Introduction to

Solid StatePower Electronics

Editor: John William Motto, Jr.

Semiconductor DivisionYoungwood, Pennsylvania 15697

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AcknowledgementThe material for this updated text was originallywritten by Dr. William E. Newell, a noted authorityon power electronics with the aid of his associates atthe Westinghouse Research and DevelopmentCenter. Until his untimely death in 1976, Dr. Newellhad devoted much of his life to helping practicingengineers and students alike better understand thefield of power electronics. As a “teacher” withinWestinghouse, as well as at Carnegie MellonUniversity, Dr. Newell tested the materials in this textwell, with over five years of use and refinement.

We dedicate this text to the memory of Dr. Newellfor his unselfish contributions to the field of powerelectronics. In order that future generations ofengineers and students can profit from the efforts ofDr. Newell, The Westinghouse Electric Corporationhas engaged his friends and colleague John W.Motto, Jr., to edit this work. John brings to this taskover 17 years of applications and ratings experienceon power semiconductor devices.

We hope you find this text a worthwhile addition toyour reference library. We appreciate any commentsyou might have so that your ideas can be reflected insucceeding editions of this text.

POWEREX, Inc.200 Hillis StreetYoungwood, PA 15697-1800

February 1977COPYRIGHT© 1977 BY POWEREX, INC.

All Rights Reserved

Information furnished in this text is believed to beaccurate and reliable. However, Powerex can assumeno responsibility for its use; nor any infringements ofpatents, or other rights of third parties which mayresult from its use. No license is granted byimplication or other use under any patent or patentrights of Powerex.

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ForewordAlthough power electronics has a considerablylonger history than many better known areas ofelectrical engineering, its widespread significance isonly now beginning to gain recognition. In a sense,power electronics is the marriage of techniqueswhich are characteristically power. But theincompatibilities which have traditionally separatedthese two specialties makes the consummation of thismarriage far from simple. Power engineers find itdifficult to adapt their intuitive thought patterns to amicrosecond time scale, and electronics engineershave similar problems in adjusting to the demands ofa megawatt power scale. Thus there is a growingneed for electrical engineers whose capabilitiestranscend the two fields.

At least three current trends dictate that in spite ofits slow rise to prominence, power electronics willsoon emerge as a discipline and profession ofmajor importance:

1. The growing shortage of oil and gas and growingenvironmental concern will stimulate theutilization of clean electrical power in countlessnew areas now predominantly served by otherforms of energy.

2. Efficiency in the manipulation and control ofelectrical power will have increasing priority asthe rising cost of power forces the abandonmentof techniques which are short-term cheap butlong-term wasteful.

3. The evolution of present applications and thecreation of new applications will cause increasingdemands for speed, precision, and reliability inpower control that can be satisfied by no othertechnology.This text seeks to introduce powerelectronics in a way which emphasizes both theshared and the unique aspects of a wide varietyof circuits for power conversion and control. Italso seeks to build on elementary fundamentalswith which all electrical engineers should befamiliar. A review of these fundamental principalsneeded to establish a basic understanding ofpower electronics is concisely stated in AppendixI as the “Ten Cornerstones of Power Electronics.”

This text is divided into the following majorchapters:

I. IntroductionII. Diodes and Uncontrolled RectifiersIII. Thyristors, AC/DC Converters, and OtherNaturally Commutated CircuitsIV. Turn-off Devices and Self-commutated CircuitsV. Power Semiconductor Device Protection

Each of these chapters is further divided into concisediscussions of the topics and ideas that are felt to be ofgreatest importance in appreciating both thecapabilities and limitations of power electronics. Thisformat should allow the reader to quickly andselectively locate the section(s) of greatest interest tohim.

Although a familiarity with the vocabulary of powerelectronics may result from reading or listening,experience has shown that relatively little usefullearning (defined as the acquisition of new skills)occurs in this way. The problems at the end of eachchapter are the “meat” of the text. These problemsseek to illustrate important results while encouragingthe reader to think and analyze in terms of broadlyapplicable physical principles. They seek to avoidboth routine “formula plugging” and excessivemathematical grinding.

Once power electronics has penetrated your“awareness barrier,” hopefully you will be aroused tocontinue your involvement with the field, becominga contributor to the profession as well as abenefactor from the technology.

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Table of ContentsCHAPTER 1INTRODUCTION TO POWER ELECTRONICS

Section1. What is power electronics?2. Power Converters as Switching Matrices3. The Role of Power Filters4. Types of Solid-State Switches5. Commutation6. The Internal Functional Sections of a

Generalized Power Converter7. Power/Frequency Domains of Power Electronics8. Specialized Technologies of Power Electronics9. Converter Terminology Defined

10. Operation of an AC/DC Converter11. Functional Categories of Applications12. Power Electronics – Lab.. Curiosity or

Competitive Necessity?13. Power Electronics Books14. Problems

CHAPTER 2 DIODES AND UNCONTROLLED RECTIFIERS

Section1. Type 1 Switch or Diode2. Steady-State V-I Characteristic of a Diode3. Transient V-I Characteristic of a Diode4. Pulse Number5. Simplifying Assumptions6. Single-Phase Half-Wave Diode Rectifier7. Single-Phase Half-Wave Diode Rectifier – R/L Loads8. Single-Phase Half-Wave Diode Rectifier – Other Loads9. Single-Phase Full-Wave Diode Rectifiers and

Continuous Current10. Average Voltage Across an Inductance11. Two-Pulse Rectifier with Inductance Filter12. Power Relationships13. Three-Phase Diode Rectifiers14. Generalized Center-Tap Rectifier15. Generalized Bridge Rectifier16. Applications of Diode Rectifiers17. Transformers and Nonsinusoidal Waveforms18. Problems

CHAPTER 3THYRISTORS, AC/DC CONVERTERS ANDOTHER NATURALLY COMMUTATED CIRCUITS

Section1. Type 2 Switch Thyristor2. Steady-State V-I Characteristic of a Thyristor3. Two-Transistor Analog for Explaining Thyristor Turn On4. Gate Characteristics5. Transient V-I Characteristic of a Thyristor6. Thyristor Ratings – Design Tradeoffs and

Commercial Availability7. Circuit Functions Performed by Type 2 Switches8. Two-Pulse AC/DC Converter Operating as a

Controlled Rectifier9. Two-Pulse AC/DC Converter Operating as a

Synchronous Inverter10. AC/DC Converter as a linear Amplifier11. AC/DC Converter as a DC Motor Drive12. Two-Pulse Semiconverter13. Two-Pulse Dual Converter14. Waveform Derivation for a 3-Phase Bridge

AC/DC Converter15. Naturally Commutated Cycloconverter16. Bilateral Solid-State Switches17. Triac18. AC Switches and Regulators19. Varegulators20. Brushless Machines21. Summary22. Problems

CHAPTER 4TURN-OFF DEVICES AND SELF COMMUTATED CIRCUITS

Section1. Power Transistor2. Gate Controlled Switch3. DC Switching and Regulation4. Chopper Using Thyristors5. Fundamentals of Inverters6. Full-Bridge Inverter7. Half-Bridge Inverter with an Inductive Load8. Thyristor Inverters with Forced Commutation9. Types of Forced Commutation

10. Series Inverter11. Parallel Inverter12. Complementary Impulse – Commutated Inverter13. Inverter Output Voltage Control14. Inverter Output Waveform Improvement15. Frequency and Power Factor Changers16. Problems

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CHAPTER 5POWER SEMICONDUCTOR DEVICE PROTECTION

Section1. The Power Module2. Importance of Junction Temperature3. Components of Dissipation4. Average Dissipation at 60 Hz Switching Rate5. Average Dissipation at 40 KHz Switching Rate6. Simplified Geometry Assumed for the “Typical” Device7. Junction-to-Case Thermal Resistance8. Heat Sink Calculation9. Selection of Device Package and Type of Heat Sink

10. Natural Connection Air Cooling11. Forced Air Cooling12. Water Cooling13. Using the Data Sheet for Heat Sink Calculations14. Thermal Capacity15. Transmission Line for Transient Thermal Analysis16. Initial Temperature Rise17. Transient Thermal Impedance of Each Section18. Calculation of Thermal Time Constant19. Junction-to-Case Transient Thermal Impedance20. Using Transient Thermal Impedance21. Calculation of Pulse and Surge Current Capabilities22. Turn-On di/dt23. Transient Overvoltage24. Off-State dv/dt25. Series/Parallel Arrays for High-Power Applications26. Summary27. Problems

APPENDIX ITen Cornerstones of Power Electronics

APPENDIX IISelected Symbol List

APPENDIX IIIPower Electronic Circuit Configurations — Voltage andCurrent Relationships

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Chapter 1INTRODUCTION

TO POWER ELECTRONICS

Section1. What is power electronics? . . . . . . . . . . . . . . . . . . . . .1-22. Power Converters as Switching Matrices . . . . . . . . . . . .1-33. The Role of Power Filters . . . . . . . . . . . . . . . . . . . . . .1-34. Types of Solid-State Switches . . . . . . . . . . . . . . . . . . . .1-45. Commutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-56. The Internal Functional Sections of a

Generalized Power Converter . . . . . . . . . . . . . . . . . . .1-57. Power/Frequency Domains of Power Electronics . . . . . . .1-68. Specialized Technologies of Power Electronics . . . . . . . . .1-79. Converter Terminology Defined . . . . . . . . . . . . . . . . . .1-7

10. Operation of an AC/DC Converter . . . . . . . . . . . . . . .1-811. Functional Categories of Applications . . . . . . . . . . . . .1-812. Power Electronics – Lab. Curiosity or

Competitive Necessity? . . . . . . . . . . . . . . . . . . . . . . . .1-913. Power Electronics Books . . . . . . . . . . . . . . . . . . . . . .1-1014. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-12

IntroductionThe objectives of this chapter are to establish aperspective for the entire text which will enable the reader:

a. To relate and distinguish power electronics fromother fields of electronics and electrical engineering;

b. To recognize the importance of power efficiencyand to understand how it is achieved;

c. To distinguish between the basic types of solid-state switching devices;

d. To define the basic functional sections within apower converter, the specialized technologiesinvolved in converter design, and the functionalapplications which converters serve.

1. What is Power Electronics? (Figure 1.1)

For descriptive purposes, it is frequently useful todivide the overall field of electrical engineering intothree areas of specialization: electronics, power, andcontrol. The electronics area deals primarily withdevices and circuits for the processing ofinformation; the power area deals with both rotatingand static equipment for the generation,transmission, distribution, and utilization of vastquantities of electrical power; and the control areadeals with the stability and response characteristics ofclosed-loop systems using feedback on either acontinuous or sampled-data basis.

Interstitial to all three of these areas is powerelectronics, which deals with the use of electronics forthe control and conversion of large amounts ofelectrical power.

The origins of power electronics can be traced backmany years, at which time mercury-arc devices wereutilized for the rectification of AC to DC or theinversion of DC to AC. However, today’s rapidlygrowing usage of power electronics has resulted fromthe development of solid-state power devices.

Specifically, then, we will limit the use of the namepower electronics to those applications in which

electrical power flows through and is controlled byone or more solid-state power devices. (Note that thisdefinition excludes such applications as those wherea low-level electronic control circuit actuates anelectromechanical relay in the power circuit.)

All of the important parameters of the electricalwaveform are subject to regulation or conversion bysolid-state power devices, including effective voltage,effective current, frequency, and/or power factor.Often the control of electrical power is desiredsimply as a means for controlling some non-electricalparameter. For example, drives for controlling thespeed of a motor. In other applications, powerelectronics is used to control the temperature of anoven, the rate of an electrochemical refining process,the intensity of lighting, etc.

The design of power electronics equipment involvesinteractions with the source and the load, andutilizes small-signal electronic control circuits as wellas power devices. Therefore power electronics drawsupon, and indeed depends upon all of the otherareas of electrical engineering. Obviously, thepotential scope of this field is quite vast!

Despite the growing importance of powerelectronics, very few universities presently offercourses devoted to this area, and few courses in theother areas deals significantly with it either. Thereinlies the motivation for this text. To keep the textwithin bounds, we will assume that you have a basicknowledge of the other areas of electricalengineering. We will concentrate most heavily onthose important topics which are either ignored orgiven inadequate emphasis in available texts in theother areas.

Probably the single most important distinctionbetween power electronics and small-signalelectronics is the importance attached to overallpower efficiency. In most cases, other factorsoutweigh power efficiency in small-signal electroniccircuits, and power efficiency is calculated as aninteresting after thought, if at all. But in powerelectronics, power efficiency is critical because boththe cost of dissipating the heat and the cost of thewasted power are significant compared to the totalcost of the equipment.

The importance of power efficiency dictates that thebasic control element in power electronics must be aswitch, NOT a continuously variable element.

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POW

ER

STATIC

EQU

IPMEN

T

RO

TATING

EQU

IPMEN

T

ELEC

TRO

NIC

SD

EVIC

ESC

IRC

UIT

S

POWER

ELECTRONICS

CONTROL

CONTINUOUS SAMPLEDDATA

Figure 1.1Power Electronics…Interstitial to all of the major disciplines ofelectrical engineering.

2. Power Converters as Switching Matrices (Figure 1.2)

Newcomers to power electronics are frequentlybewildered by the hundreds of different circuits whichconfront them when they begin to search typicalreferences in the field. The essential differencesbetween the various circuits are seldom clearlydistinguished from their basic similarities. And if youattempt to understand each of these circuitsindividually, you have a lifetime project before you.

The first fundamental principle which you shouldremember forever is that power efficiency dictatesthat switches be used as control devices. The secondfundamental principle, which is of equal importance,is that all high-power controls or converters aresimply switching matrices. Like a telephoneexchange, a general switching matrix permits anyincoming line to be connected for a specified periodof time to any outgoing line.

If there are m incoming lines and n outgoing lines,the matrix has m x n switches. Because of the factthat most solid-state switches are unilateral, thegeneral solid-state switching matrix has 2 x m x nswitches, one switch of each polarity for eachintersection.

Specific converter circuits may have one linecommon to the input and output, in which case the 2 switches associated with the corresponding

intersection degenerate to a short circuit. Othercircuits require only one-way conduction so that 1 ofthe 2 switches at each intersection can be discarded.In summary, the number of solid-state switches in aparticular converter circuit may be less than or equalto 2 x m x n, but any converter circuit can be derivedfrom the general switching matrix.

A power switching matrix synthesizes the desiredvoltages on the output lines from selected “chunks”of the input voltages as dictated by the switchingcontrol signals.* The functions which a givenconverter circuit can perform are dictated by

(a) which of the switches are omitted from thegeneral matrix,

(b) the switching control signals, and

(c) the types of solid-state switches used.

*In a sense, a power converter is a combination analog/digital circuit. Ituses switches to perform the required processing, like a digital circuit butunlike an analog circuit. On the other hand, the power “signal” that isprocessed is continuous as in an analog circuit, not quantized as in adigital circuit.

3. The Role of Power Filters

The voltage waveforms which are synthesized by theswitching matrix are usually only an approximation tothe desired output waveforms. Although theapproximation could be improved by usingcontinuously variable devices in place of the switches,this would cause an intolerable loss in powerefficiency. Because of the repetitive nature of theswitching action, the discrepancy in the waveformsappears as spurious frequencies which can beremoved by a filter in the output lines. In otherwords, the output filter absorbs the unwanted ripplevoltages from the output of the matrix.

A secondary effect is that the input currentwaveforms to the matrix contain spuriousfrequencies which may be undesirable in the supplylines or the source. Therefore another filter may beneeded on the input lines to provide a local path forthe unwanted ripple currents required by the matrix.

These discrepancies may be viewed as an undesiredripple in the instantaneous power flow through thematrix. The filters reduce this ripple in the inputand output lines by storing energy during intervals of

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Switching ControlSignals

n OutputLines

m InputLines

Figure 1.2mxn Switching Matrix

excess power flow and returning energy to the circuitduring intervals of insufficient power flow.

Filters contribute significantly to the cost, size, andweight of power electronics equipment, as well as toits performance. The cost and weight of the filtersdepend on the maximum energy storage which isrequired, which depends in turn on both the powerrating of the circuit and the frequency of theundesired ripple. The lower the ripple frequency, thelonger the interval during which the filter must beable to absorb or supply a significant portion of therated power. Conversely, if the ripple frequency canbe increased, comparable performance can beobtained from smaller and less costly filters.

These considerations, together with the relatedsavings on transformers, motivate the selection of apower frequency greater than 60Hz in someapplications. In other applications which are limitedto 60Hz, the ripple frequency can be increased bythe selection of a circuit configuration with a higher“pulse number”, as we shall see later.

4. Types of Solid-State Switches (Figure 1.3)

Types of Switches:

Type 1 DIODES - Conduct Automatically whenForward Polarity is Applied.

Type 2 THYRISTORS or SCR’S - Begin toConduct in the Forward Direction UponCommand of a Control Signal andContinue to Conduct Until the NextCurrent Zero Crossing.

Type 3 TRANSISTORS, GATE CONTROLLEDSWITCHES, FORCE-COMMUTATEDTHYRISTORS - Forward Conduction can beInitiated and Interrupted by Control Signals

As stated previously, one of the factors whichdetermine the performance of a power switchingmatrix is the type of switch that is used. For presentpurposes we don’t need to get into the physics ofsolid-state switches, but we should distinguishbetween three main types of switches that areavailable to us. All three types are UNILATERAL.They can conduct current in one direction, known asthe forward direction, but they very rapidly ceaseconduction when the reverse polarity is applied. Theother characteristics of these switches are as follows:

Type 1 Switches always conduct whenever aforward polarity of voltage is applied totheir terminals. Type 1 switches are knownas rectifiers or diodes.

Type 2 Switches do not conduct forward currentuntil commanded to do so by a controlsignal. Therefore Type 2 switches behave asType 1 switches and continue to conduct aslong as forward current flows. Type 2switches are known as thyristors (the fullofficial name is reverse blocking triodethyristor) or silicon controlled rectifiers(SCR’s).

Type 3 Switches can not only turn on forwardconduction when commanded by thecontrol signal, but can also interruptconduction upon command without waitingfor reverse polarity to be applied. Powertransistors and gate controlled switchesexhibit Type 3 behavior. By artificial circuitmeans, thyristors can also be “forcecommutated” to operate as Type 3 switches.

It is probably obvious that the 3 types of switcheshave been listed in the order of increasing generality.That is, with appropriate control signals a Type 3

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Switching ControlSignals

n OutputLines

m InputLines

InputFilter

OutputFilter

Figure 1.3Power Electronic Converter as a Switching Matrix

switch can replace either a Type 2 or a Type 1 switch,and a Type 2 switch can replace a Type 1 switch. Butthey are not interchangeable in the opposite order.

5. Commutation

Definitions Regarding Commutation:

COMMUTATION – Current Interruption orTransfer to an Alternate Path.

NATURAL OR LINE COMMUTATION – OccursBecause of the Changing AC Line Voltages.

SELF OR FORCED COMMUTATION – OccursBecause of Inherent or Artificial Turn-off Ability.

LOAD COMMUTATION – Depends on Inherent orArtificially Produced Load Characteristics.

In the discussion of switches, we have just noted thatit is the ability to interrupt conduction whichdistinguishes the third type from the first two types.The process by which current in a switch is caused tocease is called “commutation”, and the availablemeans of commutation play an important role indetermining the capabilities of any given circuit.

Classical rectifier or inverter circuits based on mercury-arc devices utilize what is known as “natural” or “line”commutation. Commutation occurs “naturally”because of the changing AC line voltages, either:

(a) because the circuit current goes to zero, or

(b) because the circuit current is transferred toanother switch which is connected to a higherpotential. The analysis of these circuits dates backmore than 30 years, although solid-state powerdevices have made many new applicationstechnically and economically feasible.

In a turn-off circuit, commutation can be caused tooccur at arbitrary times in one of two ways. The mostobvious way is to use a Type 3 switch having inherentturn-off ability. The second way is to “forcecommutate” a Type 2 switch, and we willsubsequently explore various methods of doing this.Either of these ways is known as “self commutation”.

When commutation depends on the load havingcertain characteristics, the process is known as “loadcommutation”. Load commutation may be similar toeither line or forced commutation according to

whether the required characteristics are inherent inthe load or must be produced artificially.

Most applications of turn-off circuits were notfeasible until the advent of solid-state power devices.This technology is developing rapidly and isproviding many new opportunities for applying solid-state power electronics.

6. The Internal Functional Sections of a GeneralizedPower Converter (Figure 1.4)

We stated previously that power electronicsencompasses a vast amount of technology from boththe electronics and power fields, plus considerabletechnology not ordinarily covered by either of thosefields. In the time available, this text will focus onthose topics which will help you to understand howtypical converter circuits operate, what limitationsare encountered in designing such circuits, and howthese limitations are dealt with.

To clarify the emphasis of this text, the componentsof a generalized power converter can be grouped intosix main internal functional sections: power devices,power modules, the power circuit, the switchingcontrol signal generator, and input and output filters.

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Power Circuit

Power Modules

PowerDevice

InputFilter

OutputFilter

OutputLines

To Load

Switching ControlSignal

Generator

Input LinesFrom Power

Source

ExternalControl

Signals

1

2

n

Figure 1.4Internal Functional Sections of a Generalized Power Converter

The power devices are the solid-state devices whichhave made modern power electronics possible. Wewill emphasize their important terminalcharacteristics, but the details of semiconductorphysics and device fabrication are outside the scopeof this text.

Each power device or group of devices is embeddedwithin a power module which interfaces it with thesurrounding power circuit. The power moduleimplements the function of a single switch in theswitching matrix. In addition to one or more powerdevices, each power module contains one or moreheat sinks, fuses and other components to protectthe power devices, and sometimes local circuitry forgating and forced commutation.

The power circuit interconnects the power modulesinto the appropriate switching matrix, and includesthe transformers needed for isolation, changingvoltage levels, etc. Starting from simple bridge andcenter-tap arrangements and wye or deltatransformer connections, the number of powercircuit configurations is seemingly endless. Forexample, the ANSI Standard C34.2-1968, “Practicesand Requirements for Semiconductor PowerRectifiers,” designates some 70 standard circuitconfigurations for rectifiers alone.

The output filter smooths the discrepancies betweenthe voltage waveform synthesized by the powercircuit and the desired output voltage waveform.

The input filter provides a path for harmonic currentswhich are required by the power circuit but whichare undesirable in the input supply lines.

The switching control signal generator receives signalsfrom the input lines, the output lines, from withinthe power circuit, and/or from external control linesand processes them to generate appropriate gatingpulses to actuate each Type 2 or Type 3 switch. Thesegating pulses usually have a periodicity which isdetermined in one of three ways:

(a) In the simplest case, the gating signals aresynchronized in frequency with the AC supply,but with adjustable relative phase.

(b) When the converter is powered from a DC source,internal time constants or an internal referenceoscillator determine the periodicity of switching.

(c) In a frequency changer, the gating signals mustbe properly synchronized with both the ACsource frequency and the desired outputfrequency. In effect, the smaller of these twofrequencies modulates the larger frequency.

The design of the switching control signal generatorutilizes all of the applicable circuit technology fromboth analog and digital electronics.

This text will concentrate most heavily on variouscommon configurations for the power circuit and onsome aspects of the power module. Appropriatetiming of the switching control signals will bediscussed, but the design of the control circuits whichgenerate these signals is outside the scope of the text.

7. Power/Frequency Domains of Power Electronics (Figure 1.5)

The three-dimensional space of voltage-current-frequency provides another interesting way tocategorize power electronics. Within this space, thefirst and by far the most widely useful domain isdefined by voltage and current limits which arewithin the capability of a single solid-state device perpower module. These limits are rather arbitrary, andchange with time as new solid-state devices aredeveloped, but for present purposes they will beshown at 2kV and 1000A. This domain can be calledthe “Medium Power Domain”, although at its upperlimit it extends to a power of 2 MW!

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HighFrequency(FastDevices)

MediumPower

HighCurrent(ParallelDevices)

HighPower

100k

10k

10

0

100

1k

1GW10MW

100kW

1kW

10k

1k

10

100

20 200 20k2k

Voltage

In A

mps

Cur

rent

SwitchingFrequencyin Hz

200k

100k

HighVoltage(Series

Devices)

Figure 1.5Power Frequency Domains of Power Electronics

Power devices can be cascaded in series to increasethe voltage limit. When this is necessary, theassociated technology falls within the “High VoltageDomain”. Similarly power devices can be paralleledto increase the current limit, thereby defining a“High Current Domain”. For some applications,series/parallel arrays are required, leading to a “High Power Domain”.

Conventional solid-state power devices are capable ofmuch faster switching action that a mechanicalswitch, but the switching time is not zero.Consequently there is an upper limit on the switchingfrequency of conventional power devices, which forpresent purposes will be arbitrarily assumed to be 1kHz. However, power devices especially designed forfast switching action are available. The technologyassociated with the use of these fast devices defines a“High Frequency Domain”.

8. Specialized Technologies of Power Electronics (Figure 1.6)

A pinwheel can be used to further extend thespecialized technological categories of powerelectronics without trying to show an n-dimensionalpicture. The “Core Discipline” incorporates the“Medium Power Domain”, with the additionalrequirements of natural commutation and other

requirements being “nominal”. Power electronicsequipment for a particular application incorporatestechnology from the Core Discipline plus technologyfrom one or more of the specialized categories.

The High Voltage, High Current, High Power, andHigh Frequency Domains just discussed constitutesome of these specialized technologies. ForcedCommutation is another category of increasingimportance. Numerous other categories ofspecialized technology could be noted. For example,in some applications the power devices must bespecified on the basis of their transient surge ratingsrather than on their steady-state ratings. In someapplications, such as those for the consumermarket,cost reduction is a critical consideration. Inother applications, such as those for the consumermarket, cost reduction is a critical consideration. Inother applications, such as aerospace, size and weightare critical considerations. And so on and on.

This introductory text will touch lightly upon anumber of these categories, but the Core Disciplineand Forced Commutation form the “keyhole” whichwe hope will unlock a basic understanding of powerelectronics for you.

9. Converter Terminology Defined

Power Converter Terminology:

CONVERTER – General Term

SWITCH – Full Off or Full On

REGULATOR – Intermediate Control of AC or DC

RECTIFIER – Converts AC to DC

INVERTER – Converts DC to AC

FREQUENCY CHANGER – Changes Frequency of AC Power

POWER FACTOR CHANGER – Changes PowerFactor of an AC Load

AC/DC CONVERTER – Operates as Rectifier or Inverter

Unfortunately the field of power electronics aboundswith inconsistent and ambiguous usage ofterminology. To maintain internal consistency, we willadopt the following definitions:

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TechnologyAdvancement

Research

<2kV

<1kHz

<1000A

Natural Commutation,Other Requirements

“Nominal”

Core Discipline

SpecialControl

Strategies

HighVoltage>2kV

HighCurrent>1000A

High Power>2kV &>1000A

HighFrequency

>1kHZ

Pulse &Surge-Limited

Applications

Cost-CriticalApplications

Size/Weight-Critical

Applications

CriticalWaveform

or EMI

Critical LoadInteractions

ForcedCommutation

Figure 1.6Specialized Technologies of Power Electronics

CONVERTER – A general term which can be usedto describe any one of the following circuit types, acombination of several types, or all types as a group.

SWITCH – A device or circuit which has two extremestates - full off or full on.

REGULATOR – A circuit which permitsintermediate control of AC or DC power. Thedesired output frequency is the same as the inputfrequency. If implemented by switching devices,regulation is achieved by time - averaging with asuitable time constant. A DC regulator is often calleda chopper.

RECTIFIER – Converts AC power to DC power.

INVERTER – Converts DC power to AC power.

FREQUENCY CHANGER – A converter in whichthe desired AC output frequency is generallydifferent from the AC input frequency.

POWER FACTOR CHANGER – A converter whichmanipulates the power factor of an AC load withoutchanging frequency.

In contrast to the previous discussion of the internalfunctions and the specialized technologies of powerelectronics, these terms form the basis forcategorizing power electronics according to externalor application-oriented functions. However, as will beseen, rectifiers and inverters are not exclusive typesof circuits. Certain circuits can perform eitherfunction. Therefore we will define anotherintermediate category:

AC/DC CONVERTER – A circuit which may operateeither as a rectifier or as an inverter.

10. Operation of an AC/DC Converter (Figure 1.7)

As the name implies, an AC voltage appears at oneset of terminals of an AC/DC converter, and a DCvoltage appears at the other set of terminals. Theoperation of the converter can be described in termsof 4 quadrants as viewed from the DC terminals. Thehorizontal axis represents the average DC current,and the vertical axis represents the average DCvoltage. In quadrant I, where average voltage andaverage current are both “positive”, net power flow isfrom the AC terminals to the DC terminals; theprocess is known as positive rectification.

Similarly, if both the average voltage and averagecurrent are negative, net power flow is still from theAC terminals to the DC terminals and the process isknown as negative rectification.

If either the average voltage or the average currentreverses (but not both), power flow also reverses.That is, in Quadrants II and IV, power flows from theDC terminals to the AC terminals. In AC/DCconverters, the AC line provides the current zerosneeded to naturally commutate Type 2 switches. Inquadrants II and IV the process is known assynchronous inversion to distinguish it from the typeof inversion which requires forced commutation.

As will be seen later, some AC/DC converters canoperate only in certain quadrants. If operation isconfined to Quadrant I, the circuit is called a 1-Quadrant Converter. A 2-Quadrant Converter canoperate in either Quadrant I or Quadrant IV. And a4-Quadrant Converter can operate in any of the 4 quadrants.

11. Functional Categories of Applications (Figure 1.8)

Prior to the advent of solid-state power devices,power converters were limited almost exclusively torectifiers and synchronous inverters. We will nowattempt to meaningfully categorize the variety offunctional applications which are possible with solid-state power devices.

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Positive Average DC Voltage

PositiveRectification

Positive AverageDC Current

Negative AverageDC Current

SynchronousInversion

SynchronousInversion

Negative Average DC Voltage

1-Quadrant Converter - Operates in I Only2-Quadrant Converter - Operates in I or IV3-Quadrant Converter - Operates in I through IV

NegativeRectification

II I

III IV

Figure 1.7Regions of Operation of an AC/DC Converter

As we have discussed, 1-Quadrant Rectifiers arelimited to net power flow from the AC to DCterminals. These rectifiers can be implemented usingonly Type 1 switches (diodes).

2- and 4-Quadrant AC/DC Converters permit powerflow in either direction, but require Type 2 switches.Since power can flow either way, AC/DC convertersbridge the gap between the traditional “Rectifier”and “Inverter” categories.

It is also possible to implement AC Switches andRegulators and certain types of Frequency Changerswith naturally commutated Type 2 switches.

With Type 3 switches (or force-commutated Type 2switches), other types of inverters, DC Switches andRegulators (i.e. choppers), and Frequency and PowerFactor Changers become feasible.

The text will explain the basic principles ofoperation of circuits which serve each of thesefunctions.

12. Power Electronics - Laboratory Curiosity orCompetitive Necessity? (Figure 1.9)

The potential advantages of solid-state power electronicsover electromechanical equipment are well known:increased reliability and service life, and reduced sizeand maintenance. In some cases, there is a sizable costadvantage, while in other cases the performancecapabilities can be achieved in no other way.

For these reasons, power electronics is penetratingmany new application areas and outmoding manytraditional design approaches. Product engineering

departments accustomed to well established designprocedures involving Ward-Leonard drives, or relaylogic, or selenium rectifiers are faced with extinctionovernight if they cannot adapt to the higher levels ofcomplexity and analytical sophistication both madepossible and made necessary by solid-state electronics.

In assessing business risks, a company must try toguard against squandering development funds on apremature attempt to market a laboratory curiosity.

1-9

1-QuadrantRectifiers

Type 1Switches

Type 2Switches

(Nat

ural

Com

mut

atio

n)

AC DC AC DC DC AC

AC AC AC AC DC DC

AC Switches& Regulators

DC Switches& Regulators

Type 3 Switches(Forced Commutation)

Frequency & PowerFactor Changers

Force-CommutatedInverters

2- & 4-QuadrantAC/DC Converters

Figure 1.8Types of Power converters

CompetitiveNecessity

% of Annual Market

Maximum Time AllowableTo Defend Your Share of the Market

(Typically 2-4 Years)

Time

LaboratoryCuriosity

100%

80%

60%

20%

40%

Figure 1.9Universal S-Curve Showing the Capture of a Marketby a Product Which Takes Advantage of PowerElectronics

It must also guard against being caught withoutadequate competence in a rapidly evolvingtechnology when that technology becomes acompetitive necessity. Many technologies evolve overa period of decades, leaving adequate reaction timefor business planning.

One aspect of power electronics which is not widelyappreciated is that when it captures a market, itfrequently does so with a bang. Solid-state typicallymoves from an insignificant share of the market to adominate share in a period of only 2 to 4 years. Thusa company which does not maintain a nucleus ofcompetence in this new technology can easily lose asecure market position because of inadequatereaction time.

Power electronics has already become a competitivenecessity in many applications, and will soon emergefrom its laboratory curiosity status in many others.Therefore this text will not dwell on the details ofexisting applications, but will seek to convey anunderstanding of basic principals and techniques whichare equally applicable to countless new applications.

To supplement this brief introduction to power electronics, the followingreview articles are recommended:

A. Kusko, “Solid-State Motor-Speed Controls,” IEEESpectrum, pp. 50-55; October 1972.

H.F. Storm, “Solid-State Power Electronics in the U.S.A.,”IEEE Spectrum, pp. 49-59; October 1969.

F.W. Gutzwiller, “Thyristors and Rectifier Diodes — TheSemiconductor Workhorses,” IEEE Spectrum, pp. 102-111;August 1967.

R.G. Hoft, “Static Power Converters in the U.S.A.,” IEEEIntnl. Semiconductor Power Converter Conference Record,pp. 2-8-1 to 8; May 1972.

J.J. Bates and R.E. Colyer, “The Impact of SemiconductorDevices on Electrical Power Engineering,” Radio &Electronic Engineer, vol. 43, pp. 115-124; Jan./Feb. 1973.

W.E. Newell, “Power Electronics - Emerging from Limbo,”IEEE Trans., vol. IA-10, pp. 7-11; Jan./Feb. 1974.

A.M. Curry and R.V. Wachter, “A User Looks at theSemiconductor Converter,” IEEE Trans., vol. IGA-4, pp. 41-46; Jan./Feb. 1968.

A. Ludbrook, “Basic Economics of Thyristor AdjustableSpeed Drive Systems,” IEEE Industry and GeneralApplications 4th Annual Meeting Record, pp. 591-596; Oct.1969.

D.W. Borst, et. al., “Voltage Control by Means of PowerThyristors,” IEEE Trans., vol. IGA-2, pp. 102-124; Mar./Apr.1966.

13. Power Electronics Books

A. Power Device Manufacturer’s Handbooks

1. SCR Designers Handbook (Second Edition, 1970)Westinghouse Semiconductor Div., Youngwood, PA.

2. SCR Manual (Fifth Edition, 1972) General Electric Co.,Syracuse, NY.

3. Solid-State Power Circuits (1971) RCA Technical Series SP-52. RCA Solid-State Division, Somerville, NJ.

4. SCR Applications Handbook (First Printing) InternationalRectifier Corporation; September 1974.

5. Semiconductor Power Circuits Handbook (1968) MotorolaSemiconductor Products Div., Phoenix, Ariz.

6. Power Semiconductor DATAbook D.A.T.A., Inc., Orange,NJ (A worldwide tabulation of the ratings of commercialpower diodes, transistors, and thyristors, revised twice ayear.)

B. Textbooks and Special Issues on Power Devices

1. Semiconductor Controlled Rectifiers: Principals and Applications ofPNPN Devices (1964) by F.E. Gentry, et. al. (General Electric)Prentice-Hall, Englewood Cliffs, NJ.

2. Special Issue on High-Power Semiconductor Devices IEEETransactions on Electron Devices, vol. ED-17, no. 9;September 1970.

3. Special Issue on High-Power Semiconductor Devices Proceedings ofthe IEEE, vol. 55, no. 8; August 1967.

C. Textbooks on Power Circuit Analysis

1. The Fundamental Theory of Arc Convertors (1939) by H. Rissik.Chapman & Hall, London. Facsimile reprint available fromUniversity Microfilms, Ann Arbor, Mich. (A classic book,equally applicable today if “thyristor converter” is substitutedfor “arc convertor”.)

2. Power Diode and Thyristor Circuits (1971) by R.M. Davis.Cambridge University Press, London.

3. Rectifier Circuits: Theory and Design (1965) by J. Schaefer. JohnWiley, New York.

4. Principles of Inverter Circuits (1964) by B.D. Bedford andR.G. Hoft (General Electric) John Wiley, New York.

5. Thyristor Phase-Controlled Converters and Cycloconverters (1971)by B.R. Pelly (Westinghouse) Wiley-Interscience, New York.

6. The Theory and Design of Cycloconverters (1972) by W.McMurray (General Electric) MIT Press, Cambridge, Mass.

7. Line-Commutated Thyristor Converters (Second Edition, 1972)by G. Molgen (Siemens) Pitman Publishing, London.

1-10

8. Semiconductor Devices in Power Engineering (1968) Edited by J.Seymour. Sir Isaac Pitman, London.

9. Power Engineering Using Thyristors: Vol. 1 - Techniques ofThyristor Power Control (1970) Edited by M.J. Rose (Mullard)Mullard Ltd., London.

10. Thyristor Control (1973) by F.F. Mazda (ITT). John Wiley, NewYork.

11. Power Electronics: Thyristor Controlled Power for Electric Motors(1973) by R. Ramshaw (Univ. Waterloo, Ontario) Chapmanand Hall, London. (Distributed in USA by John Wiley, NY.)

12. Power Semiconductor Circuits (1975) by S.B. Dewan and A.Straughen Wiley-Interscience, New York.

13. Theory of SCR Circuit and Application of Motor Control (1968) byT.J. Takeuchi (Tokyo EE College) Tokyo ElectricalEngineering College Press, Tokyo. (This book is unique inits application of Laplace transforms to the analysis of powercontrol circuits.)

14. Rectifier Circuits (1972) Edited by W.F. Waller. MacmillanPress, Basingstoke, Hampshire, U.K.

15. Transistor Inverters and Converters (1963) by T. Roddam. VanNostrand, Princeton, NJ.

D. Textbooks on Applications

1. Solid-State DC Motor Drives (1969) by A. Kusko, MIT Press,Cambridge, Mass.

2. Design of Solid-State Power Supplies (1971) by E.R. Hnatek(National Semiconductor) Van Nostrand Reinhold, NewYork.

3. Direct Current Transmission - Vol.1 (1971) by E.W. Kimbark(Bonneville Power Admin.) Wiley-Interscience, New York.

4. High Voltage Direct Current Convertors and Systems (1965)Edited by B.J. Cory, Macdonald, London.

5. Electric Power Utilization by N.N. Hancock (Univ.Manchester) Sir Isaac Pitman, London.

6. Power Semiconductor Applications: Vol. I - General Considerations;Vol. II - Equipment and Systems (1972) Edited by J.D. Harndenand F.B. Golden IEEE Press, New York. (An extensive andinexpensive collection of selected papers reprinted fromliterature.)

7. Thyristor Control of AC Motors (1973) by J.M.D. Murphy,Pergamon Press, Oxford, England.

E. Elementary Books and Experimenters’ Guides

1. Understanding Silicon Controlled Rectifiers (1968) by S. Heller(Voorhees Technical Institute) Hayden, New York.

2. RCA SCR Experimenter’s Manual (1967) RCA Technical SeriesKM-71 (Experimenters Kit KD2105 also available.) RCADistributor Products, Harrison, NJ.

3. 110 Thyristor Projects Using SCR’s and Triacs (1972) by R.M.Marston, lliffe, London.

4. Thyristors (1971) by L.W. Owens (Mullard) Mullard,Mitcham, Surrey, U.K.

5. Thyristors and Their Applications (1972) by P. Atkinson(University of Reading) Mills & Boon, London.

1-11

14. Problems

Problem 1.1 Power Efficiency of a Series Rheostat as aPower Control

Assume that a series rheostat is used to control the flowof power to a 100-kW oven as shown by the circuitbelow. When RR = 0, full power is delivered to the oven.As RR is increased, power to the oven decreases andultimately approaches zero as RR becomes infinite.

(a) On the axes provided, plot the source power, theoven power, and the power dissipated in therheostat as functions of the oven voltage.

(b) Plot the power efficiency, defined as the ratio ofoven power to source power.

(c) How much power must the heat sink attached tothe rheostat be capable of dissipating?____ watts.

(d) Making whatever assumptions seem reasonable toyou, estimate the value of the power wasted by thiscontrol over its useful life, neglecting theadditional cost of dissipating the heat. If a simplebut more efficient control can be purchased for$15 per kW of controlled power, how long will ittake for the better control to completely pay foritself by reducing wasted power? (Assume thiscontrol has a power efficiency of 95%.)

Problem 1.2 Switching Power Efficiency

After working Problem 1.1 and being impressed by thepoor efficiency of a rheostat as a power control, anengineer invents a way to improve the efficiency. Henotes that the rheostat dissipates no power when it is ineither of its limiting positions, zero or infinite resistance.Therefore he devises a mechanism which causes therheostat to spend nearly all of the time in one or theother of these limiting positions, with negligible time atintermediate positions. Average power to the oven iscontrolled by varying the relative time (duty cycle) spentin the two positions. For instance, if half of the time isspent in each position, the average oven voltage is 500 Vand the average oven power is 50 kW. Repeat (a), (b),and (c) of Problem 1.1 for this modified control, andplot results on the axes below. (d) Is oven powerproportional to the square of average oven voltage?Explain.

You have probably observed that the “rheostat” is no longer arheostat but a switch, and you have just established for yourselfwhy high-power controls use a switching mode of operation.

1-12

0100

75

50

25

0

Oven Voltage

Source=1000V Oven =

10 ohms

Rheostat = RR

Pow

er in

KW

Pow

er E

ffici

ency

in %

Current

250 500 750 1000

0 25 50 75 100

Problem 1.1Power Efficiency

0100

75

50

25

0

Average Oven Voltage

Source=1000V Oven =

10 ohms

Rheostat = RR

Pow

er in

KW

Pow

er E

ffici

ency

in %

% Duty Cycle

250 500 750 1000

0 25 50 75 100

Problem 1.2Switching Efficiency

2-1

Section1. Type 1 Switch or Diode . . . . . . . . . . . . . . . . . . . . . . .2-22. Steady-State V-I Characteristic of a Diode . . . . . . . . . . .2-23. Transient V-I Characteristic of a Diode . . . . . . . . . . . .2-24. Pulse Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-35. Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . .2-36. Single-Phase Half-Wave Diode Rectifier . . . . . . . . . . . .2-47. Single-Phase Half-Wave Diode Rectifier – R/L Load . . .2-58. Single-Phase Half-Wave Diode Rectifier – Other Loads . .2-69. Single-Phase Full-Wave Diode Rectifiers and

Continuous Current . . . . . . . . . . . . . . . . . . . . . . . . .2-710. Average Voltage Across an Inductance . . . . . . . . . . . . .2-811. Two-Pulse Rectifier with Inductance Filter . . . . . . . . . .2-812. Power Relationships . . . . . . . . . . . . . . . . . . . . . . . . .2-913. Three-Phase Diode Rectifiers . . . . . . . . . . . . . . . . . . .2-1014. Generalized Center-Tap Rectifier . . . . . . . . . . . . . . . .2-1215. Generalized Bridge Rectifier . . . . . . . . . . . . . . . . . . .2-1216. Applications of Diode Rectifiers . . . . . . . . . . . . . . . . .2-1317. Transformers and Nonsinusoidal Waveforms . . . . . . .2-1318. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-15

Chapter 2DIODES AND

UNCONTROLLED RECTIFIERS

IntroductionThe objectives of this chapter are:

a. To name and describe the important steady-state and transient terminal characteristics of solid-state diodes.

b. To list the simplifying assumptions usually madein rectifier analysis.

c. To explain the operation of a one-pulse rectifier and sketch the voltage and current waveforms forvarious types of loads.

d. To work simple problems related to voltage-averaging by an inductor in the DC circuit.

e. To explain the real, reactive, and distortion components of total apparent power for the non-sinusoidal AC line current drawn by a rectifier.

f. To describe the operation of single-phase and three-phase center-tap and bridge rectifiers.

g. To list several common applications ofuncontrolled rectifiers.

2-2

1. Type 1 Switch or Diode (Figure 2.1)

A conventional solid-state diode is a Type 1 switch. Ithas two terminals, known as the anode and thecathode. Internally, a simple diode consists of asingle PN junction within a crystal of silicon. Theanode terminal connects to the P side of thejunction, and the cathode connects to the N side.When the anode or P side is positive with respect tothe cathode or N side, the diode conducts forwardcurrent with a relatively low voltage drop. Thetriangle in the schematic symbol for a diodeindicates the polarity of forward current flow.

When the polarity is reversed, a large reverse voltagecan be applied but only a small reverse leakagecurrent will flow.

2. Steady-State V-I Characteristic of a Diode(Figure 2.2)

The highly nonlinear behavior of a diode which hasjust been discussed can best be illustrated by thesteady-state V-I characteristic. The primary steady-state parameters which describe a diode are shown.The forward voltage drop is determined primarily bythe semiconductor material from which the diode isfabricated, independent of the current rating of thediode. The material is nearly always silicon, and thevoltage drop is about one volt.

The internal or junction temperature is an importantfactor in the operation of all solid-state devices.Power diodes are usually rated for operation atjunction temperatures up to 200°C. The forwardpower dissipation of a device is proportional to theforward voltage drop and the forward current. Sincethe voltage drop is nearly constant, it is the ability todissipate the heat without exceeding the maximum

junction temperature which determines the forwardcurrent rating of a device. Larger currents cause moredissipation, leading to physically larger devices to getthe heat out.

The reverse leakage current of a diode depends on itssize and internal design, and also depends onleakage at the junction surfaces. Reverse leakagecurrent increases rapidly with junction temperature.

In addition to the forward current rating of a diode,another parameter of extreme practical importanceis the reverse breakdown voltage. Together these twoparameters determine the maximum load powerwhich the diode can control. Although the reversebreakdown voltage depends on the internal design(but not the physical size) of the diode, practicalfabrication processes lead to a spread in each batchof devices. Consequently this rating is determined bytesting each individual diode.

3. Transient V-I Characteristic of a Diode (Figure 2.3)

The switching action of a solid-state device is muchfaster than that of a mechanical switch of the samepower capacity. Nevertheless it is not instantaneous.

When forward voltage is applied to a diode, a shortturn-on time elapses before the charges at the PNjunction reach equilibrium to carry full forwardcurrent. This time is measured in nanoseconds.

AnodeTerminal

CathodeTerminal

PN JunctionP

N

Figure 2.1Type 1 Switch or Diode

I

V

Reverse Leakage Current

Reverse Breakdown Voltage

Forward Voltage Drop

Rated Forward Current Dependson Heat Sink and Waveform

Figure 2.2Steady State V-I Characteristic of a Diode

2-3

Of greater practical importance is the transient whichoccurs when reverse voltage is applied to turn off adiode which has been conducting forward current.The charges which had been carrying this currentmust be “swept out” from the junction region beforethe reverse non-conducting state can be established.These charges are called “stored charge”, and dependon the internal design of the diode and on theforward current which has been flowing. Theinstantaneous reverse current which flows is usuallylimited by the circuit external to the diode, but theamp-seconds of the reverse current transient dependon the stored charge. The duration of this transient iscalled the reverse recovery time or turn-off time. Thistime is also nanoseconds for fast low voltage computerdiodes, but is measured in tenths of microseconds andmicroseconds for high power diodes.

Elementary descriptions of the physics of solid-statediodes are contained in many textbooks onelectronics. More advanced information on powerdiodes is contained in:

F. E. Gentry, et al, Semiconductor Controlled Rectifiers:Principles and Applications of PNPN Devices, Chapter 1;Prentice-Hall; 1964.

Y. C. Kao, “The Design of High-Voltage, High-PowerSilicon Junction Rectifiers,” IEEE Trans., vol. ED-17, pp. 657-660; September 1970.

4. Pulse Number

Among the important factors which influence thechoice and design of rectifiers are the magnitudeand frequency of the ripple voltage at the DCterminals. The smaller the magnitude and the higherthe frequency, the cheaper it is to filter the ripple tospecific tolerances.

The ratio of the fundamental frequency of the DCripple to the AC supply frequency is commonlycalled the pulse number. Most single-phase rectifiersare 2-pulse, and most three-phase rectifiers are 6-pulse.

5. Simplifying Assumptions

The analysis of a rectifier circuit usually begins withthe study of the idealized version of the circuit. Inmost cases the characteristics of the idealized circuitcan be read directly from design tables as containedin Appendix III. Practical deviations from theidealized circuit can then be added as needed.

The simplifying assumptions involved in theidealized circuits are as follows:

(a) The voltage drop across switching devices isneglected while they are conducting, and theleakage current is neglected while they areblocking.

(b) The turn-on and turn-off times of the switchingdevices are negligible.

(c) The AC line voltage is sinusoidal and there is nostray impedance (line impedance, transformerreactance, ect.).

(d) The DC terminals are connected to an ideal filter(an infinite inductance) which maintains the DCcurrent constant over each cycle at its averagevalue. We will look at what is involved in this assumption shortly.

Figure 2.3Transient V-I Characteristic of a Diode

Reverse Recovery Timeor Turn-off Time

Time

Time

Stored Charge

ReverseLeakage Current

Turn-onTime

Forward Current

Forward Voltage

VoltageReverse

Pulse NumberFundamental Ripple Frequency

AC Supply Frequency

=

2-4

In review the assumptions are:

Simplifying Assumptions

(a) Switches have no voltage drop or leakage current.

(b) Instantaneous switching.

(c) Sinusoidal voltage source.

(d) Constant DC current over each cycle.

It was stated in the Introduction that the ideal voltagewaveforms at the output of the switching matrixconsist of segments selected from the input voltagewaveforms as dictated by the switching control signals.If the input voltages and the switching control signalsare specified, it would seem that the output voltagewaveform from the matrix could easily bedetermined. Once the voltage waveform is known, the current waveform in a specified load can bedetermined by conventional analysis.

Fortunately an adequate analysis is sometimes asstraight-forward as the above discussion implies.However, subtle and unexpected factors can oftenhave significant effects. For example, the loadcurrent waveform may influence the commutationpoints, which in turn modify the voltage waveform.This interaction can lead to an analysis that issomewhat less straightforward. Also, thecommutation of current from one switch to anotherrequires a finite time, even if the dynamic responseof the switching devices is instantaneous. This“commutation overlap” further complicates the idealvoltage waveforms.

Therefore the best approach to an analyticalunderstanding of the operation of switched powercircuits is to proceed slowly from basic fundamentals,being careful not to be misled by implicitassumptions which are invalid.

The problems encountered in converter analysis arediscussed in:

S.B. Dewan and G. R. Slemon, “Analysis Techniques forThyristor Converters,” IEEE Power ElectronicsSpecialists Conference Record, pp. 141-148; June 1973.

S. B. Dewan and J. B. Forsythe, “Techniques of Analysisfor SCR Converters,” IEEE Industry & GeneralApplications 4th Annual Meeting Record, pp. 453-468;Oct. 1969.

6. Single-Phase Half-Wave Diode Rectifier (Figure 2.4)

Although the half-wave diode rectifier is not a usefulcircuit for high power applications, it neverthelesspermits a number of useful principles to be explainedin their simplest terms. For the moment we will adoptsimplifying assumptions (a), (b), and (c), butassumption (d) is not valid for this one-pulse circuit.*We will use this circuit to introduce the study ofrectifier waveforms and the effects which the load hason these waveforms.

If the load is purely resistive, the output voltagewaveform consists of half-cycles of a sine waveseparated by half-cycles of zero output voltage, forwhich the average value is:

Vp Vpπ

Vout

Vout

Vave=

Vout

Vave 0=

Vp

Vp

R

2Vp

L

Vp

Rπ

ω

Lω

I

I

R

Iave=

Vout

I

L

Vp

I

Iave=

Figure 2.4Single-Phase Half-Wave Diode Rectifier

V V t dtV

ave pp

= ∫ =ωπ

ωπ

πω

2 0

sin

2-5

The current waveform is identical in shape to thevoltage waveform, and the average current is:

On the other hand, if the load is purely inductive, thewaveforms change considerably. During the first halfcycle, the current builds up from zero to a peak value,Ip = 2Vp/ωL. During the half cycle, energy has beentransferred from the AC source to the inductor, and(1/2) LI2

p watt-seconds are stored in the magneticfield. However, the diode cannot interrupt the non-zero current which exists when the source voltagereverses polarity. The diode must wait for the currentto go to zero by itself before conduction ceases. Infact, the diode will continue to conduct throughoutthe second half-cycle, during which the outputvoltage is negative and the total energy stored in theinductor is returned to the AC line. If the inductor islossless, the diode will conduct continuously, and atthe end of each full cycle the total net energy transferwill be zero. Note that although the instantaneousoutput voltage goes negative, the current never does.If the current did go negative, it would violate theassumed behavior of the ideal diode.

Since the diode conducts continuously, the loadvoltage is identical to the AC source voltage, and hasan average value of zero. However, the average value ofthe current is Vp/ωL. This example illustrates thataverage voltages and currents must be manipulatedwith care. In general, average current does NOTequal average voltage divided by impedance!

Thought Question: In steady state, the diode in thisideal circuit never becomes reverse biased, and thecurrent is the same as if the diode were shorted.Would the circuit operate in the same way if thediode were not present?

*The terminology “one-pulse circuit” indicates that the circuit has a pulsenumber of one.

7. Single-Phase Half-Wave Diode Rectifier – R/L Load (Figure 2.5)

In practice, the inductor is not lossless, so we mustconsider at least some series resistance. At the end ofthe first half cycle, the current will be less than thepreviously calculated Ip because of the voltage dropand consequent power loss in R. During the secondhalf cycle, R will continue to dissipate power as longas current flows. Since there is less total energy toreturn to the AC line, current will always ceasebefore the second half cycle is completed. That is,commutation does occur, but is delayed until afterthe zero crossing of the AC source voltage. Netpower flow over the full cycle will be from the ACline to the load, as it must be to account for the lossin R.

During the time when the instantaneous voltage andcurrent are both of the same polarity, power flow is

IV

Ravep

=π

Vout

I

Vout

I

R

L

Vout

Vout

Vp

2Vp

Vp

LωI

I

R

L

Figure 2.5Single-Phase Half-Wave Diode Rectifier – R/L Load

2-6

from the AC line to the load, and the circuit acts as arectifier. During the time when the instantaneousvoltage and current are not of the same polarity,power flow is from the load to the AC line, and thecircuit acts as a synchronous inverter. Since the loadis passive, the net power flow over a full cycle mustbe from the AC line to the load. Since the diodemust conduct for the full half cycle of rectification,but ceases conduction part way through theinversion half-cycle, the circuit has positive averagevoltage and cannot invert net power from its DCterminals to its AC terminals.

If the L/R time constant of the load is smallcompared to a half-period of the AC source,commutation is delayed very little, and the operationof the circuit approaches that of a circuit with aresistive load. As L/R increases, the delay incommutation increases and the current waveformapproaches a sine wave (plus a DC component).However, since the peak current cannot exceed2Vp/ωL, the inductance cannot be increasedsufficiently to hold the current constant as requiredby simplifying assumption (d).

If the intent of the rectifier is to get maximum powertransfer into the load, the flow of power back intothe AC line on alternate half cycles is undesirable.This reverse flow occurs because of the reversal ofthe polarity of the instantaneous output voltage.Both can be prevented by connecting a“freewheeling” or “bypass” diode across the output toconduct whenever the output voltage tries to gonegative. The rectifier diode then commutates off atthe zero-crossing of the AC source voltage. The loadcurrent continues to flow, but is transferred to theloop formed by the freewheeling diode. This diode isforward biased by the “kickback” voltage of theinductance. The energy stored in L then dischargesinto R rather than being returned to the AC line.The freewheeling diode helps to prevent the loadcurrent from ever going to zero, and thereby reducesthe ripple.

8. Single-Phase Half-Wave Diode Rectifier –Other Loads (Figure 2.6)

Two other types of loads should also be noted briefly – capacitive loads and back-emf loads. If acapacitive load can be represented by R and C inseries, the steady-state load voltage charges up to Vpand thereafter current ceases.

However, if the equivalent circuit is a parallelcombination of R and C, the capacitor tends tosmooth the ripple in output voltage, but the current(in the diode) occurs in brief spikes. Note that incontrast to an inductive load which tends topostpone commutation after the voltage zerocrossing, a capacitive load causes commutation tooccur before the voltage zero crossing.

Similarly, a back-emf load of the polarity shown, suchas a battery or a DC motor, also causes current spikesand early commutation. No current flows until theAC line voltage exceeds the back-emf. If the polarityof this load emf is reversed, conduction beginsbefore the first zero crossing of the AC sourcevoltage and commutation is delayed until after thesecond zero crossing. As in the case of the inductiveload, the AC source receives power during part ofeach cycle. But the net power flow over a full cycle inthis and all other diode rectifier circuits is alwaysfrom the AC to the DC terminals. Hence they are 1-Quadrant rectifiers.

I

R

I

R

VBatt

C

Vout

Vout

VBatt

I

I

Figure 2.6Single-Phase Half-Wave Diode Rectifier - Other Loads

2-7

9. Single-Phase Full-Wave Diode Rectifiers andContinuous Current (Figure 2.7)

Single-phase full-wave diode rectifiers are two-pulsecircuits since the fundamental ripple frequency istwice the AC supply frequency. With a resistive orpartially inductive load, the current in the DC circuitis continuous,* and the diodes always commutate atthe zero crossings of the AC source voltage. Hencethe DC voltage waveform consists of successive half-cycles of a sine-wave, independent of the load. Ineffect, the two diodes that are tied to the positive DC

terminal connect it to the AC terminal which isinstantaneously more positive. Similarly, the twodiodes that are tied to the negative DC terminal (in the bridge circuit) connect it to the morenegative AC terminal.

If the load is purely resistive, the DC current has thesame waveform as the voltage except that theamplitude is multiplied by 1/R. The average value ofthese waveforms constitutes the DC or zero-frequency component, and is given by:

The AC line current is the same as the DC currentexcept that alternate half cycles are reversed inpolarity. Thus the AC line current is purelysinusoidal with no harmonic content.

Two-pulse circuits have limited practical value forhigh-power applications because of the large ripplewhich is contained in the DC output voltage. It isreadily seen that the ripple has a peak-to-peak valueof Vp and contains all even harmonics of the AC source frequency.

The ripple current is ordinarily reduced by addingseries inductance in the DC circuit to pass the DCcomponent of current while attenuating theharmonics. In practice, part or all of this inductanceis frequently in the load itself, as in the case of amotor armature or field. However for the sake ofdiscussion, the inductance will be considered as afilter which is part of the rectifier, and the resistancewill be considered as the load.

The presence of this inductance will change thecurrent waveform and consequently the voltagewaveform across the resistance. However, as long asthe current is continuous, the presence or the valueof the inductance does not influence thecommutation points of the diodes, and hence doesnot affect the output voltage waveform from thediodes. In particular, the inductance does notchange the average value of this waveform calculatedabove. This average voltage must be equal to the sumof the average voltages across the inductance and theresistance. As we shall see in the next section, theaverage voltage across the inductance must be zero.Therefore, the average voltage across the resistanceis (2/π)Vp and the average current is (2/π) (Vp/R)

Figure 2.7Two-Pulse Diode Rectifier Circuits and Waveforms

Vp

R

VpVout

Vin

IoutIave =

Vave =

t

Vout

Iout +

–

RVin

Vp sin vt

Iin

VoutVin

Vin

Iout +

–

R

2Vp

π

2Vp

πR

Iin

RI V V t dt Vave ave p p= = =∫ωπ

ωπ

πω

0

2 sin

2-8

independent of the inductance.

It should also be noted that rectifier analysis normallyassumes steady-state conditions. Obviously there is atransient with an L/R time constant which can alterthe waveforms when a rectifier is first started.

*The DC current is said to be discontinuous if it contains finite intervalsof zero current. With a capacitive or back-emf load, the current will bediscontinuous unless the series inductance exceeds some minimum criticalvalue. (See R.J. Distler and S.G. Munshi, “Critical Inductance andControlled Rectifiers,” IEEE Trans., vol. IECI-12, pp. 34-37; March1965.) With discontinuous current, the commutation points and the DCvoltage waveform from the diodes are load-dependent.

10. Average Voltage Across an Inductance

A fundamental principle which should be remembered isthat in steady-state, the voltage across an inductance,averaged over a full cycle, is always zero.

This principle may be verified as follows. The voltageacross an inductance averaged over a cycle is defined as:

We also know that the defining relationship betweenvoltage and current for an inductance is:

Integrating this equation over a full cycle gives thechange in current:

But, again by definition, the change in current over afull cycle in steady-state is zero. Hence VLave mustalso be zero.

11. Two-Pulse Rectifier with Inductance Filter(Figure 2.8)

Returning to the two-pulse diode rectifier, we canconsider the effects of inductance in series with theload in two cases. In the first case, the inductance is“nominal”. That is, it is big enough to have anappreciable smoothing effect but small enough thatthe ripple is still significant. An approximate valuefor this inductance is R/ω.

The load current waveform no longer consists of halfsine waves,* but the average current is still the sameas before, 2Vp/πR. The AC line current is no longersinusoidal, but approximates a poor square wave withsuperimposed ripple. The inductance has reducedthe harmonic content of the load current byincreasing the harmonic content of the AC source current.

V V t dtL

diLave Li

i

= = =∫ ∫ωπ

ωπ

πω π

2 20

0

2

0

2

( )

V V t dtLave L= ∫ωπ

πω

2 0

2

( )

V t Ldi

dtor di

LV t dtL L( ) ( )= = 1

∆i diL

V t dtL

Vi

i

L Lave= = =∫ ∫0

22

1 2

0

ππ

ω πω

( )

Vp

Vp

VoutVR = RIout

Vin

Iin

VoutVR = Vd = RId = 2Vp/p

Vp

Id

Vin

Iin

Waveforms forL >> R/ω

Waveforms forL ' R/ω

Figure 2.8Effects of Inductance on the Waveforms of a Two-Pulse Diode Rectifier (Voltage Across Inductance Shown Shaded)

2-9

The voltage which appears across the inductance isshown shaded. In order for the average voltageacross the inductance to be zero, the positive shadedareas must be equal to the negative shaded areas.

In the second case, the series inductance is increaseduntil it is much larger than R/ω. The ripple acrossthe load is now insignificant, and simplifyingassumption (d) is valid. The load current is constantat its average value. These constant values of loadvoltage and current are designated Vd and Id:

The inductance has totally removed the harmoniccontent of the load voltage and current, but the ACsource current waveform has become a square waveof value Id: Unless otherwise stated, most rectifieranalysis assumes these idealized conditions.

*In the one-pulse circuit, the single diode could not commutate until theload current went to zero. Note that in this two-pulse circuit the loadcurrent never goes to zero. Commutation occurs, not because the loadcurrent goes to zero, but because this current is “captured” when thevoltage in some other diode loop exceeds the voltage in the loop of thediode previously conducting.

12. Power Relationships (Figure 2.9)

The purpose of a rectifier is to convert AC powerinto DC power. The fundamental nature of powertransfer in power electronics makes it essential for usto understand the relationships between the variouscomponents of power. Although these relationshipsare probably quite familiar when the waveforms arepurely sinusoidal, they must be modified to includethe nonsinusoidal waveforms encountered inrectifiers and other types of converters.

The two-pulse rectifier which we have just discussedpresents an opportunity to explain these modifiedrelationships. Under simplifying assumptions (a),there is no power dissipation in the switchingdevices. Therefore the net power flow at the ACterminals averaged over a full cycle must be exactlyequal to the net power flow at the DC terminals.

With a large inductance in the DC circuit, the DCcurrent is constant at Id, and the voltage across theload resistance is constant at Vd. Therefore the realDC power is easily calculated as:

According to simplifying assumptions (c), the ACsource voltage is sinusoidal. Therefore its RMS value,V, is related to its peak value by:

However. as we have seen, the AC current waveformis a square wave of amplitude Id , in phase with theAC voltage. The RMS value of this square wave is Id,and there is no apparent power factor phase angle.We might be led to conclude that the AC power isthe product of the RMS AC voltage, V, and the RMSAC current, I = I:

It is apparent that this “total apparent power” is notequal to the real DC power calculated above. Wehave been misled by applying relationships derivedfor sinusoidal waves to a case where the current isnonsinusoidal.

V RI VV

d aveP= = = 1

π

VI1P = Real Power

φ = DisplacementAngle

∞Σ

δ = Distortion Angle

VI1Q = Reactive orQuadrature Power

(VIh)2 = Distortion Power

Real PowerTotal Apparent Power

I2 = I1P2 + I1Q

2 + Ih2

Power Factor = cos θ cos δ =

h = 2∞Σ

h = 2

VI = TotalApparent

Power

VI1 = ApparentFundamental Power

Figure 2.9Power Relationships

P V I V Id d d p d= = ( / )2 π

V Vp= / 2

P VI V IT p d= = ( / )2

2-10

To reconcile this situation, expand the currentwaveform into a Fourier series where the RMS valueof harmonic “h” is Ih. The RMS value of thewaveform is then:

or

In general, the fundamental current, I1, will consistof an in-phase component, I1p, and a quadraturecomponent, I1Q (if the fundamental current is not inphase with the voltage). Then:

Multiplying both sides of this equation by the squareof the RMS voltage gives:

The left side of the equation is the square of the totalapparent power, but of the three terms on the rightside, only the in-phase component of the samefrequency as the voltage represents real power(watts). The second term represents the square ofthe reactive power (VAR’s) which results if thefundamental current is out of phase with the voltage,and the third term is a “wattless” power caused by theharmonic components (i.e., distortion) of thenonsinusoidal waveform.

For sinusoidal waves, a right triangle displaysgeometrically the mathematical relationship betweenreal power, reactive power, and apparent power. Thephase angle, ∅ , is one of the angles of this triangle,and the power factor (ratio of real power to apparentpower) is equal to cos ∅ .

Similarly two right triangles can be used to give ageometric representation of the equation above. Thephase angle between the voltage and thefundamental component of current is known as the“displacement angle,” φ, and cos φ is called the“displacement factor”. The harmonic term isassociated with a second right triangle and an

angle, δ, which has little significance. However, cos δis known as the “distortion factor”, and is equal tothe ratio of the apparent fundamental power to thetotal apparent power. The two triangles may becascaded as shown in the figure, and it is apparentthat the power factor is now equal to the product ofthe displacement factor and the distortion factor.

For the two-pulse diode rectifier, the fundamentalcomponent of current is in phase with the voltage.Therefore, φ = 0, and the displacement factor isunity. However, the RMS value of the fundamentalcomponent of a square wave is equal to 4/π=2 timesthe RMS value of the square wave. Therefore thedistortion factor is 4/π=2.

The real AC power is then the product of thedisplacement factor, the distortion factor, and thetotal apparent power:

The real AC power is seen to be equal to the real DCpower calculated previously, and the desired powerbalance has been verified.

13. Three-Phase Diode Rectifiers (Figure 2.10)

The pulse number of a single-phase rectifier cannotbe increased beyond two unless some artificial meansis available for generating out-of-phase voltages. Onthe other hand, the pulse number of a polyphaserectifier may be made arbitrarily high by variousinterconnections of transformer windings. The mostcommon high-power rectifiers are supplied from athree-phase source and use a three-pulse or six-pulse circuit.

The easiest way to derive the waveforms of a three-phase bridge rectifier is to consider each halfseparately with respect to the neutral. The upperthree diodes constitute a three-pulse center-tapcircuit, and each diode conducts for 120°. Thevoltage from the positive DC terminal to the neutral,V+N, has a positive average value of (3=3/2π) VpN,where VpN is the peak phase voltage to the neutral.The waveform is as shown. If a load is connectedbetween the positive DC terminal and the neutral(including enough inductance to maintain thecurrent constant), a current Id+ will flow.

I I I I= + + +12

22

32 ....

I I Ihh

212 2

2

= +=

∞

∑

I I I Ip Q h

h

212

12 2

2

= + +=

∞

∑

( ) ( ) ( ) ( )VI VI VI VIp Q h

h

21

21

2 2

2

= + +=

∞

∑

VIV I

V Ipp d

p d1 14

2 22+

=( )π π

2-11

Similarly the lower group of three diodes generates anegative three-pulse waveform between the negativeDC terminal and the neutral, and a load will cause acurrent Id- to flow. As long as the neutral isconnected, the two groups of diodes operateindependently as three-pulse rectifiers, and thecurrents Id+ and Id- need not be equal. Howeverthere will be a net DC component of current in eachwinding, tending to cause DC magnetization of thecores, if these currents are unequal.

If the load is connected directly between the positiveand negative DC terminals without a neutralconnection, the average DC voltage doubles and Id+

and Id- must be equal because there is no return pathfor a difference current. Therefore coremagnetization is no longer a problem. However,another significant advantage also occurs. Note thatcommutation occurs alternately, not simultaneously,in the two groups of diodes. When the two three-pulse groups operate together, a six-pulseoutput waveform results.

The current in each wye-connected secondarywinding supply AC to the bridge is a square wave ofamplitude Id, consisting of a 120° positive pulse, a60° period to zero current, a 120° negative pulse,and another 60° period of zero current. If theprimaries of the transformer are also wye-connected,the AC line currents will have the same waveform.On the other hand, if the primaries are delta-connected, the AC line current becomes different inwaveform although the harmonic content remainsthe same.

Circuits having higher pulse numbers furtherincrease the ripple frequency, making filtering easierand reducing the harmonic content of the ACcurrent. However, these circuits may also reduce theconduction angle of each diode, thereby reducingthe utilization of its capacity. Therefore, circuitshaving pulse numbers higher than six are usuallyused only for special purposes.

We will now state, without proof, several general laws

V+N

V-N

(V+Nave = VPN)

Vout

VCAVBCVAB

t

t

t

VCNVBNVAN

3=­32p

(Voutave = Vd = VPN) 3=­3

p

Wye-Connected Primaries

Delta-ConnectedPrimaries

Current inLine A

Id- =­ VPN3

VPN

Id

Id+

N

+

–

V+N

Vout

V+N

Figure 2.10Waveforms for a Three-Phase Bridge Rectifer

2-12

concerning the harmonics generated by ideal diode rectifiers:

(1) The ripple voltage at the output has afundamental of pfs, where p is the pulse numberand fs is the source frequency. Because of thelack of symmetry of the positive and negativeparts of the ripple waveform, all harmonics ofthis fundamental ripple frequency will bepresent: npfs, n = 1, 2, 3, ... The peak amplitudeof any harmonic of the source frequency whichis present, relative to the uncontrolled DCvoltage, is 2/(n2p2 - 1). That is, the pulsenumber determines which harmonics offs arepresent in the output, but does not change therelative amplitude of a given harmonic if it ispresent.

(2) The only harmonics present in the inputcurrent (in addition to the fundamental currentat fs) occur at (np ± 1) fs, n = 1, 2, 3, ... Theamplitude of each harmonic current, relative tothe fundamental current, is inverselyproportional to its order (i.e., to np ± 1).

References:

J. Schaefer, Rectifier Circuits: Theory and Design, p. 257 and p.298, John Wiley, New York; 1965.

B. R. Pelly, Thyristor Phase-Controlled Converters andCycloconverters, pp. 94-107, John Wiley, New York; 1971.

C. J. Amato, “A Simple and Speedy Method for Determiningthe Fourier Coefficients of Power Converter Waveforms,”IEEE Industry & General Applications 4th Annual MeetingRecord, pp. 477-483; Oct. 1969.

I. K. Dortort, “Phase Shifting of Harmonics in AC Circuits ofRectifiers,” IEEE Trans., vol. IGA-4, pp. 655-658; Nov./Dec.1968.

14. Generalized Center-Tap Rectifier (Figure 2.11)

The two-pulse and three-pulse center tap circuitswhich we have discussed are characterized by the factthat one terminal of the DC circuit is connecteddirectly to a center-tap or neutral point of thetransformer windings which couple the rectifier tothe AC supply. The generalized center-tap rectifierhas q secondaries, with adjacent secondariesseparated by 360°/q. The required number of diodesis q, one for each secondary. The pulse number isalso equal to q.

For q = 1, the circuit is a single-phase half-waverectifier. Although widely used for low-current powersupplies, this circuit is not useful at high power levelsbecause of its large ripple voltage and because of theDC current which it draws from the AC line. Circuitsfor which q = 2, 3, or 6 are of greater practical importance.

Center-tap rectifiers are also known as “single-way”circuits because the current in each transformerwinding is unidirectional. If q is even, currents ineach pair of windings flow in opposite directions sothat the net DC magnetization of transformer coresis zero. However, if q is odd, there is no cancellationand DC magnetization of transformer cores is aproblem. In this case, magnetization is usuallyavoided by a “zig-zag” series connection of pairs oftransformer secondaries, arranged so that the effectof each winding carrying unidirectional current iscancelled by another winding which carries currentin the opposite direction.

15. Generalized Bridge Rectifier (Figure 2.12)

Bridge rectifiers do not ordinarily have a directconnection between the DC terminals and thetransformer secondaries, but each winding isconnected to two oppositely polarized diodes so thatboth polarities of current flow in each winding.Hence DC magnetization of the transformer cores isnot a problem. Another name for bridge circuits is“double-way” circuits.

360oq

qDiodes

PulseNumber= q

DCTerminalsq Windings

CommonNeutral

Figure 2.11Generalized Center-Tap Rectifier

2-13

The generalized bridge circuit also contains qsecondaries separated by 360°/q in phase, butrequires 2q diodes. Because of the lack of a commonconnection between input and output terminals, qmust be greater than one. For even values of q, thepulse number is equal to q. However, for odd valuesof q, diodes on opposite sides of the bridgecommutate alternately (not simultaneously), and theripple frequency is doubled. Thus for odd q, thepulse number is 2q.

The most useful values for q are 2 and 3, for single-phase and three-phase AC supplies respectively.

Since the advent of solid-state rectifiers, bridgecircuits are much more widely used than center-tapcircuits because of their better utilization of thetransformer windings. Previously, this advantage hadto be balanced against the disadvantage of the largerforward voltage drop of two mercury-arc rectifiers in series.

16. Applications of Diode Rectifiers

Diode rectifiers are most commonly used in thoseapplications where control of the DC voltage isunnecessary or can be achieved by other means.These rectifiers are used in the largest quantities inlow-power DC supplies for radios, television sets, andother electronic equipment. Solid-state diodes alsoled to the changeover from DC generators toalternators in automative electrical systems. These

diodes permitted the elimination of the commutator,and the lack of rectifier control is no problembecause regulation is achieved in the field circuit.

Other applications of diode rectifiers include batterychargers, power supplies for DC welding, and DCtraction drives. Important applications for very high-power diode rectifiers are brushless exciters forlarge turbine-generators (to 5 MW or more) and DCpower supplies for electrochemical process lines (to100 MW or more). A summary of diode applicationis:

Applications of Diode Rectifiers• Radio, TV, & other Electronic Equipment• Automative Electrical Systems• Battery Chargers & DC Welding Supplies• DC Traction Drives• Brushless Exciters for Turbine-Generators• DC Supplies for Electrochemical Process Lines

Diode rectifiers are seldom analyzed separately butare usually treated as a special case of the moregeneral AC/DC converters which use thyristors.Separate consideration has been given here in orderto introduce some of the basic concepts of rectifiersin the simplest possible context.

An excellent reference which uses a similarapproach, but goes into all aspects of the subjectmuch more thoroughly than is possible here, is:

J. Schaefer, Rectifier Circuits: Theory and Design, Wiley (1965).

17. Transformers and Nonsinusoidal Waveforms(Figure 2.13)

In elementary electrical courses, transformers areusually studied in the context of sinusoidalwaveforms. Therefore questions frequently ariseconcerning the behavior of transformers (and othermagnetic components) when exposed to thenonsinusoidal current and/or voltage waveforms inpower electronics circuits. These questions are bestanswered by a brief review of fundamentals.

Consider a two-winding transformer having N1 turnson the primary and N2 turns on the secondary, asshown. Assume a magnetic core with no hysteresis,no eddy-current losses, and constant permeability.Let L11 be the inductance of the primary and L22 be

360oq

2q Diodes

DCTerminals

q Windings Pulse Number

= q for q even

= 2q for q odd

FloatingNeutral

Figure 2.12Generalized Bridge Rectifier

2-14

the inductance of the secondary. Then:

and

where Lm is the “mutual inductance” and k is knownas the coefficient of coupling.

When a “magnetizing” current im is applied to theprimary, the flux generated in the core is given by:

The relationship between φm and im is linear as longas φm is below the value where saturation begins.From the differential relationship for an inductance,the induced voltage is:

Integration gives the inverse relationship:

In general, the coefficient of coupling is less thanone, and the response of the transformer isdetermined by solving the loop or node equationsinvolving one of the equivalent circuits shown.(Winding resistances R1 and R2 are indicated, butwinding capacitance is neglected.) However, mostpower transformers are designed with a coefficient ofcoupling near unity. Thus the “leakage inductances”in the windings are usually small compared to themutual inductance. Frequently both the leakageinductances and the winding resistances areneglected in power circuit analysis.

To the extent that these assumptions are valid, theprimary voltage must be equal to the induced (or“mutual” or “magnetizing”) voltage (i.e., V1 = Vm).Under the same assumptions:

independent of waveforms. If the transformer is

nN

N

L

L= ≈2

1

22

11

L k L Lm = 11 22

Φm mm

m

kL

Ni

L

nNi= =11

1 1

V kLdi

dt

L

n

di

dtN

d

dtmm m m m= = • =11 1

Φ

Φmm

m

V

Ndt

kL

Ni= =∫

1

11

1

i2

i1 R1ni2

i1

V1 V2

V1 VmLm/n = kL11

im = N1

kL11

Equivalent Circuit Referred to Primary

Equivalent Circuit Referred to Secondary

V2n

N1 Turns N2 Turns

N2N1

N1 im

In Linear Region:

n =

φm

φm

φm =

φm

∫im

kL11N1

=Vm dt

1N1

Lm

nL-kk

R2

n2

i1n2R1

i2

nV1 nVmnLm = kL22im V2n

nnLm

L-kk R2

Figure 2.13Basic Transformer Relationships

V nV nVm2 1= =

2-15

driven from a voltage source, the flux and themagnetizing current are proportional to ∫ V1 dt, or“volt-second area”. When a load is connected acrossthe secondary, the secondary AC ampere-turns N2i2must be compensated by an equal number ofprimary ampere-turns to maintain the flux changeand the induced voltage constant. (DC ampere-turnsinduce no voltage and therefore cannot becompensated by primary current. Instead they offsetthe flux swing.) Hence:

or

Stated in another way, the applied voltagedetermines N1im which is the net difference inprimary and secondary ampere-turns.

Under these idealized assumptions, it is seen thatthere is no difficulty in principle when nonsinusoidalvoltage and/or current waveforms are applied to atransformer. Specifying the ratings of thetransformer must, however, take the waveforms intoaccount.

First, if the secondary current contains a DCcomponent, the transformer core must be designedto accommodate the flux offset and prevent excessivesaturation. Otherwise, the current rating of atransformer depends on the heating in the windingresistances, which is proportional to the RMS valueof the current waveforms.

If the voltage waveform is nonsinusoidal, theinsulation must be specified according to the peakvoltage, and the core area must be specified tohandle the peak flux without excessive saturation.The flux depends on the volt-second area of thevoltage waveform which is influenced by voltageamplitude, waveform, and frequency.

Sometimes a transformer is deliberately designedwith appreciable leakage inductance (i.e., with k<1)to limit peak currents into a capacitive or back-emfload. In this case the validity of the previousassumptions should be examined, and the fullequivalent circuit used if necessary. If core losses areappreciable, these may be represented by addingresistance in parallel with Lm.

18. Problems

Problem 2.1 Comparison of Diode Rectifier Circuits

Diodes are available which have a reverse breakdownvoltage rating of VRRM = 1000V and an averageforward current rating of IT = 100A. Therefore theyeach have the capacity to supply a nominal DC loadof VRRMIT = 100kW. Their actual load capacitydepends on the rectifier circuit which is used,derating factors for overload capacity requirements,ect. The purpose of this problem is to compare thevarious rectifier circuits with respect to theirutilization of diode capacity and transformercapacity.

The attached table lists five commonly used rectifiercircuit configurations.

(a) Sketch each circuit in the form of a switchingmatrix to verify the generality of this concept.

(b) Perform the calculations necessary to designeach configurations to utilize the diodesspecified above as fully as possible, and therebyobtain the information needed to complete thetable. Assume that the four simplifyingassumptions discussed in the text are valid, andalso assume that the diodes will not overheat aslong as their average current does not exceedtheir rated value. Transformer ratings are basedon RMS voltage vs and current IL.* Ignore theeffect of the DC magnetizing current on thetransformer core of the center-tapped three-pulse circuit.

(c) Based on the data in the table alone, is any onerectifier circuit clearly superior?

*The following information is useful in calculating RMS values:

N i N i N im1 1 1 2 2= +

i i Nim1 2= +

fT T

f t dtRMST

T

=

∫1

2 1

2

12

1

2

( – )( )

2-16

Problem 2.2 Average vs. RMS Current Ratings for Diodes

In the previous problem, the diodes were rated interms of average current while transformer windingswere rated in terms of RMS current. The data sheetson solid-state power devices frequently contain bothaverage and RMS current ratings. The purpose ofthis problem is to explore how the two ratings differand why both are used.

The steady-state current rating of a diode depends onthe average amount of heat which the diode candissipate. For the purposes of this problem, assumethat dissipation occurs only when the diode isconducting forward current. Also assume that theload current, Id, is smooth, and that each diodecarries this current for a time ηT during each cycle,where h is the duty cycle and T is the period of theAC source. For a specified circuit configuration, theduty cycle decreases as the pulse number is increased.

The diode to be considered has an average currentrating of 100A at a 50% duty cycle, an RMS currentrating of 140A, and its V-I characteristic is as shownin the graph.

First, calculate the average diode current, IAVE andthe RMS diode current, IRMS, as functions of the loadcurrent, Id, and the duty cycle, η.

Second, assume that Id varies with duty cycle in sucha way that Iave is held constant at 100A. Use the V-Icharacteristic to calculate and plot the actual averagediode dissipation vs. duty cycle. Is the averagecurrent rating valid for low duty cycles?

Third, assume that Id varies with duty cycle in such away that IRMS is held constant at 140A, and repeatthe calculation and plot of dissipation vs. duty cycle.Is the RMS current rating valid for low duty cycles?

And last, attempt to interpret the two plots so as toreach a general conclusion concerning theusefulness and limitations of average and RMScurrent ratings for the more complicated currentwaveforms found in other types of converter circuits.Can you give a simple physical explanation of whysolid-state devices are more difficult to rate than suchcomponents as transformers?

2

3

6

2

6

Center-Tap

Configuration PulseNo.

2

n

3

6

4

6

No. ofDiodes

Diode Factors Transformer Secondary Factors

Comparison of Diode Utilization and Transformer Utilization for Varoius Rectifier Circuit Configurations.

Vs, Is = RMS Winding Voltage, CurrentVd, Id = Smooth DC Load Voltage, Current

q

2

3

6

2(FloatingNeutral)

3

No. ofWindings

Bridge

VdVRRM

VoltageRatio

CurrentRatio

IdnIT

DiodeUtilization

VdIdnVRRMIT

SecondaryVoltageRatio

VdVs

SecondaryCurrentRatio

IdqIL

TransformerUtilization

VdIdqVsIL

Problem 2.1Comparison of Diode Utilization and Transformer Utilization for Various Rectifier Circuit Configurations.

2-17

(A rating which attempts to avoid the shortcomingsof both average and RMS current ratings for diodesand thyristors is proposed in:

W.E. Newell, “Dissipation in Solid-State Devices — TheMagic of I1+N, IEEE Power Electronics SpecialistsConference Record, pp. 162-173; June 1974.)

Problem 2.3 Commutation Overlap

Inductance in the DC circuit of a rectifier is an assetin reducing the ripple in the DC load current. Thepurpose of this problem is to investigate the verydifferent effect of inductance in the AC circuit.Although the AC inductance is frequently ignored inrectifier analysis, in practice it is always present in theform of AC-line impedance, transformer leakagereactance, ect., and as will be seen, it causessignificant changes in the idealized waveforms.

Consider a two-pulse center-tap diode rectifierhaving an inductor Ls in series with each of the twodiodes and constant current Id in the load. Each timethe load current commutates from one diode to theother, the current in one of these inductors mustdecrease from Id to zero while the current in theother increases from zero to Id. Since the current inan inductor cannot change instantaneously, there is atime during which both diodes are conducting. Thistime is known as the “commutation overlap”.

(a) Show that the output voltage duringcommutation overlap is equal to the average ofthe two phase voltages to which the diodes areconnected. Sketch the actual waveforms ofoutput voltage and diode currents, and compare them with the ideal waveformsobtained for Ls = 0.

(b) The effect of Ls is to decrease the average DCoutput voltage by an amount which isproportional to Id. Therefore, its effect on the regulation of the rectifier is equivalent to a(lossless) resistor in the DC circuit. Determinethe value of this resistor.

(c) Discuss the generalization of the above results to rectifiers of higher pulse numbers andcontrolled rectifiers in which the diodes arereplaced by thyristors.

Problem 2.2Average vs. RMS Current Ratings for Diodes

1.0100

200

500

1000

1.5

Volts

Am

pere

s

2.0 2.5

0 0.2 0.4 0.6 0.8 1.00

50

100

150

200

250

Duty Cycle

Ave

rage

Dis

sipa

tion

in W

atts

2-18

Problem 2.4 Critical Inductance

The current in a rectifier without a filter inductancebecomes discontinuous as the AC source voltagedrops below the DC load voltage. This will happenwith a capacitor filter or a back-emf load such as abattery or a motor. Usually it is assumed that thefilter inductance is large enough that the DC currentis not only continuous but constant. The purpose ofthis problem is to determine the minimum or“critical” inductance, LC, required to prevent the DCcurrent from becoming discontinuous. In particular,calculate LC, in terms of load resistance R and sourcefrequency ω for a 2-pulse diode rectifier. C in thefollowing circuit may be assumed to be so large thatthere is negligible ripple voltage across R.

(Hint: Determine and integrate VL (t) to obtain ∆I.For L = LC,iL will be zero at w as indicated.Furthermore, it can be shown from the symmetry ofthe waveforms (for α = 0) that (iL)ave = 0.5 DI.)

Vout

2 - PulseDiode

Rectifier Vd

VL

iL(iL)AVE = Id =

VdR

Lc

+ +

–

–

C R

iL

IId

Vd

Vout

∆

ψ

Problem 2.4Critical Inductance

3-1

IntroductionThe objectives of this chapter are:

a. To name and describe the important steady-stateand transient terminal characteristics of thyristorsand triacs.

b. To analyze the effect of phase delay on average DCcurrent for an AC/DC converter in both therectifier and synchronous inverter modes ofoperation.

c. To explain and distinguish between 1-, 2-, and 4-quadrant converters.

d. To compare two-step (rectifier/inverter) and one-step (cycloconverter) AC frequency changers.

e. To show several ways of implementing a bilateralAC switch.

f. To describe and compare the four modes ofoperation of an AC regulator.

Section1. Type 2 Switch Thyristor . . . . . . . . . . . . . . . . . . . . . . .3-22. Steady-State V-I Characteristic of a Thyristor . . . . . . . . .3-23. Two-Transistor Analog for Explaining

Thyristor Turn On . . . . . . . . . . . . . . . . . . . . . . . . . .3-34. Gate Characteristics . . . . . . . . . . . . . . . . . . . . . . . . .3-45. Transient V-I Characteristic of a Thyristor . . . . . . . . . .3-56. Thyristor Ratings – Design Tradeoffs and

Commercial Availability . . . . . . . . . . . . . . . . . . . . . . .3-67. Circuit Functions Performed by Type 2 Switches . . . . . .3-78. Two-Pulse AC/DC Converter Operating as a

Controlled Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . .3-79. Two-Pulse AC/DC Converter Operating as a

Synchronous Inverter . . . . . . . . . . . . . . . . . . . . . . . . .3-910. AC/DC Converter as a Linear Amplifier . . . . . . . . . .3-1011. AC/DC Converter as a DC Motor Drive . . . . . . . . . . .3-1112. Two-Pulse Semiconverter . . . . . . . . . . . . . . . . . . . . .3-1313. Two-Pulse Dual Converter . . . . . . . . . . . . . . . . . . . .3-1414. Waveform Derivation for a Three-Phase Bridge

AC/DC Converter . . . . . . . . . . . . . . . . . . . . . . . . . .3-1415. Naturally Commutated Cycloconverter . . . . . . . . . . . .3-1816. Bilateral Solid-State Switches . . . . . . . . . . . . . . . . . .3-1917. Triac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1918. AC Switches and Regulators . . . . . . . . . . . . . . . . . . .3-2119. Varegulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-2220. Brushless Machines . . . . . . . . . . . . . . . . . . . . . . . . .3-2321. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-2522. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-25

Chapter 3THYRISTORS

AC/DC CONVERTERS AND OTHER NATURALLY

COMMUTATED CIRCUITS

1. Type 2 Switch or Thyristor (Figure 3.1)

The thyristor is a Type 2 switch containing threeinternal PN junctions in series. When the anode isnegative with respect to the cathode, the centerjunction is forward biased, but the two outerjunctions are reversed biased. Therefore a thyristorblocks the flow of reverse current until thebreakdown voltage of the two other junctions isexceeded.

If the anode is positive with respect to the cathode,the two outer junctions are forward biased but thecenter junction is reverse biased. Therefore forwardcurrent is also blocked until the breakdown voltageof the center junction is exceeded.

However, the third terminal of a thyristor isconnected to the P layer adjoining the cathode Nlayer. If this terminal is made positive with respect tothe cathode, forward current flows across thecathode PN junction. If the anode terminal is alsopositive, this forward current across the cathodejunction initiates a regenerative internal actionwhich then permits current flow across the reversebiased center junction. In other words, this thirdterminal allows the thyristor to be turned on.Because of the control which this terminal exercisesover current flow, it is known as the “gate”. Onceforward current flow is initiated, the thyristor latchesin its on state, the gate loses its control, and currentcontinues until it is commutated by some externalmeans.

2. Steady-State V-I Characteristic of a Thyristor (Figure 3.2)

The V-I characteristic of a thyristor and its primaryparameters are shown below. The reverse blockingstate is similar to that of a diode, as is the forwardblocking state, and each is characterized by a leakagecurrent and a breakdown voltage.

When a small current is applied to the gate, theforward breakover voltage decreases. If the appliedforward voltage exceeds this decreased breakovervoltage, the thyristor switches to its on-state. Thisstate is characterized by a forward voltage drop,

3-2

AnodeTerminal

GateTerminal

P

P

N

NCathodeTerminal

Figure 3.1Type 2 Switch or Thyristor

Forward Voltage Drop(On-State)

Forward LeakageCurrent (Forward

Blocking orOff-State)

ReverseLeakageCurrent

Rated On-State CurrentDepends on Dissipationand Heat Sink.

Forward BreakoverVoltage with

No Gate Current

Holding Current

Reverse Breakdown Voltage

V

Forward BreakoverVoltage with Small

Gate Current

Figure 3.2Steady-State V-I Characteristic of a Thyristor

typically about 1.4 volts. The forward current ratingof a thyristor in its on-state is determined primarilyby the internal dissipation and the thermal efficiencyinvolved in getting the heat out.

Internal junction temperature is even more criticalin a thyristor than it is in a diode. Leakage currentincreases rapidly with temperature. Since leakageacross the center junction into the gate layer isindistinguishable from external gate current, theforward blocking state will disappear if the junctionbecomes overheated, and irreversible damage mayoccur. Most thyristors cannot be relied upon to blocktheir rated maximum forward blocking voltage if thejunction temperature exceeds 125o C.

Once the thyristor is in its on-state, it will remain thereuntil the current is reduced (by external means) to avalue which is less than a minimum value known as theholding current. It then reverts to the blocking state.

Since the forward voltage drop of a thyristor ishigher and the maximum allowable junctiontemperature is lower than for a diode, a thyristor hasa lower current rating than a diode of the samephysical size.

3. Two-Transistor Analog for Explaining Thyristor Turn-On (Figure 3.3)

The regenerative turn-on behavior of a thyristor maybe explained qualitatively by splitting the fourinternal layers into two equivalent transistors asshown above. With positive bias between the anodeand cathode, the anode junction acts as the emitter

of a PNP transistor and the cathode acts as theemitter of an NPN transistor. The collector of eachtransistor feeds current to the base of the othertransistor.

The analysis of turn-on follows from the gainequation for each transistor, which states that:

where IC and IE are the collector and emittercurrents, ICBO is the collector-to-base leakagecurrent, and α is common-base current gain.Applying this relationship to each transistor in termsof the currents labeled in the Figure:

Substituting the second equation into the first toeliminate I1, and then solving for anode current IA gives:

To interpret this equation, it should first beremembered that α is much less than one for verysmall collector currents, but increases as the currentincreases. In forward blocking state with zero gatecurrent IG, the leakage currents are so small that αNand αP are both small. Therefore their sum is less thanone, the denominator in the above equation is positive,and the anode current is stable at a small value.

The stable forward blocking state can be destroyed ina number of ways. The normal way is to apply gatecurrent, which increases the base current to the NPNtransistor and thereby increases αN. The increasedNPN collector current increases the PNP basecurrent, thereby also increasing αP. As αN + αPapproaches one, the denominator of the anodecurrent equation approaches zero. The transistorsare regeneratively driven into saturation, and theanode current rises to a value determined by theexternal load circuit. The load current maintains αN + αP > 1, and the thyristor is latched into its onstate even if the gate current is removed.

Turn-on can also be initiated by other effects whichincrease the off-state current to the point where αN + αP > 1. If the forward voltage is increased above itsmaximum rating, avalanche multiplication increasesthe leakage current. This mode of turn-on will destroysome devices. Increasing the temperature increases the

3-3

IA - I1

IA + IG

Cathode

IA

I1

IG

P

P

N

N

Gate

Anode

PNP

NPN

Figure 3.3Two-Transistor Analog of a Thyristor

I I IC E CBO= +α

II I I

AN G CBON CBOP

N P

= + ++

αα α1 – ( )

PNP

NPN

: –

: ( )

I I I I

I I I I

A p A COBP

N A G CBON

1

1

= +

= + +

αα

leakage current, and above 125o C turn-on can occur atless than the rated blocking voltage. If the anodevoltage is positive and increases rapidly, the C dv/dtcurrent through the capacitance of the center PNjunction can cause unexpected turn-on. In place ofgate turn-on, some thyristors are designed to be firedby a pulse of light which generates electrons andholes in the depletion region of the blockingjunction, thereby increasing the leakage current.

Once latched, the thyristor remains in its on-stateuntil something occurs which causes αN + αP todecrease below unity. The anode current belowwhich this occurs is known as the “holding current”.

Although the simple two-transistor analog is quiteuseful explaining many aspects of turn-on, othersrequire that the thyristor be viewed as a two-dimensional array of filamentary devices. Theexternal gate lead does not connect directly to eachof these devices. Gate current must pass through thelateral resistance of the gate layer, so the gatepotential changes as the distance from the gatecontact increases.

One result of this lateral gate resistance is that thegate current does not turn the entire thyristor onsimultaneously. Only the area immediatelysurrounding the gate contact conducts initially, andthe size of this initial area depends on the amplitudeand rise time of the gate pulse.

Another result is that the gate of an ordinarythyristor cannot cause it to turn-off. In the two-transistor analog, shorting the gate to thecathode or reverse biasing it would decrease thecurrent, thereby decreasing αN and αP and causingturn-off to occur. In an actual thyristor, the lateralgate resistance prevents an appreciable part of theload current from being withdrawn through the gatelead, and the α’s are not significantly affected.

4. Gate Characteristics (Figure 3.4)

The V-I characteristic of the gate is similar to a diode,but varies considerably between units. The circuitwhich supplies firing signals to the gate must bedesigned:

(1) to accommodate these variations,

(2) not to exceed the maximum voltage and powercapabilities of the gate,

(3) to assure that triggering does not occur fromspurious signals or noise, and

(4) to assure that triggering does occur when desired.

These requirements are all summarized in a designchart which is usually provided by the devicemanufacturer to characterize the gates of a specifiedthyristor type.

3-4

VgMaximumGate Voltage

MaximumInstantaneousGate Power

High-CurrentLimit of Vg – IgCharacteristic

Minimum GateCurrent Required to Trigger allUnits at Minimum Rated Temperature

Minimum Gate VoltageRequired to Trigger all

Units at MinimumRated Temperature

Ig

6V

00 0.5A

Minimum Gate Voltagethat will not Trigger

any Unit at MaximumRated Temperature

Low-Current Limitof Vg – Ig Characteristic

Figure 3.4Design Chart for the Gate Triggering Circuit of a Thyristor

All possible safe operating points for the gate arebounded by the low and high current limits for the V-I characteristic, the maximum gate voltage, and thehyperbola representing maximum gate power. Withinthese boundaries there are three regions ofimportance.

The first region lies near the origin (shown shaded)and is defined by the maximum gate voltage whichwill not trigger any device. This value is obtained atthe maximum rated junction temperature (usually125o C). The gate must be operated in this regionwhenever forward bias is applied across the thyristorand triggering is not wanted. In other words, thisregion sets a limit on the maximum spurious signalsthat can be tolerated in the gate firing circuit.

The second region is further defined by theminimum values of gate voltage and current requiredto trigger all devices at the minimum rated junctiontemperature. This region contains the actualminimum firing points of all devices. In a sense, it is aforbidden region for the firing circuit because asignal in this region cannot be depended upon eitherto always fire all devices or to never fire any devices.

The third region is the largest and shows the limitson the gate signal for reliable firing. Ordinarily afiring signal in the lower-left part of this region isadequate. For applications where fast turn-on isrequired, a “hard” firing signal in the upper rightpart of the region may be needed.

A typical firing circuit for a 100A thyristor shoulddeliver about 6V (no load) with a source impedanceof 12 ohms.

5. Transient V-I Characteristic of a Thyristor (Figure 3.5)

The transient characteristics of a thyristor are similarto those of a diode in that turn-on and turn-off arenot instantaneous, and a pulse of reverse currentflows when the device is reverse biased immediatelyafter conducting forward current. However, the factthat a thyristor has a forward blocking state notpossessed by a diode causes several additionalparameters to be significant.

First consider the turn-on transient. Since a diodeturns on automatically as soon as it becomes forwardbiased, its entire area turns on simultaneously. Onthe other hand, the turn-on of a thyristor begins at avery small area adjacent to the gate electrode, andconduction must then propagate laterally from thisinitial area. The propagation velocity isapproximately 0.1mm/µ sec. Therefore a device 2 cm. in diameter with the gate electrode in thecenter will require some 100µ sec. before the fullarea is conducting, although the forward voltage willapproach its steady-state value much sooner thanthis. Because the area which is conducting is so smallduring the early part of this transient, the current

3-5

Time

Forward BlockingVoltage

Forward LeakingCurrent

Forward Current

Forward Voltage Dropor On-State Voltage

Reverse LeakageCurrent

SweepoutCurrent

Reverse BlockingVoltage

Reverse BlockingState

Conducting orOn-State

Turn-On Time

Turn-Off orRecovery

Time

Forward BlockingState

dv/d

t dv/d

t

Time ofGate Trigger

Figure 3.5Transient V-I Characteristics of a Thyristor

density can become extremely high if the rate ofincrease of current is not limited. Excessive currentdensity causes a hot spot which can destroy thedevice. Consequently, high power thyristors have amaximum di/dt rating at turn-on.

Another parameter of great importance is dv/dt. Aswe have seen, the center junction of a thyristor isreverse biased when the device is in its forwardblocking state, thereby limiting the steady-statecurrent to a small leakage value. However, thisjunction is shunted by a junction capacitance whichpasses appreciable current into the gate layer if thedevice is subjected to a rapidly changing voltagewhile it is in its blocking state. If the device is forwardbiased and dv/dt is positive, this current can turn thedevice on when it is supposed to remain in itsblocking state. Therefore a thyristor also has amaximum dv/dt rating, and the circuit may fail tooperate properly if this rating is exceeded.

If a thyristor is naturally commutated, the primaryfactors in the turn-off transient are the peak valueand duration of the reverse sweepout current, as inthe case of a diode. Naturally commutated thyristorsare normally reverse biased for a time which isconsiderably greater than the recovery time. Forsome types of forced commutation, the thyristor isreverse biased for a specified time, and must thenblock forward voltage. Under these circumstances,other factors which will be discussed under forcedcommutation must be taken into account.

The literature on the physics, characterization,rating, and application of thyristors is voluminous.The most comprehensive reference is:

F. E. Gentry, et al., Semiconductor Controlled Rectifiers:Principles and Applications of PNPN Devices, Prentice-Hall; 1964.

Manufactures’ handbooks also contain much useful information:

General Electric SCR Manual (Fifth Edition), GeneralElectric Semiconductor Products Dept., Syracuse, N.Y.; 1972.

Westinghouse SCR Designers Handbook (Second Edition),Westinghouse Semiconductor Div., Youngwood, Pa.; 1970.

The history of thyristor development up to1965 is interestingly summarized in:

J. D. Harnden, Jr., “Present-Day Solid-State PowerSwitches,” IEEE Spectrum, vol. 2, no. 9, pp. 107-112; 1965.

6. Thyristor Ratings – Design Tradeoffs andCommercial Availability (Figures 3.6 & 3.7)

The design of commercial thyristors involvestradeoffs between power controlling capability andspeed. In an optimum device, both can be increasedsimultaneously only by increasing the silicon area.The development of larger and larger devicesaccounts for much of the progress that has beenmade in thyristors, but the largest area that is feasibleat any specified time is limited by the technology ofgrowing silicon crystals of adequate quality, by thedifferential expansion between silicon and packagematerials, and by the heat density which can bewithdrawn through the package. Thyristors having asilicon diameter of 50mm are now readily available,and 75-100mm devices are being developed.

Although the subject of thyristor design tradeoffs andtheir interactions with circuit design tradeoffs is quitecomplex, some simple relationships between thevoltage, current, and turn-off time ratings can beillustrated. A survey of high-power commerciallyavailable thyristors is summarized in Figure 3.6 andFigure 3.7. The interaction between the ratings ofstate-of-the-art thyristors is plotted in Figure 3.6 I Tave vs. VRM, with tq shown as a running parameter.

3-6

400 8001000 1400

Voltage (VDRM/VRRM) - Volts

Cur

rent

(Iav

g) -

Am

pere

s

3000500 2000 400020

30

50

100

200300

500

1000

200030004000

500kVA

10µS

5µS

150µS

20µS15µS

40µS

30µS

5mVA

>300µS

200kVA100kVA50kVA

1mVA2mVA

Figure 3.6Average Current Ratings vs. Blocking Voltage Ratings forCommercial Thyristors. (Fastest Turn-Off Time, in µsAlso Shown for Each Curve.)

Figure 3.7 shows tq plotted vs. the product of VRMand ITave. Most devices cluster about the line whoseequation is:

Thus we have deduced an empirical relationshipbetween the power controlling capability and thespeed of commercially available thyristors.

Converters Using Type 2 Switches

• Utilize an AC Source for Natural Commutation

• Can Perform

– Controlled Rectification

– Synchronous Inversion

– Frequency Changing

– AC Switching and Regulation

7. Circuit Functions Performed by Type 2 Switches

Type 2 switches allow forward current to be turnedon at arbitrary times but do not permit arbitrary

turn-off or commutation. Consequently, almost allcircuits using Type 2 switches must be connected toan AC source which periodically reverses polarity andcauses commutation to occur “naturally”.

The functions which these circuits can performinclude:

CONTROLLED RECTIFICATION – thecontrollable conversion of AC power to DC power.

SYNCHRONOUS INVERSION – supplying powerfrom a DC source into an AC line.

FREQUENCY CHANGING – Conversion of ACpower at one frequency into AC power of asomewhat lower frequency.

AC SWITCHING AND REGULATION – on/offswitching and “continuous” regulation of theeffective value of AC power.

The basic circuits for all of these functions are thesame, since all are based on the switching matrix. Thecircuits are distinguished primarily by which switchesare omitted from the general matrix and by thetiming of the switching control signals, andsecondarily by the presence of freewheeling diodes,various interconnections of transformer windings, ect.

The circuits for performing controlled rectificationand/or synchronous inversion are called AC/DCconverters. These circuits have an obviousrelationship to diode rectifier circuits and will bediscussed first.

8. Two-Pulse AC/DC Converter Operating as aControlled Rectifier (Figure 3.9)

Consider what can be done if all of the diodes in atwo-pulse bridge (or center-tap) rectifier arereplaced by thyristors. If each thyristor receives afiring pulse at the instant when its anode becomespositive with respect to its cathode, no change occursand the circuit operates exactly as it did as a dioderectifier. But if these firing pulses are uniformlydelayed by an angle α, somewhat different resultsoccur. This change is easily implemented by buildinga controllable phase delay into the switching controlsignal generator.

Remember that all four simplifying assumptions areimposed on the analysis of this circuit. In particular,the DC current is assumed to be constant over each

3-7

V I tRM T qave( ) ( )kVA mS= 130

} AC/DC Conversion

50 200 1000 5000500100 2000

VRM x IT(avg) (kVA)

Mic

rose

cond

s

5

10

20

50

100

200

500

ImprovedTechnology

130

tq=

­

Figure 3.7Turn-Off Time Ratings Vs. Rated Power Capability forCommercial Thyristors.

cycle.* Therefore a continuous current path mustexist at all times between AC and the DC terminals.If firing of one set of thyristors is delayed,commutation is also delayed and the other set mustcontinue conduction in the meantime. If the ACsource reverses polarity, the forward bias on thethyristors to maintain this conduction will besupplied by the kickback voltage of the filterinductance.

As a typical example, let α = 45o. The voltage andcurrent waveforms are shown in the figure for thiscase. The ripple voltage appearing across the filterinductance is shaded, and as before its average valueover a full cycle must be zero. Therefore the averageload voltage is:

It is seen that the phase delay has reduced theaverage load voltage from the corresponding valuefor a diode rectifier. In general, Vd may be set to anyintermediate value by proper choice of α, accordingto the relationship:

Note also that the AC source current waveformremains a square wave of amplitude Id, but it is nowdisplaced from the voltage sine wave by an angle φ = α. Therefore the converter has an increasinglylagging power factor at the AC terminals as theoutput DC voltage is reduced.

Since the DC current is constant, the instantaneouspower dissipated in the load resistor is the samethroughout each half cycle. However, the powerinterchange between the other parts of the circuitchanges continually. When Vout exceeds Vd, thevoltage across the inductance is positive, and both theinductance and the load resistor are absorbing powerfrom the AC source. When Vout drops below Vd, theinductance starts to supply part of the power to theresistor. Then when the source voltage reversespolarity, the inductance supplies instantaneous powerto the load resistor and to the AC source. As with theone-pulse diode rectifier having an inductive load,the AC source receives instantaneous power duringpart of each cycle even through the net power flowover a cycle is from the AC source to the loadresistance.

*Note that this assumption does not prevent Id from changing as afunction of α.

3-8

IinVpsin ωt

VinVout

Vout

Vin

Iin

FundamentalComponent of Iin

Id

-Vin

Vd

Vd = Cos a

a

2Vp

π

VpIdL

R

φ

Figure 3.9Two-Pulse AC/DC Converter(Waveforms shown for α=45°)

V V sin tdtV

cos tV

d pp p( ) 45

2

4

54

4

54° = = −[ ] =∫ω

πω

πω

ππω

πω

πω

πω

VV

cosdp( )α

πα=

2

9. Two-Pulse AC/DC Converter Operating as aSynchronous Inverter (Figure 3.10)

The previous discussion is valid as long as the phasedelay is less than 90o. When α = 90o, the average DCvoltage as given by the previous equation drops to zero.Therefore the current in the load resistor also drops toa very low value, ideally zero. The circuit is “coasting”with the entire output voltage appearing across thefilter inductance and with no net power flow.

Actually this case is only of hypothetical interest witha passive load because as soon as the current dropsbelow the holding current, the thyristors refuse tofire. But now assume that the passive load is replacedby an active load which is independently capable ofmaintaining a constant flow of current greater thanthe holding current. This current must be in thepositive direction or the thyristors will not be able tocarry it. For the sake of discussion, assume that theload is a DC current source of value Id.

The AC source current is still a square wave ofamplitude Id, but now it is displaced 90o from the ACvoltage. Therefore, both the average DC voltage andthe net power are zero. The instantaneous powerflow from the AC voltage source to the DC currentsource during one part of the cycle is just balancedby the instantaneous power flow in the oppositedirection during another part of the cycle.

However, we have now established a set of conditionsunder which α can be allowed to exceed 90o. If, for

example, α = 135o, the waveform are as shown. Theaverage DC is negative:

and net power flow is from the DC current sourceinto the AC voltage source. The AC/DC converter isperforming synchronous inversion. In other words,since the average DC voltage has reserved polaritybut the average DC current has not, operation hasentered Quadrant IV. The net power flow in thisdirection may be controlled by changing α just as itwas in the other direction when α was less than 90o.One difference, however, is that there is a practicalupper limit on α at about 165o. If this limit isexceeded, the thyristor which is commutating offdoes not have sufficient time to regain its blockingability before it is subjected to forward voltage.Therefore it continues to conduct and a“commutation failure” occurs.

The operation of the AC/DC converter which wehave been discussing depends on the ability of a Type 2 switch to block forward voltage. As a result,the converter permits the average DC voltage to becontrolled and the net power flow to be reversed,neither of which is possible in a rectifier using onlyType 1 switches. These converters, in which all of theType 1 switches have been replaced by Type 2switches, are known as 2-quadrant or fully-controlledAC/DC converters.

3-9

VV V

dp p( ) cos135

2135

2° = ° = −

π π

Vout

IinId

Vd = 0a = 90o

Vp

IinId

Vout

Vd

a = 135o

Vp

Figure 3.10Waveforms for a Two-Pulse AC/DC Converter for α = 90° and 135°

An application for this type of converter which isdestined to become very important in the future ishigh-voltage DC (HVDC) transmission. A converterat each end of a DC transmission line links it to twoAC systems. The AC systems need not besynchronized (in fact, they can operate at differentfrequencies), and power can be caused to flow ineither direction.

Much literature exists on HVDC converters andsystem considerations. Introductory articles include:

F. D. Kaiser, “Solid-State HVDC Transmission Technology,”IEEE Spectrum, vol. 3, no. 11, pp. 48-51; 1966.

J. Kauferle, “Using DC Links to Enhance AC SystemPerformance,” IEEE Spectrum, vol. 9, no. 6, pp. 31-37; 1972.

Textbooks devoted to the subject include:

E. W. Kimbark, Direct Current Transmission, vol. I, Wiley-Interscience; 1971.

B. J. Cory (Editor), High Voltage Direct Current Convertors andSystems, MacDonald (London); 1965.

10. AC/DC Converter as a Linear Amplifier (Figure 3.11)

We have seen that the output voltage of an AC/DCconverter varies according to the phase delay angle.The general form of this relationship for a fullycontrolled AC/DC converter is:

where P is the pulse number and Vp is the peak of theunfiltered output voltage. Obviously Vd is a nonlinearfunction of α. However, in many applications it isdesirable for the output to be a linear function of aninput control signal. One way of implementing thislinear relationship in the firing control circuit isknown as the “cosine crossing” method.

At the heart of this method is a circuit known as azero crossing detector. This circuit compares twoinput signals and emits a pulse when they cross eachother (that is, when they are instantaneously equal).One input to the zero crossing detector is the inputcontrol signal, Vi. The other input is an internallygenerated sinusoidal signal having the samefrequency as the power frequency but displaced inphase so that it peaks at the α = 0 commutation

3-10

Vp

k1

“End Stops”

Vp

Vp

ViZero

CrossingDetector

cos ωt

α = cos-1

Output ofZero-Crossing Detector

ωt

k1

k1VI

Vp

2k1

Vd = AViViA =

π

Vd = Vi cos α =2k1π

2Vp

π

Figure 3.11Two-Pulse AC/DC Converter Using Cosine-Crossing Method to Achieve Linear Amplification

VP

sinp

V cosd p( ) απ

π α=

point. Assume that the peak value of this signal isVp/k1, where k1 is a constant. Then pulses willappear at the output of the zero crossing detector at:

The pulses which occur when the sinusoidal signalcrosses Vi in a negative-going direction can then beused to initiate one set of firing pulses.* The outputvoltage from the AC/DC converter is then:

where:

Thus the AC/DC converter operates as a linearvoltage amplifier with a gain of A.

In this analysis we have assumed that the inputvoltage, Vi, is a constant. The analysis remains validfor a time-varying input, Vi (t), as long as itsfrequency spectrum lies well below the fundamentaloutput ripple frequency, Pω. Because of thenonlinearities inherent in the switching action, amore exact analysis which is valid for rapidly varyinginputs requires advanced techniques such asdescribing functions or z-transforms. Thesetechniques are described in the following references:

G. De and A. K. Mandal, ”Incremental DescribingFunction Analysis of Subharmonic Oscillations inControl Systems with Thyristor Converters,” IEEETrans., vol. IECI-20, pp. 229-235; Nov. 1973.

G. De and A. K. Mandal, ”Impulse Analysis ofSubharmonic Oscillations in Control Systems withThyristor Converters,” Proc. IEE, vol. 120, pp. 1030-1037; Sept. 1973.

J. P. Sucena-Paiva, R. Hernandez and L. L. Freris,“Stability Study of Controlled Rectifiers Using a NewDiscrete Model,” Proc. IEE, vol.119, pp. 1285-1293;Sept. 1972.

P. A. Hazell and J. O. Flower, “Stability Properties ofCertain Thyristor-Bridge Control Systems: Part 1–TheThyristor Bridge as a Discrete Control System; Part2–The Inter-relationships of Discrete-and Continuous-Design Methods,” Proc. IEE, vol. 117, pp. 1405-1420;July 1970.

*When the sinusoidal signal crosses Vi in a positive-going direction, theoutput pulses have the opposite polarity and should be ignored. Thisscheme is repeated P times with the sinusoidal signal shifted by 2π/P togenerate the firing pulses required by other thyristors.

11. AC/DC Converter as a DC Motor Drive(Figure 3.12)

AC/DC converters are widely used in closed-loop DCmotor drives, supplying controllable power either tothe armature or to the field. Usually the other timeconstants in the loop are much greater than the timeconstant of the AC/DC converter. Hence the linearanalysis given in Section 10 is frequently valid.

The basic elements of a closed-loop speed controlsupplying the armature of a DC motor are shownabove. The AC/DC converter is represented by alinear voltage amplifier with a gain A. The insidecurrent-limiting loop serves to override the outsidespeed-control loop if necessary to prevent thecurrent from exceeding a safe value. Otherwise thisinside loop may be ignored.

The analysis of the dynamics of the speed-controlloop can be performed with standard lineartechniques, such as the Laplace transform. Theelectrical equation of the armature is:

where s = d/dt is the complex frequency operator, Lαand Rα are the armature circuit inductance andresistance, and Eα is the back emf generated in thearmature. Assuming a constant field flux, Eα isproportional to the angular speed of the armature, ωα:

where kα is the same constant which enters thetorque equation:

The electrical analog of the mechanical “circuit” ofthe armature may be derived as follows. If J themoment of inertia and B is a linear friction load(torque proportional to ωα), then:

or

where Cm is an electrical capacitance analogous tomoment of inertia, and Rm is a parallel resistanceanalogous to the dissipative load.

3-11

α = −cosV

V ki

p

1

1

VP

sinP

V cosP

sinP

VV

V kAVd p p

i

pi=

=

=

ππ α

ππ

1

Ak P

sinP

= 1

ππ

AV V L S R I Ed a a d a1 = = +( ) =

E k ork

Ea a a aa

a= =ω ω 1

T k Ia d=

T Js B Js BE

kk Ia

a

aa d= + = + =( ) ( )ω

I

E

J

ks

B

kC s

Rd

a a am

m

= + = +2 2

1

These equations can now be used to eliminate Id andEα, yielding the overall forward transfer function:

The back emf of the armature, Eα, could be used asan accurate indicator of armature speed except thatit is physically inaccessible. The applied armaturevoltage, Vd, is sometimes used as an approximateindicator of speed where precise speed control isunnecessary. Better regulation is achieved by meansof a tachometer which generates a feedback voltageproportional to armature speed. In this case, thetransfer function of the feedback path is a constant:

Examination of the loop gain, GH, reveals that this isa second-order Type 0 or proportional controlsystem. Such systems are analyzed in any textbook onfeedback control. The steady-state armature speed isdirectly proportional to the input variable, Vω, andthe speed of response is related to the loop gain andthe armature time constants, Lα/Rα and J/B.

Since the motor torque is proportional to Id, and sinceId cannot be reversed in a two-quadrant AC/DCconverter, this type of motor drive cannotregeneratively brake or reverse the direction of rotationunless a polarity reversing switch is incorporated intoeither the armature of the field circuit. We will soon see

that this limitation can be removed through the use ofa four-quadrant AC/DC converter.

References:

1. Solid-State DC Motor Drives (Chapter 2), A. Kusko;MIT Press, Cambridge, Mass.; 1969.

2. Feedback Control System Analysis and Synthesis, J. J.D’Azzo and C. H. Houpis; McGraw-Hill, New York.

3. R. Cushman, “Powerful SCR’s, Connected in Parallel,Control Industry’s Biggest Machinery,” Electronics, pp.110-120; Oct. 4, 1965.

4. R. H. Neer and R. M. Brenza, “SCR’s in Large MotorDrives,” Control Engineering, vol. 20, pp. 42-45; Feb.1973.

5. A.S.A. Farag, et al, “Studies on an SCR ControlledVariable Speed DC Shunt Motor,” IEEE Trans., vol. PAS-93, pp. 785-792; May/June 1974.

6. K. H. Bezoid, et al., “Thyristor Converters forTraction DC Motor Drives,” IEEE Trans., vol. IA-9, pp.612-617; Sept./Oct. 1973.

7. C. E. Robinson, “Redesigning of DC Motors forApplications with Thyristor Power Supplies,” IEEETrans., vol. IGA-4, pp. 508-514; Sept./Oct. 1968.

8. N.A.O. Demerdash, “Effect of Complex Forms onCopper Losses in Large DC Motors,” IEEE Industry &General Applications 5th Annual Meeting Record, pp.77-81; Oct. 1970.

9. W. C. Carter, “Mechanical Factors Affecting ElectricalDrive Performance,” IEEE Trans., vol. IGA-5, pp.282-290; May/June 1969.

3-12

Vi

Vi

H = kt

Vd

Id

T =

La Ra Ea

ka

Aka

G = (La s + Ra) (Js + B) +

kt

2ka

2ka

2ka

2ka

Id

1

J

B

ωa

ωa

AC/DC ConverterGain = A

Current Transducer DC Motor Armature and Load

Tachometer

Current-LimitLoop

Speed-Control Loop

Et

Vω

Vω +

–

+

–

Figure 3.12AC/DC Converter in a DC Motor Drive Using Type O or Proportional Speed Control in the Armature Circuit

GV

Ak

L s R Js B Ka

i

a

a a a

= =+ + +

ω( )( ) 2

HE

kt

at= =

ω

12. Two-Pulse Semiconverters (Figure 3.13)

In the previous two-pulse 2-quadrant converter, all of thediodes in the diode bridge were replaced by thyristors.

There are a number of similar two-pulse converterswhich have controllable DC output voltages, but areconfined to 1-quadrant operation. That is, they canoperate as controlled rectifiers, but since the outputvoltage cannot reverse polarity, they cannot operateas inverters. These circuits are known assemiconductors or half-controlled converters.

The most obvious way to change a full converter intoa semiconverter is to connect a freewheeling diodeacross the DC output terminals of the bridge. Duringthe interval from 180o to (180 + α)o, load currentcirculates through the freewheeling diode ratherthan through the bridge. But the same effect may beachieved more inexpensively by replacing only two ofthe four diodes in the original diode bridge bythyristors. Two possible configurations are shown. Ineach case, there is always a freewheeling path whichprevents the output voltage from going negative.

Voltage and current waveforms are shown for α = 90o. Note that Vd is controllable from 2VP/π tozero, but the corresponding range of α is from zeroto 180o rather than zero to 90o as in the fullycontrolled converter. In general, the relationship fora two-pulse semiconverter is:

Note also that the displacement angle of the AC sourcecurrent is now α/2, giving an improved power factor.

Semiconverters are widely used for DC motor driveswhere reversing is not necessary or can be provided by areversing contactor. For these applications,semiconverters are preferred to fully controlledconverters because:

(a) they require fewer thyristors which are moreexpensive than diodes and

(b) the input displacement factor is increased.

Depending on other factors, the harmonic content ofthe input current and output voltage may be reduced orincreased.

Reference:

D. L. Duff and A. Ludbrook, “Semiconverter RectifiersGo High Power,” IEEE Trans., vol. IGA-4, pp. 185-192;Mar./Apr. 1968.

3-13

VV

cosdp= +

πα( )1

Id

VdVp sin ωt

Iin Vp

IdIin

Fundamental Component of Iin

Vp(1 + Cos a)

pVd =

a = 90o

a2

φ =

Figure 3.13Two-Pulse Semiconverter Circuits and Waveforms for α = 90°

13. Two-Pulse Dual Converter (Figure 3.14)

Some types of DC motor drives require that themotor be accelerated, controlled, and regenerativelybraked in both directions. This complete 4-quadrantoperation is readily provided by a pair of fullycontrolled AC/DC converters connected in parallelso as to be capable of either controlled rectificationor synchronous inversion with either polarity at theDC terminals. One of these converters is called the“positive converter” and the other the “negativeconverter,” and together they constitute a “dualconverter”. Note that the bridge form of dualconverter implements the complete generalswitching matrix.

Usually firing pulses are supplied to only one of thetwo converters at a time. If the motor load is to besoft-started in the forward direction, firing pulsesmust be supplied to the positive converter. Initially, αshould be nearly 90o to limit the in-rush current, butthen decrease to nearly zero as the motorapproaches full speed.

When the motor is to be reversed, its current mustbe reversed. The positive converter cannot conductthis reverse current, so the firing pulses aretransferred to the negative converter. However, themotor will continue to supply positive back-emf untilits direction of rotation actually reverses. In themeantime, α can be greater than 90o so that thenegative converter acts as a synchronous inverter.The motor acts as a generator during regenerativebraking, and the energy stored in its inertia isreturned to the AC line. Then as α is reduced below90o , the motor accelerates in the reverse direction.

Note that although the inductance in the DC circuithelps to smooth the current waveform, it alsohinders rapid current reversal. Therefore its value

must be a compromise between the large valuerequired for smoothing and a small value forminimum response time.

14. Waveform Derivation for a Three-Phase BridgeAC/DC Converter (Figures 3.15 & 3.16)

The Three-Phase bridge is the most widely used high-power AC/DC converter connection. Hence athorough understanding of the operation of thiscircuit is a fundamental part of power electronics.The derivation of the voltage and current waveformswhich occur throughout the bridge under a specifiedmode of operation quickly tests whether or not thatunderstanding has been achieved. Intuitive attemptsto derive these waveforms usually lead to utterconfusion, but a methodical analysis can beperformed very easily once a few simple procedureshave been learned.

The following analysis assumes that the three-phaseAC source voltage is balanced and sinusoidal, the DCcurrent is ideally filtered so that Id is constant, theswitching devices are ideal (zero on-state voltage andoff-state current, and instantaneous switching), andcommutation overlap is negligible. If theseassumptions are not valid, corrections can usually bemade in an obvious way. We will also assume a fullycontrolled (two-quadrant), six-pulse converter,although the method is equally applicable to avariety of semiconductor, dual converter, and cycloconverter circuits.

The symmetries which are evident in the operation of asix-pulse bridge tempt us to neglect the completelabeling of the circuit and to skip apparently obvioussteps. Yielding to this temptation is a sure route tofrustration.

The bridge itself has a total of 5 nodes or terminals: 3 AC terminals labeled A, B, C, and 2 DC terminalslabeled X, Y. In addition, the filtered DC node hasbeen labeled D, and the AC neutral has been labeledN. (If the windings are delta-connected, a “virtualneutral” can be defined as a reference even though itdoesn’t exist physically.) Using these 7 labels, we canprecisely specify any voltage that can exist anywherein the converter. For instance, VAN is the voltagedrop from A to N, and is positive when A is positivewith respect to N.

Furthermore, the AC phase windings have beendrawn so as to represent the phasors of the phase

3-14

PositiveConverter

NegativeConverter

Figure 3.14Two-Pulse Dual Converter

voltages. A reference axis is indicated which isconsistent with the electrical angle θ = (180o/π)ωτdefined at the bottom of the figure. As time passesfrom t = 0, the phase windings may be pictured asrotating counterclockwise with angular velocity ω.The instantaneous phase voltage is proportional tothe projection of the length of the winding onto the

reference axis. Alternately, the windings can be heldfixed if the reference axis is rotated clockwise withangular velocity ω.

The two three-pulse halves of the bridge shouldalways be analyzed separately before combining themto determine the overall six-pulse waveforms. It is

3-15

180o0o 120o 240o 360o 480o π

ωtu =

DC

Line

s

AN BN CN ANVp

Vp

+

+

Vp

0

–

=­3

Vp–

=­3

AB AC BC BA CA CB AB AC

XY

ConductionLogic

AC

Lin

e-to

-Lin

e V

olta

ges

AC

Pha

se V

olta

ges

(Lin

e to

Neu

tral

)

Id = Constant

(Vxy)ave = Vd

IA

IB

IC

Id

A

References Axis forProjecting Phasors

X

Y

B

C

N

D

Three-PhaseSinusiodal Voltage

SourceThree-Phase

BridgeConstant-Current

DC Load or Source

Figure 3.15Guide for Drawing 3-Phase AC/DC Bridge Waveforms

obvious that each thyristor in each three-pulse groupmust conduct for 120o of each cycle under normaloperating conditions. As the conduction state ofeach phase is determined as a function of time, thisis recorded on the “conduction logic” diagram. The three-pulse voltage waveforms are then easily derivedfrom the AC phase (or line-to-neutral) voltages, andfinally the six-pulse waveforms from the AC line-to-line voltages. The figure can be used repeatedly as a

waveform sketching guide if the waveforms areactually drawn on tracing paper.

The process is most easily described in terms of anexample (Figure 3.16). The firing delay angle of α ofeach thyristor is measured from the point where adiode (substituted for the thyristor) would begin toconduct. Let us begin by determining the waveformsfor α = 0.

3-16

VYN

Fund. of IA(u = 0)

VXY

VAX

180o0o 120o 240o 360o 480o π

ωtu =

DC

Line

s

VXNAN BN CN ANVp

Vp

+

+

Vp

0

–

=­3

Vp–=­3

IAXIAY

A

AB AC BC BA CA CB AB AC

B C AB C A B C

XY

Id+ –

Id

ConductionLogic

AC

Lin

e-to

-Lin

e V

olta

ges

AC

Pha

se V

olta

ges

(Lin

e to

Neu

tral)

0+ –

a = 0

Id = Constant

(Vxy)ave = Vd

IA

IB

IC

Id

A

References Axis forProjecting Phasors

X

Y

B

C

N

D

Three-PhaseSinusiodal Voltage

SourceThree-Phase

BridgeConstant-Current

DC Load or Source

Figure 3.16Three-Phase AC/DC Bridge Waveform Example

In the three-pulse group of thyristors connected to X,thyristor AX is connected to the most positive phasevoltage over the interval 0o < θ < 120o. A diode in thisposition would conduct over this interval, and with α = 0 a thyristor will do the same. This fact is recordedon the conduction logic diagram by showing that X istied to A over 0o < θ < 120o. Similarly it can be arguedthat X is tied to B by the conduction of thyristor BXover the interval 120o < θ < 240o, and X is tied to Cover the interval 240o < θ < 360o. Thereafter the cyclerepeats. The VXN waveform should now be obvious.

In the three-pulse group connected to Y, phase A ismost negative and Y is tied to A over the interval 180o < θ < 300o. Similarly Y is tied to B over 300o < θ < 420o (or -60o < θ < 60o), and to C over 60o < θ < 180o. In this way, the conduction logicdiagram is completed and the VYN waveform isderived.

The unfiltered VXY waveform can now be derivedfrom the relationship:

since the VXN and VYN waveforms are both known.Alternately, VXY can be derived from the conductionlogic diagram by taking one interval of θ at a time.Thus for 0 < θ < 60o, X is tied to A, Y is tied to B, andVXY = VAB. For 60o < θ < 120o, VXY = VAC; ect. In eachcase, VXY is equal to one of the AC line-to-line voltages,and the corresponding segment may be sketched.

The average of the VXY waveform is equal to thefiltered DC voltage VDY = Vd. The instantaneousvoltage across the filter inductor is the differencebetween these two voltages:

The peak-to-peak ripple current in the DC circuitcan be evaluated from the volt-second area imposedon the inductor:

where the limits on the integral are the points whereVXY crosses Vd (i.e. where VXD = 0).

The only remaining voltage waveforms are thosewhich appear across the various thyristors. These,too, may be derived directly from the conductionlogic diagram and the AC line-to-line voltages. The

voltage across thyristor AX, for instance, is simply VAX. Thus:

(Since this thyristor isconducting and theconduction logicdiagram shows that Xis tied to A)

(Since X is tied to B)

(Since X is tied to C)

Having completed the analysis of the voltagewaveforms, we can turn our attention to the currentwaveforms. Under the assumptions of continuous,constant DC current Id and no commutation overlap,at any instant the current in any part of the convertermust be either +Id or zero. The conduction logicdiagram tells the interval during which each thyristorconducts Id. The AC line currents are then obtainedby adding the two thyristor currents in each phase:

From symmetry, the angular position of thefundamental component of AC line current canusually be determined and compared with thecorresponding AC phase voltage to establish thedisplacement angle φ.

This complete analysis of the converter for α = 0probably did not reveal any unexpected waveformsbecause of the relative ease with which a six-pulsediode rectifier bridge can be analyzed. However, thebeauty of the analytical process that has beendescribed is that it gives the waveforms for othervalues of α with equal ease. The entire conductionlogic is simply shifted to the right by angle α. Thecorresponding changes in the voltage waveforms maysurprise you, but these are left as an exercise.

The graphical approach to converter analysis has twomajor advantages:

(1) it is easy to perform, and

(2) it leads to a good physical understanding of whatis happening at any time at any point in thecircuit.

Once the graphical analysis has been completed,appropriate mathematical expressions can bederived, if necessary, for each segment of eachwaveform. However, an attempt to derive theseexpressions without first thinking through the details

3-17

V V VXY XN YN= −

V V V Ldl

dtXD XY dd= − =

∆lL

V dtd XD= ∫1

0 120 0° < < ° =θ VAX

120 240° < < ° =θ V VAX AB

240 360° < < ° =θ V VAX Ac

I I I I IA AB AY AX YA= + = −

of the physical behavior of the converter isguaranteed to produce false results.

Various aspects of this method of analysis have beenadapted from a number of sources. The followingreferences are recommended for further study:

“A Simplified Technique for Analyzing the three-PhaseBridge Rectifier Circuit,”A. Ludbrook and R. M. Murray, IEEE Trans. on Industry& General Applications, vol. IGA-1, pp. 182-187;May/June 1965.

Solid-State DC Motor Drives (Chapters 3 and 5), A. Kusko;MIT Press, Cambridge, Mass.; 1969.

Rectifier Circuits: Theory and Design,J. Schaefer, John Wiley, New York; 1965.

Principles of Inverter Circuits (Section 3.4),B. D. Bedford and R. G. Hoft, John Wiley, New York;1964.

Thyristor Phase-Controlled Converters and Cycloconverters, B.R. Pelly, John Wiley, New York; 1971.

The Fundamental Theory of Arc Converters,H. Rissik, Chapman & Hall, London; 1939. Facsimilereprints available from University Microfilms, AnnArbor, Michigan.

“The Rectifier Calculus,” W. M. Goodhue, AIEE Trans.,vol. 59, pp. 687-691; 1940. (Discussion on pp. 1073-1076.)

For historical research on the early development ofthe power electronics field, an invaluable aid is the

“Bibliography on Electric Power Converters,” AIEEPublication No. S-35, February 1950. Nearly 1300publications between 1903 and 1947 are listed.

15. Naturally Commutated Cycloconverter (Figure 3.17)

We have examined the two-pulse dual converter andshown that it is capable of supplying a DC outputvoltage of either polarity, and can also supportcurrent of either polarity in the DC circuit. If thetiming of the firing control signals is modified, thissame circuit can operate as a one-step frequencychanger. In this mode of operation it is known as anaturally commutated cycloconverter, and does notrequire an intermediate DC link such as thatinvolved when a controlled rectifier and asynchronous inverter are used together as afrequency changer.

In essence, a cycloconverter synthesizes the desiredlow-frequency AC waveform as a succession ofperiodically varying “DC” values. Since these “DC”values are functions of the corresponding firingdelays, this synthesis is achieved by periodically

3-18

VoutFundamental component of Vout

V

Iφ = Load Displacement Angle

Filtered Load Current

Inversion Rectification

NegativeConverter

Positive Converter Negative Converter

Inversion Rectification

Figure 3.17Voltage and Current Waveforms for a Three-Pulse Cycloconverter with a 5:1 Frequency Reduction Ratio

modulating α for each thyristor. But the derivation ofthe firing control signals is not as straightforward asit might sound from this elementary description.

Two facts should be emphasized:

(1) It is the desired output voltage waveform whichdetermines the proper value of α each time athyristor is fired, but

(2) it is the current waveform which determineswhether the positive or negative converter shouldbe activated.

Of course the phase and amplitude relationshipsbetween the voltage and current waveforms are load-dependent. Therefore the firing control logic usuallyincludes a feedback loop which maintains propersynchronization in spite of varying load conditions.

Specifically, the firing signals cause the cycloconverterto operate as follows: If the current is positive, thepositive converter is activated. If the voltage is alsopositive, then α<90o and the positive converter acts asa rectifier. But if the voltage is negative, then α>90o

and the positive converter acts as a synchronousinverter. Similarly, when the current is negative thenegative converter rectifies when the voltage isnegative and inverts when the voltage is positive.

Naturally commutated signal-phase cycloconvertersare limited to output frequencies less than about onethird of the source frequency. Because of the greaterchoice of source voltages from which to synthesizethe desired output waveform, three-phasecycloconverters have less output ripple and operate atsomewhat higher output frequencies, although stillgenerally confined to frequencies below the sourcefrequency.*

The inherent constraints of natural commutationcause this type of cycloconverter to draw a laggingsource current even for resistive load. The spuriousfrequencies present in the voltage and currentwaveforms are also usually not integer harmonics orsubharmonics of source frequency.

The primary application for naturally commutatedcycloconverters is for variable speed AC motor drives,especially at low speeds where the low outputfrequency which is obtainable helps to eliminategearing. A cycloconverter can also be used inconjunction with a wound-rotor induction motor tovary the rotor frequency and thereby provide speedvariation about the nominal speed of the motor.

The most comprehensive reference on cycloconvertersis:

B.R. Pelly, Thyristor Phase-Controlled Converters andCycloconverters, Wiley-Interscience; 1971.

Other interesting references include:

L. J. Lawson, “The Practical Cycloconverter,” IEEETrans. on Industry & General Applications, vol. IGA-4,pp. 141-144; 1968.

T. M. Hamblin and T. H. Barton, “CycloconverterControl Circuits,: IEEE Trans. on Industry Applications.vol. IA-8, pp. 443-453; 1972.

*A novel cycloconverter which operates at 3 times the source frequency isdescribed by J. E. Jenkins in “A Frequency Tripling Cycloconverter,” IEEEIndustry Applications 7th Annual Meeting Record, pp. 81-83; Oct.1972.

16. Bilateral Solid-State Switches (Figure 3.18)

The converter circuits which have been consideredutilize unilateral solid-state switches. However, thereare many AC applications where bilateral conduction isrequired. A bilateral or AC switch can be implementedin a variety of ways. The most obvious way is to use apair of unilateral devices connected in parallel withopposite polarity as was done in the general switchingmatrix. In fact, this approach has already been used inthe dual converter and cycloconverter circuit, asexamination of the circuit will reveal. Each thyristor inthe positive converter is paralleled by one of theopposite polarity in the negative converter.

Several other ways of implementing single-phase andthree-phase AC switches are also shown below. Insome cases it is feasible to replace some of thethyristors by diodes which are less expensive, and donot require a firing circuit.

17. Triac (Figure 3.19)

The two antiparallel thyristors may also be integratedinto a single device structure, commonly known as atriac. Triacs are widely used in AC switches formoderate power levels because they tend to cost lessthan a pair of thyristors, and also eliminate the needfor the second firing circuit. However, in mostapplications, one half of the device must blockforward current immediately after the other half hasceased conduction, a very difficult requirement.Because of the interaction of the two halves, it has

3-19

not been possible to build triacs with voltage,current, or frequency ratings as high as those readilyobtainable in thyristors. Therefore, high power ACswitches must be implemented with thyristors.

References on triacs include:

General Electric SCR Manual (Fifth Edition), Section 7,General Electric Semiconductor Products Dept.,Syracuse, N. Y.: 1972.

F. E. Gentry, et al, “Bidirectional Triode PNPNSwitches,” Proc. IEEE, vol. 53, pp. 355-369; April 1965.

3-20

Gate

N

N

N

I

V

On-State #1

On-State #2

Off-State

P

P

Figure 3.19Triac Junction Geometry, Symbol, and V-I Characteristic

Three-Phase AC Switches

Single-Phase AC Switches

Figure 3.18Bilateral Solid-State Switch Configurations

18. AC Switches and Regulators (Figure 3.20)

The deceptive simplicity of AC switching andregulation makes it easy to underestimate itsusefulness and overlook potentially valuableapplications. In fact, solid-state AC switches offermany advantages over electromechanical switches.For example:

a. The elimination of moving parts and arcing leadsto quiet operation, longer life without wear,greater reliability, and safety in hazardousenvironments (such as explosive atmospheres).

b. A solid-state switch is easy to actuate remotely bya logic-level signal, or it can be interfaced withother electronic circuits and transducers forsensing and control. Photo-isolation of the input,by means of a light-emitting diode and aphotodetector, eliminates the possibility of surgesin the power circuit finding their way back intosensitive control circuits.

c. The speed of a solid-state switch is nearlyinstantaneous compared to the period of 60 Hzpower.

A number of manufacturers offer general-purpose“solid-state relay” modules which are rated for standardAC line voltages to 480V and currents to 200A.

Obviously these solid-state switches can be used foroccasionally turning an AC load off or on. But

because of their speed and their capability for anearly infinite number of switching operations, theyhave much greater flexibility to achieve control on acycle-by-cycle basis that is impossible with anelectromechanical switch. This control permitsefficient regulation of the effective AC power tointermediate values as well as full off or full on.

Solid-state AC regulators can be designed to operatein various modes. If the initiation of conduction isphase delayed each half cycle, the regulation isequivalent to the phase-delay control of a rectifier.This mode is easy to implement and is widely used inlamp dimmers and speed controls for small tools andappliances. The fundamental power frequencyremains 60 Hz so that flicker or pulsations are not aproblem. However, the fact that turn-on does notoccur at zero voltage causes significant high-frequency interference to be generated. Appropriatefilters are therefore needed.

If the load time constant is somewhat longer than acycle of the power frequency, an alternate mode ofregulation which eliminates high-frequencyinterference can be used. In this mode, turn-onalways occurs at zero voltage. Of course, with linecommutation, turn-off always occurs at zero current.Regulation is achieved by simply omitting one ormore complete cycles of conduction as appropriate,and is therefore known as integral-cycle control. (Ifthe existence of an unbalanced DC component is nota problem, half-cycle control can be used.) This

3-21

Phase-Delay

Tota

l

Integral-Cycle

Diff

eren

tial

Figure 3.20Four Modes of Operation for an AC Regulator

mode of control is feasible, for example, with waterheaters, room heaters, ovens, and the like.

Phase control and integral-cycle control as describedabove permit “total” control of the effective outputvoltage from full line voltage to zero. Both generatetransients which can be troublesome in someapplications. These transients can be minimized by“differential” control if a reduced range of control isadequate. The minimum required voltage isobtained from a tapped transformer winding, andswitching occurs from this minimum value to fullvalue rather than from zero. Intermediate values maybe obtained either by phase delay or by integral-cycleswitching.

The voltage waveforms resulting from these four modesof AC regulation are compared above for a resistiveload. Inductive loads cause commutation to be delayedafter the zero crossing of voltage, and the design of thecontrol circuits must take this fact into account.

References:

W. Shepherd, “Steady-State Analysis of the SeriesResistance-Inductance Circuit Controlled by SiliconControlled Rectifiers,” IEEE Trans., vol. IGA-1, pp.259-265; July/Aug. 1965.

R. Thompson, “A Thyristor Alternating VoltageRegulator,” IEEE Trans., vol. IGA-4, pp. 162-166;Mar./Apr. 1968.

19. Varegulators (Figure 3.21)

The preceding power circuits have been “two-port”circuits which accept real power from a source andconvert or control it as they transmit it to a load.Recently another “one-port” class of power circuits isbeginning to appear. These circuits incorporate aninternal “load” which acts as a reservoir tocontrollably store and return energy. If the inputpower is averaged over an appropriate interval, thereis ideally no net flow of real power or watts but onlyan interchange of reactive power or vars. Hence wewill call these circuits “varegulators”.

An obvious type of varegulator is shown in Figure3.21 (a). An AC switch controls the portion of eachvoltage half-cycle which is applied to an inductor. Ifthe firing of the AC switch is phase delayed untilafter the peak of the source voltage, the conductioninterval will be less than 180o, and the effectivecurrent will be a function of the delay angle. Hence,the reactive power drawn from the source can becontinuously controlled by varying the delay angle.Unfortunately, this circuit can supply only inductiveor “lagging” vars, while most practical applicationsrequire capacitive or “leading” vars to compensatefor a lagging power factor elsewhere in the system.

Circuit (b) illustrates the use of a capacitive load togenerate leading vars. However, if the firing angle isarbitrarily delayed to control the vars, the currentthrough the AC switch occurs in huge pulses as the

3-22

ACLine

(a) (b)

(d)

Wound-RotorInduction Motor

Flywheel

AC Line60 Hz

(c)

Cyclo-Converter

0 to ±6.5 Hz

Figure 3.21Types of Varegulators

capacitor voltage equalizes with the source voltage.These pulses destroy the switching devices. Hencefiring must be synchronized to occur when theinstantaneous source voltage is equal to the capacitorvoltage. Var control is achieved by splitting thecapacitor into sections and controlling each sectionby an independent AC switch. Only enough sectionsare fired to obtain the desired vars. This circuit hasthe disadvantages of requiring numerous AC switchesand their associated firing circuits while achievingonly stepwise rather than continuous var control.

Circuit (c) combines circuits (a) and (b). Anunswitched capacitor provides the maximum leadingvars that will be required, while phase delay of theAC switch in series with the inductor subtracts acontinuously controllable number of lagging vars.

These static varegulators are beginning to replacerotary synchronous condensers for preventinglighting flicker on power systems which supply highlypulsating loads, such as industrial arc furnaces.

To appreciate the versatility of the varegulatorconcept, consider circuit (d) which was developed tomeet the needs of a rather unique application. Hugeelectrically powered draglines were to be operated ata remote mining site supplied only by a relativelysmall-capacity transmission line. Normal operation ofthe draglines involves large cyclic pulsations duringwhich the peak power can reach several times theaverage power level for up to 30 sec. During theremainder of the cycle, comparable peaks of powercan be regenerated back into the power system.Obviously the system voltage will swing widely andcannot be stabilized from the generating stationbecause of the impedance of the transmission line.Not only the draglines but all other loads on thepower system would be severely affected.

The problem has been solved by installing avaregulator near the draglines. Because of the largeamount of energy storage needed to smooth thepower pulsations (some 600 MW-seconds), mechanicalstorage in a large flywheel was chosen over electricalstorage. Rapid control of the coupling between theflywheel and the power system was achieved by meansof a wound-rotor induction machine. The rotor is fedfrom a cycloconverter whose frequency and phasingrelative to the system voltage determine themagnitude and direction of power flow.

Much of the peak power to the draglines is drawnfrom the flywheel, and regenerated power is returned

to the flywheel. The power system must supply onlynominal variations about the average power level.

References:

J. A. Mulcahy and A. Ludbrook, “A New FlickerCorrecting System for Arc Furnaces,” Jour. of Metals,pp. 63-66; April 1967.

H. Frank and B. Landstrom, “Power-Factor Correction withThyristor-Controlled Capacitors,” ASEA Jour., vol. 44 pp.180-184; 1971.

K. Thorborg, “A Three-Phase Inverter with ReactivePower Control,” IEEE Trans., vol. IA-9, pp. 473-481;July/Aug. 1973.

L. A. Kilgore and D. C. Washburn, “Energy Storage atSite Permits Use of Large Excavators on Small PowerSystems,” Westinghouse Engineer, vol. 30, pp. 162-167;Nov. 1970.

P. T. Finlayson and D. C. Washburn, “Cycloconverter-Controlled Synchronous Machines for LoadCompensation on AC Power Systems,” IEEE Intnl.Semiconductor Power Converter Conference Record,pp. 2-12-1 to 8; May 8-10, 1972.

M. S. Erlicki and A. Emanuel-Eigeles, “New Aspects ofPower Factor Improvement: Part I - Theoretical Basis;Part II - Practical Circuits,” IEEE Trans., vol. IGA-4, pp.441-455; July/Aug. 1968.

W. Shepherd and P. Zakikhani, “Power FactorCorrection in Nonsinusoidal Systems by the Use ofCapacitance,” Jour. Physics D; Applied Physics, vol. 6,pp. 1850-1861; 1973.

J.W. Motto, D. P. Orange and R. A. Otto, “The StaticVar Generator and Alternative Approaches To PowerSystem Var Compensation,” Proceedings: AmericanPower Conference, 1975.

20. Brushless Machines (Figure 3.22)

Increasing attention is being given to the use of solid-state switches, not only as drives and exciters forconventional DC and AC machines, but also asintegral parts of these machines to improve theiroverall performance in various ways. In particular,solid-state devices make it possible to eliminatemechanical commutators and/or slip rings togetherwith their well-known limitations and maintenanceproblems. Brushless excitation of large alternatorsand brushless motors are two typical examples ofbrushless machines.

The DC field excitation of today’s largest powerstation alternators may require 600V at 10,000A ormore. Traditionally this excitation power was

3-23

supplied through slip rings to the rotor fieldwindings from a DC generator with a commutator.Solid-state has now done away with both the sliprings and the commutator. The principles of abrushless excitation system are shown above. Therotors of all machines are mounted on a single shaftdriven by a turbine or other prime mover. Rotatingwindings are interconnected along the shaft, butrequire no electrical connections to the stator.

The smallest machine is a pilot alternator whichgenerates the excitation for the exciter alternator. Thepilot alternator has a permanent magnet rotor. TheAC output from its stator windings is coupled to thestator field windings of the exciter alternator througha controlled rectifier. Gate control of this rectifier isused to provide regulation of the main AC output.

The rotor of the exciter alternator generate ACwhich is rectified by diodes mounted on the shaft.The components of this rotating rectifier must bedesigned to withstand high centrifugal forces (up to8000g). The DC output supplies the rotor fieldwindings of the main alternator. The main ACoutput is obtained from the stator windings. (SeeReference 1 for further information.)

Various modifications of this basic approach utilizethyristor control on the shaft (2,3,4), or rectificationof the voltages induced in the field windingsthemselves (5).

The development of brushless motors is making itpossible to achieve the many advantages of DCmotors without the disadvantages of a commutator.Such motors are being used in applications rangingfrom miniature instrument drives to hundreds ofhorsepower. Most motors of this type resemble asynchronous motor in that the main polyphase ACwindings are on the stator and the DC field windingsor permanent magnets for small sizes are on therotor, as shown above. To avoid the loss of torque atspeeds well below synchronous speed (which ischaracteristic of AC motors), the stator windings aresupplied by a solid-state AC converter which switchesin synchronism with the actual rotor position. Thatis, the AC frequency is determined by rotor-positionsensors (photoelectric, magnetic, etc.), while thephasing, is determined by the torque required.

If the power source is DC, the AC converter is aninverter. For an AC power source, the AC converter

3-24

PilotExciter

Alternator

ExciterAlternator

DiodeRectifier

Main Alternator

ACOutput

To DC FieldDC Field

GateControl

ControlledRectifier

Regulator

Brushless Exciter for a Large Alternator

AC or DCSupply

Solid-StateAC

Converter

Variable Freq. ACTo Stator Windings

Synchronous MotorLoad

Synchronizing SignalsFrom Rotor-Position Sensors

GateControl

Torque ControlSignal

StatorWindings

Prime Mover

Rotors

Figure 3.22Brushless Machines

may be a cycloconverter, a self-commutatedfrequency changer, or a rectifier/inverter.

The DC rotor field is usually supplied through sliprings. However, the use of a solid “claw-pole” rotorallows the field windings (as well as the AC windings)to be placed on the stator. Further information iscontained in References 6, 7, and 8.

References:

1. T. L. Dillman, et al., “Brushless Excitation,” IEEE Spectrum,pp. 58-66; March 1972.

2. W. F. Wright, et al., “Brushless Thyristor ExcitationSystems,” IEEE Trans., vol. PAS-91, pp. 1848-1854;Sept./Oct. 1972.

3. E. C. Hartung, et al., “A Rotating Thyristor ExcitationSystem for Hydroelectric Generators,” IEEE Trans., vol.PAS-91, pp. 2171-2178; Sept./Oct. 1972.

4. I. R. Smith and C. D. Manning, “BrushlessSynchronous Generator with Rotating ThyristorExcitation,” Proc. IEE, vol. 120, pp. 901-902; Aug. 1973.

5. B. J. Chalmers, et al., “General Principle for BrushlessSynchronous Machines and Its Application in anInvertor-Fed Drive,” Proc. IEE, vol. 119, pp. 1641-1642;Nov. 1972.

6. J. Inagaki, et al., “Commutators Get the Brushoff,” IEEESpectrum, pp. 52-58; June 1973.

7. T. Maeno and M, Kobata, “AC Commutatorless andBrushless Motor,” IEEE Trans., vol. PAS-91, pp. 1476-1484;July/Aug. 1972.

8. N. Sato, “A Brushless DC Motor with ArmatureInduced Voltage Commutation,” IEEE Trans., vol. PAS-91, pp. 1485-1492; July/Aug. 1972.

9. T. Tsuchiya, “Basic Characteristics of Cycloconverter-Type Commutatorless Motors,” IEEE Trans., vol. IGA-6,pp. 349-356; July/Aug. 1970.

10. A. H. Hoffmann, “Brushless Synchronous Motors for LargeIndustrial Drives,” IEEE Trans., vol. IGA-5, pp. 158-162;Mar./Apr. 1969.

11. J. M. D. Murphy, Thyristor Control of AC Motors, PergamonPress, Oxford; 1973. See Chapter 8, Commutatorless DCMotors, pp. 140-149.

SUMMARY

• Most established applications utilize Type 2 Switchesand Natural Commutation: AC/DC Converters,Cycloconverters, and AC Switches and Regulators.

• Many newer applications utilize Type 3 Switches withTurn-Off Capability or Forced Commutation: DCSwitches and Regulators, Inverters, and GeneralizedFrequency Changers.

21. Summary

In summary, we have seen that naturally commutatedcircuits can perform a variety of powerconversion/control functions in many applications.The number of naturally commutated circuits in usefar exceeds the number of circuits which caninterrupt current. Part of the reason for this fact ishistorical; many present applications of solid-statepower electronics have evolved from mercury-arcconverters where the cost of forced commutation wasprohibitive. Nevertheless, there is an increasingnumber of applications which cannot be served bynaturally commutated circuits but where turn-offsolid-state switching devices is feasible. Most of theseapplications involve a DC power source and lack aconnection to an AC source which could causecommutation to occur naturally. In the next sectionwe will examine the technology and applications ofcircuits using Type 3 or turn-off switches.

22. Problems

Problem 3.1 Inductive Loading of a Controlled Rectifier

To test your understanding of the effects ofinductance and delayed conduction, consider a one-pulse controlled rectifier which is loaded by a pureinductance. The waveforms for this circuit with nophase delay (α = 0) were discussed in the coursenotes under uncontrolled rectifiers. If conduction isdelayed (α > 0), current flow will not be continuousbut will cease to 360o.

(a) If the extinction angle is δ(α < δ < 360o), sketchthe voltage and current waveforms for somearbitrary value of α and calculate δ and theaverage load current as functions of α.

(b) Repeat (a) if the circuit is modified by connectinga free-wheeling diode across the load inductance.

3-25

(c) Describe what would happen if a two-pulse rectifierwere loaded by a pure inductance. First considerthe case when α < 90o and then the case for α > 90o.

Problem 3.2 Capacitor Switching

Capacitors are frequently used to correct the powerfactor of lagging loads. An engineer proposes using asolid-state AC switch in series with such a capacitor topermit rapid compensation for changes in powerfactor. Furthermore, control of the phase delay ofthe AC switch allows continuous variation of theeffective leading KVAR’s rather than the usualstepped variation as individual capacitors areconnected or disconnected by mechanical switches.

When the engineer tries the circuit in the lab, hefinds that the AC switches constantly burn out,although the average current calculated from theKVA rating of the capacitor is well within the currentrating of the solid-state devices.

(a) Examine the operation of a phase-delayed ACswitch loaded by a capacitor and explain thecause of this engineers problem.

(b) Are solid-state devices totally unsuited forswitching capacitors or can they be used withcertain restrictions? What restrictions do youthink are necessary?

Problem 3.3 Two-Pulse Battery Charger

The two-pulse AC/DC converter shown here is usedas a battery charger.

(a) Calculate and plot the charging current Id as afunction of the phase delay angle α. First assumethat L is zero, and then assume L is large enoughthat the ripple current is negligible. Does Linfluence the average value of Id? The waveformof Id?

(b) If the battery is inadvertently connected into thecircuit with the opposite polarity to that shown,calculate and plot Id as a function of α.

(c) Discuss what would happen in the circuit if Rwere negligibly small.

Problem 3.4 three-Phase Bridge AC/DC Converter

(a) The results of a complete analysis of a three-phase bridge AC/DC converter are shown in thetext for α = 0. Repeat the analysis, showing thewaveforms for VXN, VYN, IAX, IAY, IA, VXY, Vd, VAX,and VYA, and showing the conduction logicdiagram for α = 45o. What is the displacementangle, φ? What is the pulse number?

(b) Assuming that there is a “load” capable ofmaintaining continuous current, repeat (a) for α = 90o.

(c) Repeat (b) for α = 135o?

(d) Assume that the three thyristors whose anodesare tied to Y are replaced by diodes so the circuitbecomes a semiconverter. Repeat the analysis for α= 90o, and compare the results with those of (b).Have the pulse number and displacement anglechanged?

(e) Assume that the arc is supplied through atransformer having its secondaries connecteddelta and its primaries connected wye (withneutral floating). Determine the currentwaveforms in the AC lines for α = 45o, andcompare them with (a).

(f) Repeat (c) if there is a commutation overlapangle of 15o. Show where dv/dt firing might beexpected to occur.

Problem 3.5 Motor Drive Dynamics

Note: This problem assumes previous knowledge of thetheory of simple feedback systems.

A six-pulse bridge dual converter draws power from a60-Hz, 240V RMS line-to-line three-phase AC supplyand drives the armature of a 240V DC motor rated100 hp at 1750 rpm. Constant field excitation isprovided by another power supply. The moment ofinertia of the armature and load is J = 25 Nms2, andthe armature circuit constants are Rα = 0.07 ohms

3-26

VRMS = 111V

VB = 50V

R = 5V

+

–

IdL

and Lα = 5mH. Assume that the load can berepresented by constant B, and that the friction andwindage losses are 10 hp at 1750 rpm. The peakvalue of the cosinusoidal reference signal into thezero-crossing detector is 5V. The current-limitingloop is set to operate at ± 400A. (The footnote*contains reference information that may be useful.)

(a) Assume kt = 0 (i.e., no feedback) and that themotor is operating at 1750 rpm with full load.What voltage Vω is needed in steady state? Find αand the AC source power factor. Calculate Rmand Cm (the electrical analogs of the mechanicalcircuit), and compare the electrical (Lα/Rα) andmechanical (J/B) time constants. Determine thenatural frequency ωo, damping factor δ, andspeed regulation η of the motor.

*1 hp = 745.7W

Sinusoidal coefficients of a Fourier seriesexpansion are given by:

For a quadratic characteristic equation:

Speed regulation = η = (no-load speed – full-loadspeed)/(full-load speed).

(b) Assume kt = 0.15Vs/rad and re-evaluate thosefactors in (a) that are affected.

(c) The motor is operating in steady-state as in (b)when Vω is suddenly changed to -25V. Determine indetail how the system responds. In particular, plot Idand ωα as functions of time. (You are sure to runinto trouble if your analysis isn’t guided by physicalreasoning.)

(d) If the motor drive contained a 2-quadrantconverter instead of a dual converter, explainhow an armature reversing switch could be usedto duplicate the results of (c).

(e) Would your explanation in (d) have to bechanged to apply to a semiconverter? If so, how?

Problem 3.6 Operation of a Three-Pulse Cycloconverter

The waveforms for a three-pulse cycloconverteroperating with a 5:1 frequency reduction ratio areshown in the course notes. (a) Draw the circuit forthis converter, and also show it in the form of aswitching matrix. (b) The filtered sinusoidal loadcurrent waveform is reproduced three times belowwith the times indicated at which switching occurs.Show how the load current is divided between thethree-phase of the AC supply by sketching the three-phase-current waveform. (c) Compare thephase-current waveforms with the correspondingphase voltages and discuss how it could beestablished that the AC source sees a laggingdisplacement factor. (d) Use the switching matrixconcept to determine how many thyristors arerequired to implement a 3-φ to 3-φ cycloconverterwithout neutral connection.

3-27

b f t sin ktdtκ

π

π= ∫1

0

2( )

A s A s A

A A

A A A

22

1 0

0 0 2

1 2 0

0

2

+ + =

=

=

,

,ω

δ

Load CurrentPhase A

Phase B

Phase C

Load Current Waveforms

Problem 3.7 Qualitative Analysis of an AC Regulator

The circuit above shows a triac used in a simplephase-delay AC regulator intended for resistive loads.The purpose of this program is to qualitativelyexplore some of the effects of inductive loads andparasitic parameters. Seek to understand in detailwhat happens in the circuit with the goal of settingup a solution procedure, but an actual analyticalsolution is not required.

(a) Idealized waveforms of triac voltage and loadvoltage and current are shown for a resistive loadand α = 90o. How would these waveforms change ifthe load had a lagging power factor of 70%? Wouldload power vary as (cos α)2? What aspect of themodified waveform might cause the triac to fail tocommutate at all?

(b) Assume that the triac turns off when the currentbecomes less than a small (but non-zero) holdingcurrent, and that there is a small (but non-zero)capacitance across the triac. How would theseparasitic parameters further modify the waveformsduring the turn-off and turn-on transients? Wouldthe previous commutation problem be more orless severe?

(c) What corrective measures might be designed intothe circuit to assure proper commutation?

3-28

ParasiticCapacitance

LoadTriac

Load Voltage & Current

Triac Voltage

Waveformsfor Resistive

Load

4-1

Chapter 4TURN-OFF DEVICES AND

SELF-COMMUTATEDCIRCUITS

Section1. Power Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-22. Gate Controlled Switch . . . . . . . . . . . . . . . . . . . . . . .4-33. DC Switching and Regulation . . . . . . . . . . . . . . . . . .4-34. Chopper Using Thyristors . . . . . . . . . . . . . . . . . . . . . .4-45. Fundamentals of Inverters . . . . . . . . . . . . . . . . . . . . .4-56. Full-Bridge Inverter . . . . . . . . . . . . . . . . . . . . . . . . . .4-77. Half-Bridge Inverter with an Inductive Load . . . . . . . .4-78. Thyristor Inverters with Forced Commutation . . . . . . . .4-89. Types of Forced Commutation . . . . . . . . . . . . . . . . . . .4-9

10. Series Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1011. Parallel Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1112. Complementary Impulse – Commutated Inverter . . . . .4-1213. Inverter Output Voltage Control . . . . . . . . . . . . . . . .4-1214. Inverter Output Waveform Improvement . . . . . . . . . .4-1415. Frequency and Power Factor Changers . . . . . . . . . . . .4-1516. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-16

IntroductionThe objectives of this chapter are:

a. To describe and compare the three ways ofimplementing turn-off switches: transistors, gatecontrolled switches, and force-commutated thyristors.

b. To explain the basic ways in which a chargedcapacitor can be used to turn a thyristor off andthe factors which determine the size of capacitorrequired in each case.

c. To analyze the operation of a chopper indecreasing or increasing a DC voltage.

d. To describe and compare the basic circuitconfigurations for transistor and thyristorinverters, and discuss ways of controlling outputvoltage and waveform.

e. To discuss the functions that can be performedby a force-commutated frequency changer.

1. Power Transistor (Figure 4.1)

In contrast to a thyristor or Type 2 switch which canbe turned on by a short gate pulse and “latches” intoconduction until current is commutated externally, apower transistor is a non-latching Type 3 switch.Operated as a switch, a transistor must be held in itson-state by a continuous control signal in the basecircuit. When this signal is removed, the switchautomatically reverts to its off-state.

The junction structure, circuit symbol, and V-Icharacteristic of an NPN transistor are shown above.In the forward direction (collector positive withrespect to the emitter for an NPN transistor), thelimiting values of the V-I characteristic are similar tothose of a thyristor, with the cut-off and saturationregions corresponding to the off and on statesrespectively. Two important differences should benoted, however.

First, since the transistor is not self-latching, the transition between the two limitingstates can be stopped at any intermediate value ofcollector current. If the base current is inadequate tocarry the collector into saturation for a specified loadline, the operating point remains in a region of highdissipation, and the temperature increase maydamage the device.

Second, after the transistor reaches its saturationstate, it can easily drop out of saturation (with thesame destructive effects) if the collector current

increases above βIB or if the base current decreasesbelow IC/β, where β is the common-emitter currentgain. Therefore, the current overload capacity of atransistor is very limited.

Also contrary to a thyristor, a power transistor is notdesigned to operate in the third quadrant of its V-Icharacteristics. Normally, if the transistor is exposedto reverse currents, a shunting diode betweencollector and emitter is used to bypass these currents.

At present, transistors are usually the mosteconomical Type 3 switches for relatively low powerlevels, and because of the slower response ofthyristors, transistors must be used for operatingfrequencies above about 10kHz. However, transistorsare not available with power capacities greater thanabout 10kVA.

The following references may be consulted forfurther information on power transistor design,ratings, and circuit applications:

Solid-State Power Circuits, (Designer’s Handbook SP-52),RCA Solid State Div., Somerville, N.J.; 1971.

Westinghouse Silicon Power Transistor Handbook,Westinghouse Semiconductor Div., Youngwood, Pa.;1967.

Power Transistors, (Technical Series PM-81), RCA SolidState Div., Somerville, N.J.; 1971.

4-2

N

P

N

Collector

Base

Emitter

IB2

IB3 > IB2 > IB1

IB1

VCE

IB = 0

IC

Load Line

Saturation or On-State Voltage

Cut Off or Off-StateLeakage Current

Figure 4.1NPN Transistor Junction Geometry, Symbol, and V-I Characteristic (A Non-Latching Type 3 Switch)

2. Gate Controlled Switch (Figure 4.2)

A conventional thyristor switches from its forwardconducting state to its forward blocking state onlywhen its anode current is reduced by external meansto a value less than its required holding current.Since the holding current is normally a very weakfunction of gate current, the gate cannot interruptappreciable anode current. However, thyristors canbe specially designed to increase the dependence ofthe holding current on gate current. If the gatecurrent can cause the holding current to exceed theload current which is flowing, then the device iscapable of gate controlled turn-off. This type ofdevice is known as a turn-off thyristor or a gatecontrolled switch. The ratio of load current whichcan be interrupted to the gate current needed istypically 3 to 10.

Although the gate controlled switch offers attractivepossibilities for the future, it is presently commerciallyavailable only in current ratings up to about 10A.

References:

J. M. Goldey, et al., “Turn-Off Gain in PNPN Triodes,”Solid-State Electronics, vol. 3, pp. 119-122; 1961.

T. C. New, et al., “High-Power Gate Controlled Switch,”IEEE Trans., vol. ED-17, pp. 706-710; Sept. 1970.

E. D. Wolley, “Gate Turn-Off in PNPN Devices,” IEEETrans., vol. ED-13, pp. 590-597; July 1966.

H. F. Storm, “Introduction to Turn-Off Silicon-Controlled Rectifiers,” IEEE Trans. Comm. & Elect., pp.375-383; July 1963.

E. D. Wolley, et al., “Characteristics of a 200-Amp GateTurn-Off Thyristor,” IEEE Industry Applications Soc.8th Annual Meeting Record, pp. 251-257; Oct. 1973.

P. S. Raderecht, “The Development of a Gate AssistedTurn-Off Thyristor for Use in High FrequencyApplications,” Intnl. Jour. Electronics, vol. 36, pp. 399-416; 1974.

3. DC Switching and Regulation (Figure 4.3)

Since a Type 3 switch can both initiate and interruptconduction, it can act as a DC switch if connected inseries with a DC source and load.

The same circuit, with appropriate repetitiveswitching control signals, can also be extended tochange or regulate the average value of a DC source.The average value is determined by the relative on-time or duty cycle of the “chopped” outputwaveform. The duty cycle can be adjusted by varyingthe on-time, the off-time, or both, and all threemodes of control are used in practical choppers.

Step-down choppers, in which the average outputvoltage is less than the input voltage, are mostcommonly used. A circuit of this type is shown in theexample, including an inductance to smooth theoutput waveform and a freewheeling diode toprovide a path for the load current while the Type 3switch is off. Although the average voltage isdecreased by this circuit, the peak source current isequal to the load current, so the average sourcecurrent is less than the load current. Therefore thischopper steps up the average current.

With some rearrangement, the circuit can increasethe average voltage and decrease the averagecurrent. As shown in the second circuit, L is placedin series with the source and the Type 3 is moved to ashunt position. When the switch is closed, L storesvolt-seconds. When the switch opens, the inductivekickback voltage adds to the source voltage, causingthe diode to conduct, and raising the load voltageabove the source voltage. L performs voltageaveraging, while C performs current averaging. BothL and C are assumed large, yielding waveforms withperfect smoothing. (In fact, except for the lack of anAC source, all the assumptions made in rectifieranalysis have also been made here.)

From this description, it is seen that choppers arethe functional equivalent of a step-down or a step-upDC transformer, with the additional advantage thatthe effective turns ratio can easily be changedelectronically by changing the duty cycle of theswitching control signals. Choppers are widely usedas speed controls in DC-powered vehicles such asindustrial fork lifts, golf carts, electric cars, etc. Thesize of the smoothing inductors and capacitors isreduced if the frequency of operation of the chopperis increased. But then switching losses increase, sothe frequency chosen is a compromise.

4-3

The Gate-Controlled Switch–

A Thyristor with Turn-OffCapability.

(A Latching Type 3 Switch)

Figure 4.2The Gate Controlled Switch

References:

E. Reimers, “Design Analysis of Multiphase DC ChopperMotor Drive,” IEEE Trans., vol. IA-8 pp. 136-144;Mar./Apr. 1972.

D. Y. Chen, et al., “Computer-Aided Design andGraphics Applied to the Study of Inductor-Energy-Storage DC-to-DC Electronic Power Converters,” IEEETrans., vol. AES-9, pp. 585-597; July 1973.

G. W. Wester and R. D. Middlebrook, “Low-FrequencyCharacterization of Switched DC-DC Converters,” IEEE Trans., vol. AES-9, pp. 376-385; May 1973.

R. V. Langmuir, “Repetitively Switched Circuits,” IEEETrans., vol. AES-9, pp. 59-64; Jan. 1973.

P. W. Franklin, “Theory of the DC Motor Controlled byPower Pulses: Part I-Motor Operation; Part II-BrakingMethods, Commutation, and Additional Losses,” IEEETrans., vol. PAS-91, pp. 249-262; 1972.

4. Chopper Using Thyristors (Figure 4.4)

The choppers which were just considered usedpower transistors as Type 3 switches. For larger powerratings, a transistor can be replaced by a thyristor

with an associated circuit for forced commutation, asshown above. Assume initially that thyristor T1 is inits blocking state and that the circuit in the dashedbox is absent.

At time t1 a firing pulse is applied to the gate of T1.By t2, following a brief turn-on transient, the voltageVout has reached its steady-state value equal to thesource voltage, VB. Since T1 cannot interrupt current,conduction will continue forever. The circuit does nothave an AC source to cause natural commutation andto restore control to the gate of T1.

Now insert the circuit shown in the dashed box.Capacitor C will charge through RC to voltage VBwith the polarity indicated. When it is desired to turnT1 off, say at time t3, a firing pulse is applied to thegate of auxiliary thyristor T2, thereby connecting Cacross T1. The capacitor voltage adds to the batteryvoltage, increasing Vout to 2VB. But more importantly,it reverse biases T1 and diverts the load current. As Crecharges, the polarity on T1 again becomes positiveat t4. If the interval from t3 to t4 equals or exceedsthe turn-off time of T1 and if dv/dt is not excessive,then T1 will be able to block forward voltage andgate control is reestablished. By t5, C has charged to

4-4

VB

+

–Vout Vd

Iin

R

On Off On Off On Off

L

Step-Down ChopperId

Vout Vd

Iin

RC

L

Step-Up ChopperIout Id

VB

Id

Iin

Iinave =

Vout

Id = Constant

Id

Vd =

Idton

ton + tofftontoff

VBton

ton + toff

OnOff On Off On Off

VB

VB

Iout

Vout

Iin

Vd = (1+ )VB

VB

+

–

Figure 4.3DC Regulators or Choppers

VB with polarity opposite to that shown, and the loadcurrent begins to flow in the holding current of T2and T2 will turn off. The thyristor circuit has nowreturned to its fully off state and will remain off untilanother firing pulse is applied to T1.

The circuit in the dashed box is one of manypossible circuits for the forced commutation ofregular thyristors to give them Type 3 turn-offcapability. Other types of circuits for forcedcommutation will be discussed in later sections. Thecritical component in these circuits is thecommutation capacitor which must store enoughenergy to support load current for some minimumturn-off and must prevent excessive dv/dt beingapplied to the main thyristor. Note that because ofthe freewheeling diode, the energy stored in the DCinductance does not influence the design of thecommutating circuit.

The minimum turn-off time (t4 - t3) is an importantrating of the thyristor and is somewhat longer thanthe recovery time when the device is not immediatelyexpected to block forward voltage. Because of theinterdependence of this turn-off time and thepreceding forward current, junction temperature,dv/dt, etc., it must be determined in a tester which iscapable of dynamically simulating the conditionsencountered in the actual circuit under consideration.

References:

B. D. Bedford and R. G. Hoft, Principals of InverterCircuits (Chapter 10), John Wiley, New York; 1964.

R. M. Davis, Power Diode and Thyristor Circuits(Chapter 7), Cambridge Univ. Press, London; 1971.

K. Heumann, “Pulse Control of DC and AC Motors bySilicon-Controlled Rectifiers,” IEEE Trans. Comm. &Elect., vol. 83, pt. 2, pp. 390-399; July 1964

R. Cushman, “Powerful SCR’s, Connected in Parallel,Control Industry’s Biggest Machinery,” Electronics, pp.110-120; Oct. 4, 1965.

I. I. Ponner, “Design Criteria for DC Switch withResonant Circuit Connected Between Thyristors,” IEEETrans., vol. IECI-17, pp. 394-395; Aug. 1970.

B. H. Smith, “A simple Bilateral Variable-Ratio DC PulseConverter,” IEEE Trans., vol. IECI-15, pp. 1-11; Nov.1968.

5. Fundamentals of Inverters (Figure 4.5)

A chopper converts DC power to DC power andtherefore the output voltage maintains a constantpolarity. An inverter, on the other hand, supplies ACpower and must be able to cyclically reverse the

4-5

IdL

RC

R

T1

T2

C Vout

VT1

VB

+

–

–

+

Circuit for Forced Commutation of T1

VB

VB

VT1

-VB

Vout2VB

t1 t2 t3 t4 t5

dvdt

IdC=

Figure 4.4Chopper Using a Thyristor with Forced Commutation

output polarity. Three basic single-phase circuitconfigurations are used to achieve reversal.:

(a) A full bridge is equivalent to a DPDT reversingswitch which allows the full source to beconnected to the full load with either polarity.Four Type 3 switches are needed to implementthe DPDT switch.

(b) A center tap on the source allows the loadpolarity to be reversed when the load is switchedfrom one half of the source to the other. Onlyhalf of the source is used at a time, but a halfbridge (or SPDT switch) containing only twoType 3 switches is adequate.

(c) A center tap on the load (which is usuallyachieved by a center tapped transformerwinding) requires only two Type 3 switches andalso allows the full source voltage to be used.

If the load on any of these inverter circuits isresistive, the output voltage and current waveformsare square waves. Under these idealized conditions,the Type 3 switches must each block forward voltageduring the half cycle when they are turned off, butthey are never exposed to reverse voltage or current.

However, an inductive load is more typical. In thiscase, the inductance stores appreciable energyduring each half cycle. If the Type 3 switches turn off

rapidly, thereby interrupting the load current beforeit can reverse and be picked up by thecomplementary switches which are turning on, ahuge reverse inductive voltage is induced acrossthese switches and may destroy them. In a rectifier ora chopper, which have a DC output, this reversevoltage could be avoided by shunting the output by afreewheeling diode. But with an AC output from aninverter, the “reverse” voltage is of opposite polarityon alternate half cycles. Two freewheeling diodes ofopposite polarity would simply short the output.

The solution is to transfer the freewheeling diodesfrom the AC output circuit to the DC circuit wherethe reverse transients are always of the same polarity.These diodes are in fact connected either directly oreffectively in parallel with the Type 3 switches. Theparallel combination of a Type 3 switch and a diodecan support (conduct) current in both directions butcan support (block) voltage in only one direction.Note that the switch required by an inverter is thedual switch required by an AC/DC converter whichcan support (block) voltage in both directions butcan support (conduct) current in only one direction.

Other important aspects of transistor inverters areanalyzed in the following references:

F. C. Y. Lee, et al., “Analysis of Limit Cycles in a Two-Transistor Saturable-Core Parallel Inverter,” IEEETrans., vol. AES-9, pp. 571-584; July 1973.

4-6

DPDTSPST

Full Bridge Half Bridge(or Center-Tapped Source)

Center Tap(Center-Tapped Load)

SPST

Figure 4.5Basic Inverter Circuit Configurations

F. C. Y. Lee and T. G. Wilson, “Analysis and Modeling ofa Family of Two-Transistor Parallel Inverters,” IEEETrans., vol. MAG, pp. 414-418; Sept. 1973.

F. C. Y. Lee and T. G. Wilson, “Analysis of StartingCircuits for a Class of Hard Oscillators: Two-TransistorSaturable-Core Parallel Inverters,” IEEE Trans., vol.AES-10, pp. 100-112; Jan. 1974.

6. Full-Bridge Inverter (Figure 4.6)

The full-bridge inverter circuit has an apparentdisadvantage in that it requires four switches ratherthan only two for the other two configurations.However, it avoids the need for a center-tappedsource or a transformer coupling to the load. Withthe addition of one more leg containing two moreswitches, it is also capable of supplying a three-phase AC output.

Another advantage is less obvious. The otherconfigurations connect the load to the source witheither positive or negative polarity, but they cannotshort the load to insert segments of zero voltage intothe output waveform while continuing to carryinductive load current. However, a full bridge withshunting diodes can supply load current in eitherdirection at zero voltage if all of the switches on oneside of the bridge are turned on simultaneously. Aswill be discussed subsequently, this capability is usefulin permitting regulation of the output voltage.

Shown in Figure 4.6 is a full-bridge, three-phaseinverter with shunting diodes. Also shown is anoutput voltage waveform having segments of zero voltage.

An interesting theoretical technique using “Park vectors” for the analysis of three-phaseinverters is discussed in:

R. K. Jardan, et al., “General Analysis of Three-PhaseInverters,” IEEE Trans., vol. IGA-5, pp. 672-679;Nov./Dec. 1969.

R. K. Jardan, “Modes of Operation of Three-PhaseInverters,” IEEE Trans., vol. IGA-5, pp. 680-685;Nov./Dec. 1969.

7. Half-Bridge Inverter with an Inductive Load(Figure 4.7)

To be certain that the operation of an inverter withan inductive load is clear, let us examine the half-bridge circuit shown in more detail. Similarconclusions are also valid for the other circuit configurations.

Immediately prior to time t1, Type 3 switch T1(shown as a transistor) has been conducting and theload current Iout has reached a value +IP. At t1, basecurrent is removed from T1 to turn it off, and basecurrent is supplied to T2 to turn it on. However,current in the inductance cannot change instantly; itmust be a continuous function of time. Since Iout att1 is positive, it cannot flow through T2 but transfersfrom T1 to the parallel diode D2 instead. But sinceD2 is connected to the other half of the DC source,Vout reverses polarity immediately.

D2 continues to conduct until t2 when the currentreverses. During the interval from t1 to t2, the energystored in the load inductance is being discharged,

4-7

OutputShorted

Figure 4.6Full-Bridge Three-Phase Inverter with Shunting Diodes

partly into the load resistance and partly back intothe DC source. This flow of power back to the DCsource represents the return of the reactivecomponent of power.

At t2, Iout becomes negative and transfers from D2 toT2. Iout increases in the negative direction(asymptotically approaching VB/R) until t3 when T2is turned off and T1 is turned on. This completes the

half cycle, and the next half cycle is the same exceptthat T1 and T2 interchange roles.

The waveform of current in the DC source is thesame as that in the load, except that the relativepolarity reverses each half cycle, as shown. When thesource current waveform is multiplied by theconstant source voltage, the result is the sourcepower waveform. Positive power represents the flowof real and reactive power from the source to theload, and negative power represents the return ofthe reactive component of power.

Note that the switching control signals applied to T1and T2 determine the frequency but not the effectivevalue of Vout, which is just opposite to the effect ofthe control signals in an AC/DC converter.

Note: The Commutation Circuit in a Thyristor Inverter IsOrdinarily Not Separable from the Rest of the Circuit.

8. Thyristor Inverters with Forced Commutation

In principle, a thyristor and its associated circuit forforced commutation can be substituted directly intothe previous inverter circuits in place of eachtransistor. In practice, a number of difficulties arisewhen this is done, including the following:

(a) The shunting diodes prevent the application ofan appreciable reverse voltage to achieveminimum turn-off time.

(b) Application of a brief firing pulse at t1 isineffective because the thyristor is not forwardbiased until after t2, and the delay between t1and t2 is load dependent.

(c) Inefficient charging of the commutatingcapacitor can significantly reduce the overallefficiency of the inverter.

Therefore, in most practical inverters, the means forforced commutation become an integral part of thecircuit rather than a subcircuit which can be isolatedand studied separately. Because of this intimateinvolvement between the commutation circuit andthe operation of the inverter, we will next distinguishbetween various types of forced commutation.

References:

A. J. Humphrey, “Inverter Commutation Circuits,” IEEETrans., vol. IGA-4, pp. 104-110; Jan./Feb. 1968.

4-8

IB1

Iout

DIoutdtVout

VL = L

VR = RIout

VB

VB

IB2

+

–

+

–

D1

T1

T2

D2

T1 OnT2 Off

T1 OnT2 Off

T2 OnT1 Off

Real Power Plus Reactive Power

Return of Reactive Power

D2T1 T1

T2T2

Iout

IB1 IB2 IB1

D1 D1

Vout

t1t2

t3

+VB

-VB

+IP

-IP

+IP

-IP

VR

VL

Figure 4.7Circuit and Waveforms for a Half-Bridge Inverter with an Inductive Load

S. P. Jackson, “Three-Phase Static Inverters – A Survey,”IEEE Industry & General Applications 3rd AnnualMeeting Record, pp. 797-810; Oct. 1968.

A. Verhoef, “Basic Forced-Commutated Inverters andTheir Characteristics,” IEEE Trans., vol. IA-9, pp. 601-606; Sept./Oct. 1973.

S. B. Dewan and D. L. Duff, “Practical Considerations inthe Design of Commutation Circuits for Choppers andInverters,” IEEE Industry & General Applications 4thAnnual Meeting Record, pp. 469-475; Oct. 1969.

N. W. Mapham, “The Classification of SCR InverterCircuits,” IEEE Intnl. Convention Record, pt. 4, pp. 99-105; 1964.

9. Types of Forced Commutation (Table 4.1)

Circuits using a capacitor to force-commutate athyristor may be classified on the basis of

(a) the energy storage requirement on thecommutating capacitor, and

(b) the way in which turn-off occurs.

Because of the variety of intermediate and specialcases, the boundaries of the classes are not distinctand clear-cut. Compounding the difficulty is the lackof standard terminology: the literature freely usesdifferent names to mean the same thing and thesame names to mean different things. Nevertheless,an attempt at classification is believed to beworthwhile because of the emphasis which it placeson those aspects which are of fundamentalimportance. But you should be cautioned that theliterature is not universally consistent with theterminology adopted here.

The size of the commutating capacity is determinedby:

(a) the current which flows through it during thecommutation interval,

(b) the duration of that interval, and

(c) the change in voltage which occurs.

If the duration of the commutation interval is heldto the minimum which will assure turn-off of thethyristor, the size of the required capacitor is alsominimized, and “Impulse commutation” occurs. If thereis inductive energy stored in the load, the inductivecurrent must be routed elsewhere by bypass diodes.In this type of commutation, the capacitor sizedepends on the turn-off time of the thyristor buttends to be independent of the load phase angle.

Otherwise, the commutating capacitor must becapable of absorbing the inductive energy, andcommutation is said to be “load dependent”. In mostcases, the mode of commutation requires a muchlarger capacitor and is more difficult to design toaccommodate wide load variations. However, if theoutput waveform is important, the commutatingcapacitor may replace an equivalent amount ofcapacitance in the output filter.

Some circuits for forced commutation, such as theexample discussed in connection with the chopper,utilize an “auxiliary” thyristor to turn off the mainthyristor which carries load current. This auxiliarythyristor is not located in the main load circuit, and isprovided with some means of self-extinction.

4-9

Load Dependent

Means

Auxiliary

AuxiliaryImpulseChopper

Complementary

ParallelInverter

McMurray-BedfordInverter

Inherent

SeriesInverter(Commutating Capacitor Absorbs

Energy from Load Inductance)

ImpulseType

(Dependent on Device RecoveryTime; Uses Bypass Diodes)

Table 4-1Classes of Forced Commutation

In another class of forced commutation, the thyristorswhich carry load current occur in “complementary”groups, and the circuit is so arranged that the presentlyconducting thyristor is automatically commutated offwhen (but not until) the next load carrying thyristor isfired. This class is also known as auto or selfcommutation. Although this type of circuit avoids theneed for auxiliary commutating thyristors, it has thedisadvantage that external means must be provided toturn the equipment off. If control signals are simplyremoved from the thyristor gates, thyristors which werethen conducting will continue to conduct.

With both auxiliary and complementary commutation,turn-off of a conducting thyristor is initiated by firinganother thyristor. We will lump the remaining circuitsinto the category of “inherent” commutation, althoughthis is not a term which is used in the literature. Loadcurrent in these circuits is made self-extinguishing, as forinstance by means of a series capacitor.

Each feasible combination of these types of forcedcommutation leads to a corresponding family ofthyristor inverters. We have discussed impulsecommutation using an auxiliary thyristor in connectionwith the thyristor chopper, and the same technique canalso be used in an inverter. *We will now examine the

series, parallel, and McMurray-Bedford inverterconfigurations as typical examples of the countlesspossible inverter circuits using the other types of forcedcommutation.

References:

*W. McMurray, “SCR Inverter Commutated by an AuxiliaryImpulse,” IEEE Trans. Comm. & Elect., vol. 83, pt. I, pp.824-829; Nov. 1964.

J. B. Klaassens, “Analysis of a Forced Commutation Circuitfor Design of a Class of Thyristor Inverters,” IEEE Trans.,vol. IECI-20, pp. 125-129; Aug. 1973.

S. B. Dewan and D. L. Duff, “Optimum Design of an Input-Commutated Inverter for AC Motor Control,” IEEE Trans.,vol. IGA-5, pp. 699-705; Nov./Dec. 1969.

10. Series Inverter (Figure 4.8)

The first inverter circuit shown illustrates one form ofload-dependent, inherent commutation used in a half-bridge (i.e., center tapped source) configuration. Thecapacitor in series with the load causes inherentcommutation to occur, and also forces the average (notthe effective or RMS) load current to be zero. Becauseof this fact, the load current Iout and voltage Vout are

4-10

+VB

–VB

VB

+

–

T1

C

T2 Vout

IoutVC

12

Vout

T1 On

T1 & T2 Off

T2 On

L = 0

VC

VR

VL

VR

VB

12VB

+

–

+

– R

T1

CT2 L

R

L

Vout

Iout

VC

+VB

–VB

Vout

L <

VC

VR

VC

VL

R2C4

+VB

–VB

Vout

L >

VC

VR

VC

VL

R2C4

Figure 4.8Series Inverter and Waveforms for the Lower Circuit for Various Values of L

independent of the location of the return lead from theload. It can be returned to either end of the DC sourceinstead of to the center tap. Therefore the secondconfiguration is more commonly used, and is knownsimply as a series inverter.

First consider the operation of this inverter with aresistive load (i.e., L = O). When T1 is fired, Vout jumpsdiscontinuously to +VB and an initial current VB/Rbegins to flow in the load. As the capacitor charges, thecurrent decreases exponentially. Eventually the capacitorvoltage will approach VB with the polarity indicated, thecurrent will decrease to the holding current, and T1 willturn off. Note that turn-off is “inherent” in the behaviorof this circuit; a turn-off signal is not required as it iswith an auxiliary or complementary type of turn-offcircuit.

The circuit will now remain inactive, with a voltage ofnearly +VB appearing across the capacitor, until T2 isfired. This connects C across R, causing VR to jumpdiscontinuously to -VB, and causing C to dischargeinto R. Again, when the current decreases to theholding current, T2 turns off, and C is left withnearly zero voltage across its terminals. The entirecycle is ready to repeat when T1 is fired again.

Note that the frequency of the circuit is limited bythe RC time constant which determines how rapidlythe current decreases. If, in attempt to increase thefrequency, one thyristor is fired before the previousone turns off, both thyristors will conduct and form ashort circuit across the DC source.

Consider next the case where the load circuitcontains inductance, but it is small enough that thecircuit is overdamped (i.e., L < R2 C/4). The currentIout, and therefore voltage VR can no longer changediscontinuously, but the voltage waveform Vout ischanged relatively little, although it now has zero initial slope.

The final case to be considered is when L is largeenough that the circuit is underdamped oscillatory(i.e., L > R2 C/4). The output voltage (across L and R) now goes negative before the currentdecreases to the holding current and turn-off occurs.Consequently, the capacitor is charged to a voltage inexcess of VB. In contrast to the exponential currentwaveforms when L was small, the current waveformnow closely approximates half sine waves. It is clearthat in this resonant mode the commutatingcapacitor must be capable of absorbing most of theenergy stored by the load inductance each half cycle

(bypass diodes are not needed), and that theoperation of the circuit is highly load dependent.

To summarize the characteristics of series inverters,they have a fixed pulse width which is determined bythe time constant of the load and the commutatingcapacitor, and are limited to operation at relativelylow frequencies. However, they have the advantagesof being self-starting when firing pulses are appliedand self-stopping when they are removed.

For an analysis of the series inverter, see chapter 5 ofB. D. Bedford and R. G. Hoft, Principles of InverterCircuits, John Wiley, New York; 1964.

11. Parallel Inverter (Figure 4.9)

The inverter shown here uses a center-tapped load,and the commutation capacitor is effectively inparallel with the load. Hence, this circuit iscommonly called a parallel inverter. The inductor,LD, in the DC circuits is required to prevent excessivespikes of current from being drawn by C whenswitching occurs.

The commutating capacitor effectively resonates withthe load, thereby absorbing the energy stored by LLeach half cycle. As in the series inverter, operation ofthe circuit is highly load dependent, although thiscircuit uses complementary commutation rather thaninherent commutation. Current does not automaticallydecrease below the holding current to achieve turn-off.However, when the next load-carrying thyristor is fired,the commutation capacitor is connected so that itsvoltage reverse biases the thyristor which had beenconducting, causing it to turn-off.

4-11

Ld

VB

T2

LL

RL

T1

C+

–

Figure 4.9Parallel Inverter

Complementary commutation permits the parallelinverter to be operated with variable pulse widthsand at somewhat higher frequencies than thecorresponding series inverter. However, the circuit isnot self-starting. Special arrangements must be madeto establish an appropriate initial charge on C so thatthe first thyristor can be turned off after the first halfcycle. The circuit also cannot be turned off by simplyremoving the firing pulses.

Detailed steady-state analysis of the parallel inverterand the load dependence of its waveforms involvessolving a third-order differential equation andmatching initial and final conditions on a half cycle.Therefore general analytical solutions are notfeasible, but the waveforms can be generated byanalog or digital simulation. These are presented in:

B. D. Bedford and R. G. Hoft, Principles of InverterCircuits (Chapter 4), John Wiley, New York; 1964.

and in:

C. F. Wagner, “Parallel Inverter with Resistive Load,”AIEE Trans., vol. 54, pp. 1227-1235; Nov. 1935.

C. F. Wagner, “Parallel Inverter with Inductive Load,”AIEE Trans., vol. 55, pp. 970-980; Sept. 1936.

Another inverter, known as a current-source inverter, isrelated to the parallel inverter in that there isinductance in series with the DC source and thecommutating capacitor parallels the load. The current-source inverter is described in the following references:

B. R. Pelly, “Latest Developments in Static High-Frequency Power Sources for Induction Heating,” IEEETrans., vol. IECI-17, pp. 297-312; June 1970.

K. P. Phillips, “Current-Source Converter for AC MotorDrives,” IEEE Trans., vol. IA-8, pp. 679-683; Nov./Dec.1972.

W. Farrer and J. D. Miskin, “Quasi-Sine-Wave FullyRegenerative Inverter,” Proc. IEE, vol. 120, pp. 969-976;Sept. 1973.

R. B. Maag, “Characteristics and Application of CurrentSource/Slip Regulated AC Induction Motor Drives,”IEEE Industry & General Applications 6th AnnualMeeting Record, pp. 411-416; Oct. 1971.

G. R. Slemon, et al., “Synchronous Motor Drive withCurrent-Source Inverter,” IEEE Trans., vol. IA-10, pp.412-416; May/June 1974.

12. Complementary Impulse-Commutated Inverter(Figure 4.10)

Impulse-commutated inverters require thecommutating capacitor to reverse bias the thyristorwhich is being turned off only for a minimum time.Therefore the capacitor can be of minimum size. Itdoes not have to absorb the energy stored in the loadinductance, and therefore circuit operation is alsomore nearly independent of load variations.

Since the commutating capacitor does not absorbthe inductive load energy, provision must be made toroute this energy elsewhere. Bypass diodes directlyacross the thyristors form a freewheeling path whichserves this purpose, but because of the long timeconstant of the low impedance path, the energy doesnot usually have a chance to diminish very far in theavailable time of a half cycle. The energy becomes“trapped” in a circulating current which hinderssubsequent commutations. The circuits shown above,using tapped windings and slightly modified diodeconnections, couple this energy more quickly andefficiently back to the DC source, thereby avoidingunnecessary dissipation. The first circuit above wasderived from the parallel inverter which has a centertapped load winding. The second circuit isequivalent, but was derived from the half bridge orcenter-tapped source configuration. These circuits,together with similar complementary impulse-commutated circuits having other minormodifications, and known as “McMurray-Bedfordinverters” after their inverters.

References:

B. D. Bedford and R. G. Hoft, Principles of InverterCircuits (Chapter 7 by W. McMurray), John Wiley, NewYork; 1964.

W. McMurray and D. P. Shattuck, “A Silicon ControlledRectifier Inverter with Improved Commutation,” AIEETrans., vol. 80, pt. 1, pp. 531-542; Nov. 1961.

13. Inverter Output Voltage Control (Figure 4.11)

Inverters are used primarily for variable-speed ACmotor drives, for induction heating, and for reliablesources of AC power to critical loads such ascomputers (uninterruptible power supplies). Theymay be supplied either from a battery or other DCsource, or from rectified AC.

4-12

Two practical problem areas which have not beendiscussed in connection with inverters are:

(a) control of the output, and

(b) achieving a sinusoidal output voltage waveform.

Simple “phase control” of the firing signals changesthe frequency but does not appreciably affect theoutput voltage of an inverter as it did in an AC/DCconverter. One way of controlling the AC outputvoltage is to control the DC input voltage. If the DCoriginates from rectified AC, phase control of the

rectifier will serve this purpose. If the DC source isfixed, such as a battery, a chopper can be used aheadof the inverter. Although control of the DC inputvoltage is a useful technique where only a limitedcontrol range of the AC output is desired, widevariations lead to problems in achieving satisfactorycommutation. An alternate approach is to use a fullbridge configuration and insert controllablesegments of zero voltage into the output waveform.The effective value of the waveform will depend onthe relative length of these zero segments.

4-13

Ld

LL

Ld

VB12

VB12

C

+

–

+

–RL

LL

RL

Center Tap

Half Bridge

VB

+

C–

Figure 4.10McMurray-Bedford Inverters

FundamentalComponent

DC Source Control Zero-Voltage Segment Control

FundamentalComponent

Figure 4.11Ways of Controlling the Effective Output Voltage of an Inverter

14. Inverter Output Waveform Improvement(Figure 4.12)

Improvement of the basic square-wave voltage outputof an inverter may be sought in three different ways.The first, brute-force way is to filter the output so asto adequately attenuate the harmonic frequencies,leaving only the sinusoidal fundamental frequency.This approach is used where the requirements onharmonic output are not too severe, but the filtersoon becomes both large and expensive for tighterrequirements.

The other two ways seek to minimize the size of thefilter by modifying the output waveform so as tocancel the lower order harmonics which otherwiseare both largest in amplitude and most difficult toattenuate. One approach, known as pulse widthmodulation (PWM), involves repetitive switching of asingle inverter during each half cycle. The other,known as harmonic neutralization, involves anumber of series inverters staggered in phase buteach switching at the fundamental rate.

One form of PWM inverter derives its firing controlsignals by comparing a sinusoidal reference signal atthe desired frequency with a sawtooth carrier signalof a considerably higher frequency. The firing signalsare generated at the instants when the other twosignals are equal. With this type of inverter, spuriousfrequencies in the output waveform cluster aroundthe carrier frequency and its harmonics rather thanat harmonics of the desired power frequency.

In a harmonic neutralized inverter, N conventionalinverter stages which are staggered in phase by180o/N are connected in parallel across the DC

source but in series across the AC output terminals.The amplitude of each stage is adjusted so as tocancel all harmonics less than (2N-1) in the 2N-stepped approximation to a sine wave. For thewaveform shown, N=6 and the inverter stages arestaggered by 180o/6=30o. The 90o stage is degenerate(i.e., has zero amplitude) so that only five stages arerequired. The lowest order harmonic appearing inthe output is the eleventh.

References:

B. D. Bedford and R. G. Hoft, Principles of InverterCircuits, (Chapters 8 and 9), John Wiley, New York;1964.

R. R. Ott, “A Filter for Silicon-Controlled RectifierCommutation and Harmonic Attenuation in High-Power Inverters,” IEEE Trans. Comm. & Elect., vol. 82,pp. 259-262; May 1963.

B. Mokrytzki, “Pulse Width Modulated Inverters for ACMotor Drives,” IEEE Trans., vol. IGA-3, pp.493-503;Nov./Dec. 1967.

P. D. Corey, “Methods for Optimizing the Waveform ofStepped-Wave Static Inverters,” AIEE Paper 62-1147;1962. Reprinted in Power Semiconductor Applications, vol.I, pp. 321-328; IEEE Press, New York; 1972.

A. Kernick, et al., “Static Inverter with Neutralization ofHarmonics,” AIEE Trans., vol. 81, pt. II, pp. 59-68; May1962.

T. M. Heinrich and A. Kernick, “Controlled CurrentFeedback in a Static Inverter with Neutralization ofHarmonics,” IEEE Trans. Aerospace, vol. 2, pp. 985-992;April 1964.

J. W. Bates, “Sequence Amplitude Modulated Inverters,”IEEE Power Electronics Spec. Conf. Rec., pp. 222-229;June 1973.

4-14

FundamentalComponent

Harmonic Neutralization Pulse Width Modulation

FundamentalComponent

Figure 4.12Ways of Eliminating Low-Order Spurious Frequencies to Improve the Output Voltage Waveform of an Inverter

T. M. Corry, “A New Concept for Generating Three-Phase Sine Wave Voltages with Semiconductor PowerSwitches,” IEEE Power Electronics Spec. Conf. Rec., pp.230-236; June 1973.

S. B. Dewan and J. B. Forsythe, “Harmonic Analysis of aSynchronized Pulse-Width-Modulated Three-PulseInverter,” IEEE Trans., vol. IA-10, pp. 117-122; Jan./Feb.1974.

H. S. Patel and R. G. Hoft, “Generalized Techniques ofHarmonic Elimination and Voltage Control in ThyristorInverters: Part I - Harmonic Elimination,” IEEE Trans.,vol. IA-9, pp. 310-317; May/June 1973.

L. J. Penkowski and K. E. Pruzinsky, “Fundamentals of aPulsewidth Modulated Power Circuit,” IEEE Trans., vol.IA-8, pp. 584-592; Sept./Oct. 1972.

J. J. Pollack, “Advanced Pulsewidth Modulated InverterTechniques,” IEEE Trans., vol. IA-8, pp. 145-154;Mar./Apr. 1972.

R. D. Adams and R. S. Fox, “Several ModulationTechniques for a Pulsewidth Modulated Inverter,” IEEETrans., vol. IA-8, pp. 636-643; Sept./Oct. 1972.

15. Frequency and Power Factor Changers(Figure 4.13)

Frequency changers are basically dual converterswhich allow periodic modulation of the switchingcontrol signals rather than simple phase delay.Because of its dependence on natural commutation,the cycloconverter is a special case which is limited tooutput frequencies which are less than about 1/3 ofthe source frequency and to a lagging inputdisplacement factor. Converters with selfcommutation have neither of these restrictions.

If the switches can interrupt current, it is obviousthat the source waveform(s) can be chopped intobrief segments to synthesize output frequencieswhich are higher than the source frequency. The factthat independent control can be exercised over thedisplacement factor (and thereby the power factor)is not as obvious. The control logic which determineswhen each input is connected to the output and forhow long is not uniquely specified by the outputfrequency. In the cycloconverter, the control logic isconstrained by the requirements of naturalcommutation. With self commutation it can bechosen to implement other features. For example, itcan be chosen to minimize the ripple or deviationfrom the desired output waveform. Or by toleratingmore ripple, the converter can control thedisplacement angle seen by the source.

The mathematical theory of these generalizedconverters has only recently been explored and isbeyond the scope of this text. However, theiroperation can be explained qualitatively in terms of asingle-phase two-pulse example. Imagine a dualconverter operating as a naturally commutatedcycloconverter. The output waveforms shown aboveoccur when α = 90o and the average output is zero.The waveform on the left occurs if the output currentis positive, and the one on the right if the outputcurrent is negative. Natural commutation must occurto a phase which is more positive when the current ispositive, and to a phase which is more negative whenthe current is negative. The desired output frequencyis generated by periodically modulating α about 90°at this frequency, thereby modulating the averagevalue of each switching cycle about zero. Inherent in

4-15

Natural Commutation(Lagging Displacement Factor)

Forced Commutation(Leading Displacement Factor)

Figure 4.13The Two Complementary Waveforms of a Two-Pulse Frequency Changer Without Phase Modulation

this naturally commutated mode of operation is alagging displacement factor at the AC input.

If the dual converter is self commutated, it can bemade to commutate at any time. It can, for instance,generate the “naturally commutated” waveforms justdescribed. It can also generate waveforms which areexactly complementary to these. That is, it cancommutate to a phase which is more negative whenthe current is positive, and to a phase which is morepositive when the current is negative. In thiscomplementary mode, it would generate anequivalent output except that the displacementfactor at the input would change sign and beleading.

Since the converter can generate equivalent outputswith either a lagging or leading input displacementfactor depending on which mode is selected, it canalso generate controllable intermediate displacementfactors by interleaving the two modes. The details ofthe interleaving process are determined by thecontrol logic which characterizes the various types offrequency changers. It can, for instance, maintainthe displacement factor constant at unity for varyingload power factors. With different control logic, theconverter can be made to change the sign of theload displacement factor. This converter is called anUnrestricted Frequency Changer (UFC), and causesan inductive load to appear capacitive to the source,or a capacitive load to appear inductive.

Admittedly this explanation of frequency and powerfactor changers is brief. Hopefully it has beensuggestive of the very general functional capabilitiesof self-commutated power converters whichincorporate the tremendous computational versatilityof today’s small-signal electronic circuits. The fullimplications of being able to reliably and repeatedlyswitch megawatts in microseconds are not widelyappreciated, even among electrical engineers. Yet thetheoretical and practical possibilities offer a wealth ofchallenge to the imaginative designer.

16. Problems

Problem 4.1 Operation of a Thyristor Chopper

The circuit shown above is used as a DC regulator orchopper. Assume that the reactive components, Land C, are so large that IL and VC (= VR) areconstant over a cycle.

(a) If transistor T1 is on for 25% of each cycle,calculate and plot the waveforms for VL and iC.Also calculate the load voltage, current, andpower. Show that the average power drawn fromthe battery is equal to the load power.

(b) Repeat (a) for 75% duty cycle on T1.

(c) What can this chopper circuit do which the twostep-down and step-up chopper circuits shown inthe course notes cannot do?

Problem 4.2 Impulse Commutation of a Thyristor Chopper

The purpose of this problem is to investigate some ofthe basic relationships and constraints involved inthe design of a thyristor chopper using auxiliaryimpulse commutation, the circuit for which is shownin the course notes. Assume that the chopper is tooperate at a fixed frequency of 400 Hz and that thenominal duty cycle is to be variable from 20% to100%. For proper turn-off, the main thyristor T1must be reverse biased for at least 50 µsec. The filterinductor is 0.2 H, and it may be assumed initially thatId is constant over each cycle. The DC source voltageis 1000V, and the load resistor is 10 ohms.

(a) Calculate the minimum value of C which isrequired to give satisfactory commutation overthe reduced range of duty cycle. What is theminimum dv/dt rating for thyristor T1? Sketchthe waveform for the voltage Vout for the limitingvalues of the duty cycle.

4-16

VB = 90V

+

–+

–

R = 3V

T1 D1

CL

IL = Constant

VL VR =Constantic

(b) Calculate the maximum energy stored in L and Cand compare the two values. Note that impulsecommutation allows the ratio of these twoenergies to be quite high.

(c) Check the assumption that Id is constant over acycle by calculating the worst-case peak-to-peakripple as a percentage of Id.

(d) Investigate the influence of the value of thecharging resistor RC on circuit operation andattempt to specify a suitable value.

Problem 4.3 Operation of a Series Inverter

The purpose of this problem is to explore some ofthe important aspects of the operation of the seriesinverter which was discussed in the course notes. Thecircuit is as follows:

In order to produce a reasonably sinusoidal output, thistype of inverter is frequently designed to beconsiderably underdamped, and is driven with a verysmall conduction gap (assumed negligible) betweenthe extinction of one thyristor and the firing of theother one. Assuming sinusoidal waveforms and that thecircuit operates at its resonant frequency (f= 1/2 π=LC):

(a) Sketch the waveforms of voltage and current foreach component in the circuit. Show byappropriate shading those areas which must beequal in order that the average capacitor currentand inductor voltage are zero.

(b) The average power supplied by the battery mustbe equal to the average power dissipated by theload resistor. Use this fact to calculate the currentI. Then use I to calculate the voltage across eachcomponent, and appropriately label thewaveforms in (a).

(c) Calculate the peak energy which the commutatingcapacitor must be able to store. Calculate the peak

energy in L. How do these two values compare? Isthe comparison consistent with the discussion of“load-dependent” commutation?

(d) Compare the KVA ratings required for L and Cwith the KW rating of the load. What KVA rating(peak blocking voltage x average current) isrequired for thyristors T1 and T2? Are thecomponents utilized efficiently?

(e) Would you expect this circuit to have good voltageregulation for changing load resistance? Explain.

(f) Aside from the small conduction gap which hasbeen assumed to simplify the analysis, is there adanger that turning on one thyristor will causethe other one to turn on also? Explain. Discusshow the waveforms and analysis of the circuitwould change if the switching frequency weredecreased to permit a longer conduction gap.

Problem 4.4 Resonant Impulse Commutation

Shown below is a half-bridge auxiliary impulsecommutated inverter. Commutation of mainthyristors T1 and T2 is achieved by the resonantcharge reversal on C via L and auxiliary thyristors A1and A2. The initial voltage on C depends on Q andon the load, but for the purposes of this problemassume that VC (0) = 250V with the polarity shown,and that T1 is initially conducting. Assume that theload current has reached equilibrium at ILOAD = VB / RL = 100A and that the load inductanceis large enough to hold ILOAD constant during thecommutation interval.

To commutate T1 off, A1 is fired. As ILC increases,IT1 decreases by a corresponding amount. When ILCexceeds ILOAD, IT1 becomes zero and the excesscurrent flows through D1. When ILC decreases belowILOAD again, T1 has regained its blocking ability,ILOAD transfers to D2, and the polarity of the loadvoltage reverses. C recharges to -VC (0) before A1

4-17

VB = 628V

+

– R = 4V

T1 C = 40mF

T2L = 0.1H

I

VB =100V

+

–

RL = 1V ILOAD

VB =100V

+

–

ILC

T2

T1

Load

C =20mF

L =50mH

A2

A1

D2

D1

ceases conduction, and is therefore ready tocommutate T2 off one-half cycle later.

Determine the maximum rated turn-off time (tq) for T1 and T2.

Problem 4.5 Harmonic Neutralization

The amplitudes of the square-wave components of aharmonic neutralized inverter are given by:

where VRMS is the desired RMS value of thefundamental output frequency and N is the numberof inverter stages. (a) Calculate the amplitudes of thecomponents for a six-stage inverter used to synthesizea sine wave having VRMS = 120 V. (b) Plot the desiredsinusoidal output waveform, and superimpose uponit the actual stepped output waveform. (c) Sketch thedifference between these two waveforms whichconstitutes the remaining ripple. What is its peakamplitude as a percentage of VRMS? What is thefundamental frequency of the ripple relative to thedesired output?

Problem 4.6 Operation of a Frequency Changer

The Unrestricted Frequency Changer (UFC) is aparticular type of force-commutated frequencychanger which has the following desirableproperties:

1. The time between successive commutations isconstant and is given by:

where fin and fout are the input and outputfrequencies and P is the pulse number.

2. The phases always conduct in the same sequenceand for the same length of time.

(a) Using these properties of the UFC, constructthe output waveform for a three-pulse UFChaving fout/fin = 1/5.

(b) If the load draws a 30o lagging current,indicate which of the commutations occur“naturally”, and which must be “forced”.

(c) Using the same commutation frequency asbefore, construct an output waveform whichcould be achieved with a naturallycommutated cycloconverter. Compare theconduction sequence with that in (a).

(d) Repeat (a) for fout/fin = 5/1.

4-18

VV

N

n

NnRMS= π π2

cos

TP F FC

in out

=+1

( )

5-1

Section1. The Power Module . . . . . . . . . . . . . . . . . . . . . . . . . .5-22. Importance of Junction Temperature . . . . . . . . . . . . . .5-23. Components of Dissipation . . . . . . . . . . . . . . . . . . . . .5-34. Average Dissipation at 60 Hz Switching Rate . . . . . . . .5-45. Average Dissipation at 40 KHz Switching Rate . . . . . . .5-46. Simplified Geometry Assumed for the “Typical” Device . .5-47. Junction-to-Case Thermal Resistance . . . . . . . . . . . . . .5-58. Heat Sink Calculation . . . . . . . . . . . . . . . . . . . . . . .5-59. Selection of Device Package and Type of Heat Sink . . . .5-5

10. Natural Convection Air Cooling . . . . . . . . . . . . . . . . .5-611. Forced Air Cooling . . . . . . . . . . . . . . . . . . . . . . . . . .5-612. Water Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-713. Using the Data Sheet for Heat Sink Calculations . . . . . .5-714. Thermal Capacity . . . . . . . . . . . . . . . . . . . . . . . . . .5-1415. Transmission Line for Transient Thermal Analysis . . .5-1416. Initial Temperature Rise . . . . . . . . . . . . . . . . . . . . .5-1517. Transient Thermal Impedance of Each Section . . . . . .5-1518. Calculation of Thermal Time Constant . . . . . . . . . . .5-1619. Junction-to-Case Transient Thermal Impedance . . . . . .5-1620. Using Transient Thermal Impedance . . . . . . . . . . . . .5-1721. Calculation of Pulse and Surge Current Capabilities . .5-1722. Turn-On di/dt . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-1923. Transient Overvoltage . . . . . . . . . . . . . . . . . . . . . . .5-2024. Off-State dv/dt . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-2125. Series/Parallel Arrays for High-Power Applications . . .5-2226. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-2327. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-23

IntroductionThe objectives of this chapter are to provideinformation which will enable the reader:

a. To evaluate and compare the variouscomponents of dissipation within a solid-statedevice in a variety of applications.

b. To calculate the requirements and select a heatsink for a specified device dissipation.

c. To estimate the steady-state and transient thermalimpedance and the thermal time constant of asimple device geometry.

d. To calculate the rise in junction temperaturecaused by a current pulse or surge.

e. To explain the physical basis for voltage, current,dv/dt and di/dt ratings on devices and theapproaches that are used to protect solid-statedevices from stresses in excess of these ratings.

f. To discuss the maximum voltage and currentcapabilities of presently available power devicesand the precautions that are required in thedesign of series/parallel arrays of devices toextend the maximum power capabilities of solid-state equipment.

Understanding of these topics is vital in the design of solid-state power equipment if its full potentialreliability is to be achieved. It is in this area thatintuition developed from design experience intraditional power or electronic equipment is mostmisleading and most likely to cause unforeseenproblems in the design.

Chapter 5POWER SEMICONDUCTOR

DEVICE PROTECTION

1. Power Module

The Power Module Protects the Power Device from:• Junction Temperature• Voltage• Current• dv/dt• di/dt

in Excess of its Ratings.

The economic feasibility of a power converter may bedetermined either by the transformers, inductorsand capacitors or by the solid-state components. Onthe other hand, technical feasibility is nearly alwayslimited by the current state of the art of availablesolid-state devices. The function of the powermodule is to interface the solid-state power deviceand its limitations with the requirements andpeculiarities of the power circuit so as to achievetechnical feasibility, reliability, and overall minimumcost. In particular, the purpose of the power moduleis to protect the power device from junctiontemperature, voltage, current, dv/dt, and di/dt in excess of its ratings. Each of these factors isassociated with a corresponding potential failure mechanism.

Unfortunately, many factors interact in determiningwhether or not a solid-state power device will fail tooperate properly or be damaged under a particularset of conditions. In arriving at the ratings shown onthe specification sheet, a device manufacturer testshis devices under set of conditions which may or maynot be representative of a particular application.(And since the test conditions are not uniformbetween manufacturers, comparisons are not alwayseasy to make.) One manufacturer may beconservative and rate his devices under worst-caseconditions, but this puts him in an apparent pricedisadvantage compared to another manufacturerwho may for example specify typical values. Typicalvalues can not be used for design purposes becauseprocesses do change and the typical values may bealtered significantly. It is not enough to simplycompare device data sheets the actual capability ofthe device should be carefully evaluated. The powersemiconductor device selected should consider thetrade-offs of price, performance, and availability of acontinuing supply of reliable devices. Frequently onerating is of crucial concern to a designer and hewould like to know whether he can safely exceed thatrating if he stays well below other ratings, or whetherhe must buy a premium device with apparent excess

capacity on most of it’s ratings. There are noinfallible rules, but skepticism is advisable. Theneophyte designer is cautioned that the reliability ofsolid-state power devices tends to be a precipicefunction. Within their capabilities, they can lastmuch longer than the equipment they arecontrolling. But if their capabilities are exceeded,their lifetime may literally be measured inmicroseconds! All other avenues should beexhausted before trying to squeeze the last cent outof the cost of the solid-state power devices which are selected.

With these forewarnings, we will proceed to explorethe fundamentals of device protection, beginningwith thermal considerations. Device protectionprinciples will be described primarily using thyristorscharacteristics and ratings; however, most of theseprinciples apply equally well to all powersemiconductor devices.

Additional information on device rating andprocurement can be found in:

J. S. Read and R. F. Dyer, “Power Thyristor RatingPractices,” Proc. IEEE, vol. 55, pp. 1288-1301; 1967.

M. C. Sardi, “Semiconductor Yield and Life Cycles,”Electro Procurement, November 1973.

2. Importance of Junction Temperature (Table 5.1)

Nearly all of the characteristics of solid-state devicesare sensitive to the internal junction temperature.Since ratings are merely design limits oncharacteristics, they too are temperature dependent. Therefore thermal protection will be considered first.

Although the dissipation in switching devices canoften be ignored in the analysis of the power circuit(as we have done), it is a vital factor in the choice ofthe device itself and its heat sink. Since thedissipation is highly concentrated at the junction of adevice where temperature is critical, and since we donot have access to the junction to measure itstemperature directly, thermal design must beperformed analytically. Both the average junctiontemperature when the device is nearly in thermalequilibrium and the instantaneous peak junctiontemperature resulting from a transient areimportant, but each involves a different analytical approach.

5-2

In order to illustrate what is involved in thermalcalculations, we will analyze a “typical” thyristorwhich, although grossly oversimplified, neverthelessyields values which are representative of actualdevices. The assumed characteristics of this thyristorare listed at the beginning of this section. Theimportant characteristics include the forward voltagedrop, the leakage current, and the switching times,all of which have previously been neglected.

3. Components of Dissipation (Figure 5.1)

Junction temperature depends both on thedissipation or heat input to the device and on thethermal resistance encountered by the heat inflowing to a cooling medium. In normal cyclicoperation, the dissipation is not constant but variesas a function of time, depending on the state of theswitch. There are five distinct components ofdissipation corresponding to the forward blockingstate, the reverse blocking state, the conducting state,the turn-on interval, and the turn-off interval. (Power

dissipated by gating signals will be neglected here,but may be significant in some applications.) Forpresent purposes, the two blocking states areassumed to have equal dissipations, and it is assumedthat the device is loaded to its full ratings, 600V and100A, thereby controlling a 60 KW load.

In either blocking state, the device dissipates 6000V x 10 mA = 6W, while in the conducting state itdissipates 1.4V x 100A = 140W. For simplicity we willassume that these values are constant rather thanvariable with time, as they are with sinusiodalwaveforms.

During the switching transitions we cannot avoid thefact that the dissipation changes rapidly. During the 3µs turn-on interval, the voltage across the devicemust decrease from 600V to 1.4V, while the current isincreasing from 10 mA to 100A. The instantaneouspower is the instantaneous product of voltage andcurrent, and hence it depends on the dynamiccurves followed by the voltage and current. Again forsimplicity we will assume that these transistors arelinear, as shown in figure 5-1. The power waveformduring turn-on is then an inverter parabola having apeak value of 15 kW! Obviously, when subjected tothis rate of dissipation, the junction temperaturerises very rapidly. The energy dissipated in the switchduring the turn-on transient is the area under thepower parabola, (2/3) x 15 kW x 3µs = 30 mJ.

Assuming the same linear transitions during theturn-off interval, the dissipation again peaks at 15kW, although the energy increases to 200 mJ becauseof the longer turn-off time.

5-3

100AVoltage = 600V

Current = 10mA 1.4V

Turn-OffTime

= 20msReverseBlocking

State

ForwardBlocking

State

ConductingState

15 kW Peak

Turn-On Time= 3ms

Turn-On Energy= 30 mJ

Turn-Off Energy= 200 mJ

PowerDissipation

= 6W

10mA

600V

6W140W

Figure 5.1Assumes Linear Switching Transitions Showing Components of Dissipation in Switching Device

Table 5.1Assumed Characteristics of a “Typical” Thyristor

IT(RMS) Current Rating 100AVRRM, VDRM Blocking Voltage Rating 600VVT Forward Voltage Drop at 100A 1.4VIDRM, IRRM Leakage Current at 600V 10mAtON Turn-On Time 3mstOFF Turn-Off Time 20ms

4. Average Dissipation at 60 Hz Switching Rate(Table 5.2)

To calculate the average junction temperature, wemust first calculate the average dissipation underspecified conditions. For example, with equal on-and-off-times at 60 Hz, the various components ofdissipation per cycle are calculated above, and theaverage dissipation is 87W. For low switching rates,the switching times are very short compared to thetotal period, and the average dissipation isdetermined primarily by the dissipation in theconducting state.

5. Average Dissipation at 40 KHz Switching Rate(Table 5.3)

To emphasize what happens at high switching rates,let us consider an extreme case. Judging by theswitching times alone, it should be possible tocomplete a full cycle in 25µs, corresponding to aswitching rate of 40 kHz. Again the individualcomponents of dissipation and the averagedissipation are calculated above. The switchingcomponents of dissipation are now dominant, and itis seen that the average dissipation has risen to aphenomenal 9.2 kW! When the small physical size ofa solid-state device with a 100A rating is considered,it is obviously impossible to dissipate this much heatat a tolerable temperature.

Therefore the maximum frequency at whichswitching is feasible depends on switching dissipationas well as on switching speed, and the current ratingof a device must be decreased at high frequencies.Since the switching dissipation tends to be a constant

amount of energy per cycle while the conductingand blocking components tend to be constant power,the “corner frequency” above which switchingdissipation is a dominant factor can be estimatedfrom:

For our “typical” thyristor operating at 100A with a50% duty cycle, the conducting and blocking poweraveraged over a cycle is (140 + 6)/2 = 73W, and theswitching energy per cycle is 0.23 J. Thus fC ' 73/0.23 or about 300 Hz.

Note: Although this simplified analysis seems to indicatethat the dissipation during turn-off is somewhat moreimportant than during turn-on, the opposite is usually truein practice. The turn-on dissipation is concentrated nearthe periphery of the gate electrode of a thyristor, causing ahot spot whose average and peak temperatures can greatlyexceed those elsewhere. In addition, the turn-off waveformsare highly oversimplified.

6. Simplified Geometry Assumed for the “Typical” Device(Figure 5.2)

In addition to average dissipation, the other factorwhich determines average junction temperature isthe thermal resistance between the junction and thecooling medium. Part of this resistance is internal tothe packaged device as purchased, and part isdetermined by the heat sink and the flow rate of thecoolant. To facilitate calculations, we will assume the

5-4

On-Time = Off-Time = 8.3ms

Conducting Dissipation 140W x 8.3ms = 1.17JBlocking Dissipation 6W x 8.3ms = 0.05JTurn-On Dissipation 0.03JTurn-Off Dissipation 0.20JTotal Dissipation per Cycle 1.45J

Average Power Dissipation = 87W1.45J

16.7ms

Table 5-2Comparison of the Components of Dissipation for a60Hz Switching Rate with a 50% Duty Cycle

On-Time = Off-Time = 0.5 (25 – 3 – 20) = 1µs

Conducting Dissipation 140.W x 1µs = 0.140mJBlocking Dissipation 6.W x 1µs = 0.006mJTurn-On Dissipation 30. mJTurn-Off Dissipation 200. mJ

Total Dissipation per Cycle 230. mJ

Average Power Dissipation = 9.2kW230mJ

25µs

Table 5-3Comparison of the Components of Dissipation for a40Hz Switching Rate with a 50% Duty Cycle

f

Conducting Blocking PowerAveraged Over a Cycle

Switching Energy Per CycleC ≈

&

simplified device geometry shown above. Thedissipation is assumed to occur at one surface of thesilicon wafer than at the junction within the silicon,and the device package is assumed to be equivalentto a cylindrical copper slug. We would like tocalculate the heat sink which is required for aspecified average junction temperature and aspecified average dissipation. First we must calculatethe “junction-to-case” thermal resistance.

7. Junction-to-Case Thermal Resistance

Calculation of Steady-State Thermal Resistance for theSilicon Wafer and Copper Slug

The thermal resistance of the cylindrical silicon wafer isequal to the thickness (d) divided by the product of thethermal conductivity (k) and the cross-sectional area(A)- (in a consistent set of units). For the dimensionsassumed previously, this resistance is calculated aboveand is found to be 0.0135oC/W. That is, in steady state,100W of dissipation will cause a temperaturedifferential across the silicon of only 1.35oC.

Similarly the thermal resistance of the copper packageis calculated to be 0.146oC/W. The sum of these two

thermal resistances gives the “junction-to-case”thermal resistance. This value is listed by the devicemanufacturer on the data sheet, so that it need not becalculated. We have gone through the calculationsimply to show what is involved.

8. Heat Sink Calculation

Heat Sink Calculation Assuming Switch Must Be Capableof Carrying 100A Continuously (140W) in a 35°CAmbient with a 120°C Junction Temperature

Assume that the switch must be capable of carrying100A continuously, that the maximum temperatureof the cooling ambient is 35oC, and that the averagejunction temperature is to be 120oC. The total“junction-to-ambient” thermal resistance cannotexceed 0.607oC/W, as calculated above. Subtractingthe junction-to-case thermal resistance gives themaximum thermal resistance of the heat sink (andthe interface between the device and the heat sink)as 0.447oC/W.

Obviously, this calculation may be reversed to obtainthe average junction temperature for a specified heatsink and average dissipation.

9. Selection of Device Package and Type of Heat Sink

Early semiconductor rectifiers and thyristors were stud-mount devices. They were easy to mount and could beinstalled by almost any manufacturer. This packagedesign was suitable for many years, since the currentrating/device never exceeded 200 to 300A. As thedemand for higher current devices evolved, the discpackage design was developed. (Trade names for disc-type power semiconductors include “Pow-R-Disc”, “Hockey Puck” and “Press Pak”.)

Unlike the stud-mount device example we have beenconsidering, where all of the heat is dissipated throughthe stud base of the package to a single heat sink, thedisc-packaged device dissipates heat through bothupper and lower pole faces to two separate heat sinks.

5-5

R Silicond

kA

m

W m C m

C W

θ π( )

( / )( . )( / )

. /

= = ×° ×

= °

−

−2 10

84 1 5 10 4

0 0135

4

2 4 2

R Copperd

kA

m

W m C m

C W

θ π( )

( / )( . )( / )

. /

= = ×° ×

= °

−

−1 10

388 1 5 10 4

0 146

2

2 4 2

HeatSink

HeatInput

Silicon Water 1.5 cmDia., 0.02 cm Thick

Copper Slug,1.5 cm dia., 1 cm Thick

Figure 5.2Simplified geometry assumed for a silicon switchingdevice mounted on a heat sink

R TotalT T

Ave Dissipation

C

WC WJ A

θ ( ).

( ). /= − = − ° = °120 35

1400 607

R Heat Sink R Total R Silicon R Copper

C Wθ θ θ θ( ) ( ) ( ) ( )

. . . . /

= − −= − − = °0 607 0 014 0 146 0 447

This effectively cuts junction-to-case thermal impedancein half. Thus, a disc rectifier or SCR with double-sidedcooling offers higher output current ratings thanconventional stud-mount rectifiers or SCR’s ofequivalent element sizes. For example a disc unit canoffer up to 80% more current capability and cost lessthan the same size silicon element in a stud package.

The selection of the best type of heat sink for a givenapplication depends upon many factors such asvolume & weight restrictions, economics, and thecapacity required compared to available devices. Themain categories of heat sinks are natural(convection) cooling, and forced convection cooling.For natural convection, air is normally the mediumalthough oil may be used in high voltageapplications. Forced convection air cooling is mostwidely used for power semiconductors applicationsalthough there has been an increased use of watercooling in recent years.

Another technique which is being developed but isnot yet commercially available is the heat-pipe devicepackage. This package places a cooling liquid inintimate contact with the silicon wafer. As the liquidabsorbs its heat of vaporization, it becomes a gaswhich flows to another part of the package where theheat is exchanged with an outside cooling medium.The very high thermal efficiency of this type ofcooling results from the fact that the vapor transportcarries heat away with a very small temperaturegradient compared to normal conduction (i.e., diffusion) of heat.

10. Natural Convection Air Cooling (Figure 5.3)

The graph in Figure 5.3 shows the thermal resistanceof one manufacturer’s optimized line of naturalconvection cooled, finned heat sinks plotted againstvolume. For a specified thermal resistance, theclosest larger standard heat sink should be selected.The thermal resistance of 0.447oC /W resulting fromthe previous example requires a heat sink having avolume of about 150 in.3.

11. Forced Air Cooling (Figures 5.4 & 5.5)

The thermal resistance of a heat sink may bedecreased below its ratings with natural convection byusing forced air cooling. Figure 5.4 illustrates thedecrease that can be expected for several typical heat

sinks. Figure 5.5 illustrates the low heat sink thermalresistance that can be realized for disc type devicesand forced air cooling. Note that two of the specifiedheat sinks are used for double size cooling (solid line).

5-6

0.10.1

1

10

100

1 10

Volume in Cubic InchesS

tead

y-st

ate

The

rmal

Res

ista

nce

in o

C/W

att

100 1000

Figure 5-3Thermal Resistance vs. Volume for an Optimized Lineof Natural Convection Cooled, Finned Heat Sinks (at 50° Rise Above Ambient)

5" x 5" x 1/16"Aluminum Plate

7" x 7" x 1/8"Aluminum Plate

6" x 6" x 9" Finned Extrusion

4" x 4" x 5"Finned Extrusion

00

1

2

3

200

Air Velocity in ft./min.

The

rmal

Res

ista

nce

in o

C/W

400100 300 500

Figure 5-4Forced Air Cooling of Stud Type Devices

12. Water Cooling

Water cooling is the most efficient type of cooling ingeneral use today. While it may not be a practicalsolution to every design problem, more and moresystems are being converted to water cooling eachyear. Inherent advantages in size, weight and costsimply cannot be overlooked.

A typical example of a water-cooled assembly isshown at the beginning of this chapter.

13. Using the Data Sheet for Heat Sink Calculations(Figure 5.6)

Figure 5.6 the data sheet for the Powerex Type T500thyristor will be used to illustrate how the devicemanufacturer attempts to ease standard thermalcalculations. For low frequencies (400 Hz or less)where switching losses are small, and for standardcurrent waveforms such as square waves and phase-delayed sine waves, curves are given which show themaximum average power which will be dissipatedand the maximum allowable case temperature (for125oC junction temperature) versus the averagecurrent and the conduction angle. Another curvegives the forward voltage drop as a function ofcurrent.

Using the T500–80 (Figure 5.6) with a square wave ofcurrent having an amplitude of 120A and a duty cycleof 50%. (The average current is then 60A and theconduction angle is 180o.) Figure 5.6-A shows that the

voltage drop at 120A is 1.5 volts. The dissipationduring conduction is 120A x 1.5V = 180W, but sincethere is little dissipation during the other half cycle,the average dissipation is approximately 180/2 = 90W.Figure 5.6-I shows the average dissipation for thissquare wave as 94W. Under “Thermal and MechanicalCharacteristics,” we find the junction-to-case thermalresistance listed as 0.28oC/W. Therefore, the averagejunction temperature is 94W x 0.28oC/W = 26°C abovethe case temperature. For an average junctiontemperature of 125oC, the corresponding casetemperature would be 99oC. Figure 5.6-J shows themaximum allowable case temperature for 60A averageand 180o conduction as 92oC. This curve has a 7oCmargin to allow for the fact that the peak junctiontemperature will exceed the average temperature.

Figures 5.6-E and 5.6-F give dissipation andmaximum case temperature for phase-delayed half-wave sinusoidal waveforms.

Note that although this thyristor is rated 125A RMSor 80A half-wave average, its average current capacitymust be derated if the conduction angle is less than180o or if the heat sink is inadequate to maintain thecase temperature at or below 76oC.

5-7

0 500 750250100 1000 1250 15000

0.1

0.2

0.3

0.4

0.5

Air Velocity in ft/min

The

rmal

Res

ista

nce

in o

C/W

2.7" x 5" x 6"Finned AL Extrusion

2.7" x 10" x 8"Finned AL Extrusion

Single side coolingDouble side cooling

Figure 5-5Forced Air Cooling of Stud Type Devices

5-8

Figure 5-6T500 Data Sheet

5-9

Figure 5-6 (Continued)T500 Data SheetFigure 5-6 (Continued)T500 Data Sheet

5-10

Figure 5-6 (Continued)T500 Data Sheet

5-11

Figure 5-6 (Continued)T500 Data Sheet

5-12

Figure 5-6 (Continued)T500 Data Sheet

5-13

Figure 5-6 (Continued)T500 Data Sheet

14. Thermal Capacity

Calculation of Thermal Capacity for the Silicon Waferand Copper Slug

To this point, the discussion on thermalconsiderations has dealt only with average junctiontemperature and steady-state thermal resistance. Fornormal, low frequency waveforms, this type ofanalysis is adequate. But for calculating peakjunction temperatures resulting from low duty cycle(pulsed) waveforms or from transient overloads,steady-state analysis is not adequate.

We are all familiar with the fact that electricalequipment can be temporarily overloaded withoutpermanent damage, sometimes to hundreds of timesits steady-state ratings. The reason is that in a pieceof equipment such as a motor or a transformer, thereis a considerable mass of copper and iron whichmust rise in temperature before damage fromoverheating occurs. The energy of the transientoverload is absorbed by the large thermal capacity ofthe equipment without excessive temperature rise.(Of course different considerations prevail if theheating is localized, as when an arc develops.)

The same generalization holds true for a solid-statepower device except that the maximum temperature(at which control can be maintained) is more criticaland the thermal capacity is orders of magnitudesmaller (than for a motor or transformer ofcomparable power rating). Therefore an engineerwho is accustomed to traditional electricalequipment must be very cautious until he retrains hisintuition on solid-state devices. Although the excessthermal capacity of these devices is small, it must befully utilized if reliability is to be achieved withoutthe prohibitive cost of an overly conservative design.

Let us return to the simple device geometry whichwas previously defined and calculate its thermalcapacity. Thermal resistance is the temperature rise

per watt of heat flow. Thermal capacity is the heatenergy which can be stored per degree oftemperature rise. Consequently thermal capacity isequal to the product of the specific heat of thematerial (c) and its mass (density (ρ) times volume(Ad)). The thermal capacities of the silicon waferand the copper slug are calculated above. Note thattogether they can absorb only a little more than 6watt-seconds (i.e., joules) per oC of temperature rise.Remember that the peak dissipation duringswitching was 15kW. Obviously the junctiontemperature will change very rapidly under thesecircumstances.

15. Transmission Line for Transient Thermal Analysis (Figure 5-7)

Thermal resistance and thermal capacity are now“lumped” elements but are both distributedthroughout the device being considered. The deviceis, in fact, a thermal transmission line in which heatflow is governed by the diffusion equation. For thecylindrical geometry we have assumed, the one-dimensional diffusion equation applies, and there isan exact analogy with a corresponding RC electricaltransmission line. Heat power is then analogous tocurrent, and temperature difference is analogous tovoltage. The complete transmission line between thejunction and the cooling ambient is shown above. Alltemperatures (voltages) are measured relative to theambient. Since the heat is ultimately dissipated atambient temperature, the output of the transmissionline is “short circuited”. Each of the three sections ofthe line, representing the silicon wafer, the copperslug, and the heat sink, respectively, may be assumedto be uniform transmission lines.

The analytical solution for the junction temperaturefor a specified power input (as a function of time) isextremely tedious, even for this simplified geometry,and involves solving the diffusion equation for eachof the three sections and matching the boundaryconditions between sections. Fortunately the “timeconstants” of the three sections differ widely so thatthey can be analyzed separately. Both the individualanalysis and the combination of the separate resultslend themselves to a very simple graphical approach.

5-14

C Silicon pc Ad

kg m J kgC m m

J C

θ ( )

( . / )( / )( . )( )

. /

=

= × ° × ×

= × °

− −

−

2 34 10 735 1 77 10 2 10

6 1 10

3 3 4 2 4

2

C Copper pc Ad

kg m J kgC m m

J C

θ ( )

( . / )( / )( . )( )

. /

=

= × ° ×= °

− −

8 9 10 393 1 77 10 10

6 2

3 3 4 2 2

16. Initial Temperature Rise

Initial Increase in Junction Temperature as a Function ofTime for a Homogeneous Cylinder

Consider one section of a uniform RC transmissionline (thermal or electrical). Assume initially that thesection is infinite in length so that we do not need toworry about the boundary condition at the outputend. Also assume that the heat power input is zeroprior to t = 0 and constant at a value P thereafter. Thatis, the power input is assumed to be a step function.

At t = 0, the junction temperature, Tj, will begin torise above its previous value. We will not solve thediffusion equation, but it can be shown that thesolution for the junction temperature rise as afunction of time takes the simple form shown above.The left side of this equation is the temperature riseper watt of power input, and therefore has the sameunits as thermal resistance. This quality is known asthe “transient thermal impedance.”

The right side of the equation consists of a constanttimes the square root of time. Note first that theconstant depends only on the thermal conductivity,density, and specific heat of the material and its cross-sectional area. With a little algebraicmanipulation, this constant can be shown to be equalto Rθ/=π/4 RθCθ.

Next, note that temperature rises in proportion to=t, not t itself. As the heat propagates down thetransmission line, more and more material must beheated for each degree of temperature rise at thejunction. Since the power input is constant, the rateof temperature rise decreases. If this relationship isplotted on log-log graph paper, the graph will be astraight line with a slope of one half.

This solution for an infinite transmission lineaccurately represents the initial temperature rise of afinite line (that is, a finite thickness of material). Butwhen the heat penetrates to the other face of thematerial and the temperature at the face begins toincrease, the solution rapidly becomes inaccurate.Stated concisely, this is the asymptotic solution forvery small values of time.

17. Transient Thermal Impedance of Each Section(Figure 5-8)

Next consider that the section of transmission line isterminated in a short circuit, corresponding to theactual piece of material being mounted on aninfinite heat sink which can absorb any amount ofheat with no temperature increase. Steady-state willhave been reached when the input temperature risereaches DTJ = PRθ, or when the transient thermalimpedance reaches DTJ/P = Rθ. In other words, thesteady-state thermal resistance forms the asymptoteof transient thermal impedance for very long times.

The two asymptotic solutions are shown above. It isseen that they intersect at t = (π/4) RθCθ. This is theeffective time for the heat to penetrate a specifiedthickness of material, and is therefore an effectivethermal “time constant”. For values of time near thetime constant, neither asymptotic solution isaccurate, and the actual solution “rounds the corner”between them.*

The asymptotic solutions for the individual sectionsof the overall thermal transmission line are uniquelydetermined by the steady-state thermal resistanceand the time constant for each section.

*At t = (π/4) RθCθ, the actual value of DTJ/PRθ is 0.883. Note thesimilarity between this asymptotic approach to transient thermalimpedance and the well-known Bode plot of frequency response which hasan actual value of 1/2 = 0.707 at the corner frequency of a first-ordernetwork.

5-15

∆T

P A kpct R

t

R Cj = =2

4π θ π

θ θ

Rθ(Silicon)

Cθ(Silicon)

Cθ(Copper)

Cθ(HeatSink)

Rθ(Copper)

Rθ(Heat Sink)

Heat PowerInput

JunctionTemperature

(AboveAmbient)

Ambient Temperature = Reference

Figure 5-7Thermal Transmission Line for AnalyzingTemperature Rise of Junction

18. Calculation of Thermal Time Constant

Calculation of Thermal Time Constants for the SiliconWafer and Copper Slug

The thermal time constants of the silicon wafer andthe copper slug are calculated above from thepreviously calculated values of Rθ and Cθ. It is seen that the time constant of the package is more thanthree orders of magnitude larger than that of thesilicon wafer.

19. Junction-to-Case Transient Thermal Impedance(Figure 5-9)

If the asymptotic solutions for the silicon wafer andthe copper slug are each plotted on the same log-loggraph, and the curve rounded at the intersections asshown in Figure 5-9, the junction-to-case transientthermal impedance plot is the result. The asymptoteon the right is actually the junction-to-case thermalresistance, but on a logarithmic scale the differencefrom the thermal resistance of the copper alone isnot perceptible. When the packaged device ismounted onto a heat sink, the transient thermalimpedance of the heat sink is added in the same way.Since the heat sink will have an even longer timeconstant, only the right portion of the curve will be changed.

Note that when the transient thermal impedance isplotted on log-log paper, the time constants of theconstituent parts are recognizable from the bumps inthe curve. For transients which are short comparedto the time constant of the silicon wafer, the

5-16

π πθ θ4

0 0135R C (Silicon) =4

C /W) (0.061 J/C ) 0.64ms( . ° ° =

π πθ θ4

0 146 71R C (Copper) =4

C /W) (6.2 J/C ) 0 s( . .° ° =

10-5 10-4 10-3 10-2 10-1 100 10110-3

10-2

10-1

101

Time in Seconds

Tra

nsie

nt T

herm

al Im

peda

nce

∆TJ/

P in

°C

/W

4π RθCθ (Silicon) =

6.4 x 10-4 sec

4π RθCθ (Copper) =

0.71 sec

Rθ (Silicon) =0.0135 Co/W

Rθ (Copper) =0.146 Co/W

Figure 5-9Log-log Plot of the Transient Thermal Impedance of the “Packaged” Silicon Wafer (Not Including the Heat Sink)

∆TJ

log

P

Rθ

Slope = 1/2

∆TJ

t =

log t

4π

= Rθt

RθCθ

P

∆TJ = RθP

4π RθCθ

Figure 5-8Transient Junction Temperature Rise as a Function ofTime Following the Application of Constant Power Input

transient thermal impedance is determined solely bythe characteristics of the silicon, independent ofwhether the silicon is mounted on anything. Forlonger transients which are still shorter than thepackage time constant, the transient thermalimpedance is independent of the heat sink. But fortransients having a duration greater than thepackage time constant, the heat sink does play a role.

This analysis has assumed uniform distribution of theheat over the cross-sectional area of the device. Thisassumption is not valid during the turn-on transientwhen the conducting area is less than the total area.Therefore transient thermal impedance, as presentedhere, cannot be used for transients which are shorter than 10 to 100µs, depending on the size ofthe device.

In spite of the fact that the curve of transientthermal impedance is easier to plot and interpret onlog-log scales, it is the universal practice of devicemanufacturers to give this curve on log-linear scales(see Figure 5.6-B of the Type T500 data sheet).

20. Using Transient Thermal Impedance (Table 5.4)

We are now in a position to use the transient thermalimpedance curve to calculate instantaneous junctiontemperature or peak temperature for specifiedcurrent surges, or to calculate the allowable currentsurges for a specified peak junction temperature. Toillustrate, refer to Figures 5.6-A and 5.6-B on the datasheet for the Powerex T500 thyristor.

First, let us calculate the fluctuation in junctiontemperature about its average value when thecurrent is a 120A square wave with a 50% duty cycle.This corresponds to the example previouslyconsidered, where we found the dissipation duringconduction to be 180W. If the frequency of thesquare wave is 60 Hz, the duration of each half cycleis 8.3 ms. From Figure 5.6-B we see that the transientthermal impedance for t = 8.3 ms is 0.04oC/W.Therefore, the temperature rise during each halfcycle of conduction is 180W x 0.04oC/W = 7oC. Ofcourse the junction must cool by an equal amountduring the half cycle of nonconduction. But it maybe surprising to you to find that there is anappreciable temperature fluctuation of the junctioneven during normal 60 Hz operation. Since theduration is less than the time constant of the device,this fluctuation cannot be reduced by an improvedheat sink.

21. Calculation of Pulse and Surge Current Capabilities (Figure 5-10)

As another example, let us determine the rise injunction temperature when the Type T500 issubjected to a 1000A pulse lasting 1.5 ms. FromFigure 5.6-A on the data sheet, the forward voltagedrop is 3.0V, and from Figure 5.6-B the transientthermal impedance is about 0.016oC/W. Therefore:

If the device is operated with an average junctiontemperature of 77oC, this pulse will not cause thepeak junction temperature to exceed 125oC. Statedin another way, when operating with a junctiontemperature of 77oC, a Type T500 can carry a 1.5 mscurrent pulse equal to 8 times (1000/125 = 8) itsRMS current rating without losing its blocking ability.

Repetition of this calculation for a constant DTj = 50oC yields the normalized graph shownabove. It is evident that a thyristor can toleratesubstantial overload currents without hindering itsoperation if:

(a) the duration of the overload is sufficiently short,

and

(b) the junction is operating somewhat below itsrated maximum temperature.

The current capacity calculated above is a“repetitive” value for an arbitrarily selectedtemperature rise in that the thyristor can besubjected to such currents an unlimited number of

5-17

Junction Temperature Fluctuatuion at 60 Hz (Square Wave with 50% Duty Cycle Assumed)

Dissipation During Half-Cycle of Conduction P = 180W

Duration of Half-Cycle at 60 Hz 8.3 ms

Transient Thermal Impedanceat 8.3 ms Zu = 0.04oC/W

DTj = P Zu = 180W x 0.04oC/W = 7oC

∆Tj = × × ° °3 0 1000 0 016. . /V A C W = 48 C

Table 5.4Fluctuation of Junction Temperature

times without shortening its life expectancy. Currentratings such as the surge current are “non-repetitive”in the sense that a thyristor should not be exposed tomore than 100 of these overloads during its entireuseful life. Under these overloads, the junctiontemperature will exceed 125oC and the thyristor maynot immediately block its rated peak forward voltage.

The surge current rating gives the peak half-wavesinusoidal current 60 Hz which the thyristor canwithstand for the specified number of cycles. If gatingpulses are removed immediately when a fault occurs, theType T500 can withstand 1800A peak for one-half cycle(8.3 ms) until the current passes through zero. If thefault is cleared by a circuit breaker which acts moreslowly, the allowable peak fault current drops to 1300Afor 3 cycles (i.e., 3 half-cycles of conduction separated by3 half-cycles of non-conduction) or 1170A for 10 cycles.These peak values have been converted to RMS valuesand are shown for comparison on the following graph.

In many cases, the circuit in which a thyristor is usedis so “stiff” that a short-circuit fault can causecurrents much higher than these values. If thethyristor is to be protected, either additionalimpedance must be designed into the circuit to limitthe current to a safe value until a circuit breaker can

open, or a special fast-acting fuse must be placed inseries with the device. The “I2t” rating of a thyristor isprovided as a guideline in selecting an appropriatefuse which will prevent permanent damage but willnot blow unnecessarily. This is a non-repetitiverating, and is equal to the square of the RMS currentcapacity times the duration of the surge. The I2

rating may be used to calculate the maximum non-repetitive surge current for less than one-half cycle(i.e., 8.3 ms). The corresponding curve is shownabove for the Type T500 which has an I2t rating of13,495 A2s.

In most applications, the power device is selected onthe basis of its steady-state current ratings andprotected so that its surge ratings are not exceeded.However, in other applications where shutdown ofthe equipment cannot be tolerated, oversize deviceswith spare steady-state capacity must be used toachieve the required surge capacity.

References on Pulse and Surge Current Capacity include:

William E. Newell, “Dissipation In SolidState Devices - The Magic of I1 + N,” IEEE Transactions on

Industry Application, July/Aug. 1976.

5-18

0.1 1 10 100 10001

10

100

Duration in Milliseconds

I/Rat

ed I M

RS

Pulsed CurrentCapability for TJ = 50oC

I2t Non - Repetitive Surge Ratingfor Fuse Coordination

1/2 Cycle Surge Rating (Non - Repetitive)(Converted to RMS)

3 Cycles10 Cycles

Figure 5-10Current Carrying Capability (Repetitive and Non-Repetitive) for a Westinghouse Type T500 Thyristor (Rated IRMS = 125A; Rated I2t = 13,495 A2s)

John W. Motto, “A New Quality to Describe PowerSemiconductor Subcycle Current Ratings,” IEEE Transaction onIndustry Applications, July/Aug. 1971.

William E. Newell, “Transient Thermal Analysis of Solid-StatePower Devices-Making a Dreaded Process Easy,” IEEETransactions on Industry Applications, July/Aug. 1976.

John W. Motto, “Computer Aided Thyristor Overload CurrentRatings for Motor Drive and Other Applications RequiringRepetitive Overloads,” IEEE Industry and General Applications,Sept./Oct. 1969.

Additional information on thermal analysis, heat sinkselection, and current ratings may be found in:

Westinghouse SCR Designers Handbook (Second Edition),Sections 4.3 to 4.6 and 10, Westinghouse SemiconductorDivision, Youngwood, PA; 1970.

General Electric SCR Manual (Fifth Edition), Section 3.1 to 3.6,15, and 18, General Electric Semiconductor Products Dept.,Syracuse, New York; 1972.

Wakefield Heat Sink Product Bulletin; Wakefield Eng. Inc.,Wakefield, Mass; 1966.

E. T. Schonhoizer, “Fuse Protection for Power Thyristors,” IEEETrans. on Industry Applications, vol. IA-8, pp. 301-309; 1972.

W. E. Newell, “Dissipation in Solid-State Devices - The Magic ofI1+N,” IEEE Power Electronics Specialists Conference Record, pp. 162-173; June 1974.

Westinghouse Thyristors Surge Suppression Ratings AD 54-550;Oct. 1976.

W. Savage, “How to Get the Most Out of High-Power Rectifiersand SCR’s,” EDN; August 1974.

The effects of maximum temperature and temperature cyclingon the life expectancy of power devices are discussed in:

D. E. Piccone and I. Somos, “Are You Confused by High di/dtSCR Ratings?,” Electronic Engineer, pp. 89-92; Jan. 1969.

D. E. Piccone and I. Somos, “Accelerated Life Tests for Determining the Life Expectancy of a Thyristor Due to di/dt Failure Modes,”IEEE Industry Applications 7th Annual

Meeting Record, pp. 469-476; Oct. 1972.

R. E. Locher, “Large Diameter Rectifier Diode and Thyristor In-Service Reliability,” IEEE Industry Applications 9th Annual Meeting Record, pp.463-465; Oct. 1974.

M. Morozowich, “Are Semiconductors Reliable?,” ElectronicProducts Magazine; July 1976.

22. Turn-On di/dt (Figures 5-11 & 12)

Turn-On di/dt• Is Determined by the Lateral Rate of Spreading

of the Turned-On Area,• Can be limited by increasing Circuit Inductance,• Capability is Increased by Special Device

Geometries and by “Hard” Gate Drive.

The preceding discussion has emphasized theintimate relationship between the thermalcharacteristics of a thyristor and its steady-state andtransient current-carrying ability. The correspondingdata sheet ratings are a concise statement of thebasic information needed to choose a heat sink andto coordinate fuses and circuit breakers with thyristorcapabilities. We will now examine some of the otherfactors which influence the design of the powermodule to protect the power device.

When the transient characteristics of thyristors werediscussed earlier in the text, it was stated that thegate turns on only a small part of the total thyristorarea. The time required for the conducting area topropagate laterally, typically 0.1mm per microsecond,across the full device necessitates a correspondinglimitation on the rate of increase of load current atturn on. This limitation is expressed as a maximumdi/dt rating. Exceeding this rating causes excessivecurrent density at turn-on, and the resulting hot spotwill degrade or destroy the thyristor.

In some applications, high di/dt occurs at turn-on butis not essential. For this case, the solution is usually todesign additional inductance (either linear orsaturating) into the circuit to limit di/dt to a safe value.

The di/dt capability of any thyristor is decreased ifminimum gate drive is used and increased if the gateis driven “hard”, thereby increasing the initiallyconducting area. Most thyristors today incorporatean integrated amplifier termed an amplifying ordi/namic gate design which includes a small pilotthyristor in the gate circuit which generates a verystrong pulse to turn on the main thyristor rapidly.This is illustrated in Figure 5.11.

In other applications, high di/dt is essential and mustbe accommodated by the power devices. For thisreason, fast turn-on devices with increased di/dtratings are available and development is continuing.Originally, the gate contact on thyristors was located atone edge, and conduction had to propagate acrossthe entire diameter. Later turn-on time was reduced

5-19

by moving the gate contact to the center, cutting themaximum distance for propagation in half. Still newerthyristor geometries use interdigitated gate electrodesto initiate conduction over a large percentage of thetotal area. This is illustrated in Figure 5.12.

References on di/dt and gating considerations forfast turn-on include:

Westinghouse SCR Gate Turn-on Characteristics, A.D. 54-540, pp. 1-4;December 1976.

General Electric SCR Manual (Fifth Edition), Section 3.7 and 5.5,General Electric Semiconductor Products Dept., Syracuse, N.Y.; 1972.

H. F. Storm and J. G. St. Clair, “An Involute Gate-EmitterConfiguration for Thyristors,” IEEE Trans., vol. ED-21, pp. 520-522; Aug. 1974.

N. Mapham, “The Rating of Silicon Controlled Rectifiers whenSwitching into High Currents,” IEEE Trans. on Communicationsand Electronics, vol. 83, pp. 515-519; September 1964.

D. A. Paice and P. Wood, “Nonlinear Reactors as ProtectiveElements for Thyristor Circuits,” IEEE Trans. on Magnetics, vol.MAG-3, pp. 228-232; September 1967.

S. Ikeda, et al., “The Current Pulse Ratings of Thyristors,” IEEETrans. on Electron Devices, vol. ED-17, pp. 690-693; September1970.

23. Transient Overvoltage

Transient Overvoltage• Is Usually Caused by Inductive Kickback.

Damage Can Be Prevented By• Circuit Redesign,• Devices Having Adequate Voltage Ratings,• RC Suppressor Circuits,• Nonlinear Suppressor Devices,• Solid-State “Crowbar” Which Rapidly Shorts

the Voltage Transient.

Unanticipated voltage transients which exceed therated blocking voltages of solid-state power devicesare probably the most frequent single cause ofunreliability in these devices. Depending on theseverity of the overvoltage, the energy which itrepresents, and its frequency of repetition, the devicemay fail immediately or its characteristics mayprogressively deteriorate. Because voltage breakdowntends to occur at the surface of the device ratherthan within the silicon, the energy required to causepermanent damage may be relatively small.

Except for random disturbances such as lightning,most voltage transients can be traced to an inductor,either within or external to the equipment, in whichcurrent has been initiated or interrupted abruptly.Obvious cases include removing the load on arectifier having a large smoothing inductor in theDC circuit, for example, by blowing a fuse. Caseswhich are much more subtle involve hiddeninductance such as the leakage reactance of atransformer, or unintentional current such as therapid cessation of reverse sweepout current in adiode or thyristor (which occurs each cycle). Othercauses of transient overvoltage are the capacitivecoupling from a high voltage circuit to a low voltagecircuit and the energizing of a RLC circuit where thecapacitor will charge to two times the peak linevoltage.

Protection against destructive overvoltage transientscan be achieved in four general ways:

(a) Redesign the circuit operation or physical locationto remove or minimize the cause of the transient.

(b) Suppress the transient by absorbing its energy inan appropriately designed RC circuit locatedacross the source of the transient.

(c) Shunt the power devices by nonlinear resistiveelements which clip the voltage transient at a safe

5-20

Anode

Cathode

Gate

Figure 5-11Equivalent Circuit of an Integrated Gate Amplifier

Edge-FiredGate

Center-FiredGate

DynamicGate

DynamicSno-Flake

GateFigure 5-12Various Gate Cathode Geometrics

level. Selenium transient suppressors (known bythe tradenames Voltrap, Thyrector, ect.) andzener diodes have been used for this purpose,and ceramic suppressors (known as ZincNonlinear Resistors, ZNR’s and Metal OxideVaristors MOV’s) have recently become available.However currently available ceramic suppressorsdevices are relatively low in energy absorbingcapability compared to selenium transientsupressors.

d. Occasional severe transients can sometimes bestbe limited by a solid-state “crowbar” circuit whichshorts the line and absorbs the transient energy,at least until an auxiliary circuit breaker can open.

References on overvoltage protection of devices include:

Westinghouse SCR Designers Handbook (Second Edition), Sections9.3 to 9.5, Westinghouse Semiconductor Div., Youngwood, Pa.; 1970.

General Electric SCR Manual (Fifth Edition), Section 16,General Electric Semiconductor Products Dept., Syracuse, N.Y.;1972.

F. D. Martzoff, “Surge Voltages in Residential and IndustrialPower Circuits,” IEEE Trans. on Power Apparatus and Systems,vol. PAS-89, pp. 1049-1055; July/Aug. 1970.

T. Liao and T. H. Lee, “Surge Suppressors for the Protection ofSolid-State Devices,” IEEE Transactions on Industry and GeneralApplications, vol. IGA-2, pp. 44-52; Jan./Feb. 1966.

D. C. Kay, et al., Transient Voltage Suppression Manual GeneralElectric Semiconductor Products Dept., Syracuse, N.Y.; 1976.

Voltrap Surge Suppressor Westinghouse TD-16435 Westinghouse Electric Corporation, Buffalo, N.Y.14240.

24. Off-State dv/dt

Off-State dv/dt Capability• Is Determined by Capacitive Current Into

Gate Layer,• Can Be Limited to a Safe Value by an

RC Snubber,• Is Increased by Shorted Emitter Design.

If a thyristor is in its forward blocking state, it is thereverse-biased center junction which prevents the flow ofappreciable current. As we have seen, gate currentwhich is greater than some critical minimum valuecauses regenerative action which turns the device onand thereafter prevents this junction from blockingcurrent.

However, this center junction is shunted by capacitance.If the device is subjected to a rapidly changing voltage(anode increasingly positive with respect to thecathode), this capacitance causes a current

to flow across the blocking junction into the gatelayer. This current is equivalent to externallysupplied gate current, and will turn the device on ifit exceeds the critical value. Thus, if the device is toremain in its blocking state, it cannot be subjected todv/dt in excess of a critical value.

The dv/dt seen by a device can be reduced byconnecting a capacitor across the device. Used alone,this capacitor will cause excessive di/dt when thedevice turns on and shorts the capacitor. Therefore aresistor is needed in series with the capacitor to limitthe current transient at turn-on. The RC dv/dtlimiter is known as a “snubber”, and suitable valuesfor R and C are frequently recommended by thethyristor manufacturer. The values chosen are acompromise between dv/dt reduction, tolerabledi/dt, and minimum added dissipation.

Some applications, such as high-frequency inverters,have an inherently high dv/dt. The cost of snubberscan be eliminated in other applications if the deviceshave adequate dv/dt capability. Therefore there arecontinuing efforts to develop devices having evenhigher dv/dt ratings. One noteworthy approach tothe design of such devices involves the use of a“shorted emitter”. In this type of device, the cathodeelectrode partially shorts the gate P-layer to thecathode N-layer, providing an alternate path for thedv/dt current. Although this design modificationincreases the gate drive needed to turn the deviceon, it permits substantial increases in dv/dt ratings.

References on dv/dt effects, snubber design, andshorted emitter devices include:

Westinghouse SCR Designers Handbook (Second Edition),Sections 4.7, 5.3.1 and 7.2.1, Westinghouse Semiconductor Div.,Youngwood, Pa.; 1970.

General Electric SCR Manual (Fifth Edition), Sections 1.5, 3.10,5.4, and 16.3.2, General Electric Semiconductor Products Dept.,Syracuse, N.Y.; 1972.

International Rectifier SCR Application Handbook, P. G. Hoff etal., International Rectifier Corp., Semiconductor Div., ElSegundo, Ca.; 1974.

5-21

i Cdv

dtj=

J. B. Rice and L. E. Nickles, “Commutation dv/dt Effects inThyristor Three-Phase Bridge Converters,” IEEE Trans. onIndustry and General Applications, vol. IGA-4, pp. 665-672;Nov./Dec. 1968.

J. B. Rice, “Design of Snubber Circuits for Thyristor Converters,”Conference Record of the 4th Annual Meeting of the IEEE Industryand General Applications Group, pp. 485-489; Oct. 1969.

W. McMurray, “Optimum Snubbers for Power Semiconductors,”IEEE Trans., vol. IA-8, pp. 593-600; Sept./Oct. 1972.

C. K. Chu, “Geometry of Thyristor Cathode Shunts,” IEEE Trans.on Electron Devices, vol. ED-17, pp. 687-690; Sept. 1970.

P. S. Raderacht, “A Review of the ‘Shorted-Emitter’ Principle asApplied to PNPN Silicon Controlled Rectifiers,” Intnl. Jour. ofElectronics, vol. 31, pp. 541-564; 1971.

25. Series/Parallel Arrays for High-Power Applications (Figures 5-13)

The maximum power which can be controlled by asingle solid-state power device is determined by itsrated blocking voltage and by its rated forwardcurrent. To maximize either of these ratings requiressome compromise of the other rating. Also laboratorydevices can always be cited with capabilitiesconsiderably beyond those of devices which arecommercially available at any particular time.

Present limits on diode ratings are about 5 kV and3000A individually, with a maximum product ofabout 5 MW. Similar limiting ratings for thyristorsare about 4 kV, 2500A, and 4 MW.

Applications requiring voltages which are higherthan the capabilities of individual devices can beserved by series strings of devices, and highercurrents can be achieved by devices connected inparallel. For example, electrochemical process linesmay require 100,000A or more, necessitating manyparallel devices. High-voltage DC converters requirehigh voltage and high current, leading toseries/parallel arrays of devices.

If all of the characteristics of solid-state devices of agiven type were closely matched, series/parallelarrays of devices could be routinely assembled. Insome cases, screening of devices to select matchedsets is an economical alternative. However, in mostcases the spread of characteristics of productiondevices necessitates an equalizing network to assurethat no individual device in the array is overstressed.

Because of the rapid change in on-state current withforward voltage, parallel devices tend to sharecurrent unequally. The best device with the lowestforward voltage monopolizes the load current,becomes overheated, and is destroyed. Similarly,when parallel thyristors are first turned on, thedevice with the shortest turn-on time is subjected to avery high di/dt which may destroy it.

Analogous problems occur in series strings ofdevices. Because of the wide variation in off-stateleakage currents between production devices of atype, those devices having the lowest leakage currentsare subjected to excessive blocking voltages when thedevices are in equalibrium. Also, mismatches injunction capacitances and stored charge causeunequal distribution of transient voltage stresses.

The design of series/parallel equalization networksto achieve maximum overall power rating, reliability,minimum dissipation, and minimum cost is beyondthis text. Suffice it to say that parallel devices can beforced to share current by a resistor in series witheach device or by equalizing reactors (which aremore costly but also dissipate less power). Seriesdevices can be forced to share static blockingvoltages by means of a resistor in parallel with eachdevice. Transient blocking voltages are wqualized bya series RC circuit in parallel with each device. Seriesand parallel assemblies are available from somemanufacturers.

The problem of equalizing network design can oftenbe avoided by purchasing standard multiple-deviceassemblies complete with heat sinks. In this case, thedevice manufacturer assumes responsibility for theselection of properly matched devices or for theadequacy of the equalization network.

Further information on series/parallel arrays of devices and onthe design of equalization networks can be found in:

Westinghouse SCR Designers Handbook (Second Edition),Section 9, Westinghouse Semiconductor Div., Youngwood, Pa.;1970.

General Electric SCR Manual (Fifth Edition), Section 6, GeneralElectric Semiconductor Products Dept., Syracuse, N.Y.; 1972.

High Voltage Rectifier Stacks TD 54-200 12-76, Powerex, Inc.,Youngwood, Pa.

Air & Water Cooled Pow-R-Disc Assemblies Data Book, Powerex,Inc., Youngwood, Pa.

5-22

A, R. Mulica, “How to Use Silicon Controlled Rectifiers in Seriesor Parallel,” Control Engineering, vol. 11, pp. 95-99; May 1964.

G. Karady and T. Gilsig, “The Calculation of Transient VoltageDistribution in a High Voltage DC Thyristor Valve,” IEEE Trans.,vol. PAS-92, pp. 893-899; May/June 1973.

G. Karady and T. Gilsig, “The Calculation of the Turn-OffOvervoltages in a High Voltage DC Thyristor Valve,” IEEE Trans.,vol. PAS-91, pp. 565-574; Mar./Apr. 1972.

G. Karady and T. Gilsig, “The Calculation of Turn-OnOvervoltages in a High Voltage DC Thyristor Valve,” IEEE Trans.,vol. PAS-90, pp. 2802-2811; Nov./Dec. 1971.

J. Reeve and R. Marttila, “Calculation of Turn-On Stresses in aHVDC Valve Having Parallel Thyristor Strings,” IEEE Trans., vol.PAS-92, pp. 864-870; May/June 1973.

G. Sherbondy A., Graphical Approach to Paralleling PowerSemiconductors Tech-Tip 5-6, Westinghouse SemiconductorDivision, Youngwood, Pa.; Dec. 1976.

26. Summary

The technical feasibility of a new application ofpower electronics technology is nearly alwaysdetermined by the state of the art in solid-state powerdevices. Many potential applications are well withinthe direct capabilities of existing devices withoutextensive interfacing, and the design of theassociated circuitry is simple and routine. In othercases, peculiar requirements or economic trade-offsmay dictate that very careful attention be given tothe design of the power module to protect the powerdevice from excessive stresses and to squeeze out thefull performance of which it is capable.

Each aspect of the design of the power module tointerface device limitations with system requirementsis the subject of considerable specialized technology.Hopefully this text has helped you to appreciate boththe immense capabilities of the devices that areavailable and the general approaches that areavailable when needed to overcome particular devicelimitations.

27. Problems

Problem 5.1 Switching Dissipation

The “typical” thyristor considered in the text wasassumed to have a 3µs turn-on time during whichboth the voltage and the current changed from theirblocking values to their conducting values. Under

5-23

Figure 5-14The Power Module is the Pivot of Balanced Power Electronics Design

Equalization ofParallel Devices

EqualizingReactor

EqualizingResistor

For DynamicEqualization

For StaticEqualization

Equalization ofSeries Devices

Figure 5-13Voltage and Current Balancing Techniques

these conditions, the turn-on energy for 100A offorward current was calculated to be 30 mJ.

Assume that the voltage across the thyristor dropslinearly in 3µs as before, but that inductance isadded to the circuit so that the current requirestwice as long (i.e., 6µs) to build up to 100A. Howdoes this change the energy dissipated in the switchduring the turn-on interval?

The peak dissipation in the “typical” thyristor during the switchinginterval was found to be 15 KW. Because this dissipation is concentratedin a very small volume surrounding the junction, the correspondingpower density is of the order of 1 MW/cm3, and the energy flux is aboutthe same as that at the surface of the sun, 6 KW/cm2! Why doesn’t thethyristor light up like the sun (or does it)?

Problem 5.2 Steady-State Thermal Resistance and Average Current Rating

Six Type T500 thyristors are used in a 6-pulse bridgeAC/DC converter. If they are to operate with a casetemperature of 90oC and an average junctiontemperature of 120oC, how much DC load currentcan the converter tolerate? If they are cooled byambient air at 40oC, what is the maximum thermalresistance of each heat sink?

Problem 5.3 Transient Thermal Impedance

The I2t rating of a thyristor is a non-repetitive rating.Therefore it can be expected that a thyristor which isstressed to the limit allowed by this rating will have apeak junction temperature considerably higher than125oC and that an appreciable delay is requiredbefore forward blocking ability is regained.

Use the plot of transient thermal impedance for theType T500 to estimate the increase in junctiontemperature which occurs when a matched fuse (I2t = 13,495 A2s) blows. Assume that the clearingtime for the fuse is 2 ms and that the current isconstant over this interval.

Problem 5.4 Transient Thermal Impedance

A graphical process is determined in the text forcombining the transient thermal impedances of twodissimilar cylinders, namely silicon and copper. Testyour understanding of this process by using it todetermine the increase in thermal time constant whichtakes place when the length of a cylinder is doubled.In other words, use the process to combine thethermal impedance graphs of two identical cylinders.

Verify your result by showing that it is consistent withthe analytical expression for the time constant.

The plot of junction-to-case thermal impedance shows the rise in junctiontemperature which would occur (as a function of time) if the device weremounted on an infinite heat sink. Use physical reasoning to predict howthe plot would change for a device which is insulated from thermalcontact with any other body.

Problem 5.5 Using Superposition with Transient Thermal Impedance

Idealized heat conduction is a linear process.Therefore the rise in junction temperature for anarbitrary dissipation waveform can be obtained bythe superposition of step-function responses.Determine the peak junction temperature rise of aType T500 thyristor caused by a 700A, 20 ms pulsefollowed after an interval of 40 ms by a 500A, 40 mspulse.

Problem 5.6 Current Sharing of Parallel Thyristors

Type T500 thyristors have an RMS current rating of125A. Assume that 2 of these thyristors are to beparalleled in order to control a 250A load current.The forward V-I characteristic may be approximatelyby a linear relationship of the form:

where the nominal values are Eo = 1.2v and R = 0.003ohms. Assume that production devices have a ± 10%

spread in both Eo and Ro.

(a) Calculate the worst case distribution of currents iftwo production devices are paralleled to carry250A. Compared to the ± 10% tolerance on Eoand Ro, what is the corresponding percentagetolerance on current?

(b) Calculate the value of the equalizing resistorwhich must be connected in series with eachthyristor to hold the current tolerance within ± 10%. Calculate the power dissipated inthe equalizing resistors and compare it with thepower dissipated in the thyristors.

5-24

V E R I= +0 0

A-1

APPENDICES

APPENDIX ITen Cornerstones of Power Electronics

The fundamental principles needed to understandthe repetitively switched power circuits used in powerelectronics are not new to electrical engineers. Onceyou gain self confidence in applying these oldprinciples, mixed with patience and preserverance,you will find that the mysteries of the new field ofpower electronics disappear.

In power electronics, concise conclusions which areboth useful and general are as rare as the circuitconfigurations are diverse. Therefore, anunderstanding of how the fundamentals can beapplied to reach conclusions which are both specificand correct is the only safe foundation. Ifconclusions are remembered, the necessaryrestrictions and qualifications on their validity aresoon forgotten. And intuitive guessing is even riskier.

The ten cornerstones needed to establish a basicunderstanding of power electronics can be stated insimple terms:

1. Kirchhoff’s Voltage Law: The sum of theinstantaneous voltages around a closed loop is zero. (eN = 0

2. Kirchhoff’s Current Law: The sum of theinstantaneous currents into a junction is zero. (iN = 0

3. Ohm’s Law: eR(t) = RiR(t)

4.

5.

(Note: Because of the nonsinusoidal waveforms inrepetitively switched circuits, the relationships in (4)and (5) must be used in place of impedancerelationships based on ZL = j ωL and ZC = 1/j ωC. Ingeneral, instantaneous and average values are moreuseful in switched circuits than the RMS values whichare used almost exclusively in sinusoidal analysis.)

6. Instantaneous Power: P(t) = e(t) x i(t)

7.

8.

By definition, there is no net change in a periodicallyvarying function in steady-state over a full cycle. From(4) and (7):

Therefore:

9. The average voltage across an inductor over a fullcycle in steady-state is zero.

Similarly from (5) and (7):

10. The average current through a capacitor over afull cycle in steady-state is zero.

Regardless of the apparent simplicity of theseprinciples and the apparent complexity of manypower circuits, lack of understanding of the circuitoperation can nearly always be traced to a lack ofunderstanding of the implications of one or more ofthese principles.

e t Ldi t

dt Le t dtL

LL( )

( )( )= = ∫ or iL∆ 1

[ ( )] ( )f tT T

f t dtT

T

AVE =− ∫1

2 1 1

2

i t Cde t

dt Ci t dtC

CC( )

( )( )= = ∫ or eC∆ 1

f tT T

f t dtT

T( ) ( )[ ] =

−

∫RMS

1

2 1

2

12

1

2

e tT

e t dtL

TiL LT L T

( ) ( )[ ] = = =∫AVE

10∆

i tT

i t dtC

TeC CT C T

( ) ( )[ ] = = =∫AVE

10∆

APPENDIX IISelected Symbol List

Standard Circuit Symbols

Vdo = DC or average voltage with nophase control and no load.

Vd = DC or average voltage with phasecontrol and/or load.

Id = DC or average current.

Pd = VdId = Real Power (W).

Pa = VdoId = Apparent fundamental power (VA)neglecting harmonics.

fs = Source frequency.

α = Phase control angle.

φ, cos φ = Displacement angle, displacementfactor.

µ = Commutation overlap angle.

Other Circuit Symbols

Vs = RMS line-to-line sinusoidal sourcevoltage (secondary).

VP = =2VS = Peak line-to-line sinusoidal sourcevoltage (secondary).

IL = RMS AC line current (secondary).

IL1 = RMS fundamental AC line current (secondary).

P = Pulse number.

Standard Device Rating Symbols

VRRM, VDRM = Thyristor reverse and forward off-state blocking voltages (repetitive).

VT = On-state voltage.

IT = On-state current.

ITSM = Maximum on-state thyristor surgecurrent (non-repetitive).

IRRM, IDRM = Reverse and forward off-state thyristorblocking (leakage) currents.

tq = Circuit-commutated turn-off time.

trr = Junction reverse recovery time.

TJ = Junction temperature. (Subscripts c = Case A = Ambient).

RθJC = Junction-to-case thermal resistance.

ΖθJC (t) = “Transient thermal impedance”Unit step function thermalresponse with device mounted onfixed-temperature heat sink.

Characteristic – An inherent and measurableproperty of a device, or a set of related values, forstated or recognized conditions.

Rating – A maximum or minimum limiting capabilityor condition for a device determined for specifiedoperating and environmental conditions.

A-2

­­

A-3

APPENDIX IIIPower Electronic Circuit Configurations — Voltageand Current Relationships

Half-Wave Resistive Load

1.

2.

3.

4.

5.

6.

7.

Half-Wave InductiveLoad with FreeWheeling Diode

Center Tap FullWave ResistiveLoad or InductiveLoad with FWD

Center Tap Inductive Load

Center Tap Inductive Loadwith FWD Thyristorin DC Circuit

Single Phase Bridge Half Control Resistive or Inductive FWD Load

Single Phase Bridge Full Control Resistive or Inductive FWD Load

Name WaveformsMax.

ThyristorVoltage

Circuit PRV

Notation Schematic SCR Diode

1.4ERMS

1.4ERMS

2.8ERMS

2.8ERMS

2.8ERMS

1.4ERMS

1.4ERMS

1.4ERMS

EP

EP

2EP

2EP

EPEP

EP

0

EP

. . .

. . .

. . .

. . .

1-1-1-H

2-1-1-C

2-1-1-C

2-1-1-CwithThyristorin DCCircuit

HalfControl4-1-1-Band FWD

FullControl4-1-1-B

1-1-1-Hwith FreeWheelingDiode

ERMS Ra

O

EP

ERMS

L

R

a

O

EP

ERMSR

a

O

EP

ERMS

RL

a

O

EP

ERMS

R L SCR

a

O

EP

ERMS

CR1CR1

CR2

R

L

a

O

EP

ERMS

R

L

a

O

EP

A-4

<180o

FWD

180o

180o

180o

180o

180o

180o

180o

360o

<180o

FWD

180o

<180o

210o

Maximum LoadVoltage

Ed = AverageEa = RMS

Load Voltage with Delayed Firing

Maximum AverageThyristor Current

Maximum AverageDiode Current

Average Amps.Cond.Period

AverageAmps.

Cond.Period

. . .. . .

. . .. . .

. . .. . .

Ed =E p

π

Ea =E p

2

Ed =E p

π

Ed =2E p

π

Ed =2E p

π

Ed =2E p

π

Ed =2E p

π

Ed =2E p

π

Ed =E p

2π(1+ cosα)

Ea =E p

2 ππ −α + 1

2sin2α

Ed =E p

2π(1+ cosα)

Ed =E p

π(1 +cosα)

Ed =2E p

dc

π(cosα)

Ed =E p

π(1 +cosα)

Ed =E p

π(1 +cosα)

Ed =2E p

π(cosα)

E p

πR

E p

2πR

E p

πR

E p

πR

E p

πR

2E p

πR

E p

πR

E p

πR

.54E p

πR

.262Ep

πR

E p

πR

.5 Ep( )πR

E p

πR

.26 2Ep( )πR

(I = K)

dc(I = K)

HighlyInd. Load

Convent-ional

( )

A-5

Three Phase HalfWave InductiveLoad

Three Phase HalfWave ResistiveLoad or InductiveLoad with FreeWheeling Diode

Three Phase BridgeFull ControlInductive Load

Inverse ParallelThyristors ResistiveLoad

Three Phase BridgeHalf ControlResistive Load orInductive Load withFree Wheeling Diode

Three Phase BridgeFull ControlResistive Load orInductive Load withFree Wheeling Diode

9.

10.

11.

12.

13.

14.

Name WaveformsMax.

ThyristorVoltage

Circuit PRV

Notation Schematic SCR Diode

EP

EP

2.45ERMS

2.45ERMS

2.45ERMS

2.45ERMS

2.45ERMS

2.45ERMS

2.45ERMS

2.45ERMS

2.45ERMS

2.45ERMS

1.4ERMS

2.45ERMS

2.45ERMS

. . .

. . .

. . .

3-1-1-Y

3-1-1-Ywith FreeWheelingDiode

6-1-1-Bwith FullControl

AC Switch

6-1-1-BFull Controland FWD

6-1-1-BHalf Controland FWD

ERMS

IDRL

ERMS

CR1 IDRL

ERMS

RL ID

CR1

ERMS

RL

CR2

ERMS

CR1(3)

RL ID

ERMS

IRMS

RL

a

O

EP

a

O

EP

a

O

EP

a

O

EP

a

O

EP

a

O

EP

8. Single Phase Bridge with Thyristorin DC and FreeWheeling Diode

1.4ERMS

EPEP4-1-1-BwithThyristorControland FWD

ERMS

CR1CR1

CR2

SCRR

L

a

O

EP

A-6

360o

120

120o

120o

120o

120o

180o

180o

<180o

<120o

FWDSize to .33 ID

Size to .33 ID

Size to .33 ID

.33 ID

<120o

FWD

<120o

FWD

120o

Maximum LoadVoltage

Ed = AverageEa = RMS

Load Voltage with Delayed Firing

Maximum AverageThyristor Current

Maximum AverageDiode Current

Average Amps.Cond.Period

AverageAmps.

Cond.Period

. . .. . .

. . .. . .

. . .. . .

.33 ID

.33 ID

.33 ID

.33 ID

.33 ID

Ed =2E p

πEd =

E p

π1 + cosα

2E p

πR

E p

πR

.5Ep

πR

Ed =3 3E p

2π

Ed = 3 3E p

2πcosα

Ed =3 3E p

2π

Ed = 3 3E p

2πcosα

Ed = 3E p

2πi + cos(a + 30)[ ]

Ed =3 3E p

π

Ed = 3 3E p

πcosα

Ed =3 3E p

π

Ed = 3 3E p

πcosα

Ed = 3 3E p

π1+ cosα

2- 3

2sinα

Ed =3 3E p

πEd =

3 3E p

2πi + cosα( )

Ea =Ep

1.4Ea =

E p

2ππ - α + 1

2sin )( 2α

dc(I

(0<α<30o)

(0<α<60o)

(30o<α<150o)

= K)

E p

πR

( )

( )

( )

( )

( )

(60o<α<120o)