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# Schawm's Electric Machines and Electromechanics

(Parte **1** de 5)

and ELECTROMECHANICS

Second Edition

SYED A. NASAR, Ph.D.

Professor of Electrical Engineering University of Kentucky

SCHAUM'S OUTLINE SERIES McGRAW-HILL

New YorkSt. LouisSan FranciscoAuckland BogotaCaracusLisbon LondonMadrid Mexico CityMilanMontreal New Delhi

San Juan Singapore Sydney Tokyo Toronto v,' rya fl.

SYED A. NASAR is Professor of Electrical Engineering at the University of Kentucky.Having earned the Ph.D. degree from the

University of California at Berkeley, he has been involved in teaching, research, and consulting in electric machines for over 40 years. He is the author, or co-author, of five other books and over 100 technical papers.

Schaum's Outline of Theory and Problems of ELECTRIC MACHINES AND ELECTROMECHANICS

Copyright © 1998, 1981 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America.Except as permitted under the Copyright Act of 1976, no part of this publication may be reproduced or distributed in any forms or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

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ISBN 0-07-045994-0

Sponsoring Editor: Barbara Gilson Production Supervisor: Tina Cameron Editing Supervisor: Maureen B. Walker

Library of Congress Cataloging-in-Publication Data

Nasar, S. A. Schaum's outline of theory and problems of electric machines and electromechanics / Syed A. Nasar. - 2nd ed.

p. cm. - (Schaum's outline series)

Includes index. ISBN 0-07-045994-0 (pbk.) 1. Electric machinery.2. Electric machinery-Problems, exercises, etc.3. Electric machinery-Outlines, syllabi, etc. 1. Title.I. Series. TK2181.N38 1997 621.31'042'076-dc2l 97-23529 CIP

McGraw-Hill A Division of

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Preface

A course in electric machines and electromechanics is required in the undergraduate electrical engineering curriculum in most engineering schools. This book is aimed to supplement the usual textbook for such a course. It will also serve as a refresher for those who have already had a course in electric machines or as a primer for solo study of the field. In each chapter a brief review of pertinent topics is given, along with a summary of the governing equations. In some cases, derivations are included as solved problems.

The range of topics covered is fairly broad. Beginning with a study of simple dc magnetic circuits, the book ends with a chapter on electronic control of dc and ac motors. It is hoped that the presentation of over 400 solved or answered problems covering the entire range of subject matter will provide the reader with a better insight and a better feeling for magnitudes.

In this second edition, the theme of the first edition is retained. Major additions and deletions include an expanded discussion of the development of equivalent circuits of transformers and the addition of a section on instrument transformers in Chapter 2; and the addition of a section on energy-efficient induction motors. In the revision, Chapter 7 has undergone major changes: Sections on linear induction motors, electromagnetic pumps, and homopolar machines are deleted. The chapter focuses on small electric motors. Thus, sections on starting of single-phase induction motors, permanent magnet motors, and hysteresis motors are added. In Chapter 8, the section on power semiconductors is completely rewritten. Finally, new problems are added to every chapter.

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Chapter 1 MAGNETIC CIRCUITS | 1 |

1.1 Introduction and Basic Concepts | 1 |

1.2 Permeability and Saturation | 2 |

1.3Laws Governing Magnetic Circuits | 3 |

1.4AC Operation and Losses | 41.5Stacking Factor .......................................... 5 |

1.6 Fringing | 5 |

1.7Energy Stored in a Magnetic Field | 5 |

1.8 Inductance Calculations | 6 |

1.9Magnetic Circuits with Permanent Magnets | 7 |

Chapter 2POWER TRANSFORMERS | 24 |

2.1Transformer Operation and Faraday's Law | 24 |

2.2EMF Equation of a Transformer | 25 |

2.3 Transformer Losses | 25 |

2.4Equivalent Circuits of Nonideal Transformers | 25 |

2.5Tests on Transformers | 27 |

2.6 Transformer Connections | 29 |

2.7 Autotransformers | 30 |

2.8 Instrument Transformers | 31 |

Chapter 3ELECTROMECHANICAL SYSTEMS | 47 |

3.1 Electromechanical Energy Conversion | 47 |

3.2Force and Torque Equations | 49 |

3.3 Electromechanical Dynamics | 50 |

3.4 Electromechanical Analogies | 52 |

Chapter 4DC MACHINES | 71 |

4.1 Operating Principles | 71 |

4.2 Commutator Action | 72 |

4.3Armature Windings and Physical Features | 73 |

4.4 EMF Equation | 74 |

4.5 Torque Equation | 74 |

4.6 Speed Equation | 75 |

4.7 Machine Classification | 75 |

4.8Airgap Fields and Armature Reaction | 76 |

4.9Reactance Voltage and Commutation | 78 |

4.10Effect of Saturation on Voltage Buildup in a Shunt Generator | 79 |

4.11Losses and Efficiency | 80 |

4.12Motor and Generator Characteristics | 81 |

4.13DC Motor Dynamics | 81 |

Contents V

F-' 0.4

Chapter 5POLYPHASE INDUCTION MOTORS | 98 |

5.1 General Remarks | 98 |

5.2MMFs of Armature Windings | 98 |

5.3Production of Rotating Magnetic Fields | 100 |

5.4Slip: Machine Equivalent Circuits | 101 |

5.5Calculations from Equivalent Circuits | 103 |

5.6 Energy-Efficient Induction Motors | 104 |

5.7Approximate Equivalent Circuit Parameters from Test Data | 105 |

Chapter 6 SYNCHRONOUS MACHINES | 123 |

6.1Types and Constructional Features | 123 |

6.2Generator and Motor Operation; The EMF Equation | 123 |

6.3 Generator No-Load, Short-Circuit, and Voltage-RegulationCharacteristics | 125 |

6.4Power-Angle Characteristic of a Round-Rotor Machine | 126 |

6.5Performance of the Round-Rotor Motor | 127 |

6.6Salient-Pole Synchronous Machines | 128 |

6.7Transients in Synchronous Machines | 130 |

Chapter 7SINGLE-PHASE MOTORS AND PERMANENT MAGNETMACHINES | 153 |

7.1 Small AC Motors | 153 |

7.2Analysis of Single-Phase Induction Motors | 153 |

7.3Starting of Single-Phase Induction Motors | 153 |

7.4 Permanent Magnet Machines | 156 |

7.5 Hysteresis Motors | 162 |

Chapter 8ELECTRONIC CONTROL OF MOTORS | 174 |

8.1 General Considerations | 174 |

8.2 Power Solid-State Devices | 175 |

8.3RMS and Average Values of Waveforms | 179 |

8.4Control of DC Motors | 180 |

8.5Control of AC Motors | 185 |

8.6 SCR Commutation | 189 |

Appendix A Units Conversion | 209 |

Appendix BCharacteristics of Single-Film-Coated, Rounded, Magnet Wire | 210 |

Appendix CCharacteristics of Magnetic Materials and Permanent Magnets | 211 |

vi CONTENTS INDEX .......... ................... ................................215

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Chapter 1

Magnetic Circuits

1.1 INTRODUCTION AND BASIC CONCEPTS

Electric machines and electromechanical devices are made up of coupled electric and magnetic circuits. By a magnetic circuit we mean a path for magnetic flux, just as an electric circuit provides a path for the flow of electric current. Sources of magnetic fluxes are electric currents and permanent magnets. In electric machines, current-carrying conductors interact with magnetic fields (themselves arising from electric currents in conductors or from permanent magnets), resulting in electromechanical energy conversion. Consider a conductor of length I placed between the poles of a magnet. Let the conductor carry a current I and be at right angles to the magnetic flux lines, as shown in Fig. 1-1.It is found experimentally that the conductor experiences a force F, the direction of which is shown in Fig. 1-1 and the magnitude of which is given by

F=BIl (1.1)

Here, B is the magnitude of the magnetic flux density B, whose direction is given by the flux lines. The SI unit of B or B is the tesla (T). (Another, equivalent unit will be introduced shortly.) Notice from (1.1) that B could be defined as the force per unit current moment. Equation (1.1) is a statement of Ampere's law; the more general statement, which holds for an arbitrary orientation of the conductor with respect to the flux lines, is

F = 1 x B(1.2) where 1 is a vector of magnitude 1 in the direction of the current. Again the force is at right angles to both the conductor and the magnetic field (Fig. 1-2). Ampere's law, (1.1) or (1.2), providing as it does for the development of force, or torque, underlies the operation of electric motors.

0, Direction of force (outward)

Fig. 1-1Fig. 1-2

The magnetic flux, 4, through a given (open or closed) surface is the flux of B through that surface;

=f B -dSf B ndS S s where n is the unit outward normal to the elementary area dS of the surface (Fig. 1-3). In case B is constant in magnitude and everywhere perpendicular to the surface, of area A, (1.3) reduces to

Direction of force;F = Bllsin B

Flux lines

in'

2MAGNETIC CIRCUITS[CHAP. I from which = BA(1.4)

The SI unit of magnetic flux is the weber (Wb). We see from (1.5) that B or B may be expressed in Wb/m2, i.e.,1 T = 1 Wb/m2.

dl-----+.)1- +-- --

Fig. 1-3 IAN turnsFig. 1-4

The mutual relationship between an electric current and a magnetic field is given by Ampere's circuital law, one form of which is

=I (1.6a) where H is defined as the magnetic field intensity (in A/m) due to the current I.According to (1.6a), the integral of the tangential component of H around a closed path is equal to the current enclosed by the path.

When the closed path is threaded by the current N times, as in Fig. 1-4, (1.6a) becomes

=NI=. "(1.6b) in which -7 (or NI) is known as the magnetomotive force (abbreviated mmf). Strictly speaking, ."has the same units, amperes, as I.However, in this book we shall follow the common convention of citing .7 in ampere turns (At); that is, we shall regard N as carrying a dimensionless unit, the turn.

Magnetic flux, magnetic flux density, magnetornotive force, and (see Section 1.2) permeabilityare the basic quantities pertinent to the evaluation of the performance of magnetic circuits. The flux, 0, and the mmf, J I-, are related to each other by where 91 is known as the reluctance of the magnetic circuit.

1.2 PERMEABILITY AND SATURATION

In an isotropic, material medium, H, which is determined by moving charges (currents) only, and B, which depends also on the properties of the medium, are related by

B = µH(1.8)

CHAP. 1]MAGNETIC CIRCUITS3 where t is defined as the permeability of the medium, measured in henries per meter (Him). (For the henry, see Section 1.8.) For free space, (1.8) gives

B = p0H(1.9) where po, the permeability of free space, has the value 4ic x 10'' H/m.

The core material of an electric machine is generally ferromagnetic, and the variation of B with H is nonlinear, as shown by the typical saturation curve of Fig. 1-5(a).It is clear that the slope of the curve depends upon the operating flux density, as classified in regions I, I, and II.This leads us to 2.

i.Sheet steel 1.6

1.4 -

FIIIro 1.2 I 1.0

H, A/m Fig. 1-5(a)

Fig. 1-5(b) the concept of different types of permeabilities. We rewrite (1.8) as

B=pH=pgoH (1.10) in which g is termed permeability and R. = p/po is called relative permeability (which is dimensionless). Both ji and p, vary with H along the B-H curve. In the following, relative permeability is assumed; that is, the constant po is factored out. The slope of the B-H curve is called differential permeability;

_ 1 dB Rdpo dH

The initial permeability is defined as µ; µo H-m H(1.12)

The (relative) permeability in region I is approximately constant and equal to the initial permeability. In all three regions, the ratio of B to H at a point on the curve is known as amplitude permeability:

1BpQ=-po -H Different ferromagnetic materials have different saturation curves, as shown in Fig. 1-5(b).

1.3 LAWS GOVERNING MAGNETIC CIRCUITS

In some respects, a magnetic circuit is analogous to a dc resistive circuit; the similarity is summarized in Table 1-1.

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Table 1-1.Analogy between a do electric circuit and a magnetic circuit

Electric CircuitMagnetic Circuit

Ohm's law, I = V/R.F/93 resistance, R = 11aAreluctance, 91 = 1/µA current, Iflux, voltage, Vmmf, Jr conductivity, apermeability, p conductance, Gpermeance, 9

In the table, 1 is the length and A is the cross-sectional area of the path for the flow ofcurrent in the electric circuit, or for the flux in the magnetic circuit. In a magnetic circuit, however, 1 is themean length of the flux path. Because 4 is analogous to I and 91 is analogous to R, the laws of resistors in seriesor parallel also hold for reluctances. The basic difference between electrical resistance, R, and magneticreluctance, 91, is that the former is associated with an energy loss (whose rate is 12R), while the latter isnot. Also, magnetic fluxes take leakage paths (Fig. 1-6), whereas electric currents normally do not.

Fig. 1-6. Path of leakage flux, 4),.Fig. 1-7. Deltamax tape-wound core 0.002-in strip hysteresis loop.

1.4 AC OPERATION AND LOSSES

If the mmf is ac, then the B-H curve of Fig. 1-5 is replaced by the symmetrical hysteresisloop of

Fig. 1-7. The area within the loop is proportional to the energy loss (as heat)per cycle; this energy loss is known as hysteresis loss.

Eddy currents induced in the core material (Fig. 1-8) constitute another feature of the operationof a magnetic circuit excited by a coil carrying an alternating current. The losses due to hysteresis and eddy currents-collectively known as core losses or iron losses are approximately given by the following expressions:

eddy-current loss:Pe = K(W/kg)(1.14)

hysteresis loss:Ph = Kh fB,5 to 2.5(W /kg) 5

In (1.14) and (1.15), B,,, is the maximum flux density, f is the ac frequency, K. is a constant depending upon the material conductivity and thickness, and K. is another proportionality constant. In addition, in (1.14), t is the lamination thickness (See Sec. 1.5).

1.5 STACKING FACTOR

To reduce eddy-current loss, a core may be constructed of laminations, or thin sheets, with very thin layers of insulation alternating with the laminations. The laminations are oriented parallel to the direction of flux, as shown in Fig. 1-8(b). Eddy-current loss is approximately proportional to the square of lamination thickness, which varies from about 0.05 to 0.5 m in most electric machines. Laminating a core increases its volume. The ratio of the volume actually occupied by the magnetic material to the total volume of the core is known as the stacking factor; Table 1-2 gives some values.

Table 1-2

Lamination Thickness, mmStacking Factor

(a) UnlaminatedFig. 1-8 (b) Laminated

Because hysteresis loss is proportional to the area of the hysteresis loop, the core of a machine is made of "good" quality electrical steel which has a narrow hysteresis loop. Tape wound cores. also have lower losses. Magnetic properties of some core materials are given in Appendix C.

1.6 FRINGING

Fringing results from flux lines appearing along the sides and edges of magnetic members separated by air, as shown in Fig. 1-9; the effect increases with the area of the airgap. Fringing increases with the length of the airgap.

1.7 ENERGY STORED IN A MAGNETIC FIELD

The potential energy,stored in a magnetic field within a given volume, v, is defined by the volume integral

6MAGNETIC CIRCUITS

1 Air gap

Fig. 1-9

Fringing flux fB2dv

1.8 INDUCTANCE CALCULATIONS Inductance is defined as flux linkage per unit current; i l

The unit of inductance is the henry (H). From (1.17) it is seen that 1 H = 1 Wb/A.

For a magnetic toroid wound with n distinct coils, as shown in Fig. 1-10, n2 inductancesmay be defined:

_flux linking the pth coil due to the current in the qth coil_N (keg)(1.18)Lnq current in the qth coiliq where kpq, the fraction of the flux due to coil q that links coil p, is called the coupling coefficient between the two coils. By definition, kpq <_ 1; a value less than 1 is attributable to leakage flux between the locations of coil p and coil q. When the two subscripts in (1.18) are equal, the inductance is termed self-inductance; when unequal, the inductance is termed mutual inductance between coils p and q.Inductances are symmetrical; that is, for all p and q, en = kqandLen _ Lnq(1.19)

To express Lpq in terms of the magnetic-circuit parameters, we substitute 0g = Ngiq/3i in (1.18), obtaining k N negLnq =91= k qp E(1.20) where 91 is the reluctance of the magnetic circuit and P is its permeance. We may replace 'R in (1.20)by l/µA (for a circuit for which 1 and A can be defined) to get

C]. orb t.7 r''001°ao

CHAP. 1 ]MAGNETIC CIRCUITS7

Fig. 1-10

Lpq =(1.21)

Equations (1.17) through (1.21) can be used for inductance calculations.Alternatively, we may express the energy stored in an inductance L, carrying a current i, as

W =. Lie(1.2)inf and then obtain L by equating the right side of (1.2) to the right side of (1.16). For an n-coil system, the general relationship is

1pgtplq =1 f B- H dv(1.23) 2 p=1q=12v

1.9 MAGNETIC CIRCUITS WITH PERMANENT MAGNETS

In Section 1.1 we mentioned that a permanent magnet is the source of a magnetic field.In a magnetic circuit excited by a permanent magnet, the operating conditions of the magnet are largely determined by its position in the circuit. The second-quadrant B-H characteristics (demagnetization curves) of a number of Alnico permanent magnets are shown in Fig. 1-1; Fig. 1-12 shows the characteristics of several ferrite magnets. Commercially available characteristics are still expressed in CGS units (which, if desired, may be converted to SI units by use of Appendix A). Through a point (Hd,Bd) of a demagnetization curve there pass a hyperbola giving the value of the energy product, BdHd, and a ray from the origin (of which onlythe distal end is shown) giving the value of the permeance ratio, BdHHd. The significance of the energy product is apparent from (1.16), and a permanent magnet is used most efficiently when the energy product is maximized.

Example 1.1 The remanence, B,, of a permanent magnet is the value of B at zero H after saturation; the coercivity,.

H, is the value of H to reduce B to zero after saturation. Using Fig. 1-1, find B,, H, and the maximum energy product, (B).,, for Alnico V. Compare with the value listed in Appendix C, Table C-1.

We read B, and H from the vertical and horizontal intercepts, respectively, of the demagnetization curve.

(Parte **1** de 5)