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SYNCHRONOUS MACHINES Theory and Performance

General Electric Series SYNCHRONOUS MACHINES by Charles Concordia TRANSIENTS IN POWER SYSTEMS by Harold A. Peterson SERVOMECHANISMS AND REGULATING SYSTEM DESIGN, VOLUME I by Harold Chestnut and Robert W. Mayer TRAVELING WAVES ON TRANSMISSION SYSTEMS by L. V. Bewley TRANSFORMER ENGINEERING by the late L. F. Blume, A. Boyajian, G. Camilli, T. C. Lennox, S. Minneci, and V. M. Montsinger, Second Edition CIRCUIT ANALYSIS OF A-C POWER SYSTEMS, TWO VOLUMES by Edith Clarke CAPACITORS FOR INDUSTRY by W. C. Bloomquist, C. R. Craig, R. M. Partington, and R. C. Wilson PROTECTION OF TRANSMISSION SYSTEMS AGAINST LIGHTNING by W. W. Lewis MAGNETIC CONTROL OF INDUSTRIAL MOTORS by Gerhart W. Neumann POWER SYSTEM STABILITY Volume I—Steady State Stability; Volume II— Transient Stability; by Selden B. Crary FIELDS AND WAVES IN MODERN RADIO by Simon Ramo and John R. Whinnery MATERIALS AND PROCESSES edited by J. F. Young MODERN TURBINES by L. E. Newman, A. Keller, J. M. Lyons, and L. B. Wales; edited by L. E. Newman ELECTRIC MOTORS IN INDUSTRY by D. R. Shoults and C. J. Rife; edited by T. C. Johnson A SHORT COURSE IN TENSOR ANALYSIS FOR ELECTRICAL ENGINEERS by Gabriel Kron TENSOR ANALYSIS OF NETWORKS by Gabriel Kron MATHEMATICS OF MODERN ENGINEERING Volume I by the late Robert E. Doherty and Ernest G. Keller; Volume II by Ernest G. Keller VIBRATION PREVENTION IN ENGINEERING by Arthur L. Kimball

SYNCHRONOUS MACHINES Theory and Performance CHARLES CONCORDIA Analytical Engineering Department General Electric Company Schenectady, New York One of a series written fay Genera/ Electric authors for the advancement of engineering knowledge JOHN WILEY & SONS, INC., NEW YORK CHAPMAN & HALL, LTD., LONDON, 1951

Copyright, 1951, by General Electric Company All rights reserved. This book or any part thereof must not be reproduced in any form without the written permission of the publisher Printed in the United States of America

PREFACE The primary object of this book is to present a unified development of the fundamental circuit theory of the transient performance of synchronous machines as currently used by the engineers directly concerned with the prediction of machine performance. The material was written for a synchronous machines course that has been given at the General Electric Company for the past three years. The general equations developed are applied to the calculation of transient short-circuit currents and torques; steady-state power, torque, and current, both in synchronous operation and during starting; and the voltage disturbances occasioned by sudden application of load. Emphasis is on a more or less rigorous mathematical development and on obtaining a fundamental physical understanding of the machine so that the reader will be best equipped to extend the theory as he needs it. It is presumed that the reader is acquainted, but not necessarily familiar, with 1. The ordinary steady-state and transient theory of static circuits including the elementary law of electromagnetic induction in circuits. 2. The general physical appearance of a synchronous machine. 3. Ordinary differential equations, and operational calculus in at least one of its various forms. 4. A usual undergraduate course in round-rotor a-c rotating electric machinery, covering only the steady-state performance. 5. Symmetrical components. 6. The per-unit system of representation of machine and power system parameters. Thus, the book is intended primarily for the practicing engineer who wants to learn something about the transient theory of synchronous machines and who has heretofore been obliged to dig through the technical literature of the past twenty-five years to do so. It is not intended as a reference book, even though formulas for many specific cases can be found in it. On the contrary, it is intended to be read as a whole from the beginning. It will be evident to those familiar with the literature of synchronous machines that for the sake of unity many of the derivations and the notation, and in some cases the form of the results, have been revised.

vi PREFACE In particular, the method of deriving the general equations and the treatment of single-phase short circuits had to be considerably revised. Also, certain new material has been added. These new items are: the treatment of the double-line-to-ground short circuit, of the unidirectional components of short-circuit torque, of the starting torque, and of voltage dip on application of load. The theory presented in this book is the culmination of the work of many engineers over a period of about twenty-five years. Acknowledgment of sources can therefore be made only through the list of references and the bibliography, as they are otherwise literally too numerous to mention. However, I want to acknowledge specifically the continued encouragement and support of Mr. S. B. Crary and the contributions to the point of view made by Mr. Gabriel Kron. I must remark further my conviction that an essential factor in the achievement of such quality as this book may have is the atmosphere of a large industrial corporation, that combines the necessity for keeping in direct contact with the latest practice with the opportunity for specialization afforded by its size. CHARLES CONCORDIA March 1951

CONTENTS CHAPTER PAGE 1 PHYSICAL DESCRIPTION OF A SYNCHRONOUS MACHINE 1 2 MATHEMATICAL, DESCRIPTION OF A SYNCHRONOUS MACHINE 6 Voltage Relations, 8 Flux-Linkage Relations, 9 Inductance Relations, 10 Transformations of Equations, 13 Armature Voltage Equations, 16 The Operational Impedances, 18 Per-Unit Quantities, 20 Slip Test for xd and xq, 23 Short-Circuit Test for xd, 24 ZeroSequence Reactance, 25 Power Output, 25 Torque, 28 Summary, 30 Problems, 31 3 STEADY-STATE, BALANCED, SYNCHRONOUS OPERATION 32 The Steady-State Vector Diagram, 32 Field Flux Linkage, 34 Power Output, 36 Power-Angle Characteristics, 37 Stability, 40 Reactive Volt-Amperes, 44 Power-Angle Characteristics for Two Machines, 46 Summary, 52 Problems, 52 4 THREE-PHASE SHORT-CIRCUIT CURRENT 54 Synchronizing Currents, 58 Steady-State Components of Short-Circuit Current, 59 Short-Circuit Test, 66 Short Circuit with Armature Re sistance, 67 Field Current, 72 Summary, 74 Problems, 75 5 SINGLE-PHASE SHORT-CIRCUIT CURRENT 76 Line-to-Line Short Circuit, 76 Phase Quantities, 81 Line-to-Neutral Short Circuit, 81 Open-Phase Voltage for Line-to-Line Fault, 84 Harmonic Components of Voltage and Current, 85 Decrement Factors, 90 Field Current, 95 Summary, 97 Problems, 98 6 DoUBLE-LINE-TO-GROUND SHORT CIRCUIT AND SEQUENTIAL FAULTS . . 100 Symmetrical Components, 103 Rotor Decrement Factors, 108 Stator or Armature Decrement Factors, 109 Field Current, 112 Open-Phase Voltage, 113 Sequential Application of Faults, 114 Summary, 117 Problems, 118 7 SHORT-CIRCUIT TORQUES 119 Three-Phase Short Circuit with All Resistances Neglected, 119 ThreePhase Short Circuit—Effect of Armature Resistance, 121 Three-Phase Short Circuit—Effect of Rotor Resistance, 123 Discussion of ThreePhase Short-Circuit Torques, 127 Line-to-Line Short Circuit, 134 Torque, 135 Other Types of Short Circuit, 142 Harmonic Components of Line-to-Neutral Torque, 145 Unidirectional Components of Torque, 147 Unidirectional Component of Torque Due to D-C Component of Current, 157 Approximate Torque Equations, 160 Summary, 163 Problems, 164 vii

viii CONTENTS CHAPTER PAGE 8 STARTING TORQUE 165 Equivalent Circuit, 171 Relation to Approximate Torque Equation, 171 Comparison of "Exact" and Approximate Methods, 175 Average Torque (d- and g-Axis Method), 177 Field Excitation, 180 Summary, 183 Problems, 184 9 VOLTAGE DIP 185 Effect of Voltage Regulator, 191 Minimum Voltage, 193 Required Exciter Ceiling, 195 Saturation, 196 Exciter Response, 197 Voltage Recovery Time, 200 Effect of Initial Load, 200 Field Current, 201 Summary, 202 Problems, 203 APPENDIX A FOURIER SERIES FOR CURRENTS, AND FUNDAMENTAL-FREQUENCY COMPONENTS OF id AND lq, FOR DoUBLE-LINE-TO-GROUND FAULT 205 B TORQUE 212 REFERENCES 216 BIBLIOGRAPHY 217 INDEX ..........,,....,,............ 221

1 PHYSICAL DESCRIPTION OF A SYNCHRONOUS MACHINE A synchronous motor or generator consists essentially of two elements: the first to produce a magnetic field, the second ajset of armature coils in which voltages are produced by the relative motion of the two elements. In the usual modern machine the field structure rotates within a stator which supports and provides a magnetic-flux path for the armature windings. The exciting magnetic field is ordinarily produced by a set of coils (the field winding) on the moving element or rotor. Since most electric power is generated (and in the case of large blocks of power, consumed) as three-phase power, there are ordinarily three armature coils, disposed around the stator at 120° intervals so that, with uniform rotation of the magnetic field, voltages displaced 120° in phase will be produced in the coils. Two observations about this last statement are in order here. 1. It only applies without qualification to a two-pole (i.e., one pair of poles) machine. In a machine with, e.g., two pairs of poles, there must be correspondingly two complete sets of armature coils 180° apart, the three coils of each being set 60° apart. Figure 1 illustrates the disposition of the coils and the magnetic-flux paths for a two- and fourpole machine. More generally, the three coils of each set must be (120/n) degrees apart, and the sets (360/n) degrees apart, where n is the number of pairs of poles. It is usual and convenient to measure the distance between coils in "electrical degrees" where 360 electrical degrees corresponds to the angle included in one pole pair and 360 actual (or mechanical) degrees equals 360n electrical degrees. In terms of electrical degrees, then, the three coils of each set are always 120° apart. 2. The steady-state voltages produced (with balanced load) are always 120° apart in phase regardless of the speed of rotation of the field. That is, since (1/n) revolution (a displacement equal to the space occupied by one pole pair) will always correspond to one cycle of the gen1

PHYSICAL DESCRIPTION OF A SYNCHRONOUS MACHINE STATOR COIL SIDES Fio. 1. Arrangement of coils in 2- and 4-pole machines

PHYSICAL DESCRIPTION OF A SYNCHRONOUS MACHINE 3 erated voltage, i.e., the fundamental frequency will always be exactly n times the speed of rotation, and, since with constant rate of rotation the time required for the rotor to move any given distance is proportional to the distance moved, the time required for the field to move from any given position with respect to one coil to the corresponding position with respect to the next coil is just one third of a cycle or 120 electrical degrees. Of course, the machine is ordinarily connected to a three-phase bus to which voltage is also being supplied by other synchronous machines, so that this applied voltage will not correspond to the rotational speed unless the machine is running in synchronism with the rest of the machines on the system. Hence, the term "synchronous machine" means one that ordinarily runs in synchronism with other machines of the same general type, and the terms synchronous operation, out of synchronism, and out of step have meaning only for a machine connected to a system, and not for a single machine operating alone. Since in normal operation the magnetic flux produced by the field winding is rotating with respect to the stator windings and its supporting magnetic structure, voltages are produced in the iron as well as in the coils, and it is necessary to laminate the stator iron in order to break up the eddy-current paths and thus minimize the i2r losses and shortcircuiting effect which would otherwise result. The field structure or rotor on the other hand sustains principally only a constant flux and so does not have to be laminated throughout. When balanced three-phase armature currents of speed frequency are flowing, the mmf's produced by these currents tend to combine to give a resultant mmf that rotates at the same speed as the field. It is found that the best way to study the effects of this armature mmf is to resolve it into its space harmonics, upon which it may be discovered that the fundamental component rotates at rotor speed and so is stationary with respect to the field, while some of the space harmonics rotate at different speeds and so are moving with respect to the field. Since the armature coils are distributed along the stator surface so as to tend to minimize all harmonics other than the space fundamental, these harmonic effects may be regarded as secondary from the standpoint of performance. They contribute to the armature leakage reactance (i.e., to components of armature flux which do not link any of the rotor windings) and to rotor surface eddy-current losses which make it desirable to laminate at least the surface of the rotor iron whenever possible. In general, practically all machines except high-speed, two- or four-pole turbine-generators have laminated pole faces and are constructed with salient poles as shown in Fig. 2, whereas the rotors of two- and four-

4 PHYSICAL DESCRIPTION OF A SYNCHRONOUS MACHINE pole machines may be made as a single solid piece of steel. In Fig. 2 can also be seen a damper winding or amortisseur consisting usually of a set of copper or brass bars set in pole-face slots and connected together Fig. 2. Rotor for salient-pole machine .j^isE! A-^|g gHNMHj [.uuu,e*w ,n , '. § T^&W • ■ m< W Fig. 3. Rotors for solid-rotor machines at the ends of the machine. This amortisseur has several useful functions: x to permit starting of synchronous motors as induction motors using the amortisseur as equivalent to the squirrel cage of an inductionmotor rotor, to assist in damping rotor oscillations, to reduce overvoltages under certain short-circuit conditions, and to aid in synchronizing the machine. 1 Superscripts refer to items in the list of references at the end of the book.

PHYSICAL DESCRIPTION OF A SYNCHRONOUS MACHINE 5 Figure 3 shows two rotors for two-pole solid-rotor turbine-generators. In this case the solid steel rotor itself serves the purpose of the amortisseur. From the brief description given above it is evident that the stator and rotor of a synchronous machine differ in these respects: The stator is more or less standardized and relatively simple in form for any type of synchronous machine and is, moreover, completely symmetrical with respect to any of the three phases. On the other hand, the rotor presents a considerable variety of forms, ranging from the simplest case of a single field winding on an otherwise symmetrical laminated rotor to a salient-pole rotor with an amortisseur having several windings, or to a solid rotor, which, although symmetrical except for the field, is still complex in that the solid steel rotor may be considered as equivalent to an amortisseur of infinitely many circuits.

2 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE We have pointed out in Chapter 1 that a synchronous machine consists of two major components, the stator and the rotor, that are in relative motion and that are rather different in structure. Regardless of this it is of course possible to write down the circuit-voltage equations simply in terms of the self- and mutual inductances of all the windings. In order to do this, we must first decide what the rotor windings are. We shall assume that the rotor magnetic paths and all of its electric circuits are symmetrical about both the pole and interpole axes as shown in Fig. 4 for a salient-pole machine. The field winding is of course separate from the others and has its axis in line with the pole axis. The amortisseur bars are all connected together in a more or less continuous mesh, but, if they are arranged symmetrically, current paths may be chosen which are also symmetrical about both the pole and interpole axis. Figure 4 shows the circuits used. The bars are numbered starting from the direct axis, which is in line with the pole axis. The direct axis circuits are then numbered Id, 2d, etc., to correspond with these bars. In the quadrature axis, which is taken as 90 electrical degrees ahead of the direct axis in the direction of normal rotor rotation, the circuits are numbered Iq, 2q, etc., starting outward from this axis. This symmetrical choice of the rotor circuits has the virtue of making all mutual inductances and resistances between direct- and quadrature-axis rotor circuits equal to zero. In some machines the amortisseur bars are not connected between poles, but even in these cases current may flow between poles at the ends of the machine through the rotor iron itself since the bars are not insulated. This lack of insulation means also that the circuit equations are only approximations to the actual case, in which some current may spread through the iron. This effect is small except where the interpole iron path is concerned, and except in turbinegenerators wherein the currents in the iron form the whole amortisseur 6

MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE 7 effect. Since the turbine-generator is so different in this respect from the salient-pole generator, it will be treated separately after we have seen how the salient-pole case turns out. All mutual inductances between stator and rotor circuits are periodic functions of rotor angular position. In addition, because of the rotor saliency, the mutual inductances between any two stator phases are also periodic functions of rotor angular position. We thus arrive at a POLE I "> •> d Id dl It I j^^M-.-^-.-MNUMBERING OF ROTOR CIRCUITS FIQ. 4. Diagram of amortisseur circuits set of differential equations most of whose coefficients are periodic functions of rotor angle, so that even in the case of constant rotor speed (when the equations are linear if saturation is neglected) they are awkward to handle and difficult to solve. However it is found that, if certain reasonable assumptions are made, a relatively simple transformation of variable will eliminate all these troublesome functions of angle from the equations. The first assumption is that the stator windings are sinusoidally distributed along the air gap as far as all mutual effects with the rotor are concerned. This assumption of sinusoidal distribution of the stator windings may be justified from the standpoint that in practically all synchronous machines the windings are distributed so as to minimize all harmonics as much as is feasible.2 The principal justification comes from the comparison of performance calculated on that basis with actual performance obtained by test.

8 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE The second assumption is that the stator slots cause no appreciable variation of any of the rotor inductances with rotor angle. This assumption is evident for machines with a large number of slots per pole, but for machines with a very small number (especially an integral number) of slots it is not so evident. However, a small number of slots occurs principally in machines having a large number of poles, and thus any effect of the slots may be made to average out over the whole machine. Again the final justification comes from comparison of theory and test results. A third assumption which will be made in this book at least for the present is that saturation may be neglected. The effects of various assumptions regarding saturation will be shown later. The electrical performance of a synchronous machine may now be described by the following equations. Voltage Relations ARMATURE OR STATOR ea ~ Pta — ria tb = ptb — rib (1) ec = p\l/c — ric where ea = terminal voltage of phase a fa = total flux linkage in phase a ia = current in phase a. Note that the direction of positive armature current is taken as opposite to what might have been expected in a static network in order to have positive current correspond to generator action. a, b, c, are the three phases lettered in the direction of rotor rotation as shown in Fig. 5 r = resistance of each armature winding, assumed to be the same for a, b, c p = the derivative operator d/dt, t = time FIELD + Wd (2) where here and in all the following equations the symbols e, t, i have the same meaning as above, the subscripts denoting the circuit in question.

FLUX-LINKAGE RELATIONS 9 DIRECT-AXIS AMORTISSEUR 0 = ptid + rudiid + ri2di2d -\ ---0 = pfad + r2idiid + r22di2d H ---- (3) etc. Here the subscripts I2d and 2ld denote mutual effects between circuits Id and 2d (see Fig. 4). It may be noted that the amortisseur circuits are resistar 2e-coupled as well as inductance-coupled and that there is no coupling between direct- and quadrature-axis circuits because of the rotor .symmetry about the direct and quadrature axes. QUADRATURE-AXIS AMORTISSEUR 0 = ptiq + rnqiiq + rl2qi2q -\ ---0 = pt2q + r2iqiiq + r22qi2g -\ ---- (4) etc. Flux-Linkage Relations ARMATURE — XbbH ~ Xbcic + Xbfdifd + + xc2di2d -\ ----- h xclqilq + xc2qi2q H ---- (5) where the x's are inductances to be defined later and the subscripts refer, as before, to the circuits in question. FIELD ^/d = —Xfadia ~ Xfbdtb ~ Xfcdlc + Xffdifd + Xf idild + Xf2di2d H ----- h Xfiqiiq + xf2qi2q -\ ---- (6)

10 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE DIRECT-AXIS AMORTISSEUR tld = — xiadia — Xlbq = — Pq sin 0J where Pd and Pq are proportional to effective permeance coefficients in the direct and quadrature axes, respectively. Space-harmonic components of flux are also produced, but, since they do not link the stator, they do not concern us now. The linkage with phase a caused by this flux is then proportional to (see Fig. 5) d cos 0a —^g sin 0a = Pd cos2 0a + Pq sin2 0a = .Pd + P" + Pd~PqCOS26a = A + BCOS26a 22 (10) There is also some flux linking phase a that does not link the rotor. This flux adds only to the constant term A of equation 10, and so the inductance remains of the form of equation 9. Similarly, Xbb = XaaO + Xaa2 COS 26b (11) Xee = XaaO + Xaa2 COS 20c where 06 = 6 - 120° (12) 0c = 0 + 120° ARMATURE MUTUAL INDUCTANCES To determine the form of the mutual inductance between, e.g., phases a and b, we may recognize first that there may be a component of mutual flux that does not link the rotor and is thus independent of angle. Then

12 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE considering current in phase a, the components of air-gap flux are, as before, proportional to d = Pd cos 6a and g = -Pg sin 6a (13) and the linkage with phase 6 due to these components is proportional to d cos 6b — q sin 0(, = Pd cos 0a cos 6b + Pq sin 0a sin 6b = Pd cos 6 cos (6 - 120) + Pq sin 6 sin (0 - 120) -P q + - - cos 2(6 - 60°) 42 = - JA + B cos 2(0 - 60°) = -[£A + £ cos 2(0 + 30°)] The total mutual inductance is thus of the form, Xab = — [XabO + Xaa2 COS 2(0 + 30°)] Note that the variable part of the mutual inductance is of exactly the same magnitude as that of the variable part of the self-inductance and that the constant part has a magnitude of very nearly half that of the constant part of the self-inductance. Now that we know the answer, it may seem obvious from symmetry considerations that the mutual inductance ab should have a (negative) maximum when the pole axis is lined up 30° behind phase a or 30° ahead of phase b, and a (negative) minimum when it is midway between the two phases. We might also have reasonably taken a chance that higher harmonic terms could not appear in the mutual inductance since they dropped out of the selfinductance. Finally, we can write all the stator mutual inductances as Xab = Xba = ~[XabO + Xaa2 COS 2(0 + 30°)] Xbc = Xcb = — [XaW + Xaa2 COS 2(0 — 90°)] (14) ab0 + £aa2 COS 2(0 + 150°)] ROTOR SELF-INDUCTANCES Since we are neglecting the effects of stator slots and of saturation, all the rotor self-inductances, £//d, xnd, x22d, £119, etc., are constants.

TRANSFORMATIONS OF EQUATIONS 13 ROTOR MUTUAL INDUCTANCES All mutual inductances between any two circuits both in the direct axis and between any two circuits both in the quadrature axis are constant, and of course xfld = xifd, etc. Because of the rotor symmetry there is no mutual inductance between any direct- and any quadratureaxis circuit. Thus: Xflq ~ xf2q = Xldlq = xld2q = xlqfd = ^l«ld = xlq2d = 0, etc. (15) MUTUAL INDUCTANCES BETWEEN STATOR AND ROTOR CIRCUITS By considering current in each rotor winding in turn and recalling that only the space-fundamental component of the flux produced will link the sinusoidally distributed stator, we see that all stator-rotor mutual inductances vary sinusoidally with angle and that they are maximum when the two coils in question are in line. Thus: Xald = Xfad = Xafd COS 6 Xbfd = Xfbd = Xafd COS (0 — 120) Xcfd = Xfcd = Xafd COS (0 + 120) Xal d = Xiad = Xald COS 6 Xbl d - Xibd = Xald COS (6 — 120) (16) xcld = xUd = Xau cos (6 + 120), etc. Xalq = Xlaq = — Xalq SU1 6 Xblq = XUq = — Xalq SU1 (6 - 120) Xciq = xUq = -xalq sin (6 + 120), etc. Transformations of Equations Utilizing the mutual-inductance relations equations 16, we may rewrite the rotor flux-linkage equations 6, 7, and 8 as FIELD tfd = — xafd[ia cos 6 + ib cos (0 — 120) + ic cos (6 + 120)] + Xffdifd + xfidiid + xndiid + • • • (17)

14 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE DIRECT-AXIS AMORTISSEUR tid = —xaid[ia cos 6 + ib cos (6 - 120) + ie cos (6 + 120)] + xndiid + xi2di2d -\ ---- , etc. (18) QUADRATURE-AXIS AMORTISSEUR \h9 = +xaig[ia sin 6 + ib sin (6 - 120) + ic sin (6 + 120)] + Xi2qi2q H ---- , etc. (19) The form of these equations suggests that they may be simplified by the substitution of new variables id, iq, IQ defined by the relations : s id = f ta cos 6 + ib cos (6 - 120) + ic cos (6 + 120)] iq = - f [*'a sin 6 + i b sin (6 ~ 12O) + ic sin (0 + 120)] (20) By reference to Fig. 5, or for that matter by inspection of equations 17, 18, and 19, we see that id and iq are proportional to the components of mmf in the direct and quadrature axes, respectively, produced by the resultant of all three armature currents ia, ib, and ic. The factor % is introduced so that, for balanced phase currents of any given (maximum) magnitude, the maximum values of id and iq as the phase of the currents is varied will be of the same magnitude. The maximum magnitude of any one of the phase currents under these balanced conditions will be given by V i2d + i2q and will also be the same. The current i0 is introduced since, if three currents ia, 4, and ic are to be eliminated, in general three substitute variables will be required. i0 is the conventional zero-phase-sequence current of symmetrical-component theory, and, if only i0 exists (i.e., if ia = ib = ie), equations 17, 18, and 19 show that no flux will link the rotor. By substituting the relations 20 in equations 17, 18, and 19, we obtain, for the rotor-circuit flux linkages: = — ^Xafdid + Xffdifd + Xfidild. + ' '' = —%Xaidid + xlfdifd + xndiid + xi2di2d -\ ---- (21) = —fcalqiq + Xllqilg + Xl2qi2g H ---- , etC.

TRANSFORMATIONS OF EQUATIONS 15 Now, equations 9, 11, 14, and 16 are substituted for equations 5 for the armature flux linkages, to obtain ia = —XooO^'o + Xabo(h + te) - X^ia COS 26 + Xaa2ib COS 2(6 + 30°) + Xaa2ic cos 2(6 + 150°) + (XafcCifd + Xaldild + Xa2di2d H ) COS 6 - (Xalqilq + Xa2qi2q H ) SHI 0 ib = — Xaaolb + Xabo(ic + 4) + Xaa2ia cos 2(6 + 30°) - Xaa&b cos 2(6 - 120°) + «c cos 2(6 - 90°) (22) + (Xafdifd + Xaidild + xa2a42d H ) cos (6 - 120°) - (Xalqilq + Xa2qi2q -\ ) sin (6 - 120°) ic = —Xaaoic + Xabo (la + ib) + Xaa2ia cos 2(6 + 150°) + x^h cos 2(6 - 90°) - xoo2ic cos 2(0 + 120°) + (Xafdifd + Xaidild + Xa2dkd H ) cos (6 + 120°) - (Xalqilq + xa2qHq H ) sin (6 + 120°) In these flux-linkage equations the armature phase currents ia, ib, and ic may be eliminated in favor of the new variables id, iq, and i0, which does not, however, eliminate the trigonometric functions of rotor angle in this case. The form of the new equations suggests that a simplification can be effected by defining, similarly to id, iq, and i0, three new flux linkages id, iq, and i0. id = f [ia cos 6 + ib cos (6 - 120) + ic cos (6 + 120)] iq = -l\ia sin 6 + ib sin (9 - 120) + ic sin (5 + 120)] (23) io = \(ia + ib + ic) Now, if equations 22 are substituted in equations 23 and the proper trigonometric reductions are made, we obtain the relatively simple relations: id = —(XaaO + XabO + ^Xaa2)id + Xafdifd + XaldHd + Xa2di2d H . iq = —(Xaa0 + xabo — %xaa2)iq + xaiqiiq + xa2ai2a + • • • (24) io = —(XaaO — 2xabo)io

16 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE In equations 24, fa and \{/q may be regarded as corresponding to flux linkages in coils moving with the rotor and centered over the direct and quadrature axes, respectively. The equivalent direct-axis moving armature circuit has the self-inductance: Xd = ZooO + ZoW + f Zoo2 (25) and the equivalent quadrature-axis moving armature circuit has the self-inductance: Xq = XaaO + XabO ~ fZoo2 (26) There is also an equivalent zero-sequence axis coil which has the selfinductance: ZO = XaaO ~ ^abO (27) and which is completely separated magnetically from all the other coils. Armature Voltage Equations Finally, we can eliminate the phase quantities ia, ib, ie, and fa, \l/b, tc from equations 1, as they now occur nowhere else. This may most easily be done by denning new voltages ed, eq, and e0 in the same manner as the currents (equations 20) and flux linkages (equations 23). ed = f [ea cos 6 + eb cos (6 - 120) + ec cos (6 + 120)] eq = -f[ea sin 6 + eb sin (6 - 120) + ec sin (0 + 120)] (28) By substituting equations 1 in equations 28 and utilizing the relations 20, we obtain ed = f [cos 6pta + cos (6 - 120) ptb + cos (6 + 120) p^c] - rid eq = -f [sin 6 pfa + sin (6 - 120) p+b + sin (6 + 120) ptc] - riq e0 = pt0 — ri0 (29) The bracketed expressions in equations 29 may be evaluated by differentiating the first two of equations 23, whence ptd = f [cos 6 pta + cos (6 - 120) ptb + cos (6 + 120) ptc] - I [fa sin 6 p6 + tb sin (6 - 120) p6 + tc sin (6 + 120) p6] or, by substituting \l/q from equations 23, Ptd = f [COS pta + COS (6 - 120) ptb + COS (6 + 120) ptc] + tqp6 .. J (30)

ARMATURE VOLTAGE EQUATIONS 17 and similarly, piq= - f [sin 0 pia + sin (0 - 120) pib + sin (0 + 120) pic] - idp6 (31) Thus, equations 29 become ed = Pid - iqpQ - rid eq = ppa + +dP0 ~ riq (32) eo = Pio - ri0 We note that these equations 32 are just like the original relations 1 but with the addition of generated- or speed-voltage terms iqp6 and idp6 in the direct- and quadrature-axis voltages. From a physical viewpoint our algebraic manipulations have corresponded to the specification of the armature quantities along axes fixed in the rotor and thus rotating with speed, p6, with respect to the stator axes. We should therefore naturally expect to find generated voltages as well as induced voltages produced by these rotating flux linkages. The complete set of machine-performance equations now consists of the circuit voltage equations 32, 2, 3, and 4, and the flux-linkage equations 24 and 21. At constant rotor speed these equations are linear differential equations with constant coefficients, and even with variable rotor speed they are considerably simpler than the original set of equations. The phase quantities ia, H, ic, ea, eb, ec, ia, ib, and ^c in any particular problem may be found from the substitute variables id, iq, and t'o, etc., by solving the relations 20, 23, and 28 to obtain ia = id cos 0 — iq sin 0 + i0 ib = id cos (0 - 120°) - iq sin (0 - 120°) + i0 (33) ic = id cos (0 + 120°) - iq sin (0 + 120°) + to ia = id cos 0 — \pq sin 0 + "A0 ib = id cos (0 - 120°) - iq sin (0 - 120°) + i0 (34) ic = id cos (0 + 120°) - iq sin (0 + 120°) + &, ea = ed cos 0 — eq sin 0 + e0 eb = ed cos (0 - 120°) - eq sin (0 - 120°) + e0 (35) ec = ed cos (0 + 120°) - eq sin (0 + 120°) + e0 s

18 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE 'We have stated previously that the armature-voltage equations 32 are in a form especially suited to the solution of salient-pole synchronousmachine problems. However, in order to keep clear our concept of what we are doing, it should be pointed out that equations 32 in no way imply the existence of a salient-pole machine or even of any rotating machine. Equations 1 apply to any three coils of equal resistance, and the relations 33, 34, and 35 may be used to obtain equation 32, regardless of the nature of any other coils magnetically linked with these three coils. In the salient-pole machine case we have chosen 6 to be the angle of the direct axis of the rotor ahead of the axis of phase a, but this choice is not at all a necessary condition for the validity of equations 32. It is selected solely to simplify the flux-linkage relations. In case, for example, of a completely symmetric static network containing no capacitors, all choices of 6 lead to identical flux-linkage relations (in all cases simpler than the original relations in terms of threephase quantities) so that we could take 6 equal to anything from 6 = 0 to 6 = angle of any machine that we might later want to connect to our three coils. The Operational Impedances Since in many important problems one is interested primarily in the results as viewed from the machine armature terminals, as, e.g., in computing short-circuit currents, it is convenient to write the machine equations in a more compact form by eliminating the rotor currents. This may be done by (1) substituting the rotor flux-linkage relations 21 into the rotor-circuit voltage equations 2, 3, and 4, (2) solving these for the rotor currents in terms of the field voltage e/d and the armature currents id, iq, and (3) substituting the resulting relations in the armature flux-linkage relations 24. This may be a more or less difficult job of solving several simultaneous equations depending on the complexity of the amortisseur, but it is evident that, if we treat the derivative operator p = d/dt algebraically, as will be legitimate for many problems since all the flux-linkage relations and all the rotor-circuit voltage relations are linear, we shall arrive finally at a result of the form td = G(p)efd - xd(p)id tg = -xq(p)iq (36) where G(p), xd(p), xq(p) are operators expressed as functions of the derivative operator p. In the case of i/y, it is further evident that

THE OPERATIONAL IMPEDANCES 19 G(p) can be obtained by solving for fa as a function of efd with id = 0 and that Xd(p) may similarly be found by solving for fa as a function of id with efd = 0. We shall conform to the usual practice and call Xd(p), xq(p), and x0 the direct, quadrature, and zero-sequence axis operational impedances of the synchronous machine, even though it appears from their definitions that a more logical name would be "operational inductance." It has been stated previously that the direct- and quadrature-axis fluxes may be thought of as linking coils moving with the rotor and centered over the direct and quadrature, axes of the machine. This, together with the general form of equations 2, 3, 4, and 21, seems to suggest that at least in certain cases we should be able to regard the whole group of direct- (or quadrature-) axis circuits as representable by some sort of equivalent static electric circuit. For example, except for the mutual resistances, they are very similar to the equations of a manywinding transformer.4 In that event the calculation of xd(p) and xq(p) could be considerably simplified. However, an essential condition for the existence of a static equivalent circuit is the reciprocity of the mutual-inductance coefficients, and this condition is not completely satisfied in the present instance. That is, in equations 21 the mutualinductance coefficients between armature currents and rotor flux linkages are — %Xafd, —%Xaid, ~V2Xa2d, •", —%Xalq, — %Xa2q, '• ', but the mutual-inductance coefficients between armature flux linkages and rotor currents are xafd, xaid, xa2d, • • •, xaiq, xa2q, ■ ■ • • That is, they are of only two-thirds magnitude and of opposite sign. This difficulty arises because of the transformation used for both current and flux linkage, which was chosen merely to keep the magnitudes of \p(t = V^ + t2q ) and of i(i = V i2d + i2q) unity for balanced unit flux linkages fa, fa, fa and currents ia, ib, and ic, respectively. It could easily have been avoided by other choices6 of transformation equations, but it seemed desirable to preserve the property of equal magnitudes. In any event, the difficulty is easily resolved by also changing over the rotor currents by a % factor, to obtain the flux-linkage relations: DIRECT AXIS ^d = —Xdid + Xafdlfd + Xaldhd + Xa2dhd H fyd = —Xafdid + Xffdlfd + Xfidhd + Xf2dhd H (37) iu = —Xaidid + Xfidlfd + Xudhd + Xi2dhd H , etc.

20 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE QUADRATURE AXIS tq ~ — Xqiq ~T" Xaigliq + Xa2qI2q ~\~ • '' tiq = — Xalqiq + Xnqliq + Xl2qI2q + • • • (38) faq = —Xa2qiq + XiZqllq + X22ql2q + ' ' ', 6tC. and the rotor-circuit voltage relations: DIRECT AXIS (ROTOR CIRCUITS) efd = Pt/d + Rfdlfd 0 = pt!d + Rudlid + Ri2dI2d +••• (39) 0 = Pt2d + Ri2dlid + R22dl2d + ' ' ', etc. QUADRATURE AXIS (ROTOR CIRCUITS) 0 = Ptlq + Rnqhq + #124/29 H 0 = Pt2q + Rl2qllq + R22qhq H , etC. These equations 37, 38, 39, and 40, together with the armature-circuit voltage equations 32, now constitute the complete set of machine-performance equations. The operational expressions of equations 36, G(p), xd(p), and xq(p), may be found from equations 37-40 in exactly the same way (and with exactly the same results) as from equations 2, 3, 4, 21, and 24. In addition, now an equivalent circuit may be used to represent and visualize these quantities,6"9 as will be shown later. All the inductances and resistances represented by upper-case symbols are three halves times the corresponding lower-case quantities of equations 2, 3, 4, 21, and 24 (e.g., Xafd = Y^afd), and all the rotor currents represented by upper-case symbols are two thirds times the corresponding lower-case currents (e.g., Ifd = %ifd). This nomenclature is followed in this chapter and in Chapter 3; but in following chapters we shall return to the use of lower-case letters, even though we shall at all times be concerned only with reactances as denned by equations 37 and 38 (i.e., the reciprocal system) and shall never again have occasion to use the nonreciprocal system. Thus it will not be necessary to use a nomenclature that distinguishes between the two systems. Per-Unit Quantities 7 Actually this multiplication of all rotor currents by % is not such a drastic procedure as it may at first glance seem, since the inductance (or reactance) and resistance coefficients of synchronous machines will usually be specified as per-unit values rather than as ohms or henrys

PER-UNIT QUANTITIES 21 anyway. The base values of armature current and voltage will ordinarily be determined by the machine rating, whereas the base values of the rotor currents are chosen so as to make the self-inductances of the armature, field, and outermost (in each axis) amortisseur circuits of about the same order of magnitude, as in the usual transformer equivalent circuit. It is not obvious in the present case, however, that base currents should be chosen inversely as the turns of each circuit, as is the case with transformers, because of the effect of the distribution of each winding in modifying its flux-producing effectiveness. The base field current may be taken as that value that will produce the same space-fundamental component of air-gap flux as is produced by base armature current id, that is, by per-unit balanced three-phase armature currents ia = cos 6 ib = cos (0 - 120°) », = cos (0 + 120°) Similarly, the base amortisseur current may be taken at that value that will produce the same space-fundamental component of air-gap flux as is produced by unit armature current id, when this amortisseur current flows in a direct-axis amortisseur winding of full (180°) pitch. It is usually found most convenient' to use the same base value for all amortisseur currents in both the direct and quadrature axes. The two-thirds factor which had to be introduced into the rotor currents now makes its appearance simply by the fact that the effective turn ratio which must be used is calculated from the ratio of base field (or amortisseur) current to three halves times base armature phase current. One might be led to expect such a ratio from another point of view from the fact that unit id produces an air-gap, space-fundamental mmf exactly 1J^ times as big as, e.g., unit ^'a acting alone in the direct axis. This may be seen from equations 9, 10, and 11, as follows. When the direct axis (pole axis) is lined up with phase a, unit ia produces a flux: 4>d = Pd cos 6a = Pd g = -P, sin 6a = 0 On the other hand, by equations 33 unit id corresponds to armature current ia = cos 0a, 4 = cos 06, ic = cos 0c which, regardless of rotor position, produce fluxes d = Pd(cos2 6a + COs2 06 + cos2 6c) = f Pd and q = —P9(cos 0a sin 0a + cos 06 sin 06 + cos 0c sin 0c) = 0

22 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE Now let us designate as (x^d) the space-fundamental component of air-gap flux linkage produced by an armature current id. Then xad is a quantity slightly less than xd, and both the mutual inductance Xafd between field and armature direct axis and the mutual inductance Xand between the (perhaps imaginary) full-pitch, direct-axis amortisseur circuit and the armature direct axis are approximately equal to xad in per unit. In a large number of problems it has been customary to lump all the amortisseur circuits into one equivalent full-pitch circuit in each axis. In this case, the direct-axis circuits have generally been used with only a single mutual inductance xad among all three circuits. Now instead of considering the machine air-gap fluxes, which are after all not directly measurable, we may see what these definitions mean in terms of the terminal voltages. From equations 32, if the machine is running steadily at synchronous speed (p6 = 1.0) * and open circuit (id = iq = 0), then \l/g — ^0 = 0 also, and e^ = 0, eq = \[/d. Also from the rotor-circuit equations 39 and 40, all the amortisseur currents lid, I2d, etc., /j9, I2q, etc., are zero and From equations 37, the direct-axis armature flux linkages are td = Xafdlfd (41) and, since eq = td, it is evident that for normal armature terminal voltage eq = 1.0, the required per-unit field current is Ifd = l/(Xafd), while the required per-unit field voltage is e/d — Rfd/Xafd = rfd/xafd. Now, if we know the required no-load rated-voltage field voltage and current (neglecting saturation) in volts and amperes, we have direct relations to calculate the base field quantities. That is, if when the actual field current is //0 amperes the per-unit field current is l/Xafd, the base field current is //6 = (Xafdlf0) amperes. Similarly, if the actual field voltage is e/0 volts when the per-unit field voltage is Rfd/Xafd, the base field voltage is e/6 = (Xafdef0/Rfd) volts. On the other hand, if the base quantities are known, the per-unit machine impedances may be calculated as a A /0 and * The unit of time is that required for the rotor to move one electrical radian at synchronous speed. For example, for a normal system frequency of 60 cycles per second, the unit of time is 1/2ir60 — %^^ second.

SUP TEST FOR xrf AND x, 23' Methods for calculating all the per-unit quantities from design data are discussed in detail in references 7. Slip Test for x and x If balanced steady-state armature currents IB = » COS t «6 = * cos (t - 120°) (42) ic = i cos (t + 120°) are applied and the rotor is again at synchronous speed so that 0 = 00 + i (00 is the rotor position at zero time), then Ja = I COS (0 — 00) *6 = *cos(0 -00 - 120°) (43) ic = i cos (0 - 00 + 120°) and, by equations 20, id = * cos 00 iq = —i sin 00 (44) i0 = 0 The armature flux linkages are fa = —xdi cos 0o ^a = +xqi sin 00 (45) ^0 = 0 and the terminal voltages, with armature resistance neglected t. arc, by equations 32, ed = —tq — —Xyi sin 00 eg = +td = — Xdi cos 00 (46) e0 = 0 From equations 35, the voltage of phase a is ea = —xqi sin 00 cos 0 + x,ii cos 00 sin 0 (47) Thus, if we change 60 slowly from zero to 90°, ca changes from ea =• -\-Xdi sin t to ea = +xqi sin t. The other two phase voltages, eb and ec, t The armature resistance is usually loss than 1 per cent, while the steady-state armature reactances are of the order of magnitude of 100 per cent.

24 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE also vary in the same way, and the flux linkages (by equations 45 and 34) as well. The direct- and quadrature-axis steady-state, or synchronous, reactances of a synchronous machine may be measured in this way by supplying the terminals of a synchronously driven and unexcited machine with balanced voltages and slowly varying the rotor phase angle 00. This is the so-called slip test, but we must note that because of the very small field resistance the slip must be practically negligible for it to be successful. Short-Circuit Test for xd Another way of measuring the direct-axis armature reactance Xd is from measurements of the steady-state armature open-circuit voltages and short-circuit currents. With fixed field excitation voltage e/d, the field current is, by equation 2, or, by equation 39, //' = — Rfd All amortisseur currents are evidently equal to zero. On open circuit, the armature flux linkages are, by equations 24, X Yd = Xa y or, by equations 37, / v T Xafd Yd = .X.afdlfd = — — «fd Rfd The armature terminal voltages are, by equations 32, ed = 0 ^afd Xafd eq = - e/d = -r— efd = E (48) Tfd Kfd e0 = 0 where E is introduced for convenience and may be considered as the field excitation measured in terms of the terminal voltage that it would produce on open-circuit, normal-speed operation. Note also that E is the field excitation as normally used in the steady-state vector diagram, and even for many transient problems in those cases where the effect

POWER OUTPUT 25 H=0 Xafd r Xafd 6fd - f? E Ifd = Xd Xd Rfd Xd Xd of the amortisseur may be either neglected or considered only approximately as simply a more or less arbitrarily added damping effect. If the field excitation is given as E rather than e/d, it is evident that the first equation 36 may still be written in the form ^ = G(p)E — Xd(p)id, where now G(0) = 1.0 instead of xafd/rfd. For unity terminal voltage (eq = E = 1.0) the field voltage must be e/d = Tfd/xafd. On short circuit e^ = eq = 0, and with negligible armature resistance td = tq = 0. Then equations 37 and 38 give and id = —//d = —^ = - = - (49) Xd Xd Kfd Xd Xd Thus Xd may be found from the ratio of the steady-state open-circuit voltage to the steady-state short-circuit current, neglecting saturation. Actually it is found convenient to calculate Xd as the ratio of the field current Ifd = Xd/Xafd required to produce unit armature current (by equation 49) on short circuit to the field current Ifd = l/Xafd required to produce unit terminal voltage on open circuit. This ratio is directly the per-unit reactance. Zero-Sequence Reactance The zero-sequence armature reactance may be measured by impressing zero-sequence currents ia = ib = ic = i cos t (and by equations 20, id = iq = 0, and i0 = i cos t). Then, by equations 36, td = "A« = 0, and \p0 = —xqi cos t. Equations 32 now result in ed = eq = 0, and eo = +x0i sin t + ri cos t. Note that now armature resistance may not be negligible (although it probably will be) since the zero-sequence reactance x0 is small (3 to 10 per cent) compared to Xd or xq. Power Output The instantaneous per-unit power output of a three-phase synchronous machine is given by P = \(eaia + ebib + ecic) (50) where the factor % is introduced so that with balanced operation at unity power factor and with voltages and currents of unit magnitude the power output is unity. The power is output rather than input because of the original definition of the sign of armature current (see equation 1).

26 MATHEMATICAL DESCRIPTION OF A SYNCHRONOUS MACHINE Now eliminate the phase quantities by substituting from equations 33 and 35. We obtain the power in terms of direct-, quadrature-, and zero-axis quantities as P = edid + eqiq + 2eoi0 (51) For balanced current and voltage of unit magnitude (V e2d + e2q = ~Vi2d + i2q =1.0 and eo = to = 0) and of unity power factor (iq/id — eq/ed), the power is again unity. As an example we shall consider the steady-state power fed into a network of negligible impedance (an infinite bus) and with a voltage of magnitude e. Suppose the open-circuit voltage E [E = (xafd/rfd)efd as in equation 48] of the synchronous machine is ahead of the corresponding bus voltage by a constant angle 5. That is, if by equations 35 the open-circuit machine voltages are ea = — E sin 0 eb = -E sin (0 - 120°) (52) ec = -E sin (0 + 120°) Then the system voltages are ea = —e sin (0 — 5) = + e sin (5 — 0) eb = -e sin (0 - 8 - 120°) = +e sin (8-0 + 120°) ec = -e sin (0-8 + 120°) = +e sin (8 - 6 - 120°) or, expanding the sines, ea = e sin 8 cos 0 — e cos 8 sin 0 eb = e sin 8 cos (0 - 120°) - e cos 8 sin (0 - 120°) ec = e sin 5 cos (0 + 120°) - e cos 5 sin (0 + 120°) whence, by comparing with equations 35, we see that ed = e sin 8 eq = e cos 8 (53) e0 = 0 In the steady state the currents id and iq may be found from equation 32: ed = -tq - rid eq = +fa — nq (54)

POWER OUTPUT 27 where, by equations 37 and 38, fa = +Xafdlfd — Xdid By equation 39, By equation 48, Then or /Xafd\ fa = + [ -jt- J efd - x&d fa = E — Xdid fa = —Xqtq eq = E — x^d — riq —red + xq(E — eq) id = to = XdXq + r2 +xded + r(E - eq) xdxq 4- r2 (55) (56) (57) In terms of the bus voltage e, as given by equations 53, the currents are ■re sin 5 + xq(E — e cos 5) U= t0 = XdXq + r2 +Xde sin 5 + r(E — e cos 5) (58) XdXq + r2 The power output is, by equation 51, P = e = tan-1 — = 36.8° 0.8 sin (« + )i (114) [pxq(p) + r]iq = whence . _ [pxq(p) + r]eM + [pxd(p) + r][pxq(p) + r] + 7—7 *d (H8) A(p) * Note that in these equations, as in all the rest of this book, only the reciprocal system of inductance coefficients is used, but we have returned to the use of lower case x's, as remarked in Chapter 2.

56 THREE-PHASE SHORT-CIRCUIT CURRENT By comparing equation 118 with the first equation 36, it becomes evident that _ _ p(XlldXafd - XfidXaid) + X«f _ 2 2 P(XlldRfd (120) Similarly, for the quadrature axis, we may start from equations 40 (with I2q = Izq = 0) ar>d equation 38. The final expression for the quadrature-axis operational impedance is *9