Electric Machinery [2]

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PROPERTY OF

»

8

i

7~R T E S

7

SCIENT1 A" VERITAS

ELECTRIC MACHINERY BY

MICHAEL LIWSCHITZ-GARIK,

Dr-Ing.

Polytechnic Institute of Brooklyn; Consulting Westinghouse Electric Corporation; Lecturer Westinghouse Advanced Engineering School

at the

Professor Engineer

to

at

the

the

ASSISTED

CLYDE

C.

BY

WHIPPLE, E.E.

Professor at

of

the

the Polytechnic Institute of Brooklyn; Chairman Undergraduate Electrical Engineering Department

Volume

II

A-C MACHINES

D.

NEW YORK

VAN NOSTRAND COMPANY, Inc. 250

FOURTH AVENUE 1946

TK

Copyright,

1946

BT

D. VAN NOSTRAND COMPANY, Inc.

All Rights Reserved or any parts thereof, may not be reproduced in any form without written per mission from the authors and the publishers. This

PRINTED

book,

IN THE UNITED

STATES OF AMERICA

To My Wife OLGA LIWSCHITZ-GARIK

I

dedicate my work

PREFACE This work, Electric Machinery, is presented in two volumes: I, Funda mentals and Direct-Current Machines; II, Alternating-Current Machines, for undergraduate courses in electrical engineering. The principal aim of these two volumes is to provide the fundamental link between the basic laws of electrodynamics and the performance characteristics of the electric machines, since experience shows that this connection is most difficult for the average student to understand. To achieve this objective, the ma chines are treated from a general point of view and the features common to all of them are discussed in separate chapters. Such chapters are : the magnetic circuit of the main flux, the magnetic circuits of the leakage fluxes, losses and cooling, windings, the emfs induced in a winding, and the mmfs produced by a winding carrying current. Furthermore, in the treat ment of the machines operating on the transformer principle (induction motors and synchronous machines), of the commutator machines with salient poles (d-c machines and single-phase series motor), and of the a-c commutator machines operating with a rotating field (repulsion motors and polyphase commutator motors), the features common to each of these groups are emphasized. The correct sequence in the classification of ma chines in groups, according to their common features, requires that the d-c machines be treated after the synchronous machines, since the d-c machine is a synchronous machine with a special device — the commutator. Con sidering the fact that in most colleges d-c machines are taught first, these machines have been discussed immediately after the introductory chap ters. However, in their treatment, attention is called to the features which are common with a-c machines; in the treatment of a-c machines, the features which are common with those of d-c machines are pointed out. The treatment in separate chapters of the features common to all machines

and the emphasis of the features common to the different groups

machines make it possible to handle the subject matter in less space to teach it in a shorter time than is possible when each machine is treated as a separate entity. The saving in time is especially important these days when extensive developments in the field of electronics make it necessary to devote more teaching time in this field. Some suggestions for using the book in a two-, three-, or four-semester course are given in the booklet containing the answers to the problems. Although the book has

of

and

PREFACE

vi

been written primarily for students, it also contains considerable material of value to practicing engineers. The information on a-c circuits necessary for the treatment of a-c machines is given in two short chapters. Certain chapters, considered to be somewhat difficult for undergradu ate study, are placed in Appendices at the end of the book. On the other hand, the inductances of the leakage fluxes, which influence the per formance of all electric machines considerably, are discussed in one of the first chapters dealing with the fundamentals. Special consideration is given to the demands encountered in practice, on the basis of the experi ence of the author as a lecturer at the Westinghouse Advanced Engineer ing School. The basis for this work on Electric Machinery is a book in three volumes published in German by the author during the years 1926 to 1934 and translated into several languages. In order to achieve suitability as a text book, the material has been brought up to date and adapted to the con ditions of this country.

Brooklyn,

July,

N. Y.

1946

Michael Liwschitz-Gakik.

ACKNOWLEDGEMENTS The publication of this work has been greatly furthered by the assis tance of my colleague Professor C. C. Whipple, and of Professor R. T. Weil, Jr., of Manhattan College. Professor Whipple contributed the chapter on control in the first volume, the chapters on application and control and articles 13, 17, 18 and 19 of the chapter on transformers in the second volume, and has read the manuscript and proofs through all stages. In addition he contributed the examples and problems to the chapters on d-c machines, transformers, and synchronous machines. Professor Weil has read the greater part of the manuscript and has also checked the formulae of the Appendices. The problems to Chapter 5 of Volume I and Chapters 7 have been worked out by my son S. Garik. Professor and 8 of Volume A. B. Giordano, of the Polytechnic Institute of Brooklyn, has read the chapter on d-c machines. To all of these gentlemen I express my sincere appreciation. I am also indebted to Dr. H. S. Rogers, President of the Polytechnic Institute of Brooklyn, and to Dr. E. Weber, Head of the Department of Electrical Engineering at the Polytechnic Institute of Brooklyn, for their kind assistance in offering the facilities of the Institute necessary for the completion of this work. The Westinghouse Electric Corporation gener ously supported my efforts and supplied information and photographs am deeply grateful. for which wish to express my gratitude to the General Electric Com Finally, pany, the National Electric Coil Company, the Allis Chalmers Corpora tion, the Century Electric Company, Robbins and Meyers, Inc., the Wagner Electric Corporation, and the Cutler-Hammer Company for

II

I

I

their

contribution of cuts, data, and photographs. M. L.-G.

vu

CONTENTS CHAPTER Application

of Faraday's

1

Law to the Generation of Polyphase Emfs page

1-1. 1-2. 1-3.

Generation of Polyphase Emfs Power in Single-Phase and Polyphase Systems Representation of Sinusoidal Voltages and Currents by Complex Quantities

CHAPTER

1

5 8

2

The Transformer 2-1 . 2-2. 2-3. 2-4. 2-5. 2-6. 2-7. 2-8.

Transformer Construction The No-Load Voltage Diagram of the Transformer The Transformer Under Load The Secondary Leakage The Voltage Diagram of the Loaded Transformer — Reduction to the Primary Winding Voltage and Current Relations — Equivalent Circuit The Mutual Inductance of the Transformer Voltage Regulation — Kapp Diagram

13

21 26 28 29

32 36 38

The Magnetizing Current Im The Leakage Reactance No-Load and Short-Circuit Tests

40

48

2-13.

The 3-Phase Transformer The Autotransformer

2-14.

Parallel Operation of Transformers

53

2-15.

Shape of the No-Load and Load-Current Wave Efficiency of the Transformer Transformer Polarity. Three-Phase Connections for Transformers Phase Transformation Instrument Transformers Pcr-Unit Calculation Examples Problems

56

2-9. 2-10. 2-11. 2-12.

2-16. 2-17. 2-18. 2-19. 2-20. 2-21. 2-22. 2-23.

42 46 50

ix

58 59

60 68 72 75

'77 80

CONTENTS

X

CHAPTER

3

Mechanical Elements of A-C Machines PAGI 3-1.

3- 2.

Induction Motor Synchronous Machine

81 94

CHAPTER

4

A-C Windings 4- 1.

Polyphase Windings

106

(a) Single-Layer 3-Phase Winding (6) Two-Layer 3-Phase Winding: Lap and Wave Winding. (c) Fractional-Slot 3-Phase Winding

4-2. 4-3. 4-4.

4-5.

106 . . .

113

Squirrel-Cage and Damper Windings Insulation of A-C Windings Slot Leakage of a Fractional-Pitch Winding

116

114

Emf Induced in an

5-3.

5-4. 5-5. 5-6.

5-7.

117 118 123

CHAPTER

5-2.

112

(d) Two-Phase Windings Single-Phase Windings

4-6. Problems

6-1.

108

5

A-C Winding

Influence of the Shape of the Field Distribution Curve in the Syn chronous Machine — Pitch Factor Effect of Distributing the Winding — The Distribution Factor . .

Effect of the Harmonics (a) Resulting Emf (6) Effect of the 3rd Harmonic The Emf Induced in a Direct-Current Armature Winding Time and Space Harmonics

124 129 135 135 137 138 138

Examples

139

Problems

140

CHAPTER

6

The Mmf and Field of an A-C Winding 6- 1.

The Mmf Curve of a Single-Phase Field

Winding — The Alternating 142

XI

CONTENTS

PAQE 5-2.

The Mmf of a Polyphase Winding — The Rotating Field

148

6-4.

Influence of the Harmonics The Form of the Resultant Mmf Curve

157

6-5.

Harmonic (Differential) Leakage Reactance

158

6-6.

Examples Problems

164

6-3.

6-7.

156

166

CHAPTER

7

The Polyphase Induction Motor 7-1.

7-2. 7-3.

7-4.

7-^5.

7-6. 7-7. 7-8. 7-9. 7-10.

7-11. 7-12.

The Polyphase Induction Motor as a Transformer. (a) The Induction Motor at Standstill — The Slip (6) The Induction Motor When Running Equivalent Circuit and Vector Diagram — Primary and Second ary Currents Torque and Basic Relations of the Polyphase Induction Motor — Slip at Rated Output No-load and Short-Circuit (Locked-Rotor) Tests (a) No-Load (6) Short Circuit Determination of the Machine Constants from the No-Load and Short-Circuit Tests The Circle Diagram for n = 0 (Heyland Diagram) Construction of the Circle Diagram for n = 0 from No-Load and Short-Circuit Tests — Pull-Out Torque Operation of the Polyphase Machine as Generator and Brake . . . The Exact-Circle Diagram — Torque Line and Slip Line Influence of the Resistances and Leakage Reactances on the Oper ation of the Polyphase Induction Motor — Influence of Voltage and Frequency Variation (a) Power Factor (6) Pull-Out Torque (c) Efficiency (d) Starting Current (Inrush)' (e) Starting Torque (J) Effect of Voltage and Frequency Variation (0) Influence of Saturation The Squirrel Cage as a Polyphase Winding Starting the Polyphase Induction Motor with Slip-Ring Rotor...

168 168 170 173 177 184 184 185 187 187 192 196

200

207

207 210 211 211 211 211 211

215 220

CONTENTS

XU

PAGE

7-13.

Speed Control of the Polyphase Induction Motor (a) Speed Regulation by Inserting Resistance in the Rotor

222

Cir

cuit

222

Regulation by Changing the Number of Poles of Speed With the Aid of a Special Regulating Regulation (c) — Power Factor Correction Set Starting of the Polyphase Induction Motor with Squirrel-Cage Rotor — Skin-Effect Rotors Influence of the Harmonics on the Speed-Torque Curve Influence of Skewing Special Induction Motor (a) Synchronous Induction Motor (6) Electromagnetic Coupling (c) Self-Synchronizers and Position Indicators (Selsyns, Synchrotie Apparatus, Autosyn, etc.) The Induction Voltage Regulator Per-Unit Calculation Example — Calculation of the Performance of a Polyphase In duction Motor Problems (b) Speed

7-14. 7-15. 7-16. 7-17.

7-18. 7-19. 7-20. 7-21.

CHAPTER The Single—Phase 8-1. 8-2. 8-3. 8-4. 8-5. 8-6. 8-7. 8-8.

8-9. 8-10.

223

223 231

236 241 245

245 246 246

257 260 262 267

8

Induction Motor

Two-Rotating-Field Theory and Cross-Field Theory Rotor Currents and Torques Equivalent Circuit and Circle Diagram Normal Slip and Pull-Out Slip Determination of the Reactances and Resistances from the NoLoad and Locked-Rotor Tests Consideration of the Iron Losses Influence of Secondary Resistance and Reactance on the SpeedTorque-Curve Starting of the Single-Phase Motor (a) Split-Phase Motor (fc) Capacitor-Start and Capacitor Motor (c) Shaded-Pole Motor (d) Repulsion-Start Induction Motor Example — Calculation of the Performance of a Split- Phase

271.

272 276 281 281

284 287 288 289 290 293 293

Motor

294

Problems

297

Xlll

CONTENTS

CHAPTER

9

The Synchronous Machine PAGE

The Synchronous Machine as a Current Transformer 9-2. The No-Load Characteristic 9-3. The Voltage and Mmf Diagram of the Non-Salient-Pole Syn

300

chronous Generator Armature Reaction Synchronous Machine Characterises (a) No-Load and Air-Gap Charr ct eristics (b) Short-Circuit Characteristic (c) Load Characteristics and Potier Triangle (d) Regulation Curve and External Characteristic (e) Voltage Regulation (J) Short-Circuit Ratio Determination of Regulation. . . (a) From the Vector Diagram (b) By Empirical Method Voltage Diagram of the Nou -Salient-Pole Synchronous Machine Based on Its Synchionous Reactance The Salient-Pole Synchronous Generator — Two-Reaction Theory (a) Resolution cf the Armature Mmf into Direct-Axis and

303

9-1.

9-5.

9-6.

9-7. 9-8.

9-9. 9-10. 9-11.

9-12. 9-13.

9-14.

Quadrature-Axis Components (6) The Armature Mmf in the Direct Axis (c) The Armature Mmf in the Quadrature Axis The Voltage Diagram of the Salient-Pole Synchronous Generator Characteristics and Synchronous Reactances of the SalienWole Machine The Voltage Diagram of the Salient-Pole Mac .line on the Basis of Its Synchronous Reactances Torque and Power of the Synchronous Macb ne The Single-Phase Generator (a) Resolution of the Alternating Mmf cf the Armature Into Two Oppositely Rotating Mmf Wave i (6) Influence of the Inverse Rotating Fie d (c) Suppression of the Inverse Rotating Field by Means of a Damper Winding (d) Comparison of the Single-Phase and the 3-Phase Machine. Parallel Operation and Synchronizing of Synchronous Machines. (a) Influence of the Angle 5 on the Power Developed Circu (6) Influence of Excitation on the Parallel Operation. lating Current Synchronizing of Synchronous Machines (c)

9-4.

301

304 ^T9 30^

309 310 312 313 313 314

314 315 316 318 318 319 321

322 326 329 332 336 336 336 338 340 341 341

342 346

XIV

CONTENTS

PAGE 9-15.

9-16.

9-17.

9-18.

The Synchronous Motor (a) The Voltage Diagram of the Synchronous Motor (b) The Circle Diagram of the Synchronous Motor for Constant Terminal Voltage and Constant Field Current (c) Construction of the Torque Line (d) Influence of the Excitation on the Overload Capacity of the Synchronous Motor (c) The Circle Diagram of the Synchronous Motor for Constant Terminal Voltage and Constant Power Developed (/) Synchronous Motor V-Curves (j) Starting of Synchronous Motors (h) The Synchronous Motor as a Synchronous Capacitor The Hunting of the Synchronous Machine — Nat (a) Synchronizing Torque of the Synchronous Machine ural Oscillations as a Consequence of Sudden Loads (b) Forced Oscillations as a Consequence of the Irregularity of the Prime-Mover Impulses (c) The Ratio of the Amplitude of Oscillation in Parallel Opera tion to the Amplitude of Oscillation of the Single Machine — The Amplification Factor — (d) The Natural Frequency of the Synchronous Machine The Danger of Resonance (e) Improvement of Parallel Operation by Means of a Damper Winding Examples Problems

CHAPTER

347 347 348 350 351 353 355 357 360 361 361 363

365 368 369 370 375

10

The Rotary Converter 10-1. 10-2. 10-3. 10-4. 10-5. 10-6.

10-7.

Cancellation of the D-C Mmf by the A-C Mmf The Voltage and Current Ratio in the Synchronous Converter . . The Copper Losses in the Synchronous Converter Ratio of the Power Output of the Synchronous Converter to That of a Direct-Current Machine of the Same Dimensions Commutation of the Synchronous Converter Voltage Regulation of Synchronous Converters (a) Voltage Regulation by Means of a Choke Coil (6) Voltage Regulation by Means of the Induction Voltage

379

Regulator Starting and Parallel Operation of Converters

393

382 384 390 391

392 392

394

CONTENTS

CHAPTER

XV 11

The Alternating-Current Commutator Motor page 11-1.

The Single-Phase Commutator Motor. The Direct-Current Arma ture in an Alternating Field (a) The Emf of Rotation in the Armature Winding (6) The Transformer Emf in the Armature Winding (c) The Torque (d) The Compensating Winding (e) The Transformer Emf of a Short-Circuited Winding Ele ment and the Commutating Fields in the Single-Phase Com

mutator Motor 11-2.

11-3. 11-4.

11-5..

397 397 398 400 401

402

The Single-Phase Series Motor (a) The Voltage Diagram of a Single-Phase Series Motor (6) Commutation of the Single-Phase Series Motor The Repulsion Motor The Polyphase Commutator Motor. The Direct-Current Arma ture in a Rotating Field (a) The Operation of the Commutator (b) Commutation of the Polyphase Commutator Motor The 3-phase Shunt Motor (a) Speed Regulation (b) Power-Factor Correction (c) Commutation

CHAPTER

12

407 407 408 411 417

417 421

423 423

427 428

.

Motor Application 12-1. 12-2. 12-3. 12-4. 12-5. 12-6. 12-7. 12-8. 12-9.

Characteristics of Loads Motor Types, Sizes, and Costs Applications of Various Types of Motors NEMA Classifications of Induction Motors Wound-Rotor Induction Motor Application Polyphase Induction Motors as Multi-Speed Motors Synchronous Motor Characteristics and Applications Fractional-Horsepower Motors Definitions

CHAPTER

430 432 434 442 443 446 447 452 455

13

Starting A-C Motors and Motor Protection 13-1.

Starting Induction Motors (a) Line Starters

464 464

CONTENTS

XVI

PACE (6)

13-2. 13-3. 13-4. 13-5.

Reduced Voltage Starters

466

(c) Controllers for Wound-Rotor Motors (d) Starters for Fractional Horsepower and Small-Horsepower Motors Starting Synchronous Motors Factors Governing the Type of Controller or Starter To Be Used — Manual or Magnetic ." Motor Protection Definitions

APPENDIX Fractional-Slot Al-1. Al-2. Al-3. Al-4.

472

473 474 47»'i

478 483

1

Lap Windings

Number of Recurrent Groups and Parallels The Slot Star The Layout of a Fractional-Slot Lap Winding Distribution and Pitch Factor

APPENDIX

48(1

486 490 492

2

Harmonics and Parasitic Torques in Poyphase Induction Motors A2-1. A2-2. A2-3. A2-4.

Stator Mmf and Field Harmonics Rotor Mmf and Field Harmonics Parasitic Tangential Forces — Parasitic Torques Parasitic Radial Forces — Vibration and Noise

APPENDIX The Exact-Circle A3-1. A3-2. A3-3.

A3-4. A3-5. A3-6.

Diagram

494 49S 503 506

3

of the Polyphase Induction Motor

The Rotating Polyphase Induction Motor as a Stationary Trans former — Voltage Equations Primary and Secondary Current — Power Factor Geometric Loci (a) Straight Line (6) Circle Circle Diagram of the Polyphase Induction Motor — Slip Line. . The Torque Line Influence of Variation of Leakage Reactances and Secondary Re sistance on the Shape of the Geometric Locus of the Primary Current

509 51

1

512 512 513 515 517

519

CONTENTS

APPENDIX Equivalent

Circuit

and

Starting

XV11

4

Torque

Induction Motor

of

the

Single-Phase PAGE

A4-1. A4-2. A-3.

Reduction of the Secondary Constants to the Primary Voltage Equations and Equivalent Circuit Starting Torque

APPENDIX

520 522 523

5

Capacitor Motor A5-1. A5-2.

Capacitor Motor Under Balanced Conditions Capacitor Motor Under Unbalanced Conditions

APPENDIX Pulling Into A6-1. A6-2. A6-3.

A6-4.

Step

526 529

6

of Synchronous Motors

Equation of Motion of the Synchronous Motor Pulling Into Step Solution of the Equation of Motion for a = 0 (Most Favorable Pole Position) Solution of the Equation of Motion for a ^ 0 Example

APPENDIX

533 535 539 543

7

Transient and Subtransient Reactances — Sudden Short-Circuit of Synchronous Generator

A7-1. A7-2. A7-3. A7-4. A7-5.

Transient Currents with Resistances of Both Windings Neglected Transient Currents with Resistances Not Neglected Maximum Transient Currents Transient and Subtransient Reactance in the Quadrature Axis. . . Determination of the Subtransient Reactances From a Locked Test

APPENDIX Design

Principles

a

544 547

548 551

552

8

of an Electric Machine — Specific Tangential Force

A8-1. A8-2. A8-3.

Magnitude of the Tangential Force Output Constant Range of Tangential Force

553 555 557

xviii

CONTENTS

APPENDIX

9

Harmonic Analysis page

A9-1. A9-2. A9-3.

Determination of the Coefficients of the Fourier Series Special Wave Forms Step-by-step Method of Determining the Coefficients of the Fou rier Series

558 560 561

LIST OF SYMBOLS A, a

A A

ampere-conductors

a a a,

number of parallel paths in a d-c armature

per unit of circumference

area

N1/N2 ratio of transformation for the transformer amplitude of the v-th harmonic of the mmf wave reduction factor for the secondary current of the induction motor reduction factor for impedance integral part of q (fractional slot windings) ratio of the effective turns of the main winding to those of the starting winding

ax

at a a

B,b B

B

flux density

or Bg

flux density in the gap

Bi

B„ Br

flux density in the air gap of the interpole

Bj

'amplitude of the v-th (v'-th) stator field harmonic amplitude of the p'-th rotor field harmonic

6

width

6,

equivalent pole arc

6,

width of radial ventilating duct width of pole shoe slot width width of the interpole ' slot width at the air gap brush width brush width referred to the armature main flux susceptance, per phase, (transf. and ind. mot.) instantaneous value of the v-th (v'-th) stator field harmonic instantaneous value of /i'-th rotor field harmonic

bp b,

bi fcb bfc

b„

bm

b„

br-

C

c

C,

bj

Cd

capacitance armature mmf factor in the direct axis for salient pole syn chronous machine xix

LIST OF SYMBOLS armature mmf factor in the quadratic axis for the salient pole synchronous machine

output constant number of parallel groups specific heat

D, d outside diameter of a d-c armature core (inside diameter of stator of induction motors

bore diameter

and synchronous machines) difference between 2 slots which correspond to two adjacent vectors in the slot-star

E,

e

induced emf (for a-c effective value) emf induced in the primary winding by the main flux, per phase, (transformer and ind. motor) emf induced in the secondary winding, per phase, by the main flux (transformer and ind. motor) same as E2 but referred to primary emf induced in the armature of synchronous rotor flux at open circuit

machine

by

the

internal armature emf, synchronous machine emf induced in the armature by the cross flux in the quad rature axis of the salient pole synchronous machine direct emf of the rotary converter alternating emf of the rotary converter emf due to rotation (a-c commutator motor) emf due to transformation (a-c comm. motor) emf induced in the field winding (a-c comm. motor) induced emf, instantaneous value emf of self-induction transformer

emf in the short circuited

winding element

(a-c

comm. motor) resultant voltage in the short circuited winding element (a-c comm. motor) per unit voltage drops in the transformer

F,f force

amplitude of the fundamental an alternating

mmf

of a rotating mmf wave or

of

list

Of symbols

xxi

amplitude of the «*-th mmf harmonic of the stator amplitude of the ji-th mmf harmonic of the rotor

Ft

Ff

/ /

force

f

force on a single conductor frequency in cycles per second

fn

frequency of the n-th emf harmonic

/

instantaneous value of the mmf of an a-c winding

fi

line frequency rotor frequency

fz /„

(induction motor and a-c commutator motor) frequency of rotation (repulsion motor) instantaneous value of the n-th mmf harmonic of the rotor

ft

radial force in the gap slot frequency

fn

natural frequency of oscillation

/r fr

0 g

air gap length

gi gm

interpole air gap length main flux conductance

g

gap corresponding

per phase,

(transformer

and

ind.

motor) to the joints in the transformer

H,

H

HP

h

field intensity output in horsepower

h

height

hi h,

height of the slot occupied by the conductors depth of the slot

he

height of a coil in a transformer heat transfer coefficient

h

I la

if

hi current (for a-c effective value) total armature current in a d-c winding field current

current in neutral wire Jo

no-load current

J*

reactive component of the magnetizing and induction motor)

A+«

active component

(transformer,

induction

of the magnetizing

and induction motor)

motor) current current

(transformer (transformer

xxii

11

current

primary current (transformer, induction motor) secondary current (transformer, induction motor) same as referred to the primary

I

short-circuit current current at locked rotor

z,

h

line current

I

l'2

12

magnetizing

I2

Im

LIST OF SYMBOLS

Ihp

I

Li

Ix

Ib

reactive component of the primary ourrent active component of the primary current bar current (squirrel cage)

Ir

Ii„

current corresponding to the power output ideal locked rotor current

ring current (squirrel cage)

72/

forward rotor current of the single phase motor backward rotor current of the single phase motor same as I2/ referred to the stator

I2b

IM

same as I2b referred to the stator current in the main winding

Is

current in the starting winding

Ia

armature current

In

rated current

Id

component of armature

Iq

l'2f

pole synchronous machine) component of armature current

Idc

IT

Iac

I2,

Is

If l's

Iu

(synchronous machine) current

Ia in phase with E\ (salient /„

1%,

in quadrature with

E\

(sa

lient pole synchronous machine) direct current of the rotary converter alternating current of the rotary converter slip ring current of the rotary converter effective value of the current induced by the v-th stator har monic in the rotor current in the starting winding current in the main winding same as Is referred to the main winding forward symmetrical two-phase current system (capacitor system (capacitor

motor) current, instantaneous value current per path in the d-c armature winding instantaneous value of the slip ring current of the rotary con instantaneous value of the armature current

i ia

two-phase current

ir

symmetrical

ia

/i,

motor) backward

if

verter instantaneous value of the field current

LIST OF SYMBOLS

xxiii

J

J

moment of inertia of flywheel mass

K,k

K

number of commutator bars skew factor of the fundamental

K,i

K,k,

wave

skew factor of the v-th harmonic

Carter factor

he kg

kt

ratio of total length L to equivalent armature length ratio of slot width to tooth width

ki k,

stacking factor ratio of the total length of armature

kzco

for slot leakage reduction factor for slot leakage permeance (part of slot oc

kxt

cupied by the conductors) reduction factor for slot leakage permeance (part above the

kpn = kpr kdn = kdr kp, kpi kd,

kdi

k kdpi

= kd,kpr = kdikpi

k, kdpi

kips kdPtf

ki kz

to length effective

conductors) pitch factor for the n-th or v-th harmonic distribution factor for the n-th or v-th harmonic pitch factor of the fundamental wave distribution factor of the fundamental wave average reduction factor for slot leakage constant

Ay

kdpw

L

le

winding factor of the v-th harmonic winding factor of the fundamental wave saturation factor

and kdvi

winding factors of the fundamental for primary and secondary winding factor of the starting winding (single phase motor) winding factor of the main winding (single phase motor) integer which determines the order of the stator harmonics integer which determines the order of the rotor harmonics n

v

L, I

L L

coefficient of self-inductance total length of armature core

L,

coefficient of self-inductance

L„

coefficient of self-inductance

for the slot leakage flux of the top coil side for the slot

leakage flux coefficient of self-inductance

of the bottom coil side for the

Lu

L,

slot leakage flux coefficient of self-inductance

for the tooth top leakage flux coefficient of self-inductance for the end winding leakage flux

xxiv

LIST OF SYMBOLS total coefficient of self-inductance of the primary winding, including main and leakage flux total coefficient of self-inductance of the secondary winding, including main and leakage flux same as L2 but referred to the primary coefficient of self-inductance of the primary winding due to the primary leakage flux coefficient of self-inductance of the secondary winding due to

Li Z/2

Li

Lu L2i

La

the secondary leakage flux same as L2 but referred to the primary coefficient of self-inductance of the harmonics

Lm

self-inductance

L'2i

of the main flux in the direct axis (salient pole

machine

Li

self-inductance

I I

of the leakage flux of the rotor

length

I, I,

length of armature core without radial ventilating ducts equivalent length of armature core effective length of the armature for the slot leakage flux length of the end winding for half a coil

It

mean length of a turn

le

M, m

M

coefficient of mutual inductance

M Ma,

= Mbt

magnetomotive force (mmf ) coefficient of mutual inductance between the top and bottom coil side for the slot leakage field mmf

M/

armature mmf (synchronous machine) resultant mmf (synchronous machine) armature mmf effective in the direct-axis of the salient pole synchronous machine

Ma

Mr Mad Maq

armature mmf effective in the quadrature pole synchronous machine

Md

resultant

axis of the salient

mmf in direct-axis of the salient pole synchronous

machine m

mi,

wi2

number of phases number of phases of the primary and secondary

N, n

N

number of turns, total for single phase windings, per phase for polyphase windings

Nc

number of turns per coH

LIST OF SYMBOLS

-V,

number

Nx

XXV

of turns per winding element

number of turns linked with a flux

x

Ar1

number of turns of the primary winding (Transformer,

N2

number of turns of the secondary winding (Transformer,

2VS

number of turns of the starting winding

In

duction Motor, per phase for polyphase windings)

In

duction Motor, per phase for polyphase windings)

Njg

number of turns of the main winding

N'a

number of armature turns, per phase in polyphase machines

Ar/

number of turns in a concentrated

iV

number of slots per phase in a recurrent group (fractional slot

N

field winding

number of turns per path in the a-c commutator motor

windings)

n n

rpm order of the space harmonics of the field curve produced by

direct-current number of conductors per slot number of conductors per coil normal or rated speed

n, ne

n»

synchronous

n» no

speed

no-load speed

n

order of time harmonics of the emf and current curves pro duced by a d-c field winding

P,V

p

power

Pre Pw

iron losses windage losses

Pf p.

P.c

^+.

friction losses no-load power input power input at short circuit hysteresis and eddy current loss

Pco

copper loss

Prf

power of the rotating field

*

m>

Pe Pm,

dev del

Ppa rot

Pl P

developed mechanical power electrical power of the rotor (induction motor) delivered mechanical power iron losses due to rotation power input at locked rotor number of full pole pitches between two slots which correspond to two adjacent vectors of the slot star

P

number of poles

P

instantaneous power

LIST OF SYMBOLS power input number of force pole pairs hysteresis loss (per pound)

Pin

p' Pa

eddy current loss (per pound) total iron loss (per pound)

p«

Ph+e

Q,q Q

total number of slots

Qp

Qi

number of slots per pole total number of slots of the stator

Q2

total number of slots of the rotor

Q q

heat produced in a body per second number of slots per pole per phase

q

number of slots per pole used for c6ils in the single phase wind

q

ing charge on a capacitor

R,

R R

resistance radius

Re

i

Rc Ra

r

R'2

R2

r0

r/

.

equivalent resistance resistance equivalent to the load of the induction motor resistance at locked rotor, equivalent resistance radius of the circle diagram starting resistance (induction motor) load resistance of the transformer same as R2 referred to the

primary

resistance resistance of d-c armature winding resistance of the field winding of the synchronous machine a

Re

R

r

r\

resistance of the primary winding (per phase, transformer

r2

induction motor) resistance of the secondary winding (per phase, transformer and induction motor)

r2

same as

rm

main flux resistance per phase (transformer

The

motor) equivalent bar resistance (squirrel cage)

r0 r2

rs r's

Tm

rc

r2

and

referred to the primary and induction

resistance per phase of the synchronous machine resistance of the armature winding (repulsion motor) resistance of the starting winding resistance of the main winding same as rs referred to the main winding resistance of the capacitor

LIST OF SYMBOLS S,

XXVU

a

skew in the same units as the rotor slot pitch T2,

short-circuit ratio (synchronous machine) cooling surface current density slip slip at rated load

pull-out slip slip of the rotor with respect to the backward rotating field (single-phase motor) slip of the rotor with respect to the v'-th stator harmonic

T,t period of a wave (1/f) period of commutation torque forward' torque (single-phase motor) backward torque (single-phase motor)

synchronizing torque thermal time constant amplitude of v-th harmonic of the torque curve of prime mover time in seconds

U, u number of conductors side by side in the slot of a d-c armature

V,

v

terminal voltage (for a-c effective value) voltage drop under all the brushes of same polarity terminal voltage between lines

primary terminal voltage per phase (transformer,

V'2

motor) secondary terminal voltage per phase (transformer) same as V2 but referred to primary

Vs

secondary voltage of an induction regulator

V2

V

l

terminal voltage at the test with locked rotor

Vn V,u

rated terminal voltage

V^

terminal voltage of the rotary converter slip ring voltage of the rotary converter choke coil voltage (rotary converter)

VT

Ve

direct terminal voltage of the rotary converter alternating

induction

LIST OF SYMBOLS

XXVU1

Vb v

v

vi v„Vy vT

Vis-p/2 v„-

voltage between the brushes (three-phase shunt motor) terminal voltage, instantaneous value

velocity speed of the fundamental wave speed of the v-th (v'-th) stator harmonic velocity of the rotor speed of the main wave velocity of the /i'-th rotor harmonic with respect to the stator W

W W

coil width in same units as the pole pitch weight

t

X, x

X

X,

reactance

X'2

equivalent reactance reactance at locked rotor, equivalent resistance leakage reactance of stator (starting winding) and rotor re ferred to the starting winding leakage reactance of stator (main winding) and rotor referred to the main winding load reactance of the transformer same as X2 referred to the primary

x

reactance

XI

Xg

Xtt X2

x

distance

x,

inductive reactance capacitive reactance leakage reactance of the primary winding (per phase, trans former and induction motor) leakage reactance of the secondary winding (per phase, trans former and induction motor) same as x2 referred to the primary main flux reactance (per phase, transformer and induction

xc

Xi x2

si x„ X\ Xb

xr Xbe

xi Xad Xd

xq

motor) harmonic (differential) leakage reactance leakage reactance of a bar (squirrel cage) leakage reactance of the ring (squirrel cage) equivalent bar reactance (squirrel cage) leakage reactance of the armature in the synchronous machine armature reaction reactance in the direct axis synchronous reactance in the direct axis synchronous reactance in the quadratic axis of the salient pole synchronous

machine

LIST OF SYMBOLS

xxix

reaction reactance in the quadrature axis of the salient pole synchronous machine reactance of the field winding (a-c commutof motor) leakage reactance of the compensating winding (a-c commu armature

tator motor) leakage reactance of the armature winding (repulsion motor) leakage reactance of the starting winding leakage reactance of the main winding (capacitor motor) same as xs referred to the main winding (capacitor motor) reactance of the capacitor

(capacitor motor) direct axis transient reactance direct axis subtransient reactance leakage reactance of the damper winding in the direct axis transient reactance in the quadrature axis subtransient reactance in the quadrature axis leakage reactance of the damper winding in the quadrature axis

Y,y admittance main flux admittance

(per phase, transformer

and induction

motor)

winding pitch of

a d-c armature

winding

Z, z impedance total number of conductors on an armature main flux impedance

(per phase, transformer

and induction

motor) impedance of primary winding (per phase, transformer

and

induction motor) impedance of secondary winding (per phase, transformer

and

induction motor) same as Z2 but referred to the primary equivalent impedance zone

width

impedance at locked rotor impedance of the forward motor (single-phase motor) impedance of the backward motor (single-phase motor) total impedance of the forward system (capacitor motor) total impedance of the backward system (capacitor motor) impedance of the capacitor synchronous impedance in the direct axis

LIST OF SYMBOLS

XXX

a a a

a a

a

'

angle for the determination

of the center of the circle diagram

(polyphase induction motor) angle between two adjacent bars of the squirrel cage rotor angle between current in starting winding and current in main

winding ratio of pole arc to pole pitch (synchronous machine) angle between brush axis and main field axis (a-c commutator motor)

a, am

angle between two adjacent slots magnetic field angle (S

P

angle

/3

number of poles per recurrent group (fractional slot windings)

A A

5

delta connection

A

thickness of laminations

5

torque

Ei in

angle, i.e., angle between terminal

voltage

and

emf

synchronous machine e

t e

t

voltage regulation voltage drop basis of the natural logarithms

r

f f

permeance per unit length for the leakage fluxes during com

mutation amplification factor of hunting n

tj

efficiency

rje

ordinate of the center of the circle diagram (polyphase induc

tion motor)

d

temperature

d,

temperature

at the surface

temperature of the cooling air 0

temperature rise

LIST OF SYMBOLS A

xxxi

X

permeance of the harmonic leakage permeance per unit length permeance per unit length for the slot leakage flux permeance per unit length of the top coil side for the slot leak age flux

permeance per unit length of the bottom coil side for the slot leakage flux permeance per unit length for the tooth top leakage flux permeance per unit length for mutual induction between bot tom and top coil sides in the slot parameter heat conductivity H

relative permeability order of the rotor space harmonics order of the slot-harmonics of the rotor of the rotor harmonics referred to a two pole funda mental wave order of the rotor slot-harmonics referred to a two pole fun order

damental wave ■ i

v

order of the stator space harmonics ratio of the copper losses in a rotary converter to the copper losses of a d-c machine of the same dimensions order of the stator slot-harmonics order of the stator harmonics referred to a two pole funda mental wave order of the stator slot-harmonics referred to a two pole funda

mental wave

abscissa of the center of the circle diagram

tion motor) P

resistivity

•

c force hysteresis loss constant eddy current loss constant

specific tangential

(polyphase induc

LIST OF SYMBOLS

XXXU

t

t

pitch or

Tp

pole pitch

t\

width of one stack of laminations plus width of one ventilating duct slot pitch slot pitch at the gap commutator pitch pole pitch of the n-th harmonic produced by a d-c field winding pole pitch of the v-th harmonic produced by an a-c winding primary leakage coefficient

t2

secondary leakage coefficient

t0

leakage coefficient of the armature winding

t/

leakage coefficient of the field winding

t„

t, tw re

rn

t,

*

PL VL

flux per pole in the air gap total primary flux (including leakage) total secondary flux (including leakage) primary leakage flux secondary leakage flux flux in the direct axis (salient-pole synchronous machine) flux in the quadrature axis (salient-pole synchronous machine) flux in the quadrature axis of the repulsion motor power factor angle power factor angle at no-load power factor angle at locked rotor line power factor angle

4>2i

angle between emf and current in the rotor of the induction

$1

$2 ij

*2J 3>d

*, #a Induced e.m.f

WWW +

xt

(6) Subtractive

.

xt

Polarity

B, +

wvwv AAAAA, A-, A*, (c) Additive Polarity Transformer Polarity

Fia. 2-41.

Polarity designation of

a transformer.

The direction of the induced emf is the same in both coils and the direc tions of the currents are opposite. Terminal Xx has the same instantane ous polarity as Hi. a, flii The polarity of a transformer is designated as subtractive when, as in Fig. 2-41b, the leads Hx and Xi are adjacent, or as additive when Hi and Xi are — — ' i : I 1 located diagonally opposite as in Fig. 2-41c. I IX, To test for polarity, connect an adjacent high\Xi Fio. 2-42. Test for the voltage and low- voltage terminal together as shown the in Fig. 2-42. Apply an a-c voltage to the high-voltage of determination of a transpolarity terminals. Measure the voltage between the other former. adjacent high- and low- voltage leads. If this is greater than the voltage applied to HXH2 the polarity is additive as in Fig. 2— 41c; if less it is subtractive as in Fig. 2-41b. 2-18. Three-Phase Connections for Transformers. Since it is common practice in electric power systems to generate at one voltage, transmit at

CI

THE TRANSFORMER

voltage, and distribute or use at another voltage, and since most power systems are 3-phase systems, it is necessary to provide 3-phase transformer connections. This may be done by using 3-phase transformers (Art. 2-12) or by using banks composed of single-phase transformers. In either case there are several methods available for connecting these transformers to form a 3-phase bank. Only the four most common another

methods

will be discussed here: (1) delta-delta

B

(2) wye-wye

B

Primary

Secondary

(6)

(O) Fig. 2-43.

(A-A),

Delta-delta

(A-A)

connection of 3-phase gram for balanced load.

transformers.

Voltage

dia

(Y-Y), (3) delta-wye (A-Y), (4) wye-delta (Y-A). In each discussion the all transformers in the bank will be assumed equal, and the mag netizing current and the impedances of the transformers will be neglected in the vector diagrams. As will be noted, each scheme possesses certain advantages and disadvantages. DeUarDeUa (A-A) Connection. Fig. 2-43 shows the connection and vector diagram for a A-A bank of transformers assuming a balanced unity-powerfactor load. The primary AB has a corresponding secondary winding ah. The polarity of terminal A is the same as that of a.. The vector diagram corresponds to the conventions of the fundamental diagram, Fig. 2-24, where primary and secondary currents and voltages are generally opposite

ratio of

A-C MACHINERY

62

in phase.

It

is observed that for balanced loads the primary line currents

are equal to the phase currents multiplied by v3 (see Art. 1-1), and that the same relation applies to the secondaries. With balanced load the currents in each transformer are of equal magnitude, provided the trans

formers have equal impedances and equal ratios of transformation. A phase angle of 30° is observed between line currents and phase currents, shown, for example, between IB and IBc, Fig. 2-43a. The line and phase voltages are equal, a characteristic of the A connection. It is also to be noted that, if the vector diagram, Fig. 2^43b, had been referred to the primary, then the corresponding primary and secondary line currents

would be in-phase. In the A connection it is important that the trans formers have equal ratios, otherwise circulating currents will exist within the A. Moreover, if the transformers are to share the load equally, they must have the same impedance, since there are parallel paths between each pair of line terminals, and the division of current between these paths will be dependent upon the impedances. Hence, for balanced loads and identical transformers, the primary and secondary phase currents and the voltages of corresponding phases are in-phase, and related to each other by the ratio a. The primary and secondary line currents and voltages are also in-phase and related by the same ratio a.

It is shown in Art. 2-15 that, with a sine wave of applied voltage, the magnetizing current wave of a transformer contains harmonics, mainly a 3rd. Considering the 3rd harmonics of current within the A it will be noticed that, while the fundamental components of the currents are dis placed 120° in phase, the 3rd harmonics will be displaced 3 X 120 = 360, which brings them all in phase with each other. The 3rd harmonic currents thus flow around the closed circuit of the A as single-phase currents, and do not appear in the lines, because at any junction point such as B the 3rd harmonic in the line would be the difference between 2 equal 3rd harmonics of the phases A and B. Thus in the A connection the termi nal voltages are sinusoidal, and the necessary 3 harmonic currents flow within the primary A, to produce the sinusoidal flux. Wye-Wye (Y-Y) Connection. Fig. 2-44 shows the scheme of connections and the vector diagram for a balanced load in a manner similar to Fig. 2-43. Here the line voltages are equal to V3 times the phase voltages, while the phase and line currents are equal in magnitude. There is a phase displacement of 30° between phase and the nearest line voltage, as shown (Fig. 2-44b) between Vcn and Vcb. The ratio of primary line quanti ties to corresponding secondary line quantities is a or 1 /a. Thus VCb = aVcb and Ib = aIB. If the vector diagram of the secondary (Fig. 2-44b)

THE TRANSFORMER

63

referred to the primary, corresponding quantities would be in-phase instead of displaced 180°. The operation of the Y-Y bank is markedly different if the neutral of were

the

primary is isolated from

Fig. 2-44.

are

Wye-wye

(Y-Y)

the neutral of the power source, unless loads

connection of 3-phase transformers. balanced load.

Voltage

diagram

for

balanced and the magnetizing requirements of the transformers are

With

the neutrals isolated, the sum of the 3 line currents must be zero. Consider first the condition of no-load on the secondary, and equal primary line voltages, and transformer A in Fig. 2-44 larger than the others. Thus it requires more magnetizing current, or it has a lower 2 primary coils xm (see Fig. 2—28) than the other two transformers. Since of it lines, follows that the transformer are in series between each 2 pairs Im, than its normal and its terminal voltage less of lower xm will receive will be less than normal, while the other transformers will be overmagnetized and have higher than normal voltages. The voltages be tween lines and neutral therefore will not be equal, and corresponding

identical.

A-C MACHINERY

64

secondary phase voltages will be unequal. This condition is shown ix Fig. 2-45 in which the vector diagram, Fig. 2-44, has been drawn with the 3 equal line voltages forming an equilateral triangle. Here the phase voltages are unequal, and the neutral point is not at the center of the triangle. Fig. 2-45a shows the condition for equal transformers, and Fig. 2-45b the condition where transformer A is larger than B and C,

which are assumed equal. The neutral point has moved from N to N', and a voltage VNN> exists between the neutral of the generator and the neutral point of the primary on the transformer. Thus, if these neutrals were connected by a wire, a current would flow in this neutral wire and

v~ Fig. 2-45.

(a) Wye-wye

vBA

(Y-Y)

(b)

connection under unbalanced load conditions.

balance the voltages by allowing each transformer magnetizing current.

to take its proper

A

similar condition of voltage unbalance would occur even though all transformers were identical, if the secondary loads were unbalanced. Thus, in Fig. 2-44, if a resistive or inductive load were connected between

n and a while the other phases were on open circuit, the primary current in phase A would increase. Since there are no corresponding load currents in the phases B and C, the load current of phase A must flow through phases B and C, thus acting as magnetizing current there. However, the iron in the transformers of phases C and B soon may become saturated and limit the amount of current possible. This phenomenon tends to increase the flux in transformers B and C. Hence their secondary voltages rise, and a condition similar to that shown in Fig. 2-45b is produced. As a matter of fact, it would not be possible to draw full-load current from phase A with

the other phases open. For these reasons the Y-Y system is operated with neutrals connected together, so that any unbalanced current may flow in the neutral, thus preventing voltage unbalance. With the neutrals con nected, the transformers act essentially as 3 separate single-phase systems. The requirement that the magnetizing current of a transformer shall

r THE TRANSFORMER contain a

65

large 3rd harmonic already has been mentioned. In the

Y-Y

with no neutral connection, this situation produces some diffi Since the sum of the 3 line currents must be zero, and since the

connection culties.

are in-phase, they are prevented from flowing in the trans primaries in the Y-Y arrangement without a neutral connection. The result of this situation is that the flux is distorted, i.e., it is no longer a sine wave, even though primary line voltages are sinusoidal. The primary 3rd harmonics former

phase voltages therefore are distorted and they contain pronounced 3rd harmonics. These harmonics usually are present in such a phase relation to the fundamental that they may produce peaked phase voltages of large magnitude, sufficient to cause a high stress on the insula tion of high-voltage transformers. This difficulty may be avoided by using a neutral connection, or by using a A-connected secondary, which allows the

3rd harmonic current to circulate in

harmonic

it,

and secondary

thus short-circuiting the 3rd

emf .

it

is

a

Delia-Wye (A-Y) Connection. Fig. 2^16 shows the connection scheme balanced unity-power-factor and vector diagram for load. For the noted that the corresponding secondary connections shown here

is

If

Ib

by

primary line and secondary line quantities are out of phase by 30°, i.e., the secondary voltage Fa& lags the corresponding primary line voltage VAB lags the primary current IB by 30°. 30°, and the secondary current the polarity connections of the secondary had been reversed, this angle would have been 150°. For the A-A or Y-Y connection this corresponding either 180° or 0° depending upon the polarity of the windings. angle

it

if

3-phase banks are to be connected in parallel on primary and Hence, would not be possible to use A-A on one bank and A-Y secondary sides, on the other, because the secondary voltages would be out of phase by 150° or 30° instead of 0° as required for parallel connections. Other combi nations also present this difficulty. While the ratio of primary line to secondary line quantities involves only the winding ratio a in a A-A or Y-Y bank, the ratio of line to line quantities in the A-Y are as follows:

FiV

is

I2

a

A

is

is

is

is

fa

3

= Iia /V3. With this connection there no diffi and culty with the 3rd harmonics in the magnetizing currents causing voltage with the primary Y. With the secondary neutral distortion, as there but little difficulty with connected to the neutral of the load, there unbalanced loads. practically the same connec Wye-Delia (Y-A) Connection. Since this not tion in reverse from that shown in Fig. 2-46, the vector diagram of the by shown. The 3rd harmonic magnetizing currents are taken care the primary from neutral connection on the secondary, or by connecting V2 =

A-C MACHINERY

66

to the neutral of the source of power. Unbalanced secondary loads cause nc difficulty provided the primary neutral is not isolated. The ratios of line to line quantities are F2 =

and

72

=

ZioV 3.

One particular advantage of A-connected secondary, and particularly the A-A connection, is its ability to operate 3-phase, with one transformer disconnected. This connection is referred to as the open-Aor V-connection.

(6) Fiq. 2-46.

Delta-wye

(A-Y) connection of

3-phasc

transformers.

Voltage diagram

for balanced load.

J

If 3 equal transformers are operated A-A, the total volt-ampere capacity is V3 V l, where V L and I l are fine values. The current within the trans formers is

I l (Vs.

However, if one of the transformers is disconnected,

I

the line current must be reduced to L /V3 in order not to overload transformers. Hence the ratio of the volt-ampere capacity of the V

J. l

lI l

the to A two

= 0.577. The total volt-ampere capacity of the is V /V^3 V transformers, if they were operated in parallel, would be 0.666. In other words, the maximum capacity is not available, due to the fact that the

THE TRANSFORMER

voltage and

phase

and 0.577 be

line

currents

67

are out of phase by 30°, since phase are identical in the V connection. Therefore only phase current

/0.666 = 0.866

= cos 30°,

of the total transformer capacity can

used. Use

of the Various Types of Connections. The A-A connection offers no

difficulty with harmonics in the magnetizing currents, and operates fairly well on some unbalanced loading. It does not make possible a neutral for grounding, and only one secondary voltage is available, unless taps are used. In comparison to the Y-Y connection the A-A connection is more suitable for relatively lower voltages and higher line currents. It has the advantage of open-A operation in the event of damage to one trans former. The Y-Y is seldom used with isolated neutrals, because of diffi culties with the magnetizing currents in case of unequal magnetizing characteristics, and because of voltage distortion due to absence of 3rd har monics in the magnetizing current. Unbalanced loads cause bad unbal• ance of voltages without the neutral connection. There may be telephone interference with the 3rd harmonics in the neutral. This connection pro vides a neutral for grounding the system. It also makes available two secondary voltages. The A-Y connection commonly is used at the generating end of trans mission fines, to reduce voltage stress in the high-voltage winding. With secondary grounded, on a 220-kv system, the voltage stress to ground is only 127 kv. It is used also for secondary distribution, where the secondary neutral is grounded and a neutral wire is used, to supply low-voltage 4- wire 3-phase networks. Here the line voltage usually is 208 and the phase voltage 120. This system is used widely for lighting and power distribu« tion, where 3-phase motors are available for 208-volt operation. There is no voltage distortion caused by 3rd harmonics in the primary, as these currents can flow in the primary A. The Y-A connection commonly is used at the receiving end of highvoltage transmission systems, and with a primary neutral connection grounded, it avoids 3rd harmonic and unbalance difficulties. The second ary can operate also on open-A. > In making 3-phase transformer connections it is important to know the transformer polarity. In connecting the primaries of 3 separate singlephase transformers, they may be connected at random, since there is no magnetic coupling between them. However, once the primaries have been connected the secondaries cannot be connected at random. The line '

must be V3 times phase voltage for the Y, and in the A the zero, even fundamental voltage between the closing points must be voltages

A-C MACHINERY

68

though a pronounced

3rd harmonic voltage may appear between

these

points. 2-19. Phase Transformation. Modern power transmission and distribu tion systems usually are 3-phase. In view of the fact that 2-phase systems were used extensively at one time, there still is considerable 2-phase equipment in use, and very often the only power available is 3-phase; this requires interconnection between 3- and 2-phase equipment. When rectifying from a-c to d-c, the equipment used in the conversion gives

Fig. 2-47.

Scott transformer connections and voltage diagram.

better performance on 6- or 12-phase than on 3-phase. It therefore becomes necessary to convert from 3-phase to 2-, 6-, or 12-phase. This can be accomplished readily with static transformers through connections as

outlined in the following paragraphs. 2-Phase and Vice Versa. The common method of chang ing 3-phase to 2-phase or vice versa is by the Scott or T connection. If the voltages of the 2 systems are about the same, autotransformers are used, otherwise 2-winding transformers are used. Three-Phase

to

Fig. 2-47a shows the Scott connection to change from a 3-phase 2400-volt line to a 2-phase 240-volt line. It consists of 2 separate singlephase transformers shown as A A and BB. Transformer A A is called the main transformer, transformer

BB the

teaser transformer.

The primary A

THE TRANSFORMER

6!)

main transformer is designed for the 3-phase line voltage 2400, has a center tap d. The secondary of A is designed for 240 volts and

of the

and

ratio of transformation is 10. The primary of BB is designed for a voltage Vit = Vab, i.e., also for the 3-phase line voltage. It has a tap point c so located that F^ = 0.866 F^. The end (d) is connected to the center point of winding ab. In this case Vdc = 2080. The terminal t is not used, but it is available for use if required. This is the case when both primary windings are built alike with 86.6% taps and center taps, in order to make the transformers interchangeable. From the relations just given, Vab = V + 0, F^ = -0.5V + 0.866F. These voltages form a balanced 0.866F and Vca = -0.5V 3-phase system between lines abc. In this case Vab = 2400 + 0, Vbc = -1200 -t-j'2080, Vca = -1200 -j"2080. The secondary B is wound for the same voltage as secondary A, in this case 240 volts. The ratio of teaser winding dc to secondary B is

the

j

-j

j

2080/240 = 8.66. In case a neutral point is required for the 3-phase system a tap point n is provided on the teaser winding. This tap n is located so that Vnc =

VebfVs = Vcil&mVz

= 0.666 FC(i or

f

of the turns from c to d are included between taps c and n, or 71.0% of the turns from t to d are included between t and n. For a balanced resistive load, secondary coils A and B carry equal currents, and each part of the primary carries equal currents. In this case

if we assume secondary currents of

100 amp, the

primary currents will be

11.53 amp in each part. Since the load is a balanced resistive load, the primary currents flow in phase with the phase voltage, i.e., with the volt age to neutral. This means that the current in the teaser is in phase with

the voltages Fnc and V&, while in the parts ad and bd the current is out of phase with voltages Vda and F6d by 30°. Thus in this example the power in part da is 1200 X 11.53 X cos 30 = 12 kw = power in db. The power

in the

teaser is 2080 X 11.53 = 24 kw. Three-Phase to 6-Phase. There are three common connection schemes used for making this transformation: (a) double-wye, (b) diametrical, (c) double-delta. The choice of the type to use depends upon the type of

upon the harmonic content of the various currents and volt ages. Generally speaking, either the double-Y with interconnected neu trals or the double-A is preferred over the diametrical, although the latter often is used on rotary converters. The double-Y connection is shown in Fig. 2-^8a. The primaries P are shown in A, although they may be in Y. It is noted that there are 2

load, and

A-C MACHINERY

70

identical secondaries associated with each primary. Secondaries Si are connected to form one Y having neutral ni and with voltages as shown in Fig. 2-48b. The secondaries s2 are connected to form another inde

and producing voltages as shown in Fig. pendent Y having neutral 2^8c. Observe that the two Y's are opposite in sense. It also is to be

p 7 rh kvwwJ rhk/vwvw aaa/vwJ p

(a)

rAA/Vi rAAAn a.

a.

a3

[A/Wi

rAAAn

s,

s,

s,

S,

b,

a,

bt

n,

'-WW-

Fig. 2-48.

Three-phase

to 6-phase

transformation. nection.

Double-wye

(double- Y)

con

observed that points a^Ci have phase displacements of 120° as is true for 036303. Also points 0262^2 are opposite in phase to aibiCi. Although the two Y's are not interconnected within the transformers, they become so when connected as shown in (d) to a symmetrical 6-phase load L. The points aic4, etc., of the load are connected to corresponding points on the transformer. Connection points a1C4, etc., may be adjacent slip rings of a 6-phase rotary converter, or any other interconnected 6-phase load. The resulting voltages between adjacent rings, shown in Fig. 2-48e, are equal

THE TRANSFORMER

71

magnitude to that of each secondary coil. Note, however, that there unless the load is will be no voltage between adjacent points such as

in

connected.

If

are joined together, the con the two neutrals nx and n2 of Fig. nections are as shown in Fig. 2-49a, and the connection to the load as well

Fig. 2-49.

Three-phase to 6-phase

transformation. connection.

Diametrical

and 6-phase

star

all voltage relations are unchanged. This connection sometimes is referred to as a 6-phase star-connection since the voltage diagram as well as the coil connection may be shown as in Figs. 2^49b and 2-49c. This con nection has an advantage over the one in Fig. 2-48 in that a common neutral point n is available which may be a ground point and also may be used for the return wire in a 3-wire d-c system on a rotary converter. It is observed that this connection is a true 6-phase system even though no load is connected, i.e., voltages aic4, cA, etc., are present without a con nected load. as

If

the interconnecting wire w between the 3 transformers is omitted, the voltage relations remain unchanged except that the system is no longer an independent 6-phase system but requires the load to be present. This connection requires but one secondary on each transformer and is called the diametr ical connection.

Instead of connecting the secondaries to form 2 independent Y's as in Fig. 2-49, the same coils may be connected so as to form 2 independent i's or a double-A as it is called. Here again the 6-phase system is not

A-C MACHINERY

72

independent but requires a load to make it complete. The connections Fig. 2-50a. If the same transformers are used as were used for the previous connections, the voltage between adjacent load points will be different from the double-Y connection. In the double-Y or starconnection the voltage between adjacent load points is equal in magnitude to that of one of the secondary coils, such as FaiC4 = Fnoi = Vnci. In the are shown in

double-A, however,

alc4 is VaUi

the voltage between adjacent

= 0.577Fa2ai as can be seen from

rA/Wi rAAAn

Fig. 2-50.

Three-phase

AA/V] fA/Wi

to 6-phase

transformation. nection.

load points, Fig. 2-50b.

such as

rVWi |A/Wi

Double-delta

(double-A)

For example consider that each secondary coil voltage is

con

volts : then adjacent ring voltages will be 100 for the double-Y, the 6-phase star, and the diametrical, while in the double-A the voltage will be 57.7 volts.100

2-20. Instrument Transformers. Instrument transformers are divided into two classes: potential transformers and current transformers. Each serves two purposes: (1) to insulate the high- voltage circuit from the measuring circuit, in order to protect the measuring apparatus and the operator; (2) to make possible the measurement of high voltages, with

THE TRANSFORMER

73

low-voltage instruments (usually 115 volts), or large currents with lowcurrent ammeters (usually 5 amp). This procedure simplifies the measur

problem greatly. These transformers are used also to operate relays, solenoids, circuit-breakers, or other controlling devices. The principle of the instrument transformer is fundamentally the same as that of the power transformer. The potential transformer has the high voltage applied to the primary, and the secondary, usually rated at 115 volts, is connected to the volt meter. From Figs. 2-24 and 2-25 it is observed that the ratio of primary ing

20

Fia. 2-51.

40

60

Percent Rated Output

80

Correction curves for a potential transformer.

to secondary terminal voltage is not the same as the turn ratio, but depends also upon the character of the load, upon the impedance of the windings and upon the magnetizing current Im. Neither is the phase angle

primary and secondary terminal voltages exactly 180°, or zero if the vector of V'2 is reversed. (See Fig. 2-29.) In order to keep the relation of primary to secondary voltage constant, or practically so, the potential

between

transformer is designed with as small a leakage reactance and resistance as possible. The flux density in the core is also lower, than that used in power transformers. Thus, in order to keep the ratio and phase-angle errors small, the instrument transformer is much larger than a power transformer of the same rating. Instrument transformers usually are rated from 25 to 500 volt-amp according to the burden or secondary load. As far as heating is concerned, the rating as power transformers would be from 2 to 4 times the rating as instrument transformers.

f

A-C MACHINERY

74

that Vi /V2 is more nearly unity, or Vi IV 2 more nearly equal to a, the smaller the load current, or the greater the load impedance. Also 1), then the number of coil groups

n6 ma rn Pole Single Layer Winding with

n„ 4

Fig. 4-1.

24 Slots

Three-phase, 4-pole, single-layer winding with 24 slots.

provided per pole is the sajne as_ the number of phases. Each phase oc cupies, per pole, 1 /m part of the total number of slots falling under a pole.

For example, if a 4-pole 3-phase machine has

36 slots, there

will

be

group will occupy 3 slots per pole. Three-phase windings will be considered first because they are more important than the other polyphase windings. (a) Single-Layer 3-Phase Winding. This type of winding is sometimes used in the stators of small induction motors (up to 10 hp) with partially closed slots, and also in the rotors of small wound-rotor induction motors. Fig. 4-1 shows this type of winding for a 4-pole machine with 24 slots. For 9 slots per pole, and each phase

106

A-C WINDINGS

107

the sake of clarity the end windings (end connections) are shown away from the iron. In reality they he directly at the iron. Fig. 4-2 shows a single-layer winding installed in a small a-c motor. All coils are identical \ andhaye a width equal to a pole pitch (full-pitch winding). The number of slots per pole per phase is 24/64 X 3) =2. The slots are 1 and 2, 7 and 8, 13 and 14, 19 and 20, lying in assigned to phase

I

Fia.

4-2.

Small 3-phase motor with a single-layer winding.

under the first, second, third, and fourtlj_pole£j-espectively. The remaining slots under each pole are assigned to the other 2 phases, in the order shown in Fig. 4-1. To the 2 slots per pole there correspond 2 coils per pair of poles, and the 2 coils constitute a coil group. Since the machine has 4 poles, each phase contains 2 such groups of 2 coils each. Therefore phase I consists of coil groups, terminating in Ia I& and Ic Id; phase of groups terminating in IIa II6 and IIC HdJ and phase of groups terminating in III0 III6 and Hie Hid. In each case subscript a or c represents the beginning of a coil group and b or d its end. According to the voltage to be produced in the case of generators, or the impressed voltage in the case of motors, the

pairs

III

II

A-C MACHINERY

108

must start in slot = start in slot

'

=

is

1

taken as the

1+4

and phase

+ 4. Ia, II0, and

from

start of

III must III0 are

3

then the beginnings of the phases. Ic, IIC, IIIC also could be taken as the beginnings of the

3

a

slot 5

if

4

II

phases must be displaced

5

then phase

9

phase

I,

electrical degrees, the beginnings of the slots. Hence each other, by 120730-=

3

is

A

is,

coil groups assigned to each phase are connected either in series or in parallel. The beginnings as well as the ends of the 3 phases must be displaced I„, II0, III°

The induction Bx (Fig. 5-2) is expanded by a Fourier series as (see App. 9) (5-4)

A-C MACHINERY

126

is the amplitude of the sine term of the nth harmonic, and is the amplitude of cosine term. Thus

where

B'n

Bx =

B{sin-a;+B2sin2-x

+ fig sin

T

T

+ B[' cos-x + B" cos

-x

2

T

3-x

+ B" cos

T

+

T

3

• • •

-x T

B"

,

+

• • • .

(5-5)

In nearly

For the coil aa (Fig. 5-2) having the width

t

it

*■)

all practical cases the field distribution curve satisfies the = — Bx (half-cycle symmetry) so that condition that B(x + con tains only odd harmonics. This case now will be given further consideration.

(full-pitch winding)

Xl = X x2 =

,

x +

(5-6)

T.

B"

n-(x

+

cos n

+

- x)

cosn

B'n'

-

(5-7) (x

sin

-x

+

(

53

sin n

r)

-

B'n

\b'„

BXt

=

E

BXl

>

Consequently

+

r)l-

is

Both of these equations are now introduced into Eq. (5-3). Taking into account the fact that n an integer (sin nr = 0), and also an odd number,

If

2Nclvl0r^Z (b'„ sin n

-

+

=

x

e

there results

B"

cos n

- xV

(5-8)

is

a

is

it

seen that the curve of compared with Eq. (5-4) the emf induced in full-pitch coil has exactly the same form as the field

this equation

]•

—

is

/

= nf,

t„ =

if

therefore

/„

harmonic

is

the nth harmonic field wave

(

distribution curve. Every harmonic in the emf curve has exactly the same phase as the harmonic in the field distribution curve which produced it. The harmonics in the emf curve are greater than the field harmonics in the ratio of 2NCM0~8 to unity. One pole pitch of the fundamental field wave contains n pole pitches of The frequency of the nth emf

the frequency

of the fundamental

emf wave. =

put in the form

V2

ZK sin nut

+

be e

Eq. (5-8) can

V2ZE"

cos nut

(5-9)

EMF INDUCED IN AN A-C WINDING

following relations

when the

X = a =

—

T

_

v

K=

r

_

rfa

r

dt

*» =

are introduced,

o>t,

_

dx

127

dt

- t„ZB^, *" r

K

4.44/^e^lO-8,

=

_

t

t =

T

r

f'

2

- tJB",

t„ = —

*

n

>

/„ = „/.

4.44/nATc