7', corresponding to the secondary current 7 1? which may be called the primary load current, and which is opposite thereto and of the same m.m.f.; that is, of the intensity I' =JiI v thus represented by vector OF 7' = al v 0/ oo; the primary exciting current, and the primary load current OP, or component of primary current corresponding to thejjecondary current, combined, give the total primary cur- rent 07 = 7 . The e.m.f. consumed by resistance in the primary is OEJ = EJ = 7 r in phase with 7 . The e.m.f. consumed by the primary reactance is OE Q " = Ef =7 x Q , 90 time degrees ahead of 07 . OE Q ' and OE" combined gives OS/", the e.m.f. consumed by the primary impedance. ALTERNATING-CURRENT TRANSFORMER. 75 Equal and opposite to the primary counter-generated e.m.f. OE l is the component of primary e.m.f., 0", consumed thereby. OE' combined with OE" f gives QE"^ = E^ the primary impressed e.m.f. , and angle # = #/)/, the phase angle of the primary circuit. Figs. 36, 37, and 38 give the polar diagrams for X = 45 or lagging current, O l = zero or non-inductive circuit, and 6 = 45 or leading current. Fig. 36. Vector Diagram of Transformer with Lagging Load Current. 63. As seen, the primary impressed e.m.f. E required to pro- duce the same secondary terminal voltage E at the same current 7 1 is larger with lagging or inductive and smaller with leading current than on a non-inductive secondary circuit; or, inversely, at the same secondary current I I the secondary terminal voltage E with lagging current is less and with leading current more, than with non-inductive secondary circuit, at the same primary impressed e.m.f. E . The calculation of numerical values is not practicable by measurement from the diagram, since the magnitudes of the different quantities are too different, Ef: Ef: E^. E Q being frequently in the proportion 1 : 10 : 100 : 2000. 76 ELEMENTS OF ELECTRICAL ENGINEERING. Trigonometrieally, the calculation is thus : In triangle OEE V Fig. 35, writing we have, also, hence, OE? = OE 2 + EE\ -20EEE l eos WE, - 7 A t = 180 - 0' + 0^ = E* + J/^ 2 + 2 EI.z, cos (5' Eig. 37. Vector Diagram of Transformer with Non-Inductive Loading. This gives the secondary e.m.f., E v and therefrom the primary counter-generated e.m.f. In triangle EOE l we have sin Ef>E -r sin E t EO = EE l ~ W^C thus, writing we have sin 6" -f- sin (0' - X ) = /^ -f- J? 1; wherefrom we get 0*, and # jE 1 0/ 1 = = 1 + /7 , ALTERNATING-CURRENT TRANSFORMER 77 the phase displacement between secondary current and secondary e.m.L In triangle 0/ 00 / we have since and * 2 - 2 0/ 00 / 00 / cos E> = 90, 0/ 00 / = 90 + + a, 07 00 = 7 00 == exciting current, Fig. 38. Vector Diagram of Transformer with Leading Load Current. calculated from the dimensions of the magnetic circuit. Thus the primary current is V = V + a'/, 2 + 2 ay oo sin (0 + a). In triangle 0/ 00 7 we have writing this becomes sin 7 00 0/ -s- sin 0/ 00 7 = 7 00 7 -f- 0/ ; 7 00 07 t 4- 7 ; sn Q -T- sn + <* therefrom we get 6", and thus 4 E'Ol* = ^ 2 = 90 - a - ". In triangle OE'E^ we have QE* - /2 + S'jBo 2 - 2 OJS' JB'So cos OE' E Q 78 ELEMENTS OF ELECTRICAL ENGINEERING. writing tan ' = ^ y T Q we have 4 OE'E, - 180 - ff + Ov E. OE' Ei == -f- ' E'OE. = thus the impressed e.m.f. is ^o 2 = % Ci In triangle OE'E 9 sin S'OSo -^ sin thus, writing we have sin 0/' ^ sin herefrom we get #/', and the phase displacement between primary current and impressed e.m.f. As seen ; the trigonometric method of transformer calculation is rather complicated. 64. Somewhat simpler is the algebraic method of resolving into rectangular components. Considering first the secondary circuit, of current I t lagging behind the terminal voltage E by angle # r The terminal voltage E has the components E cos # x in phase, E sin 9 i in quadrature with and ahead of the current / r The e.m.f. consumed by resistance r v I l r v is in phase. The e.m.f. consumed by reactance x v I l x v is in quadrature ahead of 7 r Thus the secondary e.m.f. has the components E cos d l + I l r l in phase, E sin i + /^ in quadrature ahead of the current I v and the total value, cos + /)* + (S sin tf + /x) 2 ALTERNATING-CURRENT TRANSFORMER. T9 and the tangent of the phase angle of the secondary circuit is , - E sin 6. + Lx. tan = - 1 ^ * E cos O l + //i Resolving all quantities into components in phase and in quadrature with the secondary e.m.f. E v or in horizontal and in vertical components, choosing the magnetism or mutual flux as vertical axis, and denoting the direction to the right and upwards as positive, to the left and downwards as negative, we have HORIZONTAL VERTICAL COMPONENT. COMPONENT. Secondary current, I v 7 l cos 6 7 t sin 9 Secondary e.m.f., E v E t Primary counter-generated e.m.f., T,7 "C1 p -&, &y n J fo i -. __i_ ; L, y a a Primary e.m.f. consumed thereby, TJV 77? t ~l A j& = jCn, T" U a Primary load current,- 7' a/ 17 + a7 t cos 6 + a7 x sin 6 Magnetic flux, <, $ Primary exciting current, 7 00 , con- sisting of core loss current, 7 00 sin a Magnetizing current, 7 00 cos a Hence, total primary current, 7 , HORIZONTAL COMPONENT. VERTICAL COMPONENT. a7 1 cos d l + 7 00 sin a a/ 1 sin O l + 7 00 cos a E.m.f. consumed by primary resistance r , E ' 7 r in phase with 7 , HORIZONTAL COMPONENT, VERTICAL COMPONENT. r a7 1 cos 6 + r 7 00 sin a r al t sin + r 7 00 cos a E.m.f. consumed by primary reactance rc , E Q = 7 x , 90 ahead of 7 , HORIZONTAL COMPONENT. VERTICAL COMPONENT. X a7 t sin /? + x 7 00 cos a x a7 1 cos 9 x / 00 sin a E.m.f. consumed by primary generated e.m.f., E f L a horizontal. 80 ELEMENTS OF ELECTRICAL ENGINEERING. The total primary impressed e.m.f., E Q , HORIZONTAL COMPONENT. PI + a/ 1 (r cos +X sin 6) + 7 00 (> sin a + x cos a). ci VERTICAL COMPONENT. a/ x (r sin x cos 5) + / 00 (r cos a X Q sin a), or writing tan ' = ? r o since _____________ <^ ^ Vr 2 + x 2 = 2 07 sin ' = -^ , and cos ' = - #o o Substituting this value, the horizontal component of E is ^ + a^J, cos ( - O + ^oo sin (a +(?/); tt the vertical component of J5 is og^ sin (6 - d,'} + z / 00 cos (a + fl/), and, the total primary impressed e.m.f. is #0= v F'~+azo/iC ^ _ -_ - Combining the two components, the total primary current is J v / (a/ 1 cos ^ + / 00 sin a) 3 + (al sin 0 + J K + 0). (9) The e.m.f. consumed by primary resistance r is r o/o = r o ( ai i + A) + jV (ai a + 0). (10) The horizontal component of primary current (ai\ 4- A) gives as e.m.f. consumed by reactance x a negative vertical com- ponent, denoted by jx (ai l + A). The vertical component of primary current / (ai t + gr) gives as e.m.f. consumed by reac- tance X Q a positive horizontal component, denoted by x (ai 9 +g). Thus the total e.m.f. consumed by primary reactance x is x (ai 2 + flf) jx (ai t 4- A), (11) and the total e.m.f. consumed by primary impedance is rfrtn JL, Tf]\ J_ /y (fin J_ /Tf\ J.. /|* Fy ^/yV J- /i^ y //Y1* JL /)M /*1 O^ o \ ai/ i i ri ) ' x o v at 2 -T yj "T j L" v^a y/ ^o ^i "* *vJ* \*^J Thus, to get from the current the e.m.f. consumed in reac- tance X Q by the horizontal component of current; the coefficient RECTANGULAR COORDINATES. 85 j has to be added; in the vertical component the coefficient j omitted; or, we can say the reactance is denoted by jx Q for X the horizontal and by -? for the vertical component of current. 1 In other words, if / = i -f ji r is a current, x the reactance of its circuit, the e.m.f. consumed by the reactance is jxi + xi f = xi f jxi. 67. If instead of omitting j in deriving the reactance e.m.f. for the vertical component of current, we would add j also (as done when deriving the reactance e.m.f. for the horizontal component of current), we get the reactance e.m.f. jxi fxi'j which gives the correct value jxi + xi! ', if f = -1; (13) that is, we can say, in deriving the e.m.f. consumed by reactance, x, from the current, we multiply the current by jx, and substitute f = 1. By defining, and substituting, f = 1, jx can thus be called the reactance in the representation in rectangular coor- dinates and r jx the impedance, The primary impedance voltage of the transformer in the preceding could thus be derived directly by multiplying the current, 7 = (ai, + K) + j(ai 2 + g), (9) by the impedance, Z Q = r - jx which gives o' = Z*! o - fro - Fo) [K + h) + j (ai, + g)] - f (ai t + h) + j> (ai 2 + g} - jx Q (ai t -f h) - f x, (at, + g), and substituting f 1, Ed = fro ( ai i + K) + X Q (ai 2 +g)] + j [r (ai a + g) - x (at\ 4- A)], (14) and the total primary impressed e.m.f. is thus Eo - ^ + #o' " ( ai 2+9 r ) J +? [ r o( ai 2+9 f ) -^ ( ai + ^ (15) 86 ELEMENTS OF ELECTRICAL ENGINEERING. 68* Such an expression in rectangular coordinates as / = i + jV (16) represents not only the current strength but also its phase. It means, in Fig. 39, that the total current 01 has the two rectangular components, the hori- zontal component / cos = i and the vertical component /sin = i f . Thus, i f Fig. 39. Magnitude and Phase tantf=;r ? (17) in Rectangular Coordinates. that is, the tangent function of the phase angle is the vertical component divided by the horizontal component, or the term with prefix / divided by the term with- out j. The total current intensity is obviously / = VV + i' 2 - (18) The capital letter / in the symbolic expression J = i + ji f thus represents more than the / used in the preceding for total current, etc., and gives not only the intensity but also the phase. It is thus necessary to distinguish by the type of the latter the capital letters denoting the resultant current in symbolic expres- sion (that is, giving intensity and phase) from the capital letters giving merely the intensity regardless of phase; that is ; 7 i + ji' denotes a current of intensity and phase tan = i In the following, clotted italics will be used for the symbolic expressions and plain italics for the absolute values of alternating waves. In the same way z = Vr 2 + x 2 is denoted in symbolic repre- sentation of its rectangular components by Z - r - jx. (19) RECTANGULAR COORDINATES. 87 When using the symbolic expression of rectangular coordinates it is necessary ultimately to reduce to common expressions. Thus in the above discussed transformer the symbolic expres- sion of primary impressed e.m.f. Eo =T~ + r (ai, +%) + x (ai, + g) ] + / f"r (ai 2 + g] -x,(ai, + h) J (15) means that the primary impressed e.m.f. has the intensity = V [a and the phase tan (21) This symbolism of rectangular components is the quickest and simplest method of dealing with alternating-current phenom- ena, and is in many more complicated cases the only method which can solve the problem at all, and therefore the reader must become fully familiar with this method. EXAMPLES. 69. (1.) In a 20-kw. transformer the ratio of turns is 20 : 1, and 100 volts are required at the secondary terminals at full load. What is the primary current, the primary impressed e.m.f., and the primary lag, (a) at non-inductive load, <9j= 0; (6) with O t = 60 degrees time lag in the external secondary circuit; (c) with Q l = 60 degrees time lead in the external secondary circuit? The exciting current is 7 00 ' == 0,3 + 0.4 j ampere, at e = 2000 volts impressed, or rather, primary counter-generated e.m.f. The primary impedance, Z Q = 2 5 j ohms. The secondary impedance, Z l = 0.004 0.01 j ohm. We have, in symbolic expression, choosing the secondary current Jj as real axis the results calculated in tabulated form on page 88. 88 ELEMENTS OF ELECTRICAL ENGINEERING. ^ CO g Q CO c ^ CO O T r II CX D rH r- . "^ cc 3 1>- II o C5 tr " O Cv 1 . . c .^ CD CC D CM '^ CO O5 ^ r- 1 o "~t~ oc co >"- t* l>- ^ CO 3 o , CM CO ^S | <* ci + - H CO -0 D rH O ^ - + "* H- ^ CM CO ^ ' c n* * > 3 -f- rH rH tf - O rH CO Tfl 00 Xtl 3 O OJ CM >JC CO ^ c J > CO o >o D CO CO .^ 00 rH Cft C rH OO ^^ ^ + H - + + I w J c 3 O O "* H Jf- T H If 5 ^ CD 1 CO 1 "? o rH co o CX D CO CD 1 > a \ <^ 1 CM* CO o ir ( 5 t^I CO CM c > rH CO CO ,-4 '^ ~ |>. II i -'- 1 co co cc > ?i- i be CO CC > g* '^ co CD | CC ) ^ 1> f. CM o So o TH cr cc 5 > J o co _. '"^ CO OC CM 00 OC OO O OC 3 t*- rs 3 rH 2 3 CO Q fe C\ 1 1 S"- ^sssJ. 00 ** CO ^ '7 sg I : ^^^ >o o > -* O co i + CM oo rH c c H ) ) <=> ** s JO "* 5 rH ^ d o" dcv > CO 1 CO co t-* c i CO c ) CO 1 d S ^ ^ *^-i Q Cv o l>- ZH CV ' TJH P^CO O C 1 ^ CO p, CM J> CO "^os d cc 00 || | i o -.f*-. *c^ o Xt< CM t- CO i J> CO "t" O CO O rH C + -Hx"f ^ C33 , OO O 1 COCC e3i r5 O5 CO C> O >CO CM 00 Sb d CO fc a C" 1 o c i " ? J > rH CO ^1 CN, rH 53 1 rH CO *"* CC CO ' co ^ o od 1 1 ^jT + + c 4 G > o o O f* C CO !ZJ 1 CM O i IO o CM CC CO 1 o c d 1 CM CM II ^ I a - CM. ^ **^ jl -f ll o fc**3 cf H g a 3 S w ii 4i 6q. o II J' S r" O r o 15 * II 4H CM 4J 1 v 11 kQ" p oj &"< s fi S r * II w CD CO cf S J^5 1^ o 8 CZ> ^, CD CM || fefl CJ CO d w s || a -2 ^^ II 5 ft JM " ^-> a_ cj Sk Jr. 8 .sg i|lf 8 Sll^&i 85*51;! bb&o4 g X* fc a e 1 g s i s | ^ b E ^ 2 . g s 1 K- 8 -l ! - T3 C, 'c 1 ^ S ^ c g *9 JTH 02% H b b b^s s! d - -. S^S J 'C "C "E A 44 " > - O< P^ PH H 8^|| *E ft - a -, cu r S cj M & '1 H W ^ fc* t cJ rf *c * *n ft ^ft 1 "O 1 c; 1 i RECTANGULAR COORDINATES. 89 + C coo T o o i4-?d o 1 j_ ' o CM ~r + 1 '*H* S ^ *? Sgg o o d fsss 1 o o o> odd 4- * o o o C40O fr- ** ^ r-4 . -H o QO t^ t-^ "" oo + o cn O3C3^ T ~'a3oo' iii ~' .g- g 5 O 2 o> J 52 '43 d ra w 03 S CD _Q T-J H So .D fl ^ fr *Sl fl c3 > - O> H s i-g *. . +Nf . . * - - M3 OO *"! !- : - S ..^ Cr 3 - - rt-" W * ? + ^- -TJJf- - 2 n - 0; VjE? 2 - aW substituting, 2: and expanding, gives e 3 = (r 2 + x 2 ) i 2 (8) = z 2 ! 3 ; hence, () e jrf, and -. = z. (9) V P T-= r 4 is the resistance or effective resistance of the receiving % circuit; that is, the maximum power is delivered into a non- inductive receiving circuit over an inductive line upon which is LOAD CHARACTERISTIC OF TRANSMISSION LINE. 93 impressed a constant e.m.f., if the resistance of the receiving circuit equals the impedance of the line, r l = z. In this case the total impedance of the system is Z Q = Z + r, = r + z-jx, (10) _ z = V (r + z) 2 + x*. (11) Thus the current is V (r + z) 2 + and the power transmitted is (12) that is, the maximum power which can be transmitted over a line of resistance r and reactance x is the square of the impressed e.m.f. divided by twice the sum of resistance and impedance of the line. At x 0, this gives the common formula, Inductive Load. 74. With an inductive receiving circuit of lag angle 0, or power factor p = cos Q, and inductance factor q = sin 0, at e.m.f. E = e at receiving circuit, the current is denoted by /- J(P + W); (15) thus the e.m.f. consumed by the line impedance Z ~ r jx is ^-^/-/(p+y^Cr-jx) = I [(rp + xq) + j (rq - xp)], and the generator voltage is J? - E + E, [6 + / (rp 4- xg)] + j7 (rq - xp) ; (16) 94 ELEMENTS OF ELECTRICAL ENGINEERING. or, reduced, + P(rq-xp}\ (17) and e = VX, 2 -P(rq-xp) T - I (rp + xq). (18) The power received is the e.m.f. times the power component of the current; thus P elp = /P VE, 2 - P (rq - xpY - Pp (rp + xq). (19) The curve of e.m.f., e } as function of the current / is again an arc of an ellipse. At short-circuit e = 0; thus, substituted, E I = ^S (20) 2 the same value as with non-inductive load, as is obvious. 75. The condition of rnaximum output delivered over the line is that is, differentiated, &7 - P (rq - xpY - e + I (rp + xq); (22) substituting and expanding, e 2 = P (r 2 + or 5 ) e = Iz'j or j-z. (23) /> z l j is the impedance of the receiving circuit; that is, the power received in an inductive circuit over an inductive line is a maximum if the impedance of the receiving circuit, z v equals the impedance of the line, z. In this case the impedance of the receiving circuit is Z l = z (p - jg), (24) LOAD CHARACTERISTIC OF TRANSMISSION LINE. 95 and the total impedance of the system is f7 T7 i /7 jfj s=s j 4- jfj = r - jx + z (p - jg) = (r + J)2) j (x + gs Thus, the current is J, and the power is V(r + pzY + (x + (25)' (r + + (x + 2(z + rp + xq) (26) EXAMPLES. 76. (1.) 12,000 volts are impressed upon a transmission line of impedance Z = r jx = 20 50 j. How do the voltage and the output in the receiving circuit vary with the current with non-inductive load? ^ ^ VOLTS 11000 ^ \ KW. 1000 10000 ^s \ / ^~m "X 900 9000 / \ \ 800 8000 / \ \ \ 700 7000 y / ^ \ 000 0000 / \ \ \ 500 5000 i / \ \ 400 4000 / \ \ 300 3000 / / \ \\ 200 2000 / M 100 1000 L M P. 1 ,20 40 CO 80 100 120 140 160 180 200 220 40, Non-reactive Load Characteristics of a Transmission Line. Constant Impressed e.m.f. 96 ELEMENTS OF ELECTRICAL ENGINEERING. Let e = voltage at the receiving end of the line, i = current: thus = ei = power received. The voltage impressed upon the line is then E Q = e + Zi = e + ri jxi', or ; reduced, ri) 2 Since E Q = 12,000, 12,000 - V(e + n) 2 + x 2 r = V(e + 20 i) 2 + 2500 r," e = Vi - x 2 ! 2 - n" = Vl^OOl) 3 -250iF?"- 20 1. The maximum current f or e = is thus, - Vl2,000 2 - 2,500 f - 20 i; i = 223. Substituting for i gives the values plotted in Fig. 40. V. e. p = t'i. 12,000 20 11,500 230X10 3 40 11,000 440X10 3 60 10,400 624X10 3 80 9,700 776X10 3 100 8,900 890X10 3 120 8,000 960X 10 3 140 6,940 971X10 3 160 5,750 920 X10 3 180 4,340 784X 10 3 200 2,630 526X10 3 220 400 88X 10 3 223 16. PHASE CONTROL ON TRANSMISSION LINES. 77. If in the receiving circuit of an inductive transmission line the phase relation can bo changed, the drop of voltage 1 in the line can be maintained constant at varying loads or even decreased with increasing load; that is, at constant generator voltage the transmission can be compounded for constant volt- age at the receiving end, or even over-compounded for a voltage increasing with the load. PHASE CONTROL ON TRANSMISSION LINES. 97 i. Compounding of Transmission Lines for Constant Voltage. Let r = resistance, x = reactance of the transmission line, e = voltage impressed upon the beginning of the line, e = volt- age received at the end of 88" line. Let i = power current in the receiving circuit; that is, P = ei = transmitted power, and \ = reactive current produced in the system for controlling the voltage. i l shall be considered positive as lagging, negative as leading current. Then the total current, in symbolic representation, is the line impedance is Z = r - jx, and thus the e.m.f. consumed by the line impedance is E 1 ^ ZI = (r-jx) (i + ji 1 ) = ri -f jn\ jxi fxi^ and substituting f = 1, E l = (ri + xtj) + j (n'j xi). Hence the voltage impressed upon the line E Q = e + EI = (e + ri + xij + j (n\ - o;i); (1) or, reduced, e = V(e + ri + xi^ + (n\ - xi) 2 . (2) If in this equation e and e Q are constant, t\, the reactive component of the current, is given as a function of the power component current i and thus of the load ei. Hence either e Q and e can be chosen, or one of the e.m.fs. e or e and the reactive current ^ corresponding to a given power current i. 78. If i 1 = with i = 0, and e is assumed as given, e = e, Thus, e = V(e + ri + xij 2 + (ri l - xi) 2 ; 2 e (ri + og + (r 3 + z 2 ) (t 2 - ^ 2 ) = 0. 98 ELEMENTS OF ELECTRICAL ENGINEERING. From this equation it follows that ex \/e 2 x* 2 eriz 2 Thus, the reactive current i l must be varied by this equa- tion to maintain constant voltage e = e irrespective of the load ei. As seen, in this equation, i l must always be negative, that is, the current leading. i l becomes impossible if the term under the square root becomes negative, that is, at the value iV = 0; At this point the power transmitted is This is the maximum power which can bo transmitted with- f> (%t Tnam ff J out drop of voltage in the line with a power current i = *~y '-> . z The reactive current corresponding hereto, since the square root becomes zero, is thus the ratio of reactive to power current, or the tangent of the phase angle of the receiving circuit, is z r (7) ^ * A larger amount of power is transmitted if 6 is chosen > e, a smaller amount of power if e Q < e. In the latter case i t is always leading; in the former case h is lagging at no load, becomes zero at some intermediate load, and leading at higher load. 79. If the line impedance Z r f x and the received voltage e is given, and the power current i at which the reactive PHASE CONTROL ON TRANSMISSION LINES. 99 current shall be zero, the voltage at the generator end of the line is determined hereby from the equation (2) : by substituting i 1 7 i = i , 6 = V(e + n ) 2 + x\\ (8) Substituting this value in the general equation (2) : e o " *( e + n " + X H + (n^ ^* gives (e + n* ) 2 + x\' 2 = (e + ri + xi,) 2 + (n\ - xt) 2 (9) as equation between i and i r If at constant generator voltage e , at no load i = 0, e = 6 , i x - i and at the load (10) /_i f> p I / v*o* ^ i -4 / T | 'j ' ( P -I'- 7"J i - | / y* o * (5 T -t't/Q / r v ^o V ~ O/ r -^ ''o ? or, expanded, This equation gives i Q ' as function of i , e , r, x. If now the reactive current i l varies as linear function of the power current i, as in case of compounding by rotary converter with shunt and series field, it is Substituting this value in the general equation (e + n ) 2 + AY >= (e + ri + xi^f + (n\- xi) 2 gives e as function of i; that is, gives the voltage at the receiving 100 ELEMENTS OF ELECTRICAL ENGINEERING. end as function of the load, at constant voltage e at the gener- ating end, and e = for no load, i = 0, i, = i/, and e = e for the load, i = i , ^ = 0. Between i = and i i , e > e , and the current is lagging. Above i = %, e < e 07 and the current is leading. By the reaction of the variation of e from e upon the receiv- ing apparatus producing reactive current i v and by magnetic saturation in the receiving apparatus, the deviation of e from e Q is reduced, that is, the regulation improved. 2. Over-compounding on Transmission Lines. 80. The impressed voltage at the generator end of the line was found in the preceding, e, = V(e + ri + xi^ + (n\ - xi)\ (2) If the voltage at the end of the line e shall rise proportionally to the power current i, then e = e l + ai; (15) thus, e Q = v / [e 1 + (a + r) i+ xij* + (ri, - xt) 3 , (16) and herefrom in the same way as in the preceding we get the characteristic curve of the transmission. If e v i l = at no load, and is leading at load. If e o < e v l \ ^ s Always leading, the maximum output in lens than before. , If e > e v i t is lagging at no load, becomes J^ero at some intermediate load, and leading at higher load. The maximum output is greater than at # e r The greater a, the less is the maximum output at the same e and e r The greater e &? the greater is the maximum output at the same e l and a, but the greater at the same time the lagging cur- rent (or less the leading current) at no load. PHASE CONTROL ON TRANSMISSION LINES. 101 EXAMPLES, 81. (1.) A constant voltage of e Q is impressed upon a trans- mission line of impedance Z = r jx = 10 20 j. The volt- age at the receiving end shall be 10,000 at no load as well as at full load of 75 amp. power current. The reactive current in the receiving circuit is raised proportionally to the load, so as to be lagging at no load, zero at full load or 75 amp., and lead- ing beyond this. What voltage e has to be impressed upon the line, and what is the voltage e at the receiving end at J, f, and 1J- load? Let 7 = i\ + ji 2 =- current, E e voltage in receiving circuit. The generator voltage is then E Q = e + ZI - e + (r - jx) (i, + ji 2 ) = (e + n\ + xi 2 ) + j (n' 3 xij = (e + 10 i t + 20 i 2 ) + j (10 i 2 - 20 ^); or, reduced, = (e + 10 i, + 20 g* + (10 i\ - 20 i,)\ When h ^ 75, i 2 = 0, e 10,000; substituting these values, e 2 - 10,750 2 + 1500 2 = 117.81 X 10 8 ; hence, e = 10,860 volts is the generator voltage. When ^ = 0, e = 10,000, e = 10,860, Ieti 2 = i; these values substituted give 117.81 X 10 6 - (10,000 + 20 i? + 100 i 2 = 100 X 10 6 + 400 i X 10 3 + 500 i 3 , or, i = 44.525 -1.25ZMO-" 3 ; this equation is best solved by approximation, and then gives i s, p = 42.3 amp. reactive lagging current at no load. Since ^Q 2 = (^ + Tl i + 2^* 2 ) 2 + (^"2 ^\) 2 ? 102 ELEMENTS OF ELECTRICAL ENGINEERING. it follows that e (n' 3 xij 2 (ri l 4- a^ 3 ); or, e = V117.81 X 10 6 - (10 i a - 20 i>) 2 - (10 i, + 20 Substituting herein the values of i 1 and i 2 gives e. V 2 . e. 42.3 10,000 25 28.2 10,038 50 14.1 10,038 75 10,000 100 -14.1 9,922 125 -28.2 9,803 82. (2.) A constant voltage e Q is impressed upon a trans- mission line of impedance Z = r jjc = 10 10 j. The volt- age at the receiving end shall be 10,000 at no load as well as at full load of 100 amp. power current. At full load the total current shall be in phase with the e.m.f. at the receiving end, and at no load a lagging current of 50 amp. is permitted. How much additional reactance x is to be inserted, what must be the generator voltage e 07 and what will be the voltage e at the receiv- ing end at | load and at 1,J load, if the reactive current varies proportionally with the load? Let = additional reactance inserted in circuit. Let I = i + i = current. Then e* - (e + n\ + x,^) 2 4- (n' 3 - x^) 2 ==* (e + 10 i t + x^) 3 + (10 i 2 - x^) 2 , where x t = x + x total reactance of circuit between e and e . At no load, ^ = 0, i 2 = 50, e - 10,000; thus, substituting, e 2 (10,000 + 50 x t ) 2 + 250,000, PHASE CONTROL ON TRANSMISSION LINES. 103 At full load, i t = 100, i 2 = 0, e = 10,000; thus, substituting, e, 2 = 121 x 10 6 + 10,000 x t 2 . Combining these gives (10,000 + 50 x,Y + 250,000 = 121 X 10 6 + 10,000 x, 2 ; hence, Xl = 66.5 40.8 = 107.3 or 25.7; thus X Q = Zj x = 97.3, or 15.7 ohms additional reactance. Substituting x, = 25.7 gives e * ( e + 10 1\ + 25.7 1' 2 ) 2 + (10 i a - 25.7 ^) 3 ? but at full load i x = 100, i z = 0, e = 10,000, which values substituted give e 2 - 121 X 10 6 + 6.605 X 10 6 - 127.605 X 10 6 , e = 11,300, generator voltage. Since e - \/e 2 - (10 i 2 - 25.7 i,) 2 - (10 i, + 25.7 i 2 ), it follows that 6 = V127.605 X 10 6 - (10 i 2 - 25.7 i 3 ) 2 - (10 i, + 25.7 i 2 ). Substituting for i t and i 3 gives e. v H- e. 50 10,000 50 25 10,105 100 10,000 150 -25 9,658 104 ELEMENTS OF ELECTRICAL ENGINEERING. 83. (3.) In a circuit whose voltage e fluctuates by 20 per cent between 1800 and 2200 volts, a synchronous motor of internal impedance Z Q = r jx Q = 0.5 5 j is connected through a reactive coil of impedance Z 1 = T\ jo^ = 0.5 10 / and run light, as compensator (that is, generator of reactive currents). How will the voltage at the synchronous motor terminals e v at constant excitation, that is, constant counter e.m.f. e = 2000, vary as function of e at no load and at a load of i = 100 amp. power current, and what will be the reactive current in the synchronous motor? Let / = i t + ji 2 = current in receiving circuit of voltage e v Of this current /, ji a is taken by the synchronous motor of counter e.m.f. e, and thus E l = e 4- Z Ji a - e + xfa + jr i a ; or, reduced, ^ = (e + * o g 2 + r \ 2 . In the supply circuit the voltage is E - ^ + /^ - [e + r^ + (x + x t ) ij + j [(r + r t ) i a - ^ij; or, reduced, e 2 - [e + r^ + (x + x,) t a ] + [(r + r t ) f 3 - x^] 8 . Substituting in the equations for e t 2 and e 2 the above values of r and x : at no load, f t = 0, we have 6l ( e + 5 i) 2 + 0.25 i 2 2 and e, 2 - (e + 15 r 2 ) 2 + i 2 3 ; at full load., i l = 100, we have e ? ( e + 5 i 2 ) 2 + 0.25 i 2 2 , ^ 3 - (e + 50 + 15 i* 3 ) 2 + (i 2 - 1000) 2 , and at no load, i i 0, substituting e = 2000, we have e? - (2000 + 5g 2 + 0.25i/, e 2 - (2000 4- 15 g 3 +tV; PHASE CONTROL ON TRANSMISSION LINES. 105 at full load, i l 100, we have e* = (2000 + 5g 2 + 0.25t' 2 2 , e* = (2050 + 15 g 3 + (i 2 - 1000) 2 . Substituting herein e Q = successively 1800, 1900, 2000, 2100, 2200, gives values of i v which, substituted in the equation for e i> gi ye ^e corresponding values of e { as recorded in the follow- ing table. As seen, in the local circuit controlled by the synchronous compensator, and separated by reactance from the main circuit of fluctuating voltage, the fluctuations of voltage appear in a greatly reduced magnitude only, and could be entirely eliminated by varying the excitation of the synchronous compensator. e = 2,000. No load, tj = 0, Full load, t\ = 100, C Q' 1 V e i- V e r 1,800 -13.3 1,937 -39 1,810 1,900 - 6.7 1,965 -30.1 1,850 2,000 2,000 -22 1,885 2,100 + 6.7 2,035 -13.5 1,935 2,200 + 13.3 2,074 - 6.5 1,970 17. IMPEDANCE AND ADMITTANCE. 84. In direct-current circuits the most important law is Ohm's law, e e -- ir, or e or r where e is the e.m.f. impressed upon resistance r to produce current i therein. Since in alternating-current circuits a current i through a resistance r may produce additional e.m.fs. therein, when apply- p ing Ohm's law, i - to alternating current circuits, e is the total e.m.f. resulting from the impressed e.m.f. and all e.m.fs. produced by the current i in the circuit. Such counter e.m.fs. may be due to inductance, as self-induc- tance, or mutual inductance, to capacity, chemical polarization, etc. 106 ELEMENTS OF ELECTRICAL ENGINEERING. The counter e.m.f. of self-induction, or e.m.f. generated by the magnetic field produced by the alternating current ij is repre- sented by a quantity of the same dimensions as resistance, and measured in ohms: reactance x. The e.m.f. consumed by reactance x is in quadrature with the current, that consumed by resistance r in phase with the current. Reactance and resistance combined give the impedance, z = Vr 2 + a- 2 ; or, in symbolic or vector representation, Z = r - jx. In general in an alternating-current circuit of current i y the e.m.f. e can be resolved in two components, a power component e, in phase with the current, and a wattless or reactive com- ponent e 2 in quadrature with the current. The quantity 1 power e.m.f., or e.m.f. in phase with the current ___ i current l is called the effective resistance. The quantity e^ _ reactive e.m.f., or e.m.f. in quadrature with the current _ i current l is called the effective reactance of the circuit. And the quantity z l = Vr/ + x * or, in symbolic representation, is the impedance of the circuit. If power is consumed in the circuit only by the ohmic resist- ance r, and counter e.m.f. produced only by self-inductance, the effective resistance r 1 is the true or ohmic resistance r, and the effective reactance x l is the true or inductive reactance x. By means of the terms effective resistance, effective reactance, arid impedance, Ohm's law can be expressed in alternating- current circuits in the form e e /i\ f^ SSS """" SSS rTt.-.-i.-.-._;i,.i..M-w.T.mTrmnr,nnm M ( X 1 *i Vr7 + x> IMPEDANCE AND ADMITTANCE. 107 or, e = iz l i Vr t 2 + re, 2 ; (2) or, 2, = V?7T^ 2 = J; (3) 6 or, in symbolic or vector representation, or, = 7^ = /(r,-^); (5) EI or, Z, = r x - jX = y (6) In this latter form Ohm's law expresses not only the intensity but also the phase relation of the quantities; thus e i=5 {r 1= = power component of e.m,f., 6 2 = ix 1 = reactive component of e.m.f. 6 85. Instead of the term impedance z == - with its components, % the resistance and reactance, its reciprocal can be introduced, which is called the admittance. The components of the admittance are called the conduc- tance and susceptance. Resolving the current i into a power component i l in phase with the e.m.f. and a wattless component i 2 in quadrature with the e.m.f., the quantity h power current, or current in phase with e.m.f. _ e ~ e.m.f. ~~ g is called the conductance. The quantity L reactive current, or current in quadrature with e.m.f. , "* = f ' = e e.m.f. is called the susceptance of the circuit. The conductance represents the current in phase with the e.m.f, or power current, the susceptance the current in quad- rature with the e.m.f. or reactive current. 108 ELEMENTS OF ELECTRICAL ENGINEERING. Conductance g and susceptance 6 combined give the admittance + V; (7) or, in symbolic or vector representation, Y = g + jb. (8) Thus Ohm's law can also be written in the form i = ey = e vc/ 2 + 6 2 ; (9) or, e = 1 = t ; (10) y V/ + 6 2 or, (11) or, in symbolic or vector representation, 7 = EY - # (0 + j'6) ; (12) or, * - ^r ~Y+-^> or, y + j& - 4 ' ( W ) and i t =* eg = power component of current, ^ 2 = e5 = reactive component of current. 86. According to circumstances, sometimes the use of the terms impedance, resistance, reactance, sometimes the use of the terms admittance, conductance, susceptance, is more convenient. Since, in a number of series-connected circuits, the total e.m.f., in symbolic representation, is the sum of the individual e.m.fs., it follows that in a number of series-connected circuits the total impedance, in symbolic expression, is the sum of the impedances of the individual circuits connected in series. Since, in a number of parallel-connected circuits, the total current, in symbolic representation, is the mm of the individual currents, it follows that in a number of parallel-connected cir- cuits the total admittance, in symbolic expression, is the sum of the admittances of the individual circuits connected in parallel IMPEDANCE AND ADMITTANCE. 109 Thus in series connection the use of the term impedance, in parallel connection the use of the term admittance, is generally more convenient. Since in symbolic representation r = |; (is) or, ZY=1; (16) that is, (r -jz) (g + jb) = 1; (17) it follows that (rg + xb) +j (rb - xg) = 1; that is rg + zb = 1, rb xg = 0. (19) (20) (21) / T **/ O or, in absolute values, (22) zy = 1, (23) (r 2 + z 2 ) (^ 2 + fe 2 ) = 1. (24) Thereby the admittance with its components, the conduc- tance and susceptance, can be calculated from the impedance and its components, the resistance and reactance, and inversely. If x = 0, z = r and g = -, that is, g is the reciprocal of the resistance in a non-inductive circuit; not so, however, in an inductive circuit. EXAMPLES. 87. (1.) In a quarter-phase induction motor having an impressed e.m.f. e = 110 volts per phase, the current is 7 = h + JV* 8 10 + 10 ? at standstill, the torque = > . 110 ELEMENTS OF ELECTRICAL ENGINEERING. The two phases are connected in series in a single-phase cir- cuit of e.m.f. e = 220, and one phase shunted by a condenser of 1 ohm capacity reactance. What is the starting torque D of the motor under these con- ditions, compared with D , the torque on a quarter-phase cir- cuit, and what the relative torque per volt-ampere imput, if the torque is proportional to the product of the e.m.fs. impressed upon the two circuits and the sine of the angle of phase dis- placement between them? In the quarter-phase motor the torque is D = ae 2 12, 100 a, where a is a constant. The volt-ampere input is P o = 2 e Vi* + i 2 2 = 31,200; hence, the " apparent torque efficiency/ 7 or torque per volt- ampere input, The admittance per motor circuit is the impedance is Y = = 0.91 + 0.91 f, 110 (100- 100 fl _ " 7 100 + 100 j (100 + 100 j) (100- 100 j) J ' the admittance of the condenser is F = r * o J) thus, the joint admittance of the circuit shunted by the con- denser is 7, = Y + 7 = 0.91 + 0.91 f - j = 0.91 -0.09 j; its impedance is g _ 1 1 _ 0.91 + 0.09 j __ 1 Y, 0.91- 0.09 j 0.9P + 0.09 s I - uy + - 11 ?' and the total impedance of the two circuits in series is Z 2 = Z + Z, = 0.55 - 0.55 j + 1.09 + 0.11 / - 1.64 - 0.44 /. IMPEDANCE AND ADMITTANCE. Ill Hence, the current, at impressed e.m.f. e 220, I-i + n = e - 22Q __ 220 (1.64 +0.443) 1 J 2 Z 2 1.64 - 0.44 f 1.64 2 + 0.44 2 - 125 + 33.5 j; or, reduced, / - V125 2 + 33.5 2 = 129.4 amp. Thus, the volt-ampere input, P a = el - 220 X 129.4 = 28,470. The e.m.fs. acting upon the two motor circuits respectively are E l ^IZ 1 ^ (125 + 33.5 j) (1.09 + 0.11 j) = 132.8 + 50.4 j and E' - IZ = (125 + 33.5 j) (0.55 - 0.55 j) - 87.2 - 50.4 f. Thus, the tangents of their phase angles are ^o 4- tan ff, = + -5g| = + 0.30; hence, 6, = + 21; . tan 6' = - ^ = - 0.579; hence, ff = - 30; o7.2 and the phase difference, - ^ - fl x 51. The absolute values of these e.m.fs. are BI = V132.8 + 50.4 2 = 141.5 and e' - V87.2 2 - 50.4 2 = 100.7; thus, the torque is D == ae^' sin 6 - 11,100 a; and the apparent torque efficiency is Z) 11,100 a flOQ ""P. 28,470 ""* 39a - Hence, comparing this with the quarter-phase motor, the relative torque is D = 11,100 q = D 12,100 a ' 112 ELEMENTS OF ELECTRICAL ENGINEERING. and the relative torque per volt-ampere, or relative apparent torque efficiency, is J2L * 39a _ i 005 88. (2.) At constant field excitation, corresponding to a nominal generated e.m.f. e = 12,000, a generator of synchro- nous impedance Z Q = r - jx = 0.6 GO/ feeds over a trans- mission line of impedance Z l = r l /Xj = 1218 /, and of capacity susceptance 0.003, a non-inductive receiving circuit. How will the voltage at the receiving end, e, and the voltage at the generator terminals,^, vary with the load if the line capacity is represented by a condenser shunted across the middle of the line? Let / = i = current in receiving circuit, in phase with the e.m.f., E = e. The voltage in the middle of the line is .2 - . + Tj- . = e + 6 i 9 ij. The capacity susceptance of the line is, in symbolic expression, F = 0.003 /; thus the charging current is / 2 = E 2 Y = - 0.003 / (e + 6 i - 9 ij) = - 0.027 i - / (0.003 e + 0.018 i), and the total current is / t / + / 2 0.973 i - / (0.003 6 + 0.018 i). Thus, the voltage at the generator end of the Jine is = e + 6 i-9 r? + (6 - 9 j) [0.973 i - j (0.003 e + 0.018 i)] = (0.973 e + 11.68 i) -j (17.87 i + 0.018 e), and the nominal generated e.m.f. of the generator is SO-^H-ZO/, = (0.973 e + 11.68 i) - j (17.87 i + 0.018 e) + (0.6 - 60 j) [0.973 i - j (0.003 e + 0.018 i)] = (0.793 e + 1 1.18 i) - j (76.26 i + 0.02 e) ; or, reduced, and e = 12,000 substituted, e a 2 - 144 X 10" = (0.793 e + 11.18 1) 2 + (76.26 i + 0.02 e) 2 ; thus, and at at IMPEDANCE AND ADMITTANCE. e 2 + 33 ei + 9450 1 2 = 229 X 10 6 , e = - 16.5 i + V229 X 10 - 9178 i\ e, = V (0.973 e + 11.68 i) 3 + (17.87 i + 0.018 e) i = 0, e = 15,133, ^ - 14,700; e = 0, i = 155.6, e t = 3327. 118 2 POWER CURRENT REC'D AMP \ \ \ v 2000 1000 0000 0000 10 20 30 40 50 00 70 80 SO 100 110 120 130 140 150 Fig, 41. Reactive Load Characteristics of a Transmission Line fed by j Synchronous Generator with Constant Field Excitation. Substituting different values for i gives i e. e i* i. e. c r 15,133 14,700 100 10,050 11,100 25 14,488 14,400 125 7,188 8,800 50 13,525 13,800 150 2,325 4,840 75 12,063 12,730 155.6 3,327 which values are plotted in Fig. 41. 114 ELEMENTS OF ELECTRICAL ENGINEERING. 18. EQUIVALENT SINE WAVES. 89. In the preceding chapters, alternating waves have been assumed and considered as sine waves. The general alternating wave is, however, never completely, frequently not even approximately, a sine wave. A sine wave having the same effective value, that is, the same square root of mean squares of instantaneous values, as a general alternating wave, is called its corresponding " equivalent sine wave.' 7 It represents the same effect as the general wave. With two alternating waves of different shapes, the phase relation or angle of lag is indefinite. Their equivalent sine waves, however, have a definite phase relation, that which gives the same effect as the general wave, that is, the same mean (ei). Hence if e = e.m.f. and i current of a general alternating wave, their equivalent sine waves are defined by cos 6 = Vmean (e 2 ), i = Vmean (i 2 ); and the power is Po = Vo cos Vo *** mean (ei) ; thus, mean (ei) Vmean (e 2 ) Vmean (i 2 ) Since by definition the equivalent sine waves of the general alternating waves have the same effective value or intensity and the same power or effect, it follows that in regard to inten- sity and effect the general alternating waves can be represented by their equivalent sine waves. Considering in the preceding the alternating currents as equiva- lent sine waves representing general alternating waves, the investigation becomes applicable to any alternating circuit irrespective of the wave shape. The use of the terms reactance, impedance, etc., implies that a wave is a sine wave or represented by an equivalent sine wave. EQUIVALENT SINE WAVES. 115 Practically all measuring instruments of alternating waves (with exception of instantaneous methods) as ammeters, volt- meters, wattmeters, etc., give not general alternating waves but their corresponding equivalent sine waves. EXAMPLES. 90. In a 25-eycle alternating-current transformer, at 1000 volts primary impressed e.m.f., of a wave shape as shown in Fig. 42 and Table I, the number of primary turns is 500, the length of the magnetic circuit 50 cm., and its section shall be chosen so as to give a max- imum density (B = 15,000. At this density the hys- teretic cycle is as shown in Fig. 43 and Table II. _ What is the shape of ~~ Fig. 42. Wave-shape of e.m.f. in current wave, and what Example 90. the equivalent sine waves of e.m.f., magnetism, and current? The calculation is carried out in attached table. TABLE II. OtOO^OOiOiOOO^OOiOiOOC>OOO j.o CO i>-Or-tor*i-irHtoa i icoco-O-OSO J^ ^ -^ w ^ ' CO IOCO(MOC55COCOIOCOO(MCO .^ _ OCSOO^hOJLOCM rHi-HCMCOTtiCOOTHOOO 03 a "f 7 + + p-1 o o o < co'coc 1 1 I I I 1 OTHor^Ovtc^ccoosoo6cOO'-Ht > -CO'rH 10 CO CO T-l ^ CO CO CO CO ooooooooooooooooooo rHC (in megalines per pole), / the fre- quency of rotation (in hundreds of cycles per second), E the e.m.f. generated in the armature turns. This formula assumes a sine wave of e.m.f. If the e.m.f. wave differs from sine shape, the e.m.f. is E 4.44 rfn$ 3 2 V2 where r = form factor of the wave, or times ratio of ' 7T effective to mean value of wave, that is, the ratio of the effective 125 126 ELEMENTS OF ELECTRICAL ENGINEERING. value of the generated e.m.f . to that of a sine wave generated by the same magnetic flux at the same frequency. The form factor f depends upon the wave shape of the gener- ated e.m.f. The wave shape of e.m.f. generated in a single conductor on the armature surface is identical with that of the distribution of magnetic flux at the armature surface and will be discussed more fully in the chapter on commutating machines. The wave of total e.m.f. is the sum of the waves of e.m.f. in the individual conductors, added in their proper phase relation, as corresponding to their relative positions on the armature surface. 4. In a Y or star-connected three-phase machine, if E Q = e.m.f. per circuit, or Y or star e.m.f., E = E Q \/3 is the e.m.f. between terminals or A (delta) or ring e.m.f., since two e.m.fs. displaced by 60 degrees are connected in series between terminals (\/3 = 2 cos 30). In a A-connected three-phase machine, the e.m.f. per circuit is the e.m.f. between the terminals, or A e.m.f. In a F-connected three-phase machine, the current per cir- cuit is the current issuing from each terminal, or the line current, or Y current. In a A-connected three-phase machine, if / = current per circuit, or A current, the current issuing from each terminal, or the line or Y current, is Thus in a three-pnase system, A current and e.m.f., and Y current and e.m.f. (or ring and start current and e.m.f. respec- tively) , are to be distinguished. They stand in the proportion 1 -*- V3. As a rule, wnen speaking of current and of e.m.f. in a three- phase system, under current the F current or current per line, and under e.m.f. the A e.m.f. or e.m.f. between lines is understood. 5. While the voltage wave of a single conductor has the same shape as the distribution of the magnetic flux at the arma- ture circumference and so may differ considerably from a sine, that is, contains pronounced higher harmonics, the terminal voltage is the resultant of the waves of many conductors, and, especially with a distributed armature winding, shows the higher harmonics in a much reduced degree; that is, the resultant SYNCHRONOUS MACHINES. 127 is nearer sine shape, and some harmonics may be entirely elimi- nated in the terminal voltage wave, though they may appear in the voltage wave of a single conductor. Thus, for instance, in a three-phase F-connected machine, the voltage per circuit, or Y voltage, may contain a third harmonic and multiples thereof, while in the voltage between the terminals this third harmonic is eliminated. The voltage between the terminals is the resul- tant of two Y voltages, displaced from each other by 60 degrees. Sixty degrees for the fundamental, however, is 3 X 60 = 180, or opposition for the third harmonic; that is, the third harmonics in those two Y voltages, which combine to the delta or terminal voltage, are opposite, and so neutralize each other. Even in a single turn, harmonics existing in the magnetic field and thus in the single conductor can be eliminated by fractional pitch. Thus, if the pitch of the armature turn is not 180 degrees, but less by -, the e.m.fs. generated in the two con- Tl ductors Oi a single turn are not exactly in phase, but differ by - n of a half wave for the fundamental, and thus a whole half wave for the nih harmonic, so that their nth harmonics are in opposition and thus cancel. Fractional pitch winding of a "pitch defi- ciency " of thus eliminates the nth harmonic; for instance, n with 80 per cent pitch, the fifth harmonic cannot exist. In this manner higher harmonics of the e.m.f. wave can be reduced or entirely eliminated, though in general, with a dis- tributed winding, the wave shape is sufficiently close to sine shape without special precaution being taken in the design. II. Electromotive Forces. 6. In a synchronous machine we have to distinguish between terminal voltage E, real generated e.m.f. E v virtual generated e.m.f. EV and nominal generated e.m.f. E . The real generated e.m.f. E l is the e.m.f. generated in the alter- nator armature turns by the resultant magnetic flux, or mag- netic flux interlinked with them, that is, by the magnetic flux passing through the armature core. It is equal to the terminal voltage plus the e.m.f. consumed by the resistance of the arma- 128 ELEMENTS OF ELECTRICAL ENGINEERING. ture, these two e.m.fs. being taken in their proper phase relation; thus E, = E + lr, where / = current in armature, r = effective resistance. The virtual generated e.m.f. E 2 is the e.m.f. which would be generated by the flux produced by the field poles, or flux corre- sponding to the resultant m.m.f., that is, the resultant of the m.m.fs. of field excitation and of armature reaction. Since the magnetic flux produced by the armature, or flux of armature self-inductance, combines with the field flux to the resultant flux, the flux produced by the field poles does not pass through the armature completely, and the virtual e.m.f. and the real gen- erated e.m.f. differ from each other by the e.m.f. of armature self-inductance; but the virtual generated e.m.f., as well as the e.m.f. generated in the armature by self-inductance, have no real and independent existence, but are merely fictitious com- ponents of the real or resultant generated 'e.m.f. E r The virtual generated e.m.f. is E^E, -jlx, ~ where x is the self-inductive armature reactance, and the e.m.f. consumed by self-inductance Ix is to be combined with the real generated e.m.f. E l in the proper phase relation. 7. The nominal generated e.m.f. E Q is the e.m.f. which would be generated by the field excitation if there were neither self- inductance nor armature reaction, and the saturation were the same as corresponds to the real generated e.m.f. It thus does not correspond to any magnetic flux, and has no existence at all, but is merely a fictitious quantity, which, however, is very useful for the investigation of alternators by allowing the combination of armature reaction and self-inductance into a single effect by a (fictitious) self-inductance or synchronous reactance X Q . The nominal generated e.m.f. would be the terminal voltage with open circuit and load excitation if the saturation curve were a straight line. The synchronous reactance x is thus a quantity combining armature reaction and self-inductance of the alternator. It is the only quantity which can easily be determined by experiment by running the alternator on short-circuit with excited field. If in this case L = current, P = loss of power in the armature SYNCHRONOUS MACHINES. 129 coils, E Q = e.m.f. corresponding to the field excitation at open E P circuit, -y* = Z Q is the synchronous impedance, r-f = r is the "*o ^o effective resistance (ohmic resistance plus load losses), and = V^ 2 r 3 the synchronous reactance. In this feature lies the importance of the term " nominal generated e.m.f. " E Q , the terms being combined in their proper phase relation. In a polyphase machine, these considerations apply to each of the machine circuits individually. III. Armature Reaction. 8. The magnetic flux in the field of an alternator under load xis produced by the resultant m.m.f. of the field exciting current and of the armature current. It depends upon the phase rela- tion of the armature current. The e.m.f. generated by the field Fig. 46. Model for Study of Armature Reaction. Armature Coils in Position of Maximum Current. exciting current or the nominal generated e.m.f. reaches a max- imum when the armature coil faces the position midway between the field poles, as shown in Fig. 46, A and A'. Thus, if the armature current is in phase with the nominal generated e.m.f., it reaches its maximum in the same position A, A' of armature coil as the nominal generated e.m.f., and thus magnetizes the 130 ELEMENTS OF ELECTRICAL ENGINEERING. preceding, demagnetizes the following magnet pole (in the direction of rotation) in an alternating-current generator A; magnetizes the following and demagnetizes the preceding mag- net pole in a synchronous motor A! (since in a generator the rotation is against, in a synchronous motor with the magnetic attractions and repulsions between field and armature). In this case the armature current neither magnetizes nor demag- netizes the field as a whole, but magnetizes the one side, demag- netizes the other side of each field pgle, and thus merely distorts the magnetic field. 9. If the armature current lags behind the nominal generated e.m.f., it reaches its maximum in a position where the armature coil already faces the next magnetic pole, as shown in Fig. 46, B and J3', and thus demagnetizes the field in a generator B, magnetizes it in a synchronous motor B'. If the armature current leads the nominal generated e.m.f., it reaches its maximum in an earlier position, while the arma- ture coil still partly faces the preceding magnet pole, as shown in Fig. 46, C and C", and thus magnetizes the field in a gener- ator, Fig. 46, C, and demag- netizes it in a synchronous motor C". With non-inductive load, or with the current in phase with the terminal voltage of an alternating-current generator, the current lags behind the nominal generated e.m.f., due to armature reaction and self- inductance, and thus partly de- magnetizes; that is, the voltage is lower under load than at no load with the same field exci- tation. In other words, lagg- ing current demagnetizes and leading current magnetizes the field of an alternating-current generator, while the opposite is the case with a synchronous motor. 10. In Fig. 47 let OF= $ = resultant m.m.L of field exci- Fig. 47. Diagram of m.m.fs. in Loaded Generator. SYNCHRONOUS MACHINES. tation and armature current (the m.m.f. of the field excita- tion being alternating with regard to the armature coil, due to its rotation) and 6 2 the lag of the current / behind the virtual e.m.f. E 2 generated by the resultant m.m.f. The virtual_e.m.f. E 2 lags in time 90 degrees behind the result- ant flux of Off, and is thus represented by OE 2 in Fig. 47, and the m.m.f. of the armature current ff a by 03 a , lagging by angle # 2 behind OE T The resultant m.m.f. 0$ is the diagonal of the parallelogram with the component m.m.fs. 0$ a = armature m.m.f. and Off = total impressed m.m.f. or field excitation, as sides, and from this construction Off is found. Off is thus the position of the field pole with regard to the armature. It is trigonometrically, ff = v 2 + ff a 2 + 2 ffff a sin Oy If / = current per armature turn in amperes effective, n number of turns per pole in a single-phase alternator, the arma- ture reaction is ff a = nl ampere-turns effective, and is pulsating between zero and nl V2. In a quarter-phase alternator with n turns per pole and phase in series and / amperes effective per turn, the armature reaction per phase is nl amperes-turns effective and nl ^2 ampere-turns maximum. The two phases magnetize in quad- rature, in phase and in space. Thus, at the time , correspond- ing to angle after the maximum of the first phase, the m.m.f. in the direction by angle behind the direction of the magnetiza- tion of the first phase is nl \/2 cos 2 0. The m.m.f. of the second phase is nl \/2 sin 2 0] thus the total m.m.f. or the armature reaction ff a = nl V2, and is constant in intensity, but revolves synchronously with regard to the armature; that is, it is station- ary with regard to the field. In a thrce-phaser of n turns in series per pole and phase and / amperes effective per turn, the m.m.f. of each phase is nl \/2 ampere-turns maximum; thus at angle 6 in position and angle 6 in time behind the maximum of one phase; The m.m.f. of this phase is nl \/2cos 2 #. 132 ELEMENTS OF ELECTRICAL ENGINEERING, The m.m.f. of the second phase is nl \/2 cos 2 (6 + 120) = nl V2 ( - 0.5 cos 6 - 0.5 \/3 sin <9) 3 . The m.m.f. of the third phase is nl V2 cos 2 (0 + 240) = nl V2 ( - 0.5 cos + 0.5 V3 sin 0) 2 . Thus the total m.m.f. or armature reaction, 3 a = nl V2 (cos 2 6+ 0.25 cos 2 0+ 0.75 sin 2 6 + 0.25 cos 2 + 0.75 sin 2 0) - 1.5 n/ \/2, constant in intensity, but revolving synchronously with regard to the armature; that is, stationary with regard to the field. These values of armature reaction correspond strictly only to the case where all conductors of the same phase are massed together in one slot. If the conductors of each phase are dis- tributed over a greater part of the armature surface, the values of armature reaction have to be multiplied by the average cosine of the total angle of spread of each phase. 11. The single-phase machine thus differs from the poly- phase machine: in the latter, on balanced load, the armature reaction is constant, while in the single-phase machine the armature reaction and thereby the resultant m.m.f. of field and armature is pulsating. The pulsation of the resultant m.m.f. of the single-phase machine causes a pulsation of its magnetic field under load, of double frequency, which generates a third harmonic of e.m.f, in the armature conductors. In machines of high armature reaction, as steam turbine driven single-phase alternators, the pulsation of the magnetic field may be sufficient to cause serious energy losses and heating by eddy currents, and thus has to be checked. This is usually done by a squirrel- cage induction machine winding in the field pole faces, or by short-circuited conductors laid in the pole faces in electrical space quadrature to the field coils. In these conductors, second- ary currents of double frequency are produced which equalize the resultant m.m.f. of the machine. IV* Self-Inductance. 12. The effect of self-inductance is similar to that of arma- ture reaction, and depends upon the phase relation in the same manner. SYNCHRONOUS MACHINES. 133 If E l = real generated voltage, O l = lag of current behind generated voltage E v the magnetic flux produced by the arma- ture current / is in phase with the current, and thus the counter e.m.f. of self-inductance is in quadrature behind the current, and therefore the e.m.f. consumed by self-inductance is in quadrature ahead of the current. Thus in Fig. 48, denoting OE l = E l ^.j ie g enera ted e.m.f., the current is Fig. 48. Diagram of e.m.fs. in 01 = 7, lagging 6 l behind OE p the ^ Loaded Generator. e>m ^ con sumed by self-inductance OE^j is 90 degrees ahead of, the current, and the virtual gene- rated e.m.f. E v is the resultant of OE l and OE^'. As seen, the diagram of e.m.fs. of self-inductance is similar to the diagram of m.m.fs. of armature reaction. 13. From this diagram we get the effect of load and phase relation upon the e.m.f. of an alternating-current generator. Let E terminal voltage per machine circuit, / = current per machine circuit, and = lag of the current behind the terminal voltage. Let r = resistance, x = reactance of the alternator armature. Then, in the polar diagram, Fig. 49, OE = E, the terminal voltage, assumed as zero vector. 01 = /, the current, lagging by the angle EOI = 6. The e.m.f. consumed by resistance is OS/ = Ir in phase with 01. _ The e.m.f. consumed by reactance is OE 2 f = /#, 90 degrees ahead of OL _ _ The real generated e.m.f. is found by combining OE and OE ' to The virtual generated e.m.f. is OE 1 and OE 2 ' combined to 134 ELEMENTS OF ELECTRICAL ENGINEERING. The m.m.f. required to produce this e.m.f. JE7 a is OcF= , 90 cleg, ahead of OE r It is the resultant of the armature m.m.f. or armature reaction and of the impressed m.m.f. or field excita- tion. The armature m.m.f. is in phase with the current /, and is nl in a single-phase machine, n V2 / in a quarter-phase machine, 1.5 \/2n/ in a three-phase machine, if n = number of armature turns per pole and phase. The m.m.f. of armature reaction is represented in the diagram by 0^ a = ^a in phase Fig. 49. Diagram Showing Combined Effect of Armature Reaction and Armature Self-Inductance. with 01, and the impressed m.m.f^pr field excitation OtF = # is the side of a parallelogram with 0$ as diagonal and 0$ a as other side; or, the m.m.f. consumed by armature reaction is represented by 0$d = $ a in opposition to OL Combining OSF/ and 0$ gives 0$ Q = ^ 07 the field excitation. In Figs. 50, 51, 52 are drawn the diagrams for = zero or non-inductive load, == GO degrees, or 60 degrees lag (inductive load of power-factor 0.50), and 60 deg. 7 or 60 deg. lead (anti-inductive load of power-factor 0.50). SYNCHRONOUS MACHINES. 135 Thus it is seen that with the same terminal voltage E a much higher field excitation, ^ , is required with inductive load than with non-inductive load, while with anti-inductive Fig. 50. Diagram of Generator e.m.fs. and m.m.fs. for Non-Reactive Load load a much lower field excitation is required. With open cir- cuit the field excitation required to produce the terminal voltage E E would be 77- IF = CF 00 , or less ^0 than the field excitation 9F with non-inductive load. Inversely, with constant field excitation, the voltage of an alternator drops with non-in- ductive load, drops much more with inductive load, and drops less, or even rises, with anti- inductive load. Fig, 61, Diagram of Generator e.raufs. and m.m.fs. for Lagging Reactive Load. Power Factor 0.50. Fig. 52. Diagram of Generator e.m.fs. and m.m.fs. for Leading Reactive Load. Power Factor 0.50. 136 ELEMENTS OF ELECTRICAL ENGINEERING. V. Synchronous Reactance e 14. In general, both effects, armature self-inductance and armature reaction, can be combined by the term "synchronous reactance. " In a polyphase machine, the synchronous reactance is different, and lower, with one phase only loaded, as " single-phase synchro- nous reactance, " than with all phases uniformly loaded, as "poly- phase synchronous reactance. 7 ' The resultant armature reac- tion of all phases of the polyphase machine is higher than that with the same current in one phase only, and so also the self- inductive flux, as resultant flux of several phases, and thus represents a higher synchronous reactance. Let r = effective resistance, X Q = synchronous reactance of armature, as discussed in Section II. Let E = terminal voltage, I = current, 6 = angle of lag of the current behind the terminal volt- tage. It is in polar diagram, Fig. 53, Fig. 53. Diagram Showing Effect of Synchronous Reactance. Fig 54, Diagram of Gene- rator e.m.fs. Showing Effect of Synchronous Reactance with Non- Re- active Load. OE = E = terminal voltage assumed as zero vector. 01 / = current lagging by the angle EOI = 6 behind the terminal voltage. SYNCHRONOUS MACHINES. 137 OJ?/j= Ir js__the e.m.f. consumed by resistance, in phase with 01, and OE ' = 7x the e.m.f. consumed by the synchronous reactance, OOjlegrees ahead of the current OL OE t f and OEJ combined give OE f = E' the e.m.f. consumed by the synchronous impedance. Combining OE{, 0/i T ', OE gives the nominal generated e.m.f. OE Q = E , corresponding to the field excitation & Q . In Figs. 54, 55, 56, are shown the diagrams for = or non- inductive load, 9 = 60 degrees lag or inductive load, and 6 =- 60 degrees or anti-inductive load. Fig. 55. Diagram of Generator e.m.fs. Showing Effect of Synchronous React- ance with Lagging Reactive Load. 6 = 60 degrees. Fig. 56. Diagram of Gene- rator e.m.fs. Showing Effect of Synchronous Reactance with Leading Reactive Load. Q = 60 degrees. Resolving all e.m.fs. into components in phase and in quad- rature with the current, or into power and reactive components, in symbolic expression we have: the terminal voltage E = E cos jE sin 9; the e.m.f. consumed by resistance, $/=* ir; the e.m.f. consumed by synchronous reactance, A 7 / = ]ix , and the nominal generated e.m.f., E Q ^E + Ei + EQ = (#cos + ir) -; (Ssin 9 + ix ); or, since o f j. r .0. i i / power current\ cos p = power factor of the load = *m - r) r r \ total current / and _ q = Vl p 2 = sin = inductance factor of the load \ wattless current^ total current / 138 ELEMENTS OF ELECTRICAL ENGINEERING. it is E Q - (Ep + ir) - j(Eq + 6(X Polax Diagram and synchronous reactance x , the of Synchronous Motor, polar diagram is as follows, Fig. 60. OE E is the terminal voltage assumed as zero vector. The current 01 = / lags by the angle EOI = 0. The e.m.f. consumed by resistance, isjXE/ = 7r. The e.m.f. consumed by synchronous reactance, OE ' = Ix . Thus, com-' bining OE{ and OE ' gives OE', the e.m.f. consumed by the synchronous impedance. The e.m.f. consumed by the synchro- nous impedance OE f and the e.m.f. consumed by the nominal generated or counter e.m.f. of the synchronous motor OE Q , combined, give the impressed_e.m.f. OE. Hence OE Q is one side of a parallelogram, with OE' as the other side, and OE as diagonal. OjG? 00 , (not shown) equal and opposite OE , would thus be the nominal counter-generated e.m.f. of the synchronous motor. 142 ELEMENTS OF ELECTRICAL ENGINEERING. In Figs. 61 to 63 are shown the polar diagrams of the syn- chronous motor for 6 = deg., = 60 deg., = 60 deg. It is seen that the field excitation has to be higher with lead- ing and lower with lagging current in a synchronous motor, while the opposite is the case in an alternating-current generator. Fig. 61. Polar Diagram of Synchronous Motor. 0== 0. Fig. 62. Polar Diagram of Synchronous Motor. 60 deg. Fig. 63. Polar Diagram of Synchronous Motor. 60 degrees. In symbolic representation, by resolving all e.m.fs. into power components in phase with the current and wattless components. in quadrature with the current i, we have: the terminal voltage, E = E cos 6 jE sin Ep - jEq; the e.m.f. consumed by resistance, $/ = ir, and the e.m.f . consumed by synchronous reactance, EJ ~ SYNCHRONOUS MACHINES. 143 Thus the e.m.f. consumed by the nominal counter-generated e.m.f. is E Q = E -ES -EJ = (E cos - tr) - f (# sin 5 - tx ) - (JSfp -w-)-j(#g ~iz ); or, in absolute values, ' cos - ir) 2 + (E sin 6 - hence, E = i (rp + x q) V$ 2 - ^ 2 (x p - r^) 3 . The power consumed by the synchronous motor is P - iEp; that is, the current times the power component of the impressed e.m.f. The mechanical power delivered by the synchronous motor armature is that is, the current times the power component of the nominal counter-generated e.m.f. Obviously to get the available mechan- ical power, the power consumed by mechanical friction and by molecular magnetic friction or hysteresis, and the power of field excitation, have to be subtracted from this value P . VIIL Characteristic Curves of Synchronous Motor. 17. In Fig. 64 are shown, at constant impressed e.m.f. E, the nominal counter-generated e.m.f. E Q and thus the field excita- tion # required, 1. At no phase displacement, 6 = 0, or for the condition of minimum input; 2. For = + 60, or 60 deg. lag: p = 0.5, q = + 0.866, and 3. ForS = -60, or 60 deg. lead :p 0.5, q = - 0.866; with the current / as abscissas, the constants being r = 0.1, x = 5, and E = 1000. These curves are called the compounding curves of the syn- chronous motor. I- 000 #0 20 40 60 80 100 120 140 160 180 200 Fig, 64. Synchronous Motor Compounding Curves, I 140 130 120 110 100 00 PER CENT. 100 50 40 30 20 10 20 30 40 50 00 100 110 120 130 140 Fig. 65. Synchronous Motor Load Characteristics. SYNCHRONOUS MACHINES. 145 In Fig. 65 are shown, with the power output P l = i (Ep ir) (iron loss and friction) as abscissas, and the same constants r = 0.1, = 5, E = 1000, and constant field excitation 3^; that is, constant nominal counter-generated e.m.f. E = 1109 (corresponding to p = 1, q at I = 100), the values of current / and power-factor p. As iron loss is assumed 3000 watts, as friction 2000 watts. Such curves are called load cJiaracteristics of the synchronous taotor. 18. In Fig. 66 are shown, with constant power output, P = i (Ep ir), and the same constants, r = 0.1, x = 5, E = 1000, \ / /. '/? \ \ \ ^ S = 0.1 S 5, y ISO \ \ \ * ^ y i$ & ^ ,-. f"" ^ / '/ // 100 \ \ \ ^ K" /, ^ lAO \ \ xi"" 1 V i AG<- -/ >- .HAD / r if / frequency of machine, n = L p number of armature slots per pair of poles, f l = nf. For instance, /= 33.3, n = 51, thus / x = 1700. Under the assumption, width of slots equals width of teeth = 2 X width of air gap, the dis- tribution of magnetic flux at the pole face is plotted in Fig. 100. The drop of density opposite each slot consists of two curved branches equal to those in Fig. 90, T,- , - 01 x that is, calculated by Fig. 100, Effect of Slots on Flux ' J Distribution. (g __ EL J; , . VE7TZ? The average flux is 7525; that is, by cutting half the armature surface away by slots of a width equal to twice the length of air gap, the total flux under the field pole is reduced only in the proportion 8000 to 7525, or about 6 per cent. DIRECT-CURRENT COMMUTATING MACHINES. 191 The flux (B pulsating between 8000 and 5700 is equivalent to a uniform flux (Bj = 7525 superposed with an alternating flux (B , shown in Fig. 101, with a maximum of 475 and a minimum of 1825. This alternating flux SF e is called saturation factor k s , and is plotted in Fig. 102. It is the ratio of a small percentage increase in field excitation to a corre- Iff /IL Fig. 103. Saturation Curves. spending percentage increase in voltage thereby produced. The quantity 1 is called the percentage saturation of the ma- KS chine, as it shows the approach of the machine field to magnetic saturation. 57. Of considerable importance also are curves giving the terminal voltage as function of the field excitation at load. 198 ELEMENTS OF ELECTRICAL ENGINEERING. Such curves are called load saturation curves, and can be constant current load saturation curve, that is, terminal voltage as func- tion of field ampere-turns at constant full-load current through the armature, and constant resistance load saturation curve, that is, terminal voltage as function of field ampere-turns if the machine circuit is closed through a constant resistance giving full-load current at full-load terminal voltage. A constant-current load saturation curve is shown as B, and a constant resistance load saturation curve as C in Fig. 102. X. Compounding. 58. In the direct-current generator the field excitation re- quired to maintain constant terminal voltage has to be increased with the load. A curve giving the field excitation in am- pere-turns per pole, as function of the load in amperes, at constant terminal voltage, is called the compounding curve of the machine. The increase of field excitation required with load is due to: 1. The internal resistance of the machine, which consumes e.m.f. proportional to the current, so that the generated e.m.f., and thus the field m.m.f . corresponding thereto, has to be greater under load. If kr = resistance drop in the machine as fraction nrvt of terminal voltage, = , the generated e.m.f. at load has to be e e (1 + k r ), and if & Q = no-load field excitation, and k a = satu- ration coefficient, the field excitation required to produce the e.m.f. e (1 + ft r ) is 9F (1 + k*k r ) ; thus an additional excitation of k s kr$ is required at load, due to the armature resistance. 2. The demagnetizing effect of the ampere-turns armature reaction of the angle of shift of brushes r t requires an increase of field excitation by rfta. (Section VII.) 3. The distorting effect of armature reaction does not change the total m.m.f. producing the magnetic flux. If, however, mag- netic saturation is reached or approached in a part of the mag- netic circuit adjoining the air gap, the increase of magnetic density at the strengthened pole corner is less than the decrease at the weakened pole corner, and thus the total magnetic flux with the same total m.m.f. is reduced, and to produce the same DIRECT-CURRENT COMMUTATING MACHINES. 197 total magnetic flux an increased total m.m.f., that is, increase of field excitation, is required. This increase depends upon the saturation of the magnetic circuit adjacent to the armature conductors. 4. The magnetic stray field of the machine, that is, that part of the magnetic flux which passes from field pole to field pole with- out entering the armature, usually increases with the load. This stray field is proportional to the difference of magnetic potential between field poles; that is, at no-load it is proportional to the ampere-turns m.m.f. consumed in air gap, armature teeth, and armature core. Under load, with the same generated e.m.f., that is, the same magnetic flux passing through the armature core, the difference of magnetic potential between adjacent field poles is increased by the counter m.m.f. of the armature and by saturation. Since this magnetic stray flux passes through field poles and yoke, the magnetic density therein is increased and the field excitation correspondingly, especially if the magnetic den- sity in field poles and yoke is near saturation. This increase of field strength required by the increase of density in the external magnetic circuit, due to the increase of magnetic stray field, depends upon the shape of the magnetic circuit, the armature reaction, and the saturation of the external magnetic circuit. Curves giving, with the amperes output as abscissas, the ampere-turns per pole field excitation required to increase the voltage proportionally to the current, are called over-compound- ing curves. In the increase of field excitation required for over- compounding, the effects of magnetic saturation are still more marked. XI. Characteristic Curves. 59. The field characteristic or regulation curve, that is, curve giving the terminal voltage as function of the current output at constant field excitation, is of less importance in commutating machines than in synchronous machines, since commutating machines are usually not operated with separate and constant excitation, and the use of the series field affords a convenient means of changing the field excitation proportionally to the load. The curve giving the terminal voltage as function of current out- put, in a compound-wound machine, at constant resistance in the shunt field, and constant adjustment of the series field, is, how- 198 ELEMENTS OF ELECTRICAL ENGINEERING. ever, of importance as regulation curve of the direct-current generator. This curve would be a straight line except for the effect of saturation, etc., as discussed above. XII. Efficiency and Losses. 60. The losses in a commutating machine which have to be considered when deriving the efficiency by adding the individual losses are: 1. Loss in the resistance of the armature, the commutator leads, brush contacts and brushes, in the shunt field and the series field with their rheostats. 2. Hysteresis and eddy currents in the iron at a voltage equal to the terminal voltage, plus resistance drop in a generator, or minus resistance drop in a motor. 3. Eddy currents in the armature conductors when large and not protected. 4. Friction of bearings, of brushes on the commutator, and windage. 5. Load losses, due to the increase of hysteresis and of eddy currents under load, caused by the change of the magnetic dis- tribution, as local increase of magnetic density and of stray field. The friction of the brushes and the loss in the contact resist- ance of the brushes are frequently quite considerable, especially with low-voltage machines. Constant or approximately constant losses are: friction of bearings and of commutator brushes, and windage; hysteresis and eddy current losses; and shunt field excitation. Losses increasing with the load, and proportional or approximately proportional to the square of the current: armature resistance losses; series field resistance losses; brush contact resistance losses; and the so-called "load-losses/' which, however, are usually small in commutating machines. XIII. Commutation. 61. The most important problem connected with commutating machines is that of commutation. Fig. 104 represents diagrarnmatically a commutating machine. The e.m.f. generated in an armature coil A is zero with this coil at or near the position of the commutator brush B v It rises DIRECT-CURRENT COM MUTATING MACHINES. 199 and reaches a maximum about midway between two adjacent sets of brushes, 5 t and B 2J at C, and then decreases again, reaching zero at or about 5 2J and then repeats the same change in opposite direction. The current in armature coil A, however, is constant during the motion of the coil from B l to B 2 . While the coil A passes the brush J3 2 , however, the current in the coil Fig. 104. Diagram for the Study of Commutation. A reverses, and then remains constant again in opposite direc- tion during the motion from J5 2 to 5 3 . Thus, while the armature coils of a commutating machine are the seat of a system of poly- phase e.m.fs. having as many phases as coils, the current in these coils is constant, reversing successively. 62. The reversal of current in coil A takes place while the gap G between the two adjacent commutator segments between which the coil A is connected, passes the brush B 2 . Thus, if l w = width of brushes, S peripheral speed of commutator per second in the same measure in which l w is given, as in inches per second if l w is given in inches, = - is the time during which 200 ELEMENTS OF ELECTRICAL ENGINEERING. the current in A reverses. Thus, considering the reversal as a "1 o single alternation, f is a half period, and thus/ = 57 = 57 is IQ & 1 W the frequency of commutation; hence, if L = inductance of the armature coil A, the e.m.f. generated in the armature coil during commutation is e d = 2 7r/ Li , where i = current reversed, and the energy which has to be dissipated during commutation is i 2 L. The frequency of commutations very much higher than the frequency of synchronous 1000 cycles per second, or mo: 63. In reality, however? tj mutation are not sinusoids les, and averages from 300 to changes of current during com- fcut a complex exponential func- tion, and the resistance ow the commutated circuit enters the problem as an importantreactor. In the moment when the gap G of the armature coil A^reaches the brush J3 2 , the coil A is short- circuited by tne' -hrushrand the current i in the coil begins to die out, or ratheijrop cJjfange at a rate depending upon the internal resistance and ttta iMuctance of the coil A and the e.m.f. gener- ated in the coiM)y wie magnetic flux of armature reaction and by the fieldmagi^c flux. The higher the internal resistance the faster ijs tlW cftange of current, and the higher the inductance the slower ph$ current changes. Thus two cases have to be dis- tinguished/ 1. No inagnktic flux enters the armature at the position of the brushes, tha\ is, no e.m.f. is generated in the armature coil under commutation, except that of its own self-inductance. In this case the commutation is entirely determined by the induc- tance and resistance of the armature coil A, and is called resistance commutation. 2. Commutation takes place in an active magnetic field; that is, in the armature during commutation an e.m.f. is generated by its rotation through a magnetic field. This magnetic field may be the magnetic field of armature reaction, or the reverse magnetic field of a commutating pole ; or the fringe of the main field of the machine, into which the brushes are shifted. In this case the commutation depends upon the inductance and the resistance of the armature coil and the e.m.f. generated therein by the main magnetic field, and if this magnetic field is a com- mutating field, is called voltage commutation. DIRECT-CURRENT COM MUTATING MACHINES. In either case the resistance of the brushes and their contact may either be negligible, as usually the case with copper brushes, or it may be of the same or a higher magnitude than the internal resistance of the armature coil A. The latter is usually the case with carbon or graphite brushes. In the former case the resistance of the short-circuit of arma- ture coil A under commutation is approximately constant; in the latter case it varies from infinity in the moment of beginning commutation down to minimum, and then up again to infinity at the end of commutation. 64. (a.) Negligible resistance of brush and brush contact. This is more or less approximately the case with copper brushes. Let i Q = current, L = inductance, r = resistance of armature coil, _ " S and e = e.m.f. generated in the armature coil by its rotation through the magnetic field, where e is negative for the magnetic field of armature reaction and positive for the commutating field. Denoting the current in the coil A at time t after beginning of commutation by i, the e.m.f. of self-inductance is r di Thus the total e.m.f. acting in coil A, r di e + e L = e Z/y > and the current is - e, e Ldi t Q = -~ = time of commutation, r r r dt Transposing, this expression becomes rdt di T the integral of which is rl , le . ~~- 10g.(j; where log e c = integration constant. 202 ELEMENTS OF ELECTRICAL ENGINEERING. Since at t = 0, i i , we have therefore /e , \ T - C== (r T t and, at the end of commutation, or, t = t Q , For perfect commutation, \ = - %; that is, the current at the end of commutation must have reversed and reached its full value in opposite direction. Substituting in this last equation the value ^=^0 from the pre- ceding equation, and transforming, we have taking the logarithms of both terms, e . 7*0 = log*; ; L e T and, solving the exponential equation for e, we obtain i t ~~ T*o * 1 + 1 _ fi -' It is evident that the inequality e > i Q r must be true, otherwise perfect commutation is not possible. If e = 0, we nave r that is, the current never reverses, but merely dies out more or less, and in the moment when the gap G of the armature coil leaves the brush B the current therein has to rise suddenly to full intensity in opposite direction. This being impossible, due DIRECT-CURRENT COMMUTATING MACHINES. 203 to the inductance of the coil, the current forms an arc from the brush across the commutator surface for a length of time depend- ing upon the inductance of the armature coil. Therefore, with low-resistance brushes, resistance commutation is not permissible except with machines of extremely low arma- ture inductance, that is, armature inductance so low that the magnetic energy - -- , which appears as spark in this case, is harmless. Voltage commutation is feasible with low-resistance brushes, but requires a commutating e.m.f. e proportional to current i ; that is, requires shifting of brushes proportionally to the load, or a commutating pole. In the preceding, the e.m.f. e has been assumed constant dur- ing the commutation. In reality it varies somewhat, usually increasing with the approach of the commutated coil to a denser field. It is not possible to consider this variation in general, and e is thus to be considered the average value during commutation. 65. (6.) High-resistance brush contact. Fig. 105 represents a brush B commutating armature coil A. Let r = contact resist- Fi S- 105 - Brush Commutating Coil A, ance of the brush, that is, resistance from the brush to the commutator surface over the total bearing surface of the brushes. The resistance of the commutated circuit is thus internal resistance of the armature coil 7 plus the resistance from C to B plus the resistance from StoD. Thus, if t = time of commutation, at the time t after the beginning of the commutation, the resistance from C to B is * ~ t T thus, the total resistance of the = r from B to D is t commutated coil is 204 ELEMENTS OF ELECTRICAL ENGINEERING. If i current in coil A before commutation, the total cur- rent into the armature from brush B is 2 i . Thus, if t = current in commutated coil, the current from B to D = % + i, the current from 5 to C = i i. Hence, the difference of potential from D to C is The e.m.f. acting in coil A is and herefrom the difference of potential from D to C is T ~ ' ____ j t -ir, hence, or, transposing, The further solution of this general problem becomes diffi- cult, but even without integrating this differential equation a number of important conclusions can be derived. Obviously the commutation is correct and thus sparkless, if the current entering over the brush shifts from segment to seg- ment in direct proportion to the motion of the gap between adjacent segments across the brush, that is, if the current den- sity is uniform all over the contact surface of the brush. This means that the current i in the short-circuited coil varies from + i to i c as a linear function of the time. In this case it can be represented by thus, di ^ 2 dt~ t DIRECT-CURRENT COMMUTATING MACHINES. 205 Substituting this value in the general differential equation gives, after some transformation, which gives at the beginning of commutation, t = 0, _ - /2L \ e '~* d~~ r )> at the end of commutation, t = t , f2L . that is, even with high-resistance brushes, for perfect com- mutation, voltage commutation is necessary, and the e.m.f. e impressed upon the commutated coil must increase during com- mutation from e l to e 2 , by the above equation. This e.m.f. is proportional to the current %, but is independent of the brush resistance r . RESISTANCE COMMUTATION. 66. Herefrom it follows that resistance commutation cannot be perfect, but that at the contact with the segment that leaves the brush the current density must be higher than the average. Let ki ratio of actual current density at the moment of leav- ing the brush to average current density of brush contact, and considering only the end of commutation, as the most important moment, we have For t = t Q t l this gives t 1 i = -{ + 2ki-~i QJ r o while uniform current density would require 206 ELEMENTS OF ELECTRICAL ENGINEERING. The general differential equation of resistance commutation, Substituting in this equation the value of i from the foregoing equation, expanding and cancelling t Q t, we obtain 2 rj* (fe - 1) + r (2 A* - 1) - 2 ktf - 2 kiLt = 0; hence, and for t t ot 2 (r i > + r - rt 2 - Li) that is, fc t - is always > 1. The smaller L and the larger r 07 the smaller is h; that is, the nearer it is to 1, the condition of perfect commutation, and the better is the commutation. Sparkless commutation is impossible for very large values of kij that is, when L approaches r , or when r is not much larger than - . For this reason, in machines in which L cannot ^0 be made small, r is sometimes made large by inserting resistors in the leads between the armature and the commutator, so-called " resistance" or " preventive " leads as used in alternating-current commutator motors. XIV. Types of Commutating Machines. 67. By the methods of excitation, commutating machines are subdivided into magneto, separately excited, shunt, series, and compound machines. Magneto machines and separately excited machines are very similar in their characteristics. In either, the field excitation is of constant, or approximately constant, impressed m.m.f. Magneto machines, however, are little used, except for very small sizes. By the direction of energy transformation, commutating machines are subdivided into generators and motors. DIRECT-CURRENT COMMUTATING MACHINES. 207 Of foremost importance in discussing the different types of machines is the saturation curve or magnetic characteristic; that is, a curve relating terminal voltage at constant speed to ampere-turns per pole field excitation, at open circuit. Such a curve is shown as A in Figs. 106 and 107. It has the same general shape as the magnetic flux density curve, except that the 130 120 110 100 90 80 70 60 50 10 30 20 10 / - = ^* = . ' -Ill " ..II *^ k/ / .* -" - - ^.' 55= ^S= (P ^- *-* r*^ ^ "G* ? ^ < **^ ** ^p 2 \ x **^*^ X ^ d / /t f*j ^ / / // / / D / I/ / / A / / \ i / \ \ \ \ s \ \ \ \ Alj dPER E8 \ \ b S s\ '0 10 20 30 40 00 00 70 80 00 100 110 120 130 140 150 160 Fig. 108. Separately Excited or Magneto-generator Demagnetization Curve and Load Characteristic with Constant Shift of Brushes. 100 90 70 60 40 30 20 10 10 20 30 40 50 60 70 80 90 100 110 120 130 Fig. 109. Separately Excited or Magneto-generator Demagnetization Curve and Load Characteristic with Variable Shift of Brushes. assumption of constant coefficient of armature reaction q, that is, corresponding to curve D in Fig. 106. This curve becomes zero at the current i oy which makes i Q q = 3> Subtracting from curve A in Fig. 39 the drop of voltage in the armature and commutator resistance, ac = tr, gives the external characteristic 210 ELEMENTS OF ELECTRICAL ENGINEERING. B of the machine as generator, or the curve relating the terminal voltage to the current. In Fig. 109 the same curves are shown under the assumption that the armature reaction varies with the voltage in the way as represented by curve G in Fig. 106. In a separately excited or magneto motor at constant speed the external characteristic would lie as much above the demag- netization curve A as ijb lies below in a generator in Fig. 108, and at constant voltage the speed would vary inversely pro- portional hereto. Shunt Generator. 70. The external or load characteristic of the shunt generator is plotted in Fig. 110 with the current as abscissas and the terminal voltage as ordinates, as A for constant coefficient of 50 100 150 200 250 300 350 Fig. 110. Shunt Generator Load Characteristic, armature reaction, and as B for a coefficient of armature reac- tion varying with the voltage in the way as shown in G, Fig. 106, The construction of these curves is as follows : In Fig. 106, og is the straight line giving the field excitation oh as function of the terminal voltage hg (the former obviously being proportional to the latter in the shunt machine). The open-circuit or no-load voltage of the machine is then kq. Drawing gl parallel to da (assuming constant coefficient of armature reaction, or parallel to the hypothenuse of the triangle iq, ir at voltage og, when assuming variable armature reaction), DIRECT-CURRENT COMMUTATING MACHINES. 211 then the current which gives voltage gh is proportional to gl, that is, i: i = gl:da, where i is the current at the voltage de~ As seen from Fig. 110, a maximum value of current exists which is less if the brushes are shifted than at constant position of brushes. From the load characteristic of the shunt generator the resistance characteristic is plotted in Fig. Ill; that is, the de- pendence of the terminal voltage upon the external resistance ^ terminal voltage ^ ^ -n- -. -i -. i A R ^ &. 9 Curve A in Fig. Ill corresponds to current constant, curve B to varying armature reaction. It is seen that at a certain definite resistance the voltage becomes zero, and for lower resistance the machine cannot generate but loses its excitation. The variation of the terminal voltage of the shunt generator with the speed at constant field resistance is shown in Fig. 112, at no load as A, and at constant current i as B, These curves are derived from the preceding ones. They show that below a certain speed, which is much higher at load than at no load, the machine cannot generate. The lower part of curve B is unstable and cannot be realized. Series Generator. 71. In the series generator the field excitation is proportional to the current i } and the saturation curve A in Fig. 113 can thus be plotted with the current i as abscissas. Subtracting oh = ir, the resistance drop, from the voltage, and adding bd iq, the armature reaction, gives a load saturation curve or external characteristic B of the series generator. The terminal voltage is zero at no load or open circuit, increases with the load, reaches a maximum value at a certain current, and then de- creases again and reaches zero at a certain maximum current, the current of short-circuit. Curve B is plotted with constant coefficient of armature reac- tion q. Assuming the brushes to be shifted with the load and proportionally to the load, gives curves C, Z>, and E, which are higher at light load, but fall off faster at high load. A still further shift of brushes near the maximum current value even overturns the curve as shown in F. Curves E and F correspond ICO 150 110 130 120 110 100 90 80 70 CO 50 40 30 O.I 0.2 ^. \ 200 400 (300 800 1000 1200 ULOO 1000 1800 2000n MS Fig. 114. Series Generator Resistance Characteristic. 600 500 400 400 200 100 / t / ^ ^ ^ - - ^ . - " EEE bf / -^? ^ ^ 4 ^ 4 M" &**' i^, /I tf / //, / I / 7 / /, k I / // I / i <>/ J 7 AG // Kb PER -"ru RNS situ T F| LD y 1 1000 2000 3000 4000 5000 Fig. 115. Compound Generator 6000 7000 8000 9000 Saturation Curve. At constant position of brushes the compound generator, when adjusted for the same voltage at no load and at full load, under- compounds at higher and over-compounds at lower voltage, and even at open circuit of the shunt field gives still a voltage op as series generator. When shifting the brushes under load, at lower DIRECT-CURRENT COMMUTATING MACHINES. 215 voltage a second point g is reached where the machine compounds correctly, and below this point the machine under-compounds and loses its excitation when the shunt field decreases below a certain value; that is, it does not excite itself as series generator. B. MOTORS. Shunt Motor. 73. Three speed characteristics of the shunt motor at con- stant impressed e.m.f. e are shown in Fig. 116 as A } P, Q J corre- sponding to the points d, p, q of the motor load saturation curve, Fig. 107. Their derivation is as follows : At constant impressed Fig. 116. Shunt Motor Speed Curves, Constant Impressed e.m.f. e.m.f. e the field excitation is constant and equals $ , and at current i the generated e.m.f. must be e ir. The resultant field excitation is $ iq, and corresponding hereto at constant speed the generated e.m.f. taken from saturation curve A in Fig. 107 is e r Since it must be e ir, the speed is changed in xi ,. e ir the proportion e, At a certain voltage the speed is very nearly constant, the demagnetizing effect of armature reaction counteracting the effect of armature resistance. At higher voltage the speed falls, at lower voltage it rises with increasing current. In Fig. 117 is shown the speed characteristic of the shunt motor as function of the impressed voltage at constant output, that is, constant product, current times generated e.m.f. If i = current and P =* constant output, the generated e.m.f. 216 ELEMENTS OF ELECTRICAL ENGINEERING. P must be approximately e t = T , and thus the terminal voltage 6 e = ^ + ir. Proportional hereto is the field excitation ff . The resultant m.m.f. of the field is thus $ = 9^ ?"g, and corre- sponding thereto from curve A in Fig. 108 is derived the ejn.f. e which would be generated at constant speed by the m.m.f. 5\ REV. PER M|N. 2000 1800 1GOO UQO 1200 1000 800 COO 100 200 / jS / / jr / / 1 S 7 / 4 / / * / / I / s \ y voT 1* Fig. 117. Shunt Motor Speed Curve, Variable Impressed e.m.f. Since, however, the generated e.m.f. must be e v the speed is changed in the proportion - 1 6 The speed rises with increasing and falls with decreasing im- pressed e.m.f. Still further decreasing the impressed e.m.f., the speed reaches a minimum and then increases again, but the conditions become unstable. Series Motor. 74. The speed characteristic of the series motor is shown in Fig. 118 at constant impressed e.m.f. e. A is the saturation curve of the series machine, with the current as abscissas and at ' constant speed. At current i, the generated e.m.f. must be g _. ny* e irj and the speed is thus times that, for which curve -4 6 i is plotted, where e l = e.m.f. taken from saturation curve .A. This speed curve corresponds to a constant position of brushes midway between the field poles, as generally used in railway motors and other series motors. If the brushes have a constant DIRECT-CURRENT COMMUTATING MACHINES. 217 shift or are shifted proportionally to the load, instead of the saturation curve A in Fig. 118 a curve is to be used correspond- ing to the position of brushes, that is, derived by adding to the REV. PER MIN. 2200 2000 1800 1600 1100 1200 1000 800 COO 400 200 x 110 130 120 110 100 90 80 70 60 50 40 30 20 10 Fig. 118. 100 120 Series Motor MO 160 180 Speed Curve. abscissas of A the values iq, the demagnetizing effect of arma- ture reaction. The torque of the series motor is shown also in Fig. 118, derived as proportional to A X i, that is, current X magnetic flux. Compound Motors. 75. Compound motors can be built with cumulative com- pounding and with differential compounding. Cumulative compounding is used to a considerable extent, as in elevator motors, etc., to secure economy of current in starting and at high loads at the sacrifice of speed regulation; that is, a compound motor with cumulative series field stands in its speed and torque characteristic intermediate between the shunt motor and the series motor. Differential compounding is used to secure constancy of speed with varying load, but to a small extent only, since the speed regulation of a shunt motor can be made sufficiently close, as was shown in the preceding. 218 ELEMENTS OF ELECTRICAL ENGINEERING. Conclusion. 76. The preceding discussion of commutating machine types can obviously be only very general, showing the main character- istics of the curves, while the individual curves can be modified to a considerable extent by suitable design of the different parts of the machine when required to derive certain results, as, for instance, to extend the constant-current part of the series gen- erator; or to derive a wide range of voltage at stability, that is, beyond the bend of the saturation curve in the shunt generator; or to utilize the range of the shunt generator load characteristic at the maximum current point for constant-current regulation; or to secure constancy of speed in a shunt motor at varying impressed e.m.f., etc. The use of the commutating machine as direct-current con- verter has been omitted from the preceding discussion. By means of one or more alternating-current compensators or auto-transformers, connected to the armature by collector rings, the commutating machine can be used to double or halve the voltage, or convert from one side of a three-wire system to the other side and, in general, to supply a three-wire Edison system from a single generator. Since, however, the direct- current converter and three-wire generator exhibit many fea- tures similar to those of the synchronous converter, as regards the absence of armature reaction, the reduced armature heating, etc., they will be discussed as an appendix to the synchronous converter C. ALTERNATING-CURRENT COMMUTATING MACHINES. I. General. Alternating-current commutating machines have so far be- come of industrial importance mainly as motors of the series or varying-speed type, for single-phase railroading, and to some extent also as constant-speed motors for cases as elevators or hoists, where efficient acceleration under heavy torque is neces- sary. As generators, they would be of advantage for the gen- eration of very low frequency, since in this case synchronous machines are uneconomical, due to their very low speed, resultant from the low frequency. The direction of rotation of a direct-current motor, whether shunt or series motor, remains the same at a reversal of the impressed e.m.f., as in this case the current in the armature circuit and the current in the field circuit and so the field mag- netism both reverse. Theoretically, a direct-current motor therefore could be operated on an alternating impressed e.m.f. provided that the magnetic circuit of the motor is laminated, so as to follow the alternations of magnetism without serious loss of power, and that precautions are taken to have the field re- verse simultaneously with the armature. If the reversal of field magnetism should occur later than the reversal of armature cur- rent, during the time after the armature current has reversed, but before the field has reversed, the motor torque would be in opposite direction and thus subtract; that is, the field magnetism of the alternating-current motor must be in phase with the arma- ture current, or nearly so. This is inherently the case with the series type of motor, in which the same current traverses field coils and armature windings. Since in the alternating-current transformer the primary and secondary currents and the primary voltage and the secondary voltage are proportional to each other, the different circuits of the alternating-current commutator motor may be connected with each other directly (in shunt or in series, according to the type of the motor) or inductively, with the interposition of a 219 220 ELEMENTS OF ELECTRICAL ENGINEERING. transformer, and for this purpose either a separate transformer may be used or the transformer feature embodied in the motor, as in the so-called repulsion type of motors. This gives to the alternating-current commutator motor a far greater variety of connections than possessed by the direct-current motor. While in its general principle of operation the alternating- current commutator motor is identical with the direct-current motor, in the relative proportioning of the parts a great differ- ence exists. In the direct-current motor, voltage is consumed by the counter e.m.f. of rotation, which represents^ the power output of the motor, and by the resistance, which represents the power loss. In addition thereto, in the alternating-current motor voltage is consumed by the inductance, which is wattless or reactive and therefore causes a lag of current behind the voltage, that is, a lowering of the power-factor. While in the direct-current motor good design requires the combination of a strong field and a relatively weak armature, so as to reduce the armature reaction on the field to a minimum, in the design of the alternating-current motor considerations of power-factor predominate; that is, to secure low self-inductance and there- with a high power-factor, the combination of a strong armature and a weak field is required, and necessitates the use of methods to eliminate the harmful effects of high armature reaction. As so far only the varying-speed single-phase commutator motor has found an extensive use as railway motor, this type of motor will mainly be treated in the following, and the other types 'discussed in the concluding paragraphs. II. Power-Factor. In the commutating machine the magnetic field flux generates the e.m.f. in the revolving armature conductors, which gives the motor output; the armature reaction, that is, the magnetic flux produced by the armature current, distorts and weakens the field, and requires a shifting of the brushes to avoid sparking due to the short-circuit current under the commutator brushes, and where the brushes cannot be shifted, as in a reversible motor, this necessitates the use of a strong field and weak armature to keep down the magnetic flux at the brushes. In the alternat- ing-current motor the magnetic field flux generates in the anna- ALTERNA TING-CURRENT COMMUTA TING MACHINES. 221 ture conductors by their rotation the e.m.f. which does the work of the motor, but, as the field flux is alternating, it also generates in the field conductors an e.m.f. of self-inductance, which is not useful but wattless, and therefore harmful in lowering the power-factor, hence must be kept as low as possible. This e.m.f. of self-inductance of the field, e , is proportional to the field strength , to the number of field turns n o; and to the frequency / of the impressed e.m.f. e = 2^/n 4>10- 8 , (1) while the useful e.m.f. generated by the field in the armature conductors, or "e.m.f. of rotation/' e, is proportional to the field strength , to the number of armature turns n v and to the frequency of rotation of the armature, / , e=2 7 r/ n 1 10- 8 . (2) This later e.m.f., e, is in phase with the magnetic flux <3>, and so with the current i, in the series motor, that is, is a power e.m.f., while the e.m.f. of self-inductance, e , is wattless, or in quadrature with the current, and the angle of lag of the motor current thus is given by tan 6 = 6 . , (3) e + ir ^ ' where if = voltage consumed by the motor resistance. Or ap- proximately, since ir is small compared with e (except at very- low speed), tan#=^; (4) e ^ and, substituting herein (1) and (2), =- (5) /X Small angle of lag and therewith good power-factor therefore require high values of / and n l and low values of /and n . High / requires high motor speeds and as large number of poles as possible. Low /means low impressed frequency; there- fore 25 cycles is generally the highest frequency considered for large commutating motors. High n t and low n Q means high armature reaction and low field excitation, that is, just the opposite conditions from that required for good commutator motor design. 222 ELEMENTS OF ELECTRICAL ENGINEERING. Assuming synchronism, / = /, as average motor speed, 750 revolutions with a 4-pole 25-cyele motor, an armature reac- tion n l equal to the field excitation n Q would then give tan 6 = 1 6 = 45 deg., or 70.7 per cent power-factor; that is, with an armature reaction beyond the limits of JV\ good motor design, the power-factor is still too low for use. The armature, however, also has a self-inductance; that is, the magnetic flux produced by the armature current as shown diagrammatically in Fig. 119 generates a reactive e.m.f. in the arma- ture conductors, which again lowers the power-factor. While this armature self- inductance is low with small number * YN- x -u x- * f armature turns, it becomes consid- Fig. 119. Distribution of , , , ,, , . , Main Field and Field of erable when the Dumber of armature Armature Reaction. turns n t is large compared with the field turns n . Let 61 = field reluctance, that is, reluctance of the magnetic field circuit, and the armature reluctance, that is, /n /n = .a ratio of reluctances of the armature and the field mag- . netic circuit; then, neglecting magnetic saturation, the field flux is the armature flux is $ x = L. ^ tkJ Zh 5 depends upon and is in phase with the field current, except as far as it is modified by the magnetic action of the short-circuit current in the armature coil under the commu- tator brushes. In the conductively compensated series motor, 1, the quad- rature flux is z?ero at complete compensation, and in the direc- 234 ELEMENTS OF ELECTRICAL ENGINEERING. tion of the armature reaction with undercompensation, in opposition to the armature reaction at over-compensation, but in either case in phase with the current and so approximately with the field. In the other motor types, whatever quadrature flux exists is not in phase with the main flux, but as transformer flux is due to the resultant m.m.f. of primary and secondary circuit. In a transformer with non-inductive or nearly non-inductive secondary circuit, the magnetic flux is nearly 90 deg. in time phase behind the primary current, a little over 90 deg. ahead of the secondary current, as shown in transformer diagram, Fig. 136. In a transformer with inductive secondary, the magnetic flux is less than 90 deg. behind the primary current, more than 90 deg. ahead of the secondary current, the more so the higher is the inductivity of the secondary circuit, as shown by the transformer diagram, Fig. 137. Herefrom it follows that In the inductively compensated series motor, 2, the quad- rature flux is very small and practically negligible, as very little voltage is consumed in the low impedance of the secondary circuit C; whatever flux there is, lags behind the main flux. In the inductively compensated series motor with secondary excitation, or inverted repulsion motor, 3, the quadrature flux ! is quite large, as a considerable voltage is required for the field excitation, especially at moderate speeds and therefore high currents, and this flux 4> l lags behind the field flux $, but this lag is very much less than 90 deg., since the secondary cir- cuit is highly inductive; the motor field thus corresponding to the conditions of the transformer diagram, Fig. -137. As result hereof, the commutation of this type of motor is very good, flux $ t having the proper phase and intensity required for a corn- mutating flux, as will be seen later, but the power-factor is poor. In the repulsion motor, 4, the quadrature flux is very con- siderable, since all the voltage consumed by the rotation of the armature is induced in it by transformation from the compen- sating winding, and this quadrature flux $ t lags nearly 90 deg. behind the main flux <>, since the secondary circuit is nearly non-inductive, especially at speed. In the repulsion motor with secondary excitation, 5, the quadrature flux $ x is also very large, and practically constant, ALTERNA TING-CURREN T COMMUTA TING MACHINES. 235 corresponding to the impressed e.m.f., but lags considerably less than 90 deg. behind the main flux $, the secondary circuit being inductive, since it contains the field coil F, The lag of the flux & t increases with increasing speed, since with increasing speed Figs. 136-137. Transformer Diagram, Non-inductive and Inductive Load. the e.m.f. of rotation of the armature increases, the e.m.f. of self-inductance of the field decreases, due to the decrease of cur- rent, and the circuit thus becomes less inductive. The series repulsion motors 6 and 7, give the same phase relation of the quadrature flux $ t as the repulsion motors, 5 and 6, but the intensity of the quadrature flux $ t is the less the smaller the part of the supply voltage which is impressed upon the compensating winding. 286 ELEMENTS OF ELECTRICAL ENGINEERING. V. Commutation. In the commutator motor, the current in each armature coil or turn reverses during its passage under the brush. In the armature coil, while short-circuited by the commutator brush, the current must die out to zero and then increase again to its original value in opposite direction. The resistance of the armature coil and brush contact accelerates, the self-inductance retards the dying out of the current, and the former thus assists, the latter impairs commutation. If an e.m.f. is generated in the armature coil by its rotation while short-circuited by the commutator brush, this e.m.f. opposes commutation, that is, retards the dying out of the current, if due to the magnetic flux of armature reaction, and assists commutation by reversing the armature current, if due to the magnetic flux of over-com- pensation, that is, a magnetic flux in opposition to the armature reaction. Therefore, in the direct-current commutator motor with high field strength and low armature reaction, that is, of negligible magnetic flux of armature reaction, fair commutation is pro- duced with the brushes set midway between the field poles, that is, in the position where the armature coil which is being commutated encloses the full field flux and therefore cuts no flux and has no generated e.m.f., by using high-resistance carbon brushes, as the resistance of the brush contact, increas- ing when the armature coil begins to leave the brush, tends to reverse the current. Such "resistance commutation' 7 obviously cannot be perfect; perfect commutation, however, is produced by impressing upon the motor armature at right angles to the main field, that is, in the position of the commutator brushes, a magnetic field opposite to that of the armature reaction and proportional to the armature current. Such a field is produced by over-compensation or by the use of a commutating pole or interpole. As seen in the foregoing, in the direct-current motor the counter e.m.f. of self-inductance of commutation opposes the reversal of current in the armature coil under the commutator brush, and this can be mitigated in its effect by the use of high- resistance brushes, and overcome by the commutating field of over-compensation. In addition hereto, however, in the alter- ALTERNA TING-CURRENT COMMUTA TING MACHINES. 237 nating-current commutator motor an e.m.f. is generated in the coil short-circuited under the brash, by the alternation of the magnetic flux, and this e.m.f., which does not exist in the direct- current motor, makes the problem of commutation of the alter- nating-current motor far more difficult. In the position of commutation no e.m.f. is generated in the armature coil by its rotation through the magnetic field, as in this position the coil encloses the maximum field flux; but as this magnetic flux is alternating, in this position the e.m.f. generated by the alter- nation of the flux enclosed by the coil is a maximum. This "e.m.f. of alternation' 7 lags in time 90 deg. behind the magnetic flux which generates it, is proportional to the magnetic flux and to the frequency, but is independent of the speed, hence exists also at standstill, while the "e.m.f. of rotation' 7 which is a max- imum in the position of the armature coil midway between the brushes, or parallel to the field flux is in phase with the field flux and proportional thereto and to the speed, but independ- ent of the frequency. In the alternating-current commutator motor, no position therefore exists in which the armature coil is free from a generated e.m.f., but in the position parallel to the field, or midway between the brushes, the e.m.f. of rota- tion, in phase with the field flux, is a maximum, while the e.m.f. of alternation is zero, and in the position under the com- mutator brush, or enclosing the total field flux, the e.m.f. of alternation, in electrical space quadrature with the field flux, is a maximum, the e.m.f. of rotation absent, while in any other position of the armature coil its generated e.m.f. has a com- ponent due to the rotation a power e.m.f. and a component due to the alternation a reactive e.m.f. The armature coils of an alternating-current commutator motor, therefore, are the seat of a system of polyphase e.m.fs., and at synchronism the polyphase e.m.fs. generated in all armature coils are equal, above synchronism the e.m.f. of rotation is greater, while below syn- chronism the e.m.f. of alternation is greater, and in the latter case the brushes thus stand at that point of the commutator where the voltage between commutator segments is a maximum. This e.m.f. of alternation, short-circuited by the armature coil in the position of commutation, if not controlled, causes a short- circuit current of excessive value, and therewith destructive sparking; hence, in the alternating-current commutator motor 288 ELEMENTS OF ELECTRICAL ENGINEERING. it is necessary to provide means to control the short-circuit current under the commutator brushes, which results from the alternating character of the magnetic flux, and which does not exist in the direct-current motor; that is, in the alternating- current motor the armature coil under the brush is in the position of a short-circuited secondary, with the field coil as primary of a transformer; and as in a transformer primary and sec- ondary ampere-turns are approximately equal, if n = number of field turns per pole and i = field current, the current in a single armature turn, when short-circuited by the commutator brush, tends to become i =^ i, that is, many times full-load current; and as this current is in opposition, approximately, to the field current, it would demagnetize the field; that is, the motor field van- ishes, or drops far down, and the motor thus loses its torque. Especially is this the case at the moment of starting; at speed, the short-circuit current is somewhat reduced by the self-induc- tance of the armature turn. That is, during the short time during which the armature turn or coil is short-circuited by the brush the short-circuit current cannot rise to its full value, if the speed is considerable, but it is still sufficient to cause destruc- tive sparking. Various means have been proposed and tried to mitigate or eliminate the harmful effect of this short-circuit current, as high resistance or high reactance introduced into the armature coil during commutation, or an opposing e.m.f. either from the out- side, or by a commutating field. High-resistance brush contact, produced by the use of very narrow carbon brushes of high resistivity, while greatly improv- ing the commutation and limiting the short-circuit current so that it does not seriously demagnetize the field and thus cause the motor to lose its torque, is not sufficient, for the reason that the resistance of the brush contact is not high enough and also is not constant. The brush contact resistance is not of the nature of an ohmic resistance, but more of the nature of a counter e.m.f. ; that is, for large currents the potential drop at the brushes becomes approximately constant, as seen from the volt-ampere characteristics of different brushes given in Figs. 138 and 139. Fig. 138 gives the voltage consumed by the brush contact of a copper brush, with the current density as abscissas, while Fig. 139 gives the voltage consumed by a high-resistance carbon ALTERNA TING-CURRENT COMMUTA TING MACHINES. 239 brush, with the current density in the brush as abscissas* It is seen that such a resistance, which decreases approximately inversely proportional to the increase of current, fails in limit- ing the current just at the moment where it is most required, that is, at high currents. 0.15 ao 40 co so iuo 120 no 130 iso AMP, PER SQ. IN, I 810 260 280 300 Fig. 138. E.m.f. Consumed at Contact of Copper Brush. -1-.0- -04 10 20 30 Fig. 139. E.m.f. Consumed at Contact of High-resistance Carbon Brush. 50 60 70 80 90 100 110 12Q 130 * liO AMP, PERSQ, IN A COMMUTATOR LEADS. Good results have been reached by the use of metallic resist- ances in the leads between the armature and the commutator. As shown diagrammaticaUy in Fig. 140 as C> each commutator segment connects to the armature A by a high non-inductive resist- ance Bj and thus two such resistances are always in the circuit of 240 ELEMENTS OF ELECTRICAL ENGINEERING. the armature coil short-circuited under the brush, but also one or two in series with the armature main circuit, from brush to brush. While considerable power may therefore be consumed in these high-resistance leads, nevertheless the efficiency of the motor is greatly increased by their use; that is, the reduction in the loss of power at the commutator by the reduction of the short-circuit current usually is far greater than the waste of Fig. 140. Commutation with Resistance Leads. power in the resistance leads. To have any appreciable effect, the resistance of the commutator lead must be far higher than that of the armature coil to which it connects. Of" the e.m.f. of rotation, that is, the useful generated e.m.f. , the armature resistance consumes only a very small part, a few per cent only. The e.m.f. of alternation is of the same magnitude as the e.m.f. of rotation, higher below, lower above synchronism. With a short-circuit current equal to full-load current, the resistance of the short-circuit coil would consume only a small part of the generated e.m.f. of alternation, and to consume the total e.m.f. the short-circuit current therefore would have to be about as many times larger than the normal armature current as the useful generated e.m.f. of the motor is larger than the resistance drop in the armature. Long before this value of short-circuit current is reached the magnetic field would have disappeared by the demagnetizing force of the short-circuit current, that is, the motor would have lost its torque. The ratio of the maximum e.m.f. of alternation e l at the brushes to the maximum e.m.f. of rotation midway between the brushes, is the ratio of frequency of alternation,/, to frequency of rotation, / , or e l * e g = / * / . The mean value of the 2 1 e.m.f. of rotation, per armature coil, then is - e , and, assuming - ALTERNA TING-CURRENT COMMUTA TING MACHINES. 241 as the fraction of the generated voltage consumed by the arma- ture resistance, and r = resistance per armature coil, the volt- age consumed by the resistance of the armature coil at normal current i would be -, If then r is the total resistance of the short-circuit under the brush, armature coil, brush contact, and resistance leads re- quired to limit the short-circuit current to q times the arma- ture current, the total voltage of alternation is consumed by current i = qi, in resistance r , if qir, = ej (14) hence, dividing, f 12e 2 r o """ P * *i I 2 f i4f, (15) P*f and hence, if at synchronism, / = /, the resistance drop in the arma- ture is 5 per cent, or p = - = 20, and the short-circuit cur- U.Uo rent should be limited to twice the armature current, q = 2, r = 15.7r; the resistance of the two leads and brush contact thus is 2 r l = r r = 14.7 r, and each resistance lead and brush contact thus about 7 times the resistance of the armature coil to which it connects. The current i in the resistance lead, however, Is twice the armature current t, and the heat produced in the resistance lead, ifr v therefore is about 30 times that produced in the armature coil. The space available for the resistance lead, however, is less than that available for the armature coil. It is obvious herefrom that it is not feasible to build these resistance leads so that each lead can dissipate continuously, or even for any appreciable time, without rapid self-destruction, the heat produced in it while in circuit. 242 ELEMENTS OF ELECTRICAL ENGINEERING. When the motor is revolving, even very slowly, this is not neces- sary, since each resistance lead is only a very short time in cir- cuit, during the moment when the armature coils connecting to it are short-circuited by the brushes; that is, if n l = number of 2 armature turns from brush to brush, the lead is only of the n, time in circuit, and though excessive current densities in mate- rials of high resistivity are used, the heating is moderate. In starting the motor, however, if it does not start instantly, the current continues to flow through the same resistance leads, and thus they are overheated and destroyed if the motor does not start promptly. Hence care has to be taken not to have such motors stalled for any appreciable time with voltage on. The most serious objection to the use of high-resistance leads, therefore, is their liability to self-destruction by heating if the motor fails to start immediately, as for instance when putting the voltage on the motor before the brakes are released, as is done when starting on a steep up-grade to keep the train from start- ing to run back. Thus the advantages of resistance commutator leads are the improvement in commutation resulting from the reduced short- circuit current, and the absence of a series demagnetizing effect on the field at the moment of starting, which would result from an excessive short-circuit current under the brush, and such leads are therefore extensively used ; their disadvantage, however, is that when they are used the motor must be sure to start immediately by the application of voltage, otherwise they are liable to be destroyed. It is obvious that even with high-resistance commutator leads the commutation of the motor cannot be as good as that of the motor on direct-current supply; that is, such an alternating- current motor inherently is more or less inferior in commutation to the direct-current motor, and to compensate for this effect far more favorable constants must be chosen in the motor design than permissible with a direct-current motor, that is, a lower voltage per commutator segment and lower magnetic flux per pole, hence a lower supply voltage on the armature, and thus a larger armature current and therewith a larger com- mutator, etc. The insertion of reactance instead of resistance in the leads ALTERNA TING-CURREN T COMMUTA TING MACHINES. 2 connecting the commutator segments with the armature coils of the single-phase motor also has been proposed and used for limiting the short-circuit current under the commutator brush. Reactance has the advantage over resistance, that the voltage consumed by it is wattless and therefore produces no serious heat- ing and reactive leads of low resistance thus are not liable to self- destruction by heating if the motor fails to start immediately. On account of the limited space available in the railway motor considerable difficulty, however, is found in designing suffi- ciently high reactances which do not saturate and thus decrease at larger currents. At speed, reactance in the armature coils is very objectionable in retarding the reversal of current, and indeed one of the most important problems in the design of commutating machines is to give the armature coils the lowest possible reactance. Therefore the insertion of reactance in the motor leads interferes seriously with the commutation of the motor at speed, and thus requires the use of a suitable commutating or reversing flux, that is, a magnetic field at the commutator brushes of sufficient strength to reverse the current, against the self-inductance of the arma- ture coil, by means of an e.m.f. generated in the armature coil by its rotation. This commutating flux thus must be in phase with the main current, that is, a flux of over-compensation. Reactive leads require the use of a commutating flux of over- compensation to give fair commutation at speed. COUNTER E.M.FS. IN COMMUTATED COIL. Theoretically, the correct way of eliminating the destructive effect of the short-circuit current under the commutator brush resulting from the e.m.f. of alternation of the main flux would be to neutralize the e.rn.f. of alternation by an equal but oppo- site e.m.f. inserted into the armature coil or generated therein. Practically, however, at least with most motor types, consider- able difficulty is met in producing such a neutralizing e.m.f. of the proper intensity as well as phase. Since the alternating current has not only an intensity but also a phase displace- ment, with an alternating-current motor the production of com- mutating flux or commutating voltage is more difficult than with direct-current motors in which the intensity is the only variable. 244 ELEMENTS OF ELECTRICAL ENGINEERING. s s' JLL J-L 7FTTT By introducing an external e.m.f. into the short-circuited coil under the brush it is not possible entirely to neutralize its e.m.f. of alternation, but simply to reduce it to one-half. Several such arrangements were developed in the early days by Eicke- meyer, for instance the arrangement shown in Fig. 141, which represents the development of a commutator. The com- mutator consists of alternate live segments S and dead segments S', that is, seg- ments not connected to armature coils, and shown shaded in Fig. 141. Two sets of brushes on the com- mutator, the one, B v ahead in position from the other, jB 2 , by one commutator seg- ment, and connected to the first by a coil JV, contain- ing an e.m.f. equal in phase, but half in intensity, and opposite, to the e.m.f. of alternation of the armature coil; that is, if the armature coil contains a single turn, coil N is a half turn located in the main field space; if the armature coil A contains m turns, turns in Fig. 141. Commutation with External e.m.f. The dead segments S' as not to short- J3 2 , so the main field space are used in coil N. are cut between the brushes B 1 and circuit between the brushes. In this manner, during the motion of the brush over the com- mutator, as shown by Fig. 142 in its successive steps, in position. 1 there is current through brush B u * 2 there is current through both brushes B 1 and B 2 , and the armature coil A is closed by the counter e.m.f. of coil N, that is, the difference A N is short-circuited; 3 there is current through brush j3 2 ; 4 there is current through both brushes J5 X and B 2 , and the coil N is short-circuited ; 5 the current enters again by brush B t ; thus alternately the coil N of half the voltage of the armature ALTERNATING-CURRENT COMMUTATING MACHINES. 245 coil A, or the difference between A and N is short-circuited, that is, the short-circuit current reduced to one-half. Complete elimination of the short-circuit current can be pro- duced by generating in the armature coil an opposing e.m.f. This e.m.f. of neutralization, however, cannot be generated by the alternation of the magnetic flux through the coil, as this -3 4 Fig. 142. Commutation by External e.m.f. would require a flux equal but opposite to the full field flux traversing the coil, and thus destroy the main field of the motor. The neutralizing e.m.f., therefore, must be generated by the rotation of the armature through the commutating field, and thus can occur only at speed; that is, neutralization of the short-circuit current is possible only when the motor is revolving, but not while at rest. The e.m.f. of alternation in the armature coil short-circuited under the commutator brush is proportional to the main field <>, to the frequency /, and is in time quadrature with the main field, being generated by its rate of change; hence, it can be represented by ^ = 2 xf$ 10~ 8 f. (17) The e.m.f. e l generated by the rotation of the armature coil through a commutating field < l )/ is, however, in time phase with the field which produces it; and since e t must be equal and in phase with e to neutralize it, the commutating field ^> 7 , therefore, must be in time phase with e , hence in time quadrature with $; that is, the commutating field <' of the motor must be in time quadra- ture with the main field $ to generate a neutralizing voltage e 1 of the proper phase to oppose the e.m.f. of alternation in the short-circuited coil. This e.m.f. e l is proportional to its gen- erating field $ 7 , and to the speed, or frequency of rotation, / 0> hence is c 4 - 2 nf$' 10- 8 , (18) 246 ELEMENTS OF ELECTRICAL ENGINEERING. and from e t = e it then follows that $' =J -/; (19) /o that is, the commutating field of the single-phase motor must be in time quadrature behind and proportional to the main field, proportionl to the frequency and inversely proportional to the speed; hence, at synchronism, / = /, the commutation field equals the main field in intensity, and, being displaced there- from in quadrature both in time and in space, the motor thus must have a uniform rotating field, just as the induction motor. Above synchronism, /,, > /, the commutating field <&' is less than the main field ; below synchronism, however, / < /, the commutating field must be greater than the main field to give complete compensation. It obviously is not feasible to in- crease the commutating field much beyond the main field, as this would require an increase of the iron section of the motor beyond that required to do the work, that is, to carry the main field flux. At standstill <' should be infinitely large, that is, compensation is not possible. Hence, by the use of a commutating field in time and space quadrature, in the single-phase motor the short-circuit current under the commutator brushes resulting from the e.m.f. of alter- nation can be entirely eliminated at and above synchronism, and more or less reduced below synchronism, the more the nearer the speed is to synchronism, but no effect can be pro- duced at standstill. In such a motor either some further method, as resistance leads, must be used to take care of the short-circuit current at standstill, or the motor designed so that its commutator can carry the short-circuit current for the small fraction of time when the motor is at standstill or running at very low speed. The main field < of the series motor is approximately inversely proportional to the speed / , since the product of speed and field strength, f $, is proportional to the e.m.f. of rotation, or useful e.m.f. of the motor, hence, neglecting losses and phase displace- ments, to the impressed e.m.f., that is, constant. Substituting therefore $ = ~ $ , where $ = main field at synchronism, into /o equation (19), ALTERNA TING-CURRENT COM.MUTA TING MACHINES. 247 (20) that is, the commutating field is inversely proportional to the square of the speed; for instance, at double synchronism it should be one-quarter as high as at synchronism, etc. Of the quadrature field $' only that part is needed for com- mutation which enters and leaves the armature at the position of the brushes; that is, instead of producing a quadrature field $' in accordance with equation (20), and distributed around the armature periphery in the same manner as the main field <&, but in quadrature position thereto, a local commutating field may be used at the brushes, and produced by a commutating pole or commu- tating coil, as shown diagrammati- cally in Fig. 143 as K t and K. The excitation of this commu- tating coil K then would have to be such as to give a magnetic air-gap density CB', relative to that of the main field, (B, by the same equations (19) and (20) : Fig. 143. Comxnutatio'n with Commutating Poles. (B' = /(B (21) As the alternating flux of a magnetic circuit is proportional to the voltage which it consumes, that is, to the voltage impressed upon the magnetizing coil, and lags nearly 90 deg. behind it, the magnetic flux of the commutating poles K can be produced by energizing these poles by an e.m.f. e, which is varied with the speed of the motor, by equation (22) where e Q is its proper value at synchronism. Since (B 7 lags 90 deg. behind its supply voltage e, and also lags 90 deg. behind (B, by equation (2), and so behind the supply current and, approximately, the supply e.m.f. of the motor, the 248 ELEMENTS OF ELECTRICAL ENGINEERING. voltage, e > required for the excitation of the comrriutating poles is approximately in phase with the supply voltage of the motor; that is, a part thereof can be used, and is varied with the speed of the motor. Perfect commutation, however, requires not merely the elimi- nation of the short-circuit current under the brush, but requires a reversal of the load current in the armature coil during its passage under the commutator brush. To reverse the current, an e.m.f. is required proportional but opposite to the current and therefore with the main field ; hence, to produce a reversing e.m.f. in the armature coil under the commutator brush a second commutating field is required, in phase with the main field and approximately proportional thereto. The commutating field required by a single-phase commutator motor to give perfect commutation thus consists of a component in quadrature with the main field, or the neutralizing component, which eliminates the short-circuit current under the brush, and a component in phase with the main field, or the reversing component, which reverses the main current in the armature coil under the brush; and the resultant commi^tating field thus must lag behind the main field, and so approximately behind the supply voltage, by somewhat less than 90 deg., and have an intensity varying approximately inversely proportional to the square of the speed of the motor. Of the different motor types discussed under IV, the series motors, 1 and 2, have no quadrature field, and therefore can be made to commutate satisfactorily only by the use of com- mutator leads, or by the addition of separate commutating poles. The inverted repulsion motor, 3, has a quadrature field, which decreases with increase of speed, and therefore gives a better commutation than the series motors, though not perfect, as the quadrature field does not have quite the right intensity. The repulsion motors, 4 and 5, have a quadrature field, lag- ging nearly 90 deg. behind the main field, and thus give good commutation at those speeds at which the quadrature field has the right intensity for commutation. However, in the repul- sion motor with secondary excitation, 5, the quadrature field is constant and independent of the speed, as constant supply volt- age is impressed upon the commutating winding (7, which pro- duces the quadrature field, and in the direct repulsion motor, 4, ALTERS A TING-CURRENT COMMUTA TING MACHINES. 249 the quadrature field increases with the speed, as the voltage consumed by the main field F decreases, and that left for the compensating winding C thus increases with the speed, while to give proper commtitating flux it should decrease with the square of the speed. It thus follows .that the commutation of the repulsion motors improves with increase of speed, up to that speed where the quadrature field is just right for commutating field, which is about at synchronism, but above this speed the commutation rapidly becomes poorer, due to the quadrature field being far in excess of that required for commutating. In the series repulsion motors, 6 and 7, a quadrature field also exists, just as in the repulsion motors, but this quadrature field depends upon that part of the total voltage which is im- pressed upon the commutating winding C, and thus can be varied by varying the distribution of supply voltage between the two circuits; hence, in this type of motor, the commutating flux can be maintained through all (higher) speeds by impressing the total voltage upon the compensating circuit and short-cir- cuiting the armature circuit for all speeds up to that at which the required copamutating flux has decreased to the quadrature flux given by the motor, and from this speed upwards only a part of the supply voltage, inversely proportional (approximately) to the square of the speed, is impressed upon the compensating circuit, the rest shifted over to the armature circuit. The differ- ence between 6 and 7 is that in 6 the armature circuit is more inductive, and the quadrature flux therefore lags less behind the main flux than in 7, and by thus using more or less of the field coil in the armature circuit its inductivity can be varied, and therewith the phase displacement of the quadra- ture flux against the main flux adjusted from nearly 90 deg. lag to considerably less lag, hence not only the proper inten- sity but also the exact phase of the required commutating flux produced. As seen herefrom, the difference between the different motor types of IV is essentially found in their different actions re- garding commutation. It follows herefrom that by the selection of the motor type quadrature fluxes ^ l can be impressed upon the motor, as com- mutating flux, of intensities and phase displacements against the main flux $, varying over a considerable range. 250 ELEMENTS OF ELECTRICAL ENGINEERING. VI. Motor Characteristics. The single-phase commutator motor of varying speed or series characteristic comprises three circuits, the armature, the com- pensating winding, and the field winding, which are connected in series with each other, directly or indirectly. The impressed e.m.f. or supply voltage of the motor then consists of the components: 1. The e.m.f. of rotation e v or voltage generated in the arma- ture conductors by their rotation through the magnetic field $. This voltage is in phase with the field < and therefore approxi- mately with the current i, that is, is power e.m.f., and is the voltage which does the useful work of the motor. It is propor- tional to the speed or frequency of rotation/ , to the field strength , and to the number of effective armature turns n v e t = 2 TT/X* 10- 8 . (23) The number of effective armature turns n v with a distributed winding, is the projection of all the turns on their resultant direc- tion. With a full-pitch winding of n series turns from brush to brush, the effective number of turns thus is + - 2 n t = m [avg cos] * = - m. (24) -a * With a fractional pitch winding of the pitch of r degrees, the effective number of turns is r +- 2 . T n l m-[avg cos] * = - m sin-' (25) 71 ~2 7T 2 2. The e.m.f. of alternation of the field, e Q , that is, the voltage generated in the field turns by the alternation of the magnetic flux $, produced by them and thus enclosed by them. This vol- tage is in quadrature with the field flux $ and thus approximately with the current /, is proportional to the frequency of the im- pressed voltage /, to the field strength , and to the number of field turns n, t __ 2 fcfr,*i(r. (26) 3. The impedance voltage of the motor, e' = IZ (27) and Z = r jx } where r = total effective resistance of field coils, armature with commutator and brushes, and compensating winding, x = total ALTERNA TING-CURRENT COMMUTA TING MACHINES. 251 self-inductive reactance, that is, reactance of the leakage flux of armature and compensating winding or the stray flux passing locally between the armature and the compensating conductors plus the self-inductive reactance of the field, that is, the reactance due to the stray field or flux passing between field coils and armature. In addition hereto, x comprises the reactance due to the quadrature magnetic flux of incomplete compensation or over- compensation, that is, the voltage generated hy the quadrature flux $' in the difference between armature and compensating conductors, n t n 2 or n 2 n r Therefore the total supply voltage E of the motor is E = e l + e + e' = 2 TT/X$ 10~ 8 - 2 jjffn^ 10" 8 + (r - jx) I . (28) Let, then, R = magnetic reluctance of field circuit, thus n J $ = -^ = the magnetic field flux, when assuming this flux as ri in phase with the excitation 7, and denoting *-*?- *. (30) as the effective reactance of field inductance, corresponding to the e.m.f. of alternation, S = &> = ratio of speed to frequency, or speed * as fraction of synchronism, . n . ^ ^ c = i = ratio of effective armature turns to n field turns; substituting (30) and (31) in (28), or, I = ~ ( ^ r> (33) and, in absolute values, i = \ (34) The power-factor is given by tan - X + ft - (35) r + cSx ^ 252 ELEMENTS OF ELECTRICAL ENGINEERING. The useful work of the motor is done by the e.m.f. of rotation, jg, -c&cj, and, since this e.m.f. E t is in phase with the current 7 , the useful work, or the motor output (inclusive friction, etc.), is P - EJ __ _ JL_ _ ? (r + cSx )-+(x + x ) 2 (36) ; -r v-fc -r ^ ; and the torque of the motor is > f*y* >" (37) n __ __ rr ft U LJO^ For instance, let e = 200 volts, c = ^ = 4, n o Z= r - jx = 0.02 - 0.06 /, x c = 0.08; then 10,000 in amp., cot e 32,0005 32,000 The behavior of the motor at different speeds is best shown by plotting ij p = cos 0, P and D as ordinates with the speed S as abscissas, as shown in Fig. 144. In railway practice, by a survival of the practice of former times, usually the constants are plotted with the current / as abscissas, as shown in Fig. 145, though obviously this arrange- ment does not as well illustrate the behavior ot the motor. Graphically, by starting with the current 7 as zero axis 07, the motor diagram is plotted in Fig. 146. The_yoltage consumed by the resistance r is OE r ir, in phase with 07; the voltage consumed by thejreactance x is OE X = ix, and 90 deg. ahead of 07. OEr and OE X combine to the voltage consumed by the motor impedance, OE f = iz. ALTERNATING-CURRENT COMMUTA.TING MACHINES. 258 0.1 0/4 0.3 0.4 0.5 O.o 0.7 0.8 0.9 1.0 1.1 1-5 1.0 JU7 1.8 l.U 2.0 2.1 Fig. 144. Single-Phase Commutator Motor Speed Characteristics. 2.0 1 D: 640 600 560 520 480 440 400 360 320 280 240 200 160 120 80 40 1 8 S: ^ -Ii7~ \ \ / l-4~ ^ S / \ f P: \ ^~" P "V / OOS^? \ X \J / 110 -1 0- \, / fi 100 -0 9- ^. *^. A / \ r-90-.-T - -0 8- / \^ -SSs / \ 80 7- , / \ ~*** y \ 70 6- / \ / /^ N \ \ 60 " "^ 5 / K \ \ \ ...gQ.inu,-,. 4 n / ^ \ \ \ 3 / \ \ OA 0*2- / \ w o 04- ^ X i= \ 100! 00*8 10 4( Si V B \ 7 >0 8( M> 9< 10 0011 00 IS 00 U K\ Fig. 145, Single-Phase Commutator Motor Current Characteristics, 254 ELEMENTS OF ELECTRICAL ENGINEERING. Combining OE' j^jiz, OE l = e v and OE = e thus gives the terminal voltage OE e of the motor, and the phase angle J5/07 - 0. In this diagram, and in the preceding approximate calcula- tion, the magnetic flux < has been assumed in phase with the current /. In reality, however, the equivalent sine wave of magnetic flux <3> lags behind the equivalent sine wave of exciting current / Fig, 146. Single-Phase Commutator Motor Vector Diagram. by the angle of hysteresis lag, and still further by the power consumed by eddy currents, and, especially in the commutator motor, by the power consumecHn the short-circuit current under the brushes, and the vector 0<$> therefore is behind the current vector 01 by an angle 6 aj which is small in a motor in which the short-circuit current under the brushes is eliminated and the eddy currents are negligible, but may reach considerable values in the motor of poor commutation. Assuming then, in Fig. 147, 0$ lagging behind^O/ by angle 6 a , OE l is in phase with 0$, hence lagging behind 01; that is, the e.m.f. of rotation is not entirely a power e.m.f., but contains a wattless lagging component. The e.m.f. of alternation, OE^ is 90 deg. ahead of 0$, hence less than 90 deg. ahead of O/, and therefore contains a power component representing the power consumed by hysteresis, eddy currents, and the short-circuit current under the brushes. Completing now the diagram, it is seen that the phase angle 6 is reduced, that is, the power-factor of the motor increased by the increased loss of power, but is far greater than corresponding thereto. It is the result of the lag of the e.m.f. of rotation, which ALTERNA TING-CURREN T COMMUTA TING .MACHINES. 255 produces a lagging e.m.f. component, partially compensating for the leading e.m.f. consumed by self-inductance, a lag of the e.m.f. being equivalent to a lead of the current. As the result of this feature of a lag of the magnetic flux , by producing a lagging e.m.f. of rotation and thus compensating E Fig. 147. Single-Phase Commutator Motor Diagram with Phase Displacement between Flux and Current. for the lag of current by self-inductance, single-phase motors having poor commutation usually have better power-factors, and improvement in commutation, by eliminating or reducing the short-circuit current under the brush, usually causes a slight decrease in the power-factor, by bringing the magnetic flux $ more nearly in phase with the current /. Inversely, by increasing the lag of the magnetic flux , the phase angle can be decreased and the power-factor improved. Such a shift of the magnetic flux in phase with the Rotation, e.m.f. ^rotation OE V and the component of current in the field OF and in the non- inductive resistance 07", in phase and in quadrature respec- tively with 0$, which combined make up the totaLeurrent. The projection of the e.m.f. of rotation OE l on 01 then is the power component of the e.m.f., which does_the work of the motor, and the quadrature projection of OE^ is the compen- sating component of the e.rn.f. of rotation, which neutralizes the wattless component of the e.m.f. of self-inductance. Obviously such a compensation involves some loss of power in the non-inductive resistance r shunting the field coils, and as the power-factor of the motor usually is sufficiently high, such compensation is rarely needed. In motors in which some of the circuits are connected in series with the others the diagram is essentially the same, except that a phase displacement exists between the secondary and the primary current. The secondary current l l of the transformer lags behind the primary current / slightly less than 180 deg. ; that is, considered in opposite direction, the secondary current leads the primary by a small angle , and in the motors with secondary excitation the field flux , being in phase with the ALTERNA TING-CURRENT COIUIUTA TING MACHINES. 257 field current I t (or lagging by angle 6 a behind it), thus leads the primary current / by angle 6 Q (or angle # a ). As a lag of the magnetic flux increases, and a lead thus decreases the power-factor, motors with secondary field excitation usually have a slightly lower power-factor than motors with primary field Fig. 149. Single-Phase Commutator Motor. Unity Power-Factor Produced by Lagging e.m.f. of Rotation. excitation, and therefore, where desired, the power-factor may be improved by shunting the field with jijion-inductive resist- ance r . Thus for instance^ in Fig. 150 07 = j>rlrnary current, 07[ = secondary current, OE V in phase with OI 13 is the e.m.f. of rotation, in the case of thejsecondary field excitation, and OE~ Q , in quadrature ahead of OT V is the^m-f-jjf alternation, while OE 7 is the impedance voltage, and OE 17 OE Q and OE' com- bined give the supply voltage OE, and EOI = 6 the angle of lag. Shunting the field by a non-inductive resistance r , and thus resolving the secondary current OI t into the components O// in the field and 01" in the non-inductive resistance, gives the diagram Fig. 151, where a = 7/0* = angle of lag of magnetic field. The action of the commutator in an alternating-current motor, in permitting compensation for phase displacement and thus allowing a control of the power-factor, is very interesting and important, and can also be used in other types of machines, as induction motors and alternators, by supplying these machines with a commutator for phase control. 258 ELEMENTS OF ELECTRICAL ENGINEERING. A lag of the current is the same as a lead of the e.m.f ., and in- versely a leading current inserted into a circuit has the same effect as a lagging e.m.f. inserted. The commutator, however, produces an e.m.f. in phase with the current. Exciting the field by a lagging current in the field, a lagging e.m.f. of rotation is Fig. 150. Single-Phase Commutator Motor Diagram with Secondary Excitation. EI Fig. 151. Single-Phase Commutator Motor with Secondary Excitation Power- Factor Improved by Shunting Field Winding with Non-inductive Circuit. produced which is equivalent to a leading current. As it is easy to produce a lagging current by self-inductance, the com- mutator thus affords an easy means of producing the equivalent of a leading current. Therefore the alternating-current com- mutator is one of the important methods of compensating for lagging currents. Other methods are the use of electrostatic or electrolytic condensers and of overexcited synchronous machines. Based on this principle, a number of designs of induction ALTERNATING-CURRENT COMMUTATING MACHINES. 259 motors and other apparatus have been developed, using the commutator for neutralizing the lagging magnetizing current and the lag caused by self-inductance, and thereby producing unity power-factor or even leading currents. So far, however, none of them has come into extended use. This feature, however, explains the very high power-factors feasible in single-phase commutator motors even with consider- able air gaps, far larger than feasible in induction motors. VII. Efficiency and Losses. The losses in single-phase commutator motors are essentially the same as in other types of machines : (a.) Friction losses, air friction or windage, bearing friction and commutator brush friction, and also gear losses or other mechanical transmission losses. (&.) Core losses, as hysteresis and eddy currents. These are of two classes, the alternating core loss, clue to the alternation of the magnetic flux in the main field, quadrature field, and arma- ture; and the rotating core loss, due to the rotation of the arma- ture through the magnetic field. The former depends upon the frequency, the latter upon the speed. (c.) Commutation losses, as the power consumed by the short- circuit current under the brush, by arcing and sparking, where such exists. (d.) fr losses in the motor circuits, the field coils, the compen- sating winding, the armature and the brush contact resistance. (e.) Load losses, mainly represented by an effective resistance, that is, an increase of the total effective resistance of the motor beyond the ohmic resistance. Driving the motor by mechanical power and with no voltage on the motor gives the friction and the windage losses, exclusive of commutator friction, if the brushes are lifted off the com- mutator, inclusive, if the brushes are on the commutator. Energizing now the field by an alternating current of the rated frequency, with the commutator brushes off, adds the core losses to the friction losses; the increase of the driving power then measures the rotating core loss, while a wattmeter in the field exciting circuit measures the alternating core loss. Thus the alternating core loss is supplied by the impressed 260 ELEMENTS OF ELECTRICAL ENGINEERING. electric power, the rotating core loss by the mechanical driving power. Putting now the brushes clown on the commutator adds the commutation losses. The ohmic resistance gives the %~r losses, and the difference between the ohmic resistance and the effective resistance, calcu- lated from wattmeter readings with alternating current in the motor circuits at rest and with the field unexcited, represents the load losses. However, the different losses so derived have to be corrected for their mutual effect. For instance, the commutation losses are increased by the current in the armature; the load losses are less with the field excited than without, etc.; so that this method of separately determining the losses can give only an estimate of their general magnitude, but the exact determination of the efficiency is best carried out by measuring electric input and mechanical output. VIII. Discussion of Motor Types. Varying-speecl single-phase commutator motors can be divided into two classes, namely, compensated series motors and repul- sion motors. In the former, the main supply current is through the armature, while in the latter the armature is closed upon itself as secondary circuit, with the compensating winding as primary or supply circuit. As the result hereof the repulsion motors contain a transformer flux, in quadrature position to the main flux, and lagging behind it, while in the series motors no such lagging quadrature flux exists, but in quadrature position to the main flux, the flux either is zero complete compensation or in phase with the main flux over- or undercompensation. A. COMPENSATED SERIES MOTORS. Series motors give the best power-factors, with the excep- tion of those motors in which by increasing the lag of the field flux a compensation for power-factor is produced, as discussed in V. The commutation of the series motor, however, is equally poor at all speeds, due to the absence of any commutating flux, and with the exception of very small sizes such motors there- fore are inoperative without the use of either resistance leads or ALTERS A TING-CURREN T COMMUTA TING MACHINES. 261 commutating poles. With high-resistance leads, however, fair operation is secured, though obviously not of the same class with that of the direct-current motor; with commutating poles or coils producing a local quadrature flux at the brushes good results have been produced abroad. Of the two types of compensation, conductive compensation, 1, with the compensating winding connected in series with the armature, and inductive compensation, 2, with the compensated winding short-circuited upon itself, inductive compensation necessarily is always complete or practically complete com- pensation, while with conductive compensation a reversing flux can be produced at the brushes by over-compensation, and the commutation thus somewhat improved, especially at speed, at the sacrifice, however, of the power-factor, which is lowered by the increased self-inductance of the compensating winding. On the short-circuit current under the brushes, due to the e.m.f. of alternation, such over-compensation obviously has no helpful effect. Inductive compensation has the advantage that the compensating winding is not connected with the supply circuit, can be made of very low voltage, or even of individually short- circuited turns, and therefore larger conductors and less insula- tion used, which results in an economy of space, and therewith an increased output for the same size of motor. Therefore inductive compensation is preferable where it can be used. It is not permissible, however, in motors which are required to operate also on direct current, since with direct-current supply no induction takes place and therefore the compensation fails, and with the high ratio of armature turns to field turns, without compensation, the field distortion is altogether too large to give satisfactory commutation, except in small motors. The inductively compensated series motor with secondary excitation, or inverted repulsion motor, 3, takes an intermedi- ary position between the series motors and the repulsion motors ; it is a series motor in so far as the armature is in the main supply circuit, but magnetically it has repulsion motor characteristics, that is, contains a lagging quadrature flux. As the field exci- tation consumes considerable voltage, when supplied from the compensating winding as secondary circuit, considerable voltage must be generated in this winding, thus giving a corresponding transformer flux. With increasing speed and therewith decreas- 262 ELEMENTS OF ELECTRICAL ENGINEERING. ing current, the voltage consumed by the field coils decreases, and therewith the transformer flux which generates this voltage. Therefore the inverted repulsion motor contains a transformer flux which has approximately the intensity and the phase re- quired for commutation; it lags behind the main flux, but less than 90 deg., thus contains a component in phase with the main flux, as reversing flux, and decreases with increase of speed. Therefore the commutation of the inverted repulsion motor is very good, far superior to the ordinary series motor, and it can be operated without resistance leads; it has, however, the serious objection of a poor power-factor, resulting from the lead of the field flux against the armature current, due to the secondary excitation, as discussed in V. To make such a motor satis- factory in power-factor requires a non-inductive shunt across the field, and thereby a waste of power. For this reason it has not come into commercial use. B. REPULSION MOTORS. Repulsion motors are characterized by a lagging quadrature flux, which transfers the power from the compensating winding to the armature. At standstill, and at very low speeds, repulsion motors and series motors are equally unsatisfactory in commu- tation; while, however, in the series motors the commutation remains bad (except when using commutating devices), in the repulsion motors with increasing speed the commutation rapidly improves, and becomes perfect near synchronism. As the result hereof, under average railway conditions a much inferior com- mutation can be allowed in repulsion motors at very low speeds than in series motors, since in the former the period of poor commutation lasts only a very short time. While, therefore, series motors cannot be satisfactorily operated without resistance leads (or commutating poles), in repulsion motors resistance leads are not necessary and not used, and the excessive current density under the brushes in the moment of starting permitted, as it lasts too short a time to cause damage to the commutator. As the transformer field of the repulsion motor is approx- imately constant, while the proper commutating field should decrease with the square of the speed, above synchronism the transformer field is too large for commutation, and at speeds ALTERNATING-CURRENT COMMUTATING MACHINES. 263 considerably above synchronism 50 per cent and more the repulsion motor becomes inoperative because of excessive sparking. At synchronism, the magnetic field of the repulsion motor is a rotating field, like that of the polyphase induction motor. Where, therefore, speeds far above synchronism are required, the repulsion motor cannot be used; but where synchronous speed is not much exceeded the repulsion motor is preferred because of its superior commutation. Thus when using a commutator as auxiliary device for starting single-phase induction motors the repulsion motor type is used. For high frequencies, as 60 cycles, where peripheral speed forbids synchronism being greatly exceeded, the repulsion motor is the only type to be seriously considered. Repulsion motors also may be built with primary and sec- ondary excitation. The latter usually gives a better commuta- tion, because of the lesser lag of the transformer flux, and therewith a greater in-phase component, that is, greater reversing flux, especially at high speeds. Secondary excitation, however, gives a slightly lower power-factor, A combination of the repulsion motor and series motor types is the series repulsion motor, 6 and 7. In this only a part of the supply voltage is impressed upon the compensating wind- ing and thus transformed to the armature, while the rest of the supply voltage is impressed directly upon the armature, just as in the series motor. As result thereof the transformer flux of the series repulsion motor is less than that of the repulsion motor, in the same proportion in which the voltage impressed upon the compensating winding is less than the total supply voltage. Such a motor, therefore, reaches equality of the transformer flux with the commutating flux, and gives perfect commutation at a higher speed than the repulsion motor, that is, above synchronism. With the total supply voltage impressed upon the compensating winding, the transformer flux equals the corn- mutating flux at synchronism. At n times synchronous speed the commutating flux should be of what it is at synchronism, 77* and by impressing : of the supply voltage upon the compen- n sating winding, the rest on the armature, the transformer flux 264 ELEMENTS OF ELECTRICAL ENGINEERING. is reduced to ~r of its value, that is, made equal to the required n* commutating flux at n times synchronism. In the series repulsion motor, by thus gradually shifting the supply voltage from the compensating winding to the arma- ture and thereby reducing the transformer flux, it can be maintained equal to the required commutating flux at all speeds from synchronism upwards; that is, the series repulsion motor arrangement permits maintaining the perfect commutation, which the repulsion motor has near synchronism, for all higher speeds. With regard to construction, no essential difference exists between the different motor types, and any of the types can be operated equally well on direct current by connecting all three circuits in series. In general, the motor types having primary and secondary circuits, as the repulsion and the series repulsion motors, give a greater flexibility, as they permit winding the circuits for different voltages, that is, introducing a ratio ^ of transformation between primary and secondary circuit. Shift- ing one motor element from primary to secondary, or inversely, then gives the equivalent of a change of voltage or change of turns. Thus a repulsion motor in which the stator is wound for a higher voltage, that is, with more turns, than the rotor or armature, when connecting all the circuits in series for direct- current operation, gives a direct-current motor having a greater field excitation compared with the armature reaction, that is, the stronger field which is desirable for direct-current operation but not permissible with alternating current. In general, the constructive differences between motor types are mainly differences in connection of the three circuits. For instance, let F = field circuit, A = armature circuit, C = com- pensating circuit, T = supply transformer, R = resistance used in starting and at very low speeds. Connecting, in Fig. 152, the armature A between field F and compensating winding C. With switch open the starting resistance is in circuit; closing switch short-circuits the starting resistance and gives the running conditions of the motor. With all the other switches open the motor is a conductively compensated series motor. Closing 1 gives the inductively compensated series motor, ALTERNA TING-CURRENT COMMUTA TING MACHINES. 265 Closing 2 gives the repulsion motor with primary excitation. Closing 3 gives the repulsion motor with secondary excitation. Closing 4 or 5 or 6 or 7 gives the successive speed steps of the series repulsion motor with armature excitation. F A C Fig. 152. Alternating-Current Commutator Motor Arranged to Operate either as Series or Repulsion Motor. Connecting, in Fig. 153, the field F between armature A and compensating winding (7, the resistance 12 is again controlled by switch 0. All other switches open gives the conductively compensated series motor. T Fig. 153. Alternating-Current Commutator Motor Arranged to Operate either as Series or Repulsion Motor. Switch 1 closed gives the inductively compensated series motor. Switch 2 closed gives the inductively compensated series motor with secondary excitation, or inverted repulsion motor. Switch 3 closed gives the repulsion motor with primary excitation. 266 ELEMENTS OF ELECTRICAL ENGINEERING. Switches 4 to 7 give the different speed steps of the series repulsion motor with primary excitation. Opening the connnection at x and closing at y (as shown in dotted line), the steps 3 to 7 give respectively the repulsion motor with secondary excitation and the successive steps of the series repulsion motor with armature excitation. Still further combinations can be produced in this manner, as for instance, in Fig. 152, by closing 2 and 4, but leaving open, the field F is connected across a constant potential supply, in series with resistance R, while the armature also receives con- stant voltage, and the motor then approaches a finite speed, that is, has shunt motor characteristic, and in starting, the main field F and the quadrature field AC are displaced in phase, so give a rotating or polyphase field (unsymmetrical). To discuss all these motor types with their in some instances very interesting characteristics obviously is not feasible. In general, they can all be classified under series motor, repulsion motor, shunt motor, and polyphase induction motor, and com- binations thereof. IX. Other Commutator Motors. Most of the development on single-phase commutator motors has taken place in the direction of varying speed motors for railway service. In other directions commutators have been applied to alternating-current motors and such motors devel- oped (a.) For limited speed, or of the shunt motor type, that is, motors of similar characteristic as the single-phase railway motor, except that the speed does not indefinitely increase with decreasing load but approaches a finite no-load value. Sev- eral types of such motors have been developed, as stationary motors for elevators, variable-speed machinery etc., usually of the single-phase type. By impressing constant voltage upon the field the magnetic field flux is constant, and the speed thus reaches a finite limit- ing value at which the e.m.f. of rotation of the armature through the constant field flux consumes the impressed voltage of the armature. By changing the voltage supply to the field different speeds can be produced, that is, an adjustable speed motor. The main problem in the design of such motors is to get the ALTERNA TING-CURRENT COMMUTA TING MACHINES. 267 field excitation in phase with the armature current and thus produce a good power-factor. (b.) Adjustable-speed polyphase induction motors. In the secondary of the polyphase induction motor an e.m.f. is gen- erated which, at constant impressed e.m.f. and therefore ap- proximately constant flux, is proportional to the slip from synchronism. With short-circuited secondary the motor closely approaches synchronism. Inserting resistance into the second- ary reduces the speed by the voltage consumed in the second- ary. As this is proportional to the current and thus to the load, the speed control of the polyphase induction motor by resistance in the secondary gives a speed which varies with the load, just as the speed control of a direct-current motor by resist- ance in the armature circuit; hence, the speed is not constant, and the operation at lower speeds inefficient. Inserting, how- ever, a constant voltage into the secondary of the induction motor the speed is decreased if this voltage is in opposition, and is increased if this voltage is in the same direction as the sec- ondary generated e.m.f., and in this manner a speed control can be produced. If c = voltage inserted into the secondary, as fraction of the voltage which would be induced in it at full frequency by the rotating field, then the polyphase induction motor approaches at no load and runs at load near to the speed (1 c) or (1 + c) times synchronism, depending upon the direction of the inserted voltage. Such a voltage inserted into the induction motor secondary must, however, have the frequency of the motor secondary currents, that is, of slip, and therefore can be derived from the full frequency supply circuit only by a commutator revolving with the secondary. If cf is the frequency of slip, then (1 c)/ is the frequency of rotation, and thus the frequency of com* mutation, and at frequency / impressed upon the commutator the effective frequency of the commutated current is / (1 c)/ = c/, or the frequency of slip, as required. Thus the commutator affords a means of inserting voltage into the secondary of induction motors and thus varying its speed. However, while these commutated currents in their resultant give the effect of the frequency of slip, they actually consist of sections of waves of full frequency, that is, meet the full station- ary impedance in the rotor secondary, and not the very much 268 ELEMENTS OF ELECTRICAL ENGINEERING. lower impedance of the low-frequency currents in the ordinary induction motor. If therefore the brushes on the commutator are set so that the inserted voltage is in phase with the voltage generated in the secondary, the power-factor of the motor is very poor. Shift- ing the brushes, by a phase displacement between the generated and the inserted voltage, the secondary currents can be made to lead, and thereby compensate for the lag due to self-inductance and unity power-factor produced. This, however, is the case only at one definite load, and at all other loads either over- compensation or under-compensation takes place, resulting in poor power-factor, either lagging or leading. Such a polyphase adjustable-speed motor thus requires shifting of the brushes with the load or other adjustment, to maintain reasonable power- factor, and for this reason has not been used. (c) Power-factor compensation. The production of an alter- nating magnetic flux requires wattless or reactive volt-amperes, which are proportional to the frequency. Exciting an induction motor not by the stationary primary but by the revolving secondary, which has the much lower frequency of slip, reduces the volt-amperes excitation in the proportion of full frequency to frequency of slip, that is, to practically nothing. This can be done by feeding the exciting current into the secondary by commutator. If the secondary contains no other winding but that connected to the commutator, the exciting current from the commutator still meets full open magnetic circuit inductance, and the motor thus gives a poor power-factor. If, however, in addition to the exciting winding, fed by the commutator, a permanently short-circuited winding is used, as a squirrel-cage winding, the exciting impedance of the former is reduced to practically nothing by the short-circuit winding coincident with it, and so by over-excitation unity power-factor or even leading current can be produced. The presence of the short-circuited winding, however, excludes this method from speed control, and such a motor (Heyland motor) runs near synchronism just as the ordinary induction motor, differing merely by the power-factor. This method of excitation by feeding the alternating current through a commutator into the rotor has been used very success- fully abroad in the so-called "compensated repulsion motor 77 of Winter-Eichberg. This motor differs from the ordinary ALTERNATING-CURRENT COMMUTATIXG MACHINES. 269 repulsion motor merely by the field coil F in Fig. 154 being replaced by a set of exciting brushes G in Fig. 155, at right angles to the main brushes of the armature, that is, located so that the m.m.f. of the current between the brushes magnetizes in the Fig. 154. Plain Repulsion Motor. Fig. 155. Winter-Eichberg Motor. same direction as the field coils F in Fig. 154. Usually the excit- ing brushes are supplied by a transformer or compensator, so as to vary the excitation and thereby the speed. This arrangement then lowers the e.m.f. of self-inductance of field excitation of the motor from that corresponding to full frequency in the ordinary repulsion motor to that of the fre- quency of slip, hence to a negative value above synchronism ; so that hereby a compensation for lagging current can be produced above synchronism, and unity power-factor or even leading currents produced. D. SYNCHRONOUS CONVERTERS. L General. no. For long-distance transmission, and to a certain extent also for distribution, alternating currents, either polyphase or single-phase, are extensively used. For many applications, however, as especially for electrolytic work, direct currents are required, and are usually preferred also for electrical railroad- ing and for low-tension distribution on the Edison three-wire system. Thus, where power is derived from an alternating system, transforming devices are required to convert from alternating to direct current. This can be done either by a direct-current generator driven by an alternating synchronous or induction motor, or by a single machine consuming alternat- ing and producing direct current in one and the same armature. Such a machine is called a converter, and combines, to a certain extent, the features of a direct-current generator and an alter- nating synchronous motor, differing, however, from either in other features. Since in the converter the alternating and the direct current are in the same armature conductors, their e.m.fs. stand in a definite relation to each other, which is such that in practically all cases step-down transformers are necessary to generate the required alternating voltage. Comparing thus the converter with the combination of syn- chronous or induction motor and direct-current generator, the converter requires step-down transformers, the synchronous motor, if the alternating line voltage is considerably above 10,000 volts, generally requires step-down transformers also, with voltages of 1000 to 10,000 volts, however, usually the synchronous motor and frequently the induction motor can be wound directly for the line voltage and stationary transformers saved. Thus on the one side we have two machines with or generally without stationary transformers, on the other side a single machine with transformers. 270 SYNCHRONOUS CONVERTERS. 271 Regarding the reliability of operation and first cost, obviously a single machine is preferable. Regarding efficiency, it is sufficient to compare the converter with the synchronous-motor-direct-current-generator set, since the induction motor is inherently less efficient than the syn- chronous motor. The efficiency of stationary transformers of large size_varies from 97 per cent to 98 per cent, with an average of 97.5 per cent. That of converters or of synchronous motors varies between 91 per cent and 95 per cent, with 93 per cent as average, and that of the direct-current generator between 90 per cent and 9-4 per cent, with 92 per cent as average. Thus the converter with its step-down transformers will give an average efficiency of 90.7 per cent, a direct-current generator driven by synchronous motor with step-down transformers an efficiency of 83.4 per cent, without step-down transformers an efficiency of 85.6 per cent. Hence the converter is more efficient. Mechanically the converter has the advantage that no transfer of mechanical energy takes place, since the torque consumed by the generation of the direct current and the torque produced by the alternating current are applied at the same armature con- ductors, while in a direct-current generator driven by a syn- chronous motor the power has to be transmitted mechanically through the shaft. II. Ratio of e.m.fs. and of Currents. in. In its structure the synchronous converter consists of a closed-circuit armature, revolving in a direct-current excited field, and connected to a segmental commutator as well as to collector rings. Structurally it thus differs from a direct- current machine by the addition of the collector rings, from certain (now very little used) forms of synchronous machines by the addition of the segmental commutator. In consequence hereof, regarding types of armature windings and of field windings, etc., the same rule applies to the converter as to all commutating machines, except that in the converter the total number of armature coils with a series-wound armature, and the number of armature coils per pair of poles with a multiple-wound armature, must be divisible by the number of phases. Regarding the wave-shape of the alternating counter-gen- 272 ELEMENTS OF ELECTRICAL ENGINEERING. erated e.m.f., similar considerations apply as for a synchronous machine with closed-circuit armature: that is, the generated e.m.f. usually approximates a sine wave, due to the multi-tooth distributed winding. Thus, in the following, only those features will be discussed in which the synchronous converter differs from the commu- tating machines and synchronous machines treated in the pre- ceding chapters. Fig. 156 represents diagrammatically the commutator of a direct-current machine with the armature coils A connected to adjacent commutator bars. The brushes are BJS V and the field poles F,F r If now two oppositely located points a t a 2 of the commutator are connected with two collector rings D^D V it is obvious that the e.m.f. between these points a t a 27 and thus between the col- lector rings -D 1 D 2 , will be a maximum . in the moment when the points a t a 2 coincide with the brushes B~B^ and is in this moment equal to the direct voltage E of this ma- chine. While the points a t a 2 move away from this position, Fig. 156. Single-Phase Converter Com- the difference of potential be- mutator * tween a l and a 2 decreases and becomes zero in the moment where a^ 2 coincide with the direction of the field poles FJF y In this moment the differ- ence in potential between a t and a 2 reverses and then increases again, reaching equality with E, but in opposite direction, when a t and a 2 coincide with the brushes S 2 and B^ that is, between the collector rings D 1 and D 2 an alternating voltage is produced whose maximum value equals the direct-current electromotive force E, and which makes a complete period for every revolution of the machine (in a bipolar converter, or n p periods per revo- lution in a machine of 2 n p poles). Hence, this alternating e.m.f. is e = E sin 2 rft, where / frequency of rotation, E = e.m.f. between brushes SYNCHRONOUS CONVERTERS. 273 of the machine; thus, the effective value of the alternating e.m.f. is E E - = ^' 112. That is, a direct-current machine produces between two collector rings connected with two opposite points of the commutator an alternating e.m.f. of i. X the direct-current v2 voltage, at a frequency equal to the frequency of rotation. Since every alternating-current generator is reversible, such a direct-current machine with two collector rings, when supplied with an alternating e.m.f. of X the direct-current voltage at v2 the frequency of rotation, will run as synchronous motor, or if at the same time generating direct current, as synchronous converter. Since, neglecting losses and phase displacement, the output of the direct-current side must be equal to the input of the alter- nating-current side, and the alternating voltage in the single-phase converter is = X Ej the alternating current must be \/2 = \/2 X /, where / = direct-current output. If now the commutator is connected to a further pair of collector rings, D 3 D 4 (Fig. 157), at the points a 3 and a 4 midway be- tween a l and a 3 , it is obvious that between D 3 and D 4 an alternating voltage of the same frequency and intensity will be produced as f;" 157 * *i r ~ ae /* J f . * Converter Commu- between L> 1 and L> 2 , but m quadrature there- tator. with, since at the moment where a 3 and a 4 coincide with the brushes BJS^ and thus receive the maxi- mum difference of potential, a 1 and a 2 are at zero points of potential. Thus connecting four equidistant points a v a 2 , a 3 , a 4 of the direct-current generator to four collector rings D v D 2 , D 37 Z> 4 , gives a four-phase converter, of the e.m.f. E l <*= E per phase, v 2t 274 ELEMENTS OF ELECTRICAL ENGINEERING. The current per phase is (neglecting losses and phase displace- ment) 7,-i, 1 V2 since the alternating power, 2 EJ V must equal the direct-current power, El. Connecting three equidistant points of the commutator to three collector rings as in Fig. 158 gives a three-phase converter. 113. In Fig. 159 the three e.m.fs. between the three collector rings and the neutral point of the three-phase system (or Y voltages) are represented by the vectors OE V OE 2 , OE Z , thus Fig. 158. Three-Phase Syn- chronous Converter. Fig. 159. E.m.f. Diagram of Three-Phase Con- verter. the e.m.f. between the collector rings or the delta voltages by vectors EJE 2 , EJB& and E 3 E r The e.m.f. OE l is, however, nothing but half the e.m.f. E 1 in Fig. 156, of the single-phase converter, that is, = . 2V 2 Hence the Y voltage, or voltage between collector ring and neutral point or center of the three- phase voltage triangle, is 7? 77f JL7 , ^ and thus the delta voltage is W ~E, 2V2 0.612 E. Since the total three-phase power 3 I l E l equals the total continuous-current power IE, it is SYNCHRONOUS CONVERTERS. 275 In general, in an n-phase converter, or converter in which n equidistant points of the commutator (in a bipolar machine, or n equidistant points per pair of poles in a multipolar machine with multiple- wound armature), are connected to n collector rings, the voltage between any collector ring and the common neutral, or star voltage, is consequently the voltage between two adjacent collector rings, or ring voltage, is sin-- n "SbSf g "o ^3 C 2 X Q S/! HOT H fa 00 H 8 Volts between col~ 1 1 1 1 1 1 1 lector ring and neutral point . . 2\/2 -0.354 2N/2 -0.354 2V2 -0 354 2^2 = 354 2V2 = 354 = 0354 2V2 = 0354 1 v V3 1 TT Volts between ad- jacent collector rin"s 1 vl = 707 2\/2 612 2V2 = 0.612 4-05 -0.354 183 sin - n Amperes per line . V2 V| 2N/2 1 V2 V2 2V2 3 3 71 1.0 = 1.414 = 817 =0 943 =0.707 = 0.472 0236 V2 2 V2 v f\/2 Amperes between adjacent lines . V2 = 1 414 = 0.817 3 V| = 0.545 - _ = 0,472 455 -"- sin - I n These currents give only the power component of alternating current corresponding to the direct-current output. Added thereto is the current required to supply the losses in the machine, that is, to rotate it, and the wattless component if a phase dis- placement is produced in the converter. III. Variation of the Ratio of Electromotive Forces. 115. The preceding ratios of e.m.fs. apply strictly only to the generated e.m.fs. and that under the assumption of a sine wave of alternating generated e.m.f. The latter is usually a sufficiently close approximation, since the armature of the converter is a multitooth structure, that is, contains a distributed winding. 278 ELEMENTS OF ELECTRICAL ENGINEERING. The ratio between the difference of potential at the commu- tator brushes and that at the collector rings of the converter usually differs somewhat from the theoretical ratio, due to the e.m.f. consumed in the converter armature, and in machines converting from alternating to continuous current, also due to the shape of the impressed wave. When converting from alternating to direct current, under load the difference of potential at the commutator brushes is less than the generated direct e.m.f. ; and the counter-generated alternating e.m.f. less than the impressed, due to the voltage consumed by the armature resistance. If the current in the converter is in phase with the impressed e.m.f., armature self-inductance has. little effect, but reduces the counter-generated alternating e.m.f. below the impressed with a lagging and raises it with a leading current, in the same way as in a synchronous motor. Thus in general the ratio of voltages varies somewhat with the load and with the phase relation, and with constant im- pressed alternating e.m.f. the difference of potential at the com- mutator brushes decreases with increasing load, decreases with decreasing excitation (lag), and increases with increasing excita- tion (lead). When converting from direct to alternating current the reverse is the case. The direct-current voltage stands in definite proportion only to the maximum value of the alternating voltage (being equal to twice the maximum star voltage), but to the effective value (or value read by voltmeter) only in so far as the latter depends upon the former, being = = maximum value with a sine wave. v2 Thus with an impressed wave of e.m.f. giving a different ratio of maximum to effective value, the ratio between direct and alternating voltage is changed in the same proportion as the ratio of maximum to effective; thus, for instance, with a flat-topped wave of impressed e.m.f., the maximum value of alternating impressed e.m.f., and thus the direct voltage depending there- upon, are lower than with a sine wave of the same effective value, while with a peaked wave of impressed e.m.f. they are higher, by as much as 10 per cent in extreme cases, In determining the wave shape of impressed e.m.f. at the con- SYNCHRONOUS CONVERTERS. 279 verter terminals, not only the wave of generated generator e.m.f., but also that of the converter counter-generated e.m.f., may be instrumental. Thus, with a converter connected directly to a generating system of very large capacity, the im- pressed e.m.f. wave will be practically identical with the gener- ator wave, while at the terminals of a converter connected to the generator over long lines with reactive coils or inductive regu- lators interposed, the wave of impressed e.m.f. may be so far modified by that of the counter e.m.f. of the converter as to resemble the latter much more than the generator wave, and thereby the ratio of conversion may be quite different from that corresponding to the generator wave. Furthermore, for instance, in three-phase converters fed by ring or delta connected transformers, the star e.m.f. at the con- verter terminals, which determines the direct voltage, may differ from the star e.m.f. impressed by the generator, by con- taining different third and ninth harmonics, which cancel when compounding the star voltages to the delta voltage, and give identical delta voltages, as required. Hence, the ratios of e.m.f s. given in Section II have to be corrected by the drop of voltage in the armature, and have to be multiplied by a factor which is \/2 times the ratio of effective to maximum value of impressed wave of star e.m.f. (V2 being the ratio of maximum to effective of the sine wave on which the ratios in Section II were based), that is, by the " form factor 33 of the e.m.f. wave. With an impressed wave differing from sine shape, there is a current of higher frequency, but generally of negligible mag- nitude, through the converter armatufe, due to the difference between impressed and counter e.m.f. wave. IV. Armature Current and Heating. 116. The current in the armature conductors of a converter is the difference between the alternating-current input and the direct-current output. In Fig. 161, a v a 2 are two adjacent leads connected with the collector rings D v D 2 in an n~phase converter. The alternating e.m.f. between a l and a 2 , and thus the power component of the alternating current in the armature section between a t and a 2 , 280 ELEMENTS OF ELECTRICAL ENGINEERING. will reach a maximum when this section is midway between the brushes B t and B 2 , as shown in Fig. 161. The direct current in every armature coil reverses at the moment when the coil passes under brush B l or B 2J and is thus a rectangu- lar alternating current as shown in Fig. 162 as /. At the moment when the power component of the alter- nating current is a maximum, an armature coil d midway between two adjacent alternating leads a and a 2 is midway between the brushes B 1 and B 2 , as in Fig. 161, and is thus in the middle of its rec- Armature Heating in Synchro' ^^ ^tinuous-CUIlBDt wave, nous Converters. and consequently in this coil the power component of the alternat- ing current and the rectangular direct current are in phase with each other, but opposite, as shown in Fig. 162 as 7 X and 7, and the actual current is their difference, as shown in Fig. 163. Fig. 162. Direct Current and Alternating Current in Armature Coil d, Fig. 161. Fig. 163. Resultant Current in Coil d, Fig. 161. In successive armature coils the direct current reverses suc- cessively; that is, the rectangular currents in successive arma- ture coils are successively displaced in phase from each other; and since the alternating current is the same in the whole sec- tion a 1 a 2 , and in phase with the rectangular current in the coil d, SYNCHRONOUS CONVERTERS. 281 it becomes more and more out of phase with the rectangular current when passing from coil d towards a l or a 2 , as shown in Figs. 164 to 167, until the maximum phase displacement be- tween alternating and rectangular current is reached at the alternating leads a l and a 2 , and is equal to - - 117. Thus, if E = direct voltage, and / = direct current, in an armature coil displaced by angle T from the position d, mid- way between two adjacent leads of the w-phase converter, the direct current is for the half period from to x, and the alter- nating current is where \/2 F sin (0 - r), T , /V2 . it n sin - n is the effective value of the alternating current. Thus, the actual current in this armature coil is ~1 4 sin (0 - r) . 7T fi sin - In a double-current generator, instead of the minus sign, a plus sign would connect the alternating and the direct current in the parenthesis The effective value of the resultant converter current thus is : W*J[V de V 1 r A sin (0r) 1 Ydft dd 2V,J . TC nsm~ n 1 I 8 , 16 cos T "2 / - - ,w - x ' IL 1 n sin - nn sin - V n n Since - is the current in the armature coil of a direct-current i 282 ELEMENTS OF ELECTRICAL ENGINEERING. Fig. 164. Alternating Current and Direct Current in Coil between d and a t or a 2) Fig. 161. Fig, 165. Resultant of Currents Given in Fig. 164. Fig. 166. Alternating Current and Djrect Current in Coil between d and c^ or a 2 , Fig. 161. Fig. 167. Resultant of Currents Shown in 166. SYNCHRONOUS CONVERTERS. 283 generator of the same output, we have 7 A 16 cos r 1 1 .> . 71 n 2 sm 2 - n nn sin - n the ratio of the power loss in the armature coil resistance of the converter to that of the direct-current generator of the same output, and thus the ratio of coil heating. This ratio is a maximum at the position of the alternating leads, r = - , and is n 16 cos - 8 | i __ 2 n 2 sin 2 - nn sin - n n It is a minimum for a coil midway between adjacent alter- nating leads, T = 0, and is = 8 ^ 16 ~ . ~ 7T . 7T n sm - n?r sin - n n Integrating over r from (coil d) to - , that is, over the whole phase or section a x a 2 , we have IF r = - r^-di- 8 ii 16 , 7t n the ratio of the total power loss in the armature resistance of an n-phase converter to that of the same machine as direct- current generator at the same output, or the relative armature heating. Thus, to get the same loss in the armature conductors, and consequently the same heating of the armature, the current in the converter, and thus its output, can be increased in the pro- portion =. over that of the direct-current generator. vT The calculation for the two-circuit single-phase converter is somewhat different; since in this in one-third of the armature 284 ELEMENTS OF ELECTRICAL ENGINEERING. the Pr loss is that of the direct-current output, and only in the 2 7T other two-thirds or an arc -- is there alternating current. o Thus in an armature coil displaced by angle r from the center of this latter section the resultant current is to- V2/'sin(0 -r)- giving the effective value thus, the relative heating is /2 11 16 2 with the minimum value at r = 0, it is r " J6 a? 3 and with the maximum value at T = , it is o y. ' II ^ _ O 10. r -~ ~"" ' the average current heating in two-thirds of the armature is - sm 3 x?\/3 3 in the remaining third of the armature, T 2 = 1 7 thus the average is _ 3 = 1.151, and therefore the rating is -^ = 0.93. vT SYNCHRONOUS CONVERTERS. 285 By substituting for n in the general equations of current heat- ing and rating based thereon, numerical values, we get the following table: Type. *i Z fc- si -circuit le-phase rcuit le-phase. i 1 * | H but - + at the one, n n ^ 6 at the other lead, and thus at the other side of the same n lead. The resultant current is thus increased at the one, de- creased at the other lead, and the heating changed accordingly. 286 ELEMENTS OF ELECTRICAL ENGINEERING. For instance, in a quarter-phase converter at zero phase dis- placement, the resultant current at the lead would be as shown in Fig. 168, - = 45 deg., while at 30 deg. lag the resultant currents i\t in the two coils adjacent to the commutator lead are displaced respectively by h = 75 deg. and by - = 15 deg., and n n so of very different shape, as shown by Figs. 169 and 170, giv- ing very different local heating. Phase displacement thus in- creases the heating at the one, decreases it at the other side of each commutator lead. Let again, / == direct current per commutator brush. The effective value of the alternating power current in the armature winding, or ring current, corresponding thereto, is r, /V2 . 7T n sin - n Let pi' total power current, allowing for the losses of power in the converter; ql' = reactive current in the converter, assumed as positive when lagging, as negative when leading, and si' = total current, where s = Vp 2 + 7Z the converter armature. r 2n C + " * r = ~ I Trdr 7T */ TT O 7T . 7T 2 - 7T 2 sin - n n SYNCHRONOUS CONVERTERS. 289 1 + 8 s 2 16 s cos 6 = 1 + 8 (p 2 + g 2 ) 16 p n' sm' - n 119. This gives for Three-phase, n = 3: Tr = 1 + 1.185 s 2 - 1.955 s cos (r - 0), r* = 1 + 1.185 s 2 - 1.955 s cos (60 0), r = 1 + 1.185s 2 - 1.620 p. Quarter-phase, n = 4: TT = 1 + S 2 _ } 795 s cog ( r _ 0^ Tm = 1 + s 2 - 1.795 s cos (45 6), r = 1 + s 2 - 1.620 p. Six-phase, n = 6: TV ~ 1 + 0.889 s 2 - 1.695 5 cos (r - 6), Tm = i + 0.889 s 2 - 1.695 s cos (30+ 0), P - 1 + 0.889 s 2 - 1.62 p, oc -phase, n == co : TT = rn = r = 1 + 0.810 s 2 - 1.62 s cos 6 = 1 + 0.810 s 2 - 1.62 p. Choosing p = 1.04, that is, assuming 4 per cent loss in friction and windage, core loss and field excitation, the tV loss of the armature is not included in p, as it is represented by a drop of direct-current voltage below that corresponding to the alternat- ing voltage, and not by an increase of the alternating current over that corresponding to the direct current we get, for dif- ferent phase angles from given below: deg. to 6 =* 60 deg., the values e - 10 20 30 40 50 60 9 s /. === 1.04 1.056 1.108 1.20 1.36 1.62 2.08 cos a = s sin react, cur. 1S4 0379 o f\n n &?fi 1 24 i sn power cur. " u.uu \J,O t \J J..<~C X .OvJ tan 6 0.176 0.364 0.577 0.839 1.192 1.732 Three-phase: 7m = 1 1.62 2,08 2.70 3.65 519 8.16 1.26 7m 7 - J 1.00 0.80 068 0.70 0.99 2.06 0.60 0.64 0.77 1.02 1.51 2.43 4.45 10 20 30 40 50 60 1.02 1.39 1.88 2.64 3.87 6.30 0.55 0.43 0.38 0,42 0.73 1.71 0.43 0.54 0.75 1.16 1.94 3.64 0.62 0.88 1.27 L86 2.85 4.85 0.31 0.24 0.25 0.38 0.75 1.79 290 ELEMENTS OF ELECTRICAL ENGINEERING. Quarter-phase: 7m =) I 076 7m' = J T 0.40 Six-phase: 7m =} \ 0.44 7m' = j I - 0.28 0.31 0.41 0.60 0.97 1.65 3.17 co -phase: 7w = 7m '=r=0.20 0.22 0.32 0.49 0.82 1.45 2.82 120. The values are shown graphically in Figs. 171 and 172, 4,1, * a reactive current , . , ,. , with tan 6 = --- - as abscissas, and rm as ordmates energy current in Fig. 171, T as ordinates in Fig. 172. As seen, with increasing phase displacement, irrespectively whether lag or lead, the average as well as the maximum arma- ture heating very greatly increases. This shows the necessity of keeping the power-factor near unity at full load and overload, and when applied to phase control of the voltage by converter, means that the shunt field of the converter should be adjusted so as to give a considerable lagging current at no load, so that the current comes into phase with the voltage at about full load. It therefore is very objectionable in this case to adjust the con- verter for minimum current at no load, as occasionally done by ignorant engineers, since such wrong adjustment would give con- siderable leading current at load, and therewith unnecessary armature heating. It must be considered, however, that above values are referred to the direct-current output, and with increase of phase angle the alternating-current input, at the same output, increases, and the heating increases with the square of the current. Thus at 60 deg. lag or lead, the power-factor is 0.5, and the alternating- current input thus twice as great as at unity power-factor, corre- sponding to four times the heating. It is interesting therefore to refer the armature heating to the alternating-current input, that is, compare the heating of the converter with that of a synchronous motor of the same alternating-current input. This is given by r / / J x. / / / :ET 3^0 y / / r / 310 / / y 320 / / / 300 / / / 2SO y / / 260 & X # f / 240,/ ^ f & ^ / / 4, / # / / 200 / / o-j / < // / lao . / / / f / / iA / f / / ^ ^y / 140 / V / / 4 y 120 V 1 / / / Di REC' r cu RRET T /1 QE ^ER> TOR HEA riNG 100 X X / ' X f 80 LX X 60 ^ " x ^ X 40 - *** "**^ IEAC TIVE cu ^REN T PF RCF ST 20 1 5 2 o a 4 , i 5FU 6 LL IJQAD [) 7,0 8 POW'EH duRR 9,0 lOO 11 ENT V i: 1-i 1! Fig. 171. Maximum Pr Heating in Converter Armature Coil Expressed in Per Cent of Direct-Current Generator Pr Heating. INQ^ 1,0 2.0 3.0 40 CURRENT PERCE *L . LOAD POWER CU.RRE ' 7.0 Sp 30 1QQ 1 PERCENT 60 / / 200 140 1$Q /&4 Fig. 172. Average Pr Heating in Converter Armature Expressed in Per Cent of Direct-Current Generator Pr Heating. (291) 292 ELEMENTS OF ELECTRICAL ENGINEERING. and, for p = 1.04, gives the following values: 0= tan 0= 0. 10 176 20 0.364 30 .577 40 0.839 50 1.192 60 14.32 Three-phase: I\ - 0.555 0. 57 .63 .71 0. 82 0. 93 1.03 Quarter-phase: r, - 0.37 0. 385 .44 .52 0. 63 0. 74 0.84 Six-phase: r t = 0.26 0. 28 .335 .42 0. 52 0. 63 0.73 oo -phase : r, = 0.185 0. 197 .26 .34 0. 44 0. 55 0.65 It is seen that, compared with the total alternating-current input, the armature heating increases much less with increasing phase displacement, and is almost always much lower than the heating of the same machine at the same input and phase angle, when running a synchronous motor, as shown in Fig. 173. iYNCHfi >IOUS MOTOR HEATING 30 40 E CURRENT PER RRENT PER-! CENT 1 100 140 1$OI6()170 Fig. 173. Average Pr Heating in Converter Armature Expressed in Per Cent of Synchronous Motor lh Heating at the Same Power- Factor V. Armature Reaction. 121. The armature reaction of the polyphase converter is the resultant of the armature reactions of the machine as direct- current generator and as synchronous motor. If the com- mutator brushes are set at right angles to the field poles or without lead or lag, as is usually done in converters, the direct- current armature reaction consists in a polarization in quadra- ture behind the field magnetism. The armature reaction due to the power component of the alternating current in a synchro- SYNCHRONOUS CONVERTERS. 293 nous motor consists of a polarization in quadrature ahead of the field magnetism, which is opposite to the armature reaction as direct-current generator. Let m = total number of turns on the bipolar armature or per pair of poles of an n-phase converter, / = direct current, then the ~" tji number of turns in series between the brushes = , hence the M wil total armature ampere-turns, or polarization, = . Since, how- Zi ever, these ampere-turns are not unidirectional, but distributed over the whole surface of the armature, their resultant is = avg. cos 2 and, since avg. cos we have SF == direct-current polarization of the converter 7T (or direct-current generator) armature. In an n-phase converter the number of turns per phase = n The current per phase, or current between two adjacent leads (ring current), is . 7T n sin- n hence, the ampere-turns per phase, ml' _ \/2mI n ~~ 2 . x n 1 sin - n These ampere-turns are distributed over - of the circumference n of the armature, and their resultant is thus cr a cos n and, since * ( + n n . it avg. cos < = - sin - , ( _i * n n 294 ELEMENTS OF ELECTRICAL ENGINEERING, we have 5: _. - resultant polarization. 1 xn, * > in effective ampere-turns of one phase of the converter. The resultant m.m.f. of n equal m.m.fs. of effective value of SFp thus maximum value of g^x/iJ, acting under equal angles 2 r 1 v , and displaced in phase from each other by - of a period, or n n r ' 2 x phase angle , is found thus : n Let 5^= SFjX/^ sinf ^ ^-j = one of the m.m.fs. of phase 2 { 7T . 2 i/T angle 5 = - , acting in the direction T = - - ; that is, the 71 71 zero point of one of the m.m.fs. 5^ is taken as zero point of time (9, and the direction of this m.m.f. as zero point of direc- tion r. The resultant m.m.f. in any direction r is thus g: cos(r+ \ n 2 T I n ^ + r -- ~)+ n sin (0 r) n l and, since we have /7) sm (fl ^j that is, the resultant m.m.f. in any direction r has the phase = r, and the intensity ? SYNCHRONOUS CONVERTERS. 295 thus revolves in space with uniform velocity and constant in- tensity, in synchronism with the frequency of the alternating current. Since in the converter, ^ \ /7 2mI jr _ , 1 7T71 we have the resultant m.m.f. of the power component of the alternating current in the n-phase converter. This m.m.f. revolves synchronously in the armature of the converter; and since the armature rotates at synchronism, the resultant m.m.f. stands still in space, or, with regard to the field poles, in opposition to the direct-current polarization. Since it is equal thereto, it follows that the resultant armature reac- tions of the direct current and of the corresponding power component of the alternating current In the synchronous con- verter are equal and opposite, thus neutralize each other, and the resultant armature polarization equals zero. The same is obviously the case in an inverted converter, that is, a machine changing from direct to alternating current. 122. The conditions in a single-phase converter are different, however. At the moment when the alternating current = 0, the full direct-current reaction exists. At the moment when the alternating current is a maximum, the reaction is the differ- ence between that of the alternating and of the direct current; and since the maximum alternating current in the single-phase converter equals twice the direct current, at this moment the resultant armature reaction is equal but opposite to the direct- current reaction. Hence, the armature reaction oscillates with twice the fre- quency of the alternating current, and with full intensity, and since it is in quadrature with the field excitation, tends to shift the magnetic flux rapidly across the field poles, and thereby tends to cause sparking and power losses. This oscillating reaction is, however, reduced by the damping effect of the magnetic field structure. It is somewhat less in the two-circuit single-phase converter. 296 ELEMENTS OF ELECTRICAL ENGINEERING. Since in consequence hereof the commutation of the single- phase converter is not as good as that of the polyphase con- verter, in the former usually voltage commutation has to be resorted to; that is, a commutating pole used, or the brushes shifted from the position mid -way between the field poles; and in the latter case the continuous-current ampere-turns inclosed by twice the angle of lead of the brushes act as a demagnetizing armature reaction, and require a corresponding increase of the field excitation under loacL 123. Since the resultant main armature reactions neutralize each other in the polyphase converter, there remain only 1. The armature reaction due to the small power component of current required to rotate the machine, that is, to cover the internal losses of power, which is in quadrature with the field excitation or distorting, but of negligible magnitude. 2. The armature reaction due to the wattless component of alternating current where such exists. 3. An effect of oscillating nature, which may be called a higher harmonic of armature reaction. The direct current, as rectangular alternating current in the armature, changes in phase from coil to coil, while the alternating current is the same in a whole section of the armature between adjacent leads. Thus while the resultant reactions neutralize, a local effect remains which in its relation to the magnetic field oscillates with a period equal to the time of motion of the armature through the angle between adjacent alternating leads; that is, double frequency in a single-phase converter (in which it is equal in magnitude to the direct-current reaction, and is the oscillating armature reaction discussed above), sextuple frequency in a three-phase converter, and quadruple frequency in a four- phase converter. The amplitude of this oscillation in a polyphase converter is small, and its influence upon the magnetic field is usually neg- ligible, due to the damping effect of the field spools, which act like a short-circuited winding for an oscillation of magnetism. A polyphase converter on unbalanced circuit can be con- sidered as a combination of a balanced polyphase and a single- phase converter; and since even single-phase converters operate quite satisfactorily, the effect of unbalanced circuits on the SYNCHRONOUS CONVERTERS. 297 polyphase converter is comparatively small, within reasonable limits. Since the armature reaction of the direct current and of the alternating current in the converter neutralize each other, no change of field excitation is required in the converter with changes of load. Furthermore, while in a direct-current generator the arma- ture reaction at given field strength is limited by the distortion of the field caused thereby, this limitation does not exist in a converter; and a much greater armature reaction can be safely used in converters than in direct-current generators, the dis- tortion being absent in the former. The practical limit of overload capacity of a converter is usu- ally far higher than in a direct-current generator, since the arma- ture heating is relatively small, and since the distortion of field, which causes sparking on the commutator under overloads in a direct-current generator, is absent in a converter. The theoretical limit of overload that is, the overload at which the converter as synchronous motor drops out of step and comes to a standstill is usually far beyond reach at steady frequency and constant impressed alternating voltage, while on an alternating circuit of pulsating frequency or droop- ing voltage it obviously depends upon the amplitude and period of the pulsation of frequency or on the drop of voltage. VI. Reactive Currents and Compounding. 124. Since the polarization due to the power component of the alternating current as synchronous motor is in quadrature ahead of the field magnetization, the polarization or magnetizing effect of the lagging component of alternating current is in phase, that of the leading component of alternating current in opposition to the field magnetization; that is, in the converter no magnetic distortion exists, and no armature reaction at all if the current is in phase with the impressed e.m.f., while the armature reaction is demagnetizing with a leading and mag- netizing with a lagging current. Thus if the alternating current is lagging, the field excitation at the same impressed e.m.f. has to be lower, and if the alter- nating current is leading, the field excitation has to be higher, than required with the alternating current in phase with the 298 ELEMENTS OF ELECTRICAL ENGINEERING. e.m.f. Inversely, by raising the field excitation a leading current, or by lowering it a lagging current, can be produced in a converter (and in a synchronous motor). Since the alternating current can be made magnetizing or demagnetizing according to the field excitation, at constant impressed alternating voltage, the field excitation of the con- verter can be varied through a wide range without noticeably affecting the voltage at the commutator brushes; and in con- verters of high armature reaction and relatively weak field, full load and overload can be carried by the machine without any field excitation whatever, that is, by exciting the field by armature reaction by the lagging alternating current. Such converters without field excitation, or reaction converters, must always run with more or less lagging current, that is, give the same reaction on the line as induction motors, which, as known, are far more objectionable than synchronous motors in their reaction on the alternating system, and therefore they are no longer used. Conversely, however, at constant impressed alternating volt- age the direct-current voltage of a converter cannot be varied by varying the field excitation (except by the very small amount due to the change of the ratio of conversion), but a change of field excitation merely produces wattless currents, lagging or magnetizing with a decrease, leading or demagnetizing with an increase of field excitation. Thus to vary the continuous- current voltage of a converter usually the impressed alternating voltage has to be varied. This can be done either by potential regulator or compensator, that is, transformers of variable ratio of transformation, or by the effect of wattless currents on self- inductance. The latter method is especially suited for con- verters, due to their ability of producing wattless currents by change of field excitation. The e.m.f. of self-inductance lags 90 deg. behind the current; thus, if the current is lagging 90 deg. behind the impressed e.m.f., the e.m.f. of self-inductance is 180 deg. behind, or in opposition to, the impressed e.m.f., and thus reduces it. If the current is 90 deg. ahead of the e.m.f., the e.m.f. of self-inductance is in phase with the impressed e.m.f., thus adds itself thereto and raises it. Therefore, if self-inductance is inserted into the lines between converter and constant-potential generator, and a watt- less lagging current is produced by the converter by a decrease SYNCHRONOUS CONVERTERS. 299 of its field excitation, the e.m.f. of self-inductance of this lagging current in the line lowers the alternating impressed voltage at the converter and thus its direct-current voltage; and if a watt- less leading current is produced by the converter by an increase of its field excitation, the e.m.f. of self-inductance of this leading current raises the impressed alternating voltage at the converter and thus its direct-current voltage. 125. In this manner, by self-inductance in the lines leading to the converter, its voltage can be varied by a change of field excitation, or conversely its voltage maintained constant at constant generator voltage or even constant generator excita- tion, with increasing load and thus increasing resistance drop in the line; or the voltage can even be increased with increasing load, that is, the system over-compounded. The change of field excitation of the converter with changes of load can be made automatic by the combination of shunt and series field, and in this manner a converter can be compounded or even over-compounded similarly to a direct-current generator. While the effect is the same, the action, however, is different; and the compounding takes place not in the machine as with a direct-current generator, but in the alternating lines leading to the machine, in which self-inductance becomes essential. As the reactance of the transmission line is rarely sufficient to give phase control over a wide range without excessive reac- tive currents, it is customary, especially at 25 cycles, to insert reactive coils into the leads between the converter and its step- down transformers, in those cases in which automatic phase control by converter series fields is desired, as in power trans- mission for suburban and interurban railways, etc. Usually these reactive coils are designed to give at full-load current a reactance voltage equal to about 15 per cent of the converter supply voltage, and therefore capable of taking care of about 10 per cent line drop at good power factors. VII. Variable Ratio Converters (" Split Pole ") Converters. 126. With a sine wave of alternating voltage, and the com mutator brushes set at the magnetic neutral, that is, at right angles to the resultant' magnetic flux, the direct voltage of a converter is constant at constant impressed alternating voltage. It equals the maximum value of the alternating voltage between 300 ELEMENTS OF ELECTRICAL ENGINEERING. two diametrically opposite points of the commutator, or "dia- metrical voltage, " and the diametrical voltage is twice the voltage between alternating lead and neutral, or star or F voltage of the polyphase system. A change of the direct voltage, at constant impressed alter- nating voltage, can be produced Either by changing the position angle between the commu- tator brushes and the resultant magnetic flux, so that the direct voltage between the brushes is not the maximum diametrical alternating voltage but only a part thereof, Or by changing the maximum diametrical alternating voltage, at constant effective impressed voltage, by wave-shape distortion by the superposition of higher harmonics. In the former case, only a reduction of the direct voltage below the normal value can be produced, while in the latter case an increase as well as a reduction can be produced, an increase if the higher harmonics are in phase, and a reduction if the higher harmonics are in opposition to the fundamental wave of the diametrical or Y voltage. A, VARIABLE RATIO BY A CHANGE OF THE POSITION ANGLE BETWEEN COMMUTATOR BRUSHES AND RESULTANT MAG- NETIC FLUX. 127. Let, in the commutating machine shown diametrically in Fig. 174, the potential difference, or alternating voltage between one point a of the armature winding and the neutral (that is, the Y voltage, or half the diametrical voltage), be represented by the sine wave, Fig, 176. This potential differ- ence is a maximum, e, when a stands at the magnetic neutral, at A or B. If, therefore, the brushes are located at the magnetic neutral, A and 5, the voltage between the brushes is the potential difference between A and 5, or twice the maximum Y voltage, 2 e, as indicated in Fig. 176. If now the brushes are shifted by an angle r to position C and D, Fig. 175, the direct voltage between the brushes is the potential difference between C and D, or 2 e cos r with a sine wave. Thus, by shifting the brushes from the position A, B, at right angles with the magnetic flux, to the position E, F, in line with the magnetic flux, any direct SYNCHRONOUS CONVERTERS. 301 voltage between 2 e and can be produced, with the same wave of alternating voltage a. As seen, this variation of direct voltage between its maximum value and zero, at constant impressed alternating voltage, is inde- pendent of the wave shape, and thus can be produced whether the alternating voltage is a sine wave or any other wave. Fig. 174. Diagram of Commutating Machine with Brushes in the Magnetic Neutral. s Fig. 175. E.in.f. Variation by Shifting the Brushes. Fig. 176. Sine Wave of e.m.f. It is obvious that, instead of shifting the brushes on the com- mutator, the magnetic field poles may be shifted, in the opposite direction, by the same angle, as shown in Fig. 177, A, B, C. Instead of mechanically shifting the field poles, they can be shifted electrically, by having each field pole consist of a num- ber of sections, and successively reversing the polarity of these sections, as shown in Fig. 178, A, B, C, D. 302 ELEMENTS OF ELECTRICAL ENGINEERING. Instead of having a large number of field pole sections, obviously two sections are sufficient, and the same gradual change can be brought about by not merely reversing the sec- tions but reducing the excitation down to zero and bringing it up again in opposite direction, as shown in Fig. 179, A, B, C, D, E. Fig. 177. E.m.f. Variation by Mechanically Shifting the Poles. A B C D' 'Fig. 178. E.m.f. Variation by Electrically Shifting the Poles. A Fig. 179. E.m.f . Variation by Shifting Flux Distribution* SYNCHRONOUS CONVERTERS. 308 In this case, when reducing one section in polarity, the other section must be increased by approximately the same amount, to maintain the same alternating voltage. When changing the direct voltage by mechanically shifting the brushes, as soon as the brushes come under the field pole faces, self-inductive sparking on the commutator would result if the iron of the field poles were not kept away from the brush position by having a slot in the field poles, as indicated in dotted line in Fig. 175 and Fig. 177, J5. With the arrangement in Figs. 175 and 177, this is not feasible mechanically, and these arrangements are, therefore, unsuitable. It is feasible, how- Fig. 180. Variable Ratio or Split-Pole Converter. ever, as shown in Figs. 178 and 179, that is, when shifting the resultant magnetic flux electrically, to leave a commutating space between the polar projections of the field at the brushes, as shown in Fig. 179, and thus secure as good commutation as in any other commutating machine. Such a variable-ratio converter, then, comprises an armature A, Fig. 180, with the brushes B, B' in fixed position and field poles Pj P' separated by interpolar spaces C, C r of such width as required for commutation. Each field pole consists of two parts, P and P v usually of different relative size, separated by a nar- row space, DD', and provided with independent windings. By 304 ELEMENTS OF ELECTRICAL ENGINEERING. varying, then, the relative excitation of the two polar sections P and P t an effective shift of the resultant field flux and a corresponding change of the direct voltage is produced. As this method of voltage variation does not depend upon the wave shape, by the design of the field pole faces and the pitch of the armature winding the alternating voltage wave can be made as near a sine wave as desired. Usually not much atten- tion is paid hereto, as experience shows that the usual distributed winding of the commutating machine gives a sufficiently close approach to sine shape. Armature Reaction and Commutation. 128. With the brushes in quadrature position to the resultant magnetic flux, and at normal voltage ratio, the direct-current generator armature reaction of the converter equals the syn- chronous motor armature reaction of the power component of the alternating current, and at unity power-factor the converter thus has no resultant armature reaction, while with a lagging or leading current it has the magnetizing or demagnetizing reaction of the wattless component of the current. If by a shift of the resultant flux from quadrature position with the brushes, by angle r, the direct voltage is reduced by factor cos r, the direct current and therewith the direct-current armature reaction are increased, by factor - , as by the law COS T of conservation of energy the direct-current output must equal the alternating-current input (neglecting losses). The direct- current armature reaction $ therefore ceases to be equal to the armature reaction of the alternating energy current JF , but is greater by factor & J COS T cosr The alternating-current armature reaction # , at no phase dis- placement, is in quadrature position with the magnetic flux. The direct-current armature reaction 9r, however, appears in the position of the brushes, or shifted against quadrature position by angle r; that is, the direct-current armature reaction is not in opposition to the alternating-current armature reaction, but SYNCHRONOUS CONVERTERS. 305 differs therefrom by angle r, and so can be resolved into two components, a component in opposition to the alternating-cur- rent armature reaction & w that is, in quadrature position with the resultant magnetic flux, F" = tf cos r = $ , that is, equal and opposite to the alternating-current armature reaction, and thus neutralizing the same; and a component in quadrature position with the alternating-current armature reac- tion 3^, or in phase with the resultant magnetic flux, that is, magnetizing or demagnetizing, &' = F sin r = 5^ tan T; that is, in the variable-ratio converter the alternating-current armature reaction at unity power-factor is neutralized by a component of the direct-current armature reaction, but a result- ant armature reaction & remains, in the direction of the result- ant magnetic field, that is, shifted by angle (90 r) against the position of brushes. This armature reaction is magnetizing or demagnetizing, depending on the direction of the shift of the field r. ' It can be resolved into two components, one at right angles with the brushes, ?/ == $' cos T = F sin r, and one, in line with the brushes, $2 = &' sin r = F sin 2 r = $F sin T tan T, as shown diagrammatically in Figs. 181 and 182. There exists thus a resultant armature reaction in the direc- tion of the brushes, and thus harmful for commutation, just as in the direct-current generator, except that this armature reac- tion in the direction of the brushes is only SF/ ^ sin 2 r, that is, sin 2 r of the value of that of a direct-current generator. The value of $ 2 ' can also be derived directly, as the difference between the direct-current armature reaction $ and the com- ponent of the alternating-current armature reaction, in the direction of the brushes, $ cos T, that is, cy 2 ' = $ _ = m.m.f. of field excitation), in the direction of the commutator brushes, and either in the direction of armature reaction, thus interfering with commutation, or in opposition to the armature reaction, thus improving commutation. If the magnetic flux is shifted in the direction of armature rotation, that is, that section of the field pole weakened towards which the armature moves, as in Fig. 181, the component 3y" of field excitation at the brushes is in the same direction as the armature reaction, 3 2 ', thus adds itself thereto and impairs the commutation, and such a converter is hardly operative. In this case the component of armature reaction, $', in the direction of the field flux is magnetizing. If the magnetic flux is shifted in opposite direction to the armature reaction, that is, that section of the field pole weakened which the armature conductor leaves, as in Fig. 182, the com- ponent Sy" of field excitation at the brushes is in opposite direction to the armature reaction S^', therefore reverses it, if sufficiently large, and gives a commutating or reversing flux 1 represents an over-normal, p < 1 a subnormal direct voltage. The direct current, and thereby the direct-current armature reaction, then is changed from the value which it would have at normal voltage ratio, by the factor - , as the P product of direct volts and amperes must be the same as at normal voltage ratio, being equal to the alternating power input minus losses. With unity power-factor, the direct-current armature reac- tion ff in a converter of normal voltage ratio is equal and opposite, and thus neutralized by the alternating-current armature reac- tion 9r , and at a change of voltage ratio from normal, by factor p, and thus change of direct current by factor - - The direct- current armature reaction thus is hence, leaves an uncompensated resultant. As the alternating-current armature reaction at unity power- 316 ELEMENTS OF ELECTRICAL ENGINEERING. factor is in quadrature with the magnetic flux, and the direct- current armature reaction in line with the brushes, and with this type of converter the brushes stand at the magnetic neutral, that is, at right angles to the magnetic flux, the two armature reactions are in the same direction in opposition with each other, and thus leave the resultant, in the direction of the commutator brushes, ^ 3F - The converter thus has an armature reaction proportional to the deviation of the voltage ratio from normal. 135. If p > 1, or over-normal direct voltage, the armature reaction is negative, or motor reaction, and the magnetic flux produced by it at the commutator brushes thus a commutating flux. If p < 1, or subnormal direct voltage, the armature reaction is positive, that is, the same as in a direct-current gen- erator, but less in intensity, and thus the magnetic flux of arma- ture reaction tends to impair commutation. In a direct-current generator, by shifting the brushes to the edge of the field poles, the field flux is used as reversing flux to give commutation. In this converter, however, decrease of direct voltage is produced by lowering the outside sections of the field poles, and the edge of the field may not have a sufficient flux density to give commuta- tion, with a considerable decrease of voltage below normal, and thus a separate commutating pole is required. Preferably this type of converter should be used only for raising the voltage, for lowering the voltage the other type, which operates by a shift of the resultant flux, and so gives a component of the main field flux as commutating flux, should be used, or a combination of both types. With a polar construction consisting of three sections, this can be done by having the middle section at low, the outside sections at high excitation for maximum voltage, and, to de- crease the voltage, raise the excitation of the center section, but instead of lowering both outside sections, leave the section in the direction of the armature rotation unchanged, while lowering the other outside section twice as much, and thus produce, in addition to the change of wave shape, a shift of the flux, as represented by the scheme Fig. 189. SYNCHRONOUS CONVERTERS. Magnetic Density. Pole section . . . 1 2 3 V Max. voltage ". . ^ -(B + (B -CB H - f (B + J(B + (B ~| (B H r J (B +f l f 7 (1 + t) cos r a 7 n \/2 (1 + t) cos r q . _ (5) nsm - n Due to the loss pi in the converter, this current is increased by (1 + p^ in a direct converter, or decreased by the factor (1 pi) in an inverted converter. 320 ELEMENTS OF ELECTRICAL ENGINEERING. The power component of the alternating current thus is 7, -/'(! + pi) ___ j V2 (1 + Q (1 + pi) cos r a nsin ' (6) n where pi may be considered as negative in an inverted converter. With the angle of lag V the reactive component of the current is 7 2 I t tan lf and the totaZ alternating ring current is 7 t cos 6 1 = /pV^q+O (l + P/)cosr a ^ (?) 7T n sin - cos 6. n or, introducing for simplicity the abbreviation , _(1 +Q (l+?)/)COST g ^ 7 > (o; it is 7 J~ A/0 (9) 138. Let, in Fig. 191, A'OA represent the center line of the magnetic field structure. The resultant magnetic field flux 0$ then leads OA by angle _ The resultant m.m.f. of the alternating power current I I is OT V at right angles to 0$, and the resultant m.m.f. of the alternating reactive current 7 2 is 0/^in opposition to 0 ? while the total alternating current 7 is 07, lagging by angle 6 l behind 07 r The m.m.f. of direct-current armature reaction is in the direc- tion of the brushes, thus lagging by angle n behind the position OB, where BOA = 90 deg. ; and given by 07 . The angle by which the direct-current m.m.f. 07~ lags in space behind the total alternating m.m.f. 07 thus is, by Fig. 191, r = 6 l - r a - r b . (10) SYNCHRONOUS CONVERTERS. 321 If the alternating m.m.f. in a converter coincides with the direct-current m.m.f., the alternating current and the direct cur- rent are in phase with each other in the armature coil midway , Jo Fig. 191, Diagram of Variable Ratio Converter. between adjacent collector rings, and the current heating thus a minimum in this coil. Due to the lag in space, by angle T O , of the direct-current m.m.f. behind the alternating current m.m.f., the reversal of the Fig. 192. Alternating and Direct Current in a Coil Midway between Adjacent Collector Leads. direct current is reached in time before the reversal of the alter- nating current in the armature coil; that is, the alternating current lags behind the direct current by angle 6 Q = T O in the armature coil midway between adjacent collector leads, as shown by Fig. 192, and in an armature coil displaced by angle r from the middle position between adjacent collector leads the alternating current thus lags behind the direct current by angle (T + ), where r is counted positive in the direction of armature rotation (Fig. 193). 322 ELEMENTS OF ELECTRICAL ENGINEERING. The alternating current in armature coil r thus can be ex- pressed by i== j^^ (d _ r _ ^. (11) hence, substituting (9), i= _2M sin( ^_ T _ 5)) (12) . r wsm- Fig. 193. Alternating and Direct Current in a Coil at the Angle r from the Middle Position. and as the direct current in this armature coil is -A and opposite to the alternating current i, the resultant current in the arma- ture coil r is r J-* sin (0 ~ r - ) - 1 nsin- n (13) and the ratio of heating, of the resultant current i compared with the current - of the same machine as direct-current gen- erator of the same output, thus is -U . (i)' sin - T - J -: ^ (14) Averaging (14) over one half-wave gives the relative heating of the armature coil r as 1 ! r" AL /7/j ! r" *~*f. X*"-* f. \2 . 7T nsm- 71 sin (0 - r - ) - 1 9. (15) SYNCHRONOUS CONVERTERS. 323 Integrated, this gives ft 2 sin 2 - xn sin - n n 139. Herefrom follows the local heating in any armature coil T, in the coils adjacent to the leads by substituting r = - n and also follows the average armature heating by averaging TV from T = -tor=+-- ft n The average armature heating of the ^-phase converter there- fore is 2 or, integrated, n p ir -, K cos . 2 2 7T 7T" n sm - n This is the same expression as found for the average armature heating of a converter of normal voltage ratio, when operating with an angle of lag O l of the alternating current, where k denotes the ratio of the total alternating current to the alternating power current corresponding to the direct-current output. In an n-phase variable ratio converter (split-pole converter), the average armature heating thus is given by n L (1 + Q (! + ?/) cos rq *" ^ ' (8) e o -^-Ta-T 6 ; (10) and t = ratio of third harmonic to fundamental alternating voltage wave; pi = ratio of loss to output; d t ** angle of lag of alternating current; r a = angle of shift of the resultant mag- netic field in opposition to the armature rotation, and T& = angle of shift of the brushes in the direction of the armature rotation. 324 ELEMENTS OF ELECTRICAL ENGINEERING. 140. For a three-phase converter, equation (18) gives (n = 3) r== ~2T + 1 ~ LG21 * G S (19) - 1.185 k 2 + 1 - 1.621 k cos . For a six-phase converter, equation (18) gives (n = 6) r--| + 1-1.621 teas*. (2Q) - 0.889 k 2 + 1 - 1.621 k cos . For a converter of normal voltage ratio, using no brush shift, T6-0; when neglecting the losses, it is COS and equations (19) and (20) assume the form: Three-phase: . - oc . r _ l^i _ Q.62L cos 2 19 1 Six-phase : n Q o Q r _JiL- 0.621. cos 2 5 t The equation (18) is the most general equation of the relative heating of the synchronous converter, including phase displace- ment 9 V losses pi, shift of brushes r&, shift of the resultant mag- netic flux r a , and the third harmonic t. While in a converter of standard or normal ratio the armature heating is a minimum for unity power-factor, this is not in gen- eral the case, but the heating may be considerably less at same lagging current, more at leading current, than at unity power- factor, and inversely. 141. It is interesting therefore to determine under which con- ditions of phase displacement the armature heating is a minimum so as to use these conditions as far as possible and avoid con- SYNCHRONOUS CONVERTERS. 325 ditions differing very greatly therefrom, as in the latter case the armature heating may become excessive. Substituting for k and 6 Q from equations (8) and (10) into equation (18) gives r i . i = i -j- n 2 sin 2 -cos 2 6, n 16 (1 + (1 + pi) cos r q cos f fl, T a 75) X,QV Substituting - sin - = m, (20) n n which is a constant of the converter type, and is for a three- phase converter, m 3 = 0.744; for a six-phase converter, m e = 0.955 and rearranging, gives -- (1+ i) (1+ pO cos r a cos (r a + T 6 ) 3T . 8 / g ~i ; ;; tan = 1-0.877- (31) COS r a f or Ta = o, or no flux shift, this gives / 00 _ A KAfl ) C 3 U.OUU, / ^oo 0.176 .$ Since CQS fa + rb "> < i f or brush shift in the direction of cos r a 328 ELEMENTS OF ELECTRICAL ENGINEERING. armature rotation, it follows that shifting the brushes increases the third harmonic required to carry out the voltage regulation without increase of converter heating, and thus is undesirable. It is seen that the third harmonic, t, does not change much with the flux shift r a , but remains approximately constant, and positive, that is, voltage raising. It follows herefrom that the most economical arrangement regarding converter heating is to use in the six-phase converter a third harmonic of about 17 per cent to 18 per cent for raising the voltage (that is, a very large pole arc), and then do the regula- tion by shifting the flux, by the angle r c , without greatly reduc- ing the third harmonic, that is, keep a wide pole arc excited. As in a three-phase converter the required third harmonic is impracticably high, it follows that for variable voltage ratio the six-phase converter is preferable, because its armature heating can be maintained nearer the theoretical minimum by propor- tioning t and r a . VIII. Starting. 143. The polyphase converter is self-starting from rest; that is, when connected across the polyphase circuit it starts, acceler- ates, and runs up to complete synchronism. The e.m.f. between the commutator brushes is alternating in starting, with the fre- quency of slip below synchronism. Thus a direct-current volt- meter or incandescent lamps connected across the commutator brushes indicate by their beats the approach of the converter to synchronism. When starting, the field circuit of the converter has to be opened or at least greatly weakened. The starting of the polyphase converter is largely a hysteresis effect and entirely so in machines with laminated field poles, while in ma- chines with SDlid magnet poles or with a short-circuited winding (squirrel-cage) in the field poles, secondary currents in the latter contribute to the starting torque, but at the same time reduce the magnetic starting flux by their demagnetizing effect. The torque is produced by the attraction between the alternating currents of the successive phases upon the remanent magnetism and secondary currents produced by the preceding phase. It is necessarily comparatively weak, and from full-load to twice full-load current at from one-third to one-half of full voltage is required to start from rest without load. For larger converters, SYNCHRONOUS CONVERTERS. 329 low- voltage taps on the transformers are used to give the lower starting voltage. While an induction motor can never reach exact synchronism, but must even at no-load slip slightly to produce the friction torque, the converter or synchronous motor reaches exact syn- chronism, due to the difference of the magnetic reluctance in the direction of the field poles and in the direction in electrical quadrature thereto; that is, the field structure acts like a shuttle armature and the polar projections catch with the rotating magnet poles in the armature, in a similar way as an induction motor armature with a single short-circuited coil (synchronous induction motor, reaction machine) drops into step. Obviously, the single-phase converter is not self-starting. At the moment of starting, the field circuit of the converter is in the position of a secondary to the armature circuit as primary ; and since in general the number of field turns is very much larger than the number of armature turns, excessive e.m.fs. may be generated in the field circuit, reaching frequently 4000 to 6000 volts, which have to be taken care of by some means, as by breaking the field circuit into sections. As soon as synchronism is reached, which usually takes from a few seconds to a minute or more, and is seen by the appearance of continuous voltage at the commutator brushes, the field circuit is closed and the load put on the converter. Obviously, while starting, the direct- current side of the converter must be open-circuited, since the e.m.f. between commutator brushes is alternating until syn- chronism is reached. When starting from the alternating side, the converter can drop into synchronism at either polarity; but its polarity can be reversed by strongly exciting the field in the right direction by some outside source, as another converter, etc., or by momen- tarily opening the circuit and thereby letting the converter slip one pole. Since when starting from the alternating side the converter requires a very large and, at the same time, lagging current, it is occasionally preferable to start it from the direct-current side as direct-current motor. This can be done when connected to storage battery or direct-current generator. When feeding into a direct-current system together with other converters or con- verter stations, all but the first converter can be started from 330 ELEMENTS OF ELECTRICAL ENGINEERING. the continuous current side by means of rheostats inserted into the armature circuit. To avoid the necessity of synchronizing the converter, by phase lamps, with the alternating system in case of starting by direct current (which operation may be difficult where the direct voltage fluctuates, owing to heavy fluctuations of load, as rail- way systems), it is frequently preferable to run the converter up to or beyond synchronism by direct current, then cut off from the direct current, open the field circuit and connect it to the alternating system, thus bringing it into step by alternating current. If starting from the alternating side is to be avoided, and direct current not always available, as when starting the first converter, a small induction motor (of less poles than the con- verter) is used as starting motor. IX. Inverted Converters. 145- Converters may be used to change either from alter- nating to direct current or as inverted converters from direct to alternating current. While the former use is by far the more frequent, sometimes inverted converters are desirable. Thus in low-tension direct-current systems an outlying district may be supplied by converting from direct to alternating, transmitting as alternating, and then reconverting to direct current. Or in a station containing direct-current generators for short-distance supply and alternators for long-distance supply, the converter may be used as the connecting link to shift the load from the direct to the alternating generators, or inversely, and thus be operated either way according to the distribution of load on the system. Or inverted operation may be used in emergencies to produce alternating current. When converting from alternating to direct current, the speed of the converter is rigidly fixed by the frequency, and cannot be varied by its field excitation, the variation of the latter merely changing the phase relation of the alternating current. When converting, however, from direct to alternating current as the only source of alternating current, that is, not running in multiple with engine- or turbine-driven alternating-current generators, the speed of the converter as direct-current motor depends upon the SYNCHRONOUS CONVERTERS. 331 field strength, thus it increases with decreasing and decreases with increasing field strength. As alternating-current generator, however, the field strength depends upon the intensity and phase relation of the alternating current, lagging current reducing the field strength and thus increasing speed and frequency, and leading current increasing the field strength and thus decreasing speed and frequency. Thus, if a load of lagging current is put on an inverted converter, as, for instance, by starting an induction motor or another con- verter thereby from the alternating side, the demagnetizing effect of the alternating current reduces the field strength and causes the converter to increase in speed and frequency. An increase of frequency, however, may increase the lag of the current, and thus its demagnetizing effect, and thereby still further increase the speed, so that the acceleration may become so rapid as to be beyond control by the field rheostat and endanger the machine. Hence inverted converters have to be carefully watched, especially when starting other converters from them; and some absolutely positive device is necessary to cut the inverted converter off the circuit entirely as soon as its speed exceeds the danger limit. The relatively safest arrangement is separate excitation of the in- verted converter by an exciter mechanically driven thereby, since an increase of speed increases the exciter voltage at a still higher rate, and thereby the excitation of the converter, and thus tends to check its speed. This danger of racing does not exist if the inverted converter operates in parallel with alternating generators, provided that the latter and their prime movers are of such size that they cannot be carried away in speed by the converter. In an in- verted converter running in parallel with alternators the speed is not changed by the field excitation, but a change of the latter merely changes the phase relation of the alternating current supplied by the converter; that is, the converter receives power from the direct-current system, and supplies power into the alter- nating-current system, but at the same time receives wattless current from the alternating system, lagging at under-excitation, leading at over-excitation, and can in the same way as an ordinary converter or synchronous motor be used to compensate for watt- less currents in other parts of the alternating system, or to regu- late the voltage by phase control. 332 ELEMENTS OF ELECTRICAL ENGINEERING. X. Frequency. 146. While converters can be designed for any frequency, the use of high frequency, as 60 cycles, imposes such severe limita- tions on the design, especially that of the commutator, as to make the high-frequency converter inferior to the low-frequency or 25-cycle converter. The commutator surface moves the distance from brush to next brush, or the commutator pitch, during one-half cycle, that is, sV second with a 25-cycle, T^T second with a 60-cycle converter. The peripheral speed of the commutator, however, is limited by mechanical, electrical, and thermal considera- tions, centrifugal forces, loss of power by brush friction,^ and heating caused thereby. The limitation of peripheral speed limits the commutator pitch. Within this pitch must be included as many commutator segments as necessary to take care of the voltage from brush to brush, and these segments must have a width sufficient for mechanical strength. With the smaller pitch required for high frequency, this may become impossible, and the limits of conservative design thus may have to be exceeded. In a converter, due to the absence of armature reaction and field distortion, a higher voltage per commutator segment can be allowed than in a direct-current generator. Assuming 15 volts as limit of conservative design would give for a 600-volt con- verter 40 segments from brush to brush. Allowing 0.25 inch for segment and insulation, as minimum conservative value, 40 segments give a pitch of 10 inches. Estimating 4200 feet per minute as conservative limit of commutator speed gives 70 feet or 840 inches peripheral speed per second, and with 10 inches pitch this gives 84 half cycles, or 42 cycles, as limit of the fre- quency, permitting conservative commutator design. At 60 cycles higher voltage per segment, narrower segments and higher commutator speeds thus becooe necessary than represent best design, and the 60-cycle converter does not permit as conservative commutator design, especially at higher voltage, as a low-frequency converter, and a lower self-inductance of commutation thus must be aimed at than permissible in a 25-cycle converter, the more so as the frequency of commutation (half the number of commutator segments per pole times fre- quency of rotation) necessarily is higher in the 60-cycle converter. SYNCHRONOUS CONVERTERS. 333 Thus shallower armature slots become necessary at the higher frequency. Somewhat similar considerations also apply to the armature construction: the peripheral speed of the armature, even if chosen higher for the 60-cycle converter, limits the pitch per pole at the armature circumference, and thereby the ampere conductors per pole and thus the armature reaction, the more so as shallower slots are necessary. The 60-cycle converter cannot be built with anything like the same armature reaction as is feasible at lower frequency. On the armature reaction, however, very largely depends the stability of a synchronous motor or converter, and machines of low armature reaction tend far more to surging and pulsation of current and voltage than machines of high armature reaction. The 60-cycle converter therefore cannot be made as stable and capable of taking care of violent fluctuations of load and of ex- cessive overloads as 25-cycle converters can, and in this respect must remain inferior to the lower-frequency machine, though under reasonably favorable conditions regarding variations of load, variations of supply voltage, and overload they can be built to give good service. It is this inherent inferiority of the 60-cycle converter which has largely been instrumental in introducing 25 cycles as the frequency of electric power generation and distribution. XL Double-Current Generators. 147. Similar in appearance to the converter, which changes from alternating to direct current, and to the inverted converter, which changes from direct to alternating current, is the double- current generator; that is, a machine driven by mechanical power and producing direct current as well as alternating current from the same armature, which is connected to commutator and col- lector rings in the same way as in the converter. Obviously the use of the double-current generator is limited to those sizes and speeds at which a good direct-current generator can be built with the same number of poles as a good alternator; that is, low-fre- quency machines of large output and relatively high speed ; while high-frequency low-speed double-current generators are unde- sirable. The essential difference between double-current generator and 334 ELEMENTS OF ELECTRICAL ENGINEERING. converter is, however, that in the former the direct current and the alternating current are not in opposition as in the latter, but in the same direction, and the resultant armature polarization thus the sum of the armature polarization of the direct current and of the alternating current. Since at the same output and the same field strength the arma- ture polarization of the direct current and that of the alternating current are the same, it follows that the resultant armature polari- zation of the double-current generator is proportional to the load regardless of the proportion in which this load is distributed between the alternating- and direct-current sides. The heating of the armature due to its resistance depends upon the sum of the two currents, that is, upon the total load on the machine. Hence, the output of the double-current generator is limited by the current heating of the armature and by the field distortion due to the armature reaction, in the same way as in a direct-current generator or alternator, and is consequently much less than that of a converter. In double-current generators, owing to the existence of arma- ture reaction and consequent field distortion, the commutator brushes are more or less shifted against the neutral, and the direction of the continuous-current armature polarization is thus shifted against the neutral by the same angle as the brushes. The direction of the alternating-current armature polarization, however, is shifted against the neutral by the angle of phase displacement of the alternating current. In consequence thereof, the reactions upon the field of the two parts of the armature polar- ization, that due to the continuous current and that due to the alternating current, are usually different. The reaction on the field of the direct-current load can be overcome by a series field. The reaction on the field of the alternating-current load when, feeding converters can be compensated for by a change of phase relation, by means of a series field on the converter, with self- inductance in the alternating lines, or reactive coils at the con- verters. Thus, a double-current generator feeding on the alternating side converters can be considered as a direct-current generator in which a part of the commutator, with a corresponding part of the series field, is separated from the generator and located at a distance, connected by alternating leads to the generator. Ob- SYNCHRONOUS CONVERTERS. 335 viously, automatic compounding of a double-current generator'is feasible only if the phase relation of the alternating current changes from lag at no load to lead at load, in the same way as produced by a compounded converter. Otherwise, rheostatic control of the generator is necessary. This is, for instance, the case if the voltage of the double-current generator has to be varied to suit the conditions of its direct-current load, and the voltage of the converter at the end of the alternating lines varied to suit the conditions of load at the receiving end, independent of the voltage at the double-current generator, by means of alternating potential regulators or compensators. Compared with the direct-current generator, the field of the double-current generator must be such as to give a much greater stability of voltage, owing to the strong demagnetizing effect which may be exerted by lagging currents on the alternating side, and may cause the machine to lose its excitation altogether. For this reason it is frequently preferable to excite double-current generators separately. XII. Conclusion. 148. Of the types of machines, converter, inverted converter, and double-current generator, sundry combinations can be devised with each other and with synchronous motors, alternators, direct- current motors and generators. Thus, for instance, a converter can be used to supply a certain amount of mechanical power as synchronous motor. In this case the alternating current is increased beyond the value corresponding to the direct current by the amount of current giving the mechanical power, and the armature reactions do not neutralize each other, but the reaction of the alternating current exceeds that of the direct current by the amount corresponding to the mechanical load. In the same way the current heating of the armature is increased. An inverted converter can also be used to supply some mechanical power. Either arrangement, however, while quite feasible, has the disadvantage of interfering with automatic control of voltage by compounding. Double-current generators can be used to supply more power into the alternating circuit than is given by their prime mover, by receiving power from the direct-current side. In this case a part of the alternating power is generated from mechanical power, 336 ELEMENTS OF ELECTRICAL ENGINEERING, and the other converted from direct-current power, and the machine combines the features of an alternator with those of an inverted converter. Conversely, when supplying direct-current power and receiving mechanical power from the prime mover and electric power from the alternating system, the double-current generator combines the features of a direct-current generator and a converter. In either case the armature reaction, etc., are the sum of those corresponding to the two types of machines combined. 149. A combination of the converter with the direct-current generator is represented by the so-called " motor converter" which consists of the concatenation of a commutating machine with an induction machine. If the secondary of an induction machine is connected to a second induction or synchronous machine on the same shaft, and of the same number of poles, the combination runs at half synchronous speed, and the first induction machine as frequency converter supplies half of its power as electric power of half frequency to the second machine, and changes the other half as motor into mechanical power, driving the second machine as generator. (Or, if the two machines have different number of poles, or are connected to run at different speeds, the division of power is at a different but constant ratio.) Using thus a double- current generator as second machine, it receives half of its power mechanically, by the induction machine as motor, and the other half electrically, by the induction machine as frequency converter. Such a machine, then, is intermediate between a converter and a direct-current generator, having an armature reaction equal to half that of a direct-current generator. Such motor converters are occasionally used on high-fre- quency systems, as their commutating component is of half frequency, and thus affords a better commutator design than a high-frequency converter. They are necessarily much larger than standard converters, but are smaller than motor generator sets, as half the power is converted in either machine. One advantage of this type of machine for phase control is that it requires no additional reactive coils, as the induction machine affords sufficient reactance. The motor converter, however, is a syn- chronous machine, that is, it cannot replace induction motor generator sets at the end of very long transmission lines, where synchronous machines tend to surging. SYNCHRONOUS CONVERTERS. 387 The use of the converter to change from alternating to alter- nating of a different phase, as, for instance, when using a quarter- phase converter to receive power by one pair of its collector rings from a single-phase circuit and supplying from its other pair of collector rings the other phase of a quarter-phase system, or a three-phase converter on a single-phase system supplying the third wire of a three-phase system from its third collector ring, is outside the scope of this treatise, and is, moreover, of very little importance, since induction or synchronous motors are superior in this respect. APPENDIX. XIII. Direct-Current Converter. 150. If n equidistant pairs of diametrically opposite points of a commutating machine armature are connected to the ends of n compensators or auto-transformers, that is, electric circuits interlinked with a magnetic circuit, and the centers of these com- pensators connected with each other to a neutral point as shown diagrammatically in Fig. 194 for n = 3, this neutral is equidis- Fig. 194. Diagram of Direct-Current Converter. tant in potential from the two sets of commutator brushes, and such a machine can be used as continuous-current converter, to transform in the ratio of potentials 1 : 2 or 2 : 1 or 1 : 1, in the latter case transforming power from one side of a three-wire system to the other side. Obviously either the n compensators can be stationary and connected to the armature by 2 n collector rings, or the compen- 338 ELEMENTS OF ELECTRICAL ENGINEERING. sators rotated with the armature and their common neutral con- nected to the external circuit by one collector ring. The distribution of potential and of current in such a direct- current converter is shown in Fig. 195 for n = 2, that is, two compensators in quadrature. With the voltage 2 e between the outside conductors of the system, the voltage between the neutral and outside conductor is e, that on each of the 2 n compensator sections is e sin iin(0-0 - ), * -0,1,2 . . . 2n-l. V n/ Neglecting losses in the converter and the compensator, the currents in the two sets of commutator brushes are equal and of the same direction, that is, both outgoing or both incoming, and Fig. 195. Distribution of e.m.f. and Current in Direct-Current Converter. opposite to the current in the neutral; that is, two equal currents i enter the commutator brushes and issue as current 2 i from the neutral, or inversely. From the law of conservation of energy it follows that the cur- rent 2 i entering from the neutral divides in 2 n equal and constant n branches of direct current, - , in the 2n compensator sections, and n hence enters the armature, to issue as current i from each of the commutator brushes. SYNCHRONOUS CONVERTERS. 339 In reality the current in each compensator section is where i Q is the exciting current of the magnetic circuit of the compensator, and a the angle of hysteretic advance of phase. At the commutator the current on the motor side is larger than the current on the generator side, by the amount required to cover the losses of power in converter and compensator. In Fig. 195 the positive side of the system is generator, the negative side motor. This machine can be considered as receiv- ing the current i at the voltage e from the negative side of the system, and transforming it into current i at voltage e on the positive side of the system, or it can be considered as receiving current i at voltage 2 e from the system, and transforming it into current 2 i at the voltage e on the positive side of the system, or of receiving current 2 i at voltage e from the negative side, and returning current i at voltage 2 e. In either case the direct- current converter produces a difference of power of 2 ie between the two sides of the three-wire system. The armature reaction of the currents from the generator side of the converter is equal but opposite to the armature reaction of the corresponding currents entering the motor side, and the motor and generator armature reactions thus neutralize each other, as in the synchronous converter; that is, the resultant armature reaction of the continuous-current converter is prac- tically zero, or the only remaining armature reaction is that corresponding to the relatively small current required to rotate the machine, that is, to supply the internal losses in the same. The armature reaction of the current supplying the electric power transformed into mechanical power obviously also remains, if the machine is used simultaneously as motor, as for driving a booster connected into the system to produce a difference between the voltages of the two sides, or the armature reaction of the currents generated from mechanical power if the machine is driven as generator. IS i. While the currents in the armature coils are more or less sine waves in the alternator, rectangular reversed currents in the direct-current generator or motor, and distorted triple-fre- quency currents in the synchronous converter, the currents in the 340 ELEMENTS OF ELECTRICAL ENGINEERING. armature coils of the direct-current converter are approximately triangular double -frequency waves. Let Fig. 196 represent a development of a direct-current con- verter with brushes B 1 and B v and C one compensator receiving current 2 i from the neutral. Consider first an armature coil a l yyyy a'"a"a B B: B 31 yroyyyyyyyyyyyyyyyyyy Fig. 196. Development of a Direct-Current Converter. adjacent and behind (in the direction of rotation) a compensator lead 6 r In the moment when compensator leads b l 6 2 coincide with the brushes B 1 5 2 the current i directly enters the brushes and coil a t is without current. In the next moment (Fig. 196.A) the total current i from 6 t passes coil a i to brush B v while there is yet practically no current from b t over coils o! a", etc., to brush B r But with the forward motion "of the armature less and less of the current from b : passes through a t a v etc., to brush B l and more over of of', etc., to brush J3 2 , until in the position of a t midway between 6 t and & 2 (Fig. 1965), one-half of the current from fcj passes a l a 2 , etc., to B v the other half a ; a" ', etc., to B r With the further rotation the current in a i grows less and be- comes zero when ~b l coincides with 5 2 , or half a cycle after its coincidence with J3 r That is, the current in coil a l approxi- SYNCHRONOUS CONVERTERS. 841 mately has the triangular form shown as i l in Fig. 197, changing twice per period from to It is shown negative, since it is against the direction of rotation of the armature. In the same way we see that the current in the coil a! ', adjacent ahead of the lead b lf has a shape shown as i f in Fig. 197. The current in coil Fig. 197. Current in the Various Coils of a Direct-Current Converter. a midway between two commutator leads has the form i , and in general the current in any armature coil a x , distant by angle r from the midway position a , has the form i x , Fig. 197. All the currents become zero at the moment when the com- pensator leads b 1 b 2 coincide with the brushes B l B v and change by i at the moment when their respective coils pass a commu- tator brush. Thus the lines A and A' in Fig. 198 with zero values at B t B 2 , the position of brushes, represent the currents in the individual armatiire coils. The current changes from A to A f at the moment = r when the respective armature coil passes the brush, twice per period. Due to the inductance of the armature coils, which opposes the change of current, the current waves are not perfectly triangular, but differ somewhat therefrom. With n compensators, each compensator lead carries the current -, which passes through the armature coils as triangular current, 342 ELEMENTS OF ELECTRICAL ENGINEERING. Fig. 198. Current in Individual Coils of a Direct-Current Converter with One Compensator. Fig. 199. Current in a Single Coil of a Direct-Current Converter with Three Compensators. SYNCHRONOUS CONVERTERS. 343 n changing by - in the moment the armature coil passes a commu- tator brush. This current passes the zero value in the moment the compensator lead coincides with a brush. Thus, the different currents of n compensators which are superposed in an armature coil ax have the shape shown in Fig. 199 for n = 3. That is, each compensator gives a set of slanting lines A^A/, A 2 A/, A 3 A 3 ', and all the branch currents i v i v i 3 , superposed, give a resultant current ix } which changes by i in the moment the coil passes the brush. ft n ix varies between the extreme values - (2 p 1) and - (2 p + 1), if the armature coil is displaced from the midway position between two adjacent compensator leads by angle r, and p = - TC p varies between and + 2n 2n Thus the current in an armature coil in position p = - can be denoted in the range from p to 1 + p, or r to x + T, by where The effective value of this current is x = - 7T 7 = Since in the same machine as direct-current generator at volt- * age 2 e and current i y the current per armature coil is -, the ratio of current is / 1 2 and thus the relative Pr loss or the heat developed in the armature coil * /A 2 344 ELEMENTS OF ELECTRICAL ENGINEERING. with a minimum, P = o, r <> = i and a maximum, 1 _ = n 3 n* The mean heating or 7 2 r of the armature is found by integrating over f from _j_ to _, JL 2ft 2n' as /2ti ... = 1 JL ^ 1 + n 2 3 3ft 2 3^ 2 This gives the following table, for the direct-current converter, of minimum current heating, ^ , in the coil midway between adjacent commutator leads, maximum current heating, f mj in the coil adjacent to the commutator lead, mean current heating, r, and rating as based on mean current heating in the armature, JL. vf ' DIRECT-CURRENT CONVERTER Pr RATING No. of compensators, n= d.c gen. 1 2 3 4 n oo Minimum current heat- ing y>=0, 7j= 1 i 1 i 1 i i Maximum current heat- 1 infif, 59 = M ynisr-. ' r n 1 f I 7 , 1 H H n* J Mean current heating, 1 r- 1 1 A I? \l 1 1 Rating, ^7= 1 1.225 1.549 1.643 1.681 V/rr* 1.732 SYNCHRONOUS CONVERTERS. 845 As seen, the output of the direct-current converter is greater than that of the same machine as generator. Using more than three compensators offers very little advantage, and the difference between three and two compensators is comparatively small, also, but the difference between two and one compensator, especially regarding the local armature heating, is considerable, so that for most practical purposes a two-compensator converter would be preferable. The number of compensators used in the direct-current con- verter has a similar effect regarding current distribution, heat- ing, etc., as the number of phases in the synchronous converter. Obviously these relative outputs given in above table refer to the armature heating only. Regarding commutation, the total current at the brushes is the same in the converter as in the gen- erator, the only advantage of the former being the better com- mutation due to the absence of armature reaction. The limit of output set by armature reaction and corresponding field excitation in a motor or generator obviously does not exist at all ifi a converter. It follows herofrom that a direct-current motor or generator does not give the most advantageous direct- current converter, but that in the direct-current converter just as in the synchronous converter, it is preferable to proportion the parts differently, in accordance with above discussion, as, for instance, to useless conductor section, a greater number of con- ductors in series per pole, etc. XIV. Three- Wire Generator and Converter. 152. A machine based upon the principle of the direct-current converter is frequently used to supply a three-wire direct-current distribution system (Edison system). This machine may be a single generator or synchronous converter, which is designed for the voltage between the outside conductors of the circuit (the positive and the negative conductor), 220 to 280 volts, while the middle conductor of the system, or neutral conductor, is connected to the generator by compensator and collector rings, or, in the case of a synchronous converter, is connected to the neutral of the step-up transformers, and the latter thus used as compensators. A three-wire generator thus is a combination of a direct- current generator and a direct-current converter, and a three- 346 ELEMENTS OF ELECTRICAL ENGINEERING. wire converter is a combination of a synchronous converter and a direct-current converter. Such a three-wire machine has the advantage over two separate machines, connected to the two sides of the three-wire direct-current system, of combining two smaller machines into one of twice the size, and thus higher Fig. 200. Three- Wire Machine with Single Compensator. Fig. 201. Three-Wire System with Two Machines. space- and operation-economy and lower cost, and has the further advantage that only half as large current is commutated as by the use of two separate machines; that is, the positive brush of the machine on the negative and the negative brush of the machine on the positive side of the system are saved, as seen by the diagrammatic sketch of the machine in Fig. 200 and the two separate two-wire machines in Fig. 201. The use of three-wire 220-volt machines on three-wire direct-current systems thus has practically displaced that of two separate 110-volt machines. SYNCHRONOUS CONVERTERS. 347 A. THREE-WIRE DIRECT-CURRENT GENERATOR. 153. In such machines, either only one compensator or auto- transformer is used for deriving the neutral, as shown diagram- matically in Fig. 200, or two compensators in quadrature, as shown in Fig. 202, but rarely more. As the efficiency of conversion of a direct-current converter with two compensators in quadrature (Fig. 202) is higher than that of a direct-current converter with single compensator (Fig. Fig. 202. Three-Wire Machine with Two Compensators. 200), it is preferable to use two (or even more) compensators where a large amount of power is to be converted, that is, where a very great unbalancing between the two sides of the three- wire system may occur, or one side may be practically unloaded while the other is overloaded. Where, however, the load is fairly distributed between the two sides of the system, that is, the neutral current (which is the difference between the currents on the two sides of the system) is small and so only a small part of the generator power is converted from one side to the other, and the efficiency of this conversion thus of negligible influence on the heating and the output of the machine, a single compensator is preferable because of its simplicity. In three-wire distribu- tion systems the latter is practically always the case, that is, the load fairly balanced and the neutral current small The size of the compensators depends updn the amount of un- balanced power, that is, the maximum difference between the load on the two sides of the three-wire system, and thus equals the product of neutral current i and voltage e between neutral 348 ELEMENTS OF ELECTRICAL ENGINEERING. and outside conductor; that is, in the three-wire system of volt- age e per circuit, voltage 2 e between the outside conductors, and maximum current i in the outside conductors, the generator power rating is p = 2 ei. Let now 2 = maximum unbalanced current in the neutral usually not exceeding 10 to 20 per cent of i and using a single compensator, connected diametrically across the armature, Fig. 200, the maximum of the alternating voltage which it receives is 2e, and its effective voltage therefore e V2. As the neutral current i divides when entering the compensator, the current in the compensating winding is ^ (neglecting the small excit- ing current), and the volt-ampere capacity of the compensator thus is e i and _ 2 V2 i 0.354 V Even with the neutral current equal to the current in the out- side conductor, or the one side of the system fully loaded, the other not loaded, the compensator thus would have only 35.4 per cent of the volt-ampere capacity of the generator, and as a compensator of ratio 1 * 1 is half the size of a transformer of the same volt-ampere capacity, in this case the compensator has, approximately, the size of a transformer of 17.7 per cent of the size of the generator. n With the maximum unbalancing of 20 per cent, or -? = 0.2, % the compensator thus has 7 per cent of the volt-ampere capacity of the generator, or the size of a transformer of only 3.5 per cent of the generator capacity, that is, is very small, and this method is therefore the most convenient for deriving the neutral of a three-wire distribution system. When using n compensators, obviously each has of the size fit which a single compensator would have. SYNCHRONOUS CONVERTERS. 349 The disadvantage of the three-wire generator over two sepa- rate generators is that a three-wire generator can only divide the voltage in two equal parts, that is, the t\vo sides of the system have the same voltage at the generator. The use of two separate generators, however, permits the production of a higher voltage on one side of the system than on the other, and thus takes care of the greater line drop on the more evenly loaded side. Even in the case, however, where a voltage difference between the t\vo sides of the system is desired for controlling feeder drops, it can more economically be given by a separate booster in the neu- tral, as such a booster would require only a capacity equal to the neutral current times half the desired voltage difference between the two sides, and with 20 per cent neutral current and 10 per cent voltage difference between the tw r o sides, thus would have only one per cent of the size of the generator. B. THREE-WIRE CONVERTER. 154. In a converter feeding a three-wire direct-current system the neutral can be derived by connection to the transformer neutral. Even in this case, however, frequently a separate com- pensator is used, connected across a pair of collector rings of the converter, since, as seen above, with the moderate unbalancing usually existing, such a compensator is very small. When connecting the direct-current neutral to the transformer neutral it is necessary to use such a connection that the trans- former can operate as compensator, that is, that the direct current in each transformer divides into two branches of equal m.m.f., otherwise the direct-current produces a unidirectional magneti- zation in the transformer, which superimposed upon the magnetic cycle raises the magnetic induction beyond saturation, and thus causes excessive exciting current and heating, except when very small For instance, with Y connection of the transformers supplying a three-phase converter, Fig. 203, each transformer secondary receives one-third of the neutral current, and if this current is not very small and comparable with the exciting current of the trans- former which can rarely be the magnetic density in the transformer rises beyond saturation by this unidirectional m.m.f. This connection thus is in general not permissible for deriving the neutral. 350 ELEMENTS OF ELECTRICAL ENGINEERING. Fig. 203. Neutral of Y-Connected Transformers Connected to Neutral of Three-Wire System Supplied from a Three-Phase Converter. Fig. 204. Quarter-Phase Converter with Transformer Neutral Connected to Direct-Current Neutral. Fig. 205. Three-Phase Converter with Neutral of the T-Connected Trans- formers -as Direct-Current Neutral. Fig. 206. Three-Phase Converter with Transformer Neutral Connected to Direct-Current Neutral. SYNCHRONOUS CONVERTERS. 351 In a quarter-phase converter, as shown In Fig. 204, the transformer neutral can be used as direct-current neutral, since in each transformer the direct current divides into two equal branches, which magnetize in opposite direction, and so neu- tralize. The T connection, Fig. 205, can be used for three-phase con- verters with the neutral derived from a point at one-third the height of the teaser transformer, since the m.m.fs. of the direct current \ balance in the transformers, as seen in Fig. 205. Delta connection on three-phase and double delta on six- phase converters cannot be used, as it has no neutral, but in this case a separate compensator is required. The diagrammatical connections of transformers can, however, be used on six-phase converters, and the connection shown in Fig. 206, which has two coils on each transformer, connected to different phases, on three-phase converters. E. INDUCTION MACHINES. I. General. 155. The direction of rotation of a direct-current motor, whether shunt- or series-wound, is independent of the direction of the current supplied thereto ; that is, when reversing the current in a direct-current motor the direction of rotation remains the same. Thus theoretically any continuous-current motor should operate also with alternating currents. Obviously in this case not only the armature but also the magnetic field of the motor must be thoroughly laminated to exclude eddy currents, and care taken that the currents in the field and armature circuits reverse simultaneously. Obviously the simplest way of fulfilling the latter condition is to connect the field and armature circuits in series as alternating-current series motor. Such motors are used to a considerable extent, mainly for railroading. Their disad- vantage for many purposes is the use of a commutator, and also that their speed is not constant but depends upon the load. The shunt motor on an alternating-current circuit has the objection that in the armature winding the current should be power current, thus in phase with the e.m.f., while in the field winding the current is lagging nearly 90 deg., as magnetizing current. Thus field and armature would be out of phase with each other. To overcome this objection either there is inserted in series with the field circuit a condenser of such capacity as to bring the current back into phase with the voltage (Stanley), or the field may be excited from a separate e.m.f. differing 90 deg. in phase from that supplied to the armature. The former ar- rangement has the disadvantage of requiring almost perfect constancy of frequency, and therefore is not practicable. In the latter arrangement the armature winding of the motor is fed by one, the field winding by the other phase of a quarter-phase system, and thus the current in the armature brought approxi- mately into phase with the magnetic flux of the field. Such an arrangement obviously loads the two phases of the system unsymmetrically, the one with the armature power 352 INDUCTION MACHINES, 35S current, the other with the lagging field current. To balance the system two such motors may be used simultaneously and combined in one structure, the one receiving power current from the first, magnetizing current from the second phase, the second motor receiving magnetizing current from the first and power current from the second phase. The objection that the use of the commutator, in an alter- nating-current motor tends to vicious sparking and therefore greatly limits the design, can be entirely overcome by utilizing the alternating feature of the current; that is, instead of leading the current into the armature by commutator and brushes, producing it therein by electromagnetic induction, by closing the armature conductors upon themselves and surrounding . the armature by a primary coil at right angles to the field exciting coil. Such motors have been built, consisting of two structures each containing a magnetizing circuit acted upon by one phase and a primary power circuit acting upon a closed -circuit armature as secondary and excited by the other phase of a quarter-phase system (Stanley motor). Going still a step further, the two structures can be com- bined into one by having each of the two coils fulfill the double function of magnetizing the field and producing currents in the secondary which are acted upon by the magnetization produced by the other phase. Obviously, instead of two phases in quadrature any number of phases can be used. This leads us by gradual steps of development from the con- tinuous-current shunt motor to the alternating-current polyphase induction motor. In its general behavior the alternating-current induction motor is therefore analogous to the continuous-current shunt motor. Like the shunt motor, it operates at approximately constant magnetic density. It runs at fairly constant speed, slowing down gradually with increasing load. The main difference is that in the induction motor the current in the secondary does not pass through a system of brushes, as in the continuous-current shunt motor, but is produced in the secondary as the short- circuited secondary of a transformer; and in consequence thereof the primary circuit of the induction motor fulfills the double. 354 ELEMENTS OF ELECTRICAL ENGINEERING. function of an exciting circuit corresponding to the field circuit of the continuous-current machine and a primary circuit producing a secondary current in the secondary by electromagnetic induction. 156. Since in the secondary of the induction motor the cur- rents are produced by induction from the primary impressed currents, the induction motor in its electromagnetic features is essentially a transformer; that is, it consists of a magnetic cir- cuit or magnetic circuits interlinked with two electric circuits or sets of circuits, the primary and the secondary circuits. The difference between transformer and induction motor is that in the former the secondary is fixed regarding the primary, and the electric energy in the secondary is made use of, while in the latter the secondary is movable regarding the primary, and the me- chanical force acting between primary and secondary is used. In consequence thereof the frequency of the currents in the secondary of the induction motor differs from, and as a rule is very much lower than, that of the currents impressed upon the primary, and thus the ratio of e.m.fs. generated in primary and in secondary is not the ratio of their respective turns, but is the ratio of the product of turns and frequency. Taking due consideration of this difference of frequency be- tween primaiy and secondary, the theoretical investigation of the induction motor corresponds to that of the stationary trans- former. The transformer feature of the induction motor pre- dominates to such an extent that in theoretical investigation the induction motor is best treated as a transformer, and the elec- trical output of the transformer corresponds to the mechanical output of the induction motor. The secondary of the motor consists of two or more circuits displaced in phase from each other so as to offer a closed sec- ondary to the primary circuits, irrespective of the relative motion. The primary consists of one or several circuits. In consequence of the relative motion of the primary and secondary, the magnetic circuit of the induction motor must be arranged so that the secondary while revolving does not leave the magnetic field of force. That means, the magnetic field of force must be of constant intensity in all directions, or, in other words, the component of magnetic flux in any direction in space be of the same or approximately the same intensity but differing in phase. Such a magnetic field can either be considered as the INDUCTION MACHINES. 355 superposition of two magnetic fields of equal intensity in quad- rature in time and space, or it can be represented theoretically by a revolving magnetic flux of constant intensity, or rotating field, or simply treated as alternating magnetic flux of the same intensity in every direction. 157. The operation of the induction motor thus can also be considered as due to the action of a rotating magnetic field upon a system of short-circuited conductors. In the motor field or primary, usually the stator, by a system of polyphase impressed e.m.fs. or by the combination of a single-phase impressed e.m.f. and the reaction of the currents produced in the secondary, a rotating magnetic field is produced. This rotating field produces currents in the short-circuited armature or secondary winding, usually the rotor, and by its action on these currents drags along the secondary conductors, and thus speeds up the armature and tends to bring it up to synchronism, that is, to the same speed as the rotating field, at which speed the secondary currents would disappear by the armature conductors moving together with the rotating field, and thus cutting no lines of force. The secondary therefore slips in speed behind the speed of the rotating field by as much as is required to produce the secondary currents and give the torque necessary to carry the load. The slip of the induction motor thus increases with increase of load, and is approximately proportional thereto. Inversely, if the secondary is driven at a higher speed than that of the rotating field, the field drags the armature conductors back, that is, consumes mechanical torque, and the machine then acts as a brake or induction generator. In the polyphase induction motor this magnetic field is pro- duced by a number of electric circuits relatively displaced in space, and excited by currents having the same displacement in phase as the exciting coils have in space. In the monocyclic motor one of the two superimposed quad- rature fields is excited by the primary power circuit, the other by the magnetizing or teaser circuit. In the single-phase motor one of the two superimposed mag- netic quadrature fields is excited by the primary electric circuit, the other by the secondary currents carried into quadrature position by the rotation of the secondary. In either case, at or near synchronism the magnetic fields are practically identical 356 ELEMENTS OF ELECTRICAL ENGINEERING. The transformer feature being predominant, in theoretical investigations of induction motors it is generally preferable to start therefrom. The characteristics of the transformer are independent of the ratio of transformation, other things being equal; that is 7 dou- bling the number of turns for instance, and at the same time reducing their cross-section to one-half, leaves the efficiency, regulation, etc., of the transformer unchanged. In the same way, in the induction motor it is unessential what the ratio of primary to secondary turns is, or, in other words, the secondary circuit can be wound for any suitable number of turns, provided the same total copper cross-section is used. In consequence hereof the secondary circuit is mostly wound with one or two bars per slot, to get maximum amount of copper, that is, minimum resist- ance of secondary. The general characteristics of the induction motor being inde- pendent of the ratio of turns, it is for theoretical considera- tions simpler to assume the secondary motor circuits reduced to the same number of turns and phases as the primary, or of the ratio of transformation 1 to 1, by multiplying all secondary currents and dividing all secondary e.m.fs. by the ratio of turns, multiplying all secondary impedances and dividing all secondary admittances by the square of the ratio of turns, etc. Thus in the following under secondary current, e.m.f., impe- dance, etc., shall always be understood their values reduced to the primary, or corresponding to a ratio of turns 1 to 1, and the same number of secondary as primary phases, although in prac- tice a ratio 1 to 1 will hardly ever be used, as not fulfilling the condition of uniform effective reluctance desirable in the start- ing of the induction motor* II. Polyphase Induction Motor. 1, INTRODUCTION. 158. The typical induction motor is the polyphase motor. By gradual development from the direct-current shunt motor we arrive at the polyphase induction motor. The magnetic field of any induction motor, whether supplied by polyphase, monocyclic, or single-phase e.m.f., is at normal condition of operation, that is, near synchronism, a polyphase INDUCTION MACHINES. S57 field. Thus to a certain extent all induction motors can be called polyphase machines. When supplied with a polyphase system of e.ni.fs. the internal reactions of the induction motor are simplest and only those of a transformer with moving second- ary, while in the single-phase induction motor at the same time a phase transformation occurs, the second or magnetizing phase being produced from the impressed phase of e.m.f. by the rota- tion of the motor, which carries the secondary currents into quadrature position with the primary current. The polyphase induction motor of the three-phase or quarter- phase type is the one most commonly used, while single-phase motors have found a more limited application only, and especially for smaller powers. Thus in the following more particularly the polyphase induc- tion machine shall be treated, and the single-phase type discussed only in so far as it differs from the typical polyphase machine. 2. CALCULATION, 159. In the polyphase induction motor, Let Y = g + jb = primary exciting admittance, or admit- tance of the primary circuit with open secondary- circuit ; that is, ge = magnetic power current, be = wattless magnetizing current, where e = counter-generated e.m.f. of the motor; Z = r G ~~]x Q = primary self-inductive impedance, and Z l = r l jx v = sec- ondary self-inductive impedance, reduced to the primary by the ratio of turns.* All these quantities refer to one primary circuit and one corre- sponding secondary circuit. Thus in a three-phase induction motor the total power, etc., is three times that of one circuit, in the quarter-phase motor with three-phase armature 1| of the three secondary circuits are to be considered as corresponding to each of the two primary circuits, etc. Let e = primary counter-generated e.m.f., or e.m.f. generated in the primary circuit by the flux interlinked with primary and * The self-inductive reactance refers to that flux which surrounds one of the electric circuits only, without being interlinked with the other circuits. S58 ELEMENTS OF ELECTRICAL ENGINEERING. secondary (mutual induction) ; s = slip, with the primary fre- quency as unit; that is, s = denoting synchronous rotation, s = 1 standstill of the motor. We then have 1 s = speed of the motor secondary as fraction of syn- chronous speed, sf = frequency of the secondary currents, / = frequency impressed upon the primary ; 7 se = e.m.f. generated in the secondary. The actual impedance of the secondary circuit at the frequency s/is JV- hence, the secondary current is se se - = e where 9 sr __ s x t r the primary exciting current is I w -eY-e\g + ]T>], and the total primary current is /, = e [(i + ?) + J K + &)] = (&, + j6,), where 6, = a, + 0, 6 2 = a 2 + 6. The e.m.f. consumed in the primary circuit by the impedance Z is 7 Z , the counter-generated e.m.f. is e, hence, the primary terminal voltage is where, ~~ i 4. A _j_ /> /q h K Eliminating complex quantities, we have hence, the counter-generated e.m.f. of motor, c _ E V Cl 2 + c 2 2 ' w ere ^^ _ j m p resgec [ e .m.f., absolute value. INDUCTION MACHINES. 359 Substituting this value in the equations of I v 7 00 , 7 , etc., gives the complex expressions of currents and e.m.fs., and elim- inating the imaginary quantities we have the primary current, 7 = e Vb* + 6 2 2 , etc. The torque of the polyphase induction motor (or any other motor or generator) is proportional to the product of the mutual magnetic flux and the component of ampere-turns of the sec- ondary, which is in phase with the magnetic flux in time, but in quadrature therewith in direction or space. Since the generated e.m.f. is proportional to the mutual magnetic flux and the num- ber of turns, but in quadrature thereto in time, the torque of the induction motor is proportional also to the product of the gen- erated e.m.f. and the component of secondary current in quadra- ture therewith in time and in space. Since 1 1 = e (a 1 + ja 2 ) is the secondary current correspond- ing to the generated e.m.f. e, the secondary current in the quad- rature position thereto in space, that is, corresponding to the e.m.f. je, is . r , f . x J Ji-(- a + J a i)> and a t e is the component of this current in quadrature in time with the e.m.f. e. Thus the torque is proportional to e X a^, or D = e\ This value D is in its dimension a power, and it is the power which the torque of the motor would develop at synchronous speed. 160. In induction motors, and in general motors which have a definite limiting speed, it is preferable to give the torque in the form of the power developed at the limiting speed, in this case synchronism, as "synchronous watts, 77 since thereby it is made independent of the individual conditions of the motor, as its number of poles, frequency, etc., and made comparable with the power input, etc. It is obvious that when given in synchronous watts, the maximum possible value of torque which could be reached, if there were no losses in the motor, equals the power input. Thus, in an induction motor with 9000 watts power 360 ELEMENTS OF ELECTRICAL ENGINEERING. input, a torque of 7000 synchronous watts means that | of the maximum theoretically possible torque is realized, while the Statement, "a torque of 30 Ib. at one foot radius," would be meaningless without knowing the number of poles and the fre- quency. Thus, the denotation of the torque in synchronous watts Is the most general, and preferably used in induction motors. Since the theoretical maximum possible torque equals the power input, the ratio torque in synchronous watts output power input that is, actual torque maximum possible torque is called the torque efficiency of the motor, analogous to the power efficiency or power output . power input ; that is, power output maximum possible power output Analogously torque in synchronous watts volt-amperes input is called the apparent torque efficiency. The definition of these quantities, which are of importance in judging induction motors, are thus : The "efficiency" or "power efficiency" is the ratio of the true mechanical output of the motor to the output which it would give at the same power input if there were no internal losses in the motor. The " apparent efficiency" or t apparent power efficiency" is the ratio of the mechanical output of the motor to the output which it would give at the same volt-ampere input if there were neither Internal losses nor phase displacement in the motor. The "torque efficiency" is the ratio of the torque of the motor to the torque which it would give at the same power input if there were no internal losses in the motor. INDUCTION MACHINES. 361 The "apparent torque efficiency" is the ratio of the torque of the motor to the torque which it would give at the same volt- ampere input if there were neither internal losses nor phase dis- placement in the motor. The torque efficiencies are of special interest in starting where the power efficiencies are necessarily zero, but it nevertheless is of importance to find how much torque per watt or per volt- ampere input is given by the motor. Since D = e 2 a t is the power developed by the motor torque at synchronism, the power developed at the speed of (1 s) X synchronism, or the actual power output of the motor, is p = (i _ s ) D = e 2 a, (1 - s) _ e 2 r,s (1 - s) r* The output P includes friction, windage, etc.; thus, the net mechanical output is P friction, etc. Since, however, fric- tion, etc., depend upon the mechanical construction of the individual motor and its use, it cannot be included in a general formula. P is thus the mechanical output, and D the torque developed at the armature conductors. The primaiy current Jo =(&!+&) has the quadrature components eb t and eb 2 . The primary impressed e.m.f. E 9 *e (c t + yc 2 ) has the quadrature components ec l and ec r Since the components eb l and ec 2 , and eb 2 ancf ec v respectively, are in quadrature with each other, and thus represent no power, the power input of the primary circuit is eb l X ec l + eb z X ec z , or P = * (&A + &A)- The volt-amperes or apparent input is obviously, 161. These equations can be greatly simplified by neglecting the exciting current of the motors, and approximate values of 362 ELEMENTS OF ELECTRICAL ENGINEERING. current, torque, power, etc., derived thereby, which are suffi- ciently accurate for preliminary investigations of the motor at speeds sufficiently below synchronism to make the total motor current large compared with the exciting current. In this case the primary current equals the secondary current, that is, o/? where and = e\l + Z, s J Z^ - .frr+ gr n)-f g fof + ^o) c/ . and, in absolute values, e *=e hence, and the torque, in synchronous watts, is hence, substituting for 6, * ^ and the power is P_ * C 1 - *) ^Q V 1 If the additional resistance r is inserted into the armature circuit, and the total armature resistance thus becomes r l + r, instead of r u substituting (r^ + r) in above equations we have and INDUCTION MACHINES. 363 Neglecting also the primary self-inductive impedance, Z = T Q jx , which sometimes can be done as first approximation, especially at large values of r, these equations become 162. Since the counter-generated e.m.f. e (and thus the im- pressed e.m.f. jB ) enters in the equation of current, magnetism, etc., as a simple factor, in the equations of torque, power input and output, and volt-ampere input as square, and cancels in the equation of efficiency, power-factor, etc., it follows that the current, magnetic flux, etc., of an induction motor are propor- tional to the impressed e.m.f. ; the torque, power output, power input, and volt-ampere input are proportional to the square of the impressed e.m.f., and the torque and power efficiencies and the power-factor are independent of the impressed voltage. In reality, however, a slight decrease of efficiency and power- factor occurs at higher impressed voltages, due to the increase of resistance caused by the increasing temperature of the motor and due to the approach to magnetic saturation, and a slight decrease of efficiency occurs at lower voltages when including in the efficiency the loss of power by friction, since this is inde- pendent of the output and thus at lower voltage, that is, lesser output, a larger percentage of the output, so that the efficiencies and the power-factor can be considered as independent of the impressed voltage, and the torque and power proportional to the square thereof only approximately, but sufficiently close for most purposes. 3. LOAD AND SPEED CUBVES. 163. The calculation of the induction motor characteristics is most conveniently carried out in tabulated form by means of above-given equations as follows : Let Z Q =*r Q jx =0.1 0.3 f=primary self-inductive impe- dance. Z^r L ^=0.1 0.3 /= secondary self-inductive impe- dance reduced to primary. Y = gr+f&=0.01+(Xl j = primary exciting admittance. J3? =110 volts primary impressed e.m.f. 364 ELEMENTS OF ELECTRICAL ENGINEERING. It is then, per phase, - ^ p % H d o* & I ,** 11 . .. -: II ^ rfS n + n ^ 1 i . _l_ J?. 'I o> - h S ''Mis I + -f" II i C l! 5*3 O n N H tfc ^ 3 o *- j- *! Q - e A e 1 = ^i f > 0100 01 0.10 1 031 -hO 007 1 031 106.6 1010 10.8 01 0.0100 100 0.003 0.11 0.103 1.042 -0.023 1.042 105 7 0.1507 15.9 0.02 0100 0.200 0.012 0.21 0.112 1.055 -0.052 1.056 104.3 0.238 24.8 0.05 0102 0.490 0.073 0.50 173 1.102 -0.133 1.110 99 2 0.522 51.8 1 0109 0.920 276 0.93 376 1.206 -0 241 1.230 89.5 1.003 89 7 0.15 0120 1.25 0.563 1.26 663 1.325 -0.308 1.360 80.9 1.424 115 0.2 0136 1.47 0.883 1.48 0.983 1.443 -0.354 1.485 74.2 1 777 132 3 0181 1.66 1 49 1.67 1.50 1.617 -0.351 1.654 66 6 2.245 149 0.5 0.0325 1.54 2.31 1.55 2.41 1.878 -0.224 1.891 58.2 2 865 167 1.0 0.1000 1,00 3.00 1.01 3 10 2.031 + 0.007 2.031 54.1 3.261, 176 D P** P = jP- eff.- app. eff. = paw.fae.-s* s. e-. * a ^i c i +* 2 ~~ P P A, e- ai . d-s)D .Bo/. 6^2- ep. P Q Pa * 11,360 1.19 Oil 0.125 10.5 0.01 11,170 1.117 1.106 1.75 0.112 1.249 88.5 63 2 71 5 0.02 10,880 2 176 2.133 2.73 0.216 2.350 91.0 78,3 86.2 0,05 9,840 4.82 4.58 5 70 0.528 5.20 88 3 80 5 91.3 0.1 8,010 7.38 6.64 9 87 1.030 8.25 80.7 67.3 83 5 0.15 6,540 8 20 6.97 12 65 1.466 9.60 72 5 55.0 76 0.2 5,510 8.10 6 48 14.52 1.782 9.80 66 44.6 67.5 0.3 4,440 7.36 5.15 16 4 2.154 9.55 53.8 31 5 58.3 0.5 3,390 5,23 2.61 18 4 2 370 8 04 32 3 14.2 43.8 1.0 2,930 2.93 19 4 2.072 6 08 31.3 Diagrammatically it is most instructive in judging about an Induction motor to plot from the preceding calculation 1st. The load curves, that is, with the load or power output as abscissas, the values of speed (as a fraction of synchronism), of current input, power-factor, efficiency, apparent efficiency, and torque. 2d. The speed curves, that is, with the speed, as a fraction of synchronism, as abscissas, the values of torque, current input, power-factor, torque efficiency, and apparent torque efficiency. The load curves are most instructive for the range of speed near synchronism, that is, the normal operating conditions of the motor, while the speed curves characterize the behavior of the motor at any speed. In Fig. 207 are plotted the load curves, and in Fig. 208 the INDUCTION MACHINES. 365 speed curves of a typical polyphase induction motor of moderate size, having the following constants: e = 110; F= 0.01 0.1 /; Z l = 0.1 - 0.3 j, and Z - 0.1 - 0.3 /. As sample of a poor motor of high resistance and high admit- tance or exciting current are plotted in Fig. 209 the load curves si 0.3 ER OU TPUT 0.01 -hO.lj JL l\ 1000 2000 3000 4000 5000 6000 7000 Fig. 207. Induction Motor Load Curves. of a motor having the following constants: e Q == 110; Y = 0.04 + 0.4 j ; Z 1 - 0.3 - 0.3 j, and Z Q = 0.3 - 0.3 /, showing the overturn of the power-factor curve frequently met in poor motors. 164. The shape of the characteristic motor curves depends entirely on the three complex constants, F, Z v and Z , but is essentially independent of the impressed voltage. Thus a change of the admittance Y has no effect on the char- 366 ELEMENTS OF ELECTRICAL ENGINEERING. 1.0 0.9 0.7 0*6 0.5 ~ a = auxiliary magnetic flux, 6 = phase displacement in time between a and <3> p , r = phase displacement in space between a and p , the torque is D = $ p $ a sin T cos 0. In general the starting torque, apparent torque efficiency, etc., of the single-phase induction motor with any of these de- vices are given in per cent of the corresponding values of the same motor with polyphase magnetic flux, that is, with a mag- netic system consisting of two equal magnetic fluxes in quad- rature in time and space. 173. The infinite variety of arrangements proposed for start- ing single-phase induction motors can be grouped into three classes. 1. Phase-Splitting Devices. The primary system is composed of two or more circuits displaced from each other in position, and combined with impedances of different inductance factors so as to produce a phase displacement between them. When using two motor circuits, they can either be connected in series between the single-phase mains, and shunted with impedances of different inductance factors, as, for instance, a condensance and an inductance, or they can be connected in shunt between the single-phase mains but in series with impe- dances of different inductance factors. Obviously the impe- dances used for displacing the phase of the exciting coils can either be external or internal, as represented by high-resistance winding in one coil of the motor, etc. In this class belongs the use of the transformer as a phase- splitting device by inserting a transformer primary in series with one motor circuit in the main line and connecting the other motor circuit to the secondary of the transformer, or by feeding one of the motor circuits directly from the mains and the other from the secondary of a transformer connected across the mains with its primary. In either case it is, respectively, the internal impedance, or internal admittance, of the transformer* which is combined with one of the motor circuits for displacing its phase, and thus this arrangement becomes most effective by using 382 ELEMENTS OF ELECTRICAL ENGINEERING. transformers of high internal impedance or admittance as con- stant power transformers or open magnetic circuit transformers. 2. Inductive Devices. The motor is excited by the combina- tion of two or more circuits which are in inductive relation to each other. This mutual induction between the motor circuits can take place either outside of the motor in a separate phase- splitting device or in the motor proper. In the first case the simplest form is the divided circuit whose branches are inductively related to each other by passing around the same magnetic circuit external to the motor. In the second case the simplest form is the combination of a primary exciting coil and a short-circuited secondary coil on the primary member of the motor, or a secondary coil closed by an impedance. In this class belong the shading coil and the accelerating coil. 3. Monocyclic Starting Devices. An essentially wattless e.m.f . of displaced phase is produced outside of the motor, and used to energize a cross magnetic circuit of the motor, either directly by a special teaser coil on the motor, or indirectly by combining this wattless e.m.f. with the main e.m.f. and thereby deriving a system of e.m.fs. of approximately three-phase or any other relation. In this case the primary system of the motor is supplied essentially by a polyphase system of e.m.fs. with a single-phase flow of energy, a system which I have called "monocyclic." The wattless quadrature e.m.f. is generally produced by con- necting two impedances of different inductance factors in series between the single-phase mains, and joining the connection between the two impedances to the third terminal of a three- phase induction motor, which is connected with its other two terminals to the single-phase lines, as shown diagrammatically in Fig. 215, for a conductance a and an inductive susceptance fa. This starting device, when using an inductance and a conden- sance of proper size, can be made to give an apparent starting torque efficiency superior to that of the polyphase induction motor. Usually a resistance and an inductance are used, which, though not giving the same starting torque efficiency as avail- able by the use of a condensance, have the advantage of greater simplicity and reliability. After starting, the impedances are disconnected. INDUCTION MACHINES. 383 For a complete discussion and theoretical investigation of the different starting devices, the reader must be referred to the paper on the single-phase induction motor, "American Institute of Electrical Engineers' Transactions, February, 1898. " Fig. 215. Connections for Starting Single-Phase Motor. 174. The use of the resistance-inductance, or monocyclic, starting device with three-phase wound induction motor will be discussed somewhat more explicitly as the only method not using condensers which has found extensive commercial appli- cation. It gives relatively the best starting torque and torque efficiencies. In Fig. 215, M represents a three-phase induction motor of which two terminals, 1 and 2, are connected to single-phase mains, and the terminal 3 to the common connection of a conduc- tance a /that is, a resistance-) and an equal susceptance ja \ w (j\ thus a reactance - I connected in series across the mains. al Let 7 = g + jb = total admittance of motor between termi- nals 1 and 2 while at rest. We then have $ Y = total admit- tance from terminal 3 to terminals 1 and 2, regardless of whether the motor is delta- or Y-wound. If e = e.m.f. in the single-phase mains and E = difference of potential across conductance a of the starting device, then we have the current in a as I ml = Ea, and the e.m.f. across ja as e - E] thus, the current in ja is I 2 = ja (e - #), and the current in the cross magnetizing motor circuit from 3 to The e.m.f. 7? of the cross magnetizing circuit is, as may be seen 384: ELEMENTS OF ELECTRICAL ENGINEERING. from the diagram of e.m.fs., which form a triangle with e, E and e E as sides, and since / = J YE W we have Ea - ja (e- E) - S Y (2 #- e). This expression solved for E becomes 3a+3ja-8F' which from the foregoing value of E Q gives 3a+3jo-8F' or, substituting F = g + ]b, expanding, and multiplying both numerator and denominator by (3a-80)-/ (3o-86), gives i; ea (a f gj T\U,-~'-SUJ and the imaginary component thereof, or e.m.f. in quadrature to e in time and in space, is r, j 2 a f (q + V) E Q = lea No ^ ~- . ( a ~~ I 9) + (a f 6)^ In the same motor on a three-phase circuit this quadrature e.m.f. is the altitude of the equilateral triangle with e as sides, thus = je , and since the starting torque of the motor is pro- It portional to this quadrature e.m.f., the relative starting torque of the monocyclic starting device, or the ratio of starting torque of the motor with monocyclic starting device to that of the same motor on three-phase circuit, is / EJ 2a 2o-f -5 INDUCTION MACHINES. 385 A starting device which has been extensively used is the condenser in the tertiary circuit. In its usual form it can be considered as a modification of the monocyclic starting device, by using a condensance as the one impedance and making the other impedance infinite, that is, omitting it. It thus comprises a three-phase induction motor, in which two terminals are con- nected to the single-phase supply and the third terminal and one of the main terminals to a condenser. Usually the con- denser is left in circuit after starting, and made of such size that its leading current compensates for the lagging magnetizing- current of the motor, and the motor thus gives approximately unity power-factor. For further discussion of this subject the reader is referred to the paper on "Single-phase Induction Motors, 77 mentioned above, and to the "Theory and Calculation of Alternating- Current Phenomena/' fourth edition. 4. ACCELEKATION WITH STARTING DEVICE. 175. The torque of the single-phase induction motor (without a starting device) is proportional to the product of main flux, or magnetic flux produced by the primary impressed e.m.f., and the speed. Thus it is the same as in the polyphase motor at or very near synchronism, but falls off with decreasing speed and becomes zero at standstill. To produce a starting torque, a device has to be used to im- press an auxiliary magnetic flux upon the motor, in quadrature with the main flux in time and in space, and the starting torque is proportional to this auxiliary or quadrature flux. During acceleration or at intermediate speed the torque of the motor is the resultant of the main torque, or torque produced by the primary main flux, and the auxiliary torque produced by the auxiliary quadrature or starting flux. In general, this result- ant torque is not the sum of main and auxiliary torque, but less, due to the interaction between the motor and the starting device. All the starting devices depend more or less upon the total admittance of the motor and its power-factor. With increasing speed, however, the total admittance of the motor decreases and its power-factor increases, and an auxiliary torque device 386 ELEMENTS OF ELECTRICAL ENGINEERING. suited for the admittance of the motor at standstill will not be suited for the changed admittance at speed. The currents produced in the secondary by the main or pri- mary magnetic flux are carried by the rotation of the motor more or less into quadrature position, and thus produce the quad- rature flux giving the main torque as discussed before. This quadrature component of the main flux generates an e.m.f . in the auxiliary circuit of the starting device, and thus changes the distribution of currents and e.m.fs. in the starting device. The circuits of the starting device then contain, besides the motor admittance and external admittance, an active counter e.m.f., changing with the speed. Inversely, the currents produced by the counter e.m.f. of the motor in the auxiliary circuit react upon the counter e.m.f., that is, upon the quadrature component or main flux, and change it. Thus during acceleration we have to consider L The effect of the change of total motor admittance and its power-factor upon the starting device. 2. The effect of the counter e.m.f. of the motor upon the start- ing device and the effect of the starting device upon the counter e.m.f. of the motor, 1. The total motor admittance and its power-factor change very much during acceleration in motors with short-circuited low-resistance secondary. In such motors the admittance at rest is very large and its power-factor low, and with increasing speed the admittance decreases and its power-factor increases greatly. In motors with short-circuited high-resistance second- ary the admittance also decreases greatly during acceleration, but its power-factor changes less, being already high at stand- still. Thus the starting device will be affected less. Such motors, however, are inefficient at speed. In motors with varia- ble secondary resistance the admittance and its power-factor can be maintained constant during acceleration by decreasing the resistance of the secondary circuit in correspondence with the increasing counter e.m.f. Hence, in such motors the starting device is not thrown out of adjustment by the changing admit- tance during acceleration. In the phase-splitting devices, and still more in the inductive devices, the starting torque depends upon the internal or motor admittance, and is thus essentially affected by the change of INDUCTION MACHINES. 387 admittance during acceleration, and by the appearance of a counter e.m.f. during acceleration, which throws the starting device out of its proper adjustment, so that frequently while a considerable torque exists at standstill, this torque becomes zero and then reverses at some intermediate speed, and the motor, while starting with fair torque, is not able to run up to speed with the starting device in circuit. Especially is this the ease where capacity is used in the starting device. With the mono- cyclic starting device this effect is small in any case and absent when a condenser is used in the tertiary circuit, and therefore the latter may advantageously be left in the circuit at speed. IV. Regulation and Stability. 1. LOAD AND STABILITY. 176. At constant voltage and constant frequency the torque of the polyphase induction motor is a maximum at some definite speed and decreases with increase of speed over that correspond- ing to the maximum torque, to zero at synchronism; it also de- creases with decrease of speed from that at the maximum torque point, to a minimum at standstill, the starting torque. This maximum torque point shifts towards lower speed with increase of the resistance in the secondary circuit, and the start- ing torque thereby increases. Without additional resistance inserted in the secondary circuit the maximum torque point, however, lies at fairly high speed not very far below synchro- nism, 10 to 20 per cent below synchronism with smaller motors of good efficiency. Any value of torque between the starting torque and the maximum torque is reached at two different speeds. Thus in a three-phase motor having the following constants: impressed e.m.f v e = 110 volts; exciting admittance, Y = 0.01 + 0.1 y; primary impedance, Z = 0.1 0.3 f, and secondary impedance, Z l 0.1 0.3 j, the torque of 5.5 synchronous kilowatts is reached at 54 per cent of synchronism and also at the speed of 94 per cent of synchronism, as seen in Fig. 216. When connected to a load requiring a constant torque, irre- spective of the speed, as when pumping water against a constant head by reciprocating pumps, the motor thus could carry the load at two different speeds, the two points of intersection of the 388 ELEMENTS OF ELECTRICAL ENGINEERING. horizontal line L in Fig. 216, which represents the torque con- sumed by the load and the motor torque curve D. Of these two points d and c, the lower one, d y represents unstable con- ditions of operation; that is, the motor cannot operate at this speed, but either stops or runs up to the higher speed point c, at which stability is reached. At the lower speed d a momen- tary decrease of speed, as by a small pulsation of voltage, load, Fig, 216. Speed Torque Characteristics of Induction Motor and Load for Determination of the Stability Point. etc., decreases the motor torque D below the torque L required by the load, thus causes the motor to slow down, but in doing so its torque still further decreases, and it slows down still more, loses more torque, etc., until it comes to a standstill. Inversely, a momentary increase of speed increases the motor torque D beyond the torque L consumed by the load, and thereby causes an acceleration, that is, an increase of speed. This increase of speed, however, increases the motor torque and thereby the speed still further, and so on, and the motor increases in speed up to the point c, where the motor torque D again becomes equal to the torque consumed by the load. A momentary in- INDUCTION MACHINES. 389 crease of speed beyond c decreases the motor torque D } and thus limits itself, and inversely a momentary decrease of speed below c increases the motor torque D beyond L, thus accelerates and recovers the speed ; that is, at c the motor speed is stable. With a load requiring constant torque the induction motor thus is unstable at speeds below that of the maximum torque point, but stable above it; that is, the motor curve consists of two branches, an unstable branch, from standstill t to the max- imum torque point m } and a stable branch, from the maximum torque point m to synchronism. 177. It must be realized, however, that this instability of the lower branch of the induction motor speed curve is a function of the nature of the load, and as described above applies only to a load requiring a constant torque L. Such a load the motor could not start (except by increasing the motor torque at low speeds by resistance in the secondary), but when brought up to a speed above d would carry the load at speed c in Fig. 216. If, however, the load on the motor is such as to require a torque which increases with the square of the speed, as shown by curve C in Fig. 216, that is, consists of a constant part p (friction of bearings, etc.) and a quadratic part, as when driving a ship's propeller or driving a centrifugal pump, then the induc- tion motor is stable over the entire range of speed, from standstill to synchronism. The motor then starts, with the load repre- sented by curve C, and runs up to speed c. At a higher load, represented by curve JS, the motor runs up to speed 6, and with excessive overload, curve A, the motor would run up to low speed, point a, only, but no overload of such nature would stop the motor, but merely reduce its speed, and inversely, it would always start, but at excessive overloads run at low speed only. Thus in this case no unstable branch of the motor curve exists, but it is stable over the entire range. With a load requiring a torque which increases proportionally to the speed, as shown by C in Fig. 217, that is, which consists of a constant part p and a part proportional to the speed, as when driving a direct-current generator at constant excitation, connected to a constant resistance as load as a lighting system the motor always starts, regardless of the load, provided that the constant part of the torque, p, is less than the starting torque. With moderate load C the motor runs up to 890 ELEMENTS OF ELECTRICAL ENGINEERING. a speed c near synchronism. With very heavy load A the motor starts, but runs up to a low speed only. Especially interesting is the case of an intermediary load as represented by line B in Fig. 217. B intersects the motor torque curve D in three points, &i? & 2 > &sJ ^at is, three speeds exist at which the motor gives the torque required by the load, 24 per cent, 60 per cent, and 88 per cent of synchronism. The speeds b l and 6 3 are stable, the speed 6 2 unstable. Thus, with this load the motor starts from stand- still, but does not run up to a speed near synchronism, but accel- erates only to speed b v and keeps revolving at this low speed Fig. 217. Speed Torque Characteristics of Induction Motor and Load for Determination, of the Stability Point. (and a correspondingly very large current). If, however, the load is taken off and the motor allowed to run up to synchronism or near to it, and the load then put on, the motor slows down only to speed 6 3 , and carries the load at this high speed; hence, the motor can revolve continuously at two different speeds, 6 t and & 3 , and either of these speeds is stable; that is, a momen- tary increase of speed decreases the motor torque below that required by the load, and thus limits itself, and inversely a decrease of motor speed increases its torque beyond that corresponding to the load, and thus restores the speed. At the INDUCTION MACHINES. 391 intermediary speed, & 2 , the conditions are unstable, and a momen- tary increase of speed causes the motor to accelerate up to speed & 3 , a momentary decrease of speed from 6 3 causes the motor to slow down to speed b v where it becomes stable again. In the speed range between b 2 and 6 3 the motor thus accelerates up to 6 3 , in the speed range between 6 2 and b l it slows down to b r For this character of load, the induction motor speed curve D thus has two stable branches, a lower one, from standstill t to the point n, and an upper one, from point m to synchronism, where m and n are the points of contact of the tangents from the required starting torque p on- to the motor curve D; these two stable branches are separated by the unstable branch, from n to m, on which the motor cannot operate. 178. The question of stability of motor speed thus is a func- tion not only of the motor speed curve but also of the character of the load in its relation to the motor speed curve, and if the change of motor torque with the change of speed is less than the change of the torque required by the load, the condition is stable, otherwise it is unstable; that is, it must be -^ < to give ao ao stability, where L is the torque required by the load at speed S. Occasionally on polyphase induction motors on a load as repre- sented in Fig. 217 this phenomenon is observed in the form that the motor can start the load but cannot bring it up to speed. More frequently, however, it is observed on single-phase induction motors in which the maximum torque is nearer to synchronism, with some forms of starting devices which de- crease in their effect with increasing speed and thus give motor speed characteristics of forms similar to Fig. 218. With a torque speed curve as shown in Fig. 218, even at a load requiring con- stant torque, three speed points may exist of which the middle one is unstable. In polyphase synchronous motors and con- verters, when starting by alternating current, that is, as induction machines, the phenomenon is frequently observed that the ma-, chine starts at moderate voltage, but does not run up to syn- chronism, but stops at an intermediary speed, in the neighborhood of half speed, and a considerable increase of voltage, and thereby of motor torque, is required to bring the machine beyond the dead point, or rather " dead range/' of speed and make it run up to synchronism. In this case, however, the phenomenon is 392 ELEMENTS OF ELECTRICAL ENGINEERING. complicated by the effects due to varying magnetic reluctance (magnetic locking), inductor machine effect, etc. Instability of such character as here described occurs in elec- tric circuits in many instances, of which the most typical is the electric arc in a constant-potential supply. It occurs whenever the effect produced by any cause increases the cause and there- by becomes cumulative* When dealing with energy, obviously J 3PE D nls 0,6 A Fig. 218. Speed Torque Characteristic of Single-Phase Induction Motor. the effect must always be in opposition to the cause (Lenz's Law), as result of the law of conservation of energy. When dealing with other phenomena, however, as the speed-torque relation or the volt-ampere relation, etc., instability due to the effect assisting the cause, intensifying it, and thus becoming cumulative, may exist, and frequently does exist, and causes either indefinite increase or decrease, or surging or hunting. 2. VOLTAGE REGULATION AND OUTPUT. 179. Load and speed curves of induction motors are usually calculated and plotted for constant supply voltage at the motor terminals. In practice, however, this condition usually is only approximately fulfilled, and due to the drop of voltage in the step-down transformers feeding the motor, in the secondary INDUCTION MACHINES. 393 and the primary supply lines, etc., the voltage at the motor terminals drops more or less with increase of load. Thus, if the voltage at the primary terminals of the motor transformer is constant, and such as to give the rated motor voltage at full load, at no load the voltage at the motor terminals is higher, but at overload lower by the voltage drop in the internal impedance of the transformers. If the voltage is kept constant in the center of distribution, the drop of voltage in the line adds itself to the impedance drop in the transformers, and the motor supply volt- age thus varies still more between no load and overload. With a drop of voltage in the supply circuit between the point of constant potential and the motor terminals, assuming the cir- cuit such as to give the rated motor voltage at full load, the voltage at no load and thus the exciting current is higher, the voltage at overload and thus the maximum output and maximum torque of the motor, and also the motor impedance current, that is, current consumed by the motor at standstill, and thereby the starting torque of the motor, are lower than on a constant-poten- tial supply. Hereby then the margin of overload capacity of the motor is reduced, and the characteristic constant of the motor, or the ratio of exciting current to short-circuit current, is in- creased, that is, the motor characteristic made inferior to that given at constant voltage supply, the more so the higher the voltage drop in the supply circuit. Assuming then a three-phase motor having the following constants: primary exciting admittance, Y = 0.01 0.1 j; primary self-inductive ijnpedance, Z Q = 0.1 0.3 j; secondary self-inductive impedance, Z t = 0.1 0.3 j; supply voltage, e = 110 volts, and rated output, 5000 watts per phase. Assuming this motor to be operated 1. By transformers of about 2 per cent resistance and 4 per cent reactance voltage, that is, transformers of good regulation, with constant voltage at the transformer terminals. 2. By transformers of about 2 per cent resistance and 15 per cent reactance voltage, that is, very poorly regulating transformers, at constant supply voltage at the transformer primaries. 3. With constant voltage at the generator terminals, and about 8 per cent resistance, 40 per cent reactance voltage in line and transformers between generator and motor. 394 ELEMENTS OF ELECTRICAL ENGINEERING. This gives, in complex quantities, the impedance between the motor terminals and the constant voltage supply: 1. Z 0.04 -0.08/, 2. Z = 0.04 - 0.3 j, 3. Z - 0.16 ~0.8f. It is assumed that the constant supply voltage is such as to give 110 volts at the motor terminals at full load. The load and speed curves of the motor, when operating under these conditions, that is, with the impedance Z in series between the motor terminals and the constant voltage supply e v then can be calculated from the motor characteristics at constant termi- nal voltage e as follows : At slip 5 and constant terminal voltage e Q the current in the motor is i , its power-factor p = cos 0. The effective or equiva- & lent impedance of the motor at this slip then is z = T, and, in i /j complex quantities, Z = -^ (cos dj sin 6), and the total \ impedance, including that of transformers and line, thus is Z, = Z + Z= (^cos ff + r}-] f^sin 9 + A U / U / or, in absolute values, 171 \2 71 \2 z = V PCOS 6+r) + psin 8 + x , v \ I \ I and, at the supply voltage e v the current thus is and the voltage at the motor terminals is t> / _ ^ _ ^ 6 o s 6 t ~ e r ^i If ^ is the voltage required at the motor terminals at full load, and i the current, z* the total impedance at full load, it is hence, the required constant supply voltage is . ^ On" INDUCTION MACHINES. 395 and the speed and torque curves of the motor under this con- dition then are derived from those at constant supply voltage e e ' by multiplying all voltages and currents by the factor -& , that ^o is, by the ratio of the actual terminal voltage to the full-load terminal voltage, and the torque and power by multiplying with the square of this ratio, while the power-factors and the efficiencies obviously remain unchanged. Fig. 219. Induction Motor Load Curves Corresponding to 110 Volts at Motor Terminals at 5000 Watts Load. In this manner, in the three cases assumed in the preceding, the load curves are calculated, and are plotted in Figs. 219, 220, and 221. 180. It is seen that, even with transformers of good regu- lation, Fig. 219, the maximum torque and the maximum power are appreciably reduced. The values corresponding to constant terminal voltage are shown, for the part of the curves near maxi- mum torque and maximum power, in Figs. 219, 220, and 221. OWER OUTPUT lOpO 20)0 SOW 4000 5000 6000 Fig. 220. Induction Motor Load Curves Corresponding to 110 Volts at Motor Terminals at 5000 Watts Load. CIRCUIT IMPED/ -<&& -.8J . worrs PER CENT Fig. 221. Induction Motor Load Curves Corresponding to HO Volts at Motor Terminals at 5000 Watts Load. (390) INDUCTION MACHINES. 397 1.0 0,9 O f 8 07 ION OF SYNCHRON 0}5 0|4 0, HROfJsM 3 Q f 2 O f l Fig. 222, Induction Motor Speed Torque Characteristics with Short- Circuited Secondary. Fig. 223. Induction Motor Speed Torque Characteristics with a Resistance of 0.15 Ohm in Secondary Circuit. In Figs. 222, 223, 224, and 225 are given the speed-torque curves of the motor, for constant terminal voltage, Z 0, and the three cases above discussed; in Fig. 222 for short-circuited secondaries, or running condition; in Fig. 223 for 0.15 ohm; in Fig. 224 for 0.5 ohm; and in Fig. 225 for 1.5 ohms additional re- 398 ELEMENTS OF ELECTRICAL ENGINEERING. sistance inserted in the armature. As seen, the line and trans- former impedance very appreciably lowers the torque, and especially the starting torque, which, with short-circuited arma- SLIP FRACTION OF SYNCHRONISM OJ6 0,5 OJ4 OROUE SYN. \ 4000 3000 Fig. 224. Induction Motor Speed Torque Characteristics with a Resistance of 0.5 Ohm in Secondary Circuit. Fig. 225. Induction Motor Speed Current Characteristics with a Resistance of 1.5 Ohms in Secondary Circuit. ture, in the case 3 drops to about one-third the value given at constant supply voltage. It is interesting to note that in Fig. 224, with a secondary resistance giving maximum torque in starting, at constant ter- minal voltage, with high impedance in the supply, the starting INDUCTION MACHINES. I P O o - I 3&1 Sc 11 ~ " o it OS t> O 5 1- sl^fl - CO J< (M O -l r- 1 i-< C5 CO O5 TH rH J> to eo o J> IO -* t>- sT "IM &C 1C t- O ^ i '.III S,s H O i-l rH CO . o" 1 "- XO 1C iQ O CO ^h oooo O CN O MD C<> C5 r- 1 1C 00 CO O O CO CO ^ CO rH 5 i" JQ O II T O O 1C >O IO CM CO CO O> " c * 11 < a* -3 T s&i 0000 ua o to 3 CM U3 -r|< -^ 00 t- CSI O5 CO TH . o o n ^S CO -^ O O> 5 i-H TT< O OO "Cf< T-I O> O | ^ o g fi M ^2 1! 000 o oo t> Tf< > O J>- O5 1>- XO -^ CS ,2 d d CJ . Ii 1O O OS C3 CO T-I O5 00 CO 1C CO O O"<*< OO rH O5 t>. * rH S ! o 3" l ^ g oooo ^ Si CO O CO 00 Tfl r-4 r-l r-t CO iO O O5 l> W THOOO s w a a IS r-< iO OTiT-< -rf< rH H di ' it may be called the stability coefficient of the motor. If k s is positive, an increase of i } caused by an increase of slip Sj that is, by a decrease of speed, increases the torque Z>, and thereby checks the decrease of speed, and inversely, that is, the motor is stable. If, however, k s is negative, an increase of i causes a decrease of Dj thereby a decrease of speed, and thus further increase of i and decrease of D ; that is, the motor slows down with increas- ing rapidity, or inversely, with a decrease of i, accelerates with increasing rapidity, that is, is unstable. For the motor used as illustration in the preceding, of the constants e - 110 volts; 7 - 0.01 - 0.1 j; Z = 0.1 -0.3 j, Z t = 0.1 0.3 j, the stability curve is shown, together with speed, current, and torque, in Fig. 227, as function of the output. As seen, the stability coefficient k s is very high for light load, decreases first rapidly and then slowly, until an output of 7000 watts is approached, and then rapidly drops below zero; that is, the motor becomes unstable and drops out of step, and speed, torque, and current change abruptly, as indicated by the arrows in Fig. 227. The stability coefficient k s characterizes the behavior of the motor regarding its load-carrying capacity. Obviously, if the terminal voltage of the motor is not constant, but drops with the load, as discussed in 1, a different stability coefficient results; which intersects the zero line at a different and lower torque* 183, If the induction motor is supplied with constant termi- INDUCTION MACHINES. 403 nal voltage from a generator of close inherent voltage regulation and of a size very large compared with the motor, over a supply circuit of negligible impedance, so that a sudden change of motor current cannot produce even a momentary tendency of change of the terminal voltage of the motor, the stability curve &* Fig. 227. Induction Motor Load Curves. of Fig. 227 gives the performance of tjie motor. If, however, at a change of load and thus of moW current the regulation of the supply voltage to constancy at the motor terminals requires a finite time, even if this time is very short, the maximum out- put of the motor is reduced thereby, the more so the more rapidly the motor speed can change. Assuming the voltage control at the motor terminals effected by hand regulation of the generator or the potential regulator in the circuit supplying the motor, or by any other 404 ELEMENTS OF ELECTRICAL ENGINEERING. method which is slower than the rate at which the motor speed can adjust itself to a change of load, then, even if the supply voltage at the motor terminals is kept constant, for a momen- tary Quotation of motor speed and current, the supply voltage momentarily varies, and with regard to its stability the motor corresponds not to the condition of constant supply voltage but to a supply voltage which varies with the current, hence the limit of stability is reached at a lower value of motor torque. At constant slip s the motor torque D is proportional to the square of the impressed e.m.f. e\ If by a variation of slip caused by a fluctuation of load the motor current i varies by di, if the terminal voltage e remains constant the motor torque D varies by the fraction k s - -,- , or the stability coefficient of Do/i the motor. If, however, by the variation of current di the impressed e.m.f. e of the motor varies, the motor torque D, being proportional to 6 2 , still further changes, proportional to the change e 2 , that is, by the fraction k r ^ -5 -JT = - j. , and the e Qfl/ e Oj\/ total change of motor torque resultant from a change di of the current i thus is k Q = k 8 + k r . Hence, if a momentary fluctuation of current causes a momen- tary fluctuation of voltage, the stability coefficient of the motor is changed from k s to & = k s + k r , and as k r is negative, the voltage e decreases with increase of current i, the stability coefficient of the system is reduced by the effect of voltage regu- lation of the supply, fc, and k r thus can be called the regulation coefficient of the system. r== ? z? thus represents the change of torque produced by e (fo the momentary voltage change resulting from a current change di in the system; hence, is essentially a characteristic of the supply system and its regulation, but depends upon the motor only in so far as ~ depends upon the power-factor of the load. U/b In Fig. 227 is shown the regulation coefficient kr of the supply system of the motor, at 110 volts maintained constant at the motor terminals, and an impedance Z = 0.16 0,8 j between motor terminals and supply e.m.f. As seen, the regu- INDUCTION MACHINES. 405 lation coefficient of the system drops from a maximum of about 0.03, at no load, down to about 0.01, and remains constant at this latter value, over a very wide range. The resultant stability coefficient, or stability coefficient of the system of motor and supply, = k s + k r , as shown in Fig. 227, thus drops from very high values at light load down to zero at the load at which the curves k s and k r in Fig. 227 intersect, or at 5800 kw., and there become negative; that is, the motor drops out of step, although still far below its maximum torque point, as indicated by the arrows in Fig. 227. Thus, at constant voltage maintained at the motor terminals by some regulating mechanism which is slower in its action than the retardation of a motor speed change by its mechanical momentum, the motor behaves up to 5800 watts output in exactly the same manner as if its terminals were connected directly to an unlimited source of constant voltage supply, but at this point, where the slip is only 7 per cent in the present instance, the motor suddenly drops out of step without previous warning, and comes to a standstill, while at inherently constant terminal voltage the motor would continue to operate up to 7000 watts output, and drop out of step at 8250 synchronous watts torque at 16 per cent slip. By this phenomenon the maximum torque of the motor thus is reduced from 8250 to 6300 synchronous watts, or by nearly 25 per cent. 184. If the voltage regulation of the supply system is more rapid than the speed change of the motor as retarded by the momentum of motor and load, the regulation coefficient of the system as regards to the motor obviously is zero, and the motor thus gives the normal maximum output and torque. If the regulation of the supply voltage, that is, the recovery of the terminal voltage of the motor with a change of current, occurs at about the same rate as the speed of the motor can change with a change of load, then the maximum output as limited by the stability coefficient of the system is intermediate between the minimum value of 6300 synchronous watts and its normal value of 8250 synchronous watts. The more rapid the recovery of the voltage and the larger the momentum of motor and load, the less is the motor output impaired by this phenomenon of instability. Thus, the loss of stability is greatest with hand 406 ELEMENTS OF ELECTRICAL ENGINEERING. regulation, less with automatic control by potential regulator, the more so the more rapidly the regulator works; it is very little with compounded alternators, and absent where the motor terminal voltage remains constant without any control by prac- tically unlimited generator capacity and absence of voltage drop between generator and motor. Comparing the stability coefficient k s of the motor load and the stability coefficient k of the entire system under the assumed conditions of operation of Fig. 227, it is seen that the former intersects the zero line very steeply, that is, the stability remains high until very close to the maximum torque point, and the motor thus can be loaded up close to its maximum torque without impairment of stability. The curve & , however, intersects the zero line under a sharp angle, that is, long before the limit of stability is reached in this case the stability of the system has dropped so close to zero that the motor may drop out of step by some momentary pulsation. Thus, in the case of instability due to the regulation of the system, the maximum output point, as found by test, is not definite and sharply defined, but the stabil- ity gradually decreases to zero, and during this decrease the motor drops out at some point. Experimentally the difference between the dropping out by approach to the limits of stability of the motor proper and that of the system of supply is very marked by the indefiniteness of the latter. In testing induction motors it thus is necessary to guard against this phenomenon by raising the voltage beyond normal before every increase of load, and then gradually decrease the voltages again to normal. A serious reduction of the overload capacity of the motor, due to the regulation of the system, obviously occurs only at very high impedance of the supply circuit; with moderate impe- dance the curve fo- is much lower, and the intersection between k r and k s occurs still on the steep part of k s , and the output thus is not materially decreased, but merely the stability somewhat reduced when approaching maximum output. This phenomenon of the impairment of stability of the induc- tion motor by the regulation of the supply voltage is of prac- tical importance, as similar phenomena occur in many instances. Thus, with synchronous motors and converters the regulation of the supply system exerts a similar effect on the overload INDUCTION MACHINES. 407 capacity, and reduces the maximum output so that the motor drops out of step, or starts surging, due to the approach to the stability limit of the entire system. In this case, with synchro- nous motors and converters, increase of their field excitation frequently restores their steadiness by producing leading cur- rents and thereby increasing the power-carrying capacity of the supply system, while with surging caused by instability of the synchronous motor the leading currents produced by increase of field excitation increase the surging, and lowering the field excitation tends towards steadiness. V. Induction Generator. 1. INTRODUCTION. 185. In the range of slip from s = to s = 1, that is, from synchronism to standstill, torque, power output, and power input of the induction machine are positive, and the machine thus acts as a motor, as discussed before. Substituting, however, in the equations in paragraph 1 for 5 values > 1, corresponding to backward rotation of the ma- chine, the power input remains positive, the torque also remains positive, that is, in the same direction as for s < 1, but since the speed (1 s) becomes negative or in opposite direction, the power output is negative, that is, the torque in opposite direc- tion to the speed. In this case the machine consumes electrical energy in its primary and mechanical energy by a torque oppos- ing the rotation, thus acting as brake. The total power, electrical as well as mechanical, is con- sumed by internal losses of the motor. Since, however, with large slip in a low-resistance motor, the torque and power are small, the braking power of the induction machine at backward rotation is, as a rule, not, considerable, excepting when using high resistance in the armature circuit. Substituting for 5 negative values, corresponding to a speed above synchronism, torque and power output and power input become negative, and a load curve can be plotted for the induc- tion generator which is very similar, but the negative counter- part of the induction tnotor load curve. It is for the machine shown as motor in Fig. 207, given as Fig. 228, while Fig. 229 408 ELEMENTS OF ELECTRICAL ENGINEERING. -4000 -5000 -6000 -7000 -8000 -9000 -10000 Fig. 228. Induction Generator Load Curves. .SJCKJO" ,_> \ (UP RACTION OF YNC ION OF * Ola o's 15 1.1 1. 1.8 1, I A 01 0(6. L 0. 0.3 0,2 0. Fig. 229. Induction Machine Speed Curves. INDUCTION MACHINES. 409 gives the complete speed curve of this machine from s = 1.5 to s - -1. The generator part of the curve, for s < 0, is of the same char- acter as the motor part, s > 0, but the maximum torque and maximum output of the machine as generator are greater than as motor. Thus an induction motor when speeded up above synchronism acts as a powerful brake by returning energy into the lines, and the maximum braking effort and also the maximum electric power returned by the machine will be greater than the maxi- mum motor torque or output. 2. CONSTANT-SPEED INDUCTION OR ASYNCHRONOUS GENERATOR. 186. The curves in Fig. 229 are calculated at constant fre- quency /, and thus to vary the output of the machine as gener- ator the speed has to be increased. This condition may be ELECTRICAL OUTPUT, P n WATTS 2000 3000 4000 5000 6QOO 7000 8000 90(>0 10000 Fig, 230, Induction Generator Load Curves. realized in case of induction generators running in parallel with synchronous generators under conditions where it is desirable that the former should take as much load as its driving power permits; as, for instance, if the induction generator is driven by 410 ELEMENTS OF ELECTRICAL ENGINEERING. a water power while the synchronous generator is driven by a steam engine. In this case the control of speed would be effected on the synchronous generator, and the induction gen- erator be without speed-controlling devices, running up beyond synchronous speed as much as required to consume the power supplied to it. Conversely, however, if an induction machine is driven a constant speed and connected to a suitable circuit as load, the frequency given by the machine will not be synchronous with the speed, or constant at all loads, but decreases with increasing load from practically synchronism at no load, and thus for the induction generator at constant speed a load curve can be con- structed as shown in Fig. 230, giving the decrease of frequency with increasing load in the same manner as the speed of the induction motor at constant frequency decreases with the load. In the calculation of these induction generator curves for con- stant speed the change of frequency with the load has obviously to be considered, that is, in the equations the reactance x has to be replaced by the reactance x (1 s), otherwise the equa- tions remain the same. 3. POWEK-F ACTOR OF INDUCTION GENERATOR. 187. The induction generator differs essentially from a syn- chronous alternator (that is, a machine in which an armature revolves relatively through a constant or continuous magnetic field) by having a power-factor requiring leading current; that is, in the synchronous alternator the phase relation between current and terminal voltage depends entirely upon the external circuit, and according to the nature of the circuit connected to the synchronous alternator the current can lag or lead the ter- minal voltage or be in phase therewith. In the induction or asynchronous generator, however, the current must lead the ter- minal voltage by the angle corresponding to the load and voltage of the machine, or, in other words, the phase relation between current and voltage in the external circuit must be such as required by the induction generator at that particular load. Induction generators can operate only on circuits with lead- ing current or circuits of negative effective reactance. In Fig. 231 are given for the constant-speed induction gen- INDUCTION MACHINES. 411 erator in Fig. 230 as function of the impedance of the external circuit z -^ as abscissas (where = terminal voltage, i = current in external circuit), the leading power-factor p = cos d required in the load, the inductance factor q = sin #, and the frequency. Hence, when connected to a circuit of impedance z this induc- tion generator can operate only if the power-factor of its circuit is p; and if this is the case the voltage is indefinite, that is, the Fig. 231. Three-phase Induction Generator Power Factor and Inductance Factor of External Circuit. circuit unstable, even neglecting the impossibility of securing exact equality of the power-factor of the external circuit with that of the induction generator. Two possibilities thus exist with such an induction generator circuit. 1st. The power-factor of the external circuit is constant and independent of the voltage, as when the external circuit consists of resistances, inductances, and capacities. In this case if the power-factor of the external circuit is higher than that of the induction generator, that is, the leading current less, the induction generator fails to excite and generate. If the power-factor of the external circuit is lower than that of the induction generator, the latter excites and its voltage rises until by saturation of its magnetic circuit and the consequent increase of exciting admittance, that is, decrease of internal power- factor, its power-factor has fallen to equality with that of the external circuit 412 ELEMENTS OF ELECTRICAL ENGINEERING. In this respect the induction generator acts like the direct- current shunt generator, and gives load characteristics very similar to those of the direct-current shunt generator as dis- cussed in B; that is, it becomes stable only at saturation, but loses its excitation and thus drops its load as soon as the voltage falls below saturation. Since, however, the field of the induction generator is alter- nating, it is usually not feasible to run at saturation, due to excessive hysteresis losses, except for very low frequencies. 2cL The power-factor of the external circuit depends upon the voltage impressed upon it. This, for instance, is the case if the circuit consists of a syn- chronous motor or contains synchronous motors or synchronous converters. In the synchronous motor the current is in phase with the impressed e.m.f. if the impressed e.in.f. equals the counter e.m.f. of the motor plus the internal loss of voltage. It is leading if the impressed e.m.f. is less, and lagging if the impressed e.m.f. is more. Thus when connecting an induction generator with a synchronous motor, at constant field excitation of the latter the voltage of the induction generator rises until it is as much below the counter e.m.f. of the synchronous motor as required to give the leading current corresponding to the power-factor of the generator. Thus a system consisting of a constant-speed induction generator and a synchronous motor at constant field excitation is absolutely stable. At constant field excitation of the synchronous motor, at no load the synchronous motor runs practically at synchronism with the induction generator, with a terminal voltage slightly below the counter e.m.f. of the syn- chronous motor. With increase of load the frequency and thus the speed of the synchronous motor drops, due to the slip of frequency in the induction generator, and the voltage drops, due to the increase of leading current required and the decrease of counter e.m.f. caused by the decrease of frequency. By increasing the field excitation of the synchronous motor with increase of load, obviously the voltage of the generator can be maintained constant, or even increased with the load. When running from an induction generator, a synchronous motor gives a load curve very similar to the load curve of an induction motor running from a synchronous generator; that is, INDUCTION MACHINES. 413 a magnetizing current at no load and a speed gradually decreas- ing with the increase of load up to a maximum output point, at which the speed curve bends sharply down, the current curve upward, and the motor drops out of step. Fig. 232. Induction Generator and Synclironous Motor Load Curves. The current, however, in the case of the synchronous motor operated from an induction generator is leading, while it is lag- ging in an induction motor operated from a synchronous gener- ator. ' In either case it demagnetizes the synchronous machine 414 ELEMENTS OF ELECTRICAL ENGINEERING. and magnetizes the induction machine, that is, the synchronous machine supplies magnetization to the induction machine. In Fig. 232 is shown the load curve of a synchronous motor operated from the induction generator in Fig. 230. In Fig. 233 is shown the load curve of an over-compounded synchronous converter operated from an induction generator, LOAD. CU 1 0,2 O f 3 0,4 0,5 O f 6 OJ7 QJi f /7( 13 14 15 PER CENT 100 Fig. 233. Induction Generator and Synchronous Converter Phase Control, no Line Impedance. the over-compounding being such as to give approximately constant terminal voltage e. 188. Obviously when operating a self-exciting synchronous converter from an induction generator the system is unstable also if both machines are below magnetic saturation, since in this case in both machines the generated e.m.f. is proportional to the field excitation and the field excitation proportional to the voltage; that is, with an unsaturated induction generator the synchronous converter operated therefrom must have its mag- netic field excited to a density above the bend of the saturation curve. Since the induction generator requires for its operation a cir- cuit with leading current varying with the load in the manner determined by the internal constants of the motor, to make an induction or asynchronous generator suitable for operation on a INDUCTION MACHINES. 415 general alternating-current circuit, it is necessary to have a syn- chronous machine as exciter in the circuit consuming leading current, that is, supplying the required lagging or magnetizing current to the induction generator; and in this case the voltage of the system is controlled by the field excitation of the syn- chronous machine, that is, its counter e.m.f. Either a syn- chronous motor of suitable size running light can be used herefor as exciter of the induction generator, or the exciting current of the induction generator may be derived from synchronous motors or converters in the same system, or from synchronous alter- nating-current generators operated in parallel with the induc- tion generator, in which latter case, however, these currents can be said to come from the synchronous alternator as lagging currents. Electrostatic condensers, as an underground cable system, may also be used for excitation, but in this case besides the condensers a synchronous machine is required to secure stability. The induction machine may thus be considered as consuming a lagging reactive magnetizing current at all speeds, and con- suming a power current below synchronism, as motor, supplying a power current (that is, consuming a negative power current) above synchronism, as generator. Therefore, induction generators are best suited for circuits which normally carry leading currents, as synchronous motor and synchronous converter circuits, but less suitable for cir- cuits with lagging currents, since in the latter case an additional synchronous machine is required, giving all the lagging currents of the system plus the induction generator exciting current. Obviously, when running induction generators in parallel with a synchronous alternator no synchronizing is required, but the induction generator takes a load corresponding to the excess of its speed over synchronism, or conversely, if the driving power behind the induction generator is limited, no speed regulation is required, but the induction generator runs at a speed exceed- ing synchronism by the amount required to consume the driv- ing power. The foregoing consideration obviously applies to the polyphase induction generator as well as to the single-phase induction generator, the latter, however, requiring a larger exciter in consequence of its lower power-factor. Therefore, even in 416 ELEMENTS OF ELECTRICAL ENGINEERING. a single-phase induction generator, preferably polyphase excita- tion is used, that is, the induction machine and its synch- ronous exciter wound as polyphase machines, but the load connected to one phase only of the induction machine. The curves shown in the preceding apply to the machine as poly- phase generator. The effect of resistance in the secondary is essentially the same in the induction generator as in the induction motor. An increase of resistance increases the slip, that is, requires an in- crease of speed at the same torque, current, and output, and thus correspondingly lowers the efficiency. Induction generators have been proposed and used to some extent for high-speed prime movers, as steam turbines, since their squirrel-cage rotor appears mechanically better suited for very high speeds than the revolving field of the synchronous generator. The foremost use of induction generators will probably be for collecting small water powers in one large system, due to the far greater simplicity, reliability, and cheapness of a small induction generator station feeding into a big system compared with a small synchronous generator station. The induction generator station requires only the hydraulic turbine, the induction machine, and the step-up transformer, but does not even require a turbine governor, and so needs practically no attention, as the control of voltage, speed, and frequency takes place by a synchronous generator or motor main station, which col- lects the power while the individual induction generator stations feed into the system as much power as the available water happens to supply. The synchronous induction motor, comprising a single-phase or polyphase primary and a single-phase secondary, tends to drop into synchronism and then operates essentially as reaction machine. A number of types of synchronous induction gener- ators have been devised, either with commutator for excitation or without commutator and with excitation by low-frequency synchronous or commutating machine, in the armature, or by high-frequency excitation. For particulars regarding these very interesting machines, see " Theory and Calculation of Alternating- Current Phenomena," fourth edition. INDUCTION MACHINES. 417 VI. Induction Booster* 189. In the induction machine, at a given slip s, current and terminal voltage are proportional to each other and of constant phase relation, and their ratio is a constant. Thus when con- nected in an alternating-current circuit, whether in shunt or in series, and held at a speed giving a constant and definite slip s, either positive or negative, the induction machine acts like a constant impedance. The apparent impedance and its components, the apparent resistance and apparent reactance represented by the induction machine, vary with the slip. At synchronism apparent impe- dance, resistance, and reactance are a maximum. They decrease with increasing positive slip. With increasing negative slip the apparent impedance and reactance decrease also, the apparent resistance decreases to zero and then increases again in negative direction as shown in Fig. 234, which gives the apparent impe- dance, resistance, and reactance of the machine shown in Figs. 207 and 208, etc., with the speed as abscissas. The cause is that the power current is in opposition to the terminal voltage above synchronism, and thereby the induction machine behaves as an impedance of negative resistance, that is, adding a power e.m.f. into the circuit proportional to the current. As may be seen herefrom, the induction machine when inserted in series in an alternating-current circuit can be used as a booster, that is, as an apparatus to generate and insert in the circuit an e.m.f. proportional to the current, and the amount of the boost- ing effect can be varied by varying the speed, that is, the slip at which the induction machine is revolving. Above synchronism the induction machine boosts, that is, raises the voltage; below synchronism it lowers the voltage; in either case also adding an out-of-phase e.m.f. due to its reactance. The greater the slip, either positive or negative, the less is the apparent resistance, positive or negative, of the induction machine. The effect of resistance inserted in the secondary of the induc- tion booster is similar to that in the other applications of the induction machine; that is, it increases the slip required for a certain value of apparent resistance, thereby lowering the effi- 418 ELEMENTS OF ELECTRICAL ENGINEERING. ciency of the apparatus, but at the same time making it less dependent upon minor variations of speed; that is, requires a lesser constancy of slip, and thus of speed and frequency, to give a steady boosting effect. Fig. 234, Effective Impedance of Three-Phase Induction Machine, VIL Phase Converter. 190. It may be seen from the preceding that the induction machine can operate equally well as motor, below synchronism, and as generator, above synchronism. In the single-phase induction machine the motor or generator action occurs in one primary circuit only, but in the direction in quadrature to the primary circuit there is a mere magnetizing INDUCTION MACHINES. 419 current either in the secondary, in the single-phase motor proper, or in an auxiliary field-circuit, in the monocyclic motor. The motor and generator action can occur, however, simul- taneously in the same machine, some of the primary circuits acting as motor, others as generator circuits. Thus, if one of the two circuits of a quarter-phase induction machine is con- nected to a single-phase system, in the second circuit an e.m.f. is generated in quadrature with and equal to the generated e.m.f. in the first circuit; and this e.m.f. can thus be utilized to produce currents which, with currents taken from the primary single- phase mains, give a quarter-phase system. Or, in a three-phase motor connected with two of its terminals to a single-phase system, from the third terminal an e.m.f. can be derived which, with the single-phase system feeding the induction machine, com- bines to a three-phase system. The induction machine in this application represents a phase converter. The phase converter obviously combines the features of a single-phase induction motor with those of a double transformer, transformation occurring from the primary or motor circuit to the secondary or armature, and from the secondary to the ter- tiary or generator circuit. Thus, in a quarter-phase motor connected to single-phase mains with one of its circuits, if Y = g + jb primary polyphase exciting admittance, Z Q r jx = self-inductive impedance per primary or ter- tiary circuit, ^i ^ r i ~~ i x i ~ resultant single-phase self-inductive impe- dance of secondary circuits. Let e = e.m.f. generated by the mutual flux and Z = r j x = impedance of the external circuit supplied by the phase converter as generator of second phase. We then have -, current of second phase produced by phase converter, eZ e = JZ = - =- = = terminal voltage at generator circuit of phase converter. 420 ELEMENTS OF ELECTRICAL ENGINEERING. The current in the secondary of the phase converter is then where /> / = load current ; & Y ^ exciting current of quadrature magnetic flux, es current required to revolve the machine, and the primary current is *o-/i + r> where /' = eY -= exciting current of main magnetic flux. From these currents the e.m.fs. are derived in a similar manner as in the induction motor or generator. Due to the internal losses in the phase converter, the e.m.fs. of the two circuits, the motor and generator circuits, are prac- tically In quadrature with each other and equal only at no load, but shift out of phase and become more unequal with increase of load, the unbalancing depending upon the constants of the phase converter. It is obvious that the induction machine is used as phase con- verter only to change single-phase to polyphase, since a change from one polyphase system to another polyphase system can be effected by stationary transformers. A change from single- phase to polyphase, however, requires a storage of energy, since the power arrives as single-phase pulsating, and leaves as steady polyphase flow, and the momentum of the revolving phase con- verter secondary stores and returns the energy. With increasing load on the generator circuit of the phase converter its slip increases, but less than with the same load as mechanical output from the machine as induction motor. An application of the phase converter is made in single-phase motors by closing the tertiary or generator circuit by a condenser of suitable capacity, thereby generating the exciting current of the motor in the tertiary circuit. The primary circuit is thereby relieved of the exciting current of the motor, the efficiency essentially increased, and the power- factor of the single-phase motor with condenser in tertiary cir- cuit becomes practically unity over the whole range of load. INDUCTION MACHINES. 421 At the same time, since the condenser current is derived by double transformation in the multitooth structure of the induc- tion machine, which has a practically uniform magnetic field, irrespective of the shape of the primary impressed e.m.f. wave, the application of the condenser becomes feasible irrespective of the wave shape of the generator. Usually the tertiary circuit in this case is arranged on an angle of 60 deg. with the primary circuit, and in starting a powerful torque is thereby developed, with a torque efficiency superior to any other single-phase motor starting device, and when combined with inductive reactance in a second tertiary circuit, the appar- ent starting torque efficiency can be made even to exceed that of the polyphase induction motor (see page 385). For further discussion hereof, see A. I. E. E. Transactions, 1900, p. 37. VIII. Frequency Converter or General Alternating-Current Transformer. 191. The e.m.fs. generated in the secondary of the induction machine are of the frequency of slip, that is, synchronism minus speed, thus of lower frequency than the impressed e.m.f. in the range from standstill to double synchronism; of higher frequency outside of this range. Thus, by opening the secondary circuits of the induction machine and connecting them to an external or consumer's circuit, the induction machine can be used to transform from one frequency to another, as frequency converter. It lowers the frequency with the secondary running at a speed between standstill and double synchronism, and raises the fre- quency with the secondary either driven backward or above double synchronism. Obviously, the frequency converter can at the same time change the e.m.f. by using a suitable number of primary and secondary turns, and can change the phases of the system by having a secondary wound for a different number of phases from the primary, as, for instance, convert from three-phase 6000- volts 25-cycles to quarter-phase 2500-volts 62.5-cycles. Thus, a frequency converter can be called a "general alter- nating-current transformer. 77 422 ELEMENTS OF ELECTRICAL ENGINEERING. For its theoretical discussion and calculation, see "Theory and Calculation of Alternating-Current Phenomena." The action and the equations of the general alternating-cur- rent transformer are essentially those of the stationary alter- nating-current transformer, except that the ratio of secondary to primary generated e.m.f. is not the ratio of turns but the ratio of the product of turns and frequency, while the ratio of sec- ondary current and primary load current (that is, total primary current minus primary exciting current) is the inverse ratio of turns. The ratio of the products of generated e.m.f. and current, that is, the ratio of electric power generated in the secondary, to electric power consumed in the primary (less excitation), is thus not unity but is the ratio of secondary to primary fre- quency. Hence, when lowering the frequency with the secondary revolving at a speed between standstill and synchronism, the secondary output is less than the primary input, and the differ- ence is transformed into mechanical work; that is, the machine acts at the same time as induction motor, and when used in this manner is usually connected to a synchronous or induction generator feeding preferably into the secondary circuit (to avoid double transformation of its output) or to a synchronous con- verter, which transforms the mechanical power of the frequency converter into electrical power. When raising the frequency by backward rotation, the sec- ondary output is greater than the primary input (or rather the electric power generated in the secondary greater than the primary power consumed by the generated e.m.f.), and the differ- ence is to be supplied by mechanical power by driving the fre- quency changer backward by synchronous or induction motor, preferably connected to the primary circuit, or by any other motor device. Above synchronism the ratio of secondary output to primary input becomes negative; that is, the induction machine gen- erates power in the primary as well as in the secondary, the pri- mary power at the impressed frequency, the secondary power at the frequency of slip, and thus requires mechanical driving power. The secondary power and frequency are less than the primary INDUCTION MACHINES. 423 below double synchronism, more above double synchronism, and are equal at double synchronism, so that at double syn- chronism the primary and secondary may be connected in multi- ple or in series and the machine is then a double synchronous alternator further discussed in the " Theory and Calculation of Alternating-Current Phenomena/ ' fourth edition. As far as its transformer action is concerned, the frequency converter is an open magnetic circuit transformer, that is, a transformer of relatively high magnetizing current. It com- bines therewith, however, the action of an induction motor or generator. Excluding the case of over-synchronous rotation, it is approximately (that is, neglecting internal losses) electrical input -r- electrical output ~- mechanical output ~ primary fre- quency -T- secondary frequency -~ speed or primary minus sec- ondary frequency; that is, the mechanical output is negative when increasing the frequency by backward rotation. Such frequency converters are to a certain extent in com- mercial use, and have the advantage over the motor-generator plant of requiring an amount of apparatus equal only to the out- put, while the motor-generator set requires machinery equal to twice the output. An application of the frequency converter when lowering the frequency is made in concatenation or tandem control of induc- tion machines, as described in the next section. In this case the first motor, or all the motors except the last of the series are in reality frequency converters. IX. Concatenation of Induction Motors. 192. In the secondary of the induction motor an e.m.f. is generated of the frequency of slip. Thus connecting the sec- ondary circuit of the induction motor to the primary of a second induction motor, the latter is fed by a frequency equal to the slip of the first motor, and reaches its synchronism at the frequency of slip of the first motor, the first motor then acting as fre- quency converter for the second motor. If, then, two equal induction motors are rigidly connected together and thus caused to revolve at the same speed, the speed of the second motor, which is the slip s of the first motor at no load, equals the speed of the first motor: 5 = 1 5, and thus 424 ELEMENTS OF ELECTRICAL ENGINEERING. s = 0.5, That is, a pair of induction motors connected this way in tandem or in concatenation, that is, "chain connection, " as commonly called, or in cascade, as called abroad, tends to approach s = 0.5, or half synchronism, at no load, slipping below this speed under load; that is, concatenation of two motors reduces their synchronous speed to one-half, and thus offer as means to operate induction motors at one-half speed. In general, if a number of induction machines are connected in tandem, that is, the secondary of each motor feeding the primary of the next motor, the secondary of this last motor being short-circuited, the sum of the speeds of all motors tends towards synchronism, and with all motors connected together so as to revolve at the same speed the system operates at - synchronous speed, when n = number of motors. If the two induction motors on the same shaft have a different number of poles, they synchronize at some other speed below synchronism, or if con- nected differentially, they synchronize at some speed above syn- chronism. Assuming the ratio of turns of primary and secondary as 1:1, with two equal induction motors in concatenation at standstill, the frequency and the e.m.f. impressed upon the second motor, neglecting the drop of e.m.f. in the internal impedance of the first motor, equal those of the first motor. With increasing speed, the frequency and the e.m.f. impressed upon the second motor decrease proportionally to each other, and thus the mag- netic flux and the magnetic density in the second motor, and its exciting current, remain constant and equal to those of the first motor, neglecting internal losses; that is, when connected in concatenation the magnetic density, current input, etc., and thus the torque developed by the second motor, are approxi- mately equal to those of the first motor, being less because of the internal losses in the first motor. Hence, the motors in concatenation share the work in approx- imately equal portions, and the second motor utilizes the power which without the use of a second motor at less than one-half synchronous speed would have to be wasted in the secondary resistance; that is, theoretically concatenation doubles the tor- que and output for a given current, or power input into the motor system. In reality the gain is somewhat less, due to the INDUCTION MACHINES. 425 second motor not being quite equal to a non-inductive resistance for the secondary of the first motor, and due to the drop of volt- age in the internal impedance of the first motor, etc. At one-half synchronism, that is, the limiting speed of the con- catenated couple, the current input in the first motor equals its exciting current plus the transformed exciting current of the second motor, that is, equals twice the exciting current. 193. Hence, comparing the concatenated couple with a single motor, the primary exciting admittance is doubled. The total impedance, primary plus secondary, is that of both motors, that is, doubled also, and the characteristic constant of the con- catenated couple is thus four times that of a single motor, but the speed reduced to one-half. Comparing the concatenated couple with a single motor re- wound for twice the number of poles, that is, one-half speed also, such rewinding does not change the self-inductive impe- dance, but quadruples the exciting admittance, since one-half as many turns per pole have to produce the same flux in one-half the pole arc, that is, with twice the density. Thus the character- istic constant is increased fourfold also. It follows herefrom that the characteristic constant of the concatenated couple is that of one motor rewound for twice the number of poles. The slip under load, however, is less in the concatenated couple than in the motor with twice the number of poles, being due to only one-quarter the internal impedance, the secondary impedance of the second motor only, and thus the efficiency is slightly higher. Two motors coupled in concatenation are in the range from standstill to one-half synchronism approximately equivalent to one motor of twice the admittance, three times the primary impedance, and the same secondary impedance as each of the two motors, or more nearly 2.8 times the primary and 1.2 times the secondary impedance of one motor. Such a motor is called the equivalent motor. 194. The calculation of the characteristic curve of the concat- enated motor system is similar to, but more complex than, that of the single motor. Starting from the generated e.m.f. e of the second motor, reduced to full frequency, we work up to the im- pressed e.m.f. of the first motor e , by taking due consideration of the proper frequencies of the different circuits, Herefor the 426 ELEMENTS OF ELECTRICAL ENGINEERING. 235. Comparison of Concatenated Motors with a Single Motor of Double the Number of Poles. reader must be referred to "Theory and Calculation of Alternat- ing-Current Phenomena, " fourth edition. The load curves of the pair of three-phase motors of the same constants as the motor in Figs. 207 and 208 are given in Fig. 235, the complete speed curve in Fig. 236. Fig. 235 shows the load curve of the total couple, of the two individual motors, and of the equivalent motor. INDUCTION MACHINES. 427 As seen from the speed curve, the torque from standstill to one-half synchronism has the same shape as the torque curve of a single motor between standstill and synchronism. At one-half synchronism the torque reverses and becomes negative. It reverses again at about two-thirds synchronism, and is positive between about two-thirds synchronism and synchronism, zero at synchronism, and negative beyond synchronism. Thus, with a concatenated couple, two ranges of positive torque and power as induction motor exist, one from standstill -h 1.4 1.3 Fig. 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0,5 0.4 0.3 0.2 0.1 -0.1 -0.2 -0.3 -0.4 -0.5-0.6-0.7 236. Concatenation of Induction Motors, Speed Curves. to half synchronism, the other from about two-thirds synchro- nism to synchronism. In the ranges from one-half synchronism to about two-thirds synchronism, and beyond synchronism, the torque is negative, that is, the couple acts as generator. The insertion of resistance in the secondary of the second motor has in the range from standstill to half synchronism the same effect as in a single induction motor, that is, shifts the max- imum torque point towards lower speed without changing its value. Beyond half synchronism, however, resistance in the secondary lengthens the generator part of the curve, and makes the second motor part of the curve more or less disappear, as 428 ELEMENTS OF ELECTRICAL ENGINEERING. seen In Fig. 236, which gives the speed curves of the same motor as Fig. 235, with resistance in circuit in the secondary of the second motor. The main advantages of concatenation are obviously the abil- ity of operating at two different speeds, the increased torque and power efficiency below half speed, and the generator or braking -8000 6000 - % ^^ _i \ 2 rvo .1-0.3 3 Y = 0.01 + >.1j s P... \ z Q ** -*.^ \ <0 \\ o z > ___ -2000 v X*-' 4000 v "/ / -6000 1.0 9 8 7 6 \ 5 0, _/ 4 3 I 1 0.( Fig. 237. Concatenation of Induction Motors" Speed Curve with Resistance in the Secondary Circuit. action between half speed and synchronism, and such concatena- tion is therefore used to some extent in three-phase railway motor equipments, while for stationary motors usually a change of the number of poles by reconnecting the primary winding through a suitable switch is preferred where several speeds are desired, as it requires only one motor. X. Synchronizing Induction Motors. 195 Occasionally two or more induction motors are operated in parallel on the same load, as for instance in three-phase railroading, or when securing several speeds by concatenation. In this case the secondaries of the induction motors may be connected in multiple and a single rheostat used for starting and speed control. Thus, when using two motors in con- catenation for speeds from standstill to half synchronism, from half synchronism to full speed, the motors may also be operated on a single rheostat by connecting their secondaries in parallel As in parallel connection the frequency of the secondaries must be the same, and the secondary frequency equals the slip, it follows that the motors in this case must INDUCTION MACHINES. 429 operate at the same slip, that is, at the same frequency of rota- tion, or in synchronism with each other. If the connection of the induction motors to the load is such that they cannot operate in exact step with each other, obviously separate resis- tances must be used in the motor secondaries, so as to allow different slips. When rigidly connecting the two rnotore with each other, it is essential to take care that the motor second- aries have exactly the same relative position to their primaries so as to be in phase with each other, just as would be necessary when operating two alternators in parallel with each other when rigidly connected to the same shaft or when driven by synchronous motors from the same supply. As in the induction motor secondary an e.m.f. of definite frequency, that of slip, is generated by its rotation through the revolving motor field, the induction motor secondary is an alternating-current generator, which is short-circuited at speed and loaded by the starting rheostat during acceleration, and the problem of operating two induction motors with their secondaries connected in parallel on the same external resistance is thus the same as that of oper- ating two alternators in parallel. In general, therefore, it is undesirable to rigidly connect induction motor secondaries me- chanically if they are electrically connected in parallel, but it is preferable to have their mechanical connection sufficiently flex- ible, as by belting etc., so that the motors can drop into exact step with each other and maintain step by their synchronizing power. It is of interest, then, to examine the synchronizing power of two induction motors which are connected in multiple with their secondaries on the same rheostat and operated from the same primary impressed e.m.f. Assume two equal induction motors with their primaries connected to the same voltage supply and with their second- aries connected in multiple with each other to a common resistance r, and neglecting for simplicity the -exciting current and the voltage drop in the impedance of the motor primaries as not materially affecting the synchronizing power. Let Z^ r l jx l = secondary self-inductive impedance at full frequency; s = slip of the two motors, as fraction of syn- chronism; e = absolute value of impressed e.m.f. and thus, when neglecting the primary impedance, of the e.m.f. generated in the primary by the rotating field. 430 ELEMENTS OF ELECTRICAL ENGINEERING. If then the two motor-secondaries are out of phase with each other by angle 2 r, and the secondary of the motor 1 is behind in the direction of rotation and the secondary of the motor 2 ahead of the average position by angle r, then E 1 = se Q (cos r j sin r) = secondary generated e.m.f. of the first motor, (1) E 2 = se Q (cos r + j sin r) = secondary generated e.m.f. of the second motor. (2) And if /! = current corning from the first, 7 2 = current coming from the second motor secondary, the total current, or current in the external resistance, r, is / - /i + /; (3) it is then, in the circuit comprising the first motor secondary and the rheostat r, ^-LZ-Ir-Q, (4) in the circuit comprising the second motor secondary and the rheostat r, fr-l^-Ir-O, (5) where Z = r 1 jsx^, substituting (3) into (4) and (5) and rearranging gives E, - I, (Z + r) - If = 0, Ei-I.f-htf + ) -0. These two equations added and subtracted give & + &-(/,+ / 2 )(Z + 2r) =0, hence, and / _ / = EI "" 4i 42 z Substituting for convenience and abbreviation, _L- - Y = g + jb, it + z r 1 -F-, z ~^ (6) (7) INDUCTION MACHINES. 481 into equation (6) and substituting (1) and (2) into (6), gives 7j + / 2 = 2se 7 cos T, 1 1 ~ /2 = -Sjse^sinr; (8) hence, /2 1 = se 1 Y cos T jY l sin r| (9) is the current in the secondary circuit of the motor, and there- fore also the primary load current, that is, the primary current corresponding to the secondary current, and thus, when neg- lecting the exciting current, also the primary motor current, where the upper sign corresponds to the first, or lagging, the lower sign to the second, or leading, motor. Substituting in (9) for Y and Y l gives 1.2 = se o {(9 cos T b l sin T) + j (6 cos r =F g l sin T)}, (10) the primary e.m.f. corresponding hereto is Ef = e Q {cos r T j sin T}, (11) where again the upper sign corresponds to the first, the lower to the second motor. The power consumed by the current // with the e.m.f. EJ- is the sum of the products of the horizontal components, and of the vertical components, that is, of the real components and of the imaginary components of these two quantities (as a horizontal component of one does not represent any power with a vertical component of the other quantity, being in quadrature therewith). p i _ I 7? l T 1 I ^2 l#2 *2 I? where the brackets denote that the sum of the product of the corresponding parts of the two quantities is taken. As discussed in the preceding, the torque of an induction motor, in synchronous watts, equals the power consumed by the primary counter e.m.f.; that is, *v - p, 1 , and substituting (10) and (11) this gives D 2 l = se* [cos T (g cos r b 1 sin r) T sin r (b cos T T g sin T ) } (12) and herefrom follows the motor output or power, by multiplying with (1 - s). 432 ELEMENTS OF ELECTRICAL ENGINEERING. The sum of the torques of both motors, or the total torque, is 2 A = D l + D 2 == se 2 1 (g l + g) (<7 t g) cos 2 r}. (13) The difference of the torque of both motors, or the synchroniz- ing torque, is 2D S = se* (b, - 6) sin 2 r, (14) where, by (7), H m, (15) r* + s\\ m - (r x + 2 r) 2 + In these equations primary exciting current and primary impedance are neglected. The primary impedance can be intro- duced in the equations, by substituting (r t + sr ) for r v and (#! + X Q } for x v in the expression of m l and m, and in this case only the exciting current is neglected, and the results are suf- ficiently accurate for most purposes, except for values of speed very close to synchronism, where the motor current is appreci- ably increased by the exciting current. It is then m ( TI + sr o + 2 r) 2 + s 2 (x, all the other equations remain the same. From (15) and (16) follows hence, is always positive. rg6. (6j &) is always positive, that is, the synchronizing torque is positive in the first or lagging motor, and negative in the second or loading motor; that is, the motor which lags in position behind gives more power and thus accelerates, while the motor which is ahead in position gives less power and thus drops back. Hence, the two motor armatures pull each other into step, if thrown together out of phase, just like two alternators. The synchronizing torque (14) is zero if T = 0, as obvious, as for r == both motors are in step with each other. The syn- chronizing torque also is zero if r = 90 deg., that is, the two motor armatures are in opposition. The position of opposition INDUCTION MACHINES. 433 is unstable, however, and the motors cannot operate in opposition, that is, for r = 90 deg., or with the one motor secondary short- circuiting the other; in this position, any decrease of T below 90 deg. produces a synchronizing torque which pulls the motors together, to r = 0, or in step. Just as with alternators, there thus exist two positions of zero synchronizing power, with the motors in step, that is, their secondaries in parallel and in phase, and with the motors in opposition, that is, their secondaries in opposition, 5 and the former position is stable, the latter unstable, and the motors thus drop into and retain the former position, that is, operate in step with each other, within the limits of their synchronizing power. If the starting rheostat is short-circuited, or r = 0, it is, by (15), &!= 6, and the synchronizing power vanishes, as is obvious, since in this case the motor secondaries are short-circuited and thus independent of each other in their frequency and speed. With parallel connection of induction motor armatures a syn- chronizing power thus is exerted between the motors as long as any appreciable resistance exists in the external circuit, and the motors thus tend to keep in step until the common starting resistance is short-circuited and the motors thereby become inde- pendent, the synchronizing torque vanishes, and the motors can slip against each other without interference by cross currents. Since the term - 1 - contains the slip, s, as factor, the syn- 2 chronizing torque decreases with increasing approach to syn- chronous speed. For T = 0, or with the motors in step with each other, it is, by (12), (15), and (16), i * _ se 2 (r t + 2 r) __ . that is, the same value as found for a single motor in paragraph 161. (As the resistance r is common to both motors, for each motor it enters as 2 r.) For T = 90 deg., or the unstable positions of the motors, it is that is, the same value as the motor would give with short- 434 ELEMENTS OF ELECTRICAL ENGINEERING. circuited armature. This is to be expected, as the two motor armatures short-circuit each other. The synchronizing torque is a maximum for r = 45 deg., and is, by (14), (15), and (16), D.-<^f^- (20) As instances are shown, in Fig. 238, the motor torque, from equation (18), and the maximum synchronizing torque, from srt ED 0.1 0,3 0,3 0,4 0,5 0,6 0,7 0,3 0.9 1.0 Fig. 238. Synchronizing Induction Motor, Motor Torque and Synchronizing Torque. equation (20), for a motor of 5 per cent drop of speed at full load and very high overload capacity (a maximum power nearly 2.5 times and a maximum torque somewhat over three timej the rated value), that is, of low reactance, as can be produced at low frequency, and is desirable for intermittent service, hence of the constants Y = 0.005 + 0.02 j, e = 1000 volts, for the values of additional resistance inserted into the armatures, r = 0; 0.75; 2; 4.5, INDUCTION AIACHINES. 435 giving the values 1 1 +2r m t = (1 + s) 2 4 s 2 , m = (1 + $ + 2 r) 2 + 4 s 2 . As seen, in this instance the synchronizing torque is higher than the motor torque up to half speed, slightly below the motor torque between half speed and three-quarters speed, but above three-quarters speed rapidly drops, due to the approach to syn- chronism, and becomes zero when the last starting resistance is cut out. XI. Self-Exciting Induction Machines. 197. In addition to the short-circuited secondary winding, in which by the rotating primary magnet field the secondary cur- rents of the frequency of slip are produced, which do the work of the induction machine, the secondary member may be supplied with a closed coil winding connected to a commutator and brushes in the same manner as in continuous-current or comnm- tating machines, except that with the three-phase system three sets of brushes displaced from each other by 120 deg., in a quarter-phase system four sets of brushes displaced by 90 deg. in position, are used. Supplying these brushes with currents of the impressed fre- quency, either directly from the primary impressed e.m.f. or by step-down transformer or compensator, these full-frequency currents are commutated so as to give in the motor armature a resultant polarization of the frequency of slip, that is, of constant relative position to the system of revolving primary magnet poles. The intensity of this secondary polarization depends upon* the voltage impressed upon the commutator, its relative position with the primary magnet field upon the position of the brushes. Setting the brushes so that the secondary polarization is in line with the primary magnet field it supplies excitation for the latter so that the primary magnetizing current is decreased, and by increasing the commutated current can be made to dis- appear or even to reverse, that is, the primary current of the 436 ELEMENTS OF ELECTRICAL ENGINEERING. induction machine becomes leading. Hence, by varying the impressed e.m.f. at the commutator, the primary current can be changed from lag to lead, and gives a V-shaped characteristic curve similar to that of the synchronous motor in Fig. 66, with exciting e.m.f. as abscissas and primary current as ordinates. By shifting the brushes from the position of coincidence of secondary polarization with the primary magnet field, the sec- ondary polarization can be resolved into two components, one in phase with the primary field giving the excitation of the machine, and one in quadrature therewith, which produces or consumes power in the machine in the same manner as in any commutat- ing machine, and thereby is of no further interest. Instead of using two secondary windings, one winding can be used connected to a commutator and shunted by resistance con- nected between the commutator segments. In this case, which, however, has the disadvantage over the double winding that the commutated circuit is. of very low voltage, one and the same winding fulfills the function of exciting winding by carrying the impressed commutated exciting current and of energy wind- ing by carrying the secondary current corresponding to the work of the machine. The advantage of this method of excitation by commutator on the secondary is that the exciting current of the induction machine is in a circuit coincident with the short-circuited second- ary, and its self-inductance is thereby reduced from the open- circuit impedance of the machine { J to the short-circuited impedance (2,), and the volt-amperes excitation reduced in the same proportion, and the power-factor correspondingly improved, and by over-excitation leading currents can be produced. The disadvantage, however, is the addition of commutator and brushes, and thereby the loss of the main advantage offered by the induction machine over other forms of machines. While the currents supplied by the commutator to the excit- ing winding give the same resultant m.m.f. as currents of the frequency of slip, they are not low-frequency currents, but have a shape of the character shown in Fig. 239, which represents the fcornmutatedj exciting current of a three-phase motor at five per cent slip. The self-inductance of these exciting currents, therefore ; is not the self-inductance corresponding to the low INDUCTION MACHINES. 437 frequency of slip, but that of full frequency; that is, the self- inductance is the same whether the current is supplied directly by permanent connections or through a commutator. The advantage resulting from such an excitation of the sec- ondary member by commutator is not due to the lowering of the self-inductance of excitation to the frequency of slip, but due to the lowering of the self-inductance by mutual inductance, Fig. 239. Self -Excited Induction Machine, Current Supplied by Commutator to the Exciting Winding. resulting from the coincidence of the exciting winding with the short-circuited winding, and the existence of a short-circuited secondary winding or its equivalent is therefore essential. With- out it the machine is a mere alternating commutator machine. Regarding the theoretical investigation of the self-excitation of induction machines, see "Theory and Calculation of Alter- nating-Current Phenomena/ 7 fourth edition. INDEX PAGE Accelerating coil, starting single-phase induction motor 282 Acceleration of single-phase induction motor starting device 385 Acyclic generator 11 machines 124 Adjustable speed polyphase induction motor 267 Admittance , 107 and impedance 1 05 polyphase induction motor 357 of single-phase induction motor "... 376 Algebraic method of transformer calculation 78 Alternating-current commutating machine 219 commutating machine, definition 122 generator, formula 16 generator, also see Synchronous machine. starting of converter 329 e.m.f 14 Alternation e.m.f. of single-phase commutator motor 237 Alternator field inductance 24 magnetic flux calculation * 8 Ampere, definition 2 -turn as magnetomotive force 2 Angle of hysteretic lead 54 of time lag 36 Apparent efficiency of motor 360 impedance of induction machine. 417 power efficiency of motor 360 power of reactive coil 59 reactance 56 resistance of induction machine 417 torque efficiency of motor 360 Armature current of converter , . 279 of direct-current converter . 340 of self-exciting induction motor . 437 of single-phase Induction motor 374 of variable ratio converter 321 reactance of alternating commutating machine . 222 reaction of commutating machine 181, 192 of converter , 292 of double-current generator 334 of reactive converter currents ..,...,. 296 439 440 INDEX PAGE Armature reaction of synchronous machine 129 of variable ratio converter 303,315 resistance of concatenated motor 427 in induction generator 416 of polyphase induction motor 368 windings 168 Asynchronous machine, see Induction machine. Auto-transformer 123 of direct-current converter 337 Auxiliary flux of single-phase induction motor 378 Average e.m.f v 12 Balancing by polyphase synchronous machine 151 Bipolar machine 167 Booster, definition 122 induction 417 synchronous 125 with three-wire generator 349 Braking action of induction generator 355 of induction machine ^409 Brush shift in commutating machine 182 Calculation of polyphase induction motor 357 of transformer 75 of transmission line 67 Capacity 59 in secondary of polyphase induction motor 372 of transmission line 61 Carbon brush, commutation 201 contact resistance 239 Cascade connection of induction motor 424 Cast-iron magnetization curve 9 Chain connection of induction motors 424 Characteristic constant of concatenated motor 425 of induction motor 367 of single-phase induction motor 376 curves of alternator 138 of polyphase induction motor 363 of synchronous motor 143 Characteristic of transmission line 91 Charge of condenser 59 Charging current of condenser 60 Circuit, electric 2 magnetic 2 Closed-circuit windings 171 Coefficient of armature reaction 207 Coiled conductor magnetic field 10 INDEX 441 PAGE Collection of water powers by induction generator 416 Combination of sine-waves in polar coordinates 46 Commutating field 201 of single-phase motor 245, 248 flux of single-phase commutator motor 233 machine, alternating current 219 definition 121 direct current 166 poles 185 of single-phase motor 247 Commutation 198 of converter, and frequency 332 of single-phase motor 336 of variable ratio converter 303, 315 Commutator induction machine 435 leads of single-phase motors 239 motors, single-phase, types 230 Compensated repulsion motor 232, 268 series motor 225, 260 commutation 248 Compensating winding of alternating commutating machine 224, 226 Compensation for phase displacement by commutator motor 257 Compensator 122, 123 of direct-current converter 337 with three-wire converter 349 of three-wire generator 347 Complete diagram of synchronous machine 135 Compound generator , 213 Compounding of commutating pole 188 of converters 297 curves of alternator 138 of commutating machine 195 of synchronous motor 143 of transmission line for constant voltage 97 Compound machines , 166 motor 217 Concatenation of induction motors 423 Condensive reactance 61 Condenser 59, 124 induction motor 420 starting of single-phase induction motor 385 Conductance 107 Conducti vely compensated series motor. , , 230, 261 Constant-current regulation in commutating machine 218 frequency induction motor 409 speed induction generator , 409 voltage compounding of transmission line 97 442 INDEX Conversion ratios of synchronous converter 277 Converter 270 definition 121 heating, general equations 324 induction frequency 421 load curves on synchronous generator 413 and motor generator efficiency 270 phase 418 phase control on induction generator 414 variable ratio 299 Copper brush, commutation 201 contact resistance 239 Core loss of transformer 81 Count er-e.m.f. of inductance 34 of resistance 35 Crank diagram in polar coordinates 47 Cross currents between alternators , 153, 158 Cumulative compounding of commutating machines 166 of motors 217 pulsation of synchronous machine . 156 Current In armature of converter 279 3 321 characteristics of single-phase commutator motor 253 electric 10 Current per phase of converter 274 ratios of converter 271 Demagnetizing armature reaction of commutating machine 193 Demagnetization curve of commutating machine 208 of synchronous machine by armature current 130 Diamagnetic material 5 Dielectric hysteresis 61 Differential compounding of commutating machine 166 of motors 217 Difference between crank diagram and polar diagram 49 Direct-current commutating machine 166 definition 121 converters 337 definition 122 generator, formula 14 starting of converter 328 Distorted wave 1 14 in the time diagram 48 Distorting armature reaction of commutating machine 193 Distortion of field by armature reaction of commutating machine 181 Distribution of magnetic flux of commutating machine 179 Double-current generator 122, 333 reentrant winding , 172 INDEX 443 PAGE Double spiral winding 171 Drop of voltage in transmission line 66 Drum winding of commutating machine 167 Dynamic condensers 146 Dynamotors, definition 121 Earth magnetic field 14 Eddy currents 57 in pole faces, by armature slots 191 Effective reactance 56, 106 resistance 53, 106 value 16 Efficiency of commutating machine 198 of motor 360 of single-phase commutator motor 269 of synchronous machine 149 of transmission line 43 Electric circuit 2 Electrolytic apparatus, definition 121 Electromagnet 11 E.m.f. of alternation of single-phase commutator motor 237 commutation of single-phase motor 244 consumed by inductance 34 consumed by resistance 35 diagram of synchronous machine 132 of direct-current machine 178 generation 12 ratios of converter : 271 of rotation of single-phase commutator motor 237 of synchronous machine 125 Electrostatic apparatus, definition 121 unit ' 61 Energy stored as magnetism 29 Equalization of load between alternators 158 Equivalent motor of concatenated couple 425 sine-wave 1 14 Excitation of induction generator 411 of self-exciting induction motor 436 Exciting admittance, polyphase induction motor 357 single-phase induction motor 376 current of magnetic circuit 54 of transformer 74 wave of transformer 119 Exciters of induction generators 125 External characteristic of commutating machine 210 Farad 6 Field characteristic of alternator 139 444 INDEX PAGE Field characteristic of commutating machine 197 distortion of double-current generator 334 Intensity 3,5 winding of alternating commutating machine 226 Fluctuating cross currents in parallel operation 155 voltage, local control 104 Force of magnet pole 1 Form factor, calculation 19 of synchronous machine 125 of wave 17 Foucault currents 57 Four-phase, also see Quarter-phase and Two-phase. converter 217 Fractional pitch armature winding of single-phase motor 228 winding 176 Frequency of commutation 200 converter 123, 421 of converters 332 pulsation with induction motor 400 and speed ratio of induction frequency converter 422 Friction, molecular magnetic 54 Full-pitch winding 175 Gas engine driven alternators 157 Generated e.m.f 12 of direct-current machine 178 of synchronous machine 127 Generator. . * 120 induction 407 saturation curve 207 Graphite brushes, commutation 201 Grounding the neutral of three-phase alternators 160 Harmonics, elimination by fractional pitch 127 of synchronous machine waves 127 of variable-ratio converter 313 Heating of converter armature 279, 283 of direct-current converter armature 344 of synchronous converter 289 general equations 324 of variable-ratio converter 317 Henry 23 High-frequency cross currents between synchronous machines 159 Horizontal component in polar coordinates 46 Hysteresis 53 current 54 law , 56 Hysteretic resistance 57 INDEX 445 PAGE Impedance 106 and admittance 105 apparent, of induction machine 417 of circuit , 36 in polar coordinates 44 polyphase induction motor 357 of single-phase induction motor 376 synchronous, of synchronous machine 129 of transmission line 62 Induced e.m.f., also see Generated e.m.f. Inductance 21 in alternating-current circuit 33 in continuous-current circuit 25 factor of induction generator 411 Induction apparatus, stationary, definition 121 booster 417 of e.m.f., also see Generation of e.m.f. frequency converter 421 generator 407 load curve on synchronous motor 413 operating synchronous motor 412 industrial use 416 phase control by converter 414 magnetic 5 machine 352 definition 121 self-excitation 435 motor concatenation 423 as shunt motor 353 starting of converter 328 synchronous operation 428 as transformer 354 phase converter 418 Inductive devices starting single-phase induction motor 382 load on transmission line , 93 Inductively compensated series motor 230, 234, 261 Inductive reactance of circuit. 34 Infinite phase converter, heating 289 Instability of induction motor 389 of single-phase induction motor speed curves. 392 Instantaneous e.m.f 13 value in polar coordinates 45 Intensity of magnetic field 1 of terrestrial magnetic field 14 of wave in polar coordinates ..,,.. 44 of wave in symbolic representation 86 Interlinkage of electric and magnetic circuit 2, 21 446 INDEX PAGE Interpoles of commutating machines 186 Inverted converter 122, 330 repulsion motor 230, 234, 262 commutation 248 Iron, hysteresis 56 magnetization curve 9 Lag of current 36 Lagging current by converter 298 in converter armature 285 of induction generator 415 e.m.f . and leading current 258 Lap winding 172 Lead, hysteretic 54 Leading current by converter 298 in converter armature 285 of induction generator 415 of self-excited induction motor 436 Leakage flux of magnetic circuit 22 reactance of transformer 72 Lentz's law 10 Limited speed alternating commutator motor 266 Lines of magnetic force 1 Load affecting magnetic flux distribution 180 characteristic of magneto machine , 209 of separately excited commutating machine 209 of series generator 213 of shunt generator 210 of synchronous motor 149 of transmission line 91 curves of alternator 139 of induction generator 408 of polyphase induction motor 363 of single-phase induction motor 376 of synchronous motor on induction generator 413 division in parallel operation of alternators 154 losses in commutating machine 198 saturation curve 207 of commutating machine 195 of synchronous machine 147 and stability of induction motor 387, 390 Local circuit controlled by synchronous compensator , 105 voltage control 104 Losses in commutating machine 198 in single-phase commutator motor , 259 in synchronous machines 149 Low-frequency generator 219 INDEX 447 PAGE Magnet pole unit 1 Magnetic characteristic of commutating machine 194 of synchronous machine 146 circuit , 2 distribution of variable ratio converter 317 field 1 of conductor 4 of electric current 2 of induction motor 354 of spiral 4 flux 1 of direct-current machine 179 induction 5 material 5 pole strength 1 power current 64 reaction , 10 Magnetization curves of iron 9, of synchronous machine by armature current, . . . 130 Magnetizing armature reaction of commutating machine 193 current 54 force 3, 5 Magneto generator 208 machines. 166 Magnetomotive force 5 diagram of synchronous machine 130 of electric current 2 Main flux of single-phase commutator motor 233 of single-phase induction motor * 378 Maximum e.m.f 13 power of transmission line 92 Megaline of magnetic force 12 Meridian, magnetic 6 Microfarad 60 Milhenry. 23 Molecular magnetic friction 54 Momentary short-circuit current of alternator 161 Monocyclic induction motor field 355 starting device, single-phase induction motor 382 Motor 120 converter , 336 on phase control 336 generator, definition 121 saturation curve 208 synchronous, also see Synchronous machine. Multiple drum winding ........ 168 reentrant winding 172 448 INDEX Multiple ring winding 168 spiral winding 172 winding 177 Multipolar machine 167 Mutual inductance 21 between lines 24 Negative resistance of induction machine 417 slip of induction machine 407 Neutral current in direct-current converter 338 in three-phase alternator 160 in three-wire generator and converter 346 Neutralizing component of single-phase motor commutating field 248 Neutral point of commutating machine 193 Nominal generated e.m.f. of synchronous machine 127 Non-inductive load on transmission line 91 Nonpolar generator 11 N-phase converter 275 variable-ratio converter, heating 323 Numerical calculation of transformer 75 of transformer by symbolic method 87 values in polar diagram 46 Ohm, definition 10 Open-circuit winding 171 coil arc machine 123 Oscillating armature reaction of converter 295, 296 Output of converter 285 of direct-current converter 345 of synchronous converter 289 and voltage regulation of induction motor. 392 Over-compensation of alternating commutator motor 225 Over-compounding curves of commutating machine 197 of transmission line 100 Overload capacity of converter 297 Parallelogram of sine waves in polar coordinates 46 Parallel operation and reactance 158 and regulation of alternator 153 of synchronous machine 152 Permeability 6 Phase characteristic of synchronous motor 145 compensator 125 control by commutator motor 257 by converter 298 by converter on induction generator 414 by inverted converter 331 INDEX 449 PAGE Phase control of transmission line 96 converter 122, 418 splitting devices starting single-phase induction motor 381 Phase of wave in polar coordinates 44 in symbolic representation 86 Polar circle of sine wave 44 coordinates 43 diagram of transformer. 75 Polarization cells 124 Pole-face losses by slots 191 strength, magnetic 1 Polygon of sine waves in polar coordinates 46 Polyphase converter on unbalanced circuit 296 field of single-phase induction motor 373 induction motor 355, 356 synchronous machine, unbalancing 150 Positive rotation in polar coordinates 47 Potential regulators .- 123 Power 16 of alternating current 41 component of current 42 of wave 107 cross currents between alternators 153, 158 efficiency of motor 360 e.m.f 41 factor 59 of alternating current commutating machine 220, 226 compensation by alternating commutator motor 268 control of single-phase commutator motor 255 of induction generator 410 of self-exciting induction motor 436 of single-phase commutator motor 254 transfer between alternators 158 of transmission line 43 Primary exciting admittance of polyphase induction motor 357 of transformer 74 Pulsating field current of short-circuited alternator 162 Pulsation of magnetic flux by armature slot , 191 of synchronous machine 155 Quadrature flux of single-phase commutator motor 233 of single-phase induction motor 378 Quarter-phase, also see Two-phase and Four-phase. alternator, armature reaction 131 converter heating 289 Racing of inverted converter 331 Radius vector 43 450 INDEX PAGE Rating of direct-current converter 344 of synchronous converter by armature heating 283, 285 of variable ratio converter 317 Ratios of conversion of synchronous converter 277 of converter 271 of transformation of transformer 73 Reactance, apparent or effective 56 of circuit 34 condensive 61 leads of single-phase commutator motors 243 and parallel operation 158 and phase control of converter 299 resistance starting device of single-phase motor 383 self-inductive, of synchronous machine 128 in starting polyphase induction motor 372 synchronous, of synchronous machine 128 of transformer 73 of transmission line, calculation 37 Reaction, magnetic 10 Reactive coils 123 calculation 57 component of current 42 of wave 107 currents in converter 297 in converter armature 285 e.m.f 41 voltage of alternating commutating machine 220 Real generated e.m.f. of synchronous machine 127 Rectangular coordinates 83 Rectifying apparatus, definition 121 machines 123 Reduction of secondary to primary in induction motor 356 Reentrant winding 172 Regulation affecting parallel operation of alternators 153 coefficient of system operating induction motor 404 curve of alternator 139 of commutating machine 197 and stability of induction motor 387 Reluctance of magnetic circuit 22 Repulsion motor 220, 225, 230, 234, 262 commutation. 248 starting of induction motor 380 Resistance 10 of armature of polyphase induction motor 368 characteristic of series generator 213 of shunt generator 211 commutation 200, 205 INDEX 451 PAGE Resistance, effective 53, 56 inductance starting device of single-phase motor 383 leads in commutating machine . 206 of single-phase commutator motor 240 negative of induction machine . 417 Resistivity 10 Reversal of current under commutator brush ... 199 Reversing component of single-phase motor commutating field 248 Ring winding of commutating machine 167 Rotary condenser 146 Rotating field of single-phase induction motor 373 magnetic field of induction motor 255 Rotation e.m.f. of single-phase commutator motor 237 Saturation affecting magnetic distribution 183 curve 207 of commutating machine . 194 of synchronous machine . . 146 effect on synchronous machine characteristic 148 factor of commutating machine 195 of synchronous machine 147 percentage of commutating machine 195 of synchronous machine 148 Secondary magnetic flux of single-phase induction motor 373 Self-excitation of induction generator 41 1 Self-exciting induction machine 435 Self-inductance 21 of alternating commutating machine armature 222 of alternating commutator motor field 221 of alternating current 33 and short-circuit current of alternator 162 of synchronous machine 132 of transmission line, calculation 18 Self-inductive armature reaction of synchronous machine 128 impedance, polyphase induction motor 357 of single-phase induction motor 376 reactance of transformer 73 induction 10 Separately excited generator 208 machine 166 Series commutator motor, commutation 248 drum winding 168 generator 211 machines 166 motor 216 repulsion motor 230, 235, 263 commutation 249 452 INDEX Series winding 177 Shading coil, starting single-phase induction motor 382 Shape of wave 114 Shift of brushes in commutating machine 182 of magnetic flux changing converter ratio 300 of flux of commutating machine 181 Short-circuit current of alternator 160 of commutation in single-phase motor 238 Shunt alternating commutator motor 266 generator 210 machines 166 motor 215 Sine wave, equivalent 114 in polar coordinates 44 Single-phase alternator, armature reaction 131 compensator motor, also see Alternating commutating machine. converter 272, 275, 419 armature reaction 295 induction motor 372 induction motor field 355 induction motor starting 380 railroading 219 short-circuit of polyphase alternator 163 Single spiral winding 177 Six-phase converter 276 converter, heating 289 variable ratio converter, heating 324 Slip of induction motor 355 of polyphase induction motor 358 Slots, effect on magnetic flux distribution 190 Speed characteristics of series motor 216 of shunt motor 215 of single-phase commutator motor 252 of concatenated motor 424 curves of induction generator 408 of polyphase induction motor 363 of single-phase induction motor 376 Speed limiting device of inverted converter 331 of prime mover, affecting parallel operation 154 Split pole converter 299 Split pole see Variable ratio converter. Squirrel cage winding in synchronous machine field 157 Stability coefficient of induction motor 402 of system operating induction motor 405 of converters and frequency 33 of induction motor 387 points of induction motor speed curve 390 INDEX 453 PAGE Stable branch of induction motor speed curve 389 Starting of continuous current 25 of converter 328 devices of single-phase induction motor 380 flux of single-phase induction motor 380 of polyphase induction motor 368 of synchronous motor 151 torque of polyphase induction motor 371 of single-phase induction motor 380 of synchronous motor , . 151 Stationary induction apparatus, definition 121 Steel, hysteresis 56 magnetization curve 9 Stopping of continuous current 27 Susceptance 107 Symbolic representation of admittance 109 of synchronous machine . .. 137 of vectors 86 Synchronizing alternators 153 induction motor 428 power of alternators 157 torque of induction motors 432 Synchronous compensator 146 cornmutating machine 122 converter 270 machines 125 definition 121 machine diagram 134 symbolic representation 137 motor 122 load curve on induction generators 412 vector diagram 141 reactance 128, 136 watts 359 Tandem connection of induction motors 424 Telephone circuit, mutual induction with transmission line 24 Terminal voltage of synchronous machine , , . 127 Three-phase alternator, armature reaction 131 converter 274 heating 289 series drum winding 171 synchronous machine 126 transmission line, voltage regulation 38, 50 variable ratio converter, heating 324 Three-section pole of variable ratio converter 317 Three-wire converter 345, 349 454 INDEX PAGE Three-wire generator 345 neutral from transformer 349 Time constant of circuit 26 diagram of polar coordinates 47 lag of current 36 Torque of concatenated induction motor 427 efficiency of motor 360 polyphase induction motor 359 of series motor 217 of single-phase induction motor 274, 279 Transformation ratio of induction motor 356 of transformer 73 Transformer 73 calculation by rectangular coordinates. 83 connections for three-wire converter 349 for variable ratio converter 310 definition *. 121 feature of induction motor 354, 356 flux of single-phase commutator motor 233 general alternating current 421 regulation and induction motor output 393 vector diagram , 74 Transmission line characteristics 91 calculation by symbolic method 90 capacity 61 constant voltage compounding , 97 impedance 62 magnetic flux calculation 7 phase control 71, 96 Trigonometric method of transformer calculation 77 Triple harmonic cross current in three-phase alternator 160 Turbo-alternators, short-circuit current 162 synchronizing power 158 Two-circuit single-phase converter 275 Two-phase, also see Quarter-phase and four-phase. Two-section pole of variable ratio converter 317 Types of commutating machines 206 Unbalanced circuit on converter 296 current in three-wire generator 348 Unbalancing of polyphase synchronous machine 150 of three-wire generator 348 Under compensation of alternating commutator motor 225 Unidirectional magnetization in three-wire transformer 349 Unipolar machine 1 1 , 124 Unit current 2 magnet pole 1 INDEX 455 PAGE Unit resistance 10 Unstable branch of induction motor speed curve 389 Unsymmetrical current waves at short-circuit of alternator 161 Variable converter ratio by shifting magnetic flux 300 by wave shape distortion 308 ratio converter 299 discussion , , . . , 326 transformer connection 310 by wave shape distortion 300 Variation of voltage of converter , 257 Varying speed single-phase commutator motor 220, 230 Vector diagram 44 of synchronous machine 135 of transformer 74 Vertical component in polar coordinates 46 Virtual generated e.m.f. of synchronous machine 127 Voltage commutation 200 control by converter 298 drop and induction motor output 393 per phase of converter 273 regulation and output of induction motor 292 and stability of induction motor 401 variation of converter 277 Wattless component of current 42 also see Reactive component . cross currents between alternators 153, 158 e.m.f 41 Wave shape of converter armature current 280 of direct-current converter armature current 340 distortion changing converter ratio 300, 308 of synchronous machine 126 of variable ratio converter 313 winding 172 Zero vector in polar coordinates 46 3 fi4fiS DOHSfl 2477 OCT 621 .3 S82th c. 1 Steinmetz, Charles Proteus, Theoretical elements 01 electrical engineering, University Libraries Carnegie-Mellon University Pittsburgh, Pennsylvania 15213 138405