Electrical Power Systems Design and Analysis by Mohamed E. El Hawary

Contents Preface Chapter I Introduction 1.1 The Development of Electric Power Systems, 1 1.2 Outline of the Text, 7 Chapter II Some Basic Principles 2.1 Introduction, 11 2.2 Power

Electrical Power Systems

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Electrical Power Systems

A JOHN WILEY & SONS, INc PUBUCATION

IEEE Press Power Systems Engineering Series

Dr Paul M Anderson, Series Editor

The Institute of Electrical and Electronics Engineers, Inc., New York

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© 1983 by Reston Publishing Company

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Library of Congress Cataloging-in-Publication Data

El-Hawary, M E

Electrical power systems: design and analysis I Mohamed E EI

-Hawary - Rev printing

p cm - (IEEE Press power systems engineering series)

Includes bibliographical references and index

derstandeth and right to them to

find knowledge Receive my instruc­

tion and not silver and knowledge

rather than choice gold For wisdom

is better than rubies, and all the

things that may be desired are not

to be compared to it I wisdom dwell

with prudence, and find out knowl­

edge of witty inventions

Proverbs 8:9-12

Createth, Createth man from a clot Read: And thy Lord is the Most Bounteous, Who teacheth by the pen, Teacheth man which he knew not

Coran 96: 1-5

Contents

Preface

Chapter I Introduction

1.1 The Development of Electric Power Systems, 1

1.2 Outline of the Text, 7

Chapter II Some Basic Principles

2.1 Introduction, 11

2.2 Power Concepts, 11

2.3 Three Phase Systems, 18

2.4 Power System Representation, 31

Some Solved Problems, 35 Problems, 39

Chapter III Power Generation and the Synchronous

3.2 The Synchronous Machine: Preliminaries, 44

3.3 Fields in a Synchronous Machine, 48

3.4 A Simple Equivalent Circuit, 52

3.5 Open-Circuit and Short-Circuit Characteristics, 55

3.6 Principal Steady-State Characteristics, 59

3.7 Power-Angle Characteristics and the Infinite Bus Concept, 63

3.B Static Stability Limit Curves, 68

3.9 Accounting for Saliency, 71

3.10 Salient-Pole Machine Power Angle Characteristics, 75

Some Solved Problems, 79 Problems, 85

Chapter IV The Transmission Subsystem

4.6 Transmission Line Models, 164

Some Solved Problems, 183

Appendix 4-A, 192

Appendix 4-B, 193

Appendix 4-C, 193

Problems, 197 Chapter V The Load Subsystem

5.1 Introduction, 217

5.2 General Theory of Transfonner Operation, 218

5.3 Transfonner Connections, 232

5.4 Three-Phase Induction Motors, 256

Some Solved Problems, 266

Problems, 277 Chapter VI Analysis of Interconnected Systems

6.1 Introduction, 283

6.2 Reduction of Interconnected Systems, 284

6.3 The Per Unit System, 296

6.4 Network Nodal Admittance Fonnulation, 299

6.5 The General Fonn of the Load-Flow Equations, 304

6.6 The Load-Flow Problem, 310

6.7 Getting Started, 315

6.8 Newton-Raphson Method, 318

6.9 The Newton-Raphson Method for Load-Flow Solution, 328

Some Solved Problems, 338

7.4 Classifications of Direct-Current Links, 363

7.5 Some Advantages of HVDC Transmission, 365

7.6 Some Economic Considerations, 370

7.7 Converter Circuits: Configurations and Properties, 376

7.8 Analysis of the Three-Phase Bridge Converter, 396

7.9 Inversion in Three-Phase Bridge Converter, 413

7.10 HVDC Link and Converter Control Characteristics, 416

7.11 Analysis of HVDC Link Performance, 420

Some Solved Problems, 427

Problems, 459

Chapter VIII Faults on Electric Energy Systems

8.1 Introduction, 469

8.2 Transients During a Balanced Fault, 470

8.3 The Method of Symmetrical Components, 474

8.4 Sequence Networks, 483

8.5 Line-to-Ground Fault, 501

8.6 Double Line-to-Ground Fault, 507

8.7 Line-to-Line Fault, 512

8.8 The Balanced Three-Phase Fault, 515

Some Solved Problems, 518

9.7 Bus Bar Protection, 582

9.8 Transmission Line Overcurrent Protection, 587

9.9 Pilot-Wire Feeder Protection, 598

Chapter X Power System Stability

10.1 Introduction, 625

10.2 The Swing Equation, 626

10.3 Electric Power Relations, 632

10.4 Concepts in Transient Stability, 641

10.5 A Method for Stability Assessment, 646

10.6 Improving System Stability, 657

Some Solved Problems, 658

Problems, 692 Chapter XI Optimal Operation of Electric

11.4 Accounting for Transmission Losses, 716

11.5 Optimal Operation of an All-Thermal System, Including Losses, 722

11.6 Optimal Operation of Hydrothermal Systems, 738

Some Solved Problems, 744 Problems, 775

Preface

This book is intended to provide an introduction to a number of important topics in engineering The book's audience consists mainly of post-secondary electrical engineering students as well as practicing en­ gineers interested in learning the fundamental concepts of power systems analysis and related designs Background requirements include a basic electric circuit course and some mathematical notions from algebra and calculus

The text material is arranged in a format which is aimed at furthering the readers' understanding by providing ample practical examples within the text to illustrate the concepts discussed In addition, each chapter contains a section that offers additional solved problems that serve to illustrate the interrelation between the concepts discussed in the chapter from a system's point of view

The text treats first models of the major components of modern day electric power systems Thus, chapters three through five provide detailed discussions of synchronous machines, transmission lines, transformers and the induction motor which is a major system load component

Chapter six deals with analysis of interconnected systems with major emphasis on load flow analysis Chapter seven is intended to present-in a reasonable amount of detail-elements of high voltage, direct current transmission which are becoming increasingly important

Chapter eight details analysis problems in systems with fault condi­ tions This is followed in Chapter nine by a treatment of system protection Chapter ten is devoted to transient stability problems at an introduc­ tory level The final chapter on optimal economic operation of power systems provides a comprehensive yet simple introduction to that im­ portant area Each of the chapters is concluded by a section of problems for drill purposes It is assumed that the reader has access to a modest computing facility such as a programmable calculator

I am indebted to my many students who have contributed immensely

to the development of this text, in particular students at Memorial Univer­ sity of Newfoundland and the Technical University of Nova Scotia, who took great interest in this project To my colleagues and friends from

electric utilities and the academe alike, my sincere appreciation for their contributions

The drafting of the manuscript involved the patient and able typing work done by many at Memorial University and lately at the Technical University of Nova Scotia Many thanks to Mrs Minnie Ewing,

Ms Marilyn Tiller, Ms Brenda Young, Mrs Ethil Pitt, and Ms Valerie Blundell of Memorial and Frances Julian of the Technical University of Nova Scotia Margaret McNeily of Kennett Square, Pennsylvania, skillfully copyedited the original manuscript I gratefully appreciate her help Also,

my thanks to Dan McCauley of Reston Publishing Company for his work

on this book Finally, the patience and understanding of my wife Ferial, and children are appreciated

Halifax, Nova Scotia

February, 1982

M E El-Hawary

CHAPTER)

Introduction

The purpose of this chapter is twofold We first provide a brief

perspective on the development of electric power systems This is not

intended to be a detailed historical review, but rather it uses historical

landmarks as a background to highlight the features and structure of the

modem power systems Following this we offer an outline of the text

material

ELECTRIC POWER SYSTEMS

Electric power is one major industry that has shaped and contributed

to the progress and technological advances of mankind over the past

century It is not surprising then that the growth of electric energy

consumption in the world has been nothing but phenomenal In the United

States, for example, electric energy sales have grown to well over 400 times

in the period between the tum of the century and the early 1970s This

growth rate was 50 t i me s as much as the growth rate in all o th e r energy forms used during the same period

Edison Electric Illuminating Company of New York pioneered the central station electric power generation by the opening of the Pearl Street station in 1881 This station had a capacity of four 250-hp boilers supplying steam to six engine-dynamo sets Edison's system used a 1l0-V dc under­ ground distribution network with copper conductors insulated with a jute wrapping The low voltage of the circuit limited the service area of a central station, and consequently central stations proliferated throughout metro­ politan areas

The invention of the transformer, then known as the "inductorium," made ac systems possible The first practical ac distribution system in the United States was installed by W Stanley at Great Barrington, Massachu­ setts, in 1866 for Westinghouse, who acquired the American rights to the transformer from its British inventors Gaulard and Gibbs Early ac distri­ bution utilized 1000- V overhead lines

By 1895, Philadelphia had about twenty electric companies with distribution systems operating at 100-V and 500-V two-wire dc and 220-V three-wire dc; single-phase, two-phase, and three-phase ac; with frequencies

of 60, 66, 125, and 133 cycles per second; and feeders at 1000-1200 V and 2000-2400 V

The consolidation of electric companies enabled the realization of economies of scale in generating facilities, the introduction of a certain degree of equipment standardization, and the utilization of the load diver­ sity between areas Generating unit sizes of up to 1300 MW are in service,

an era that was started by the 1973 Cumberland Station of the Tennessee Valley Authority A major generating station is shown in Figure 1-1, with the turbine-generator hall shown in Figure 1-2

Underground distribution at voltages up to 5 kV was made possible by the development of rubber-base insulated cables and paper-inSUlated, lead­ covered cables in the early 1900s Since that time higher distribution voltages have been necessitated by load growth that would otherwise overload low-voltage circuits and by the requirement to transmit large blocks of power over great distances Common distribution voltages in today's system are in 5-, 15-, 25-, 35-, and 69-kV voltage classes

The growth in size of power plants and in the higher voltage equip­ ment was accompanied by interconnections of the generating facilities These interconnections decreased the probability of service interruptions, made the utilization of the most economical units possible, and decreased the total reserve capacity required to meet equipment-forced outages This growth was also accompanied by the use of sophisticated analysis tools such

as the network analyzer shown in Figure 1-3 Central control of the interconnected systems was introduced for reasons of economy and safety Figure 1-4 shows the control room in a system control center The advent of the load dispatcher heralded the dawn of power systems engineering, whose

1.2 Outline of the Text 7 objective is to provide the best system to meet the load demand reliably,

safely, and economically, utilizing state-of-the-art computer facilities

Extra high voltage (EHV) has become the dominant factor in the

transmission of electric power over great distances By 1896, an ll-kV

three-phase line was transmitting 10 MW from Niagara Falls to Buffalo

over a distance of 20 miles Today, transmission voltages of 230 kV (see

Figure 1-5), 287 kV, 345 kV, 500 kV, 735 kV, and 765 kV are commonplace,

with the first llOO-kV line scheduled for energization in the early 1990s A

prototype 1200-kV transmission tower is shown in Figure 1-6 The trend is

motivated by the economy of scale due to the higher transmission capacities

possible, more efficient use of right-of-way, lower transmission losses, and

reduced environmental impact

The preference for ac was first challenged in 1954 when the Swedish

State Power Board energized the GO-mile, 100-kV dc submarine cable

utilizing U Lamm's Mercury Arc valves at the sending and receiving ends

of the world's first high-voltage direct current (HVDC) link connecting the

Baltic island of Gotland and the Swedish mainland Today numerous

installations with voltages up to BOO-kV dc have become operational around

the globe Solid-state technology advances have also enabled the use of the

silicon-controlled rectifiers (SCR) or thyristor for HVDC applications since

the late 19608 Whenever cable transmission is required (underwater or in a

metropolitan area), HVDC is more economically attractive than ac

Protecting isolated systems has been a relatively simple task, which is

carried out using overcurrent directional relays with selectivity being ob­

tained by time grading High-speed relays have been developed to meet the

increased short-circuit currents due to the larger size units and the complex

interconnections

1.2 OUTLINE OF THE TEXT

Chapter 2 lays the foundations for the development in the rest of the

book The intention of the discussion offered here is to provide a brief

review of fundamentals including electric circuit analysis and some

mathematical background, to make the treatment self-contained A student

with an introductory electric circuit background may safely omit this

chapter

Chapters 3, 4, and 5 are sequentially structured to follow the flow of

electric energy from conversion to utilization Thus Chapter 3 treats the

synchronous machine from an operational modeling point of view Empha­

sis here is on performance characteristics of importance to the electric

power specialist Chapter 4 provides a comprehensive treatment of EHV

transmission lines starting from parameter evaluation for different circuit

(Courtesy Ontario Hydro)

Figure 1-6 A Prototype 1200-kV Transmission Line Tower

(Courtesy u.s Department of Energy Bonneville Power Administration)

and conductor configurations Various transmission line performance model­ ing approaches are covered along with a unique section on the errors involved when using simplified models over the more elaborate ones Chapter

5 is entitled" The System Load" and deals with the power transformer as well as control and instrument transformers in addition to induction motor models as the latter is a major load component A brief discussion of load modeling philosophy is given at the end of the chapter

Chapter 6 treats interconnected system analysis and covers aspects of network reduction, per unit systems, and the load flow problem A com­ prehensive treatment of high-voltage direct-current transmission is given in Chapter 7 Here again emphasis is placed on analysis and control aspects that should be of interest to the electric power systems specialist

Faults on electric energy systems are considered in Chapter 8 Here we start with the transient phenomenon of a symmetrical short circuit, fol­ lowed by a treatment of unbalanced and balanced faults Realizing the crucial part that system protection plays in maintaining service integrity is the basis for Chapter 9 Here an introduction to this important area is given The transient stability problem is treated in Chapter 10 from an introductory point of view Chapter 11 introduces the subject of economic dispatch under the title "Optimal Operation of Electric Power Systems." The treatment covers thermal systems where losses are neglected, followed

by a case including losses The chapter is concluded by an introduction to hydrothermal dispatch

The text of each chapter includes a number of examples that illustrate the concepts discussed Following each chapter there is a set of solved problems that involves, in many instances, increased sophistication, and it helps to bring together the overall thrust of the concepts and techniques treated The student should have, then, no difficulty in dealing with the drill problems included at the end of each chapter

CHAPTER I)

Some Basic Principles

2.1 INTRODUCTION

The intention of this chapter is to lay the groundwork for the study

of electric energy systems This is done by developing some basic tools

involving concepts, definitions, and some procedures fundamental to electric

energy systems The chapter can be considered as simply a review of topics

that will be utilized throughout this work We start by introducing the

principal electrical quantities that we will deal with

2.2 POWER CONCEPTS

The electric power systems specialist is in many instances more

concerned with electric power in the circuit rather than the currents As the

power into an element is basically the product of voltage across and current

through it, it seems reasonable to swap the current for power without losing

any information in describing the phenomenon In treating sinusoidal

steady-state behavior of circuits, some further definitions are necessary To illustrate the concepts, we will use a cosine representation of the wavefonns Consider impedance element Z = Z L! For a sinusoidal voltage, v( t) is given by

v( t) = Vmcos wt

The instantaneous current in the circuit is

i( t) = I,,,,cos( wt -if»

where

The instantaneous power is thus given by

p(t) =v(t)i(t) = VmI m[cos(wt)cos(wt- if» ]

Using the trigonometric identity

cos a cos f3 = ! [ cos( a -f3) + cos( a + f3)]

we can write the instantaneous power as

p( t) =

V;Im

[cos if> + cos(2wt -if»]

The average power Pay is seen to be

VmIm

Since the average of cos(2wt - if» is zero, through 1 cycle, this term therefore contributes nothing to the average of p

It is more convenient to use the effective (rms) values of voltage and current than the maximum values Substituting Vm = 12 (Vnns)' and 1m = 12 (lrms)' we get

(2.2)

Thus the power entering any network is the product of the effective values

of terminal voltage and current and the cosine of the phase angle if> which is called the power factor (PF) This applies to sinusoidal voltages and currents only For a purely resistive load, cos if> = 1, and the current in the circuit is fully engaged in conveying power from the source to the load resistance When reactance as well as resistance are present, a component of the current in the circuit is engaged in conveying the energy that is periodically stored in and discharged from the reactance This stored energy, being shuttled to and from the magnetic field of an inductance or the electric field of a capacitance, adds to the current in the circuit but does not add to the average power

2.2 Power Concepts 13

The average power in a circuit is called active power, and the power

that supplies the stored energy in reactive elements is called reactive power

Active power is P, and the reactive power, designated Q, are thus*

P= VI cosq,

Q = VI sinq,

(2.3) (2.4)

In both equations, V and I are rms values of terminal voltage and current,

and q, is the phase angle by which the current lags the voltage

Both P and Q are of the same dimension, that is in watts However, to

emphasize the fact that the Q represents the nonactive power, it is mea­

sured in reactive voltampere units (var) Larger and more practical units are

kilovars and megavars, related to the basic unit by

1 Mvar = 103 kvar = 106 var Figure 2-1 shows the time variation of the various variables discussed

in this treatment

Assume that V, Vcosq" and Vsinq" all shown in Figure 2-2, are each

multiplied by I, the rms value of current When the components of voltage

V cos q, and V sin q, are multiplied by current, they become P and Q respec­

tively Similarly, if I, I cosq" and I sinq, are each multiplied by V, they

become VI, P, and Q respectively This defines a power triangle

We define a quantity called the complex or apparent power, desig­

nated S, of which P and Q are components By definition,

S=P+jQ

= VI cos q, + jVI sin q,

= VI( cosq, + jsinq,) Using Euler's identity, we thus have

*If we write the instantaneous power as

p( t) = Vnnslnns[ cos cj)(l + cos2wt)] + Vrms1nnssin </lsin2wt

then it is seen that

p( t) = P(l + cos2wt) + Qsin2wt Thus P and Q are the average power and the amplitude of the pulsating power respectively

I

I

(2.6)

Consider the series circuit shown in Figure 2-3 Here the applied

voltage is equal to the sum of the voltage drops:

with

(2.8)

(2.9)

being the individual element's complex power Equation (2.8) is known as

the summation rule for complex powers The summation rule also applies to

parallel circuits The use of the summation rule and concepts of complex

power may prove advantageous in solving problems of power system

analy-sis

The phasor diagrams shown in Figure 2-2 can be converted into

complex power diagrams by simply following the definitions relating com­

plex power to voltage and current Consider the situation with an inductive

circuit in which the current lags the voltage by the angle cpo The conjugate

of the current will be in the first quadrant in the complex plane as shown in

Z, Z2 Zn i� l

Figure 2-3 Series Circuit

Figure 2-4(a) MUltiplying the phasors by V, we obtain the complex power diagram shown in Figure 2-4(b) Inspection of the diagram as well as the previous development leads to a relation for the power factor of the circuit:

*

H

-e-c CJ)

The power factor (PF) of the R-L branch is

PFz = cos cpz = cos 57 990

=0.53

The current into the capacitance is

The input current It is

The apparent power into the circuit is

in many instances as part of a three-phase system Three-phase operation is preferable to single-phase because a three-phase winding makes more effi­ cient use of generator copper and iron Power flow in single-phase circuits was shown in the previous section to be pulsating This drawback is not present in a three-phase system as will be shown later Also three-phase motors start more conveniently and, having constant torque, run more

satisfactorily than single-phase motors However, the complications of addi­

tional phases are not compensated for by the slight increase of operating

efficiency when polyphase systems other than three-phase are used

A balanced three-phase voltage system is composed of three single­

phase voltages having the same magnitude and frequency but time-displaced

from one another by 120° Figure 2-5(a) shows a schematic representation

where the three single-phase voltage sources appear in a Y connection; a .:l

configuration is also possible A phasor diagram showing each of the phase

voltages is also given in Figure 2-5(b) As the phasors revolve at the angular

frequency iN with respect to the reference line in the counterclockwise

(positive) direction, the positive m a x i m u m value first occurs for phase a and then in succession for phases band c Stated in a different way, to an observer in the phasor space, the voltage of phase a arrives first followed by that of b and then that of c For this reason the three-phase voltage of Figure 2-5 is said to have the phase sequence abc ( order or phase sequence

or rotation are all synonymous terms) This is important for certain applica­ tions For example, in three-phase induction motors, the phase sequence determines whether the motor turns clockwise or counterclockwise

With very few exceptions, synchronous generators (commonly referred

to as alternators) are three-phase machines For the production of a set of three voltages phase-displaced by 120 electrical degrees in time, it follows that a minimum of three coils phase-displaced 120 electrical degrees in space must be used An elementary three-phase two-pole machine with one coil per phase is shown in Figure 2-6

We find it convenient for clarity of the presentation to consider representing each coil as a separate generator An immediate extension of the single-phase circuits discussed above would be to carry the power from

�

Ie

Ib -

the three generators along six wires However, for the sake of economy, instead of having a return wire from each load to each generator, a single wire is used for the return of all three The current in the return wire will be

Ia + Ib + Ie; and for a balanced load, these will cancel out as may be seen by inspecting the phasor diagram in Figure 2-7 If the load is unbalanced, the return current will still be small compared to either la' IiJ, or Ie Thus the return wire could be made smaller than the other three This connection is known as a four-wire three-phase system It is desirable for safety and system protection to have a connection from the electrical system to ground A logical point for grounding is the generator neutral point, the junction of the Y

Current and Voltage Relations

Balanced three-phase systems can be studied using techniques devel­ oped for single-phase circuits The arrangement of the three single-phase voltages into a Yor a Ll configuration requires some modification in dealing with the overall system

Y Connection

With reference to Figure 2-8, the common terminal n is called the

neutral or star ( Y) p o int The voltages appearing between any two of the line terminals a, b, and c have different relationships in magnitude and phase to the voltages appearing between any one line terminal and the neutral point n The set of voltages Vab, Vbc' and Vca are called the l ine voltages, and the set of voltages Van' ViJn, and Y::n are referred to as the

phase v oltages Analysis of phasor diagrams provide.'l the required rela­ tionships

The effective values of the phase voltages are shown in Figure 2-8 as

Van' ViJn, and Y::n Each has the same magnitude, and each is displaced 1200

from the other two phasors To obtain the magnitude and phase angle of the line voltage from a to b (i.e., Vab), we apply Kirchhoff's voltage law:

/ /

)' 1 -.1 Von = Vp l.!::L

Figure 2-8 illustrating the Phase and Magnitude Relations Between the Phase

and Line Voltage of a Y Connection

where � denotes the effective magnitude of the phase voltage Accordingly

we may write

Van=��

Vbn = l-j,/ -1200

V = en V/-2400 = V/120° p_ p Substituting Eqs (2.13) and (2.14) in Eq (2.11) yields

(2.16)

(2.17) (2.18) The expressions obtained above for the line voltages show that they

constitute a balanced three-phase voltage system whose magnitudes are 13

times the phase voltages Thus we write

(2.20)

In the above equation, IL denotes the effective value of the line current and

Ip denotes the effective value for the phase current

4 Connection

We consider now the case when the three single-phase sources are rearranged to fonn a three-phase 1.1 connection as shown in Figure 2-9 It is clear from inspection of the circuit shown that the line and phase voltages have the same magnitude:

2.3 Three-Phase Systems 25

The phase and line currents, however, are not identical, and the relationship

between them can be obtained using Kirchhoff's current law at one of the

line terminals

In a manner similar to that adopted for the Y-connected source, let us

consider the phasor diagram shown in Figure 2-10 Assume the phase

Note that a set of balanced three phase currents yields a correspond­ ing set of balanced line currents that are 13 times the phase values:

With the currents given by

P3</>(t) =va(t)ia(t) +vb(t)ib(t) +vc(t)ic(t)

This turns out to be

P3</>(t) = 2\.j,Ip[sin( wt)sin( wt-<1»

+ sin( wt - 120)sin( wt - 120 - <1»

+ sin( wt + 120 )sin( wt + 120 -cp )]

Using a trigonometric identity, we get

P3</>(t) = \.j,Ip{3 cos <I> -[cos(2 wt-<1» + cos(2wt-240 -cp)

2.3 Three-Phase Systems 27

relationship between the line and phase voltages in a Y-connected system is

/VL/= {a/VI The power equation thus reads in terms of line quantities:

P34> = {a/ VLII IL/cos</> (2.24)

We note that the total instantaneous power is constant, having a

magnitude of three times the real power per phase We may be tempted to

assume that the reactive power is of no importance in a three-phase system

since the Q terms cancel out However, this situation is analogous to the

summation of balanced three-phase currents and voltages that also cancel

out Although the sum cancels out, these quantities are still very much in

evidence in each phase We thus extend the concept of complex or apparent

power (8) to three-phase systems by defining

(2.29)

(2.30)

In specifying rated values for power system apparatus and equipment

such as generators, transformers, circuit breakers, etc., we use the magni­

tude of the apparent power 834> as well as line voltage for specification

values In specifying three-phase motor loads, we use the horsepower output

rating and voltage To convert from horsepower to watts, recall that

1 hp =746 W Since the horsepower is a mechanical output, the electrical input will be

somewhat higher due to the losses in the energy conversion process (more

on this in Chapter 5) The efficiency of a process " is defined by

p

.,,=� l';.n

The value of efficiency should be used when converting a mechanical load to

an equivalent electrical representation

Example 2.2

A Y-connected, balanced three-phase load consisting of three imped­ ances of 10/300 ohms each as shown in Figure 2-11 is supplied with the balanced line-to-neutral voltages:

Van=220�V Vbn = 220/2400 V Y.,n = 220/1200 V

A Calculate the phasor currents in each line

B Calculate the line-to-line phasor voltages

C Calculate the total active and reactive power supplied to the load