The Performance And Design Of Alternating Current Machines [Third ed.] 8123910274

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THE PERFORMANCE AND DESIGN OF ALTERNATING CURRENT MACHINES TRANSFORMERS, THREE-PHASE INDUCTION MOTORS AND SYNCHRONOUS MACHINES BY

M. G. SAY Ph.D., M.Sc., A.C.G.I., D.I.C.,

F.R.S.E.

FORMERLY PROFESSOR OF ELECTRICAL ENGINEERING, HERIOT-WATT COLLEGE, EDINBURGH

THIRD EDITION

CBS CBS PUBLISHERS & DISTRIBUTORS PVT. LTD. New Delhi • Bangalore • Pune • Cochin • Chennai (India)

ISBN: 81-239-1027-4 First Indian Edition : 1983 Reprint: 1998,1999,2000,2002

This edition has been published in India by arrangement with M/s. Pitman, London All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system without permission, in writing, from the publisher.

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PREFACE TO THE THIRD EDITION The original plan of combining performance with design has proven to be an acceptable one, particularly as the emphasis is on th« performance rather than on the design. Most readers are user: rather than designers, and they will find as comprehensive a dis cussion of a.c. machines, their basic theory of operation and applica tion, as can be concentrated into a. volume of this size. The con siderable range of a.c. machines has made it necessary to limit th< scope to transformers, three-phase induction machines and syn chronous machines. All other machines are to some degree specia and of subsidiary importance. The book is not intended as an instruction in commercial design that is an art best knowr to practising designers. An adequate study of turbo-alternator design, for example, would take a volum * in itself, at least one-half of which would be devoted to subject matter of entirely mechanical concern. In this third edition the opportunity has been taken of a complet conversion to rationalized M.K.S. units. All formulae, whcthe concerned with electrical, magnetic, mechanical or thermal quan tities, are consistent among themselves in respect of units. Only ii numerical examples are decimal multiples or submultiples employed if convenient, and where ratings are given in units of MVA. o kVA. instead of the basic volt-ampere, the fact is stated. The text has been largely re-fashioned to give emphasis to L essential unity of all electromagnetic machines, a unity that ha been obscured by the technological accretions in which each clas of machine has, in the last half-century, become separately embedded Further, an introduction is now given to the general circuit theor of machines, a powerful method of dealing with their behaviou under transient conditions. Thanks are again due to the author’s good friends in the industry whose help has been warm and unstinted. Mr. E. Openshaw Taylo> B.Sc., D.I.C., Assistant Professor of Electrical Engineering at th Heriot-Watt College, has rendered invaluable help and criticr advice during the new revision. He is also the author of a companio volume The, Performance and Design of A.C. Commutator Motors. The author has had many letters with suggestions for improve ment of the text, and will continue to welcome the comments of hi readers

M.G.8.

Heriot-Watt College, Edinburgh vii

UNIT SYSTEM Besides the usual “practical’' volt (V.), ampere (A.), ohm (O.), mho (U«)» henry (H.) and farad (F.), a self-consistent unit system with a rationalized M.K.S. basis is used throughout. The principal units are listed below, with their abbreviations:

Length, mass, time Area, volume Density Force Work and energy Power Torque Pressure Angle Velocity: angular linear Rotational speed Inertia Temperature Thermal resistivity Resistivity

*M< -netic flux, linkage Magnetic flux-density Magnetomotive force Magnetizing force Reluctance Permeance Permeability (abs.) Specific magnetic loading Specific electric loading Current density

metro, kilogram, second m., kg., «. m.2, m.3 sq. metre, cu. metro kg./m.3 kilogram por cu. metre N. newton m.-N., J. metro-newton — joule W. joule per sec. = watt N.-m. newton-metre N./m.2 newton per sq. metre rad. radian rad./s. radian per sec. m./s. metre per sec. r.p.s. revolution per sec. kg.-m.2 kilogram-sq. metre degree Celsius (centigrade) °C. thermal watt per metre per Q. sq. metre per deg. C. ohm per metre per £2-m. sq. metre weber, weber-turn Wb. VVb.-T. weber per sq. metre Wb./m.8 ampere-turn AT. amp.-turn per metre AT ./in. amp.-turn por weber AT./Wb. weber por amp.-turn Wb./AT. weber per sq. meter per amp.-turn per metro H./tn. * Wb./m. weber per sq. metro amp.-cond. per metre AC./in, A./in.‘ amp. per sq. metre

Conversions— Force; 1 N. = 1 J./m. - 0-102 kg. (force) = 0-225 lb. (force). Enenzv 2 1 J. ■= 0'738 ft,-lb, —• 0’24 Power; 1 \V. - -738 ft.-lb./s. =■ 1-34/1000 h.p. * Inertia: I kg.-rn.1 » 0-738 slug-ft? = 23-7 lb,-ft. . Torque: 1 N. *m, " 738 lb.-it.

ix

I

CONTENTS PAGE

PREFACE

.

vu ix xvii . XIX

.

UNIT SYSTEM

READING GUIDE .

•

.

.

.

.

.

LIST OF SYMBOLS CHAPTER 1

Introductory .

.

.

.

.

.

1

Alternating current machines—Alternating current system—Circuit behaviour—Transient and steady states—Steady-state convAntions —Power—Per-unit values

n Fundamental Principles.................................................................. 6 CHAPTER

The electromagnetic machine—Induction and interaction—Law of induction—Law of interaction—Generators and motors—Trans­ formers—Classification—Three-phase complexor diagram—Sym­ metrical components CHAPTER

Transformers: Theory

.

m

.

.

.

.

.11

Use of the transformer—Mutual induction—Linked electric and magnetic circuits—Theory of the power transformer—The com­ plexor diagram—Magnetizing current and core loss—The equivalent circuit—Complexor diagram of the transformer on load—Efficiency —Short-circuit loss—Efficiency on load—Regulation CHAPTER IV

Transformers: Construction ....................................................... 32 Constructional parts—Core construction—Core sections—Con­ structional framework—Windings—Insulation—Leads and terminals —Bushings—Cooling—Natural oil cooling—Forced oil cooling— Internal cooling—Tanks—Transformer oil CHAPTER V

Transformers: Operation...................................................... 57 Equivalent circuits—Installation—Noise—Effect of load-factor on losses—Connections for transformers—Nomenclature—Star/star— Delta/dolt a — Star/delta — Zig-zag/star — Three-phase/six-phase — General remarks on three-phase connections—Three-phase to twophase connections—Scott three/two-phaso connection—Le Blanc three/two-phase connection—Three/one-phase connection—Autotransformers—Equivalent circuit—Three-phase auto-connection— Simple voltage regulation by auto-transformer and reactor— Capacitor booste/—Voltage regulation and tap-changing—Tappings

xi

xii

CONTENTS —Off-circuit tap-changing—On-’oad tap-changing—Principles of control—Transformer-typo voltage regulators—Moving-coil rogtilator—Parallel operation—Voltage ratio—Impedance—Polarity— Phase-sequence—Tertiary windings—Primary current—Equivalent circuit: regulation—Tertiary windings in star/star transformer * — Rating of tertiary windings—Harmonics in transformers— Single, phase transformers—Three-phase banks of single-phase transformers —I hree-phase transformer units—Effects of transformer harmonies — Harmonic current compensation — Transients — Switching-in phenomena — Voltage surges — Surge protection — Mechanical stresses—Radial force—Axial compression—Bracing—Low-voltage t ransformers—High-voltage testing transformers—Industrial-fre­ quency testing transformers—-Transformers for other tests—Trans­ former protection: Buchholz system—Classification of transformer protection—Impedance of three-phase transformers to phasesequence component currents CHAPTER VI

Transformers: Testing............................................................... 123 Objects of tests—Phasing out—Polarity—D.C. resistance—Voltage ratio—Magnetizing current and core loss—Core-loss voltmeter— Impedance and I2li loss—Efficiency—Temperature rise—Back-toback connection—Delta/delta connection—Equivalent short-circuit run—Equivalent no-load run—Insulation tests-—Impulse tests CHAPTER VII

Transformers: Design

134

Frames—Thermal rating—Momentary load limitations—Output— Specific loadings—Allocation of losses—Design—Coils and insulation —Reactance—Cylindrical concentric coils, equal length—Cylindrical concentric coils, unequal lengths—Sandwich coils—Mechanical forces —Magnetizing current—Cooling tanks—Specification—Example: 300-kVA., 6 600/400-440-V., 50-c/s. three-phase, core-type power transformer—Example: 125-kVA., 2 000/440-V., 50-c/s., single­ phase, shell-type power transformer—Example: 25-kVA., 6 600/ 440-V., 50-c/s., three-phase, core-type transformer—Example: 1 000-kVA., 6 600/440-V., 50-c/s., three-phase, core-type transformer —Example: 15,000-kVA., 33/6-6-kV., 50-c/s., three-phase, corn-type transformer CHAPTER VIII

Electromagn^tc Machines

.

.

.

.

.

.167

Energy conversion—Operating principles—Two-polo machines— Reluctance motxirs—Machines with armature windings—Windings —Stator and rotor windings—M.m.f. of windings—Conditions for torque—E.m.f. production—Machine analysis—Equivalent circuits CHAPTER IX

Magnetic ('ircirts .

.

. ■

.

,

.

, 174

Laws of the magnetic circuit—Coro losses—Surface and pulsation os-ics—-M.igm ti; materials-—Laminated mutm inis-— Materials for Mo;ulv fluxe.t ('ab ulation of magnetic circuit--—Air gan—Gapdennity db-tnb . n.nl 1U1(1 nppliront flux-donsitv— I 1 teeth-Magnetizing current—Leak ago- Leakage of salient

xui

CONTENTS

PAGE

poles—Leakage of non-salient poles—Armature leakage—Slot leak­ age—Overhang leakage—Zig-zag leakage—Differential leakage— Phase reactance of an induction motor—Phase reactance of a synchronous machine CHAPTER X

Armature Windings.................................................

. 196

Types of windings for a.c. machines—Armature windings— Conduc­ tors, turns and coils—Connections between coil groups—Nomencla­ ture—Single-layer windings—Double-layer windings—Integral-slot windings—Fractional-slot windings—(.‘oil i.pan—Types of double layer windings— Choice of armature winding—Coil * manufacture Slots—E.m.f. of windings—Sinusoidal gap-flux—E.m.f. of windings: general ease—Distribution factor— Coil span factor—Phase connec­ tion—The e.m.f. equation—Tooth harmonies—Annature reaction— Rotating field—Magnitude of annature reaction—Field circuit equivalents of armature reaction—Phase-sequence components— Conductors—Eddy -current Josses—Conductors in sloi-—Conductors in overhang—Transformers—Insulation CHAPTER XI

237

Ventilation................................................. Dissipation of heat—Heat How—Types of enclosure—Cooling-air circuit—Quantity of cooling medium—Flow of heat to cooling gas— Exponential tempei ature-rise/time curves—Cooling coefficient— Temperature -rise—R ating CHAPTER XII

Induction Machines: Theory.................................................

249

Development—Action of the plain induct ion motor—The generalized transformer—Cage and slip-ring rotors—Cage rotor—Slip-ring rotor —Machine with constant flux—Simple theory of the induction machine with constant flux—Slip—E.m.f.’s—Currents—Power— Torque—Generator action—Torque/slip curves—Mechanical power —Simple equivalent circuit and circle diagram—Stator circle dia­ gram— No-load loss--Short-circuit loss—Input and output-—Torque —Machine with primary impedance and varying flux—Flux variation —Equivalent circuit development—Deductions from the equivalent circuit—Losses and efficiency—Calculation of performance— Current diagram—Inversion—Current locus of an induction motor — Practical current diagram—Construction from no-load and shortcircuit data—Equivalent cage rotor CHAPTER XIII

induction

Machines: Performance and Control

.

. 287

Performance—Types, ratings and applications— Motor speeds— Starting—Starting of slip-ring motors—Rotor rheostat starter— Choice of rotor voltage—Starting of cage ne-tors—Effects of reduced stator voltage—Methods of starting load phase-angle. y specific cost. angle between coil e.m.f.’s (elec.). y starter coefficient. p resistivity. d ■ current-density. pf thermal resistivity of insulation. d deflection. p resistor stepT angular velocity. A wavelength.

a, b, c'phases A, B, C. a acceleration. active component. alternating component. armature. axial. duct. ad armature direct-axis. aq armature quadrature-axis. av average. b brush. c cage-rotor. capacitor. coil. commutator. compensating. conductor. converter. copper.

Subscripts c core. critical. I2R. d conducted, direct, disturbing, duct. eddy-loss. e coil-span. electrical. e.m.f. \ eq equipotential. eu e.m.f., unsaturated. f field. form-factor, load-factor, friction. .9 gap. h harmonic, 1 hysteretic.

LIST OF SYMBOLS i

I

tn

me mm mt n o p

pd ph pl pq 9 r

rd rq «

xxi

Sunsciurrs—(contd.) current. s space. induction. starting. initial. steady-state. inlet. synchronous, synchronizing. insulation. *'c short-circuit. iron. sci ideal short-circuit. leakage. sd direct-axis synchronous. limb. direct-axis syn., unsaturated. line. xl shoe-lcakago. load. sq quadrature-axis synchronous. loss. t tensile. distribution-factor. tooth. magnetizing, torque. maximum. total. mechanical. transient. mutual. turn. mean-conductor. /J one-third distance maximum mechanical. v voltage. mean-turn. w width. normal-rated. winding-factor. harmonic-order. window. opening. x leakage, overall. reactance. overhang. !/ yoke. pth conductor. z impedance. pole. zigzag. power. 0 characteristic or surge. pressure. initial. pulsation. natural-frequency. direct-axis pulsational. no-load. phase. zero-phase-sequence. pole-leakage. 1 input. quadrature-axis pulsational. primary. quadrature-axis. stator. radial. synchronous. radiated. upper-limit. reactive component. 2 auxiliary. remanent. lower-limit. resistance. outer-cage. resultant. output. retaining-ring. rotor. rotational. secondary. direct-axis rotational. 3 inner-cage. quadrature-axis rotational. tertiary. + positive-phase-sequence. shoe. — negative-phase-sequcnce. slot.

CHAPTER I

INTRODUCTORY 1. Alternating Current Machines. The synchronous generator, con­ verter and motor, the transformer, and the induction motor, form the subjects of study in this book. They are of sufficient importance to account between them for almost all of the output of manufacturers of electrical machinery, apart from d.c. motors. 2. The Alternating Current System. The wide application of a.c. machines may be said to be due to (a) The heteropolar arrangement. (6) The transformer. Electrical energy is particularly useful by reason of the ease with which it can be distributed and converted to other desired forms. At present, the chief source of electrical energy is the mechanical form, either direct (as in hydraulic storage reservoirs or rivers) or from chemical energy via heat. A convenient direct method of converting mechanical to electrical energy is by use of the mechan­ ical force manifested when a current-carrying conductor is situated in a magnetic field. The movement of a conductor through a polar magnetic field having consecutive reversals of polarity results in the production of an induced e.m.f. which changes sign (i.e. direction) in accordance with the change in the magnetic polarity: in brief, an alternating e.m.f. Thus all ordinary electromagnetic machines are inherently a.c. machines. The wider use of electricity has necessitated the development of systems of transmission and distribution. Transmission systems are required (a) to enable sources of natural energy (e.g. waterfalls or storage lakes or coal fields), far distant from the centres of energy demand (e.g. large works, industrial areas or cities), to be economic­ ally utilized ; (6) to enable generating plant to be concentrated in a few large, favourably-situated stations; and (c) to permit of the interconnection of networks or distributing areas to ensure reliability, economy and continuity of supply. Frequent reference is made to the companion volume, The Performance and Design of Direct Current Machines, by Dr. A. E. Clayton. For brevjty, this is contracted to [A] in the text. Since references are valuable if the> Jo not involve unnecessary searching, such references are confined chiefly to a few easily-obtained textbooks, follows— [A] Performance and Design of D.C. Machines. A. E. *Clay on and N. N, Hancock (Pitman). [B] Electrical Engineering Design Manual. M. G. Say (Chapman & Hall). [Cj Performance and Design of A.C. Commutator rMbtors, E. O, Taylor (Pitman). I

2

DESIGN OF ALTERNATING CURRENT MACHINES

A high voltage is desirable for transmitting large powers in order to decrease the DR losses and reduce the amount of conductor material. A very much lower voltage, on the other hand, is required for distribution, for various reasons connected with safety and convenience. The transformer makes this easily and economically possible. Thus the fact that generation is inherently a production of alternating e.m.f.’s, coupled with the advantage of a.c. transforma­ tion, provides the basic reason for the widespread development of the a.c. system. The collateral development of a.c. motors and other methods of utilizing a.c. energy has been a natural step. It must by no means be assumed that the a.c. system represents a finality. Apart from the superiority of d.c. motors in certain cases, particularly where speed-control is desired, there are signs that a high-voltage d.c. transmission system may evolve in the near future. However, while changes, small or great, in methods of electrical production and use may be confidently expected, it is probable that a.c. machines will remain important components of electrical generation, application and utilization. 3. Circuit Behaviour. A voltage v, applied to a circuit of series resistance R and inductance L, produces a current such that v — Ri 4- L(di{dt) at every instant. If v is a constant direct voltage V, the steady-state, current is I = V/R, because di/dt — 0 in the absence of changes; but whenever the conditions are altered (e.g. by switching or by changing the parameters) the new steady state is attained through a transient condition. The instantaneous current can then be written i = i8 + it, with ie the final steady-state current and it the temporary transient component. The time-form of it depends only on the parameters, and for the RL circuit considered is the exponential it — i0E~lRlL^ = toexp ( — R/L)t. Fig. 1 (a) shows the steady-state and transient components and the total resultant current for an initiation transient when the drive­ voltage is V. As the current is zero at the switching instant (and cannot rise thereafter faster than VIL), then it must cancel is initially: hence * = L + L = h-h exP (— RIL)t = (F/7?)[l - exp (— RfL)t] This is a well-known case of practical interest in the switching of d.c. field circuits. If v is the sine voltage vm sin cut, the steady-state current is the sinewave i„ = (vm/Z) sin (art — 0), where Z = \/(R2 + co2Z2) = y/(R2 4- A2), and 0 = arc tan (X/R). Suppose the voltage to be switched on at some instant in its cycle corresponding to t = /0: the steady-state current for this instant would be (vmlZ) sin (cot0 — 0). But the current is actually zero, so that again it initially cancels is.

INTRODUCTORY

3

Subsequently it decays exponentially as before. The current following circuit switching is therefore i = i„ + iti giving i = (vJZ) [sin (cot — 0) - sin (coi0 - 0). exp (- R/L)(t - /0)]. In machine circuits R is often much smaller than A” so that Z — A and 0 ~ 90°. For these conditions, approximately, i cos cot 4- cos cotQ. exp (- R/L)(t - the rate Z?2/2 of energy abstraction from the secondary must be provided for by the introduction of energy at an equal rate E{lx (= F^d into the primary. Example. The core of a three-pnase, 50-cycle, 11 000/550-V. mesh/star, 300-kVA., core-type transformer has a r/ross cross-section

18

DESIGN OF ALTERNATING CURRENT MACHINES

of 400 cm.2 Find (a) the number of h.v. and l.v. turns per phase, (6) the e.m.f. per turn, and (c) the full-load h.v. and l.v. phase currents. The core density should not exceed 1*3 Wb./m.2 The phase-voltage ratio for mesh/star connection is 11 000/(550/^/3) = 11 000/317 V. The net core section, i.e. allowing for core-plate insulation, is 0*9 X 400 = 360 cm.2 Assuming the induction density to be 1-3 Wb./m.2, the e.m.f. per turn from eq. (6) Et = 4-44.50 . (360 x IO-4 X 1-3) = 10-4 V. The number of secondary turns is T2 = 317/10-4 = 30-5. say 31 turns, '

-

~1~T

since fractions of a turn are not possible. The adjustment to 31 turns, however, makes the e.m.f. per turn now Et = 317/31 = rr10-2 V.,a —r— and the actual core density B = 1-3 (10-2/10-4) = 1-275 Wb./m.2

The primary turns are most directly obtained from the secondary turns and the voltage ratio: T} = 31 (11 000/317) = 1 075 turns.

The rating of each phase is 100 kVA., so that the currents are Ix ’ = (100/11 000)103 = 9-1 A., and 12 = (100/317)103 = 316 A. The primary ampere-turns are I1Tl = 9-1.1 075 = 9 800 A.T.; and for the secondary LT., = 316.31 = 9 800 A.T. The magnetizing current would scarcely alter this equality: with a core-length of 40 cm. per phase and a density of 1-275 Wb./m.3, the excitation required would be about 14 A.T. per cm., or a total of 14 X 40 = 560 A.T. peak value, i.e. 400 A.T. r.m.s. This repre­ sents a magnetizing current of about 4 per cent of full-load current. 4. The Complexor Diagram. Much of the behaviour and operating characteristics of the transformer can be derived from its “vector” diagram It is also convenient for introducing the modifications due to the divergence from the “ideal” state of the actual transformer. Magnetizing Current jsnd Core Loss. Although the iron, core is highly permeable, it is not possible to generate a magnetic field in it without the application of at least a small m.m.f. Thus even

19

TRANSFORMERS: THEORY

when the secondary winding is on open circuit and giving no current, a small magnetizing current is needed to maintain the magnetic circuit or core in the magnetized state. The magnetizing m.m.f. is a function of the length, the net cross-sectional area, and the per­ meability of the iron path. The m.m.f. of the primary circuit on no load is of the order of 5 per cent of its m.m.f. on full load. The magnetizing current (symbol IOr) is to be considered as in time phase with its associated flux: a reasonable assumption, as when there is no magnetizing current, and consequently no m.m f., there will be no flux. (See Fig. 13.) For present purposes ICr is considered to be sinusoidal, although this is not the case for the saturation values normally em­ ployed in practice. The pulsation of the flux in the core is pro­ ductive of core loss, due to hysteresis and eddy-currents, as described in Chapter IX. The continuous loss of energy to the. core, which pro­ duces heat eventually dissipated from the trans­ former, requires a continuous supply from the electrical source to which the primary is connected. Since the only way in which an electric circuit can furnish energy is for a current to flow under the influence of a voltage, there must be a current component IOa which supplies the losses. The product thus represents the power FlGe 13 no-load input to supply core loss. [The voltage E1 is Comfbexor concerned (rather than VJ because this is the voltDiagram age directly associated with the^magnetic flux.) On no load, the currents ZOr for magnetization and 70a for losses will flow. The currents are in phase quadrature, since one is in phase with while the other is in the direction of Ey, which in turn is in phase-quadrature to Ow. The components 70a and ZOr have as resultant the no-wad current IQ. so that

4-

V + V)

.

(11)

The reactive component of the no-load current, 7()r, cannot physically be dissociated from the active component 7tla, for the latter is an essential part of the magnetization of the core, accounting for the hysteresis and eddy-current effects which modify and complicate the phenomenon of magnetization in ferrous metal. The no-load current makes a phase-angle Interend q mediate

Ai Une

Near earthed end

Cd) "Travelling wave front in H.V. wdg. Fig. 74. Travelling Waves

in

H.V. Winding

with the termination (e.g. earth or open-circuit). The voliage to earth existing at any point is then the resultant of the initial and travelling-wave voltage distributions at the instant considered. The voltage across an individual coil is easily seen from Fig. 74 (e) to have a pulse form due to the passage of the travelling wave. Similar considerations lead to the form of the voltage between successive coils. The voltage distribution is affected by the capaci­ tances and therefore by the design of the insulation, and it is essential to co-ordinate the insulation with the effect that it will have on the voltage distribution. Surge Protection. The avoidance of breakdown under surges includes (a) the provision of reinforced end-turn insulation; (6) the use of protective devices; and (c) special types of stress-grading construction. Method (a) has been employed in practice for many years, but is now suspect in view of greater knowledge of surge behaviour. End-turn reinforcement may actually make conditions worse by reducing c compared with C and intensifying the surge voltage build-up across the turns and coils at the line end. External or internal devices such as surge-absorbers comprise series inductors at the line end of the h.v. winding to reduce the 5-(T.a2)

108

DESIGN OF ALTERNATING CURRENT MACHINES

steepness of the wave-front, the losses absorbing surge energy and reducing the voltage reaching the h.v. winding. Method (c) aims to produce for all conditions a uniformity of voltage distribution. The two basic types are— 1. Shields. If the coil-to-earth capacitances C can be neutralized by providing new capacitances to line (on precisely the same

) Surge voltage distribution

(C) Special coils

Surge Voltage Grading

principle as is used for grading transmission-line suspensioninsulator strings), then the coil-to-earth currents through C will be directly provided by line-to-coil currents through Clt leaving the inter-coil currents through c all equal to produce a uniform distribu­ tion over the winding. This may be done by mounting, outside and around the winding, insulated metal shields connected to line, Fig. 75 (a). The effect on the electric field is shown diagrammatically in (6), the figures in­ dicating equipotentials corresponding to a surge of value 100, for an un­ shielded and for a shielded transformer. 2. Interleaved Turns. Two normal disc-type coils are shown in Fig. 75 (c). If the order of turn-connection is altered to increase the voltage between adjacent turns, their capacitance cur­ rent is increased so that the effective 1 series coil-capacitance c is raised. This tends to counter the effect of the earth Fig. 76. Components of capacitances, for it is obvious that, if Leakage Flux c >> (7, the voltage distribution under surge conditions must approach uniformity. 14. Mechanical Stresses. When a transformer is loaded, the primary and secondary ampere-turns are in magnetic opposition with reference to the core, but act cumulatively with respect to the space between them (Fig. 76). The effect of this is twofold: the magnetic­ ally-excited state of the inter-coil space gives rise to— (a) Leakage flux: i.e. flux linking one or other winding;

I

TRANSFORMERS: OPERATION

109

(6) Mutual forces between windings. * The mechanical repulsive force with normal load currents is low compared with the strength of the coils. Under fault conditions, particularly those of dead short-circuit, the forces produced may be increased several hundreds of times. Thus a transformer with a 0-05-p.u. reactance will carry a fault current initially 20 to 40 (or even occasionally more) times full-load current, with disruptive forces consequently 400 to 1 600 times those produced by full-load current.

(a) Concentric.

(c) Effect of asymmetry

Fig. 77. Mechanical Forces on Coils

The forces produced may be considered as capable of analysis into the. components— (1) Radial, tending to burst outer and crush inner windings; (2) Axial compression; and (3) Unbalanced axial forces due to asymmetry. Radial Force. Any electric circuit when carrying current develops mechanical forces, tending to enlarge its area. For a coil, this implies a tendency to become circular, provided that there is no magnetic asymmetry. The mechanical force is developed by the coil current interacting with the axial components of its own flux. Considering a transformer with concentric coils, the flux initiating the mechanical forces occupies the space betwee?,, the coils. Conse­ quently the outer coil is subjected to internal pressure tending to burst it, but the inner coil is subjected to external pressure, and tends to collapse on to the core. A circular coil section is preferable in the latter as well as in the former case, as being the strongest shape mechanically for withstanding these pressures. Fig. 77 (a) illustrates the action of the hoop-stress and counter forces. In shell-type transformers the coils are rectangular, but braced by enclosure within the core for a considerable part of their length. * Norris, "Mechanical Strength of Power Transformers in Service," Proc.l.E.E., 104(A), p. 289 (1957).

110

DESIGN OF ALTERNATING CURRENT MACHINES

For an instantaneous current i in the T turns of one winding, the flux density in the annular duct between the cylindrical primary and secondary coils of a core-type transformer (Fig. 99 (a)) is Ba — p^iTILe, Its mean value in the windings is %Ba. The mean radial force per circumferential m. length and per m. of axial depth of coil is therefore p = ^(iT/Lc)(iT/Le). For the whole area LcLmt of the cylindrical face of a coil Pr = M^T)\LmtILc) newtons. On the worst fault conditions, i may reach twice the symmetrical peak short-circuit current, i.e. 2/e times full-load current, where e is the per-unit impedance. In a 100-kVA. transformer with 0-04 p.u. impedance the radial force may reach 40 tons: in a 100-MVA. transformer with 0-1 p.u. impedance it may approach 3 000 tons. Axial Compression. So far, reference has been made only to stresses produced by the axial leakage flux. The radial components of the flux, which cross the windings chiefly at the ends (Fig. 76), will give rise to axial compressive forces tending to squeeze the windings together in the middle. With a symmetrical arrangement of wind­ ings these stresses are unimportant, even on short circuit. Sandwich coils produce axial compressive stresses as illustrated in Fig. 77 (b). In shell types, the outer coils experience repulsion which is taken up by the core in the buried length, but those parts of the coils exterior to the core will have to be braced. Stresses Due to Asymmetry. Hitherto only symmetrical cases have been considered: i.e. windings symmetrically placed with respect to each other and of the same length. Ideal symmetry is never actually attained in practice, while the demand for wide tapping ranges (involving some parts of the various windings being out of circuit) makes complete symmetry quite impossible. Considering concentric windings, Fig. 77 (c), if one is displaced relatively to the other, the currents give rise to unbalanced axial forces tending to increase the asymmetry. These stresses are transferred' directly to the end supports and clamping devices, and increase with the amount of difference in length. Thus in Fig. 78 (6) and (c), end effects wifi obtain, although con­ siderably less than in case (a). It should be observed that the forces are so directed as to tend to increase the degree of asymmetry of the geometricaLarrangement of the coils. Cases (a) and (c) in Fig. 78 are typical respectively of end and centre-tapped coils. The mechanical superiority of (c) over (a) is obvious, but at the same time the use of tappings at all is an added complexity in the mechanical design. Internal faults, involving short-circuits between parts of a winding or tapping connections, may produce very large currents in restricted parts, and will increase asymmetry and mechanical stresses in consequence.

TRANSFORMERS: OPERATION

111

The self-compression effect is related to the radial force pr in the ratio a/Lc. Additional forces due to asymmetry can be roughly estimated from the cross-flux developed by the out-of-balance m.m.f. k(iT), Fig. 79. The windings are deemed to be enclosed between iron surfaces separated by a distance 2x = 2 (a + + ^2)The cross-flux density has the maximum value [i^k[iT)j2x. For the case shown the axial force will be that produced by the mean cross­ flux density (one-half the maximum) acting on T turns, i.e.

Va = ^k{iT!2x} . Lmt. (iT) — ^k{iT}2Lmtj2{a + \ + &2) newtons.

The stress can become very large if the asymmetry is marked.

Fig. 78. Out-of-balance Cross Ampere-turns

Fig. 79. Axial Out-of-balance

Bracing. In so far as the stresses are inwards axially, they are taken by the windings themselves. Inward radial stresses are.passed to the formers, packing pieces and cores. Outward axial stresses must be withstood by the end insulation. A well-constructed trans­ former will have a suitable choice of conductor dimensions and intertum insulation, and coils well supported and braced, with the compressive stresses kept in view. .The eqd supports are not easily arranged to give good insulation and at the same time great mechani­ cal strength, so that as far as possible the need for the latter quality must be avoided by maintaining symmetry, or by a suitable distri­ bution of the unpreventable asymmetry caused by tappings, connections, etc., as indicated in Fig. 55. 15. Low-voltage Tiansformers. The use of the electric arc furnace has become increasingly common in recent years, particularly in the chemical and metallurgical industries. Such furnaces require large current at low and variable voltage, and demand special considera­ tions in transformer design. Where possible, a shell design is used, as the windings on the low-voltage side can be sectionalized with greater uniformity of impedance. A three-phase furnace requiring

J12

DESIGN OF ALTERNATING CURRENT MACHINE® J

a single turn per secondary phase may, however, be of the core type, Similar considerations apply to electro-chemical loads. The usual parallel connection of the secondary coils (on account of the large currents concerned) demands an even distribution of the leakage reactance on both primary and secondary sides: otherwise circulating currents may flow between parallel-connected sections, The secondary is frequently constructed of sheet-type single turns, kept thin to avoid excessive eddy-currents. The turns are stamped to shape from copper sheet. The phases are led out with the connections arranged for minimum leakage. The reactance of leads may be seriously large, particularly if the transformer is some distance from the furnace. Voltage regulation is provided by tapping the high-voltage side, to avoid complicated heavy-current connections. The tappings must be arranged symmetrically to avoid unbalanced mechanical forces in case of a short circuit—a not infrequent occurrence. 16. High-voltage Testing Transformers. At present h.v. testing technique requires four kinds of h.v. supply— (a) Industrial-frequency alternating voltages up to 1 000-2 000 kV., or less; (6) Constant direct voltages; (c) High-frequency alternat­ ing voltages up to about 1 000 kV. at about 100-200 kc/s.; (d) Surge or impulse voltages up to about 1 000 kV. or more, and of duration a few micro- or milli-seconds. Type (a) is most common, being applied to routine tests of materials, machines and ap­ paratus, and for use in con­ nection with Schering bridge measurements. Type (6) is employed almost exclusively Fig. 80. Cascade Arrangement for the testing of cables on site of H.V. Transformers (where portability is impor­ tant), for the production of X-rays, and feeding the anodes of high-speed cathode-ray oscillographs. High-frequency generators (c) are required for testing at radio frequencies, and for investiga­ tions on porcelain insulators. Type ( •

Destroyed paper Damaged wood Dissociated oil

can be made of the severity and continuance of the fault, while from the colour a diagnosis of the type of fault is possible. (See Table above.) A more definite diagnosis may be made by sampling the gas withdrawTi at the stop-cock. If the rate of generation of gas is small, the lower float b2 is un­ affected. When the fault becomes dangerous and the gas production violent, the sudden displacement of oil along the pipe-line tips the float b2 and causes a second mercury contact c2 to be momentarily closed, making the trip-coil circuit and operating the main switches on both h.v. and l.v. sides. Gas is not produced until the local temperature exceeds about 150° C. Thus momentary overloads do not affect the relay unless the transformer is already hot. The normal to-and-fro movement of the oil produced by the cycles of heating and cooling in service is insufficient to cause relay operation. With new transformers, especi­ ally those in tanks provided with cooling tubes, there may be a slow release of occluded air which collects subsequently in the relay chamber. Classification of Transformer Protection. The Table * on page 119 summarizes the causes of possible disturbance in trans­ former operation, with types of protection available.

18. Impedance of Three-Phase Transformers to Phase-Sequence Component Currents. The impedance of a transformer is independent of the phase-sequence of the applied voltage. Consequently its impedance to negative phase-sequence currents is the same as to those of positive phase-sequence. The impedance is that usually understood by the term in two-circuit transformers, while in trans­ formers with tertiary windings the impedance concerned is that corresponding to the equivalent circuit Fig. 64b on page 92. In determining the impedance to zero-phase-sequence currents, account must be taken of the winding connections, earthing and also in some cases the constructional type. For there to be a path for zero-phase-sequence currents implies a fault to earth and the flow' of balancing currents in both circuits of the transformer. * Szwander, Met.-Vickers Qaz., 22, p. 349 (1944).

transformer

Origin of disturbance OVKRCURRKNTS

A. Overloads exceeding permissible value

B. External Faults 1. Large tem­ porary currents 2. Lower sus­ tained fault currents C. Internal Faults •1. Large fault currents

Transformer features concerned

Deterioration of Insulating material

Typo of protection

System protection

Thermal

Transformer overload

2. Small fault currents

Kind of operation

Buchholz

Isolation of faulty network

Transformer Impedance should be large to limit fault current ■v ,

Alarm and isolation

Fuses or relays

Thermal

Remarks

(a) Winding Alarm, Iso­ Allows full use of trans­ temperature lation or former overload capacity. supervision tempera­ ture (h) Oil tempera­ indication Overload protection in­ ture supervi­ sufficient. Winding tem­ sion peratures may be exces­ sive i (c) Time-delayed Isolation Full overload capacity not over-current used. Isolation too rapid fuses or relays on high and too slow on low overloads

Mechanical

Mechanical

Protection

Restricted earth-fault

Balance ">

Gas onnation anticipates development of more serious faults Fault current must In­ crease to operate. High setting necessary for dis­ crimination Full discrimination. Re­ sponse to earth-faults only. Instantaneous. Each winding requires protection Full discrimination. Set tings independent of ex­ ternal factors

Overvoltages A. External System 1. Atmospheric End-turn reinforcement co-ordination Lowers Service continually main­ H.v. screening Non-linear surge crest tained resistors voltage Disturbs service continu­ Insulation co­ Protective gaps ordination ity Lowers Cables from steepness overhead lines Neutral-earthing Lowers Surge protection not effec­ Insulation in 2. Switching magnitude tive general of system and faults of disturb­ ance /

Jt. Internal I. Peak volt­ age due to secondary third harmonic 2. Resonance of third harmonic with system capacitance

Choice of suit­ Delta tertiary able connections System insulation

Eliminates third harm, voltages

Three-phase three-limb core-type transformers not troublesome

\

■

120

DESIGN OF ALTERNATING CURRENT MACHINES

Fig. 87(a) sets out the connections and zero-sequence conditions for the more usual types of two-circuit transformers, neglecting resistance. The reactance x)2 refers to ajj + °r

(A)

(B)

Fig. 87. Zero-phase Sequence Connections A. Two-circuit transformers B. Threc-circult transformers • Not applicable to 3-ph. core types. t Assuming primary current path complete.

(a) Yy (b0^1 neutrals isolated). There being no earth circuit, an infinite impedance is offered to z.p.s. currents. (b) Yy (secondary neutral earthed). An earth fault on the primary side will not result in zero z.p.s. current. The same is nearly true

TRANSFORMERS: OPERATION

121

for an earth fault on the secondary side where the transformer comprises an arrangement without interlinked magnetic flux (i.e. a three-phase shell type or three single-phase units) because there is no path for the primary equivalent currents. Tn the case of a three-phase core-type unit, the z.p.s. fluxes produced by secondary z.p.s. currents can find a high-reluctancc path like that shown in Fig. 69 for triplen harmonics, and for the same reason. The resulting z.p.s. reactance may be 30 per cent and upwards. (c) Yy {both neutrals earthed). Provided that the primary circuit is complete for z.p.s. currents, a secondary earth fault will produce z.p.s. components whose counterparts flow in the primary: and similarly for an earth fault in the primary circuit. The reactance per phase to z.p.s. currents is three times the reactance of the threephase windings in parallel, i.e. the reactance of one phase. It is therefore identical with the reactance to p.p.s. and n.p.s. com­ ponents. With three-phase core-type transformers the effect des­ cribed under (6) above may reduce the reactance to z.p.s. components by about one-tenth. (d) Yd {neutral isolated). No z.p.s. currents can flow to or from the primary or secondary circuits, but can circulate within the delta. (e) Yd {neutral earthed). For an earth fault in the secondary circuit the reactance to z.p.s. is infinite. For a primary circuit earth fault compensating currents can flow in the delta secondary. The impedance to z.p.s. is as for (c) above. (f) Dd. There can be no currents to or from either side to the associated circuit, and there will be no delta circulating z.p.s. currents. {g) Z earthing transformer. To z.p.s. currents the two windings on each limb have cancelling ampere-turns, and the impedance thereto is that due to leakage. For p.p.s. or n.p.s. currents, neglect­ ing magnetizing current, the connection offers infinite impedance. Fig. 87 (b) summarizes the more usual cases of three-winding trans­ formers with delta-connected tertiaries. The symbols x12, z23 and x13 have the significance given to them in §11, page 91. {a) Yy {both neutrals isolated). Reactance to z.p.s. currents is infinite. {b) Yy {secondary neutral earthed). There is no path for z.p.s. currents for a fault in the primary circuit. For a secondary circuit earth fault compensating currents can flow in the tertiary delta, the reactance being x^. (c) Yy {both neutrals earthed). Z.p.s. currents can flow with a fault on either primary or secondary circuit because the tertiary can carry compensating currents. Currents can also flow in the unfaulted primary or secondary provided that the z.p.s. circuit is complete, modifying the total z.p.s. reactance in accordance with the additional parallel component.

122

DESIGN OF ALTERNATING CURRENT MACHIN^

(d) Yd (neutral isolated). There are no paths for z.p.s. currents. (e) Dy (neutral earthed). Z.p.s. currents can flow when an earth fault occurs on the secondary circuit, with compensating currents in both primary and tertiary. The reactance to z.p.s. currents is + (xtxjxy3). For faults in the primary circuit the z.p.s. reactance is infinite.

CHAPTER Vl

TRANSFORMERS: TESTING

1. Objects of Tests. A transformer may be subjected to a range of tests for a variety of purposes, including— (a) Routine tests after manufacture; (b) Acceptance tests, heat runs, etc.; (c) Specialized investigations on particular details of design, per­ formance or operation. It is beyond the scope of this book to consider (c), and attention will be directed briefly to the testing for losses, performance, reactance, etc. The theory and explanation of many of the tests are contained in previous chapters. * 5

— Primary

MAAAAAA^T”!a2 ? Secondary

--- E, A? 0-1 (b)

‘2

a2

Fig. 88. Polarity Test

2. Phasing Out. In manufacture, where all connections are traceable, the phases on primary and secondary sides may readily be grouped. If not, all phases are short-circuited except a primary and a supposedly corresponding secondary. A small direct current is circulated in the primary and a voltmeter is connected across the secondary. A momentary deflection of the voltmeter when the primary current is made and broken confirms that the two windings concerned belong to the same phase. For this test all phase-ends should be separate. Difficulty may be experienced with internally connected zigzag (interconnected star) arrangements in three-phase windings. 3. Polarity. According to the specification No. 171: 1936 of the British Standards Institution, terminals are distinguished by suffix numbers in such a way that the same sequence of numbers represents the same direction of induced e.m.f. in both primary and secondary windings at any instant. Thus in Fig. 88 (a), if at any instant the e.m.f. jEj acts from A2 to in the h.v. winding, E2 will act from a2 Loaj in the l.v. winding: i.e. if at any instant A^ is positive and A2 • For further details of tests, see Electrical Engineering Laboratory Manual: Machines, by Parker Smith (Oxford). 123

124

DESIGN OF ALTERNATING CURRENT MACHINES

negative with respect to the applied voltage V}, then the terminal voltage in the secondarv 1 v. winding will be positive at and negative at a2. If the two windings are connected in series by joining A2 to (Fig. 88 (6) ), and an alternating voltage V' applied across the free terminals A and a9, then the markings are correct if Vt across A x./42i8 less than V'. 4. D.C. Resistance. Any suitable method may be used : preferably

Fig. 89. Coil Tester

a low-resistance bridge for large low-voltage transformers, in which the l.v. winding resistance may be very small. The measurement is made at known temperature, and may be used to check the design or to estimate eddy-current loss ratio. 5. Voltage Ratio. A direct measurement of voltage ratio by voltmeter is feasible if the voltages are reasonably low. Highvoltage transformers with large ratios present difficulties in this respect, since it is necessary to measure the two voltages with entirely different instruments, the relation between them being subject to calibration error.. The voltage ratio can be found by reference to a standard trans­ former of the same nominal ratio. The h.v. sides are connected in parallel to a supply, and a voltmeter connected to read the voltage difference between the test and standard l.v. windings.

TEA NS FOB ME RS: TESTI NG

125

Most manufacturers employ bridge methods, for checking coils oefore assembly. The connection diagram in Fig. 89 refers to an arrangement comprising a magnetic circuit of U shape with closing voke, energized from a suitable normal-frequency a.c, supply. The exciting coil is split into a large number of parallel parts in order to preserve a uniform induction density along the magnetic circuit. A standard coil of known turns, and the coil under test, are placed on the magnetic circuit, and the e.m.f.’s balanced on a bridge arrangement containing dials to which tappings are made to the sections of the standard coil, down to a single turn. By adjusting

the dials, the number of standard tapped turns is made equal to the number of turns on the test coil, as exhibited by zero deflection on the tuned vibration galvanometer. Coils checked in this way before • assembly may be safely used with certainty of accurate ratio in the transformer when built. Assembled transformers may be checked on a ratiometer, which is essentially a potential divider excited from the same supply as the transformer under test, and subdivided so as to read the l.v. voltage in terms of the h.v. Balance is obtained by connecting the ratio­ meter tapping to the l.v. winding through an ammeter and adjusting the former until the current is zero. 6. Magnetizing Current and Core Loss. This, test is made on a transformer complete with tank and oil, since it is performed at normal rated voltage and dielectric stress. One of the windings is left on open circuit and the other connected to a supply at normal voltage and frequency (Fig. 90 (tz) ). The l.v. winding is generally employed, as on the h.v. side the voltage may be difficult to manage and the no-load current rather small. Alternatively, if there are suitable l.v. and h.v. tappings, the method of Fig. 90 (6) may be used, where the voltage is increased by tapping the h.v. winding, and the current by tapping the l.v. winding. A h.v. voltmeter is neces­ sary in order to indicate when the voltage is normal. The readings of the wattmeter have to be multiplied by the ratio T^T.^. In either case readings of current and power at normal voltage and frequency are taken. The current comprises the magnetizing

1^6

DESIGN OF ALTERNATING CURRENT MACHINES

and core loss components ZOr and ZOa respectively, while the powei indicated includes the core loss, the I2R loss (which latter is usually negligible), and dielectric loss. A test made at constant normal frequency and variable voltage V is instructive. After correcting if necessary for I2R loss, Po represents chiefly core loss P{, except with transformers for very high voltages, in which the dielectric loss may assume importance.

Fig. 91. No-load Cukves

The active current component is Z0o = Po/V, the reactive (mag­ netizing) component is IOr = y/(I02~ ZOa2). A plot of these gives the general shapes shown in Fig. 91. Since, at constant frequency, the flux density Bm is directly proportional to V, and the core loss roughly proportional to Bm2, then the power/voltage curve is parabolic to a first approximation, while the active current ZOa = Pq/V is proportional to V, roughly. The magnetizing current Z0) rises sharply at first at low induction densities, until the iron reaches its maximum permeability. Thereafter it rises less steeply until, before normal voltage is reached, saturation begins and the mag­ netizing current again starts to rise more steeply. Fig. 91 has ordinates of power in per cent of normal rated kVA. at unity power factor, and of current as per cent of rated current; it is typical of transformers of medium size (e.g. 100—1 000 kVA.).

TRANSFORMERS: TESTING

127

Core-loss Voltmeter. Hysteresis accounts for 75 or 80 per cent of the core loss of modern transformers, and eddy-currents the remainder. The hysteresis loss depends only on a power of the maximum induction density Bm, so long as there is no more than one peak of density per half-cycle, but the eddy-current loss is a function of the wave-form of B, since it depends on induced e.m.f.’s. The r.m.s. value of the e.m.f. is not, however, very sensitive to harmonics. The core loss should be measured with a sinusoidal applied voltage. To avoid the error due to changes in the peak value of the test voltage that may occur without change of r.m.s. value when the voltage is not sinusoidal, an obvious precaution is to measure the mean applied voltage, to which is proportional Bm and 70-80 per cent of the core loss. This follows from the relation Bm oc fe . dt over a half­ cycle. Thus when the wave-form of the W applied voltage is in doubt, a nearly true measure of the core loss can be obtained V by measuring the mean voltage applied, and adjusting it to 2^2/77 times the rated A *AWWVWWVWW r.m.s. value. For this, it is only necessary a/ to employ a rectifier type of' voltmeter arranged to indicate the peak value. A Fig. 92. Short-cdrcuit thermionic-valve or barrier-layer voltmeter Test is of use. 7. Impedance and Z2*7i Loss. This test may be carried out with the transformer out of its tank, as the voltages are very low.' One of the windings is short-circuited, and the other (usually the h.v. winding to reduce the current and increase the voltage to be measured) con­ nected through a voltmeter, ammeter and wattmeter, duplicated where necessary for three-phase transformers, to a supply of normal frequency but only a small fraction of normal rated voltage (Fig. 92). The short-circuit connections must be carefully arranged to have as low a resistance as possible, since the l.v. windings of large transformers may have very low resistances. A voltage 7SC sufficient to circulate full-rated current in primary and secondary windings (usually of the order of 5 per cent rated voltage) permits the measurement of the full load I2R loss Pc. The wattmeter measurement includes a small core loss, usually less than 2 per cent of that on normal voltage, so that a correction is rarely necessary. The reactance, impedance, and regulation are calculable from the short-circuit test. Let I = rated current per phase; V = rated voltage per phase; p = short-circuit watts per phase corresponding to /;

128

DESIGN OF ALTERNATING CURRENT MACHINES

Pc — watts per phase corresponding to 1 in short-circuit test and corrected to 75° C.; J7SC = volts per phase used in short-circuit test for current 1; Ex = reactance e.m.f. per phase (primary and secondary equiva­ lent) at current I. Then Ex = V[7ac2- (p/I) ] * ; ex = EJV; tr = PJVI. . (27) The regulation is calculated by the expression of eqs. (19) et seq. 'Chapter III, § 6.) 8. Efficiency. In practical testing it is unusual to perform a load 'test for either efficiency or temperature-rise: the efficiency is too high to measure accurately by any method other than that of losssummation, and a load test for heating is uneconomical. Using the definitions of B.S. No. 269: 1927, the losses to be accounted in the estimation of efficiency are— Fixed loss: (a) core loss and (6) dielectric loss; measured together in a no-load test as P,-_ (corrected if necessary for no-load DR). Direct load loss: DR Joss in (c) primary and (cZ) secondary windings, computed for a mean conductor temperature of 75° C. Stray load loss: the stray loss includes eddy-current losses in (e) the conductors and (f) other parts (e.g. tank walls, constructional parts, etc.). Losses (c), (d) and (e) are given by the corrected shortcircuit test as Pc, Expressing the loss

Dp = (a) 4- (6) + (c) + (d) -i- (e) P, + P„ Zp then the efficiency isn — 1----- — P + Dp for any load P. See also Chapter III, § 7. The loss (f) is generally small enough to be ignored. Example. A 300-kVA., 11 000/440-V., mesh/star, three-phase, 50-cycle transformer gave the following test results— Open-circuit test on l.v. side at normal voltage and frequency: input 1-3 kW., 214 A. Short-circuit test on h.v. side with voltage of 600 V.: input 2-80 kW., 15-0 A.; temperature of copper, 30° C. Calculate the per-unit resistance, reactance, and impedance; and the efficiency and regula­ tion for full load at power factor 0-8 lagging. The open-circuit test gives Pt = 1-3 k W. The full-load l.v. line current is I2 = 300.103/[( V3) - 440] - 393 A. Thus the no-load current percentage is 100.214/393 -- 5-4. The full-load h.v. line current is Ix = 300 . JOVftv'S) - 11 000] = 15-73 A. The loss with 15-0 A. is 2-80 kW. and with full-load current will be p = 2-80 (15-73/15-0)2 = 3-08 kW., at 30° C. = 1-03 kW. per ph. with 15-73/y/3 = 9-1 A. per ph. The loss at 75° C. will be 934-5 -1- 75

3’08

= 3 0 kW" Or

= 3’° kW'

129

TRANSFORMERS :■ TESTING

The per-unit resistance is Pc/Q = 3-6/300 = 0-012 p.u. The voltage on short circuit for full-load current is 600 X 15-73/15 = 630 V. From eq. (27) Er = V(G302 - (1 030/9-1 )2] = 620 V.: fjr = 620/11 000 = 04)56 p.u. £r = 3-6/300 = 0-012p.u. The per-unit impedance is therefore v(0-0122 + 0-0562) = 0-058 p.u.

For full load at power factor 0-8 lagging the output is 0-8.306 = 240 kW. The losses total 1-3 -f- 3-6 = 4-9 kW. The efficiency is

The regulation is, from eq. (19) s = 0-012.0-8 + 0-056.0-6 = 0-043 p.u.

for the same load and power factor. 9. Temperature-rise. This is measured during the performance of a heat-run, intended to reproduce as far as is practicable the service conditions of continuous rated load, to observe that the temperaturerises of the various parts conform to specification. The limits allowed by B.S. 171 : 1936 are— Temperature-rise of windings by resistance, * except for windings of resistance less than 0-01 £2, and of core and oil by thermcmc' cr— Windings Type

AN, AB......................................... ON, OB, OW . OFN, OFB .... OFW.........................................

Oil Class A 55° 60° 65° 70°

C. C. C. C.

Core

Class B 75° C. 50° C. 50° C. 50° C.

As for adjacent windings

The temperature-rise limits quoted are with reference to standard ambient air or inlet water temperatures. The values adopted are: 25° C. for water; and 40° C. peak, with 35° C. as 24-hour average, for air. Transformers complying with the specification are suitable for operation continuously, 24 hours a day indefinitely, at continuous niaxiiuum rating. Some of the considerations involved in the thermal rating of transformers are discussed in Chapter VII. ♦From the expression IiO} — H0t

234-5 + 0, where 0 is in degrees centigrade. 234-5 4- 02

VbO

design of alternating current machines

ilt is noteworthy that higher limits are allowed for Class B than for Class A insulation. (See Chapter X, § 10.) ''Methods of carrying out a heat-run economically, i.e. without loading the transformer normally, are(d) Back-to-back connection; (6) Delta/delta connection; (c) Equivalent open- or short-circuit run. Back-to-back Connection. The connection is shown in Fig. 93 (a) 'and (6). Consider the single-phase arrangement, which is

Fig. 93. Back-to-baok Tests

more obvious. Two identical transformers A and B (Fig. 93 (a)), connected in parallel on both sides to a supply of normal voltage and frequency so that their e.m.f.’s are in opposition round the closed ''circuits formed by the two primaries and the two secondaries. Disregarding the effect of the auxiliary transformer, then wattmeter Wt will read the total core loss of the two transformers together. A voltmeter across the switch Sw should read zero, indicating com­ plete equality of secondary voltages and phase opposition, and when it is closed, wattmeter Wr should be unaffected. . A circulating e.m.f. is introduced into the primary circuit by means of the secondary of an auxiliary transformer C, energized from a suitable a.c. supply of any convenient frequency: it need not be vthe same as that of the main supply. The e.m.f. circulates a current, which has its counterpart in a current of equal ampere-turn value in the secondary. The e.m.f. necessary is the product of the cir­ culating current and the impedance in primary terms. The e.m.f. and the circulating current are associated with wattmeter ff2, which consequently records the I2R loss. The current is adjusted by control of the voltage applied to the primary side of the auxiliary

TRANSFORMERS: TESTING

131

transformer. By adjusting for a full-load value of the circulating current, the full-load FR loss is expended in the windings of the transformers under test, and at the same time the cores are energized to normal induction density. For a heat-run the wattmeters are not necessary: only a voltmeter to measure primary applied voltage and a secondary ammeter to indicate circulating current. If the two transformers are provided with suitable tappings, the auxiliary transformer may be dispensed with, and the circulating current produced by unbalance of the equality of e.m.f.’s when the secondaries are connected in parallel Mam supply with slightly different numbers of Auxiliary turns. The scheme of connections for testing two identical three-phase transformers, shown in Fig. 93 (6), corresponds to the case of singlephase transformers. A tapping method may be used as an alterna­ tive where suitable taps are available. Since the FR losses are provided by a circulating current, while the Fig. 94. Heat-run core losses are supplied by a current which divides into two paths in parallel, it will usually be found that one transformer is cooler than the other in a back-to-back test if the auxiliary and main supplies have the same frequency. This occurs because the phase­ angles between the circulating and core-loss currents differ in the two transformer windings: thus if, in one transformer, the circulat­ ing current is nearly in phase with the magnetizing current, it will be in phase-opposition to it in the other, and the net current will be smaller, reducing the FR loss. A difference of temperature-rise from this cause is avoidable if the circulating current has a lower frequency: this also reduces the kVA, required. Delta/Delta Connection. The essential method is indicated in Fig. 94, in which the primary side is excited at normal voltage and frequency, while the secondary, connected in “open-delta,” issupplied with a circulating current from an auxiliary single-phase supply of any convenient frequency. The method achieves the same result as the back-to-back connection without requiring two identical transformers. It is possible to apply the method whatever the normal internal connections if temporary alterations can be made where necessary. Equivalent .Shokt-cikcuit Run. Where the FR ioss on fullload current is considerably greater than the core loss, as is usual in power transformers, an approximate test of temperature-rise may be made by a short-circuit test at a current somewhat greater than normal; so that the FR loss actually occurring is equal to the sum

132

DESIGN OF ALTERNATING CURRENT MACHINES

of the core and normal DR losses. The temperature-rise and resist­ ance arc estimated, the current adjusted to a. value such that the DR loss when hot will be equal to the sum of the normal DR (hot) and core losses, and the test carried out, with adjustment of the exact value of the current near the end of the test to where I is the full-load rated current, the normal core loss, and Pc the normal DR loss. Equivalent No-load Run. Occasionally, as with high-frequency transformers, the core loss may be large compared with the full-load DR. In this case an approximation to full-load heating can be obtained by an open-circuit test at a suitable over-voltage. 10. Insulation Tests. “Megger” tests for insulation resistance are carried out between windings, between copper and core, and between the core-clamping bolts and the core. A high-voltage test is made immediately after the heat-run on a new transformer. It may be a test by direct application of a voltage from a suitable source (c.g. a testing tram former) or an inducedvoltage test in which the transformer is itself operated at a voltage and frequency suflicienlly in excess .of normal to generate in the windings a voltage of the specified magnitude. These two tests differ radically. The former applies the same voltage to all windings under test whereas in the latter a test between turns and a graduated test between windings and core is obtained. B.S. 171: 1936 distinguishes between transformers having wind­ ings with graded insulation and with fully-insulated windings; also between the. types of system on which they are to be used and their tendency to be affected by lightning. Depending upon the above and on the number of phases and the system earthing, the induced-voltage test voltages are 1 -J- 1-4F, 1 -|- Id’, 1 4- T73lz, 1 + 21’, 1 -|- 2 8 I7, and 1 d 3-461 kilovolts, where I7 — rated ser­ vice voltage in kV. 11. Impulse Tests. Voltage tests at normal frequency are not a reliable guide to the behaviour of a transformer when subjected to lightning and switching transients, for in these cases the electric stress distribution over the insulation is determined by effective capacitances whose influence at low rates-of-change is negligible. Surge phenomena have been discussed in Chapter V, § 13, with their influence on surge-voltage distribution. Methods of impulse testing * have been developed to determine the ability of a transformer to withstand the effects of high unidirectional voltages resembling the lightning surge of Fig. 72. It. is common for purchasers to specify impulse tests before a transformer is put into service. Because of its expense and complexity, which make it IJagengnth and Meador, “Impulse Testing of Power Transformers” Trans. A.l.E.P,, 71 (111), p. 697 (195:?); Taylor, Power System Transients (Newnes).

TUA NSFORMERS: TESTING

133

unsuitable for routine application, the impulse-testing procedure is applied only as a "type test. The impulse test- consists in the application of a limited number of unidirectional surge voltages to one or two phases of the h.v. and l.v. windings. Typical standard test voltages, applied to the line terminals of star-connected trans­ formers with solidly-earthed neutrals, are— Nominal voltage. kV.: 3-3 Peak test voltage, kV.: 43

6-6 11 60 75

33 66 170 250

HQ 450

132 550

275 1050

The wave of test voltage, derived from a surge-generator (such as that shown diagrammatically in Fig. 84), takes normally the forms

(C) Neutral current measurement

Fig. 95. Impulse Testing

shown in Fig. 95: (n) is a “full wave,” and (6) a “chopped wave” obtained by°paralleling the surge generator by a rod-type spark­ gap, which has a delayed breakdown. The rise time of the wavefront is arranged to be about I z./s, and the subsequent decay to half peak value for a full wave is typically about 50/zs. An impulse test schedule might consist of: (1) a full-wave test, followed by (2) two chopped-wave tests, followed by (3) one full-wave test. For each test simultaneous oscillographic records are made of the applied voltage and the neutral current. The neutral current may be measured by the voltage across a resistor connecting the transformer tank to earth. Fig. 95 (c). If an initial test is made at a voltage substantially below the expected impulse strength of a transformer, then any failure that occurs during the normal test routine will change the shape of the current oscillogram. With modern techniques, and considerable experience, failures can be detected with near certainty even when there are no other signs (such as noise or smoke). The position of such a failure in the winding is much more difficult to find, however. Fig. 95 (d) shows a pair of typical oscillograph records

CHAPTER IV

TRANSFORMERS: CONSTRUCTION 1. Constructional Parts. The transformer is a comparatively simple structure, since there are no rotating parts, or bearings. The chief elements of the construction are—(1) Magnetic Circuits, comprising limbs, yokes, and ciamping structures. (2) Electric Circuits, the primary, secondary and (if any) tertiary windings, formers, insulation and bracing devices. (3) Terminals, tappings and tapping switches, terminal insulators and leads. (4) Tank, oil, cooling devices, conservators, dryers and ancillary apparatus. Improvements are continually being made in construction, and the practice of different manufacturers depends considerably on the size of unit made, the organization of the factory, and the indivi­ duality of the designers. The substance of this chapter is to be taken only as an indication of the construction of modern trans­ formers. The practice in Great Britain and Europe is to concentrate mainly on the single- and three-phase core types. Some shell types are built for single-phase use, and somewhat rarely for three phases. A. few special constructions are sometimes employed. Attention here will be directed chiefly to the single-phase core and shell and the three-phase core types of construction. 2. Core Construction. Special alloy steel of high resistance and !ow hysteresis loss is used almost exclusively in transformer cores, ind all electrical-steel manufacturers have suitable grades. (See Chapter IX.) Induction densities up to T35-L55 Wb./m.2 are possible, the limit for 50 c/s being the loss and the magnetizing current. As the flux in the cores is a pulsating one, the magnetic circuit nust be laminated and the separate laminations insulated in order ;o retain the advantages of subdivision. Paper, japan, varnish Anna clay or phosphate may be used. The last-named is able to vjfhstand the annealing temperatures of cold-rolled steels, so that t can be applied to the whole sheet before cutting and annealing. Burring of the edges of the plates may cause a considerable ncrease in core loss by providing paths for eddy currents should he sharp edges cut through the insulation and establish contact •eiween adjacent plates. Burrs are removed before core assembly. Silicon alloy steels are hard, and cause wear of the punching tools. m2/r)10-3 whence = y/S . . 103/4 *44/). Substituting this value of O in the expression eq. (9) for the e.m.f. per turn, Et = ±44fy/S . V(r . 103/4-44/) = Ky/S . . . . . . . (33) where K = vz(4-44/r . 103),

and depends on type, material and labour costs, factory organiza­ tion, etc. The ratio r does not change with size in a range of transformers of given type. If the criterion of load for maximum efficiency is applied, K assumes values of the order given in the table below when the rating is in kVA.— Type

! i

K

I

Three-phase, shell core,. power . distribution . »> 9f Single-phase, shell core

1-3 0-6- 0-7 0-45 10- 1*2 75-0-85 * 0

These figures apply to the mean ratings of the frame-sizes in a line of transformers, so that at the limiting ratings, the value of A' will differ somewhat from the mean values. A workable design can now be built up on a basis of experience. Choosing Bni and d, the product AjAw is known from eq. (28),-

144

DESIGN OF ALTERNATING CURRENT MACHINES

Et from eq. (33), and an appropriate value of K, A< can be cal­ culated from Et and the chosen value of Rm. The core is designed to conform to a standard frame, from which d and D (Fig. 96) are obtained. The length L of the window is obtained from Av and the width between core-circumscribing circles. Typical core sections are shown in Fig. 23, with their appropriate net core areas At in terms of the diameter of the circumscribing circle. The factor 0-9 accounts for a 10 per cent loss of gross core area in plate insula­ tion. 6. Coils and Insulation. The constructive features of the several types of coils have already been outlined (Chapter IV). The insulation of individ­ ual turns is rarely called upon to withstand vol­ tages greater than 75 V. under normal conditions of service so that the insulation between successive turns is largely determined by the mechanical characteristics of the insulating material. The end turns of h.v. windings must, of course, be carefully designed and insulated to withstand overvoltages (Chapter VI, § 13). The insulation to earth, between h.v. and l.v. windings, and between coil sections, is determined by the voltage for which the transformer is designed. The clearances, thicknesses, etc., for these purposes are fixed from test combined with experience. Fig. 97 summarizes typical distances and thicknesses for (a) h.v. to l.v. winding, (c) end of h.v. winding to yoke, for oil-immersed three-phase transformers. The clearance between coil sections is of the order of 1 cm. The voltage gradient should be across boards, oil ducts and paper layers, in which direction these materials have greatest electric strength. Tappings, by B.S. 171: 1936, are designed for ± 2J and 5 per cent voltage adjustment. The reinforcement of end turns depends on the rated voltage; the total number of reinforced end-turns at each end of a winding varies from 3 per cent of the total turns in lower-voltage trans­ formers down to 1-75 per cent for 132 kV. and 0-75 for 220 kV. transformers. The practice of reinforcement, particularly for large h.v. transformers, is now closely integrated with investigations, theoretical and practical on models, into actual surge-voltage distribution.

TRANSFORMERS: DESION

145

7. Reactance. The estimation of reactance is primarily the esti­ mation of the distribution of the leakage flux and the resulting line linkages with the primary or secondary coils. An accurate solution to this problem is well-nigh impossible—as is, in fact, almost every problem of magnetic field distribution in the neighbourhood of iron masses. The distribution of the leakage flux depends on the geometrical con­

formation of the coils and of the neighbouring iron masses, and also on the permeability of the latter. The diagrams in Fig. 98 * show typical calculated distributions based on simplifying assumptions (such as constant permeability /zr = 10, two-dimensional symmetry, etc.). In case (a), that of cylindrical core-type coils of equal length, it is noteworthy how the leakage field is packed into the space between the windings, and how it runs parallel with the core for nearly the full length of the coils. Where there is an inequality in the coil-lengths, however, the field is very considerably altered, as shown in Fig. 98 (6). For the shell-type transformer with sandwich coils, Fig. 98 (c) shows a typical leakage-field dis­ tribution. In cases (a) and (c) thQ field is sufficiently symmetrical and geometrical to permit of considerable simplification for the sake of a usable approximate expression. Cylindrical Concentric Colls, Equal Length. For this case' the actual leakage field, e.g. Fig. 98 (a), is assumed to consist of a longitudinal flux of uniform and constant value in the interspace between primary and secondary; and a field crossing the conductors, reducing linearly to zero at the outer and inner surfaces. See the “axial component” in Fig. 76 and the AT distribution in Fig. 99 (a). Further, the permeance of the leakage path external to the coil length Lc is assumed to be so large as to require the expenditure of a * Based on Figs. 96, 97, and 102 of Hague’s Electromagnetic Problems in Electrical Engineerina (Oxford).

146

DESIGN OF ALTERNATING CURRENT MACHINES

negligible m.m.f.; i.e. all the m.m.f. is expended on the length Lt. The effect of the magnetizing current in unbalancing the primary and secondary ampere-turn equality is neglected. Let (AT) be the ampere-turns of either the l.v. or the h.v. coils on one limb. Then the flux density in the duct of radial width a is Ba = ^(AT)/LC. Taking one-half of the total duct leakage as linking either winding (it makes little difference to the result whether this is strictly true or not), then the duct flux linking each of the Tt primary turns is approximately [^(AT)faLmt/Lc, and the linkages are consequently ^Q(AT)T1laLmt/Lc. Here Lmt is the mean circumferential length of the annular duct, nearly the same as the mean length of a primary turn. The flux density at the radial distance x from the surface of the primary winding is xBJbv The flux in an elemental annular ring of width dx and approximate circumference Lmt is L^xBJb-^dx. This flux does not link Tr turns, but only the outer portion xTJbx: the linkages are therefore Lm^aT^xjb^dx. Summing the total linkages over the radial distance blt I

\ El

./x = 0

=

\°1/

The total linkages of the primary are \ "

o /

"c

J

\

Similarly the linkages of the secondary are

^AT)Tt

The reactance of the primary is obtained from the flux linkages per ampere X 277/, or «i =

ohms>

since, per ampere, (AT)T± — Tf. For the secondary reactance,

From eq. (13), the total reactance in primary terms is Ai =■ Xi -|- x^ —

X, = 277/rtT/

4- x-l(T^T^\ which becomes (a+ ^-±-64 ohms.

.

.

.

(34)

where Lint may be interpreted as the mean length of turn of primary and secondary together. The per-unit reactance is ex = ZiAj/Fj,

TRANSFORMERS: DESIGN

147

and using the value of

in eq. (34), / 61 + 62\1 -------------- —--------------------- I tv ~I '------------ — — \ 3 ) ^Trfp^L^AT) / ■ -|~ 62\ or (35) LcEt \ 3 J per unit substituting (AT) for the ampere-turns per limb of either coil, and Et = for the volts per turn.

Fig. 99. Pertaining to the Calculation ok Leakage PlEactance of Cylindrical Coils

Fig. 100. Analysis of Asym­ metrical Cylindrical Coils

In some cases one of the windings (usually the h.v. coil) is split into two approximately equal concentric parts. Fig. 99 (b) shows that, with similar assumptions to those made above, the percentage reactance may be written Ex

2^.,Ur)/

m

, 6; + 6.- + 6a ,

1 fl ।

2

..

I per unit (3b)

Cylindrical Concentric Coils, Unequal Lengths. Fig. 98 (6) indicates that, compared with the case of equal length, considerable difficulty will be found in making suitable simplifying assumptions. Obviously the assumption illustrated in Fig. 99 is quite inadmissible. The leal. ; field depends on the proportional difference in length, and on urn re the difference occurs (e.g. at one or both ends, or in the middle. . tc.). Ihe ease is a very common one, as it may bo prodincd by end-turn reinforcement, tappings, or by the normal smal1 difference of coil length usual in manufact ure even in the absence oi tappings, etc.

148

DESIGN OE ALTERNATING CURRENT MACHINES

The effect of divergences on the reactance requires to be investi­ gated for each individual case. Fig. 100 shows one method of treating the problem. The coils, each of ampere-turns (AT) are shown in section. The actual arrangement (a) can be considered as equal to the sum of (b), a symmetrical system of longitudinal ampere-turns amenable to the treatment leading to eq. (35), and (c) an asym­ metrical transverse system of ampere-turns producing a cross-flux. Having determined the flux distribution due to each of the systems (6) and (c) as in Fig. 76, the reactance is determined from the e.m.f.’s induced in the actual windings due to each of the flux distributions.

Per cent, reactance 5.C. stress

13-7 54 18

10

100 0

Axialstress

11 83 9

10-3 94 5’3

Fig. 101. Arrangements of Asymmetrical Cylindrical Coils

In Fig. 101, a few typical cases are given, with figures of per­ centage reactance compared with that with symmetrical concentric coils of equal length. Sandwich Coms. Fig. 102 shows the simplified case of Fig. 98 (c). It is usual to arrange the winding as in Fig. 29 (6), where the end coils have one-half the turns of the remainder. There will be n h.v. coils, (n — 1) l.v. coils of equal ampere-turns, and two l.v. half­ coils. Taking two adjacent half-coils as a unit, the reactance of such an arrangement is, by analogy with eq. (34), g /■

Y

-^ln

/

n \2 F

9 I

/

I

^1 “F ^2\

V

I ohms.

where 7’ln is the number of turns per whole section. For the n sections and 2n units, the total reactance is X, = a + L+M o

..

Tir

1/

j

, 6, + b2\

since nTin = Tx. The effect of subdivision is apparent. The perunit reactance is fc.

=

(37) per unit wnEt 6 (AT) is the full phase ampere-turns of either primary or secondary. 2

TRANSFORMERS: DESIGN

149

8. Mechanical Forces. The general discussion in Chapter V, § 14, gave for the radial force on a cylindrical coil the expression Pr =

newtons .

_ .

. (38)

The stress produced can usually be withstood by the copper provided that the coil is cylindrical. The axial force due to asymmetry was shown to be

Pa = bhk(iT}2Lmtl2(a + \ + b2) newtons .

. (39)

for an instantaneous current i. For the cases shown in Fig. 101 the resultant axial stress will be lower because the oppositely-directed terms partially neutralize. The two bottom rows of figures give stresses in terms of the radial stress in the symmetrical case. 9. Magnetizing Current. In a single-phase transformer, the ampere-turns I^Ti provided by the primary winding on no load (and by the resultant of and I2T2 on load) require to be sufficient to produce the working flux in the complete magnetic circuit. The maxi­ mum flux-density Bm in the core is chosen as a compromise between the reduction of the length per turn of the winding and the Fig. 102. Leakage saturation of the core. A density of 1 *4 Wb./m.2 of Sandwich Coils is typical of large power transformers: higher densities than this tend to require excessive magnetizing current and introduce undesirable harmonics into this current. A lower density is generally used in the yokes of core-type transformers, as here it is possible to reduce the m.m.f. and core losses without adversely affecting the length of the copper. A 25 per cent increase may be used. The calculation of the magnetizing current must take account of the small gaps between the core-plates unavoidable with the con­ structional methods adopted. If lg is the total equivalent length of such gaps, then the r.m.s. magnetizing current is /or = Wc + atjy + 800 000B Jtf)/( V2)2\

= almLJ{y/2)Ti

....

.

. (40)

where atc, atu are the ampere-turns per cm. length of core and yoke respectively with the flux-densities employed; lc, lv are the lengths of those parts; Ti is the number of primary turns; Lt is the mean core length and atm a mean value for the ampere-turns per m. length. The calculation is performed with maximum densities, and on the assumption that the current is sinusoidal (a very rough approximation), the r.m.s. value is obtained by dividing by y/2.

150

DESIGN OF ALTERNATING CURRENT MACHINES

If the core loss be Pit the active component of the no load carrent

is ^0a ~

•

•

>

(41)

It is not usual actually to calculate the no-load current or it? components. In a transformer of normal design, Zy will be of the order of 5 per cent of the full-load current, and considerable variation in it may be made without affecting the performance of the trans­ former on load. The losses, however, are important in determining the efficiency, and the harmonics in the magnetizing current may

Fig. 103. B/at and Magnetizing VA. Curves for 0-35 mm. Transformer Steed

cause trouble in transmission and distribution transformers on light loads or no-load conditions. A knowledge of the Bm/at curve for the core material, and of the total weight of iron, permits the estimation of the magnetizing current from curves. Since the primary e.m.f. is Ei = 4-44/’771BmAJ- volts, where A{ is the net core section, then the magnetizing volt-amperes will be ^i^or — volt-amperes. But from eq. (40), the term IOrl\ = atmLr2lpg')LX ohms . . . .(66) 7. Phase Reactance of an Induction Motor. * The total phase reactance of ari induction motor referred to the stator is where

= x32 =

x0 =

= X1 + X> -I" XU + Xo + Xz+ Xh • (67) stator slot reactance using the net core-length Ls in eq. (66) with the appropriate permeance from eqs. (60-62); rotor equivalent slot reactance calculated as for x31, then multiplied by KW12SJKW22S2; overhang reactance from eq. (63); zig-zag reactance from eq. (64);

xz = xh = differential reactance from eq. (65).

For calculations of motor performance from an equivalent circuit, it is quite sufficient to assume that Xj — x2' — : alternatively the total reactance may be divided in inverse proportions to the numbers of slots, giving xjx2 ~ Since the leakage flux—apart from slot-leakage—is a result of the combination of stator and rotor currents, it is not strictly possible to separate it into individual contributions. 8. Phase Reactance of a Synchronous Machine, f or a synchronous machine the leakage flux is required to be known to assess regu­ lation. Tor this purpose a close estimate is not essential. The reactance, however, enters into many problems of much greater com­ plexity, such as short-circuit behaviour and the efficacy of pole-face damping windings. It is not possible to do justice here to these important matters, whit h have occupied designers for many years.f For the calculation of voltage regulation, the following simplified formulae may lie used. The leakage flux is

= (I>s + (b,

....

(68)

where ., is the slot leakage flux and (J\ the overhang leakage flux for full-load current J per conductor. With eq. (56) a« a basis, and writing p^L^X.. for Ax, the maximum leakage flux associated with the slot portions of a coil of Tc turns is

„.-(x/2).77’c.Ao2/