First Course on
POWER ELECTRONICS ANDDRIVES
Ned Mohan Oscar A. Schott Professor of Power Electronics and Systems Department of E lectrical and Computer Engineering University ofMinnesota Minneapolis, MN USA
MNPERE Minneapolis
To my wife Mary, son Michael and daughter Tara
PSpice is a registered trademark of the OrCAD Corporation. SIMULINK is a registerd trademark ofThe Mathworks, Inc. Published by: l\1NPERE P. O. Box 14503 Minneapolis, MN 55414 USA E mail: MNPERE@AOL.COM http://www.l\1NPERE.com
Copyright © 2003 by Ned Mohan. AlI rights reserved. Reproduction or translation of any part of this book, in any form, without the written permission of the copyright owner, is unlawful.
ISBN 097152922 1 Printed in the United States of America.
•
11
PREFACE Power electronics and drives are enabling technologies but most undergraduates, at best, will take only one course in these subjects. Recognizing this reality, this book is intended to teach students both the fundamentais in the context of exciting new applications and the practical design to meet the following objectives simultaneously: Provide solid background in fundamentais to prepare students for advanced courses Teach design fundamentais so that students can be productive in industry from the very beginning ln this book, the topics listed below are carefully sequenced to maintain student interest
throughout the course and to maintain continuity as much as possible. I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Applications and Structure of SwitchMode Power Electronics Systems Practical Details ofImplementing a Switching PowerPole ~he building block) DCDC Converters: Switching Details and their Average Dynamic Models Designing the Feedback Controller in DCDC Converters Diode Rectifiers and their Design PowerFactorCorreetion Circuits (PFCs) including the Controller Design Review ofMagnetic Concepts TransfonnerIsolated SwitchMode DC Power Supplies Design ofHighFrequency Inductors and Transfonners SoftSwitching in DCDC Converters, and its applications ln HighFrequeney AC Synthesis in lnduction Heating and Compact Fluoreseent Lamps (CFLs) Electric Motor Drives Synthesis of DC and LowFrequency Sinusoidal AC in Motor Drives and Uninterruptible Power Supplies (UPS) Control ofElectric Drives and UPS Thyristor Converters Utility Applications ofPower Electronics
Instructor's Cboice ln a fast pace course with proper student background, this book with ali these topies can be covered from fronttobaek in one semester. However, the material is arranged in sueh III
a way that an instructor can either omit an entire topic or cover it quickly to provide just an overview using the PowerPointbased slides on the accompanying CD, without interrupting the flow. This book is designed to serve as a semestercourse textbook in two different curricula: 1) in a Power Electronics course where there is a separate undergraduate course offered on electric machines and drives, and 2) in a Power Electronics and Drives course where only a single course is offered on both of these subjects. The selections of chapters under these two circumstances are suggested below. Textbook in a Power Electronics Course. ln this course, since a separate undergraduate course exists on electric machines and drives, Chapter I I on Electric Motor Drives and Chapter I3 on Design of Feedback Controllers in Motor Drives can be omitted. Depending on the availability of time, the "deeper" chapters such as Chapter 6 on Power Factor Control, Chapter 9 on Design of HighFrequency lnductors and Transformers, and Chapter 10 on SoftSwitching Converters can be covered very quickly, mainly to provide an overview. Textbook in a Power Electronics and Drives Course. Since this course is intended to provide a broader coverage in a single semester by covering topics in power electronics as well in electric drives, some ofthe "deeper" chapters in power electronics listed earlier can be safely omitted. ~
Simulations ln Power Electronics, simulations using PSp ice can be extremely beneficial for
reaffuming the fundamentais and in describing the design details by making realistic problems. However, simulations are presented on the accompanying CDROM such that not to get in the way of the fundamentais.
CDROM The accompanying CD includes the following:
IV
â€˘
Extremely useful for lnstructors: PowerPointbased slides are included for every chapter to quickly prepare lectures and to review the material in class. Students can print ali the s i ides and bring to the classroom to take notes on.
â€˘
Simulations and design examples are readytoexecute, using PSp ice that is loaded on this accompanying CDROM.
CONTENTS Chapter 1 II 12 13 14 15 16 17
Power Electronics and Drives: Enabling Technologies
lntroduction to Power Electronics and Drives Applications and the Role ofPower Electronics and Drives Energy and the Environrnent Need for High Efficiency and High Power Density Structure ofPower Electronics Interface The SwitchMode LoadSide Converter Recent and Potential Advancements References Problems
Chapter2
Design of Switching PowerPole
21 22 23
Power Transistors and Power Diodes Selection of Power Transistors Selection ofPower Diodes
24
Switching Characteristics and Power Losses in PowerPoles
25
JustifYing Switches and Diodes as Ideal
26 27
Design Considerations The PWM Controller IC References Problems
Chapter3
SwitchMode DCDC Converters: Switching Analysis, Topology Selection and Design
31
DCDC Converters
32
Switching PowerPole in DC Steady State
33
SimplifYing Assumptions
34
Common Operating PrincipIes
35
Buck Converter Switching AnaIysis in DC Steady State
v
36
Boost Converter Switching Analysis in DC Steady State
37
BuckBoost Converter Switching Analysis in DC SteĂ dy State
38
Topology Selection
39
WorstCase Design
310
SynchronousRectified Buck Converter for Very Low Output Voltages
311
Interleaving of Converters
312
Regulation ofDCDC Converters by PWM
313
Dynamic Average Representation of Converters in CCM
314
BiDirectional Switching PowerPole
315
DiscontinuousConduction Mode CDCM) References Problems Appendix 3A DiscontinuousConduction Mode CDCM) in DCDC Converters Chapter4
Designing Feedback Controllers in SwitchMode DC Power Supplies
41
Objectives ofFeedback Control
42 4#44 45 46
Review ofthe Linear Control Theory Linearization ofVarious Transfer Function Blocks Feedback Controller Design in VoltageMode Control PeakCurrent Mode Control Feedback Controller Design in DCM References Problems
Chapter5 51 52 53 54 55 56
V1
Rectification ofUtility Input Using Diode Rectifiers
Introduction Distortion and Power Factor Classitying the "FrontEnd" ofPower Electronic Systems DiodeRectifier Bridge "FrontEnds" Means to Avoid Transient Inrush Currents at Starting FrontEnds with BiDirectiollal Power Flow Referellces Problems
Chapter6
61 62 63 64 65 66 67 68
â€˘
Introduction SinglePhase PFCs Control ofPFCs Designing the Inner A verageCurrentControl Loop Designing the Outer Voltage Loop Example ofSinglePhase PFC Systems Simulation Results Feedforward ofthe Input Voltage References Problems
Chapter 7 71 72 73 74 75
PowerFactorCorrection (PFC) Circuits and Designing the Feedback Controller
Magnetic Circuit Concepts
AmpereTums and Flux Inductance L Faraday's Law: Induced Voltage in a Coi I due to TimeRate of Change of Flux Linkage Leakage and Magnetizing Inductances Transformers References Problems
Chapter 8
SwitchMode DC Power Supplies
81
Applications ofSwitchMode DC Power Supplies
82
Need for Electrical Isolation
83 84 85 86 8 7 88
Classification ofTransformerlsolated DCDC Converters Flyback Converters Forward Converters FullBridge Converters HalfBridge and PushPull Converters Practical Considerations References Problems
Chapter 9 91 92
Design ofHighFrequency Inductors and Transformers
Introduction Basics ofMagnetic Design Vil
93 94 95 96 97
Inductor and Transformer C onstruction AreaProduct Method Design Example of an lnductor Design Example of a Transformer for a Forward Converter Thermal Considerations References Problems
Chapter 10
~
SoftSwitching ln DCDC Converters And Inverters For Induction Heating And Compact Fluorescent Lamps
101
fntroduction
102 103 104
HardSwitching ln the Switching PowerPoles SoftSwitching ln the Switching PowerPoles Inverters for Induction Heating and Compact Fluorescent Lamps References Problems
Chapter 11
Electric Motor Drives
111 112
lntroduction Mechanical System Requirements
113 114 115 116 117
lntroduction to Electric Machines and the Basic PrincipIes ofOperation DC Motors PermanentMagnet AC Machines Induction Machines Summary References Problems
Chapter 12
Synthesis ofDC and LowFrequency Sinusoidal AC Voltages for Motor Drives and UPS
121 122 123 J24
lntroduction Switching PowerPole as the Building Block DCMotor Drives ACMotor Drives
125 126
VoltageLink Structure with BiDirectional Power Flow Uninterruptible Power Supplies (UPS) References Problems
VIU
Chapter 13
131 132 133 134 135 136 137
Introduction Control Objectives Cascade Control Structure Steps in Designing the Feedback Controller System Representation for SmallSignal Analysis Controller Design Example of a Controller Design References Problems
Chapter 14
141 142 143 144 145
Thyristor Converters
Introduction Thyristors (SCRs) SinglePhase, PhaseControlled Thyristor Converters ThreePhase, FullBridge Thyristor Converters CurrentLink Systems References Problems
Chapter 15
151 152 153 154 155 156 157
Designing Feedback Controllers for Motor Drives
Utility Applications ofPower E1ectronics
Introduction Power Semiconductor Devices and Their Capabilities Categorizing Power Electronic Systems Distributed Generation (DG) Applications Power Electronic Loads Power Quality Solutions Transmission and Distribution (T&D) Applications References
Problems
IX
Chapter 1 POWER ELECTRONICS AND DRIVES: ENABLING TECHNOLOGIES 11
â€˘
INTRODUCTION TO POWER ELECTRONICS AND DRIVES
Power electronics is an enabling technology, providing the needed interface between the electrical source and the electrical load, as depicted in Fig. 11 [I]. The electrical source and the electrical load can, and often do, differ in frequency, voltage amplitudes and the number of phases. The power electronics interface facilitates the transfer of power from the source to the load by converting voltages and currents from one forrn to another, in which it is possible for the source and load to reverse roles. The controller shown in Fig. lI allows management of the powe r transfer process in which the conversion of voltages and currents should be achieved with as high energyefficiency and high power density as possible. Adjustablespeed electric drives represent an important application of power electronics. Power Electron ics Interface ,, Converter
Source
Load Controller
:_ L___..::_"'_"'_"'_"'_'' __ "'__'__
Figure lI Power electronics interface between the source and the load.
12
APPLICATIONS AND THE ROLE OF POWER ELECTRONICS AND DRIVES
Power electronics and drives encompass a wide array of applications. A few important applications and their role are described below.
121
Powering the Information Technology
Most of the consumer electronics equipment such as personal computers (PCs) and entertainment systems supplied from the utility mains internally need very low dc voltages. They, therefore, require power electrics in the forrn of switchmode dc power supplies for converting the input line voltage into a regulated low dc voltage, as shown in Fig. 12a. Fig. 12b shows the distributed architecture typically used in computers in which the incoming ac voltage from the utility is converted into dc voltage, for example,
11
at 24 V. This semiregulated voltage is distributed within the computer where onboard power supplies in logicIevel printed circuit boards convert this 24 V dc input voltage to a lower voltage, for example 5 V dc, which is very tightly regulated. Very large scale integration and higher logic circuitry speed require operating voltages much lower than 5 V, hence 3.3 V, 1 V, and eventually, 0.5 V leveIs would be needed. ",)1
Power
24 V (de)
5 V (de)
Utility
Converter
I I
I I
L __ U V ( d e )
(b)
(a)
Figure 12 Regulated lowvoltage dc power supplies. Many devices such as cell phones operate from low battery voltages with one or two battery cells as inputs. However, the electronic circuitry within them requires higher voltages thus necessitating a circuit to boost input dc to a higher dc voltage as shown in the block diagram of Fig. 13.
I Battery
Cell (1.5 v)

T
+ 9V (de) 
Figure 13 Boost dcdc converter needed in cell operated equipment. 122
Robotics and Flexible Production
Robotics and flexible production are now essential to industrial competitiveness in a global economy. These applications require adjustable speed drives for precise speed and position contro\. Fig. 14 shows the block diagram of adjustable speed drives in which the ac input from a 1phase or a 3phase utility source is at the line frequency of 50 or 60 Hz. The role ofthe power electronics interface, as a powerprocessing unit, is to provide the required voltage to the motor. ln the case of a dc motor, dc voltage is supplied with adjustable magnitude that controls the motor speed. ln case of an ac motor, the power electronics interface provides sinusoidal ac voltages with adjustable amplitude and frequency to control the motor speed. ln certain cases, the power electronics interface
12
may be required to allow bidirectional power flow through iI:, between the utility and the motorIoad. Eleclric Drive
,    1 I
i+<~ fixed Electric Source
'0=
I
Power Processing Unit (PPU)
f.I
(utility)
measure<l speedl position
I
~
P~ r
Signal
input command (speed I posili on)
Figure 14 Block diagram ofadjustable speed drives. lnduction heating and electric welding, shown in Fig. 15 and Fig. 16 by their block diagrams are other important industrial applications of power electronics for flexible production. Power
Electronics Interface
High Frequency
AC
Utility
Figure 15 Power electronics interface required for induction heating.
Power Electronics Interface
De
Utility
Figure 16 Power electronics interface required for electric welding. 13
ENERGY AND THE ENVIRONMENT
There is a growing body of evidence that buming of fossil fuels causes global warming that may lead to disastrous environmental consequences. Power electronics and drives can play a crucial role in minimizing the use of fossil fuels, as briefly discussed below.
13
131
Energy Conservation
It's an old adage: a penny saved is a penny earned. Not only energy conservation leads to financial savings, it helps the environment. The pie chart in Fig. 17 shows the percentages of electricity usage in the U nited States for various applications. The potentials for energy conservation in these applications are discussed below.
Lighting 19%
HVAC 16%
Motors 51%
Figure 17 Percentage use of electricity in various sectors in the U.S. 1311 ElectricMotor Driven Systems Fig. 17 shows lhat electric motors, including their applications ln heating, ventilating and air conditioning (HV AC), are responsible for consuming onehalf to twothirds of ali lhe electricity generated. Traditionally, motordriven systems run at a nearly constant speed and their output, for exarnple, flow rate in a pump, is controlled by wasting a portion of lhe input energy across a throttling valve. This waste is eliminated by an adjustablespeed electric drive, as shown in Fig. 18 by efficiently controlling lhe motor speed, hence lhe pump speed, by means of power electronics [2]. .. Outlet
Adjustable Speed Drive (ASO) ~
utility
Inlet
Pump
Figure 18 Role of adjustable speed drives in pumpdriven systems. One out of three new homes now uses electric heat pump, in which adjustablespeed drive can reduce energy consumption by as much as 30 percent [3] by eliminating onoff cycling of the compressor and running the heat pump at a speed that matches the thermal load ofthe building. The sarne is true for air conditioners. A Department of Energy report [4] estimates that by operating ali these motordriven systems more efficiently in lhe United States could annually save electricity equivalent to the annual electricity usage by the entire state ofNew York!
14
1312 Lighting As shown in the pie chart in Fig. 17, approximately onefifth of electricity produced is used for lighting. Fluorescent lights are more efficient than incandescent lights by a factor of three to four. The efficie ncy of fluorescent lights can be further improved by using highfrequency power electronic ballasts that supply 30 kHz to 40 kHz to the light bulb, as shown by the block diagram in Fig. 19, further increasing the efficiency by approximately 15 percent. Compared to incandescent light bulbs, highfrequency compact fluorescent Iam ps (CFLs) improve efficiency by a factor of four, last much longer (severa I thousand hours more), and their relative cost, although high at present, is dropping. Power Electronics Interface
CFL
Vtility Figure 19 Power e lectronics interface required for CFL. 1313 Transportation Electric drives offer huge potentia l for energy conservation in transportation. While efforts to introduce commerciallyv iable Electric Vehicles (EVs) continue with progress in battery [5] and fuel cell techno logies [6] being reported, hybrid electric vehicles (HEVs) are sure to make a huge impact [7]. According to the V.S. Environmental Protection Agency, the estimated gas mileage of the hybridelectrical vehicle shown in Fig. 110 in combined city and highway driving is 48 miles per gallon [8]. This is in comparison to the gas mileage of 22 .1 miles per gallon for an average passenger car in the V.S. in year 2001 [9]. Since a utomobiles are estimated to account for about 20% emission of ali CO z [10] that is a greenhouse gas, doubling the gas mileage of automobiles would have an enormous impact on the environment.
Figure 110 Hybrid electric vehicles with much higher gas mileage.
15
EVs and HEVs, of course, need power electronics in the forrn of adjustablespeed electric drives. Even in conventional automobiles, power e1ectronic based load has grown to the extent that it is difficult to supply ir from a 12/ 14V battery system and there are serious proposals to raise it to 36142V leveI in order to keep the copper bus bars needed to carry currents a manageable size [11]. Add to this other transportation systems, such as light rail, tlybywire planes, allelectric ships and drivebywire automobiles, and transportation represents a major application area of power electronics.
132
Renewable Energy
Clean and renewable energy that is environmentally friendly can be derived from the sun and the wind. ln photovoltaic systems, solar cells produce dc, with an iv characteristic shown in Fig. IlIa that requires a power electronics interface to transfer power into the utility system, as shown in Fig. 11 I b. ~~
>+o~
',laa...'
t+I_
i ,r:M':=~s.=:: De Input
d
Power
Electronics Interface Utility
.,
.,
(b)
(a) Figure II I Photovoltaic Systems. Wind is the fastest growing energy resource with enorrnous potential [12). Fig. 112 shows the need of power electronics in windelectric systems to interface variablefrequency ac to the linefrequency ac voltages of the utility grid.

Generator
and Power Electronics
Figure 112 Windelectric systems.
16
Utility
133
Utility Applications ofPower Electronics
Applications of power electronics and electric drives m power system s are growing rapidly. ln distributed generation, powcr clcctronics is needed to interface nonconventional energy sources such as wind, photovoltaic, and fuel cells to the utility grid. Use of power electronics allows co ntrol over the f10w of power on transmission lines, an attribute that is especially significant in a deregulated utility environment. Also, the security and the efficiency aspects of power systems operation necessitate increased use of power electronics in utility applications. Uninterruptible power supplies (UPS) are used for criticai loads that must not be interrupted during power outages. The power electronics interface for UPS , shown in Fig. 113, has linefrequency voltages at both ends, although the number of phases may be different, and a means for ene rgy storage is provided usually by batteries, which sllpply power to the load during the utility outage.
}  _  j U ninterruptible
Power Supply
Criticai
Utility
Load
Figure 113 Uninterruptible power supply (UPS) system. 134
Strategic Space and Defense Applications
Power electronics is essential for space exploration and for interplanetary traveI. Defense has always been an important application but it has become criticai in the postSept 11 th world . Power electronics will play a huge role in tanks, ships, and pla nes in which replacement of hydraulic drives by electric drives can offer significant cost, weight and reliability advantages. 14
NEED FOR IDGH EFFICIENCY AND IDGH POWER DENSITY
Power electronic systems must be energy efficicnt and reliable, have a high power density thus reducing their size and weight, and be low cost to make the overall system economically feasible. High energy efficiency is important for several reason s: it lowers operating costs by avoiding the cost ofwasted energy, contributes less to global warming, and reduces the need for cooling (by heat sinks discussed later in this book) therefore increasing power density. We can easily show the relationsh ip 10 a power electronic system between the energy efficiency, 1] , and the power den sity. The energy efficiency is defined in Eq . II in terms of the output power p" and the power loss
p,=
within the system as 17
'7=
P
(11)
o
~ +~oss
Eq. lI can be rewritten for the output power in terms of lhe efficiency and lhe power 1055 as (12)
/
01
.:: ~ Ă ::> ~
300
p.
200
~
/
 'ow ./
o
........ V
. 00
V 1,
V
1m
II o s5
o
o.,
...
/
0.84
0. 88
0.92
0.96
Efficiency Figure 114 Power output capability as a function of efficiency. ln power electronics equipment, lhe cooling system is designed to transfer dissipated power, as heat, wilhout allowing lhe internal temperatures to exceed certain limits. Therefore for a system designed to handle certain power 1055, the plots in Fig. 114 based on Eq. 12 show lhat increasing the efficiency from 84% to 94% increases the power output capability (sarne as the power rating) of lhat equipment by a factor of lhree. This could mean an increase in lhe power density by approximately the sarne factor.
15
STRUCTURE OF POWER ELECTRONICS INTERFACE
By reviewing the role of power electronics in various applications discussed earlier, we can summanze that power electronics interface is needed to efficiently control the transfer of power between dcdc, dcac, and acac systems. ln general, the power is supplied by lhe utility and hence, as depicted by the block diagram of Fig. lIS, lhe linefrequency ac is at one end. At the olhe r end, one of lhe following is synlhesized: adjustable magnitude dc, sinusoidal ac of adjustable frequency and amplitude, or high frequency ac as in lhe case of compact fluorescent lamps (CFLs) and induction heating. Applications lhat do not require utility interconnection can be considered as the subset of lhe block diagrarn shown in Fig. lIS .
18
Power Electronics Interface
Output to Load  Adjustable DC  Sinusoidal AC  Highfrequency AC
utility
Figure 115 Block diagram ofpower electronic interface. To pravide the needed functionality to the interface in Fig. 115, the transistors and diodes, which can block voltage only of one polarity, have led to the structure shown in Fig. 116.
+ conv21{
convi
ulility
Load
controller Figure 116 Voltagelink structure ofpower electronics interface. This structure consists oftwo separate converters, one on the utility side and the other on the load side. The dc ports of these two converters are connected to each other with a parallel capacitor forming a dclink, acrass which the voltage polarity does not reverse, thus allowing unipolar voltage handling transistors to be used within these converters. ln certain applications, the power flow through these converters reverses, by currents reversing direction at the dc ports. The converter structure ofFig. 116 has the following benefits that explain their popularity: â€˘
Decoupling of Two Converters: the voltageport (dclink) created by the capacitor in the middle decouples the operation of the utilityside and the loadside converters, thus allowing their design and operation to be optimized individually.
â€˘
Immunity fram Momentary Power Interruptions: energy storage in the capacitor allows uninterrupted power flow to the load during momentary powerline disturbances. Continuing developments in increasing capacitor energy density further support this structure for the power electronics interface.
ln the structure of Fig. 116, the capacitor in parallel with the two converters forms a dc voltagelink, hence this structure is called a voltagelink (or a voltagesource) structure. This structure is used in a very large power range, fram a few tens of watts to severa I
19
megawatts, even extending to hundreds of megawatts in utility applications. Therefore, we will mainly focus on this voltagelink structure in this book. At extremely high power leveis, usually in utilityrelated applications, which we will discuss in the last two chapters in this book, it may be advantageous to use a currentlink (or a currentsource) structure, where an inductor in series with the two converters acts as a currentlink. Lately in niche applications, a matrix converter structure is being reevaluated, where theoretically there is no energy storage element between the input and the output sides. Discussion of matrix converters is beyond the scope of this book on the first course on power electronics and drives. 16
THE SWITCHMODE LOADSIDE CONVERTER
[n the structure of Fig. 116, the role of the utilityside converter is to convert linefrequency utility voltages to an unregulated dc voltage. This can be done by a dioderectifier circuit like that discussed in basic electronics courses, although power quality concerns may lead to a different structure discussed in Chapter 6. We will focus our attention on the loadside converter in the structure of Fig. 116, wh<lre a dc voltage is applied as an input on one end. Applications dictate lhe functionality needed of the loadside converter. Based on the desired output ofthe converter, we can group this functionality as follows: â€˘
Group 1
Adjustable dc or a lowfrequency sinusoidal ac output in dc and ac motor drives uninterruptible power supplies regulated dc power supplies without electrical isolation
â€˘
Group 2
Highfrequencyac in compact f1uorescent lamps induction heating regulated dc power supplies where the dc output voltage needs to be electrically isolated from the input, and the loadside converter internally produ ces highfrequency ac, which is passed through a highfrequency transformer and then rectified into dc.
161
SwitchMode Conversion: Switching PowerPole as the Building BIock
Achieving high energyefficiency for applications belonging to either group mentioned above requires switchmode conversion, where in contrast to linear power electronics, transistors (and diodes) are operated as switches, either on off.
110
This switchmode conversion can be eXplained by its basic building block, a switching powerpole A, as shown in Fig. I  I 7a. It effectively consists of a bipositional switch, which forms a twoport: a voltageport across a capacitor with a voltage V;n that cannot change instantaneously, and a currentport due the series inductor through which the current cannot change instantaneously. For now, we wiII assume the switch ideal with two positions: up or down, dictated by a switching signal q A which takes on two values: I and O, respectively. The practical aspects of implementing this bipositional switch we wiII consider in the next chapter.
+ VA
JLLI,I1........11,1~[ t
(a)
(b) Figure 117 Switching powerpole as the building block in converters. The bipositional switch "chops" the input dc voltage V;n into a train of highfrequency voltage pulses, shown by the v A waveform in Fig. 117b, by switching up or down at a high repetition rate, calIed the switching frequency J,.
ControIling the pulse width
within a switching cycJe aIlows control over the average value of the pulsed output, and this pulsewidth modulation forms the basis of synthesizing adjustable dc and lowfrequency sinusoidal ac outputs, as described in the next section. Highfrequency pulses are cJearIy needed in applications such as compact fluorescent lamps, induction heating, and intemalIy in dc power supplies where electrical isolation is achieved by means of a highfrequency transformer. A switchmode converter consists of one or more such switching powerpoles. 162
PulseWidth Modulation (PWM) ofthe Switching PowerPole (constant J,)
For the applications in Group I, the objective of the switching powerpole redrawn in Fig. 118a is to synthesize the output voltage such that its average is of the desired value: dc or ac that varies sinusoidalIy at a lowfrequency. Switching at a constant switchingfrequency J, produces a train of voltage pulses in Fig. 118b that repeat with a constant switching timeperiod I;, equal to 1/ J, .
111
+
(a)
(b)
Figure 118 PWM ofthe switching powerpole. Within each switching cycle with the timeperiod T, ( = l / h) in Fig. 118b, the average value
vA
ofthe waveform is controlled by the pulse width Tup during which the switch is
in the up position, and v A equals V;. : (13) where
dA = T" p/ T,)
is defined as the dutyratio of the switching powerpole, and the
average voltage is indicated by a "" on topo The average voltage and the pole dutyratio are expressed by lowercase letters since they may vary as functions of time. The control over the average value of the output voltage is achieved by adjusting or modulating the pulse width, which later on will be referred to as pulsewidthmodulation (PWM). This switching powerpole and the control of its output by PWM set the stage for switchmode conversion with high energyefficiency. We should note that
vA and
d A in the above discussion are discrete quantities and their
values are to be calculated over each switching cycle. However in practical applications, the pulsewidth
T. p
changes very slo wly over many switching cycles, and hence we can
consider them as analog quantities
vAI} and
d A(I) that are continuous functions oftime.
For simplicity, we may not show their time dependence explicitly. 163
Switching PowerPole in a Buck DCDC Converter: An Example
As an example, we will consider the switching powerpole in a Buck converter to stepdown the input dc voltage V;., as sho wn in Fig. 119a, where a capacitor is placed across the load to form a lowpass LC filte r to provide a s mooth voltage to the load.
112
+ Vo
(a) (b)
Figure 119 Switching powerpole in a Buck converter. Tn steady state, the dc (average) input to this LC filter has no attenuation, hence the average output voltage Vo equals the average,
vA '
of the applied input voltage. Based on
Eq. 13, by control\ing d A' the output voltage can be control\ed in a range from
v'n down
to O: (14)
ln spite of the pulsating nature of the instantaneous output voltage v A(t), the series
inductance at the currentport of the pole ensures that the current i L(t) remains relatively smooth, as shown in Fig. 119b. 1631 Realizing the BiPositional Switch in a Buck Converter As shown in Fig. 120a, the bipos itional switch in the powerpole can be realized by using a transistor and a diode. When the transistor is gated on (through a gate circuitry discussed in the next chapter by a switching signal q A = 1 ), it carries the inductor current and the diode is reversed biased, as shown in Fig. 120b. When the transistor is switched off (q A = O), the inductor current "freewheels", as shown in Fig. 120c through the diode until the next switching cycle when the transistor is tumed back on. The switching waveforrns, shown earlier in Fig. 11 9b, are discussed in detail in Chapter 3. ln the switchmode circuit of Fig. 119a, higher the switching frequency, that is the frequency ofthe pulses in the VA(t ) waveforrn, smal\er the values needed ofthe lowpass
113
LC filter. On the other hand, higher switching frequency results in higher switching losses in the bipositional switch, which is the subject of the next chapter. Therefore, an appropriate switching frequency must be selected, keeping these tradeoffs in mind.
+
iL
(a)
+
iL
+1 +
+ v;, .,
"., .' ": ~....
V;n
Vo
q A1 
_iL
!
C
+
Vo
qA =0 (b)
(c)
Figure 120 Transistor and diode forming a switching powerpole in a Buck converter. 17
RECENT ANDPOTENTIALADVANCEMENTS
Given the need in a plethora of applications, the rapid growth of this field is fueled by revolutionary advances in semiconductor fabrication technology. The power electronics interface of Fig. II consists of solidstate devi ces, which operate as switches, changing from on to off at a high frequency. There has been a steady improvement in the voltage and current handing capabilities of solidstate devices such as diodes and transistors, and their switching speeds (from on to off, and vice versa) have increased dramatically, with some devices switching in tens of ns. Devices that can handle voltages in kVs and currents in kAs are now available. These semiconductor switches are integrated in a single package with ali the circuitry needed to make them switch, and to provide the necessary protection. Moreover, the costs ofthese devices are in a steady decline. Power electronics has benefited from advances in the semiconductor fabrication technology in another important way. The availability of Application Specific Integrated Circuits (ASJC~), Digital Signal Processor (DSPs), microcontrollers, and Field Programmable Gate Arrays (FPGA) at very low costs makes the controller function in the block diagram of Fig. II easy and inexpensive to implement, while greatly increasing functionality.
114
Significant areas for potential advancements in power electronic systems are in integrated and intelligent power modules, packaging, SiCbased solidstate devices, improved high energy density capacitors, and improved topologies and control. REFERENCES L 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12. 13. 14. 15. 16. 17. 18. 19.
N. Mohan, T. M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and Design, 3,d Edition, Wiley & Sons, New York, 2003. N. Mohan and R.J. Ferraro, " Techniques for Energy Conservation in AC Motor Oriven Systems", EPRlReport EM2037, Project 12011213, Septo 1981. N. Mohan and James Ramsey, "Comparative Study of AdjustableSpeed Orives for Heat Pumps", EPRlReport EM4704, Project 20334, Aug. 1986. Tuming Point Newsletter, dated November 1998, A Motor Challenge Bimonthly Publication by the U.S. Oepartment ofEnergy (DOE). www.oit.doe.gov. R. Stempel et ai, "NickelMetal Hydride: Ready to Serve", IEEESpectrum, Nov. 1998, pp. 2934. Tom Gilchrist:, "Fuel Cells to the Fore", IEEESpectrum, Nov. 1998, pp. 3540. O. Hermance and S. Sasaki, "Hybrid Electric Vehicles Take to the Streets", IEEESpectrum, Nov. 1998, pp. 4852. Toyota Motor Corporation (http://www.toyota.com/html/ shop/vehicles/prius). Energy Information Administration Monthly Energy Review, Oecember 2002 (http://www.eia.doe.gov/emeulmer/overview.html). http://www.nedo.go.jp/ itdlfellow/english/projectel7e.html. J. Kassakian, "Automotive Applications of Power Electronics", Proceedings of the NSFsponsored Faculty Workshop on Teaching of Power Electronics and Electric Orives, Tempe, AZ, Jan 35, 2002 (http://www.ece.umn.edulgroups/workshp2002). American Wind Energy Association (http://www.awea.org). Edison Electric Institute' s Statistical Yearbook Chttp://www.eei.org). "'Inlillity' VariableCapacity U nits Boast SEERS AboveI6," Air Conditioning, Heating and Refrigeration News, pp. 7980, Jan 21 , 1991. http://www.sylvania.com/home/energystar. http://www.howstuffworks.com/question417 .htm. http://www.fhwa.dot.gov/ohim/1 996/ in3.pdf. http://www.eia.doe.gov/emeu/25 0pec/sldOOO.htm. http://www.siemenswestinghouse .com/en/fuelcells/hybrids/performance/ index.cfm.
PROBLEMS Applications and Energy Conservation
II
A U.S. Oepartment ofEnergy report [4] estimates that over 122 billion kWhlyear can be saved in the manufacturing sector in the United States by using mature 115
12
and costeffective conservation technologies. Calculate (a) how many 1000MW generating plants are needed to operate constantly to supply thjs wasted energy, and (b) the annual savings in dollars ifthe cost ofelectricity is 0.10 $IkWh. ln a process, a blower is used with the flowrate profile shown in Fig. PI2a. Using the information in F ig PI 2b, estimate the percentage reduction in power consumption resulting from using an adjustab le speed drive rather than a system with (a) an outlet damper and (b) an inlet vane. ~
% Flow rate
(ff!&
H"
~~_ _ _
'O utlet dampe..
60% p r Consump(ion
{Q"/o
(%or full flow rate)
Inlet v. ne
'" 2.
20%
20%
30%
100/0
20%
% of tirne
(a)
O
100
(b)
90
80
70
60
'O
40
30
% F10w Rate
Figure PI2 13
14
ln a system, if the system si ze is based on power dissipation capacity, calculate the improvement in the power density if the efficiency is increased from 85% to 9S%. According to [13], electricity generation in the U.S. in year 2000 was
IS
approximately 3.8 x 109 MWhrs. Fig. 17 shows that 16% of that is used for heating, ventilating and air conditioning. Based on [3 , 14], as much as 30% of the energy can be saved in such systems by using adjustable speed drives. On this basis, calculate the savings in energy per year and relate that to 1000MW generating plants needed to o perate constantly to supply this wasted energy. According to [12], the total amount of electricity that could potentially be
16
17
116
generated from wind in the U nited States has been estimated at 10.8 x l 09 MWhrs annually. If onetenth of this potential is developed, estimate the number of IMW windmills that would be required, assuming the capacity factor of the windmills to be 2S%. ln problem IS, if each IMW windmill has 20% of its output power flowing through the power electronics interface, estimate the total rating of these interfaces in kW. As the pie chart of Fig. 17 shows, lighting in the U.S. consumes 19% of the generated electricity. Compact Fluorescent Lamps (CFLs) consume power one
fourth of that consumed by the incandescent lamps for the same light output [15]. According to [13], electricity generation in the U.S. in year 2000 was approximately 3.8 x I 0 9
18
19
IlO
lII
112
113
114
MWhrs.
Based on this information, estimate the
savings in MWhrs annually, assuming that ali lighting at present is by incandescent lamps, which are to be replaced by CFLs. An electrichybrid vehicle offers 52 miles per gallon in mixed (city and highway) driving conditions according to the V.S. Environrnental Protection Agency. This is in comparison to the gas mileage of 22.1 miles per gallon for an average passenger car in the V.S. in year 2001 [9], with an average of 11 ,766 miles driven. Calculate the saving s in terms of barreis of oil per automobile annually, if a conventional car is replaced by an electrichybrid vehicle. ln calculating this, assume that a barrei of oi l that contains 42 gallons yields approximately 20 gallons of gasoline [16]. Based on the estimate in [1 7] of approximately 136 million automobiles in 1995, we can project that today there are ISO million cars in the U.S. Using the results of Problem 18, calculate the total annual reduction of carbon into the atmosphere, if the consumption of each gallon of gasoline releases approximately 5 pounds of carbon [16]. Relate the savings ofbarrels ofoil annually, as calculated in Problem 19, to the imported oil, if the U .S. im ports approximately 35,000 millio n barreis of crude oil peryear [18] . Fuelcell systems that also uti lize the heat produced can achieve efficiencies approaching 80% [19], m o re than double of the gasturbine based electrical generation. Assume that 25 million households produce an average of 5 kW. Calculate the percentage of electricity generated by these fuelcell systems compared to the annual electricity generation in the V.S. of 3.8 x I 0 9 MWhrs [13] . lnduction cooking based on power electronics is estimated to be 80% efficient compared to 55% for co nventional electric cooking. If an average home consumes 2 kWhrs daily using conventional electric cooking, and that 50 million households switch to induction cooking in the V.S. , calculate the annual savings in electricity usage. Assume the average energy density of sunli g ht to be 800 W/ m' and the overall photovoltaic system efficiency to be 10%. Calculate the land area covered with photovoltaic cells needed to produce 1,000 MW, the size of a typical large centra l power plant. ln problem 113, the solar cells are distributed on top of roofs, each in area of 50 m '. Calculate the number o fhomes needed to produce the sarne power. 117
PWM ofthe Switching PowerPole 115
ln a Buck converter, the input voltage V;n = 12V.
The output voltage V
Q
is
required to be 9 V. The switching frequency J, = 200kHz. Assuming an ideal switching powerpole, calculate the pulse width T.,p of the switching signal, and the dutyratio dA of the power pole. 116
117
118
ln the Buck converter of Problem 115, assume the current through the inductor to be ripplefree with avalue of 3A (the ripple in this current is discussed later in Chapter 3). Draw the waveforrns of the voltage at the currentport and the input current at the voltageport. Using the inputoutput specifications given in Problems 115 and 116, calculate the maximum energyefficiency expected of a linear regulator where the excess input voltage is dropped across a series element that behaves as a resistor.
Chapter 2
DESIGN OF SWITCHING POWERPOLES ln the previous chapter, we examined a switching powerpole as the building block of power electronic converters, consisting of an ideal bipositional switch with an ideal transistor and an ideal diode, and pulsewidthmodulated (PWM) to achieve regulation of
the output. ln this chapter, we will discuss the availability of various power semiconductor devices, their switching characteristics and various tradeoffs in designing a switching powerpole. We will also briefly discuss a PWMIC, which is used in regulating the average output of such switching powerpoles. 21
POWER TRANSISTORS AND POWER DIODES [1)
The powerIevel diodes and transistors have evolved over decades from their signalIevel counterparts to the extent that they can handle voltages and currents in k V s and kAs respectively, with fast switching times of the order of a few tens of ns to a few fJS. Moreover, these devices can be connected in series and parallel to satisfY any voltage and current requirements. Selection of power diodes and power transistors in a given application is based on the following characteristics: 1.
Voltage Rating: the maximum instantaneous voltage that the device is required to block in its offstate, beyond which it "breaks down" and irreversible damage occurs.
2.
Current Rating: the maximum current, expressed as instantaneous, average, andJor rms, that a device can carry in its onstate, beyond which excessive heating within the device destroys it.
3.
Switching Speeds: the speeds with which a device can make a transition from its onstate to offstate, or vice versa. Small switching times associated with fastswitching devices result in low switching los ses, or considering it differently, fastswitching devices can be operated at high switching frequencies with acceptable switching power losses.
4.
OnState Voltage: the voltage drop across a device during its onstate while conducting a current. T he smaller this voltage, the smaller will be the onstate power loss. 21
22
SELECTION OF POWER TRANSISTORS [25]
Transistors are controllable switches, which are avai lable in several forms for switchmo de power electronics applications : •
MOSFETs (MetalOxideSemiconductor FieldEffect Transistors)
•
IGBTs (lnsulatedGate Bipolar Transistors)
•
lGCTs (IntegratedGateControlled Thyristors)
•
GTOs (GateTumOffthyristors)
•
Niche devices for power electronics applications, such as BJTs (BipolarJunction Transistors), SITs (Static lnduction Transistors), MCTs (MOSControlled Thyristors), and so on.
ln switchmode converters, there are two types of transistors which are primarily used: MOSFETs are typically used below a few hundred volts at switching rrequencies in excess of 100 kHz, whereas lGBTs dominatc vcry large voltagc, currcnt and powcr ranges extending to MW leveIs, provided the switching frequencies are below a few tens of kHz. lGCTs and GTOs are used in utility applications of power electronics at power leveIs beyond a few MWs. The following subsections provide a brief overview of MOSFET and lGBT characteristics and capabilities. 221
MOSFETs
ln applications at voltages below 200 volts and switching rrequencies in excess of 100 kHz, MOSFETs are clearly the device of choice because of their low onstate losses in low voltage ratings, their fast switching speeds, and a high impedance gate which requires a small voltage and charge to facilitate on/off transition. The circuit symbol of an nchannel MOSFET is shown in Fig. 21 a. It consists of three terminaIs: drain (O), source (S), and gate (G). The forward current in a MOSFET flows from the drain to the source terminal. MOSFETs can block only the forward polarity voltage, that is, a positive
V
DS
•
They cannot block a negativepolarity voltage due to an intrinsic anti
parallel diode, which can be used effectively in most switchmode converter designs. MOSFET i  v characteristics for various gate voltage values are shown in Fig. 2 1b. For gate voltages below a threshold vaI ue v GSUh )' typically in a range of 2 to 4 V, a MOSFET is completely off, as shown by the i  v characteristics in Fig. 21 b, and approximates an open switch. Beyond depends on the applied gate voltage
VGS«h)'
V eS '
the drain current iD through the MOSFET
as shown by the transfer characteristic shown in
Fig. 2lc, which is valid almost for any value of the voltage
VDS
across the MOSFET
(note the horizontal nature of i  v characteristics in Fig. 21 b). To carry iD (= lo) would
22
reqUlre a gate voltage of a va fue, at least equal to
v GS(I_ ) '
as shown in Fig. 21 e.
Typieally, a higher gate v oltage, ap proximately lO V , is maintained in arder to kcep the MOSFET in its onstate and carryi ng i,, (= l n) ' if)
R D S(on l =
iu
l /s tope k(;S = J/V 9V 7V
5V "'(iS:O;:; V( ;S(lh )
o
VJ)S
(b)
(a)
(c)
Figure 21 MOSFET: (a) symbol , (b) iv characteristics, (c) transfer characteristie. ln its onstate, a MOSFET approx imates a very small resistar
R J)S ( on"
and the drain
current that tlows through it depend s on the external circuit in whieh it is connected. The onstate resistance, inverse of the slo pe of i  v characteristics as shown in Fi g . 21 b, is a strong function ofthe blocking voltage rating VI)." ofthe device:
R
D8(o/l)
a
V 2.5fu27 /JS.""
(21 )
The relationship in Eg. 21 explain s why MOSFETs in lowvoltage applicatioh s, at less than 200 V, are an exeellent choice. The onstate resistance goes up with the junction temperature within the device and proper heatsinking must be provided to keep the temperature below the design limit. 222
IGBTs
IGBTs combine the ease of control, as in MOSFETs, with low onstate losses even at fairly high voltage ratings. Their sw itching speeds are sufficiently fast for switching freguencies up to 30 kHz. Therefore, they are used for converters in a vast voltage and power range  from a fractional kW to several MWs where s w itehing freguencies required are below a few tens of k.Hz . The circuit symbol for an IGBT is shown
III
Fig. 22a and the i  v characteristies are
shown in Fig. 22b. Similar to MOSFETs, IGBTs have a hi g h impedance gate, which requires only a small amount of en e rgy to switch thc dev ice. IGBTs havc a s mall onstate voltage, even in devices w ith large blockingvoltage rating, for example,
v,,,,
is
approximately 2 V in 1200V devices. IGBTs can be designed to bloek negative voltages, but most commercially available devices, by des ign to improve othe r properties, cannot block any apprec iable rev ersc23
polarity voltage. IGBTs have tumon and tumoff times on the order of a microsecond and are available as modules in ratings as large as 3.3 kV and 1200 A. Voltage ratings of up to 5 kV are projected.
E
(a)
(b)
Figure 22 IGBT: (a) symbol, (b) iv characteristics. 223
PowerIntegrated Modules and IntelligentPower Modules [24)
Power electronic converters, for example for threephase ac drives, require six power transistors. Powerintegrated modules (PIMs) combine several transistors and diodes in a single package. Intelligentpower modules (IPMs) include power transistors with the gatedrive circuitry. The input to this gatedrive circuitry is a signal that comes from a microprocessor or an analog integrated circuit, and the output drives the MOSFET gate. IPMs also incIude fault protection and diagnostics, thereby immensely simplif)ring the power electronics converter designo 224
Costs ofMOSFETs and IGBTs
As these devices evolve, their relative costs continue to decline. The costs of single devices are approximately 0.25 $/A for 600V devices and 0.50 $/A for 1200V devices. Power modules for the 3 kV class of devices cost approximately I $/A. 23
SELECTION OF POWER DIODES
The circuit symbol of a diode is shown in Fig. 23a and its i  v characteristic in Fig. 23 b shows that a diode is an uncontrolled device that blocks reverse polarity voltage. Power di odes are available in large ranges of reverse voltage blocking and forward current carrying capabilities. Similar to transistors, power diodes are available in several forrns as follows, and their selection must be based on their application:
24
•
Linefrequency diodes
•
Fastrecovery diodes
• •
Schottky diodes SiCbased Schottky diodes
â€˘ K A l[;i:t(a)
o (b)
Figure 23 Diode: (a) symbol, (b) iv characteristic. Rectification of linefrequency ac to dc can be accomplished by slower switching pn junction power (Iinefrequency) diodes, which have relatively a slightly lower onstate voltage drop, and are available in voltage ratings of up to 9 kV and current ratings of up to 5 kA. The onstate voltage drop across these diodes is usually on the order of I to 3 V, depending on the voltage blocking capability. Switchmode converters operating at high switching frequencies, from several tens of kHz to several hundred kHz, require fastswitching diodes. ln such applications, fastrecovery diodes, also formed by pn junction as the linefrequency diodes, must be selected to minimize switching losses associated with the diodes. ln applications with very low output voltages, the forward voltage drop of approximately a volt or so across the conventional pn junction diodes becomes unacceptably high. ln such applications, another type of device called the Schottky diode is used with a voltage drop in the range of 0.3 to 0.5 V. Being majoritycarrier devices, Schottky diodes switch extremely fast and keep switching losses to a minimum. Ali devices, transistors and diodes, discussed above are silicon based. Lately, siliconcarbide (SiC) based Schottky diodes in voltage ratings of up to 600 V have become available [6]. ln spite of their large onstate voltage drop (1.7 V, for example), their capability to switch with a minimum of switching los ses makes them attractive in converters with voltages in excess of a few hundred volts.
24
SWITCIllNG CHARACTERISTICS AND POWER LOSSES lN POWERPOLES
ln switchmode converters, it is important to understand switching characteristics of the switching powerpole. As discussed in the previous chapter, the powerpole in a Buck converter shown in Fig. 24a is implemented using a transistor and a diode, where the current through the current port is assumed a constant dc, lo, for discussing switching characteristics. We will assume the transistor to be a MOSFET, although a similar discussion applies if an lGBT is selected.
25
O L _ _ _ _ _ _ _ _ _ _ _ _ _ _ (a)
(b)
: 0[[
~~~
V;n
Vos
Figure 24 MOSFET in a switching powerpole. For the nchannel MOSFET (pchannel MOSFETs have poor characteristics and are seldom used in power applications), the typical i D versus v DS characteristics are shown in Fig. 24b for various gate voltages. The offstate and the onstate operating points are labeled on these characteristics. Switching characteristics of MOSFETs, hence of the switching powerpole in Fig. 24a are dictated by a combination of factors: the speed of charging and discharging of junction capacitances within the MOSFET, iD versus
V os
characteristics, the circuit of
Fig. 24a in which the MOSFET is connected and its gatedrive circuitry. W.e shou Id note that the MOSFET junction capacitances are highly nonlinear functions of drainsource voltage and go up dramatically by several orders of magnitude at lower voltages. ln Fig. 24a, the gatedrive voltage for the MOSFET is represented as a voltage source, which changes from O to VGG
'
approximately 10 V, and vice versa, and charges or
discharges the gate through a resistance Rgm. that is the sum of an extemal resistance ~
and the intemal gate resistance.
Only a simplitied explanation of lhe tumon and tumoff characteristics, assuming an ideal diode, is presented here. The switching details including the diode reverse recovery are discussed in the Appendix to this chapter. 241
Turnon Characteristic
Prior to tumon, Fig. 25a shows the circuit in which the MOSFET is off and blocks the input dc voltage v'n; l o "freewheels" through the diode. By applying a positive voltage to the gate, the tumon characteristic describes how the MOSFET goes from the offpoint to the onpoint in Fig. 25b. To tum the MOSFET on, the gatedrive voltage goes up rrom O to VGG
26
,
and it takes the gatedrive a tinite amount of time, called the tumon
delaytime
Id( on ) '
to charge the gatesource capacitance through the gatecircuit resistance
Rgo« to the threshold vai ue of VeSUh )'
During this tumon delay time, the MOSFET
remains off and I a continues to rreewheel through the diode, as shown in Fig. 25a.
iD I _~'Í._"" ______ .JL"",,
+
o
o
T
/ o
o G:<on) I
(a)
,;1
1
(c)
(b) Figure 25 MOSFET tumon.
For the MOSFET to tum on, first the current iD through it rises. As long as the diode is conducting a positive net current (Ia  iD ) , the voltage across it is zero and the MOSFET must block the entire input voltage
v,n'
Therefore, during the current risetime l ri
,
the
MOSFET voltage and the current are along the trajectory A in the i  v characteristics of Fig. 25b, and are plotted in Fig. 25c. Once the MOSFET current reaches l o ' the diode becomes reverse biased, and the MOSFET voltage falis. During the voltage falitime I fi
'
as depicted by the trajectory B in F ig. 25b and the plots in Fig. 25c, the gatetosource voltage remains at Ves(lo) ' Once the tumon transition is complete, the gate charges to the gatedrive voltage Vcc , as shown in Fig. 25c. 242
Turnoff Characteristic
The tumoff sequence is opposite to the tumon processo Prior to tumoff, the MOSFET is conducting Ia and the diode is reversed biased as shown in Fig. 26a. The MOSFET i  v characteristics are replotted in Fig 26b. The tumoff characteristic describes how
the MOSFET goes rrom the onpoint to the offpoint in Fig. 26b. To tum the MOSFET off, the gatedrive voltage goes down rrom Vcc to O. It takes the gatedrive a finite amount of time, called the tumoff delaytime
td (off) '
to discharge the gatesource
capacitance through the gatecircuit resistance R ,o«, rrom a voltage of Vcc to Ves(lo) ' During this tumoff delay time, the MOSFET remains on.
27
for the MOSFET to turn off, the output current must be able to " fTeewheel " through the diode . This requires the diode to become forward biased, and thus the voltage across the MOSFET must ri se as shown by the trajectory C in Fig. 26b, while the current thro ug h it remain s l u ' During this voltage ri setime
l n. '
the voltage and current are plotted in Fig.
26c, while th e gatetosource voltage rema in s at
Once the voltage across the
Vc;s(Jo )'
MOSFET reaches V;", the diode beco mes forward biased, and the MOSFET current fa lis. During the current falitime t fi ' dep icted by the trajectory D in Fig. 26b and the plo ts in Fig. 26c, the gatetosource voltage declines to
v,,;.\ÂˇUh)'
Once the turnoff trans itio n is
co mplete, the gate discharges to O.
+
, i
:.~.:
1
(a)
(b)
(c)
Figure 26 MOSFET tumoff. 243
Calculating Power Losses Within the MOSFET (assuming an ideal diode)
Power losses in the gatedrive circllitry are negligibly small except aI very hig h switching fTequencies. The primary source o f power losses is across the drain and the SOllrce, which can be divided into two categories: the conduction loss, and the switching losses. Both ofthese are di scussed in the fo llowi ng sections. 2431 Conduction Loss ln the onstate, the MOSFET conducts a drain current for an interval T,,,, during every switching tim eperiod T" with the switch dutyratio d = T,m I T,. Assllmin g this c urrent a constant l o durillg d T" the average po wer loss in the onstate resistance R IlS (",,) of the MOSF ET is: (22)
28
As pointed out earlier,
R OS(on)
varies significantly with the junction temperature and data
sheets often provide its vaI ue at the junction temperature equal to 25°C. However, it will be more realistic to use twice this resistance value that corresponds to the junction temperature equal to 120°C , for example.
Eq. 22 can be retmed to account for the
efTect of ripple in the drain current o n the conduction loss. The conduction loss is highest at the maximum load on the converter when the drain current would also be at its maximum.
2432 Switching Losses At high switching frequencies, switching power losses can be even higher than the conduction loss. The switching waveforms for the MOSFET voltage
VDS
and the current
io ' corresponding to the tumon and the tumofT trajectories in Figs. 25 and 26,
assumed linear with time, are replotted in Fig. 27. During each transition from on to ofT, and vice versa, the transistor has simultaneously high voltage and current, as seen from the switching waveforms in Fig. 27. The instantaneous power loss
p~ (t)
in the
transistor is the product of vos and io , as plotted. The average value of the switching losses from the plots in Fig. 27 is
where
I c.on
and
I c. olf
are detmed in Fig. 27 as the sum of the rise and the fali times
associated with the MOSFET voltage and current: (24) (25)
v'n
;;/\
~
O ~I~,~t<._~___ ~~"7>L5~~~~_~___~t<.~ ~____~'~I~·Figure 27 MOSFET switching losses.
29
.... Example 21
Assuming an ideal di ode, calculate losses in a MOSFET for the
following operating conditions in the powerpole of Fig. 24a: V;o
= 40 V,
10 = 5 A,
d = 0.4, and J, = 300kHz. Assume that the switching times t,.oo = t,.o./T = 25ns. The
onstate resistance
RDS(oo)
= O.3Q .
AIso calculate the percentage efficiency of the
converter based just on lhe los ses in the MOSFET. Solution
From Eq. 22, lhe average conduction power loss is p romJ = d RDS(~ /;; = 3W .
From Eq. 23, using Eqs. 24 and 25, the average switching power loss is
?'w
= 0.5 x V;oIoCt,.on + 1'.Off )J, = 1.5W . Therefore, the total power loss ~~, = 4.5W. The
output voltage of lhe converter is Vo = d V;n = 16 V. The output power Po = VJo = 80 W . From Eq. II ofChapter 1, lhe converter efficiency based on the MOSFET power losses is '7 = 94.67%. 244
Gate Driver Integrated Circuits (lCs) witb Builtin Fault Protection [2]
ApplicationSpecific ICs (ASICs) for controlling the gate voltages of MOSFETs and IGBTs greatly simplifY converter design by including various protection features, for example, lhe overcurrent protection lhat tums off the transistor under fault conditions. This functionality, as discussed earlier, is integrated into Intelligent Power Modules along with the power semiconductor devices. The IC to drive the MOSFET gate is shown ln Fig. 28 in a block diagram formo To drive the highside MOSFET in the powerpole, lhe input signal v, from the controller is referenced to the logic leveI ground, Vcc is the logic supply voltage, and Vw to drive lhe gate is supplied byan isolated power supply referenced to the MOSFET source S.
Vcc
I
Figure 28 Gatedriver IC functional diagrarn. One of lhe rcs for lhis purpose is IR2127, which can source 200 mA for gate charging and sink 420 mA for gate discharging, with typical tumon and tumoff times of 200 ns
210
and 150 ns, respectively. This lC, although it bas relatively large switching times, may be a good choice due to its easytouse fault protection capability, in which the transistor current, measured via the voltage drop across a small resistance in series with the transistor, disables the gatedrive voltage under overcurrent conditions. 25
JUSTIFYING SWITCHES AND DIODES AS IDEAL
Product reliability and high energyefficiency are two very important criteria. ln designing converters, it is essential that in the selection of semiconductor devices, we consider their voltage and current ratings with proper safety margins for reliability purposes. Achieving high energyefficiency from converters requires that in selecting power devi ces, we consider their onstate voltage drops and their switching characteristics. We also need to calculate the heat that needs to be removed for proper thermal designo Typically, the converter energy efficiency of approximately 90 percent or higher is realized. This implies that semiconductor devices have a very small loss associated with them. Therefore, in analyzing various converter topologies and comparing them against each other, we can assume transistors and di odes ideal. Their nonidealities, although essential to consider in the actual selection and the subsequent design process, represent a secondorder phenomena, that will be ignored in analyzing various converter topologies. We will use a generic symbol shown earlier for transistors, regardless of their type, and ignore the need for a specific gatedrive circuitry. 26
DESIGN CONSIDERA TIONS
ln designing any converter, the overall objectives are to optimize lheir cost, size, weight, energy efficiency, and the reliability. With these goals in mind, we will briefly discuss some important considerations as the criteria for selecting the topology and the components in a given application. 261
Switching Frequency j,
As discussed in the Buck converter example of the previous chapter, a switching powerpole results in a waveform pulsating at a high switching frequency at the currentport. The high frequency components in the pulsating waveforrn need to be filtered. lt is intuitively obvious that the benefits of increasing the switching frequencies lie in reducing the filter component values, L and C, hence their physical sizes in a converter. (This is true up to a certain value, for example, up to a few hundred kHz, beyond which, for example, magnetic losses in inductors and lhe internal inductance within capacitors reverse the trend.) Hence, higher switching frequencies allow a higher control bandwidth, as we will see in Chapter 4.
211
The negative consequences of increasing the switching frequency are in increasing the switching losses in the transistor and the diode, as discussed earlier. Higher switching frequencies also dictate a faster switching of the transistor by appropriately designed gatedrive circuitry, generating greater problems of electromagnetic interference due to higher di/ dt and dv/dt that introduce switching noise in the control loop. We can minimize these problems, at least in dcdc converters, by adopting a softswitching topology discussed in Chapter 10.
262
Selection or Transistors and Diodes
Earlier we discussed the voltage and the switching frequency ranges for selecting between MOSFETs and IGBTs. Similarly, the diode types should be chosen appropriately. The voltage rating of these devices is based on the peak voltage
V
in the
circuit (including voltage spikes due to parasitic effects during switching). The current ratings should consider the peak current j that the devices can handle and which dictates the switching power loss, the rms current 1= with
R DS(on)
for MOSFETs which behave as a resistor
in their onstate, and the average current
1 0""
for IGBTs and diodes, which
can be approximated to have a constant onstate voltage drop. Safety margins, which depend on the application, dictate that the device ratings be greater than the worstcase stresses by a certain factor.
263
Magnetic components
As discussed earlier, the filter inductance value depends on the switching frequency. Inductor design is discussed later in Chapter 9, which shows that the physical size of a filter inductor, to a first approximation, depends on a quantity called the areaproduct (Ap
)
given by the equation below:
(26) Increasing L in many topologies reduces the peak and the rms values (also reducing the transistor and the diode current stresses) but has the negative consequence of increasing the overall inductor size and possibly reducing the control bandwidth.
264
Capacito r Selection 17]
Capacitors have switching los ses designated by an equivalent series resistance, ESR, as shown in Fig. 29. They also have an internal inductance, represented as an equivalent series inductance, ESL. The resonance frequency , the frequency beyond which ESL begins to dominate C, depends on the capacitor type. Electrolytic capacitors offer high capacitance per unit volume but have low resonance frequency. On the other hand, capacitors such as metalpolypropy lene have relatively high resonance frequency. 212
Therefore, in switchmode dc power supplies, an electrolytic capacitor with high C may ofien be paralleled with a metalpolyester capacitor to form the output filter capacitor. C
ESL
ESR
1~ Figure 29 Capacitor ESR and ESL. 265
Thermal Design [8, 9]
Power dissipated in the semiconductor and the magnetic devices must be removed to limit temperature rise within the device. The reliability of converters and their life expectancy depend on the operating temperatures, which should be well below their maximum allowed values. On the other hand, letting them operate at a high temperature decreases the cost and the size of the heat sinks required. There are several cooling techniques but for generalpurpose applications, converters are ofien designed for cooling by normal air convection, without the use of forced air or liquido Semiconductor devices come in a variety of packages, which differ in cost, ruggedness, thermal conduction, and the radiation hardness if the application demands it. Fig. 2l0a shows a semiconductor device affixed to a heat sink through an isolation pad that is thermally conducting but provides electrical isolation between the device case and the heat sink. An analogy can be drawn with an electrical circuit of Fig. 210b where the power dissipation within the devi ce is represented as a dc current source, thermal resistances offered by various paths are represented as series resistances, and the temperatures at various points as voltages. The data sheets specif'y various thermal resistance values. The heat transfer mechanism is primarily conduction from the semiconductor device to its case and through the isolation pado
The resulting thermal resistances can be
represented by Rele and R e", respectively. From the heat sink to the ambient, the heat transfer mechanisms are primarily convection and radiation, and the thermal resistance can be represented by Rem. Based on the electrical analogy, the junction temperature can be calculated as follows for the dissipated power Pdl,,: (27)
213
isolation case
chip
ambicnt
(b) Figure 210 Thermal design: (a) semiconductor on a heat sink, (b) electrical analog. 266
Design Tradeoffs
As a function of switching frequency, Fig. 211 qualitatively shows the plot of physical sizes for the magnetic components and the capacitors, and the heat sink. Based on the present stateoftheart, optimum values of switching frequencies to minimize the overall size in dcdc converters using MOSFETs range fram 200 kHz to 300 kHz with a slight upward trend.
size Healsink
Magnelics and capacilors
Figure 211 Size ofmagnetic components and heat sink as a function offrequency. 27
THE PWM IC [10)
[n the pulsewidthmodulation ofthe powerpole in a dcdc converter, a high speed PWM IC such as the UC3824 fram UnitrodefTexas Instruments may be used. Functionally within this PWMIC, the control voltage vc(t) generated by the controller is compared with a ramp signal v, of amplitude
V"
at a constant switching frequency
is, as shown in
Fig. 212. The output switching signal represents the transistor switching function q(t),
214
which equals I if
VC
(I) ;::.,
V,;
otherwise O. The transistor dutyratio based on Fig. 212 is
given as d(t) = vcy) V,
(28)
Thus, the control voltage vc(t) can provide regulation of the average output voltage of the switching powerpole, as discussed further in Chapters 3 and 4 .
Figure 212 PWM IC waveforrns. REFERENCES
I. 2. 3. 4. 5. 6. 7. 8. 9. 10.
N. Mohan, T. M. Undeland, and W.P. Robbins, Power Electronics: Converlers, Applicalions and Design, 3'" Edition, Wiley & Sons, New York, 2003. Intemational Rectifier (http://www.irf.com). POWEREX Corporation (http://www.pwrx.com). Fuji Semiconductor (http: //www.fujisemiconductor.com). On Semiconductor (http://www.onsemiconductor.com). Intmeon Technologies (http://www.infineon.com). Panasonic Capacitors (http://www.panasonic.comlindustriaVcomponents/capcitor.htrn). Aavid Corporation for Heat Sinks (http://www.aavid.com). Bergquist Company for therrnal pads (http://www.bergguistcompany.com). Unitrode for PWM Controller ICs (http://www.unitrode.com).
PROBLEMS MOSFET in a PowerPole ofFig 24a
A MOSFET is used in a switching powerpole shown in Fig. 24a. Assume the diode ideal. The operating conditions are as follows: v'n = 40V, l o =5A, the switching frequency
.Is = 200kHz, and the dutyratio
d
= 0.3.
MOSFET has the following switching times:
Under the operating conditions, the
Id(on) =
100ns,
I. =
30ns,
1fr =
20ns ,
215
Id( off )
=50ns,
R DS(on)
21
I~
=25ns,
1 ft
=15ns.
The onstate resistance of the MOSFET is
= 20mn. Assume VGG as a step voltage between O V and 12 V.
Draw and label the tumon and tumoff characteristics of the MOSFET and sketch the MOSFET gateso urce voltage vGS waveform.
22
23 24
Plot the tumon and the tumoff switching power losses in the MOSFET. Calculate the average switching power loss in the MOSFET, and the percentage reduction in the converter energy efficiency due to this switching power loss. CaIculate the average conduction loss in the MOSFET. ln stead of an ideal diode, consider that the diode is actual and has a forward voltage drop VFM = 0.8 V.
Calculate the average forward power loss in the
diode. 25
ln the diode reverse recovery characteristic shown in Fig. 2AI, l a = 10ns, I" =
26
25 ns , and 1 RRM = 2 A. CaIculate the average switching power in the diode.
ln the gatedrive circuitry of the MOSFET, assume that the extemal power supply Va , in Fig. 28 has a voltage of 12 V. Each time, to tum the MOSFET on under the conditions given requires a charge Qg = 45nC. Calculate the average
27
gatedrive power loss. ln this problem, we wiII ca Iculate new values of the tumon and the tumoff delay times ofthe MOSFET, based on the gate driver IC IR2127, as described in section 244. This IC is su pplied with a voltage Vw = 12 V with respect to the MOSFET Source. Assume this voltage to equal VGG
â€˘
(a) For the tumon, assume that this driver TC is a voltage source of VGG with an internal resistance of 60n in series. CaIculate the tumon delay time assuming that the MOSFET threshold voltage VGS('h) = 3 V.
Id ( on) '
Assume the
MOSFET capacitance to be I 190 pF for this caIculation. (b) For tumingoff of the MOSFET, assume that this driver IC shorts the MOSFET gate to its source through an internal resistance of 30n. Calculate the tumoff delay ti me
I d(off) '
assuming that the MOSFET voltage
VGS(I. ) = 4.5V. Assume the MOSFET capacitance to be 1900 pF for this 28
216
calculation. ln the Buck converter 10 wh ich this switching powerpole is used, the output filter capacitor consists of a paralIel combination of an electrolytic capacitor and a metalizedpolyester capac itor. The electrolytic capacitor has the folIowing
measured values:
C = 697 fJF,
ESR = 0.037 Q, and
ESL = 16nH .
The
metalizedpolyester capacitor has the following measured values: C = 10 fJF , ESR
= 0.04Q, and ESL = 34nH.
Using any computer program, plot the
impedance magnitude of each capacitor, and of their parallel combination, on the sarne logIog plot, as a function of frequency (from 1 kHz to 1 MHz). At
.f, 29
=
200kHz, what are the relative impedance magnitudes of these two
capacitors? The MOSFET losses are the sum of those computed in Problems 22 and 23. The junction temperature must not exceed 100°C , and the ambient temperature is given as 40°C. From the MOSFET datasheet, R.)e = 1.2°C / W. The thermal pad has R.e< = I.SoC / W . Ca1culate the maximum vaI ue of R.,a that the heat sink can have.
APPENDIX 2A DIODE REVERSE RECOVERY AND POWER LOSSES ln this appendix, the diode reverse characteristic is described and the associated power losses are calculated. The increase in the MOSFET tumon switching loss due to the diode reverse recovery current is discussed in the accompany CD. 2Al
Average Forward Loss in Diodes
The current flow through a diode results in a forward voltage drop VFM across the diode. The average forward power loss
Pdiodc.F
in the diode in the circuit of Fig. 24a can be
calculated as (2AI) where d is the MOSFET dutyratio in the powerpole, and hence the diode conducts for (1 d) portion of each switching timeperiod. 2A2
Diode Reverse Recovery Characteristic
Forward current in power diodes, unlike in Schottky diodes that are majoritycarrier devices, is due to the flow of electrons as well as holes. This current results in an accumulation of electrons in the pregion and of holes in the nregion. The presence of these charges allows a flow of current in the negative direction that sweeps away these excess charges, and the negative current quickly comes to zero, as shown in the plot of Fig. 2AI. The peak reverse recovery current I RRM and the reverse recovery charge Q" shown in Fig. 2AI depend on the initial forward current l o and the rate di / dI at which this current decreases.
217
o~~
o====~~~
Figure 2A 1 Diode reverse recovery characteristic.
2A3
Diode Switching Losses
The reverse recovery current results in switching losses within the diode when a negative current is flowing beyond the interval negative voltage Vd •• , •
•
la
in Fig. 2AI , while the diode is blocking a
This switching power loss in the diode can be estimated fram the
plots of Fig. 2Al as (2A2) where /, is the switching frequency.
ln switchmode power electronics, increase in
switching losses due to the diode reverse recovery can be significant, and diodes with ultrafast reverse recovery characteristics are needed to avoid these from becoming exceSSlve.
The reverse recovery current of the diode also increases the tumon losses in the associated MOSFET ofthe powerpole, as discussed in the accompanying CD.
218
Chapter 3 SWITCHMODE DCDC CONVERTERS: SWITCHING ANALYSIS, TOPOLOGY SELECTION AND DESIGN 31
DCDC CONVERTERS [1]
ln Chapter I, we discussed the need for Buck, Boost, and BuckBoost dcdc converters, shown by the block diagram of Fig. 3la, and how the pulsewidth modulation process regulates the output voltage. The design of the feedback controller is the subject of the next chapter. The power flow through these converters is in only one direction, thus their voltages and currents remain unipolar and unidirectional, as shown in Fig. 31 b. Based on these converters, several transformerisolated dcdc converter topologies, which are used in ali types of electronics equipment, are discussed in Chapter 8.
V;n
dede
Vo
V;n' Vo
X Ijn ' l o
Vo.~f
(a)
(b)
Figure 31 Regulated switchmode dc power supplies. 32
SWITCIllNG POWERPOLE lN DC STEADY STATE
Ali the converters that we will discuss consist of a switching powerpole that was introduced in Chapter I, and is redrawn in Fig. 32a. ln these converter circuits in dc steady state, the input voltage and the output load are assumed constant. The switching powerpole operates with a transistor switching function q(t), whose waveform repeats, unchanged from one cycle to the next, and the corresponding switch dutyratio remains constant at its dc steady state value D. Therefore, ali waveforrns associated with the powerpole repeat with the switching timeperiod T, in the dc steady state, where the basic principies described below are extremely useful for analysis purposes.
31
o
+ +
q
B
(a)
(b)
Figure 32 Switching power pole as the building block of dcdc converters. First, let us consider the voltage and current of the inductor associated with the powerpole. The inductor current depends on the pulsating voltage waveform shown in Fig. 32b. The inductor voltage and current are related by the conventional differential equation, which can be expressed in the integra l form as follows: v = L diL dt L
=>
i L(t ) =
~
(31)
JV L Âˇdr
,
where r, representing time, is a variable of integration. For simplicity, we will consider the first timeperiod starting with t = O in Fig. 32b. Using the integral form in Eq . 31 , the inductor current at a time t can be expressed in terms of its initial vai ue iL (O) as : (32) ln the dc steady state, the waveforms of ali circuit variables must repeat with the
switch ing frequency timeperiod 1.
T"
resulting in the following conclusions from Eq. 32:
The inductor current waveforms repeats with
T"
and therefore in Eq. 32 (33)
2.
lntegrating over one switching timeperiod T, in Eq. 32 and us ing Eq. 33 show that the inductor voltage integral ove r T, is zero. This leads to the conclusion that the average inductor voltage, averaged over T, , is zero:
1 T,
L Jv L Âˇdr = O o
32
(34)
ln Fig. 32a, the area A in volts, that causes the current to rise, equals in magnitude the
negative area B that causes the current to decline to its initial value. The above analysis applies to any inductor in a switchmode converter circuit operating in a dc steady state. By analogy, a similar analysis applies to any capacitor in a switchmode converter circuit, operating in the dc steady state, as follows: the capacitor voltage and current are related by the conventional differential equation, which can be expressed in the integral forrn as follows: . le
dvc = C 
(35)
di
where r, representing time, is a variable ofintegration. Using the integral forrn ofEq. 35, the capacitor voltage at a time t can be expressed in terrns of its initial value vc(O) as: (36) ln the dc steady state, the waveforrns of ali circuit variables must repeat with the switching frequency timeperiod 1.
T"
resulting in the following conclusions from Eq. 36:
The capacitor voltage waveforrn repeats with
T"
and therefore in Eq. 36 (37)
2.
Integrating over one switching timeperiod that the capacitor current integral over
T, in Eq. 36 and using Eq. 37 show
T, is zero, which leads to the conclusion
that the average capacitor c urren!, averaged over
T"
is zero: (38)
ln addition to the above two conclusions, it is important to recognize that in dc steady
state, just like for instantaneous quantities, the Kirchhoff's voltage and current laws apply to average quantities as well. ln the dc steady state, average voltages sum to zero in a circuit loop, and average currents sum to zero at a node: (39) (310)
33
33
SIMPLIFYING ASSUMPTIONS
To gain a clear understanding in the dc steady state, we will first make certain simplif'ying assumptions by ignori ng the secondorder effects listed below, and later on include them for accuracy: I.
Transistors, diodes and other passive components are ali ideal unless explicitly stated. For example, we will ignore the inductor equivalent series resistance.
2.
The input is apure dc vo ltage
3.
Design specifications require the ripple in the output voltage to be very smal!. Therefore, we w ill initially assume that the output voltage is purely dc
v,n'
without any ripple, that is voeI) = VO ' and later calculate the ripple in it. 4.
The current at the currentport of the powerpole through the series inductor flows continuously, resulting in a continuous conduction mode, CCM (the discontinuous conductio n mode, DCM, is analyzed later on).
It is, of course possible to analyze a switching circuit in detail, without making the above simplif'ying assumptions as we will do in computer simulations. However, the twostep approach followed here, where the analysis is first carried out by neglecting the secondorder effects and adding them later on, provides a deeper insight into converter operation and the design tradeoffs.
34
COMMON OPERATING PRINCIPLES
ln ali three converters that we will analyze, the inductor associated with the switching powerpole acts as an energy transfer means from the input to the output. Turning on the transistor of the powerpole increases the inductor energy by a certain amount, drawn from the input source, which is transferred to the output stage during the offinterval of the transistor. ln addition to this inductive energytransfer means, depending on the converter, there may be additional energy transfer directly from the input to the output, as discussed in the following sections.
35
BUCK CONVERTER SWITCHING ANALYSIS lN DC STEADY STATE
A Buck converter is shown in Fig. 33a, with the transistor and the diode making up the bipositional switch of the powerpo le. The equivalent series resistance (ESR) of the capacitor will be ignored. Turning on the transistor increases the inductor current in the subcircuit of Fig. 33b. When the transistor is turned off, the inductor current "freewheels" through the diode, as shown in Fig. 33c.
34
q
;1
~ +
+
t
VL
VA
q
I I t:= D TS7 _ _ ~.~1
t
(a) t
I
+
+
:.:::.: v A
VL
t
!
q =1
V;n
n :;::

q=O 
(b)
.
'Lo
vL=v;,
~      
I
L
= I
t
o
,J.rvYL~~
~G ,!c VA
~Vo
=0 (d)
(c) Figure 33 Buck dcdc converter.
For a given transistor switehing funetion waveforrn q(t) shown in Fig. 33d with a switch dutyratio D in steady state, the waveforrn of the voltage v A at the eurrentport follows q(t) as shown. ln Fig. 33d, integrating v A over
T"
the average voltage VA
equals DV;n' Recognizing that the average inductor voltage is zero (Eq. 34) and the average voltages in the output loop sum to zero (Eq. 39), (311)
The inductor voltage vL pulsates between two values, (V;n Vo) and ( Vo ) as plotted in Fig. 33d. Sinee the average induetor voltage is zero, the voltseeond areas during two subintervals are equal in magnitude and opposite in signo ln de steady state, the induetor current can be expressed as the sum of its average and the ripple eomponent: (312)
where the average current depends on tbe output load, and tbe ripple component is dictated by the waveform ofthe induetor voltage v L in Fig. 33d. As shown in Fig. 33d, the ripple component eonsists of linear segments, ris ing when v L is positive and falling 35
when v L is negative. The peakpeak ripple can be calculated as follows , using either area A orB:
(313) This ripple component is plotted in F ig. 33d. Since the average capacitor current is zero in dc steady state, the average inductor current equals the output load current by the Kirchhoff's current law applied at the output node: (314)
I L =I=v" o
R
The inductor current waveforrn is shown in Fig. 33d by superposing the average and the ripple components. Next, we will calcu late the ripple current through the output capacitor. ln practice, the filter capacitor is large to achieve the output voltage nearly dc (vo(t) = Vo ). Therefore, to the ripplefrequency current, the path through the capacitor offers much smaller impedance than through the load resistance, hence justifYing the assumption that the ripple component of the inductor current tlows entirely through the capacitor. That is, in Fig.33a le (t) = IL.,;ppl. (t)
(315)
ln practice, the voltage drops across the capacitor ESR and the ESL dominate over the voltage drop
~
fledl across C. The capacitor current ie , equal to iL â€˘ripplâ€˘ in Fig. 33d, can
be used to calculate the ripple in the output voltage . The input current i;n pulsates, equal to iL when the transistor is on, otherwise zero, as plotted in Fig. 33d. An input LC filter is often needed to prevent the pulsating current from being drawn from the input dc source. The average value of the input current in Fig. 33d is (using Eq . 314 )
(3 16)
Using Eqs. 3 11 and 316, we can co nfirrn that the input power equals the output power, as it should, in this idealized converter: (3  17)
Eq. 3 1 1 shows that the voltage conversion ratio of Buck converters in the continuous conduction mode (CCM) depends on D , but is independent of the output load. If the 36
output load decreases (that is, if the load resistance increases) to the extent that the inductor current becomes discontinuous, then the inputoutput relationship in CCM is no longer valid, and, if the dutyratio D were to be held constant, the output voltage in the discontinuousconduction mode would rise above that given by Eq. 311. The discontinuous conduction mode w ill be considered fully in section 315 and in the Appendix to this chapter.
36
BOOST CONVERTER SWITCIllNG ANALYSIS lN De STEADY STATE
A Boost converter is shown in Fig. 34a. Compared to a Buck converter, it has two major differences: 1.
2.
Power flow is from a lower voltage dc input to the higher load voltage, in the opposite direction through the switching powerpole. Hence, the current direction through the series inductor of the power pole is chosen as shown, opposite to that in a Buck converter, and this current remains positive in the continuous conduction mode. ln the switching powerpole, the bipositional switch is realized using a transistor and a diode that are placed as shown in Fig. 34a. Across the output, a filter capacitor C is placed, wh ich forms the voltage port and minimizes the output ripple voltage.
t
+ C
q 
t
p
'
+
C
+
p
q
(a)
(b) Figure 34 Boost dede converter.
lt is conventional to shaw power flow from the left to the right side. To follow this convention, the circuit of Fig. 34a is flipped and drawn in Fig. 34b. The output stage consists of the output load and a larg e filter capacitor that is used to minimize the output voltage ripple. This capacitar at the output initially gets charged to a voltage equal to 1';n through the diode.
37
ln a Boost converter, tuming on the transistor in the bottom pos ition applies the input
voltage across the inductor sllch that
V I.
eqllals f';", as shown in Fig. 35a, and ii. linearly
ramps up, increasing the energy in the inductor. Tuming off the transistor forces the inductor current to flow through the diode as shown in Fig. 35b, and some of the inductively stored energy is transferred to the output stage that consists of lhe filter capacitor and the output load across it.
VL
=
+
v,,, +
qJ l
I OTIl Âˇ
1..   1,
..J.,.. ._
01'
j"
(1
I
0)T8
T t
u
Ca)
q =1
I
t
+
~n
t t q=O.J
Cc)
(b)
Figure 35 Boost converter: operation and wavefonns. The trans istor switching function is shown in Fig. 3 5c, with a steady state dutyratio D. Because of the transistor in the bottom pos ition in the powerpole, the resulting
VA
waveform is as plotted in Fig. 35c. Since the average voltage across the inductor in the dc steady state is zero, the average voltage VA equals the input voltage f';". The inductor voltage v L pulsates between two values: f';" and (Vo

f';") as plotted in Fig. 35c.
Since the average inductor voltage is zero, the voltsecond areas during the two subintervals are equal in magnitude and opposite in signo
38
The input/output voltage ratio can be obtained either by the wavefonn of v A or v L in Fig. 35c. Using the inductor voltage wavefonn whose average is zero in dc steady state, V;n
(DI;) = (Vo 
v'n )(1 D)T,
(318)
Hence, V o =  V;n ID
(319)
The inductor current wavefonn consists of its average vaI ue, which depends on the output load, and a ripple component that depends on v L
:
i L(t) = I L + i L.riPPI. (t)
where as shown in Fig. 35c,
(320) i L.,;ppl<
(= ~ Jv dr) , L .
whose average value is zero, consists
of linear segments, rising when v L is positive and falling when v L is negative.
The
peakpeak ripple can be ca1culated by using either area A or B: ML =
~V;n(DT,) = ~(Vo L
'..r' Area A
L.
(321)
V;n)(1  D)T, â€˘ . Area B
ln a Boost converter, the inductor current equals the input current, whose average can be calculated from the output load current by equating the input and the output powers: (322) Hence, using Eq. 319 and l o = Vol R, I L
=1 = Vo I =~= _I_Vo ln Vm o lD lDR
(323)
The inductor current wavefonn is shown in Fig. 35c, superposing its average and the ripple components. The current through the diode equals O when the transistor is on, otherwise it equals i L , as plotted in Fig. 35c. ln the dc steady state, the average capacitor current Ic is zero and therefore the average diode current equals the output current l o. ln practice, the filter capacitor is large to achieve the output voltage nearly dc (vo (t) = Vo )
.
Therefore, to the
ripplefrequency component in the diode current, the path through the capacitor offers much smaller impedance than through the load resistance, hence justifies the assumption
39
that the ripple component of the diode current flows entirely through the capacitor. That is,
(324) [n practice, the voltage drops across the capacitor ESR and the ESL dominate over the voltage drop
~
Jicdt across C. The plot of ic in Fig. 35c can be used to calculate the
ripple in the output voltage. The above analysis shows that vo ltage conversion ratio (Eq. 319) of Boost converters in CCM depends on 1/(1 D), and is independent of the output load, as shown in Fig. 36. If the output load decreases to the extent that the inductor current becomes discontinuous below a criticai value I L â€˘a " , the inputoutput relationship of CCM is no longer valid in DCM. If the dutyratio D were to be held constant as shown in Fig. 36, the output voltage could rise to dangerously h igh leveis in DCM, considered fully in section 315 and the Appendix to this chapter.
..,,
,,
,,
,,
]
]D 1_
,
,
+:
______ ____ ~
o r~:;=:j'~======:::;J
DCM I L,cril
CCM
Figure 36 Boost converter: voltage transfer ratio.
37
BUCKBOOST CONVERTER ANALYSIS JN DC STEADY STATE
BuckBoost converters allow the output voltage to be greater or lower than the input voltage based on the switch dutyratio D. A BuckBoost converter is shown in Fig. 37a, where the switching powerpole is implemented as shown. Conventionally, to make the power flow from lefi to the right, the BuckBoost converter is drawn as shown in Fig. 37b. As shown
ln
Fig. 38a, tuming on the transistor applies the input voltage across the
inductor such that v L equals V;n ' and the current linearly ramps up, increasing the energy in the inductor. Tuming off the transistor results in the inductor current flowing through the diode, as shown in Fig. 38b, transferring energy increase in the inductor during the previous transistor state to the output. 310
+ V;n
~
A
+ +
vL
+
lo
vA
I
I
q
+ vA 
L
+ V;n
V
O
vL
lo
1. '
I
L
Vo
+
idiode
(a)
(b)
Figure 37 BuekBoost dede converter.
u
+ (a) .,.,
+
t
t
v AO 

t
I
i (b)
t
Figure 38 BuekBoost converter: operation and waveforms. The transistor switehing funetion is shown in Fig. 38e, with a steady state dutyratio D. The resulting vA waveform is as plotted. Sinee the average voltage aeross the induetor in the de steady state is zero, the average VA equals the output voltage. voItage pulsates between two values:
v'n
The induetor
and  Vo' as plotted in Fig. 38e.
Sinee the 311
average inductor voltage is zero, the voltsecond areas during lhe !wo subintervals are equal in magn ilude and opposite in s igno The input/output voltage ratio can be obtained either by the wavefonn of v A or v[ in Fig. 38c. Using lhe v L waveform whose average is zero in the de steady state, (325) Hence, V
D
V;II
l D
o=
(326)
The induetor current eonsists of an average vai ue, whieh depends on the output load, and a ripple component that depends on v ,. : iL (I) = I L + i,..np,,,, (f) where as shown in Fig. 3Sb,
(327) iL .ripp/,
(=~ Iv,..dr),
whose average value is zero, eonsists
of linear segments, rising when v L is positive and falling when vL is negative.
The
peakpeak ripple ean be ealeulated by using eilher area A or B Ă”iL
=~V;n(DT.)=~v.,(ID)r: L
'v' Area A
L
'v~
(328)
Arca B
Applying lhe KirehhofT's eurrent law in Fig. 37a or b, the average induetor currenl equals the sum of lhe average input e urrent and lhe average outpul eurrenl (note thal lhe average capacitor eUITent is zero in d e steady slale) (329) Eguating the input and the output powers V/ti 1'" =Vo 1
(330)
(J
and using Eg. 326 (33 I) Henee, using Eqs. 329 and 33 I,
I
1
V
1 =1 +1 =   1 =    " L l_D O lDR ln
312
Q
(332)
Superposing the average and the ripple components, the inductor current waveform shown in Fig. 38c.
IS
The diode current is zero, except when it conducts the inductor current, as plotted in Fig. 38c. ln the dc steady state, the average current I c through the capacitor is zero, and therefore by the Kirchhoff's current law, the average diode current eguals the output current. ln practice, the filter capacitor is large to achieve the output voltage nearly dc (vo(t)
~~).
Therefore, to the ripplefreguency current, the path through the capacitor
ofTers much smaller impedance than througb the load resistance, hence justifies the assumption that the ripple component of the diode current flows entirely through the capacitor. That is, (333) ln practice, the voltage drops across the capacitor ESR and the ESL dominate over the voltage drop
~
ficdt across C. The plot of ic in Fig. 38c can be used to calculate the
ripple in the output voltage. The above analysis shows that the voltage converslOn ratio (Eg. 326) of BuckBoost converters in CCM depends on D /(1  D), and is independent of the output load, as shown in Fig. 39. If the output load decreases to the extent that the inductor current becomes discontinuous below a criticaI value I,'oc"," the inputoutput relationship in CCM is no longer valid in DCM. Ifthe dutyratio D were to be held constant as shown in Fig. 39, the output voltage could rise to dangerously high leveIs in DCM, considered fully in section 315 and the Appendix to th is chapter.
.,,
,
,,
D lD 1_
,,
,
,
.. _+:
r,
o ~~:~ I 1
DCM I I . ,cril
CCM
Figure 39 BuckBoost converter: voltage transfer ratio. 371
Other BuckBoost Topologies
There are two variations of the BuckBoost topology, which are used applications. These two topologies are briefly described below.
ln
certain
313
3711 SEPIC Converters (SingleEnded Primary Inductor Converters) The SEPrC converter shown in Fig. 3IOa is used in certain applications where the current drawn from the input is required to be relatively ripplefree. By applying the Kirchhoff's voitage law and the fact that average inductor voltage is zero in the dc steady state, the capacitor in this converter gets charged to an average value that equals the input voltage V;n with the polarity shown. During the oninterval of the transistor, DT" as shown in Fig. 3IOb, the diode gets reverse biased by the sum of the capacitor and the output voltages, and iL1 and i LZ flow through the transistor. During the offinterval (lD)T"
iL 1 and i LZ flow through the diode, as shown in Fig. 310c. The voltage across
4.
V
equals
c during the oninterval, and equals ( Vo ) during the offinterval. ln terms of the average value of the capacitor voltage that equal s V;. (by applying Eq . 39 in Fig. 3IOa), equating the average voltage across Lz to zero results in (334) or, V
o=
V;.
D 1 D
(335)
Unlike the BuckBoost converters, the output voltage polarity remains the sarne as that of the input.
ln
SEPIC converter
+
V;n
(a)
q
(b)
V;n
~=o
O
q=1
~) V L2 )
+
+
v;,
+
q=O Figure 310 SEPIC converter.
314
v;,
3712 Cuk Converter Named after its inventor, the Cuk converter is as shown in Fig. 3lla, where the energy transfer means is through the capacitor C between the two inductors. Using Eq. 39 in Fig. 311a, this capacitor voltage has an average value of
(v,n + VJ with the polarity
shown. During the oninterval of the transistor, Dr" as shown in Fig. 311 b, the diode gets reverse biased by the capacitor voltage and the input and the output currents f10w through the transistor. During the offinterval (I  D)r" the input and the output currents f10w through the diode, as shown in Fig. 3llc. ln terrns of the average values of the inductor currents, equating the net change in charge on the capacitor over
r,
to zero in
steady state, (336) or (337)
~L,
(a)
=::
q
~
:::
m
CL y
1,
C l

(c)
i
q=O
q =1 Figure 311 Cuk converter.
Equating input and output powers in this idealized converter leads to (338) which shows the sarne functionality as BuckBoost converters. One ofthe advantages of the Cuk converter is that it has nonpulsating currents at the input and the output but it
315
suffers from the sarne component stress disadvantages as the BuckBoost converters and produces an output voltage of the po larity opposite to that ofthe input. 38
TOPOLOGY SELECTION [2]
For selecting between the three converter topologies discussed in this chapter, the stresses listed in Table 31 can be compared, which are based on the assumption that the inductor ripple current is negligible. Table 31 Topology Selection Criteria BuckBoost Buck Boost
Criterion Transistor
V
V;n
Vo
(V,n +Vo )
Transistor
i
lo
l ;n
I;n + l o
I~,
Transistor
JDlo
JDlin
JD(I'n +1
I~g
Transistor
Dlo
DJin
D(Iin +1
(1 D)Io
(ID)Iin
(1 D)(Iin + l o )
lo
l in
lin + l o
Effect of L on C
significant
little
little
Pulsating Current
input
output
both
Diode IL
0 )
0 )
From the above table, we can c1early conclude that the BuckBoost converter suffers from severa l additional stresses. Therefore, it should be used only if both the Buck and the Boost capabilities are needed. O therwise, the Buck or the Boost converter should be used based on the desired capability. A detailed analysi s is carried out in [2]. 39
WORSTCASE DESIGN
The worstcase design should consider the ranges ln which the input voltage and the output load vary. As mentioned earlier, ofien converters above a few tens of watts are designed to operate in CeM. To ensure CCM even under very light load conditions would require prohibitively large inductance. Hence, the inductance vaI ue chosen is ofien no larger than three times the criticaI inductance (L < 3Lรง
) ,
where, as discussed in
section 315, the criticaI inductance Lรง is the value of the inductor that will make the converter operate at the border ofCCM and DCM at fullIoad. 310
SYNCHRONOUSRECTIFIED BUCK CONVERTER FOR VERY LOW OUTPUT VOLTAGES [3]
Operating voltages in computing and communication equipment have a lready dropped to an order of 1 V and even lower voltages, such as 0.5 V, are predicted in the near future.
316
At these low voltages, the diode (even a Schottky diode) of the powerpole in a Buck converter has unacceptably high voltage drop across it in comparison to the output voltage, resulting in extremely poor converter efficiency. As a solution to this problem, the switching powerpole ln a Buck converter is implemented using two MOSFETs, as shown in Fig. 312a, which are available with very low
R DS(o.)
in low voltage ratings.
The two MOSFETs are driven by almost
complimentary gate signals (some delay time, where both signals are low is necessary to avoid the shootthrough of current through the two transistors), as shown in Fig. 312b. When the upper MOSFET is off, the inductor current flows through the channel, from the source to the drain, of the lower MOSFET that has gate voltage applied to it. This results in a very low voltage drop across the lower MOSFET. At light load conditions, the inductor current may be al\owed to become negative without becoming discontinuous, flowing from the drain to the source ofthe lower MOSFET [3]. It is possible to achieve softswitching in such converters, as discussed in Chapter 10,
where the ripple in the total output and the input currents can be minimized by interleaving which is discussed in the next section.
v."
+
(a)
Figure 312 Buck converter: synchronous rectified.
311
INTERLEAVING OF CONVERTERS [4]
Fig. 313a shows two interleaved converters whose switching waveforrns are phase shifted by T, / 2, as shown in Fig. 3 l3b. ln general, n such converters can be used, their operation phase shifted by T, / n .
The advantage of such interleaved multi phase
converters is the cancel\ation of ripple in the input and the output currents to a large degree [4]. This is also a good way to achieve higher control bandwidth.
317
q\ .. 1L 1
+
)o
jr,
O~ ·+_L~t
i . 2 f,,
q2 b O
!
LI
L. r'/2 J
(a)
(b)
Figure 313 Interleaving of converters.
312
REGULA TlON OF DCDC CONVERTERS BY PWM
Almost ali dcdc converters are operated with their output voltages regulated to equal their reference values within a specified tolerance band (for example, ± I % around its nominal value) in response to disturbances in the input voltage and the output load. The average output af the switching powerpole in a dcdc converter can be controlled by pulsedwidthmodulating (PWM) the dutyratio d(t) of this powerpole. Fig. 314a shows in a blockdiagram form of a regulated dedc converter. It shows that the converter output voltage is measured and compared with its reference value within a PWM controller IC [5], brief1y described in Chapter 2. The error between the two voltages is amplified by an amplifier, whose output is the control voltage vc(t) .
zdz1z4
dede converter
topology
°t<dT": , ' Ts .E
t
~
11
q(l)! Vo"ef
: '
'.'0
O,.,..L.__'________L __+______~__:;t
(a)
(b) Figure 314 Regulation of output by PWM.
Within the PWMIC [5], the control voltage is compared with a ramp signal v,(t) as shown in Fig. 314b, where the comparator output represents the switching function q(l) whose pulsewidth d(l) can be modulated to regulate the output of the converter. The ramp signal v, has the amplitude
V"
and the switching frequency
J, constant. The
output voltage ofthis comparator represents the transistor switching function q(t), which eq ua ls I if vc(t)? v,; otherwise O. T he switch dutyratio in Fig. 314b is given as
318
d(t) = vcSt)
(339)
v,
thus the control voltage, limited in a range between O and
V"
linearly and dynamically
controls the pulsewidth d(/) in Eq . 339 and shown in Fig. 314b. 313
DYNAMIC A VERAGE REPRESENTATION OF CONVERTE RS lN CCM
ln ali three types of dcdc converters in CCM, the switching powerpole switches between two subcircuit states based on the switching function q(t). (It switches between three subcircuit states in DCM, where the switch can be considered "Stuck" between the on and the off positions during the subinterval when the inductor current is zero, discussed in detai! in the Appendix.) lt is very beneficial to obtain nonswitching average models of these switchmode converters for simulating the converter performance under dynamic conditions caused by the change of input voltage and/or the output load. Under the dynamic condition, the converter dutyratio, and the average values of voltages and currents within the converter vary with time, but relatively slowly with frequencies an order of magnitude smaller than the switching frequency. The switching powerpole is shown in Fig. 315a, where the voltages and currents are labeled with the subscript vp for the voltageport and cp for the currentport. ln the above analysis for the three converters in the dc steady state, we can write the average voltage and current relationships for the bipositional switch ofthe powerpole as (340a) (340b) Ivp
+

+ vvp
icp
Icp
â€˘
â€˘
v;,p
+
+
V:P
+ v cp
1: D
q(/)
(a)
(b)
(c)
Figure 315 Average dynamic model of a switching powerpole.
319
Relationships in Eq. 340 can be represented by an ideal transformer as shown in Fig. 315b. Under dynamic conditions, the average model in Fig. 315b of the bipositional switch can be substituted in the powerpole of Fig. 315a, resulting in the dynamic average model shown in Fig. 315 c, using Eq. 339 for d(I). Here, the uppercase letters used in the dc steady state relationships are replaced with lowercase letters with a "" on top to represent average voltages and currents, which may vary dynamically with time: D by d(I), Vcp by ,,<p (t), I cp by !;p (t) , and I "p by I"p(t). Therefore, from Eqs. 340a
and b: (341a) (341b)
I"p(l) = d(l) !;p(t)
The above discussion shows that the dynam ic average model of a switching powerpole in CCM is an ideal transformer with the turnsratio I: d(t). Using this model for the switching powerpole, the dynamic average models of the three converters shown in Fig. 316a are as in Figs. 316b in CCM. Note that in the Boost converter where the transistor is in the bottom position in the powerpole, the transformer turnratio is I: (ld), because the pole dutyratio d A is 1 minus the transistor dutyratio d(t). The average representation ofthese converters in DCM is described in the Appendix.
+ +
iL
Vo
q
+
q
p
iL Vo
.u.

Vo
.
f!;n
iL
 p 1: (ld(t))
:
~
+
J~
'Y
1 : d(t)
+
V;n
Vo
+
A
+
A

r 
r ~
1: d(t)
Figure 316 Average dynamic models ofthree converters. The ideal transformers in the circuits of Fig. 3 I 6b, being hypothetical and only a convenience for mathematical representation, can operate with ac as well dc voltages and currents, which a real transformer cannot. A symbol consisting of a straight bar with a curve below is used to remind us of this facto Since no electrical isolation exi sts between 320
the voltageport and the currentport of the switching powerpole, the two windings in the circuits ofFig. 3ISc and Fig. 316b are connected at the bottom. Moreover, the voltages in the circuit of Fig. 316 cannot become negative (however, depending on the implementation of the powerpole, the currents may become negative, that IS, reverse directions), and d is limited to a range between O to I. ln the average representation of the switching powerpole, ali the switching information is removed, and hence it provides an uncluttered understanding of achieving desired objectives in various converters. Moreover, the average model in simulating the dynamic response of a converter results in computation speeds orders of magnitude faster than that in the switching model, where the simulation time step must be smaller than at least onehundredth ofthe switching timeperiod 1:, in order to achieve an accurate resolution. 314
BIDIRECTlONAL SWITCHING POWERPOLE
ln Buck, Boost and BuckBoost dcdc converters, the implementation of the switching powerpole by one transistor and one diode dictates the instantaneous current flow to be unidirectional. As shown in Fig. 317a, by combining the switching powerpole implementations ofBuck and Boost converters, where the two transistors are switched by complimentary signals, allows a continuous bidirectional power and current capability. ln such a bidirectional switching powerpole, the positive inductor current as shown in Fig. 317b represents a Buck mode, where only the transistor and the diode associated with the Buck converter take parto Similarly, as shown in Fig. 317c, the negative inductor current represents a Boost mode, where only the transistor and the diode associated with the Boost converter take parto We will utilize such bidirectional switching powerpoles in dc and ac motor drives discussed in Chapters II and 12.
Buck
Buck Boost l r"'"]
r
Boost
o
!!
+
j! ::
l
iL
l
+
l.'
V;n
!
:y:i ,v' . rÂˇ...
q =1
q
=
+
i~,
I, ~=O
O
!
q
q = 1 or O (a)
(b) i L = positive
q = 1 or O
(c) i L = negative
Figure 317 Bidirectional power flow through a switching powerpole. ln a bidirectional switching powerpole where the transistors are gated by complimentary signals, the current through it can flow in either direction, and hence ideally, a discontinuous conduction mode does not existo The average representation of 321
the bidirectional switching powerpole in Fig. 318a is an ideal transformer shown in Fig. 318b with a tumsratio 1 :d(t), where d(t) represents the pole dutyratio that IS also the dutyratio ofthe transistor associated with the Buck mode.
Buck r··1 Boost
+
~.__._:
+ ::
::
q
iL
!,!
1: d
LI:>' q  = (1  q)
(a) Figure 318 315
(b)
Average dynamic mode! of the switching powerpole with bidirectional power flow.
DISCONTINUOUSCONDUCTION MODE (DCM)
Ali dcdc converters for unidirectional power and current flow have their switching powerpole implemented by one transistor and one diode, and hence go into a discontinuous conduction mode, D CM, below a certain output load. This operating mode for ali three converters is examined in detail in the Appendix, whereas only the criticaI loads above which converters operate in CCM are examined here. As an example as shown in Fig. 319, if we keep the switch dutyratio constant, decline in the output load results in the inductor average current to decrease until a criticaI load value is reached where the inductor current waveform reaches zero at the end of the tumoff interval. The average inductor current in this condition, we will call tbe criticaI inductor current, f L •Cril
'
For loads below this criticaI vaI ue, the inductor current cannot
reverse through the diode in any of the three converters (Buck, Boost and BuckBoost), and enters DCM where the inductor current remains zero for a tinite interval.
iL2 ____________ ILl ,i:.... "~... ~:  .       
,',
.,":.,.  ;'". 
O "
,'
'"
, "
,
;~
'",
, "
I L2 I
L cri!
•
t Figure 319 Inductor current at various loads; dutyratio is kept constant. ln DCM, the inductor current conduction becomes discontinuous for an interval during which there is no power drawn from the input source and there is no energy in the
322
inductor to transfer to the output stage. This interval of inactivity during which the inductor current remains zero generally results in increased device stresses and the ratings of the passive components. DCM also results in noise and EMI, although the diode reverse recovery problem is minimized. Based on these considerations, converters above a few tens of watts are generally designed to operate in CCM, although ali of them implemented using one transistor and one diode will enter DCM at very light loads and the feedback controller should be designed to operate adequately in both modes. It should be noted that designing the controller of some converters, such as the BuckBoost converters, in CCM is much more complicated, thus the designers may prefer to keep such converters in DCM for ali possible operating conditions. This we will discuss further in the next chapter dealing with the feedback controller designo The inductor current at the criticaI average value, although at the border of CCM and DCM, should be considered to belong to the CCM case. At this criticaI load condition, the inductor voltage v L and the current iL are drawn in Fig. 320ac for the three converters shown in Figs. 33a, 34a and 37a. As can be seen from the iL waveform in these criticaI cases in Fig. 320, the average inductor current is onehalf the peakpeak ripple.
v;,
vAj
A
t
o
i
(b)
I L:
! DT,
(I D)1$
J
o
~~+t
Cc)
"i
iL
v
V;n + Vo
iL
iL t
o 07;
(ID )Ts
Figure 320 Waveforms for the three converters at the border of CCMlDCM. ln the Buck converter of Fig. 33a, the peak inductor current in Fig. 320a can be calculated by the fact that a voltage Cv'o  Vo) is applied across the inductor for an interval
323
DT, interval when the transistor is on, causing the current to rise from zero to its peal< vaI ue. Therefore, the peal< current is [' L cril Buek . .
=
(V,n  Vo ) DT
(342)
s
L
With f, = 1/ T, ,and Vo = DV,n' the average current is (343)
I L cril Buck = V;n D(l  D) . . 2Lf, ln both the Boost and the BuckBoost converters,
v'n
is applied across the inductor for an
interval DT, interval when the transistor is on, causing the current to rise from zero to its peal< value in Figs. 320b and c, respectively. Since the average inductor current is onehalf of the peal< value in these waveforms, the criticaI value is as follows: [
.
L,cnt ,BoOSI
= [
V
L ,crlt ,Buck  Boost
(3 44)
= "'D 2 Lfs
If the average inductor current falis below its criticaI value in a converter, then its operation enters the DCM mode, which is discussed in detail in the Appendix. REFERENCES I.
2. 3. 4. 5.
N. Mohan, T. M. Undeland, and W .P . Robbins, Power Electronics: Converters,
Applications and Design, 3,d Edition, Wiley & Sons, New York, 2003 . B. Carsten, "Converter Component Load Factors; A performance Limitation of Various Topologies" PCI Proceedings, June 1988, pp. 3149. M. Walters, "An lntegrated SynchronousRectifier Power IC with ComplementarySwitching (lllP5010, lllP5011)" Technical Brief, July 1995 , TB332, Intersil Corpo Intemational Rectifier (http://www.irf.com). Unitrode for PWM Controller ICs (http://www.unitrode.com).
PROBLEMS Buck DCDC Converters ln a Buck dcdc converter, L = 50 j.JH. following conditions:
v'n
It is operating in dc steady state under the
= 40V , D = 0.3 , p" = 24W , and f, = 200kHz. Assume ideal
components. 31
Calculate and draw the waveforms as shown in Fig. 33d.
32
Draw the inductor voltage and current waveforms if Po = 12 W ; ali else is unchanged. Compare the ripple in the inductor current with that in Problem 31.
324
33
ln this Buck converter, the output load is changing. Calculate the criticai value of the output load p" below which the converter will enter the discontinuous
34
conduction mode of operation. Calculate the criticai vai ue of the inductance L below which this Buck converter will enter the discontinuous conduction mode at p" = 5 W under ali input voltage
35
36
values. Draw lhe waveforms for the variables shown in Fig. 33d for this Buck converter at the output load that causes it to operate at the border of continuous and discontinuous modes. ln this Buck converter, the input voltage is varying in a range from 24 V to 50 V. For each input vai ue, the dutyratio is adjusted to keep the output voltage constant at its nominal va lue (with v'n = 40V and D = 0.3). minimum value of the inductance L
Calculate the
lhat will keep the converter in the
continuous conduction mode at Po = 5W. Boost DCDC CQoverters ln a Boost converter, L = 50 fLH. lt is operating in dc steady state uoder the following conditions:
v'n
= 12V,
D
= 0.4,
Po = 30W,
and
J, = 200kHz.
Assume
ideal
components. 37
Calculate and draw and the waveforms as shown in Fig. 35c.
38
Draw the inductor voltage and current waveforms, if Po = 15 W; ali else is
39
unchanged. Compare the ripple in the inductor current with that in Problem 37. Ln this Boost converter, the output load is changing. Calculate the criticai value of the output load p" below which the converter will enter the discontinuous
310
conduction mode of operation. Calculate the criticai value of the inductance L below whicb this Boost converter will enter the discontinuous conduction mode of operation at Po = 5 W .
31 I
312
Draw the waveforms for the variables in Fig. 35c for this Boost converter at the output load that causes it to operate at the border of continuous and discontinuous modes. ln this Boost converter, the input voltage is varying in a range from 9 V to 15 V. For each input vai ue, the dutyratio is adjusted to keep the output voltage constant at its nominal value (with v'n = 12 V and D = 0.4 ). minimum value of the inductance L
Calculate the
that will keep the converter
10
the
continuous conduction mode at p" = 5 W under ali input voltage values.
325
BuckBoost DCDC Cooverters lo a BuckBoost converter, L = 50 JiH. following conditions: v'n
= 12V ,
D
It is operating in dc steady state under the
= 0.6,
p"
= 36W, and J, = 200kHz. Assume ideal
components. 313
Calculate and draw the waveforrns as shown in Fig. 38c.
314
Draw the inductor voltage and current waveforrns if p" = 18W; ali else is
315
unchanged. Compare the ripple in the inductor current with that in Problem 313. ln this BuckBoost converter, the output load is changing. Calculate the criticai value of the output load
316 317
318
p"
below which the converter will enter the
discontinuous conduction mode of operation. Calculate the criticaI value of the inductance L below which this BuckBoost converter will enter the discontinuous conduction mode of operation at Po = 5 W . Draw the waveforrns for the variables in Fig. 38c for this BuckBoost converter at the output load that causes it to operate at the border of continuous and discontinuous modes. ln this BuckBoost converter, the input voltage is varying in a range from 9 V to 15 V. For each input vai ue, the dutyratio is adjusted to keep the output voltage constant at its nominal value (with v'n = 12 V and D = 0.6 ). Calculate the minimum value of the inductance L that will keep the converter in the continuous conduction mode of operation at p" = 5 W under ali input voltage values.
SEPIC DCDC Cooverters 319
ln a SEPIC converter, assume the ripple in the inductor currents and the capacitor voltage to be zero. This SEPIC converter is operating in a dc steady state under the following conditions: v'n = 10V, D
= 0.333,
p"
= 50W, and J, = 200kHz.
Assume ideal components. Draw the waveforrns for ali the converter variables under this dc steadystate condition. Cuk DCDC Cooverters 320
ln a Cuk converter, assume the ripple in the inductor currents and the capacitor voltage to be zero. This Cuk converter is operating in a dc steady state under the following conditions: v'n
= 10 V,
D
= 0.333, Po = 50W, and J, = 200kHz.
Assume ideal components. Draw the waveforrns for ali the converter variables under this dc steadystate condition.
326
Interleaving ofDCDC Converters 321
Two interleaved Buck converters, each similar to the Buck converter dermed earlier before Problem 3 1, are supplying a total of 48W. Calculate and draw the waveforms of the total input current and the total current (iI. + iI, ) to the output stage. The gate signals to the converters are phase shifted by 1800
.
Regulation by PWM 322
ln a Buck converter, con sider two values of dutyratios: 0.3 and 0.4. switching frequency
J, is 200kHz and V,
The
= 1.2 V . Draw the waveforms as in
Fig 314b. Dynamic A verage Models in CCM 323 324
Draw the dynarnic average representations for Buck, Boost, BuckBoost, SEPIC and Cuk Converters in the continuous conduction rnode. ln the converters based on average representations in Problem 323, calculate the average input current for each converter in terms of the output current and the dutyratio d(t) .
Bidirectional Switching PowerPoles 325
The dcdc bidirectional converter ofFig. 318a interfaces a 121l4V battery with a 36/42V battery bank. The internal ernfs are E I = 40V (dc) and E , = 13V (dc). Both these battery sources have an internal resistance of O. Hl each. ln the dc steady state, calculate the powerpole dutyratio D A if(a) the power into the lowvoltage battery terminaIs is 140 W , and (b) the power out of the lowvoltage battery terminais is 140 W.
APPENDIX3A
DISCONTINUOUSCONDUCTION MODE (DCM) lN DCDC CONVERTERS
As briefly discussed earlier in section 315, ali dcdc converters for unidirectional power and current f10w (implemented using one transistor and one diode) enter a discontinuous conduction rnode, DCM, below a certain output load. The inductor current criticai values at the border of CCM and DCM were derived for the three converters, and are given by Eq. 343 in Buck converters, and by Eq . 344 in both the Boost and the BuckBoost converters.
327
3Al
OUTPUT VOLTAGES lN DCM
3All Buck Converters in DCM For the Buck converter in Fig. 33a, an output load less than the criticaI load (that is, higher than the criticaI load resistance) resuIts in the inductor current during the offinterval of the transistor to drop to zero, prior to the beginning of the next switching
cycle, as shown in Fig. 3AIa. The inductor current iL cannot reverse through the diode, and hence becomes discontinuous, remaining zero until the beginning of the new switching cyc1e. The normalized off interval of the transistor now consists of two subintervals: D off •1 and D off ." such that (D + D Off •1 + D off .') = I. During D off •2 ' v A jumps from O to Vo as shown in Fig. 3AIa. The extra voltseconds, shown as the area hatched in Fig. 3AIa, averaged over the switchingcycle, result in the output voltage
v;,
to go
higher than its CCM vaI ue. The output voltage is plotted in Fig. 3AI b for a dutyratio D, which shows that at very light loads in DCM, the output voltage approaches v,n' A detailed derivation is presented in the Appendix on the accompanying CD.
,,
I
,,
Do
~
+__________ IL
L _ _ _
.......
~     . . ;
DCM
IL, crit
CCM
(b)
(a)
Figure 3AI Buck converter in DCM. 3A12 Boost and BuckBoost Converters in DCM
Unlike Buck converters, Boost and BuckBoost converters represent a special challenge in DCM, where at very light loads, their output voltage, if not properly regulated can reach dangerously high values, and therefore, care should be taken to avoid this condition. Only the waveforms and the plot of the voltage conversion ratio are presented here in Fig. 3A2 and Fig. 3A3 respectively, and the detailed discussion is inc1uded in the accompanying CD.
328
VA)n~nm~mln~V..n. !
I
j !!
il· ~ii~ o~
I
'
D
~ ~i
D"",
··
i f       , ,
!:
o
•··
DoIr_'
I
T.
ID 1 
,
_________ _
~:;:=l=======:;T, iIL
o ~
1';
DCM JL.<ri, (b)
(a)
CCM
Figure 3A2 Boost converter in DCM.
•·· ,
0'+
1',
~ ID
_
··
~~~+: :
______
1 :o
, I
1L .<ri,
(a)
CCM (b)
Figure 3A3 BuckBoost converter in DCM. 3A2
A VERAGE REPRESENTA TION OF DeDC CONVERTERS lN DCM
ln the previous discussion, we saw that unlike the CCM where the output voltage was independent of the load and was dictated only by the input voltage
v,.
and the transistor
dutyratio d(t), the output voltage in DCM also depends on the converter parameters and the operating condition. The average representation of a switching powerpole in DCM can be obtained by the voltage and current wavefonns associated with the converter. ln DCM, since the output voltage at the voltageport is higher than that in the CCM case, the average model of a switching powerpole in CCM by an ideal transfonner is augmented by a dependent voltagesource v. at the currentport and a dependent currentsource i. at the voltageport. The values of these dependent sources for the three converters are calculated in the Appendix on the accompanying CD, and the results are presented below. 3A21 v. and i. for Buck and BuckBoost Converters in DCM The average representation of the switching powerpole in Buck and BuckBoost converters in DCM is shown in Fig~ 3A4 in tenns ofthe dynamic quantities, where v
 [1
k ,Buck 
2L.f,~ 
<V;n _
vo)d
jV
o
(3Al)
329
2
. lk
B llck
.
d  (Vin = 
2Lf.

) d 7lL Vo 
(3A2)
(3A3)
i k,Buck BooSI
d2 = 2Lfs ~n
_

(3A4)
diL Vk
,   <e
+
fj).,. icp
+
1: d(t) Figure 3A4 Average dynamic model for Buck and BuckBoost converters. If v. and i k are conditional such that they are both zero in CCM and are expressed by the above expressions only in DCM w here the average inductor current falIs beIow the criticai value, then the representatio n in Fig. 3A4 becomes valid for both the CCM and theDCM. ln PSpice the ideal transformer itself is represented by a dependent current source and a dependent voltage sources. The conditional dependent sources vk and i, shown in Fig. 3A4 are in addition to those used to represent the ideal transformer. 3A22
Vk
and i. for Boost Converters in DCM
The average representation of the switching powerpole in Boost converters in DCM is shown in Fig. 3A5 in terms ofthe dy namic quantities, where
2Lf,T,. V d
J(V  v ) ln
o
(3A5)
'" d2 ik,Boost
330
= ~n
2Lf,
_
diL
(3A6)
+
(ld): I Figure 3AS Average dynamic model for Boost converters. By expressing v. and i. conditionally such that they are both zero in CCM and are expressed by Eqs. 3AS and 3A6 only in DCM where the average inductor current falis below the criticaI vaI ue, the representation in Fig. 3AS becomes valid for both the CCM and the DCM.
331
Chapter 4
DESIGNING FEEDBACK CONTROLLERS lN SWITCHMODE DC POWER SUPPLIES 41
INTRODUCTION AND OBJECTIVES OF FEEDBACK CONTROL
As shown in Fig. 41, almost ali dcdc converters operate with their output voltage regulated to equal their reference value within a specified tolerance band (for example, ± 1 % around its nominal vaI ue) in response to disturbances in the input voltage and the output load. This regulation is achieved by pulsedwidthmodulating the dutyratio d(t) of their switching powerpole. ln this chapter, we will design the feedback controller to regulate the output voltages of dcdc converters.
V.
DCDC
v.
Converter
T Controller
V
Figure 41 Regulated dc power supply. The feedback controller to regulate the output voltage must be designed with the following objectives in mind: zero steady state error, fast response to changes in the input voltage and the output load, low overshoot, and low noise susceptibility. We should note that in designing feedback controllers, ali transformerisolated topologies discussed later in Chapter 8 can be replaced by their basic singleswitch topologies from which they are derived. The feedback control is described using the voltagemode control, which is later extended to include the currentmode control. The steps in designing the feedback controller are described as follows: •
Linearize the system for small changes around the dc steady state operating point (bias point). This requires dynamic averaging discussed in the previous chapter.
•
Design the feedback control\er using linear control theory.
41
â€˘
42
Confirm and evaluate the system response by simulations for large disturbances.
REVIEW OF LINEAR CONTROL THEORY
A feedback control system is as shown in Fig. 42, where the output voltage is measured and compared with a reference value
v,:.
The error between the two acts on the
controller, which produces the control voltage vc(t).
This control voltage acts as the
input to the pulsewidthmodulator to produce a switching signal q(t) for the powerpole in the dcdc converter. The average vai ue of this switching signal is d(t), as shown in Fig. 42. VoO
~
Controller
v.
Pulse Width ModuJation
d
Power Stage
and Load
Vo
PWMIC
/
~ Figure 42 Feedback control. To make use of linear control theory, various blocks in lhe power supply system of Fig. 42 are linearized around lhe steady slale dc operaling point, assuming smallsignal perturbations. Each average quantity (represented by a "" on top) associated with the powerpole of the converter topology can be expressed as the sum of its steady slate dc value (represented by an uppercase letter) and a smallsignal perturbation (represented by a",....." on top), for example, " o(t) = Vo + vo(t) d(t)=D+d(t) vo(t)
=
(41)
V, + vo(t)
where de!) is already an averaged vai ue and vc(t) does not contain any switching frequency component. Based on the smallsignal perturbation quantities in the Laplace domain, the linearized system block diagram is as shown in Fig. 43, where the perturbation in lhe reference input to this feedbackcontrolled system,
v;, is zero since
the output voltage is being regulated to its reference vai ue. ln Fig. 43, G pWM (s) is lhe transfer function of lhe pulsewidth modulalor, and G ps (s) is lhe power stage transfer function. ln the feedback path, the transfer function is of the voltagesensing network, which can be represented by a simple gain k VH ' usually less than unity.
42
Gc (s) is the
transfer function of the feedback controller that needs to be determined to satisfy the control objectives. )=0
~ , ~B
Control ler
vÂŤ s)
Gc (s)
PulseWidth Modulalor GI'JI:\I
d(s)
Power Stage
v. ( s)
+ Output Fiher
Gps(s )
(s)
~ Figure 43 Small signal contraI system representation. 421
Loop Traosfer Fuoctioo G L (s)
It is the closedIoop response (with the feedback in place) that we need to optimize.
Using linear control theory, we can achieve this objective by ensuring certain characteristics ofthe loop transfer function G L(s). ln the contraI block diagram ofFig. 43 , the loop transfer function (from point A to point B) is (42) 422
Crossover Frequeocy j , or G,. (s)
ln order to define a few necessary contraI terms, we will consider a generic Bode plot of the loop transfer function GL(s) in terms ofits magnitude and phase angle, shown in Fig. 44 as a function of frequency.
T he frequency at which the gain equals unity (that is
IGL(s)I=OdB) is defined as the crassover frequency
fc
(or me)'
This crassover
frequency is a good indicator ofthe bandwidth ofthe closedIoop feedback system, which determines the speed of the dynamic response of the control system to various disturbances. 423
Pbase and Gaio Margios
fc , the At fc, the
For the closedIoop feedback syste m to be stable, at the crassover frequency phase delay introduced by the loop transfer function must be less than 1800 phase angle L. GI. (s )1
/,
.
of the loop transfer function G I . (s), measured with respect to
180 0 , is defined as the Phase Margin (t/JPM ) as shown in Fig. 44: (43)
43
ii> ~
50 r,   ,,,
E
o  , 
~
i6'
,
~
C_50
,h,:
, ___ ____ .' __ __ ;: __
~~
4 _
: .:
~
____
l"Gain Margin _
I
I I
~ _ 100 ~~~~~,_~~~~ 2 3
10~
.3
10Â°
.
o ~_.+_._+,,
~ ~
90
Q.
10'
10
10
~~~~ ~  ~
, ,
_1 __     
Phase Margi n
c
~  180 Q.
g
,
270
,
__ ____ .1. ____ __
J
_ _ _ _
_ 1. ___ ___ _
~
10 '
10' Frequency (Hz)
Figure 44 Definitions of crossover frequency, gain margin and phase margino Note that LGL(s)1
h
is negative, but the phase margin in Eq. 43 must be positive.
Generally, feedback controllers are designed to yield a phase margin of approximately 60 0 , since much smaller values result in high overshoots and long settling times (oscillatory response) and much larger values in a sluggish response. The Gain Margin is also deĂ?med in Fig. 44, which shows that the gain margin is the value of the magnitude of the loop transfer function, measured below O dB, at the frequency at which the phase angle of the loop transfer function may (not always) cross 1800
.
If the phase angle crosses 180 0 , the gain margin should generally be in excess
of 10 dB in order to keep the sy stem response from becoming oscillatory due to parameter changes and other variations. 43
LINEARIZA TION OF V ARIOUS TRANSFER FUNCTION BLOCKS
To be able to apply linear control theory in the feedback controller design, it is necessary that ali the blocks in Fig. 42 be linearized around their dc steady state operating point, as shown by transfer functions in Fig. 43. 431
Linearizing the PulseWidth Modulator
ln the feedback control, a highspeed PWM integrated circuit such as the UC3824 [I] from Unitroderrexas Instruments may be used. Functionally, within this PWMIC shown in Fig. 45a, the control voltage v, (t) generated by the error amplifier is compared with a ramp signal v, with a constant amplitude frequency
.Is , as
V,
at a constant switching
shown in Fig. 45b. The output switching signal is represented by the
switching function q(/), which equals I if v, (/) ;0: v, ; otherwise O. The switch dutyratio in Fig. 45b is given as
44
(44)
t
â€˘
PWMIC (b)
(e)
Figure 45 PWM waveforms. ln terms of a disturbance around the dc steady state operating point, the control voltage can be expressed as (45) Substituting Eq. 45 into Eq. 44, der) = VcSt) + vcSt) V, ~
...D
(46)
d(t)
ln Eq. 46, the second term on the right side equals d(r), from which the transfer function ofthe PWMlC is (47) It is a constant gain transfer function , as shown in Fig. 45c in the Laplace domain. ~
Example 41
and instead of
V,
ln PWMICs, there is usually a dc voltage offset in the ramp voltage, as shown in Fig. 45b, a typical ValleytoPeak value ofthe ramp signal
is defined. ln the PWMlC UC3824, this valleytopeak value is 1.8 V. Calculate the linearized transfer function associated with this PWMIC.
Solution
The dc offset in the ramp signal does not change its small signal transfer
function. Hence, the peaktovalley voltage can be treated as
1 V,
1 1.8
Gp._f (s) =,; =  =0.556
""
V,.
Using Eq. 47 (48)
45
432
Linearizing the Power Stage ofDCDC Converters in CCM
To design feedback controllers, the power stage of the converters' must be linearized around the steady state de operating point, assuming a smallsignal disturbanee. Fig.46a shows the average model of the switehing powerpole, where the subscript "vp" refers to the voltageport and "cp" to the eurrentport. Each average quantity in Fig. 46a ean be expressed as the sum of its steady state dc value (represented byan uppercase letter) and a smallsignal perturbation (represented by a "" on top): d(/) = D + d(1)
""p(t) =
1"", + v",,(/)
~p(t) = ~p
(49)
+ v", (t)
""Ă‡,(t) = 1"" + J.'P (t) 7;p (t) = l cl' + lcp (t) 
T:p(l)
+
t:p(t)
+
V;.p(t)
d(/)
f.p (I)
,<c +i> :  
+ ~/I)
(a)
+
v,p (I)
vcp (t)
(b)
Figure 46 Linearizing the switehing powerpole. Uti lizing the voltage and current relationships between the two ports expressing each variable as in Eq. 49
10
Fig. 46a, and
(4IOa) and, 1,1' + l.p = (D + d)(1,p + ~p)
(4IOb)
Equating the perturbation terms on both s ides ofthe above equations (41 la) and, l.p =
D~p + 1,pd + ']i(
(4llb)
The two equations above are lineari zed by neg leet ing the produets of small perturbation terms (erossed out in Eqs. 411a and b). The resulting lin ear equations are
46
(412) and, (413) Eqs. 412 and 4 13 can be represented by means of an ideal transformer shown in Fig. 46b, which is a linear representation of the powerpole for small signals around a steady state operating point given by D ,
v.", and l cp '
TIle average representations ofBuck, Boost and BuckBoost converters are shown in Fig. 4 7a. Replacing the powerpole in each of these converters by its smallsignals linearized representation, the resulting circuits are shown in Fig. 47b, where the perturbation ii;n is zero based on the assumption of a constant dc input voltage 1I;n' and the output capacitor ESR is represented by r. Note that in Boost converters, since the transistor is in the bottom position of the switching powerpole, D in Eqs. 412 and 413 needs to be replaced by (1 D) and d by (d).
+
c=> ".
+ =
O
cil, I
Buck
I : d(t)
I: D
+ + r
Boost
(I  d(/» : 1

(I D): 1
i",
+
v"J>
+
v. 
•• r;+~"""''''r1 +
~p
BuckBoost
I : d(/) (a)
•
I:o§
r
.
d(V. + v.) !i..~
~
T > ? ~
+
I:D (b)
Figure 47 Linearizing singleswitch converters in CCM. As fully explained in the Appendix on the accompanying CD, ali three circuits for smallsignal perturbations in Fig. 47b have the sam e form as shown in Fig. 48. ln this equivalent circuit, the effective inductance L, is the same as the actual inductance L in the Buck converter, since in both states of a Buck converter in CCM, L and C are
47
always connected together.
However, in Boost and BuckBoost converters, these two
elements are not always connected, resulting in L , to be L /( I D)2 in Fig. 48: L, = L (Buck);
L,
L 2 (I D)
(Boost and BuckBoost)
(414)
L.
+
se
v..
r
Figure 48 Small signal equivalent circuit for Buck, Boost and BuckBoost converters. Transfer functions of the three converters in CCM from the Appendix on the accompanying CD are repeated below:
v~ d
"o J
(1 DL,) (I_D)2 sR L C(S2 v,.
=
(Buck)
(415)
(Boost)
(416)
I + srC (B k B ) +s(_I_ +.!:...J+_l_J ue  oost
,
RC
L,
(417)
L,C
ln the above powerstage transfer f unctions in CCM, there are several characteristics worth noting. There are two poles created by the lowpass LC filter in Fig. 48, and the capacitor ESR r results in a zero. ln Boost and BuckBoost converters, their transfer functions depend on the steady state operating value D. They also have a righthalfplane zero, whose presence can be explained by the fact that in these converters, increasing the dutyratio for increasing the output, for example, initially has an opposite consequence by isolating the input stage from the output load for a longer time. 432 J Using Computer Simulation to Obtain
"o/d
Transfer functions given by Eqs. 4 15 through 417 provi de theoreticaJ insight into converter operation. However, the Bode plots of the transfer function can be obtained with similar accuracy by means of linearization and ac analysis, using a computer
48
program sueh as PSpieeâ„˘ The converter eireuit is simulated as shown in Figs. 49 in the example below for a frequeneydomain ae analysis, using the switching powerpole average model discussed in Chapter 3 and shown in Fig. 46a. The dutyeycle perturbation d is represented as an ac souree whose frequency is swept over several deeades of interest, and whose amplitude is kept constant. ln such a simulation, PSpice first calculates voltages and currents at the dc steady state operating point, linearizes the circuit around this de bias point, and then performs the ac analysis . A Buck converter has the following parameters and is operating
..... Example 42
ln
CCM: L=IOOf.1H, C = 697f.1F , r=0.H2 , f,=IOOkHz, V,n=30V, and p,,=36W. The dutyratio D is adjusted to regulate the output voltage
v"
= 12V. Obtain both the
gain and the phase of the power stage Gps(s) for the frequencies ranging from 1 Hz to 100 kHz. Solution The PSpice circuit is shown in Fig. 49 where the de voltage source {D}, representing the dutyratio D, establishes the dc operating point. The dutyratio
perturbation J is represented as an ac source whose freq ueney is swept over several decades of interest, keeping the amplitude constant. (Sinee the circuit is linearized before the ac analysis, the best choice for the ac source amplitude is 1 v.) The switching powerpole is represented by an ideal transformer, which consists of!wo dependent sourees: a dependent eurrent source and a dependent voltage source. The circuit parameters are speeified by means of parameter bloeks within PSpice. Ideal Transformer VP1
o
{Vin}
V ou!
dutyratio D
:l
!
R_c (Re)
D
:
.
C1
:
~{C}
W<?d
.
PARAMETERS: Vin 30 Vo 12
{L}
(D)
:;:

... ...
N<:p
~
. ....,b
L1
CP1
o 3 3 o
RLoad
{R}
{VoNin}
PARAMETERS: L 100u 697uF C
Re
01
PARAMETERS: R
4
L
Figure 49 PSp ice Circuit model for a Buck converter. The Bode plot of the frequency response is shown in Fig. 410. It shows that at the crossover frequency
.fc = 1kHz selected in the next example, Example 43, the power
stage has IGps(sl = 24.66dB and LGps(s)lJ; = 138Â°.
We will make use of these
values in Example 43.
49
~ k24.66dB ~
B

>
.
.. [:8IYIY .... tlJ
~
,
\
L.GI'S(S )
.,
.,, 10011%
'"
....
I.QIOI:z
138째 3 . 01Ob:
Figure 410 The gain and the phase ofthe power
44
I
~tage
G ps (s).
FEEDBACK CONTROLLER DESIGN lN VOLT AGEMODE CONTROL
The feedback controller design is presented by means of a numerical example to regulate the Buck converter described in Example 42. The controller is designed for the continuous conduction mode (C CM) at fullIoad, which although not optimum, is stable inDCM . .... Example 43 Design the feedback controller for the Buck converter described in Example 42. The PWMIC is as described in Example 41. The output voltagesensing network in the feedback path has a gain k FB = 0.2. The steady state error is required to be zero and the phase margin of the loop transfer function should be 60째 at as high a crossover frequency as possible. ln deciding on the transfer function G c (s) of the controller, the control objectives translate into the following simultaneous characteristics of the loop transfer Solution
function G L (s), from which Gc (s) can be designed: I.
The crossover frequency fc of the loop gain is as high as possible to result in a
2.
fast respo nse ofthe closedIoop system. The phase angle of the loop transfer function has the specified phase margm, typically 60째 at the crossover frequency so that the response in the closedIoop system settles quickly without oscillations.
3.
410
The phase angle of the loop transfer function should not drop below 180째 at frequencies below the crossover frequency .
The Bode plot for the power stage is obtained earlier, as shown in Fig. 410 of Example 42. ln this Bode plot, the phase angle drops towards 180° due to the two poles of the LC tilter shown in the equivalent circuit of Fig. 48 and contirmed by the transfer function of Eq. 415. Beyond the LC tilter resonance frequency, the phase angle increases toward _90° because of the zero introduced by the output capacitor ESR in the transfer function ofthe power stage. We should not rely on this capacitor ESR, which is not accurately known and can have a large variability. A simple procedure based on the Kfactor approach [2] is presented below, which lends itself to a straightforward stepbystep designo For reasons given below, the transfer function Gc (s) of the controller is selected to be of the form in Eq. 418 whose Bode plot is shown in Fig. 41l.
(1 +s / m.)2 (l+s / mp
(418)
)'
'v' phaseboost
I...
"'::::::

IGc(s)l};

I.
I
o
v
./
)
 90 o
.......

L,
~

Figure 411 Bode plot of Gc (s) in Eq. 418. To yield a zero steady state error, GcCs) contains apoIe at the origin, which introduces 90° phase shift in the loop transfer function. The crossover frequency
h
of the loop is
chosen beyond the LC resonance frequency of the power stage, where, unfortunately, LGPS 1 /, has a large negative value. The sum of 90° , due to the pole at the origin in Gc (s) , and LGpsl/, is more negative than 180°. Therefore, to obtain a phase margin of 60° requires boosting the phase at zeroes at
h
h,
by more than 90° , by placing two coincident
to nulliJ'y the effect of the two poles in the powerstage transfer function 411
Gps(s). Two coincident poles are placed at fp (> h) to rolloff the gain rapidly much before the switching frequency. The controller gain IGc (s)lJ; is such that the loop gain equals unity at the crossover frequency. The input specifications in determining the parameters of the controller transfer function in Eq . 418 are
h, t/l""""
as shown in Fig. 41 I, and the controller gain IGc (s )1 < /
,
A step
bystep procedure for designing G c (s) is described below. Step I:
Choose the Crossover Frequency.
Choose.fc to be slightly beyond the LC
resonance frequency I /(2".J LC) , which in this example is approximately 600 Hz. Therefore, we will choose
.fc =
1 kHz.
This ensures that the phase angle of the loop
remains greater than 180° at ali frequencies. Step 2:
t/lPM
Calculate the needed Phase Boost.
= 60° . The required phase boost
t/l""""
The desired phase margin is specified as at the crossover frequency is calculated as
follows, noting that G pWM and k FB produce zero phase shift: LG!. (s)lJ; = LGps (st + LGc (s)l /<
(from Eq. 42)
(4 19)
LG L(s)l/< = 180 + t/lPM
(from Eq. 43)
(420)
(from Fig. 41 I)
(421 )
0
0
LGc (s)l J; = _90 + t/lboo." Substituting Eqs. 420 and 421 into Eq. 419,
t/lboo"
= _90
0
+ t/lPM  LG/,s (s)l J;
(422)
ln Fig. 410, LGps(s)lJ; ~ 138°, substituting which in Eq. 422 yields the required phase boost t/lb~' = 108
0 •
Step 3: Calculate the Controller Gain at the Crossover Frequency. From Eq. 42 at the crossover freq uency
h (423)
ln Fig. 410, at
.fc
= 1kHz, IGps(s)IJ;" kH' = 24.66dB = 17.1 . Therefore in Eq. 423 , using
the gain ofthe PWM block ca1culated in Example 41, (424)
412
or (425) The controller in Eq. 418 with two polezero pairs IS thoroughly analyzed in the Appendix to this chapter on the accompanying CD. According to this analys is, the phase angle of Gc (s) in Eq. 418 reaches its maximum at the geometric mean frequency
Jfz/
p
,
where the phase boost rft"oo>t , as shown in Fig. 411 , is measured with respect to
90 0 â€˘ By proper choice of the controller parameters, the geometric mean frequency is made equal to the crossover frequency fc. We introduce a factor K""">t that indicates the geometric separation between poles and zeroes to yield the necessary phase boost: (426) As shown in the Appendix, this factor can be derived in terms of <p"",,, as
K boosl = tan (45 0 + rftJxxM 4 ,
J'
(427)
ln terms of K JxxM" from Eqs 426 and 427, the controller parameters can be calculated as follows : (428) (429) (430)
Once the parameters in Eq. 418 are determined, the controller transfer function can be synthesized by a single opamp circuit shown in Fig. 412.
C2
~+
Figure 412 Implementation ofthe controller in Eq. 418 by an opamp.
413
The choice of R, in Fig. 412 is based on how much current can be drawn from the sensor output; other resistances and capacitances are chosen using the relationships derived in the Appendix and presented below: C 2 = OJ,I(k/JJpR,)
C, =C2 (OJ p /OJ, I) R,
(431 )
= l/(OJ,C,)
R, = R, /(OJ p / OJ, I) C, = l/(OJpR,) ln this numerical example with
fc
= 1kHz, tPboas' = 1080
calculate K boas' = 3 .078 in Eq. 427.
Jp = 3078Hz, and kc = 349.1.
,
and IGc(s)l r, = 0.5263, we can
J,
Using Eqs. 427 through 430,
= 324.9Hz,
For the opamp implementation, we will select
R,=IOOkn. FromEq.430, C 2 =3.0nF, C,=25.6nF, R,=19.lkn, R,=11.8kn, and C, = 4.4nF.
We can verifY the design of Example 43 in PSp ice by adding the
controller to the converter model. Notice that the op amp is modeled as a highgain differential amplifier in Fig. 413 , w here a stepchange in load is applied at 1 ms. The results are plotted in Fig. 414. Ideal trans CCM1
I' I
. ;;::
Vd
cp, Ncp
, ,
voo< C2
I
"'e5a.,~
R_<
" C
""'"
"
R1....
P
"
Rp_lod
'00
lD
RJ
.,. :T<:+
~z02
~
~.;,
I
.,.
,.,.
c,
Cl
.."'"
'j "" R2
""'"
E_Op_AmI>
"'" I.
2 /.v ::!:Vb E2
1
c:r
19. 11l
â€˘
"1>
"'"
GAlNsI.!t
â€˘
~o ~O.S56 .0.0
Figure 413 PSpice average model ofthe Buck converter with voltagemode control. Note that in the controller transfer function Gc (s) in the example above, both zeroes are at the same frequency; the same is true for both the poles. This need not be the case for possibly achieving a higher crossover frequency.
414
12.2V,       .       ,
12.1N+,Jf\~~__+__1 1LW't1'+         j lL6V!:l::1 S... IIlIns (Js
a V{V_ aut )
Figure 414 Response to a stepchange in load.
.... 45
PEAKCURRENT MODE CONTROL
Currentmode control is often used in practice due to ils many desirable features, such as simpler controller design and inherent current limiting. ln such a control scheme, an inner controlloop inside the outer voltageloop is used as shown in Fig. 415, resulting in a peakcurrentmode control system. ln this control arrangement, another statevariable, the inductor current, is utilized as a feedback signal. The overall voltageloop objectives in the currentmode control are the same as in the voltagemode control discussed earlier. However, the voltageloop controller here produces lhe reference value for the current that should flow through the inductor, hence the name currentmode control. There are two types of currentmode controlo â€˘
PeakCurrentMode Control, and
â€˘
AverageCurrentMode Control. +
l
)
v.
V., +
+ V.
Slope Compensation
Q
Controller Flipflop
Comparator
Figure 415 Peak current mode control. [n switchmode dc power supplies, peakcurrentmode control IS invariably used, and therefore we will concentrate on it here . (We will examine lhe averagecurrentmode
415
control in connection with the powerfactorcorrection circuits discussed chapter.)
ln
the next
For the current loop, the outer voltage loop in Fig. 415 produces the reference value ( of the inductor current.
This reference current signal is compared with the measured
inductor current iL to reset the flipflop when iL reaches 416a, in generating
i;, the voltage controller output
i;.
As shown in Figs. 415 and
ic is modified by a signal called the
slope compensation, which is necessary to avoid oscillations at the subharrnonic frequencies of j"
particularly at the dutyratio d> 0.5.
Generally, the slope of this
compensation signal is less than onehalf of the slope at which the inductor current falis when the transistor in the converter is tumed off. ln Fig. 416a, when the inductor current reaches the reference value, the transistor is tumed off, and is tumed back on at a regular interval T,(=l/ j,) set by the clock. For small perturbations, this current loop acts extremely fast, and it can be assumed ideal with a gain of unity in the smallsignal block diagram of Fig. 416b. The design of the outer voltage loop is described by means of the example below of a BuckBoost converter operating in CCM.
OL________~~ Clock
17; ='~ J; .(a) )=0
Yn
'[;1
7;(s)
Controller
Peak Current Mode Controller
i;.(s)
Power Stage
>. (s)
~l
Gc (s)
(b)
Figure 416 Peakcurrentmode control with slope compensation. ..... Example 44 ln this example, we will design a peakcurrentmode controller for a BuckBoost converter that has the following parameters and operating conditions:
L=100,uH, C=697,uF, r=O.Oln, j,=IOOkHz, V,n=30V. p" = 18 W in CCM and the dutyratio D
416
The output power
is adjusted to regulate the output voltage
Vo = 12 v. The phase margin required for the voltage loop is 60° . Assume that in the
voltage feedback network, k FB = 1 . Solution
ln designing the outer voltage loop in Fig. 416b, the transfer function needed
for the power stage is Vo
T
This transfer function in CCM can be obtained
/ L •
theoretically. However, it is much easier to obtain the Bode plot of this transfer function by means of a computer simulation, sim ilar to that used for obtaining the Bode plots of
Vo /
êi
in Example 42 for a Buck converter. The PSpice simulation diagram is shown in
Fig. 417 for the BuckBoost converter, where, as discussed earlier, an ideal transformer is used for the average representation ofthe switching powerpole in CCM. ln Fig. 417, the dc voltage source represents the switch dutyratio D and establishes the dc steady state, around which the circuit is linearized. ln the ac analysis, ihe frequency of the ac source, which represents the dutyratio perturbation
iL (s)
range, and the ratio of vo(s) and
êi,
is swept ove r the desired
yields the Bode plot ofthe power stage Gps (s) , as
shown in Fig. 418.
.
Vd (Vin}...=
.

VP1
o
o
CP1
o
Ncp
3 3
"
Vo
..
Idelll trans CCM1
L1
V
ou!
{L}
{O}
;1;3 _
;;;r
1V '""
o
12 (1J(VinNo+1)}
PARAMETERS:
~
R_c' {Rc)
•
C1 :
~
.
.<t>2
30
PARAMETERS: Vin
=:
{C)
L
100u
RLoad {RL) PARAMETERS:
c Rc RL
697u 0.01
8
... Figure 417 PSpice circuit for the BuckBoost converter.
417
,

....
" 
• D8IV(V outJ/ICL'I)
's
  ~.33dBl
~

~
~ .,
,~
r
V  90°
>
Figure 418 Bode plot of Vo 1"[, . As shown in Fig. 4 18, the phase angle of the powerstage transfer function leveis off at approximately _90° at  1kHz. The crossover frequency is chosen to be J; = 5 kHz , at which in Fig. 418,
LGps(s)/J; '" 90° .
As explained in the Appendix on the
accompanying CD, the powerstage transfer function
vo(s) I "[, (s)
of BuckBoost
converters contains a righthalfplane zero in CCM. The crossover frequency is chosen well below the frequency of the righthalfplane zero for reasons discussed in the Appendix. To achieve the desired phase margin of 60° , the controller transfer function is chosen as expressed as below: k Gc(s)=' S
(I+s l cv.) (l+s l cvp )
(432)
',,'
phaseboost
To yield zero steady state error, it contains apoIe at the origin which introduces a _90 0 phase angle.
The phaseboost required from this polezero combination in Eq. 432,
using Eq. 422 and LGpS(s)/fo '" 90°, is rf>boo." '" 60°. Therefore, unlike the controller transfer function of Eq. 418 for the voltagemode control, only a single polezero pair is needed to provide phase boost. ln Eq. 432, the zero and pole frequencies associated with the required phase boost can be derived, as shown in the Appendix on the accompanying CD, where Kb~" is defined the same as in Eq. 426 :
418
K """', = tan (45 0 + r/>boa<' 2 )
(433) (434) (435) (436)
At the crossover frequency, as shown in Fig. 418, the power stage transfer function has a gain IGps (s)I/, = 29.33dB. Therefore, at the crossover frequency, by definition, in Fig. 416b (437) Hence, (438) Using
the
equations
above
for
h=5kHz,
r/>,~, =60 o ,
and
IGc (s)I/, =29.27,
K """" = 3.732 in Eq. 432. Therefore, the parameters in the controller transfer function
of Eq. 431 are ca1culated as
1". = 1340Hz, f p = 18660Hz, and
kc = 246.4 X 10' .
The transfer function of Eq. 432 can be realized by an opamp circuit shown in Fig. 419.
ln the expressions derived in the Appendix, selecting R, = 10kQ and using the
transferfunction parameters calculated above, the component values in the circuit of Fig. 419 are as follows:
C2 =
30 pF
1lJ,
IlJpR,kc
C, =C2 (llJp /llJ, 1)=380pF
(439)
R, = 1I(IlJ,C,)=315kQ
c, RI
. v.
~~
~~>~
Figure 419 lmplementation of controller in Eq. 432 by an opamp circuit.
419
The PSpice circuit diagram is shown in Fig. 420 where a load change occurs at 3 ms. Using this simulation, the output voltage and its average are plotted in Fig. 421. 51
.I
....1r1:'1' m
30
L:"J
l1
IC=5A Vo
• ••
H1~ H
V1
:::::!5:
100uH
I
U1 Ro
...
\J tClose=3ms
0.01
~
8 RLoad
RLoad2 697uF : I... Com IC=12V r""
16
C2
.,
';"""
GAJN:1e5
V3
12 ,........,.
I
• Figure 420 PSpice simulation diagram ofthe peakcurrentmode control. 12 . 04 ,,,,,
12
11 .9 2 ~_+~_+~
2 . 5Oms
2 . 75ms
o AVGX (V(Vo) , lOu)
3 . 0Oms
3 . 25ms
3 . 50ms
v VIVo)
Time
Figure 421 Peak current mode control: Output voltage waveform.
46
FEEDBACK CONTROLLER DESIGN lN DCM
ln sections 44 and 45, feedback controllers were designed for CCM operation of the converters. The procedure for designing controllers in DCM is the sarne, except the average model of the power stage in PSpice simulations can be simply replaced by its model which is also valid in DCM, as described in Chapter 3. 420
REFERENCES I. 2. 3.
Unitrode PWM IC (www.unitrode.com) H . Dean Venable, "The KFactor: A New Mathematical Tool for Stability Analysis and Synthesis," Proceedings ofPowercon lO (http://www.venable.biz). N. Mohan, T. M. Undeland, and W.P. Robbins, Power Electronics: Converters. Applicalions and Design, 3'" Edition, Wiley & Sons, New York, 2003.
PROBLEMS 41
ln a voltage mode controlled DCDC converter the loop transfer function has the crossover frequency
fc
= 2 kHz. The power stage transfer function has a phase
angle of 160째 at the crossover frequency. Calculate w, and w p in the voltage controller transfer function ofEq. 418, ifthe required phase margin is 60째. 42
ln the above problem lhe power stage has a gain equal to 20 at the crossover frequency , k FB = 0.2, and G pWM = 0.6.
43
Calculate kc in the voltage controller
transfer function ofEq. 418. ln a peakcurrentmode controlled DCDC converter the loop crossover frequency in the outer voltage loop is 10 kHz. At this crossover frequency, the power stage in Fig. 416b has the gain of O. I, and the phase angle of 80째. Calculate /, and
/p
in the controller transfer function of Eq. 432 if the desired
phase margin is 60째 .
421
Chapter 5
RECTIFICATION OF UTILITY INPUT USING DIODE RECTIFIERS 51
INTRODUCTION
Power electronic systems generally get their power from the utility source, as shown by the block diagram in Fig. 51. Unless a corrective action is taken as described in the next chapter, this power is drawn by means of highly distorted currents, which have a deleterious effect on the power quality of the utility source. Furthermore, the power system disturbances in the utility source can disrupt the power electronics system's operation. Both ofthese issues are examined in this chapter. r             ,
,..,:
o
Converter Load
Source
Figure 51 Block diagram of power electronic systems. o
52
DISTORTION AND POWER FACTOR
To quantifY distortion in the current drawn by power electronic systems, it is necessary to define certain indices. As a base case, consider the linear R  L load shown in Fig. 52a which is supplied by a sinusoidal source in steady state. The voltage and current phasors are shown in Fig. 52b, where I/> is the angle by which the current lags the voltage. Using rms values for the voltage and current magnitudes, the average power supplied by the source is (51)
P = VJ, cosl/>
The power factor (PF) at which power is drawn is defined as the ratio of the real average power P to the product of the rms voltage and lhe rms current: P
PF =   = cosI/>
V,I,
(using Eq. 51)
(52)
For a given voltage, from Eq. 52, the rms current drawn is
51
J =
,
P
(53)
V, ÂˇPF
This shows that the power factor PF and the current J, are inversely proportional. The current flows through the utility distribution lines, transformers, and so on, causing losses in their resistances. This is the reason why utilities prefer unity power factor loads that draw power at the minimum value o fthe rms curren!. is
.. f7,
+ Vs
~
'V
J,
(b)
(a) Figure 52 Voltage and current phasors in simple RL circui!. 521
RMS VaI ue ofDistorted Current and the Total Harmonic Distortion (THD)
The sinusoidal current drawn by the linear load in Fig. 52 has zero distortion. However, power electronic systems with diode rectifiers as the frontend draw currents with a distorted waveforrn such as that shown by i, (t) in Fig. 53a. The utility voltage v, (t) is assumed sinusoidal. The following analysis is general, applying to the utility supply that is either singlephase or threephase, in which case the analysis is on a perphase basis .
..
1 ~
1
0 ~~ (a)
Figure 53 Current drawn by power electronics equipment with diodebridge frontend. The current wavefom1 i,Ct) in Fig. 53a repeats with a timeperiod I;.
By Fourier
analysis of this repetitive waveform , we can compute its fundamental frequency ( = 11 I; ) component i'l (I) , shown dotted in Fig. 53a. The distortion component id",oHwn (t) in the input current is the difference between i, (1) and the fundamentalfrequency component i' l (I) :
(54) 52
where
idi"""i~ (I)
using Eq. 54 is plotted in Fig. 53b. This distortion component consists
of components at frequencies that are the multiples of the fundamental frequency. To obtain the rrns vai ue of i, (I) in Fig. 53a, we will apply the basic detmition of rrns: (55)
Using Eq. 54, (56)
ln a repetitive waveforrn, the integral of the products of the two harmonic components (including the fundamental) at unequal frequencies, over the repetition timeperiod, equals zero:
ff~ (I)Âˇ g ", (I). dt = O
(57)
r,
Therefore, substituting Eq. 56 into Eq. 55, and making use of Eq. 57 that implies that the integral ofthe third terrn on the right side of Eq. 56 equals zero, 1, =
_I
Ji;, (t). dI + _I fi~i"OH;~ (t). dI + O 7; ,.
7;,
,
' I
"
I
(58)
,
or, (59)
where the rrns values of the fundamentalfrequency component and the distortion component are as follows:
I" =
_I fi;, (t). dI
7; ,,
(510)
and I (JistQrtion
=
~Ji~istOr1ion(t). dt
(511)
'r,
Based on the rms values of the fundamental and the distortion components in the input current i, (I) , a distortion index called the Total Harmonic Distortion (THD) is defined in percentage as follows:
53
% THD
=
100 X
Idisrortion
(512)
I" Using Eq . 59 into Eq. 512, '1 2 1 2 %THD=100x'" " I"
(513)
The nns value of the distortion component can be obtained based on the hannonic components (except the fundamental) as follows using Eq. 57:
where 1,. is the rms value ofthe harmonic component "h" . • Example 5 1
A current i, of square wavefonn is shown in Fig. 54a. Calculate and
plot its fundamental rrequency component and its distortion component. %THD associated with this wavefonn?
What is the
From Fourier analysis, i,(t) can be expressed as
Solution
. =4 I( SlflúV+sm . I . 3 úV+sm 1. 5 úJ/+sm 1. 7 w.t+ .... .)
(515) " 3 5 7 The fundamental rrequency component and the distortion component are plotted in Figs. 54b and 54c. I,
idislortion
I (e)  ]
Figure 54 Example 51.
54
Since the nns value 1, of the square wavefonn is equal to 1 , the nns value of the distortion component can be calculated from using Eq. 59 in Eq. 515 I dl,'O",on =
~ 1; I;,
=
~1 2  (0.91)2 = 0.4361 .
Therefore, using the definition of THD , %THD = 100 x I d",orllon = 100 x 0.4361 = 48.4%. I" 0.91 522
The Displacement Power Factor (DPF) andPower Factor (PF)
Next, we will consider the power factor at which power is drawn by a load with a distorted current wavefonn such as that shown in Fig. 53a. As before, it is reasonable to assume that the utilitysupplied linefrequency voltage v, (t) is sinusoidal, with an nns value of V, and a frequency J,(= ~ ). 8ased on Eq. 57, which states that the product 21! of the crossfrequency tenns has a zero average, the average power P drawn by the load in Fig. 53a is due only to the fundamentalfrequency component ofthe current: p=_1 fV, (/)'i, (/).dl = _1 f V, (/)'i" Ct).dl
:z; ",
:z; "
(516)
T
Therefore, in contrast to Eq. 51 for a linear load, in a load that draws distorted current, similar to Eq. 51 p =
where
~
V,I"
cos~
(517)
is the angle by which the fundamentalfrequency current component i" (I) lags
behind the voltage, as shown in Fig. 53a. At this point, another tenn called the Displacement Power Factor (DP F) needs to be introduced, where DPF = cos~
(518)
Therefore, using the DPF in Eq . 51 7, P
=
VJ" (DPF)
(519)
ln the presence of distortion in the current, the meaning and therefore the definition of the power factor, at which the real average power P is drawn, remains the same as in Eq. 52, that is, the ratio of the real power to the product of the nn s voltage and the nns current:
55
P
(520)
PF=
V,l,
Substituting Eg. 519 for Pinto Eg. 520, (521 ) ln linear loads that draw sinusoidal currents, the currentratio (1'1 I I , ) in Eg. 521 is unity, hence PF = DPF. Eg. 521 shows the following : a high distortion in the current wavefonn leads to a low power factor, even if the DPF is high. Using Eg. 513 , the ratio (1'1 I I , ) in Eg. 521 can be expressed in tenns ofthe Total Harrnonic Distortion as
_1'_1 = I,
c===I==~
(522)
I+ ( %THD)' 100
Therefore, in Eg. 521 , (523)
PF = r==I==7 Âˇ DPF I+ (%THD )' 100
The effect of THD on the power factor is shown in Fig. 55 by plotting (PF I DPF) versus THD. II shows that even if the displacement power factor is unity, a total harrnonic distortion of 100 percent (which is possible in power electro nic systems unless corrective measureS are taken) can reduce the power fact0r to approximately 0.7 (or
lz
= 0.707 to be exact), which is unacceptably low. /
0.9
0.8
PF
0.7
DPF
O.6
~
~
0.5
'" ""
"
"'
0.4 50
/00
" '..
5
%THD
,o

<50
300
Figure 55 Relation between PF/DPF and THD.
56
523
Deleterious Effects ofHarmonic Distortion and a Poor Power Factor
There are severa I deleterious effects of high distortion in the current waveforrn and the poor power factor that results due to iI. T hese are as follows: â€˘
Power loss in utility equipment such as distribution and transmission Iines, transformers, and generators increases, possibly to the point of overloading them.
â€˘
Harrnonic currents can overload the shunt capacitors used by utilities for voltage support and may cause resonance conditions between the capacitive reactance of these capacitors and the inductive reactance of the distribution and transmission lines.
â€˘
The utility voltage waveforrn will also beco me distorted, adversely affecting other linear loads, ifa significant portion ofthe load supplied by the utility draws power by means of distorted currents.
5231 Harmonic Guidelines ln order to prevent degradation in power quality, recommended guidelines Cin the forrn of the IEEE519) have been suggested by the IEEE (lnstitute of Electrical and Electronics Engineers). These guidelines place the responsibilities of maintaining power quality on the consumers and the utilities as fo llows: 1) on the power consumers, such as the users of power electronic systems, to limit the distortion in the current drawn, and 2) on the utilities to ensure that the voltage supply is sinusoidal with less than a specified amount of distortion. The limits on current distortion placed by the IEEE519 are shown in Table 51, where the limits on harrnonic currents, as a ratio of the fundamental component, are specified for various harrnonic frequencies . Also, the limits on the THD are specified. These limits are selected to prevent distortion in the voltage waveforrn of the utility supply. Table 51 Harmonic current distortion (1,/1J Odd Harm onicOrder h(in%) h < II
II"h < 17
23 " h < 35
35" h
< 20
4.0
2.0
1.5
0.6
0.3
5.0
20  50
7.0
3.5
2.5
1.0
0.5
8.0
50  100
10.0
4.5
4.0
1.5
0.7
12.0
100  1000
12.0
5.5
5.0
2.0
1.0
15.0
> 1000
15.0
7.0
6.0
2.5
IA
20.0
I se
/
I,
17 " h < 23
Tolal Harmonic Dislorlion(%)
57
Tberefore, the limits on distortion in Table 51 depend on the "stiffness" of the utility supply, which is shown in Fig. 56a by a voltage source
ii',
in series with internal
impedance Z , . An ideal voltage supply has zero internal impedance. ln contrast, the voltage supply at the end of a long distribution line, for example, will have large internal impedance.
z,
,
L_ _ _ _~_"
(a)
(b)
Figure 56 (a) Utility supply; (b) short circuit current.
To define the "stiffness" of the supply, the shortcircuit current I ," is calculated by hypothetically placing shortcircuit at the supply terminais, as shown in Fig. 56b. The stiffness of the supply must be calculated in relation to the load current. Therefore, the stiffness is defllled by a ratio called the ShortCircuitRatio (SCR): ShortCircuitRatio SCR = I,e I"
(524)
where I" is the fundamentalfrequency component ofthe load current. Table 51 shows that a smaller shortcircuit ratio corresponds to lower limits on the allowed distortion in the current drawn. For the shortcircuitratio of less than 20, the total harmonic distortion in the current must be less than 5 percent. Power electronic systems that meet this limit would also meet the limits ofmore stiffsupplies. It should be noted that the IEEE519 does not propose harrnonic guidelines for individual
pieces of equipment but rather for the aggregate of loads (such as in an industrial plant) seen from the service entrance, which is also the pointofcommoncoupling (PCC) with other customers. However, the IEEE519 is freyuently interpreted as the harmonic guidelines for speeif'ying individual pieees of equipment such as motor drives. There are other harrnonic standards, such as the IECIOOO, which apply to individual pieces of equipment.
53
CLASSIFYING THE "FRONTEND" OF POWER ELECTRONIC SYSTEMS
Interaction between the utility supply and power electronic systems depends on the "frontends" (within the powerprocessing units), which convert linefrequency ac into de. These frontends can be broadly classified as follows: 58
•
Diodebridge rectifiers (shown in Fig. 57a) in which power flows only in one direction
•
Switchmode converters (shown in Fig. 5 7b) in which the power flow can reverse and the line currents are sinusoidal at the unity power factor
•
Thyristor converters (shown in Fig. 57c) in which the power flow can be made bidirectional
(a)
(b) Figure 57 Frontend of power electronics equipment.
(c)
Ali of these frontends can be des igned to interface with singlephase or threephase utility systems. ln the following discussion, a brief description of the diode interface shown in Fig. 57a is provided, supplemented by analysis of results obtained through computer simulations. Interfaces using switchmode converters in Fig. 57b and thyristor converters in Fig. 57c are discussed later in this book.
54
DIODERECTIFIER BRIDGE "FRONTENDS"
Most power electronic systems use diodebridge rectifiers, like the one shown in Fig. 57a, even though they draw currents with highly distorted waveforms and the power through them can flow only in one direction. ln switchmode dc power supplies these diodebridge rectifiers are supplemented by a powerfactorcorrection circuit, to meet current harmonic limits, as discussed in the next chapter. Diode rectifiers rectify linefrequency ac into dc across the dcbus capacitor, without any control over the dcbus voltage. F or analyzing the interaction between the utility and the power electronic systems, the switchmode converter and the load can be represented by an equivalent resistance R,q across the dcbus capacitor. Tn our theoretical discussion, it is adequate to assume the di odes ideal. Tn the following subsections, we will consider singlephase as well as threephase diode rectifiers operating in steady state, where waveforms repeat from one Iinefrequency cycle I; ( = 1/ J;) to the next.
59
541
SinglePhase DiodeRectifier Bridge
At power leveIs below a few k W, for example 10 residential applications, power electronic systems are supplied by a singlephase utility source. A commonly used fullbridge rectifier circuit is shown in Fig. 58, in which L, is the sum of the inductance internal to the utility supply and an external inductance, which may be intentionally added in series. Losses on the ac s ide can be represented by the series resistance R, .

idr

is L,
2
id
+
R,
Vd
Cd
R,q
3
Figure 58 Fullbridge diode rectifier. As shown in Fig. 59, at the beginning of the positive halfcycle of the input voltage v" the capacitor is already charged to a dc voltage
Vd â€˘
So long as vd exceed s the input
voltage magnitude, ali diodes get reverse biased and the input current is zero . Power to the equivalent resistance R." is supplied by the energy stored in the capacitor up to ti . Beyond ti' the input current i, ( = id , } increases, flowing through diodes D I and D 2 â€˘ Beyond t" the input voltage becomes smaller than the capacitor voltage and the input current begins to decline, falling to zero at t,. Beyond t" until onehalf cycle later than ti' the input current remains zero and the power to R,q is supplied by the energy stored
in the capacitor.
o
Ivsl Figure 59 C urrent and voltage waveforms for the fullbridge diode rectifier.
510
At (t\ +
:z;) 2
during the negative halfcycle of the input voltage, the input current flows
through diodes D3 and D.. The rectifier dcside current id' continues to flow in the sarne direction as during the positive halfcycle; however, the input current i, = id' , as shown in Fig. 59. Fig. 510 shows waveforms obtained by PSpice simulations for two values ofthe acside inductance, with current THD of 86% and 62%, respectively (higher inductance reduces THD, as discussed in the next section).
",,,,.,,,,
. . . V( t.. , u
• • I ( t..1 · 3 • • V(IU:2,Cl:21
Figure 510 Singlephase diodebridge rectification. The fact that ia, flows in the sarne direction during both the positive and the negative halfcycles represents the rectification processo ln the circuit of Fig. 58 in steady state, ali waveforms repeat from one cycle to the next. Therefore, the average value of the capacitor current over a linefrequency cycle must be zero so that the dcbus voltage is in steady state. As a consequence, the average current through the equivalent loadresistance R,q equals the average of the rectifier dcside current; that is, ld = la,. 54 1 I Effects of L, and Cd on the Waveforms and the THD As Figs. 59 and 510 show, power is drawn from the utility supply by means of a pulse of current every halfcycle. The larger the "base" of this pulse during which the current flows, lower its peak vai ue and the total harmonic distortion (THD). This pulsewidening can be accomplished by increasing the acside inductance L , .
Another
pararneter under the designer' s control is the value of the dcbus capacitor C a . At its minimum, it should be able to carry the ripple current in id, and in ia (which in practice is the input dcside current, with a pulsating waveform, of a switchmode converter), and 511
keep the peaktopeak ripple in the dcbus voltage to some acceptable value, for example less than 5 percent of the dcbus average value. Assuming that these constraints are met, lower the value of Cd , lower the THD and higher the ripple in the dcbus voltage. ln practice, it is almost impossible to meet the harrnonic limits specified by the IEEE519 by using the above techniques. Rather, the powerfactorcorrection circuits described in the next chapter are needed to meet the harmonic specifications. 542
ThreePhase DiodeRectifier Bridge
It is preferable to use a threephase utility source, except at a fractional kilowatt, if such a
supply is available. A commonly used FullBridge rectifier circuit is shown in Fig. 5II a. To understand the circuit operation, the rectifier circuit can be drawn as in Fig. 51 I b. The circuit consists of a top group a nd a bottom group of diodes. Initially, L, is ignored and the dcside current is assumed to flow continuously. At least one diode from each group must conduct to facilitate the flow of id ,
.
ln the to p group, ali diodes have the ir
cathodes connected together. Therefore, the diode connected to the most pos itive voltage will conduct; the other two will be reverse biased. ln the bottom group, ali diodes have their anodes connected together. T herefore, the diode co nnected to the most negative voltage will conduct; the other two w ill be reverse biased . .!I!!:v
L ..
 "v
+
Vb
"v Ve
"v
Ls
5
3
+ +
Ls Vd
+
p
5
L, Cd
i~
ia
eq
L,
L 2
N
2 (b)
(a)
Figure 51 I Threephase diode bridge rectifier. Ignoring L, and assuming that the dcside current id , is apure dc, the waveforms are as shown in Fig. 512. I n Fig. 512a, the waveforms Cidentified by the dark po rtions of the curves) show that each diode, based on the principie described above, conducts during 1200
â€˘
The diodes are numbered so that they beg in conducting sequentially: I, 2, 3, and
so on . The waveforms for the voltages v p and vN
'
with respect to the sourceneutral ,
consist of 1200 segments of the phase voltages, as shown in Fig. 512a. The waveform ofthe dcside voltage vdC=
V I' 
vN
)
is shown in Fig. 512b. It co nsists of 60 0 segments
ofthe lineline voltages supplied by the utility . Line currents on the acside are as sho wn
512
in Fig. 512c. For example, phasea current flows for 120° during each halfcycle of the phasea input voltage; it flows through diode D, during the positive halfcycle of
Va ,
and
through diode D. during the negative halfcycle.
____.__~C=___
:1~~E=~1~2~OO_~~~.160"""ll_ _ _
mI
....J
~lr,L~~,ml
i:p
(b)
mI
(c)
Figure 512 Waveforms in a threephase rectifier (a constant id , ) . The average value of the dcside voltage can be obtained by considering only a 60° segment in the 6pulse (per linefrequency cycle) waveform shown in Fig. 512b. Let us consider the instant of the peak in the 60° segment to be the timeorigin, with
VLL
as the
peak lineline voltage. The average value Vd can be obtained by calculating the integral from
(j)(
=
ff /
6 to
(j)(
=
ff /
then dividing by the interval
1 Vd =  
6 (the area shown by the hatched area in Fig. 512b) and ff /
3:
3 J Vu. cosml ·d(m/) = V ff/3 _~/6 ~/6
A
A
LL
(525)
ff
This average value is plotted as a straight line in Fig. 512b. ln the threephase rectifier of Fig. 5lla with the dcbus capacitor filter, the input current waveforms obtained by computer simulations are shown in Fig. 513. Fig. 513a shows that the input current waveform within each halfcycle consists of two distinct pulses when L, is small. For example, in the ia waveform during the positive halfcycle, the first pulse corresponds to the f10w of dcside current through the diode pair (D" D 6 ) and then through the diode pair (D" D 2 ) . At larger values of L" within each halfcycle, the input current between the two pulses does not go to zero, as shown in Fig.513b. The effects of L, and Cd on the waveforms can be determined by a parametric analysis, similar to the case of singlephase rectifiers. The THD in the current waveform ofFig. 513b is much smaller than in Fig. 513a (23% versus 82%). The acside inductance L, is 513
required to provide a linefrequency reactance X L, ( = 2nj,L, ) that is typically greater than 2 percent ofthe base impedance 2 ba"
,
which is defined as follo ws:
v'
(526)
Base Impedance 2 ba" = 3'
P
where P is the threephase power rating of the power electronic system, and V, is the rms value of the phase voltage. Therefore, typically, the minimum acside inductance should be such that (527)
2~!:: _,;,±: _:...,,:!:_=:!._ =:! ._ I:c;:!:::,~_ D !IL!) "3 • "<1.1 , 11
(a)
2":!:: _,;,±: _:,:!:_=:! _ =::1:c;:!:::,~ _ .. nUlO) • VILIIII
(h)
Fig ure 513 Effect of L, variation (a) L, = 0. 1mH ; (b) L, = 3mH .
514
543
Comparison ofSinglePhase and ThreePhase Rectifiers
Examination of singlephase and threephase rectifier waveforrns shows the differences in their characteristics. Threephase rectification results in six identical " pulses" per cycle in the rectified dcside voltage, whereas singlephase rectification in two such pulses. Therefore, threephase rectifiers are superior in terrns of minimizing distortion in line currents and ripple across the debus voltage. Consequently, as stated earlier, threephase rectifiers should be used if a threephase supply is available. However, threephase rectifiers, just like singlephase rectifiers, are also unable to meet the harrnonic limits specified by the IEEE5) 9 unless corrective actions sueh as those deseribed in Chapter ) 2 are taken. 55
Means to Avoid Transient Inrush Currents at Starting
ln power eleetronic systems with rectifier frontends, it may be neeessary to take steps to avoid a large inrush of eurrent at the instant the system is eonnected to the utility source. ln sueh power electronic systems, the debus eapacitor is very large and initially has no voltage across it. Therefore, at the instant the switch in Fig. 514a is elosed to connect the power electronie system to the utility source, a large current flows through the diodebridge reetifier, charging the debus capacitor. This transient eurrent inrush is highly undesirable; fortunately, several means of avoiding it are available. These inelude using a frontend that consists of thyristors discussed in Chapter 14 or using a series semiconduetor switeh as shown in Fig. 514b. At the instant of starting, the resistanee aeross the switeh lets the debus capaeitor get eharged without a large inrush current, and subsequently the semiconductor switeh is turned on to bypass the resistance.
(a)
(b)
Figure 514 Means to avoid inrush eurrent.
515
56
FRONTENDS WITH BIDIRECTIONAL POWER FLOW
ln stopandgo applications such as elevators, it is costeffective to feed the energy recovered by regenerative braking of the motor drive back into the utility supply. Converter arrangements for such applications are considered in Chapter 12.
REFERENCES I.
2.
N. Mohan, T. M. Undeland, and W.P. Robbins, Power E/ectronics: Converters, Applications and Design, 3,d Edition, Wiley & Sons, New York, 2003. N. Mohan, Power Electronics: Computer Simu/ation, Ana/ysis and Education using PSpiceâ„˘, a complete simulation package available from www.mnpere.com.
PROBLEMS 51
ln a singlephase diode rectifier bridge, I , = I OA(rms) , I" = 8A(rms) , and DPF = 0.9. Calculate Idi,W.''''. and %THD.
52
ln a singlephase diode bridge rectifier circuit, the following operating condition are given: V, = 120V(rms) , P = lkW, 1,,= 10A , and THD = 80 % . Calculate the following: DPF, I di"oHion' I , and PF.
53
ln a singlephase rectifier the input current can be approximated to have a triangular waveform every half cycle with a peak of 10A and a base of 60Â°.
54 55
516
Calculate the rms current through each diode . ln the above problem, calculate the ripple component in the dcside current that will flow through the dcside capacitor. Repeat Example 51 if the current waveform is a rectangular pulse as shown in Fig 512c in a threephase rectifier, with an amplitude of 10A .
Chapter 6 POWERFACTORCORRECTION (PFC) CIRCUITS AND DESIGNING THE FEEDBACK CONTROLLER 61
INTRODUCTlON
Technical solutions to the problem of distortion in the input current have been known for a long time. However, only recently has the concem about the deleterious effects of harrnonics led to the forrnulation of guidelines and standards, which in tum have focused attention on ways of limiting current distortion. ln the following sections, powerfactorcorrected (PFC) interface, as they are often called, are briefly examined for singlephase rectification, where it is assumed that the power needs to flow only in one direction, such as in dc power supplies. The threephase frontends in motordrives applications may require bidirectional power flow capability. Such frontends, which also allow unity power factor of operation, are discussed in Chapter 12. OPERATING PRINCIPLE OF SINGLEPHASE PFCs
62
Operating principIe of a commonly used singlephase P FC is shown in Fig. 61 a where, between the utility supply and the dcbus capacitor, a Boost dcdc converter is introduced. This Boost converter consists of a MOSFET, a diode, and a small inductor Ld.
By pulsewidthmodulating the MOSFET at a constant switching frequency, the
current i L through the inductor Ld is shaped to have the fullwaverectified waveforrn Isin
wtl,
similar to Iv, (t)I, as shown in Fig. 61 b. The inductor current contains high
switchingfrequency ripple, which is removed by a small filter, and the input current i, is sinusoidal, and in phase with the supply voltage. ln the Boost converter, it is essential that the dcbus voltage Vd be greater than the peak ofthe supply voltage Vd>V,
V,: (61)
Using the average model of the Boost converter as shown in Fig. 62a and neglecting a small voltage drop across the inductor and assuming the voltage across the capacitor to be apure dc,
61
+
<f
+
+ Ld
i,
Iv,1 l
+
v,
q(t)
Cd
"' R

(a)
(b)
Figure 61 PFC circuit and waveforms. â€˘ t;. (I) +
+
(l  d )
(a)
Figure 62 A verage model and waveforms. Vd
I
(62)
Iv, 1 1 d(l) thus,
d(t) =1
V; Isin (WI )1
(63)
Vd
The switch dutyratio in the average circuit of Fig. 62a is plotted in Fig. 62b.
The
outputstage current i,;(t) can be calculated trom the ideal transformer in Fig. 62a in terms of Iv, 1=
V; Isin llJIl
and
I;. (I) = i L Isin llJIl, and by using Eq. 62, (64)
I'
1
I 1 2 llJI: . . th at m . E q. 6  4 , smWI 2 = sm â€˘ 2 llJI = cos R ecogmzong
i d
=
IV .
lV
_L I
2V
d
L
~
1.1
 ' I cos 2wt 2V L d
2
2
(65)
'~~v . ~
;.1 2(1)
Eq. 65 shows that the average current to the output stage consists of a dc component / d and a component id 2 (1) at the secondharmonic component. ln PFCs, the capacitor in the output stage shown in Fig. 63 is quite large such that it is justifiable to approximate that 62
ali the secondharmon ic ripple flows through the output capacitor, and only ld flows through the load equivalent resistor. Based on this assumption, the secondharmonic ripple in the output voltage can be calculated as
V
d2
(/)
= _1_ Jid2 . d(ml)
(66)
mC
Substituting id 2 from Eq. 65 into Eq. 66, V
d2
1ivJ
(iVJ'sm2ml
= ___ L_, cos20Jl.d(ml) =I . ., mC 2 Vd 4mCVd
(67)
'v'
v
dl
where
v _lv' 4mC V d2 
(68)
d
is the peak value of the ripple in the output capacitor. It depends inversely on the output capacitance, and therefore an appropriate value must be chosen to minimize this ripple. +
i
!
"==
v,
>1"
Figure 63 Current division in the output stage.
63
CONTROL OF PFCs
ln controlling a PFC, the main objective is to draw a sinusoidal current, inphase with the utility voltage. The reference inductor current ((I) is of the fullwave rectified form, similar to that in Fig. 61b. The requirements on the form and the amplitude of the inductor current lead to two control loops, as shown in Fig. 64, to pulsewidth modulate the switch of the Boost converter: The inner current loop ensures the form of j', (I) based on the utility voltage V,(/). The outer voltage loop determines the amplitude
iL
of
i: (I) based on the output
voltage feedback. If the inductor current is insufficient for a given load supplied by the PFC, the output voltage will drop below its preselected reference value
V;.
By measuring the output voltage and using it as the feedback signal, the
voltage loop adjusts the inductor current amplitude to bring the output voltage to
63
its reference value. ln addition to detennining the inductor current amplitude, this voltage feedback control acts to regulate the output voltage of the PFC to the preselected dc voltage. Isin
v'J
Voltage Controller
mIl
Current Loop r , i;(t), I I Current vc(t) q(/) ControlJer Power
:
Vd
Stage
I
I
I I
I I IL ____________________________ _I
Figure 64 PFC control loops. Fortunately, in Fig. 64, the inner current loop is required to have a very high bandwidth compared to the outer voltage loop. Hence, each loop can be designed separately, similar to the approach taken in the peakcurrentmode control discussed Chapter 4. 64
DESIGNING THE INNER A VERAGECURRENTCONTROL LOOP
The inner current loop is shown within the dotted box in Fig. 64. ln order to follow the reference with as little THD as possib le, an averagecurrentmode control is used with a high bandwidth, where the error between the reference ((I) , and the measured inductor current iL (t) is amplified by a current controller to produce the control voltage vo(t). This control voltage is compared w ith a ramp signal v,(/) , with a peak of switchingfrequency
V,
at the
J, in the PWM controller IC [I], to produce the switching signal
q(/) .
Just the inner current loop of Fig. 64 can be simplified, as shown in Fig. 65a. The reference input i', (I) varies with time as shown in Fig. 62b, where the corresponding v, (t) and d(/) wavefonns are also plotted. However, these quantities vary much more
slowly compared to the controlIoop bandwidth, approximately 10 kHz in the numerical example considered later on. Therefore, at each instant of time, for example at I, in Fig. 62b, the circuit of Fig. 62a can be considered in a "dc" steady state with the associated variables having values of iL(/ ,), V, (t,) and d(t,), This equilibrium condition slowly varies with time, and the current loop is designed to have a large bandwidth around each dc steady state. ln Laplace domain, this current loop is shown in Fig. 65b, as discussed below, where " " on top represents sma ll signal perturbations at very high frequencies in the range ofthe currentIoop bandwidth, for example 10kHz.
64
(L (t)
Current Controller
vc(/)
PWM
d(/)
IC
vd
Power Stage iL
(a)
Current C on trO II er Âˇ(S) +
.~
Gi(s)
Power Stage
PWM
IC
Vc(s)
d(s)
I
V,
V;n
fL(s)
sLd
(h) Figure 65 PFC current loop. 641
If
V,
d(s)1 vc(s) for the PWM ControIler
is the difference between the peak and the valley of the ramp voltage in the PWM
IC, then the smallsignal transfer function of the PWM controller, as discussed in Chapter 4 is PWM Controller Transfer Function d(s) = vc(s) 642
~ V,
(69)
IL(s)1 d(s) for the Boost Converter in the Power Stage
As described in the Appendix on the accompanying CD, in spite of the varying dc steady state operating point, the transfer function in the Boost converter simplifies as follows at high frequencies, where the current loop has Iikely to have its bandwidth: I;.(s) V    = dd(s) sLd
643
(6 I O)
Designing the Current ControIler GJs)
The transfer function in Eq. 610 is an approximation valid at high frequencies, and not a pure integrator. Therefore, to have a high loop dc gain and a zero dc steady state error in Fig. 65b, the current controller transfer function G, (s) must have apoIe at the origino ln the loop in Fig. 65b, the phase due to the pole at origin in G,(s) and that ofthe powerstage transfer function (Eq. 610) add up to 180Â°. Hence, G,(s) , as in the peakcurrent mode control discussed in Chapter 4, includes apoiezero pair that provides a phase
65
boost, and bence the specified phase margin, for example 60° at lhe loop crossover frequency: G(s)= kc I+s l úl, s 1+ s i úlp ,
(611)
',,'
phase boost
where, kc is the amplifier gain. Knowing tbe phase boost,
r/l. _" we can calculate lhe
polezero locations to provide tbe necessary phase boost, as discussed in Chapter 4: K
booSI
= tan(45° +
r/lh2OO")
(612)
fe,
1", =
(613)
K boo.st
Z
(614) where 65
fe,
is lhe crossover frequency oflhe current loop transfer function.
DESIGNING THE OUTER VOLTAGE LOOP
As mentioned earlier, lhe outer voltage loop is needed to determine the peak
,iL '
of lhe
inductor current. ln this voltage loop, lhe bandwidlh is limited to approximately 15Hz. The reason has to do with the fact lhat the output voltage across the capacitor contains a component
V d2
as derived in Eq. 67 at twice the linefrequency (at 120 Hz in 60Hz line
frequency systems). This output voltage ripple must not be corrected by the voltage loop, olherwise it will lead to lhirdharmonic distortion in lhe input current, as explained in lhe Appendix on the accompanying CD. ln view of such a low bandwidth of lhe voitage loop (approximately three orders of magnitude below the currentIoop bandwidth of  10kHz), it is perfectly reasonable to assume lhe current loop ideal at low frequencies around 15 Hz. Therefore, in lhe voltagecontrol block diagram shown in Fig. 66a, lhe current closedIoop produces its reference value
i:.
ln addition to a large dc component,
secondharmonic component controller.
i L2
i L2
due to
Vd 2
i:
iL
equal to
contains an unwanted
(Eq. 67) in the input to the voltage
at the secondharmonic component results in a thirdharmonic distortion
in lhe current drawn from lhe utility, as eXplained in the Appendix on lhe accompanying CD. Therefore, in the output ofthe voltage controller block in Fig. 66a, approximately l.5% of lhe dc component in
66
i:.
i L2
is limited to
v'd
r
(a)
i;
Voltage Controller
Closed
iL
Current
Loop
Voltage Controller
Closed Current
Loop
Power
Stage
Vd
I
Power
Stage
GJs)
(b)
Figure 66 Voltage control loop. The voltage controlloop for lowfrequency perturbations, in tbe range ofthe voltageIoop bandwidtb of approximately 15 Hz, is shown in Fig. 66b. As derived in the Appendix on tbe accompanying CD, tbe transfer function of tbe power stage in Fig. 66b at these low perturbation frequencies (ignoring tbe capacitor ESR) is:
IV,
R 2 Vd l+sRC
(615)
To achieve a zero steady state error, tbe voltagecontroller transfer function shou ld have a pole at the origino However, since tbe PFC circuit is ofien a preregulator (not a strict regulator), this requirement is waived, which otherwise would make tbe voltage controller design much more complicated. The following simple transfer function is ofien used for tbe voltage controller in Fig. 66b, where a pole is placed at the voltageloop crossover frequency Gv(s) =
(O~
(yet to be deterrnined) below 15 Hz
kv 1 + s / (O~
(616)
At fullIoad, tbe power stage transfer function given by Eq. 615 has apoIe at a very low frequency, for example of tbe order of one or two Hz, which introduces a phase lag approaching 90 degrees much beyond tbe frequency at which this pole occurs. The transfer function ofthe controller given by Eq. 616 introduces a lag of 45 degrees at the loop crossover frequency.
Therefore, these two phase lags of _135Â° at the crossover
frequency result in a satisfactory phase margin of 45Â°. By definition, at tbe crossover frequency
hv' the loop transfer function kv
1 + s/ (Ocv
1 fi; R = 1 2 Vd 1 + sRC f~
has a magnitude equal to unity (617)
67
At the secondharmonic in the voltage controller ofEq. 616,
I +slwcv
(618) J'=j(2J1'"xI20)
From Eqs. 617 and 618, the two unknowns kv and
w~
in the voltage controller transfer
function ofEq. 616 can be calculated, as described by a numerical example.
66
EXAMPLE OF SINGLEPHASE PFC SYSTEMS
The operation and control of a PFC is demonstrated by means of an example, where the parameters are as follows in Table 61, and the total harmonic distortion in the input line current is required to be less than 3 percent [I , 2]: Table 61 Parameters and Operating Values 120V Nominal input ac source voltage, ~ ,nns Line frequency,
f
60Hz
Output Voltage, Vd
250V (dc)
Maximum Power Output
250W
Switching Frequency,
J,
Output Filter capacitor, C
220 pF
ESR of the Capacitor, r
100mO
Inductor,
ImH
Ld
FullLoad Equivalent Resistance, R
661
100kHz
2500
Design ofthe Current Loop
ln Eq. 69,
V,
is given as unity. Following the procedure described in Chapter 4 for the
peakcurrentmode control of dcdc converters, for the loop crossover frequency of 10 kHz (WCi = 2n x lO' rad/s) and the phase margin of 60Â° , the parameters in the current
controller ofEq. 61 1 are as follows:
kc = 4212
w, = 1.68 x 10' rad I s wp =2.35xI0 5 rad / s
68
(619)
8ased on these parameter values given in Eq. 619 of the transfer function Gj(s) in Eq. 611, the opamp circuit is similar to Fig. 419 in Chapter 4 with the following values for a chosen value of R, = 100kn:
C 2 = 0.17nF (620)
C, = 2.2nF
R, = 27kn 662
Design oftbe Voltage Loop
ln Ihis example at fullIoad, lhe plant transfer function given by Eq. 615 has apoie at lhe frequency of 18.18 radls (2.89 Hz).
Vd2 =
6.029 V.
At fullIoad,
IL=
2.946 A, and in Eq. 68,
8ased on the previous discussion, the secondharrnonic component is
limited to 1.5 percent of
IL' such that IL2 =
0.0442A. Using Ihese values, from Eq. 617
and 618, the parameters in lhe voltage controller transfer function of Eq. 616 are calculated: kv = 0.0722, and mev = 76.634 rad / s (12.2 Hz).
This transfer function is
realized by an opamp circuit shown in Fig. 67 with lhe following values: R, = 100kn
R,
(621)
= 7.2kn
C, = 1.8,uF
c,
OUI
Figure 67 Opamp circuit to implement transfer function Gv(s).
67
SIMULATIONRESULTS
The PSpicebased simulation of the P FC system is shown in Fig. 68, where the input voltage and lhe fullbridge rectifier are combined for simplification purposes. The output load is decreased as a step at 100 ms. The resulting waveforms for the voltage and the inductor current are shown in Fig. 69.
69
D_Pwr~
11 lOM>~SIJl4J76.1il9 '~.
"'I '<X
_A
i
,
..", I w,
"
'm
"' i!
,~
.
"
"""
"1 ~~'" ln
~
IC~·
I.V\~)
"' i"
""
•• ~
..
I
." "b
,
'"
""
O. 17l1nf ~

c,
,
Figure 68 PSpice simulation diagram (the load is decreased at 100 ms). "w....~,_r__.,__.,
,~+~~~1~_+4_~~~+__4~
~L»
"w~~~~L~L~~L~~~ c
.
V(Vou~)
/\ f\
/\
/\
1/\
fi
1\
fi
1\ 1\
f\
(\/
~ o~ CIo
20
I (LI)
....
,...
....
\I
f\
f\
A
\i
\1
!\ !\
r
\1
\1 ~
!\ !\
U
Figure 69 Simulation results: output voltage and inductor current.
68
FEEDFORWARD OF THE INPUT VOLTAGE
The input voltage is fed forward as shown in Fig, 610. ln a system with a PFC interface, the output is nearly constant, independent of the changes in the nllS value of the input voltage from the utility. Therefore, an increase in the utility voltage in
i
L ,
V,
causes a decrease
and vice versa. To avoid propagating the input voltage disturbance though the
PFC feedback loops, the input voltage peak is fed forward , as shown in Fig. 610, in
deterrnining 610
i; .
v'd
,,,
,,
L ___________________________ _
..J
Figure 610 Feedforward of the input voltage. REFERENCES "UC3854 Controlled Power factor Correction Circuit Design," Philip C. Todd, Unitrode Application Note U1 34. 2. "High power Factor Switching PreRegulator Design Optimization," Lyod Dixon, Unitrode Design Application M anual. 3. N. Mohan, T . M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and D esign, 3'" Edition, Wiley & Sons, New York, 2003 . 1.
PROBLEMS 61
ln a single phase power factor correction circuit, V,
= 120V(rms) ,
Vd
= 225V,
and the output power is 100 W . Ca1culate and draw the following waveforrns, synchronized to v, wavefo rrn: iL (I) , d(t) , and the current through the diode. 62
ln the numerical example given in this chapter, calculate the rrns input current if the utility voltage is I 10V( rms) , and compare it with its nominal value when
V,
= 120V(rms).
63
ln problem 61, calculate the secondharrnonic peak voltage in the capacitor if C = 440,uF .
64
ln the numerical example g iven in this chapter, ca1culate the maximum peakpeak ripple current in the inductor. Repeat the design of the current loop in the given numerical example in this chapter, ifthe loop crossover frequency is 20kHz.
65 66
Repeat the design of the outer voltage loop in the numerical example given m this chapter, ifthe output capacitance C = 440,uF .
611
Chapter 7
MAGNETIC CIRCUIT CONCEPTS The purpose of this chapter is to revlew some of the basic concepts associated with magnetic circuits and to develop an understanding of inductors and transformers, which are needed in power electronics. 71
AMPERETURNS AND FLUX
Let us consider a simple magnetic structure ofFig. 71 consisting ofan N tum coil with a current i, on a magnetic core made up of iron. This coil applies Ni amperetums to the core. We will assume the magnetic field intensity H m in the core to be uniform along the mean path length
em'
The magnetic field intensity in the air gap is denoted as Hg.
From Ampere's Law, the closed line integral of the magnetic field intensity along the mean path within the core and in the air gap is equal to the applied amperetums: (71)
/
j

IÓm = IÓ. =IÓ
~ "./ /
=il
/ .&(
tI:
p.l d
w
/ Figure 71 Magnetic structure with air gap. ln the core and in the air gap, the flux densities corresponding to H m and Hg are as follows: (72) (73)
ln terms ofthe above flux densities in Eq. 71,
B B .!!!..e + Le g m 11m
= Ni
(74)
Jlo
71
Since flux lines form closed paths, the flux crossing any perpendicular crosssectional area in the core is the sarne as that crossing the air gap. Therefore, (75)
B
m
=..'L A
(76)
and
m
Substituting flux densities from Eq. 76 into Eq. 74,
t/J(
c
C
+ ') = Ni
m
.4.&
:!C
'R.
'R.
(77)
ln Eq. 77, the two terms within the parenthesis equal the reluctance the reluctance
~g
~m
of the core and
of the air gap, respectively. Therefore, the effective reluctance
~
of
the whole structure in the path ofthe flux lines is the sum ofthe two reluctances: (78) Substituting from Eq. 78 into Eq. 77 (79) Eq. 79 allows the flux
t/J
to be calculated for the applied amperetums, and hence Bm
and Bg can be calculated from Eq. 76. 72
INDUCTANCE L
At any instant of time in the coil of Fig. 72a, the flux linkage ofthe coil Âm' due to flux lines entirely in the core, is equal to the flux t/Jm times the number of turns N that are Iinked.
This flux linkage is related to the current i by a parameter defined as the
inductance Lm : Âm
(710)
=N'" 'I'm =Lm i
x(N) À",
~)
'~(a) 72
(b) Figure 72 Co illnductance.
where the inductance
L m (= Ă€,m /
i) is constant if the core material is in its linear operating
region. The coil inductance in the linear magnetic region can be ca1culated by multiplying ali the factors shown in F ig. 72b, which are based on earlier equations:
(711)
Eq. 711 indicates that the inductance
Lm
is strictly a property of the magnetic circuit
(i.e., the core material, the geometry, and the number ofturns), provided the operation is in the linear range of the magnetic material, where the slope of its BH characteristic can be represented by a constant f1m . 721
Energy Storage due to Magnetic Fields
Energy in an inductor is stored in its magnetic field . From the study of electric circuits, we know that at anytime, with a current i , the energy stored in the inductor is (712) where [./], for Joules, is a unit of e nergy. Initially assuming a structure without an air gap, such as in Fig. 72a, we can express the energy storage in terms of flux density, by substituting into Eq. 712 the inductance from Eq. 711 , and the current from the Ampere' s Law in Eq. 71 : [J]
(713)
where Am im = volume, and in the linear region Bm = f1mH m . Therefore, from Eq. 713, the energy density in the core is (714) Sirnilarly, the energy density in the air gap depends on f10 and the flux density
ln
it.
Therefore, from Eq. 714, the energy density in any medium can be expressed as
73
1B
2
w =   [J/m 2 f.J
3
1
(715)
ln inductors, the energy is primarily stored in the air gap purposely introduced in lhe palh of tlux lines. ln lransforrners, there is no air gap in the path of the tlux lines. Therefore, the energy
stored in the core of an ideal transforrner is zero, where the core permeability is assumed infinite, and hence Hm is zero for a finite tlux density. ln a real transforrner, the core perrneability is finite, resulting in some energy storage in the core. 73
FARADAY'S LAW: INDUCED VOLTAGE lN A COIL DUE TO TIMERATE OF CHANGE OF FLUX LINKAGE
ln our discussion so far, we have established in magnetic circuits relationships between the electrical quantity i
and the magnetic quantities H, B, rp, and À,.
These
relationships are valid under dc (static) conditions, as well as at any instant when these quantities are varying with time. We will now examine the voltage across the coil under timevarying conditions. ln the coil ofFig. 73, Faraday's Law dictates that the timerate ofchange oftluxlinkage equals the voltage across the coil at any instant: e(t)
d d = À,(t) = Nrp(t) dt
dt
(716)
Figure 73 Voltage polarity and direction offlux and curren!. This assumes that ali tlux lines link ali Ntums such lhat À, = Nrp. The polarity of the emf e(t) and the direction of rp(t) in the above equation are yet to be justified. The relationship in Eq. 716 is valid, no matter what is causing the flux to change. One possibility is that a second coil is placed on the same core. When the second coil is supplied by a timevarying current, mutual coupling causes the flux rp through the coil to change with time. The other possibility is that a voltage e(t) is applied across the coil in Fig. 73, causing the change in tlux, which can be calculated by integrating both sides of Eq. 716 with respect to time:
74
1 ' ~I) = ~O)+ Je(T).dT
(717)
No
where
~O)
is the initial flux at I = O and T is a variable of integration.
Recalling the Ohm's law, v = Ri, the current direction through a resistor is into the terminal at the positive polarity. This is the passive sign convention. Similarly, in the coil of Fig. 73, we can establish the voltage polarity and the flux direction in order to apply Faraday's law, given by Eqs. 716 and 717. Ifthe flux direction is given, we can establish the voltage polarity as follows: first determine the direction of a hypothetical current that will produce flux in the same direction as given. Then, the positive polarity for the voltage is at the terminal, which this hypothetical current is entering. Conversely, if the voltage polarity is given, imagine a hypothetical current entering the positivepolarity terminal. This current, based on how the coil is wound, for example in Fig. 73, determines the flux direction for use in Eqs. 716 and 717. Following these roles to determine the voltage polarity and the flux direction is easier than applying Lenz's law (not discussed here). The voltage is induced due to drl> l dI, regardless of whether any current flows in that coil. 74
LEAKAGE AND MAGNETIZING INDUCTANCES
Just as conductors guide currents in electric circuits, magnetic cores guide jlux in magnelic circuits. But there is an important difference. ln electric circuits, the conductivity of copper is approximately 10 20 times higher than that of air, allowing leakage currents to be neglected at dc or at low frequencies such as 60 Hz. ln magnetic circuits, however, the permeabilities of magnetic materiais are, at best, only 104 times greater than that of air. Because of this relatively low ratio, the core window in the structure of Fig. 74a has "Ieakage" flux lines, which do not reach their intended destination that may be another winding, for example in a transformer, or an air gap in an inductor. Note that the coil shown in Fig. 74a is drawn schematically. ln practice, the coil consists of multi pie layers and the core is designed to fit as snugly to the coil as possible, thus minimizing the unused "window" area. The leakage effect makes accurate analysis of magnetic circuits more difficult that requires sophisticated numerical methods, such as finite element analysis. However, we can account for the effect of leakage fluxes by making certain approximations. We can divide the total flux ri> into two parts:
75
1.
The magnetic flux </>m' which is completely confined to the core and links a li N tums, and
2.
The leakage flux, which is partially or entirely in air and is represented by an "equivalent" leakage flux </>" which a lso links a li N tums of the coi l but does not follow the e ntire magnetic path, as shown in Fig. 74b.
.pm
"
/' ~
í
+ e
=
t\ .p,
\
(a)
/
(b)
Figure 74 (a) Magnetic and leakage fluxes; (b) equivalent representation ofmagnetic and leakage fluxes .
ln Fig. 74b, </> = </>m + </>"
where </> is the equivalent flux which links ali N tums.
Therefore, the total flux linkage of the coil is À. =
N</> = _N</>m + 'v' N</>, = À.m+ À., ....
(718)
>,
The total inductance (called the selfinductance) can be obtained by dividing both sides ofEq. 718 by the current i: À.
À.
À.,
L,..
['(
(719)
~
 = m + i i i 'v" ___ L'F
(720) where Lm is often ca lled the magnetizing inductance due to </>m in lhe magnetic core, and L , is called the leakage inductance due to the leakage flux </>, . From Eqs . 719 and 720,
the total flux linkage ofthe coil in Eq. 7 18 can be written as À.
= (Lm + L, )i
(721)
Hence, from Faraday's law in Eq . 7 16,
di di e(t)=Lm+L, dI dI
e.. (l)
76
(722)
This results in the electrical circuit of Fig. 75a. ln Fig. 75b, the voltage drop due to the leakage inductance can be shown separately so that the voltage induced in the coil is solely due to the magnetizing f1ux . The coil resistance R can then be added in series to complete the rcprescntation ofthe coi l.
;m 1(1)

L di
+
+
I dI
L,

l
+
+ v (I)
e(/)
em(t)
L,
+ e(1)

i(l)
+
em(t)
Lm
./
(a)
(b)
Figure 75 (a) Circuit representation; (b) leakage inductance separated from the core. 741
Mutuallnductances
Most magnetic circuits, such as those encountered in inductors and transformers consist of multiple coils. ln such circuits, the f1ux established by the current in one coil partially links the other coil or coi Is. This phenomenon can be described mathematically by means of mutual inductances, as examined in circuit theory courses. However, we will use simpler and more intuitive means to analyze mutually coupled coils, as in a Flyback converter discussed in Chapter 8 dealing with transformerisolated dcdc converters. 75
TRANSFORMERS
ln power electronics, highfrequency transformers are essential to switchmode dc power supplies. Such transformers ofien consist of two or more tightly coupled windings where almost ali of the f1ux produced by one winding links the other windings. lncluding the leakage f1ux in detail makes the analysis very complicated and not very useful for our purposes here. Therefore, we will include only the magnetizing f1ux t/Jm that links ali the windings, ignoring the leakage f1ux whose consequences will be acknowledged separately. To understand the operating principies of transformers, we will consider a threewinding transformer shown in Fig. 76 such that this analysis can be extended to any number of windings. ln this transformer, ali windings are linked by the sarne f1ux t/Jm' Therefore, from the Faraday's law, the induced voltages at the dotted terminais with respect to their undotted terminais are as follows:
77
(723)
dtPme2 =N2 
(724)
dI
e = ,
N dtPm
(725)
' dI
â€˘
i2
+ e3
â€˘ + e2

Figure 76 Transformer with three windings. The above equations based on Faraday ' s law result in the following relationship that shows that the voltspertum induced in each winding are the sarne due to the sarne rate ofchange offlux that links thern
dtPm
e,
e
e,
dI
N,
N2
N,
2  = ==
(726)
ln accordance to the Arnpere's law g iven in Eq. 79, the flux
tPm
at any instant oftirne is
supported by the net arnperetums applied to the core in Fig. 76, (727) ln Eq. 727, the core ofFig. 76 offers reluctance
mmin the flux path and the currents are
defined positive into the dotted terminaIs of each winding such as to produce flux lines in the sarne direction. Eqs. 726 and 727 are the key to understanding transforrners: at any instant of time, applied voltage (equal to the induced voltage if the winding resistance and the leakage flux are ignored) to one of the windings dictates the flux rateofchange and hence the induced voltagepertum in other windings. The instantaneous flux
tPm IS
obtained by expressing Eq. 726 in its integral form below (with proper integral lirnits) (728)
78
that requires corresponding net amperetums given by Eq . 727 to sustain this flux, overcoming the core reluctance. It is important to note that it is immaterial to the core the apportionment ofthese winding currents. The analysis above is based on neglecting the leakage flux, assuming that the flux produced by a winding links ali the other windings. ln a simplified analysis, the leakage flux of a winding can be assumed to result in a leakage inductance, which can be added , along with the winding resistance, in series with the induced voltage e(t) in the winding in the electrical circuit representation. REFERENCE 1.
N. Mohan, T. M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and Design, 3'" Edition, Wiley & Sons, New York, 2003.
PROBLEMS
Inductors A magnetic core has the following properties: the core area Am = 0.93 1cm 2 , the magnetic path J.1,
length of Rm = 3. 76cm, and the relative permeability of the material
is
= J.1m I J.1o = 5000 .
71
Calculate the reluctance 9i ofthis core.
72
Calculate the reluctance ofan air gap oflength Rg = lmm, ifit is introduced in
73
the core of Problem 71. A coil with N = 30 tums
74
Problem 72. Calculate the inductance ofthis coi!. If the flux density in the core in Problem 73 is not to exceed 0.2 T , what is the
75
IS
wound on a core with an air gap described in
maximum current that can be allowed to flow through this inductor coil? At the maximum current calculated in Problem 74, calculate the energy stored in the magnetic core and the air gap, and compare the two.
Transformers A threewinding transformer with N, = 10 tums, N 2 = 5 tums, and N, = 5 tums uses a magnetic core that has the following properties: Am = 0.639 cm 2 length of
Rm
,
the magnetic path
= 3.l2cm, and the relative permeability ofthe material J.1, = J.1m I J.1o = 5000.
A squarewave voltage, of 30V amplitude and a frequency of 100 kHz, is applied to winding I. Windings 2 and 3 are open. Ignore the leakage inductances.
79
76
77 78 79 710
711
710
Calculate and draw the magnetizing current waveform, along with the applied voltage waveform, and the waveforms of the voltages induced in the open windings 2 and 3. Calculate the selfinductances of each winding in Problem 76. Calculate the peak flux density in Problem 76. A load resistance of 100 is connected to winding 2. Calculate and draw the currents in windings I and 2, along with the applied voltage waveform. Assuming that winding 1 is not applied a voltage, what is the peak amplitude of the squarewave voltage at 100 kHz that can be applied to winding 3 of this transformer, if the peak fl ux density calculated in Problem 78 is not to be exceeded? ln Problem 76, what is the peak amplitude of the voltage that can be applied to winding I without exceeding the peak flux density calculated in Problem 78, if the frequency of the square wave voltage is 200kHz? What is the peak value of the magnetizing current, as compared to the one at 100 kHz?
Chapter 8 SWITCHMODE DC POWER SUPPLIES 81
APPLICATIONS OF SWITCHMODE DC POWER SUPPLIES
Switehmode de power supplies represent an important power eleetronies applieation area with the worldwide market in excess of several billion dollars per year. Many of these power supplies ineorporate transformer isolation for reasons that are discussed below. Within these power supplies, transformerisolated dedc converters are derived from nonisolated dcde converter topologies already diseussed in ehapter 3. For short, we will refer to transformerisolated switehmode dc power supplies as SMPS, whose block diagram is shown in Fig. 81. As shown in Fig. 81 , these supplies eneompass the rectification of the utility supply and the voltage 17,. across a large filter capacitor is the input to the transformerisolated dcde converter, which is the focus of discussion in this ehapter. Intemally, the transformer operates at very high frequencies, upwards of a few hundred kHz are typieal, thus resulting in small size and weight, as discussed in the next chapter. input
rectifier I
60Hz ac
6)
r  , • • I~OHFac I I
+
I I I I
V;n
* I
; 
"
II I I I
....'
topology to convert de to de with isolation
r
4
.. ,r~ ~
~ ~
J''L.
I
I I I
*
;; ~:
Ou !pu!
I I I I
I HF transformerl
V.
______________________ J
I Feedback I controller
V·
•
Figure 81 Block diagram of switchmode de power supplies. 82
NEED FOR ELECTRICAL ISOLATION
Electrical isolation by means oftransformers is needed in switchmode dc power supplies for three reasons: I. 2.
Safety. It is necessary for the lowvoltage dc output to be isolated from the utility supply to avoid the shock hazard . Different Reference Potentials. The de supply may have to operate at a different potential, for example, the dc supply to the gate drive for the upper MOSFET in the powerpole is refereneed to its Souree.
81
3.
83
Voltage matehing. Ifthe dede eonversion is large, then to avoid requiring large voltage and eurrent ratings of semieonduetor deviees, it may be eeonomieal and operationally more suitable to use an eleetrieal transfonner for conversion of voltage leveis. CLASSIFICATION OF TRANSFORMERISOLATED DCDC CONVERTERS
ln the block diagram of Fig. 81 , there are following three categories of transforrnerisolated dcde eonverters, ali ofwhich are discussed in detail in this chapter:
84
•
Flyback converters derived from BuckBoost dedc converters
•
Forward converter derived from Buck dcde converters
•
FullBridge and HalfBridge eonverters derived from Buck dcde eonverters FLYBACK CONVERTERS
Flyback converters are very commonly used in applieations at low power leveis below 50 W. These are derived from the BuekBoost converter redrawn in Fig 82a, where the inductor is drawn descriptively on a low perrneability core.
+
+
v;,
(a)
(b)
(c)
Figure 82 BuekBoost and the Flyback converters. The Flyback converter in Fig. 82b consists oftwo mutuallycoupled coi Is, where the coil orientations are such that at the instant when the transistor is tumedoff, the eurrent switches to the second coil to mainta in the sarne flux in the core. Therefore, the dots on 82
coils are as shown in Fig. 82b where the current into the dot of either coil produces core flux in the sarne direction. Commonly, the circuit of Fig. 82b is redrawn as in Fig. 82c. We will consider the steady state in the incomplete demagnetization mode where the energy is never completely depleted from the magnetic core. This corresponds to the continuous conduction mode (CCM) in BuckBoost converters. We will assume ideal devices and components, the output voltage vo(t) = Vo ' and the leakage inductances to be zero. Tuming on the transistor at
t =
~.
O in the circuit in Fig. 82c applies the input voltage
across coil 1, and the core magnetizing flux tPm increases linearly from its initial value
tPm(O), as shown in the waveforms ofFig. 83. Ouring the transistor oninterval Dr;, the increase in flux can be calculated from the Faraday's law as '" Ao
'rpp
v,.
(81)
= N DTS
,
lout
0''"'t
Figure 83 Flyback converter waveforms. Oue to increasing tPm' the induced voltage (N2 / N,)
v,.
across coil 2 adds to the output
voltage Vo to reverse bias the diode, resulting in io", = O. Corresponding to the core flux, the current im can be calculated using the relationship tP = Ni / 9'l, where 9'l is the core reluctance in the flux path. Therefore, using Eq. 81, the increase in the input current during the oninterval , from its initial value 1,.(0) can be calculated using Eq. 81 as
83
I:l.i;n = 912
N,
~nD~
(82)
During the oninterval DT" the output load is entirely supplied by the energy stored in the output capacitor, and the core magnetizing flux and the input current reach their peak values at the end ofthis interval:
Jm = (Ilm(O) + ;;, DT,
(83)

(84)
9{
1,. = Ii. (O) +  , V,.DT, N,
After the oninterval, turning off the transistor forces the input current in Fig. 82c to zero. The magnetic energy stored in the magnetic core due to the flux (Ilm cannot change instantaneously, and hence the ampereturns applied to the core must be the same at the instant immediately before and after tuming the transistor off. Therefore, the current iau' in coil 2 through the diode suddenly jumps to its peak value such that
NI
I

N
201ll Âˇ _ 1",_ 0
=N1 1 1/n 
.r.
(85)
I"",=v
(86)
I 0111 = N ' Jln
,
With the diode conducting, the output voltage Va appears across coil 2 with a negative polarity.
Hence, during the offinterval (1D)T"
the core flux declines linearly, as
plotted in Fig. 83 , by tJ.(Ilp_p, where tJ.(Ilp_p = ;
, (1 
D)T,
(87)
Using Eqs. 81 and 87,
(88) The change in the current im"ct) can be calculated in a manner similar to Eq. 82, and this current is plotted in Fig. 83. Eq. 88 shows that in a Flyback converter, the dependence of the voltageratio on the dutyratio D is identical to that in the BuckBoost converter, and it also depends on the
84
coils tumsratio N 2 / N I
â€˘
Flyback converters require minimum number of components by
integrating the inductor (needed for a BuckBoost operation) with the transformer that provides electrical isolation and matching of the voltage leveIs. These converters are very commonly used in low power applications in the complete demagnetization mode (corresponding to the discontinuousconduction mode in BuckBoost), which makes their control easier. A disadvantage of the Flyback converter is the need for snubbers to prevent voltage spikes across the transistor and diode due to leakage inductances associated with the two coi Is.
85
FORWARD CONVERTERS
Forward converter and its variations derived from a Buck converter are commonly used in applications at low power leveIs up to a kW. A Buck converter is shown in Fig. 84a. ln this circuit, a threewinding transformer is added as shown in Fig. 84b to realize a Forward converter. The third winding in series with a diode D 3 , and the diode DI are needed to demagnetize the core every switching cycle. The winding orientations in Fig. 84b are such that the current into the dot of any of the windings will produce core flux in the same direction. We will consider steady state converter operation in the continuous conduction mode where the output inductor current iL flows continuously. following analysis, we will assume ideal semiconductor devices, voU) =
ln the
11", and the
leakage inductances to be zero.
D,
+ +
+ N,
V. q(l)
N,
D, q(t)
(a)
___._iL
D,
V;"
A
+
VO
(b)
Figure 84 Buck and Forward converters. Initially, assuming an ideal transformer in the Forward converter of Fig. 84b, the third winding and the diode D3 can be removed and DI can be replaced by a short circuito ln such an ideal case, the Forward converter operation is identical to that of the Buck converter, as shown by the waveform in Fig. 85, except for the presence of the transformer tumsratio N, / N I
â€˘
Therefore, in the continuous conduction mode, (89)
85
(N, I N , )V,"
( N, I N, )DV,"
o ~~U~)'~L~
.
'
•
T,
Figure 85 Forward converter operation. ln the case with a real transformer, the core must be completely demagnetized during the offinterval of the transistor, and hence the need for the third winding and the diodes DI and D" as shown in Fig. 84b. Tuming on the transistor causes the magnetizing flux in the core to build up as shown in Fig. 86. During this oninterval DT"
D, gets reverse
biased, thus preventing the current from flowing through the tertiary winding. The diode D 2 also gets reversed biased and the output inductor current flows through DI .
When the transistor is tumed off, the magnetic energy stored in the transformer core forces a current to flow into the dotted temlinal of the tertiary winding, since the current into the dotted terminal of lhe secondary winding cannot flow due to D I ' which results in
v'n
to be applied negatively across the tertiary winding, and the core flux to decline, as
shown in Fig. 86. interval
~,mag'
(The output inductor current freewheels through D 2 .)
Afier an
the core flux comes to zero and stays zero during the remaining interval,
until the next cycle begins.
1J1~
T..., T,
Fig. 86 F orward converter core fluxo To avoid the core from saturating, lhe transistor.
T d<mag
must be less lhan the offinterval (1 D)T, of
Typically, windings I and 3 are wound bifilar to provide a very tight
mutual coupling between the two, and hence, N ,
=
NI
•
Therefore, to the core is applied
an equal magnitude but opposite polarity pertum voltage during DI', and Tdemag respectively.
At the upper limit,
Td<mag
equals (\  D)T"
,
and equating it to the on
interval DT, of the transistor yields the upper limit on the dutyratio, Dmax' to be 0.5, with N, = N 3
86
•
S51
TwoSwitch Forward Converters
Singleswitch Forward converters are used in power ratings up to a few hundred watts. However, TwoSwitch Forward converters discussed below eliminate the need for a separate demagnetizing winding and are used in much higher power ratings of a kW and even higher. Fig. 87 shows the topology ofthe TwoSwitch Forward converter, where both transistors are gated on and off simultaneously with a dutyratio D
~
0.5. During the oninterval
DT, when both transistors are on, diodes DI and D 2 get reverse biased and the output
inductor current iL tlows through Do, similar to that in a singleswitch Forward converter. During the off interval when both transistors are tumed off, the magnetizing current in the transformer core tlows through the two primaryside diodes into V;., thus applying V;. negatively to the core and causing it to demagnetize. Application of V;. to the primary winding causes Do to get reversed biased and the output inductor current iL freewheels through DF . Based on the discussion regarding the demagnetization of the core in a singleswitch Forward converter, the switch dutyratio D is limited to 0.5. The voltage conversion ratio remains the sarne as in Eq. 89.
+
q(t)''
Figure 87 Twoswitch Forward converter.
S6
FULLBRIDGE CONVERTERS
FullBridge converters consist of four transistors, hence are economically feasible only at higher power leveis in applications at a few hundred watts and higher. Like Forward converters, fulIbridge converters are aIso derived from Buck converters. Unlike Flyback and Forward converters that operate in only one quadrant of the BH loop, FulIBridge converters use the magnetic core in two quadrants. A FullBridge converter consists of two switching powerpoles, as shown in Fig. 88, with a centertapped transformer secondary winding. ln analyzing this converter, we will assume the transformer ideal, although the effects of magnetizing current can be easily 87
accounted for.
We will consider steady state converter operation in the continuous
conduction mode where the output inductor current iL f10ws continuously.
As with
previous converters, we will assume ideal devices and components, and vo(t) = Vo '
t>: ,1 >
'I
+ VA
T ..
~
+
'L

>
::r:
'" D
~.,
2
Figure 88 FullBridge converter. ln the FullBridge converter of Fig. 88, the voltage v, applied to the primary winding altemates without a dc component. The waveform of this voltage is shown in Fig. 89, where v, =
T..
v,.
when transistors I; and T, are on dUTing
Dr"
and v, =
v,.
when 1", and
are on for an interval ofthe sarne duration. This waveform applies equal positive and
negative voltseconds to the transformer primary. The switch dutyratio D
ÂŤ
0.5) is
controlled to achieve the output voltage regulation by means of zero intervals between the positive and the negative applied voltages.
v,.
V,
I
o
(V..)~ 'DT,
fDT, T,
Figure 89 FullBridge converter waveforms. The way the voltage across the primary winding, hence the secondary winding, is forced to be zero classifies FullBridge converters into the following two categories: â€˘
Pulse Width Modulated (PWM), and
â€˘
PhaseShift Modulated (PSM)
PWM Control. ln PWM control, ali four transistors are tumed off, resulting in a zero voltage across the transformer windings, as discussed shortly. With ali transistors off, the output inductor current freewheels through the two secondary windings, and there are no
88
conduction losses on the primary side of the transformer. Therefore, the PWM control results in lower conduction losses, and it is the control method discussed in this chapter. PSM Control. ln phaseshift modulated control, the two transistors of each powerpole are operated at nearly 50% dutyratio, with D = 0.5. The output of each powerpole pulsates between
v'n and
O with a dutyratio of nearly 50%. The length of the zero
intervals is controlled by phaseshifting the two powerpole outputs with respect to each other, as the name of this control implies. During zero intervals, either both transistors at the top, or both transistors at the botlom are on, creating a short circuit (through one of the antiparallel diodes, depending on the direction of the current) across the primary winding, resulting in
VI
= O.
During this shortcircuited condition, the output inductor
current is reflected to the primary winding and circulates through the primaryside semiconductor devices, causing additional conduction losses. However, increased conduction losses can be offset by the reduction in switching los ses by this means of control, as we will discuss in detail in Chapter 10 on softswitching.
861
PWM Control
As shown by the block diagram of Fig. 8lOa, the PWMIC for Fu!lBridge converters provides gate signals to the transistor pairs (T,,1', and T" r.) during altemate cycles of the ramp voltage in Fig. 810b.
~,and T,
~, and
;c_. . S :..11
T,
(a) (h)
Figure 810 PWMIC and control signals for transistors. Corresponding to these PWM switching signals, the resulting subcircuits are shown in Fig. 811 for onehalf switching cycle, where the other halfcycle is symmetric . iI
'+
+~
VI~
D,
J."
+
> .!:!J.. Vi" : v A N[ ...!..._
,,
;~
L____ J
+ •'" v2
• iL
y +
Q 2t< ...I.
. ~, (0.2 1 'N
v A =.0 I
er:
+
§: v. ~
_2
(a)
"
(h)
Figure 811 FullBridge: subcircuits.
89
lnterval DT, with transistors Y; ,T, in their on state. Tuming on Y;,T, applies positive voltage V;. to the primary winding, causing D 2 to become reverse biased and iL carried through D"
IS
as shown in Fig. 81 la. During this interval, vA = (N2 / N , ) V;" as
plotted in Fig. 812. Interval (I/2D)T, with ali Transistors off.
When ali the transistors are turned off,
there is no current in the primary winding, and the output inductor current divides equally (assuming an ideal transformer) between the two output diodes as shown in the subeircuit of Fig. 811 b. This ensures that the total ampereturns acting on the transformer core equal zero beeause of iL / 2 coming out of the dotted terminal and iL / 2 going into the dotted terminal. Applying the Kirchhoff's voltage law in the loop consisting of the two seeondaries in Fig. 81 1b shows that
v2 + v; = O.
Sinee
must be individually zero, and henee also the primary voltage
v, = v;,
the two voltages
VI :
(810) During this interval, vA = O as plotted in Fig. 812. v,
I       T,       1I
v
Aif     , N,' v: N
.
N
V~V ~ 2= /,01 <> N DV
'
ln
oIfjft.:
t
Figure 812 FullBridge converter waveforms. The above diseussion completes the discussion of onehalf switching cycle. The other halfcycle with 1',,7; on applies a negative voltage (v,.) aeross the primary winding for an interval DT, and results in D2 eonducting and D, reverse biased.
During this
interval, vA = (N2 / N , ) V;. as before when the positive voltage was applied to the primary winding. The waveforms during this halfcycle are as plotted in Fig. 812. From Fig. 812, recognizing that VA = Vo in the de steady state, (81 I)
810
87
BalfBridge and PushPull Converters
Variations ofFullBridge converters are shown in Fig. 813. The HalfBridge converter in Fig. 813a consists of only two tra nsistors but requires two split capacitors to form a dc input midpoint. It is sometirnes used at slightly lower power leveis compared to the FullBridge converter. The PushPull converter in Fig. 813b has the advantage of having both transistors gates referenced to the lowside of the input voltage. The penalty is in the transformer where during the power transfer interval, only one half of the prirnary winding and one half ofthe secondary winding are utilized.
iL
+ J!;n
+ V;n V;n
• • • •
+
vA .
T,
2
(a)
(b)
Figure 813 HalfBridge and PushPull converters.
88
PRACTICAL CONSIDERATIONS
To provide electrical isolation between the input and the output, the feedback control loop should also have electrical iso lation. There are several ways of providing this isolation as discussed in Reference [1].
REFERENCE I.
N . Mohan, T . M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and Des ign, 3'" Edition, Wiley & Sons, New York, 2003.
PROBLEMS Flyback Converters: ln a Flyback converter, V;n = 30V, N, = 30 tums, and N , = IS turns.
The self
inductance of winding I is 50 flH , and ;; = 200kHz. The output voltage is regulated at Vo = 9V.
81
Calculate and draw the waveforms shown in Fig 83 along with the ripple current in the output capacitor, ifthe load is 27W .
82
For the sarne dutyratio as in Problem 81, calculate the criticai power which makes this converter ope rate at the border of incomplete and complete demagnetization modes.
811
83 84
Draw the waveforms, similar to those in Problem 81, in Problem 82. If a Flyback converter is operating in a complete demagnetization mode, derive the voltage transfer ratio in terms of the load resistance R , the switching frequency
J"
the selfinductance L., of winding I, and the dutyratio D.
Forward converter: ln a Forward converter,
v'n
= 30V , N, = lO tums, N, = S tums, and N, = lO tums. The
selfinductance ofwinding I is ISOpH, and the switching frequency
J,
= 200kHz. The
output voltage is regulated such that Vo = S V. The output filter inductance is SO pH , and the output load is 2S W . 8S
Calculate and draw the waveforms for vA' iL , iin' and iD, in Fig. 84b.
86
Ifthe maximurn duty ratio needs to be increased to 0.7, calculate N, / N,?
87
Why is diode D, necessary in Fig 84b?
TwoSwitch Forward Converters ln a twoswitch Forward converter,
v'n
= 30V , N, / N, = 2, and the switching frequency
J, = 200kHz. The output voltage is regulated such that Vo = SV. The selfinductance of winding I is ISO pH , and the output filter inductance is SO pH . 88
Calculate and draw waveforms ifthe output load is 200W.
89
Why is the dutyratio in this converter limited to O.S?
FullBridge Converters 810 ln a Fullbridge converter shown in Fig. 88, consider the output current through the filter inductor to be ripplefree dc. N, / N,
= 4.
v'n =
30V ,
The output voltage is regulated such that Vo
J,
= 200kHz,
= S V.
PWM waveforms in this converter. Assume the transformer ideal.
812
and
Calculate the
Chapter 9 Design of HighFrequency Inductors and Transformers 91
INTRODUCTION
As discussed Chapter 8, inductors and transformers are needed in switchmode dc power supplies, where switching frequencies are in excess of 100 kHz. Highfrequency inductors and transformers are generally not available offtheshelf, and must be designed based on the application specifications. A detailed design discussion is presented in Reference [Il. ln this chapter, a simple and a commonly used approach called the AreaProduct method is presented, where the thermal considerations are ignored. This implies that the magnetic component built on the design basis presented here should be evaluated for its temperature rise and efficiency, and the core and the conductor sizes should be adjusted accordingly. 92
BASICS OF MAGNETIC DESIGN
ln designing high frequency inductors and transformers, a designer is faced with countless choices. These include choice of core materiais, core shapes (some offer better thermal conduction whereas others offer better shielding to stray flux), cooling methods (natural convection versus forced cooling), and losses (Iower losses offer higher efficiency at the expense of higher size and weight) to name a few. However, ali magnetic designoptimization programs calculate two basic quantities from given electrical specifications: â€˘
The peak flux density Bm .. in the magnetic core to limit core losses, and
â€˘
The peak current density J mox in the winding conductors to limit conduction losses
The design procedure presented in this chapter assumes values for these two quantities based on the intended applications of inductors and transformers. However, they may be far from optimum in certain situations. 93
INDUCTOR AND TRANSFORMER CONSTRUCTION
Figs. 91 a and b represent the crosssection of an inductor and a transformer wound on toroidal cores. ln Fig. 9la for an inductor, the same current i passes through ali N tums of a winding. ln the transformer of Fig. 91 b, there are two windings where the 91
current i, in winding I, with N, bigger crosssection conductors, is in opposite direction to that of i 2 in winding 2 with N 2 smaller crosssection conductors. ln each winding, the conductor crosssection is chosen such that the peak current density J m~ is not exceeded at the maximum specified current in that winding. The core area
A=~
in Figs.
91 a and b a llows the flow of flux lines without exceeding the maximum flux density Bmu in the core.
~indow
(b)
(a) Figure 91 Crosssections.
94
AREAPRODUCT METHOD
The areaproduct method, based on preselected values of the peak flux density Bmu in the core and the peak current density J mu in the conductors, allows an appropriate core size to be chosen, as described below.
941
Core Window Area A,.indDw
The windows of the toroidal cores
ln
Figs. 9la and b accommodate the winding
conductors, where the conductor crosssectional area A="" depends on the maxirnum rms current for which the winding is designed. ln the expression for the window area below, the window fillfactor k w in a range from 0.3 to 0.6 accounts for the fact that the entire area of the window cannot be filled , and the subscript y designates a winding, where in general there may be more than one, like in a transformer
A,.indow = }w
L (N yA=nd.Y )
(91 )
y
ln Eq. 91, the conductor crosssectional area in winding y depends on its maxirnum rms current and the maximum allowed current density J m~ that is generally chosen to be the sarne for ali windings:
92
A cond ,y
I J
~

(92)
m~
Substituting Eq. 92 into Eq. 91,
(93)
which shows that the window area is linearly proportional to the number of tums chosen by the designer.
942
Core CrossSectional Area
A=~
The core crosssectional area in F igs. 9la and b depends on the peak flux choice ofthe maximum allowed flux density
Bm~
r/J and the
to limit core losses: (94)
Hqw the flux is produced depends if the device is an inductor or a transformer. ln an
inductor, ~ depends on the peak current, where Li equals the peak flux linkage N ~ . Hence,
' Ăš r/J=N
(inductor)
(95)
ln a transformer, based on the Faraday's law, the flux depends linearly on the applied voltseconds and inversely on the number of tums. This is shown in Fig. 92 for a Forward converter transformer with N , = N , , and the dutyratio D, which is limited to 0.5. Therefore, we Can express the peak flux in Fig. 92 as (96)
where the factor k ron" equals D in Forward converter, and typically has a maximum vaI ue of 0.5.
The factor k conv can be derived for transformers in other converter
topologies based on the specified operating conditions, for example, it equals D / 2 in a Full8ridge converter. ln general, the peak flux can be expressed in terms of any one of the windings, y for example, as (transformer)
(97)
93
v
,
v:"

o 
T,
DT, 
t
( v," )
Figure 92 Wavefonns in a transformer for a Forward converter. Substituting for
A
=
,"Ore
J from Eq. 95 or 97 into Eq. 94, Li
NB
(inductor)
(98)
(transformer )
(99)
m~
Eqs. 98 and 99 show that in both cases, the core crosssectional area is inversely proportional to the number of tums chosen by the designer.
943
Core AreaProduct A/= Aco~ A.,mdow)
The core areaproduct is obtained by multiplying the core crosssectiona l area AOON with
its window area
AwmdOl" :
(910) Substituting for
A,.;ndo w
and Aw, from the previous equations, (inductor)
(9 I I)
(transformer)
(912)
Eqs. 911 and 912 shaw that the areaproduct that represents the overall size of lhe device is independent (as it ought lO be) of the number of tums. Afier alI, lhe core and the overalI co mponent size should depend on the electrical specifications and the assumed values of Em'" and J m~ , and nol on the number of tums which is an intemal design variable.
94
944
Design Procedure Based on AreaProduct Ap
Once we pick the appropriate material and the shape for a core, the cores by various manufacturers are cataloged based on the areaproduct Ap. Having calculated the value of A p above, we can select the appropriate core. It should be noted that there are infinite combinations ofthe core crosssectional area A=,. and the window area A,."ndow that yield a desired areaproduct Ap. However, manufacturers take pains in producing cores s"ch that for a given Ap' a core has Aco" and A..mdow that are individually optimized for power density. Once we select a core, it has specific A,ore and A..,ndow ' which allow the number of tums to be calculated as follows: LI N = 
(inductor ; from Eq. 98)
(913)
(transformer ; from Eq. 99)
(914)
ln an inductor, to ensure that it has the spec ified inductance, an air gap of an appropriate length Rg is introduced in the path of flux lines. Assuming the chosen core material to have very high permeability, the co re inductance is primarily dictated by the reluctance 9l g ofthe air gap, such that
(915) where,
(916) Using Eqs. 915 and 916, the air gap length Rg can be calculated as
(917) The above equations are approximate because they ignore the effects of finite core permeability and the fringing flu x, which can be substantial. Core manufacturers generalJy specity measured inductance as a function of the number of turn for various values of the air gap length. ln this section, we used a toroidal core for descriptive purposes in which it will be difficult to introd uce an air gap. If a toroidal core must be used, it can be picked with a distributed air gap such that it has the effective air gap 95
length as calculated above. The above procedure explained for toro ida I cores is equally valid for other types of cores. The actual design described in the next section illustrates tbe introduction of air gap in a pot core.
DESIGN EXAMPLE OF AN INDUCTOR
95
ln this example, we will discuss the design of an inductor that has an inductance L = 100flH. The worstcase current through the inductor is shown in Fig. 93, where the average current 1= 5.0A , and the peakpeak ripple
M = 0.75A at the switching
frequency 1, = 100kHz. We will assume the following maximum values for the flux density and the current density: Bm~
= 0.25 T ,
this is typically in a range of 3 to 4A/ mm 2
).
and J ~
= 6.0A / mm 2
(for larger cores,
The window fill factor is assumed to be
kw = 0.5.
I L~~t
Figure 93 l nductor current waveforms. The peak value of the inductor current from Fig. 93 is j = I + M = 5.375A. The rms 2 value of the current for the waveform shown in Fig. 93 can be calculated as I~,
=
~
2 1 2 I +M = 5.0A (the derivation is left as a homework problem). 12
From Eq. 911, Area P rod uct A
= p
100 x lO6x 5.375 x 5 x 10 ' 2 0.5 x 0.25 x 6 x 106
=
3587 mm '
From the Magnetics, Inc. catalog [2], we will select a Ptype material, which has the saturation flux density of 0.5 T and is quite suitable for use at the switching fTequency of 100kHz. A pot core 26x16, which is shown in Fig. 94 for a laboratory experiment, has the core Area Aco~
= 93.1 mm 2
and the window Area A.",d~
= 39 mm 2 â€˘
Therefore, we
will select this core, which has an AreaProduct Ap = 93.1 x 39 = 363 I mm'. From Eq. 913,
96
100jl x 5.375 = 23 Tums 0.25 x93.1 x 106
N
Winding wire cross sectional area A=nd = 1,= / J =
x
= 5.0 / 6.0 = 0.83mm
2
•
We will use
tive strands of American Wire Gauge A WG 25 wires [3], each with a crosssectional area of 0.16mm 2 , in parallel. From Eq. 917, the air gap length can be calculated as
100jl
Figure 94 Pot core mounted on a plugin board.
DESIGN EXAMPLE OF A TRANSFORMER FOR A FORW ARD CONVERTER
96
Tbe required electrical specitications for the transformer in a Forward converter are as follows: j ,
= 100kHz
and V;
each winding to be 2.5 A . Bm~
= 0.25 T
and J m~
A =
k oon" k 'B J
p
wJ
$
max
= V2 = V,
A cond ,l
Assume the rms value of the current in
We will choose the following values for this designo
= 5 A/mm 2 •
From Eq. 912, where k w = 0.5 and k oon• = 0.5 ,
4 "V I = 1800 mm L. ~'.Y y
max
Y
For the pot core 22x 13 [2], Ap = 1866 mm 4
= 30 V.
A=re
= 63.9 mm 2 ,
~indow
= 29.2
mm
2
,
and therefore
For this core, the winding wire crosssectional area is obtained as
•
= I, .~, = 2.5 = 0.5 mm 2 • 5
J
m~
We will use three strands of A WG 25 wues [3] , each with a crosssectional area of
0.16mm 2 , in parallel foreach winding. From Eq. 914, N
=
,
0.5x30 =10 (63.9 X 106 ) x (100 X lO' )x 0.25 .
Hence, N,
=N2
= N,
= 10. 97
97
THERMAL CONSIDERATIONS
Designs presented here do not include eddy current losses in lhe windings, which can be very substantial due to proximity effects in inductors. These effects are carefully considered in [I]. Therefore, the areaproduct melhod is a good starting point, but the designs must be evaluated for temperature rise based on lhermal considerations. REFERENCES I.
2. 3.
N. Mohan, T. M. Undeland, and W.P. Robbins, Power E/ectronics: Converters. App/ications and Design, 3,d Edition, Wiley & Sons, New York, 2003. Magnetics, Inc. Ferrite Cores (www.maginc.com). Wire Gauge Comparison Chart (www.toolportals.comlde/umrechnung/gaugesol id.asp).
PROBLEMS
Inductor Design 91 92
Derive the expression for lhe rms current for lhe current waveform in Fig. 93. ln the design example of lhe inductor in this chapter, the core has an area Acon = 93.lmm', lhe magnetic path R. = 37.6mm, and lhe relative permeability of
the core material is 1', = I'm/1'0 = 5000. Calculate the inductance with 31 tums 93 94 95
if lhe air gap is not introduced in lhis core in the flux path. ln lhe inductor design presented in lhis chapter, what is the reluctance offered by the magnetic core as compared to lhat offered by the air gap? ln Problem 92, what is lhe maximum current lhat will cause the peak flux density to reach 0.25 T ? ln the inductor designed
10
lhis chapter, what wilĂ be the inductance and the
maximum current lhat can be passed without exceeding the
Bm~
specified, if the
air gap introduced by mistake is only onehalf of the required value R. g = 1mm . What is the stored energy with R. g = 0.5 mm compared to that at R. g = I mm ? Transformer Design 96
ln lhe design example of the transformer in lhis chapter, lhe core has an area Acon = 63.9 mm' , the magnetic path R. = 3 1.2 mm , and the relative permeability of
lhe core material is 1',
= I'm / 1'0 = 5000.
CaJculate the peak magnetizing current
97
at dutyratio ofO.5. What is the tertiary winding conductor diameter needed for the magnetizing current calculated in Problem 96?
98
Derive
98
kco~
for a transformer in a FullBridge converter.
Chapter 10 SOFT SWITCHING lN DCDC
CONVERTERSAND CONVERTERS FOR INDUCTION HEATING AND COMPACT FLUORESCENT LAMPS 101 INTRODUCTION ln converters so far, we have discussed hard switching in the switching powerpole, as described in Chapter 2. ln this chapter, we will look at the problems associated with hard switching, and some of the practical circuits where this problem can be minimized with softswitching. 102
HARDSWITCIllNG lN SWITCIllNG POWERPOLES
ln the switching powerpole repeated in Fig. 101 a, the hardswitching waveforms are as shown in Fig. 10lb, which were di scussed in Chapter 2. Because ofthe simultaneously high voltage and current associated with the transistor during the switching transition, the switching power losses in the transistor increase linearly proportional to the switching frequency and the times
t c(OO)
and
t c( ojf)
shown in Fig. 10lb, assuming an ideal diode (101)
v,o
v,o
o
V ns
lo
'x'DS
/
"rLy..;.I! ~\ s: : r" ,/['S, ,".) 1, ,
1
. ",
, • .•• '
. 
•
 ,'C"',
' 0 '
I
~
~
• o
I
~.
I
~
o L~~_?~5~~~~~~·~lc.on>1
(a)
..._ _t".o~i
(b)
Figure 101 Hard switching in a powerpole. ln hardswitching converters, in addition to the switching power losses decreasing the energy efficiency, the other problems are device stresses, thermal management of power losses, and electromagnetic interference resulting from high di/dt and dv/dt due to fast
101
transitions in the converter voltages and currents. The above problems are exacerbated by the presence of the stray capacitances and leakage inductances associated with the converter layout and the components. ln order to reduce the overall converter size, while maintaining high energy efficiency, the trend is to design dcdc converters operating at as high a switching frequency as possible (typically IDO200 kHz in small power ratings), using fastswitching MOSFETs. At high switching frequency, the switching power losses become unacceptable if hardswitching is used, and hence softswitching is often employed, as briefly described in this chapter. The problems described above, associated with hardswitching, can be minimized by means of the following: Circuit layout to reduce stray capacitances and inductances Snubbers to reduce di/dt and dv/dt Gatedrive control to reduce di/dt and dv/dt Softswitching It is always recommended to have a layout to reduce stray capacitances and inductances. Snubbers, as describe in the Appendix on the accompanying CD, consist of passive elements (R, C and possibly a di ode) to reduce di/dt and dv/dt during the switching transient, by shaping the switching trajectory. The trend in modem power electronics is to use snubbers only in transformerisolated dcdc converters, where the leakage inductance associated with the highfrequency transformer can be substantial, in spite of a good circuit layout. Generally, snubbers do not redu ce the overall losses; rather they shift some of the switching losses in the transistor to the snubber resistor.
By controlling the gate voltage of MOSFETs and IGBTs, it is possible to s low down the tumon and tumoff speed, thereby resulting in reduced di/dt and dv/dt, at the expense of higher switching losses in the transistor. The above techniques, at best, resuIt in a partial solution to the problems ofhardswitching. However, there are certain topologies and control, as described in the next section, that allow softswitching that essentially eliminate the drawbacks of hardswitching without creating new problems. 103
SOFTSWITCIDNG lN SWITCIDNG POWERPOLES
There are many such circuits and controI techniques proposed in the literature, most of which may make the problem of EM! and the overall losses worse due to large conduction losses in the switches and other passive components. Avoiding these topologies, only a few softswitching circuits are practical.
102
The goal in softswitching is that the switching transition in the powerpole occurs under very favorable conditions, that is, the switching transistor has a zero voltage and/or zero current associated with it. Based on these conditions, the softswitching circuits can be c1assified as follows: ZVS (zero voltage switching), and ZCS (zero current switching) We will consider only the converter circuits using MOSFETs, which result in ZVS (zero voltage switching). The reason is that, based on Eq. 101, softswitching is of interest at high switching frequencies where MOSFETs are used. ln MOSFETs operating at high switching frequencies, a significant reduction of switching loss is achieved by not dissipating the charge associated with the junction capacitance inside the MOSFET each time it tums on. This implies that for a meaningful softswitching, the MOSFETs should be tumed on under a zero voltage switching (ZVS) condition. As we will see shortly, the tumoff also occurs at ZVS. 1031 Zero VoItage Switching (ZVS)
To illustrate the ZVS principIe, the intrinsic antiparallel diode of the MOSFET is shown as being distinct in Figs. 102a and b, where a capacitor is used in parallel. This MOSFET is connected in a circuit such that before applying the gate voltage to tum the MOSFET on, the switch voltage is brought to zero and the antiparallel diode is conducting as shown in Fig. 102a. This results in an ideal lossIess tum on at ZVS. At tumoff, as shown in Fig. 102b, the capacitor across the switch results in an essentially ZVS tumoff where the current through the MOSFET channel is removed while the voltage across the device remains small (essentially zero) due to the parallel capacitor. This ZVS principIe is illustrated by moditying the synchronousrectified Buck dcdc converter, as discussed below.
A I
' 
j
(a)
(b)
Figure 102 ZVS in a MOSFET. 1032 Synchronous Buck Converter with ZVS
As discussed in section 310 of Chapter 3, a synchronousrectified Buck dcdc converter, used in applications where the output voltage is very low, is shown in Fig. 103a where the diode is replaced by another MOSFET and the two MOSFETs are provided
103
complimentary gate signals
q+
and
q .
The waveforms associated with this
synchronous Buck converter are shown in Fig. 103b, where the inductor is large such that the ioductorcurrent ripple is smalI (shown by the solid curve in Fig. 103b), and the inductor current remains positive in the direction shown in Fig. 103a in the continuous conduction mode.
(a)
Figure 103 Synchronousrectified Buck converter. To achieve ZVS, the circuit of Fig. 103a is modified as shown in Fig 104a by showing the internal diode of the MOSFET explicitIy, and adding small external capacitances (in addition to the junction capacitances inherent in MOSFETs). The inductance value in this circuit is chosen to be much smaller such that the inductor current has a waveform shown dotted in Fig. 103 b with a large ripple, such that the current i,. is both positive as well negative during every switching cycle. q'
T'
+
q
,1
C
0+
ir.
l O
f_...r.'Y''.,,+
~R
T
1
Vo
q
:
titIr~~~~_'~ ~
'
' dd ay
(a)
(b)
Figure 104 Synchronousrectified Buck converter with ZVS. We wilI consider the transition at lhe time when the inductor current is at its peak
iL
transistor T+, which is initialIy conducting
I =
O labeled in Fig. 103b and Fig. 104b,
in Fig. 104a. The gate signal q + of the
II. '
goes to zero, while q  remains zero, as
shown in Fig. 104b. During the transition time during which the current transfers from T+ to T  is very short, and it is reasonable to assume for discussion purposes that the
inductor current remains constant at
104
iL
as shown in Fig. 105a.
T'
+
i,
q' =
_J ~
o
'~IL...J
r q
i,
_1
(a)
(b)
Fig. 105 Transition in synchronousrectified Buck converter with ZVS. ln the circuit of Fig. 105a with both q vç (O) =
v'n'
and q  equal to zero, initially vc' (O) = O and
where from Kirchhoff's law these two voltages must add to the input
voltage (102) As the current through T + declines, equal and opposite currents flow through the two capacitors in Fig. 105a, which can be derived from Eq. 102 as follows, assuming equal capacitances C : differentiating both s ides of Eq. 102 and multiplying both sides by C d d Cvc ' +Cvç =0 di
(103)
di
(104)
As the current through T + declines, a positive ic negative
iç
causes vç
causes vc
to nse from O, and a
v,n.
If this voltage transition
to decline from its initial vaI ue of
happens slowly compared to the current fall time of the MOSFET T + , then the tumoff of T + is achieved at essentially zero voltage (ZVS). After the current through T + has gone to zero, applying the Kirchhoff's current law in Fig. 1O5a and using Eq. 104 results in (105) ln Fig. 105a, during the tumofftransition of T +, v c' rises to The voltage vç
v'n and
vç
declines to
o.
cannot beco me negative because of the diode D  (assuming an ideal
diode with zero forward voltage drop), which begins to conduct the entire iL in Fig. 105b, marking the ZVS tumoff of T + .
105
Once D  begins to conduct, the voltage is zero across T  , which is applied a gate signal
q to tumon, as shown in Fig. 104b, thus resulting in the ZVS tumon of Y. Subsequently, the entire inductor current begins to flow through the channel of T  in Fig. 105b. The important item to note here is that the gate signal to T  is appropriately delayed by an interval T"'lay shown in Fig. 1O4b, making sure that q  is applied after D  begins to conduct.
The next halfcycle in this converter is similar with the ZVS tumoff of T  , followed by the ZVS tumon of T + , facilitated by the negative peak ofthe inductor current. Although this Buck converter results in ZVS tumon and tumoff of both transistors, the inductor current has a large ripple, which will also make the size of the filter capacitor large since it has to carry the inductor current ripple. ln order to make this circuit practical, the overall ripple that the output capacitor has to carry can be made much smaller (similar ripple reduction occurs in the current drawn from the input source) by interleaving oftwo or more such converters, as discussed in section 311 of Chapter 3. 1033 PhaseShift Modulated (pSM) DCDC Converter Another practical softswitching technology is the phaseshift modulated (PSM) dcdc converter shown in Fig. 106a.
/ VB1
'/_'\~L[_
L.
VAB
c\
O
ILJ
iAB lfll_ _\\_+I _,
~
( a)
''_'T
t
(b)
Figure 106 PhaseShift Modulated (PSM) DCDC Converter. It is a variation of the PWM dcdc converters discussed in Chapter 8. The switches in each powerpole of the PSM converter operate at nearly 50 percent dutyratio and the regulation of the output voltage is provided by shifting the output of one switching powerpole with respect to the other, as shown in Fig. 106b, to control the zerovoltage intervals in the transforrner primary voltage
V
AB .
To provide the ZVS tumon and tum
off of switches, a capacitor is placed across each switch. This topology makes use of the transforrner leakage inductance and the magnetizing current. The operation ofthis circuit 106
and a superior hybrid topology shown in Fig. 107, patented by the University of Minnesota [35], are fully described in the Appendix to this chapter on the accompanying CD. Ql +
J
V in Q2
J
Q5
•
J
•
•
J •
+
Vo
J
uncontrolled full  bridge
phase  modulated ful1  bridge
Figure 107 A superior hybrid topology to achieve ZVS down to no load [35].
104
INVERTERS FOR INDUCTION HEATING AND COMPACT FLUORESCENT LAMPS
It is possible to achieve softswitching in converters for induction heating and compact
fluorescent lamps. These converters are discussed in the Appendix to this chapter on the accompanying CD.
REFERENCES N. Mohan, T. M. Undeland, and W.P. Robbins, Power Electronics: Converters. Applications and Design, 3,d Edition, Wiley & Sons, New York, 2003. 2. R. Ayyanar, N. Mohan, and E . Persson, " SoftSwitching in DCDC Converters: Principies, Practical Topologies, Design Techniques, Latest Developments," Tutorial at IEEEAPEC 2002. 3. R. Ayyanar and N. Mohan, US Patent 6,310,785, University ofMinnesota, 2001. 4. R. Ayyanar and N. Mohan, "Novel softswitching dcdc converter with full ZVSrange and reduced filter requirement  Part I: Regulated output applications", IEEE Transactions on Power Electronics, vol. 16 (2001), March 2001, p. 184192. 5. R. Ayyanar and N. Mohan, ''Novel softswitching dcdc converter with full ZVSrange and reduced filter requirement  Part II: Constantinput, variable output applications", IEEE Transactions on Power E lectronics, vol. 16 (2001), March 2001, p. 193200. I.
PROBLEMS 101
ln a synchronousrectified Buck converter with ZVS shown in Fig. 1O4a,
v'n = 12 V, v;, = 5 V, f, = 100 kHz,
and the maximum load is 20 W. Calculate
the filter inductance such that the negative peak current is at least 1.5Amps.
107
102
108
ln Problem 101 , calculate tbe capacitances across the MOSFETs if the charge/discharge time is to be no more than 0.5 JlS .
Chapter 11 ELECTRIC MOTOR DRIVES 111
INTRODUCTION
Motor drives (ac and dc) fonn an extremely important application area of power electronics with market value oftens ofbillions dollars annually in applications described in Chapter 1. Figure 111 shows the block diagram of an electricmotor drive, or for short, an electric drive. ln response to an input command, electric drives efficiently control the speed and/or the position of the mechanical load. The controller, by comparing the input command for speed and/or position with the actual values measured through sensors, provides appropriate control signals to the powerprocessing unit (PPU) consisting of power semiconductor devices. Electric Drive
1, I fixed Electric Source fonn (utility)
Power
Processing
Load
Unit
I I I I
_ _ _ 1 input command
(speed I position)
Figure 111 810ck diagram of an electric drive system. As Fig. 111 shows, the powerprocessing unit gets its power from the utility source with singlephase or threephase sinusoidal voltages of a fixed frequency and constant amplitude. The powerprocessing unit, in response to the control inputs, efficiently converts these fixedform input voltages into an output of the appropriate form (in frequency, amplitude, and the number of phases) that is optimally suited for operating the motor. The input command to the electric drive in Fig. 111 may come from a process computer, which considers the objectives ofthe overall process and issues a command to control the mechanical load. However, in generalpurpose applications, electric drives operate in an openIoop manner without any feedback. Prior to discussing the need for power electronics in electric drives for speed and position control of mechanical systems, we will brief1y examine the requirements of mechanical
111
systems, and various types of electric machines in terms of their terminal characteristics in steady state in order to determine the voltage and current ratings in designing the power electronics interface. 112
MECHANICAL SYSTEM REQUIREMENTS
Electric drives must satis1'y the requirements of torque and speed imposed by mechanical loads connected to them. Most electric motors are of rotating type. 1121 Rotational MotorLoad Systems To understand rotating systems, consider a lever, pivoted and free to move. When an external force f is applied in a perpendicular direction at a radius r from the pivot, then the torque acting on the lever is T
[Nm]
=f r [N] [m]
(III)
which acts in a counterclockwise d irection, considered here to be positive. ln a rotational system, the angular acceleration due to a net torque acting on it
IS
determined by its momentofinertia J. The net torque TJ acting on the rotating body of inertia J causes it to accelerate. Similar to systems with linear motion, Newton's Law in rotational systems becomes
T;
(112)
=Ja
where the angular acceleration a( = d llJm I dI) in rad I S 2 is dllJ TJ a =  m = dt J
(113)
ln MKS units, a torque of I Nm , acting on an inertia of I kgÂˇ m 2 results in an angular acceleration of 1 rad I S2
.
ln systems such as the one shown in Fig. 112a, the motor produces an electromagnetic torque
T.m.
The bearing friction and wind resistance (drag) can be combined with the
load torque TL opposing the rotation. ln most systems, we can assume that the rotating part of the motor with inertia J M is rigidly coupled (without tlexing) to the load inertia JL
â€˘
The net torque, the difference between the electromagnetic torque developed by the
motor and the load torque opposing it, causes the combined inertias ofthe motor and the load to accelerate in accordance with Eq. 113 :
112
Figure 112 Motor and 10ad torque interaction with a rigid coupling.
d
0
(114)
lV = dI m J,q
where the net torque
0
= T,m  TL and the equivalent combined
inertiaJ,q = J M + J L â€˘
Eq. 114 shows that the net torque is the quantity that causes acceleration, which in turn leads to changes in speed and position. Integrating the acceleration a(t) with respect to time,
, (I 15)
Speed lVm(t) = lVm(O) + Ja(T)dT o
where lVm(O) is the speed at t=0 and T is a variable of integration. Further integrating lVm (t) in Eq. 115 with respect to time yields
, Om(/) = Om(O) + JlVm(T)dT o
where 0m (O) is the position at I =
(116)
O, and
T
is again a variable of integration. Eqs. 114
through 116 indicate that torque is the fundamental variable for controlling speed and position. Eqs. 114 through 116 can be represented in a blockdiagram form, as shown in Fig. 112b. 1122 Power and Energy in Rotational Systems ln the rotational system shown in Fig. 113, if a net torque Tcauses the cylinder to rotate by a differential angle dOm' the differential work done is (117) If this differential rotation takes place in a differential time dI, the power can be expressed as dW =T __ dOm =TlV p= __ dI dI m
(I 18)
where lVm = dOm / dI is the angular speed ofrotation.
113
Figure 113 Torque, work and power. 1123 Electrical Aoalogy An analogy with electrical circuits can be very useful when analyzing mechanical
systems. A commonly used analogy, though not a unique one, is to relate mechanical and electrical quantities as shown in Table III. Table 111 TorqueCurrent Analogy Mechanical System Electrical System Torque (T) Angular speed (Ăşlm
Current (i) Voltage (v)
)
Angular displacement ( Moment of inertia (1)
em )
Flux Iinkage ( ljI Capacitance (C)
)
For the mechanical system shown in Fig. 112a, Fig. 114 shows the electrical analogy, where torques are represented by current sources. lnertias are represented by capacitors, from its node to a reference (ground) node, and the two capacitors representing the two inertias are combined to result in a single equivalent capacitor.
Jeq=JM+J L
Figure 114 Electrical Analogy. 113
INTRODUCTION TO ELE CTRIC MACHINES AND THE BASIC PRINCIPLES OF OPERATION
Electric machines, as motors, convert electrical power input into mechanical output, as shown in Fig. IIS. Machines may be operated solely as generators, but they also enter the generating mode when slowing down (during regenerative braking) where the power flow is reversed. ln this section, we will briefly look at the basic structure and the fundamental principIes of the electromagnetic interactions that govern their operation.
114
Electrical System
Motoring mode
Electrica1
MechanicaI
Machine
System
Pelec   
Generating mode
Pele<:
Pmech
Figure 115 Electric machine as an energy converter. There are two basic principies that govern electric machines' operation to convert between electric energy and mechanical work: I) A force is produced on a currentcarrying conductor when it is subjected to an externallyestablished magnetic field. 2) An emf is induced in a conductor moving in a magnetic field. 1131 Electromagnetic Force Consider a conductor of length R in Fig. 116a. The conductor is carrying a current i and is subjected to an externallyestab/ished magnetic field of a uniforrn fluxdensity B perpendicular to the conductor length. A force f.m is exerted on the conductor due to the electromagnetic interaction between the external magnetic field and the conductor current. The magnitude ofthis force
/ ,m
is given as (I 19)
f.m=Bi R y .........
~
lN)
ITJ[A}[m)
As shown in Fig. 116a, the direction of the force is perpendicular to the directions of both i and B. To obtain the direction of this force, we will superimpose the flux lines produced by the conductor current, which are shown in Fig. 116a. The flux lines add up on the right side of the conductor and subtract on the left side, as shown in Fig. I 16b. Therefore, the force f.m acts from the higher concentration of jlux /ines
10
the lower
concentralion, that is, from right to left in this case. externa! B field
v f.m
f \
"
f
"
~) ) r
f,.
/ resuhant
(b)
Figure 116 Electric force on a currentcarrying conductor in a magnetic field.
115
1132 Induced EMF ln Fig. 117a, a conductor of length I! is moving to the right at a speed u. The Bfield is uniform and is perpendicularly directed into the paper plane. The magnitude of the induced emf at any instant oftime is then given by e =B / u ...................
[VI
(1110)
[T][mllm ls )
The polarity of the induced emf can be established as follows: due to the conductor motion, the force on a charge q (positive, or negative in the case of an electron) within the conductor can be written as
fq
(1111)
=q( uxB)
where the speed and the tlux density are shown by bold letters to imply that these are vectors and their cross product determines the force. Since u and B are orthogonal to each other, as shown in Fig. 117b, the force on a positive charge is upward. Similarly, the force on an electron will be downwards. Thus, the upper end will have a positive potential with respect to the lower end. This induced emf across the conductor is independent ofthe current that would tlow ifa c10sed path were to be available (as would normally be the case). With the current tlowing, the voltage across the conductor will be the inducedemf e(t) 10 Eq. 1110 minus the voltage drops across the conductor resistance and inductance. B(into paper)
X
X
X
X
X X
X X
+
Ă6 q+
t
19 f. 
X
X
X
X
"
X
f q+ B(into paper)
X /
X
U
X
(a) (b) Figure 1 17 Conductor moving in a magnetic field. 1133 Basic Structure Based on the above basic principies, ali machines have a stationary part, called the stator, and a rotating part, called the rotor, separated by an air gap, thereby allowing the rotor to rotate freely on a shaft, supported by bearings. This is shown by the crosssection of the machine in Fig. 118a, where the stator is firmly affixed to a foundation to prevent it from tuming.
116
r=:::::~~il gap o
(a)
o (b)
Figure 118 Crosssection of the machine seen from one side. ln electric machines, a magnetic field is produced either by permanent magnets or by windings that are supplied currents to produ ce radial fluxIines through the air gap, as shown in Fig. 118b. ln order to require smalI ampereturns to create flux lines crossing the air gap, both the rotor and the stator are made up of high permeability ferromagnetic materiaIs and the length of the air gap is kept as smalI as possible. ln machines with ratings under 10 kW in ratings, a typical length of the air gap is about I mm, which is shown highly exaggerated for ease of drawing. ln the next sections, we will briefly examine commonly used machines: dc, permanentmagnet ac, and induction. 114
DC MOTORS
DC motors were widely used in the past for alI types of applications, and they continue to be used in applications to control speed and position. There are two designs of dc machines: stators consisting of either permanent magnets or a field winding. The powerprocessing units can also be c1assified into two categories: switchmode power converters that operate at a high switching frequency as discussed in the next Chapter 12, or linecommutated, thyristor converters, which are discussed later in Chapter 14. ln this chapter, our focus wilI be on smalI servodrives, which usualIy consist of permanentmagnet motors supplied by switchmode power electronic converters. 1141 Structure oCDC Machines Fig. 119 shows a cutaway view of a dc motor. It shows a permanentmagnet stator, a rotor that carries a winding, a commutator, and the brushes. ln dc machines, the stator establishes a unifonn flux
tPf
in the air gap in the radial direction (the subscript " f " is
for field). If permanent magnets like those shown in Fig. I 19 are used, the air gap flux density established by the stator remains constant. A field winding whose current can be varied can be used to achieve an additional degree of control over the air gap flux density .
117
stator
~~~ ~imagnets rotor
"",..winding
,.Ă
Figure 119 Exploded view of a dc motor; source: Engineering Handbook by ElectroCraft Corpo As shown in Fig. 119, the rotor slots contain a winding, called the armature winding, which handles electrical power for conversion to (or from) mechanical power at the rotor shaft. [n addition, there is a commutator affixed to the rotor. On its outer surface, the commutator contains copper segments, which are electrically insulated from each other by means of mica or plastic. The coils of the armature winding are connected to these commutator segments so that a stationary dc source can supply voltage and current to the rotating cummutator by means of stationary carbon brushes which rest on top of the commutator. The wear due to the mechanical contact between the commutator and the brushes requires periodic maintenance, which is the main drawback of dc machines. 1142 Operating PrincipIes ofDe Machines
As Fig. I 110 pictorially shows, the commutator and the brushes in dc machines act as a "mechanical rectifier" and convert a dc current ia supplied by a stationary dc source into a current that changes direction in the armature coil every half revolution of the rotor.
+
Figure 1110 DC machine schematic representation. 118
Ali conductors under a stator pole have currents in the sarne direction but the current direction reverses when the conduc tors reach the other pole. This is needed so that the forces produced on each conductor are in the sarne direction and add up to yield the total torque. Similarly, ali conductors under apoie have induced emfs of the sarne polarity, which changes in polarity every ha lf revolution of the rotor when the conductors reach the other pole. The above description shows that the magnitude and the direction of the electromagnetic torque depend on the armature current ia' Therefore, in a permanentmagnet dc machine, the electromagnetic torque produced by the machine is linearly related to a machine torqueconstant k T , which is unique to a given machine and is specified in its data sheets (J 112) Similarly, the induced emf ea depends only on the rotational speed "'m' and can be related to it by a voltageconstant k E
,
which is unique to a given machine and specified
in its data sheets (J 113) Equating the mechanical power ("'m T,m ) to the electrical power ( eaia )' the torqueconstant k T and the voltageconstant k E are exactly the sarne numerically in MKS units (1114) Reversing the Torgue Direction. The direction of the armature current ia determines the direction of currents through the conductors.
Therefore, the direction of the
electromagnetic torque produced by the machine ais o depends on the direction of ia ' This explains, how a dc machine wh ile rotating in a forward or reverse direction ean be made to go from motoring mode (where the speed and torque are in the sarne direction) to its generator mode (where the speed and torque are in the opposite direetion) by reversing the direetion of ia . Reversing the Direetion of Rotatio n . Applying a reversepolarity dc voltage to the armature terminais makes the armature eurrent flow in the opposite direetion. Therefore, the eleetromagnetie torque is reversed, reversing the direetion of rotation and the polarity ofindueed emfs in eonduetors, whie h depends on the direetion ofrotation. FourOuadrant Operation. The above diseussion shows that a de maehine ean easily be made to opera te as a motor or as a generator in forward or reverse direction of rotation .
119
1143 DCMachiue Equivaleut Circuit It is convenient to discuss a dc machine in tenns of its equivalent circuit ofFig. 1111, which sbows conversion between electrical and mechanical power. The annature current
ia produces the electromagnetic torque 1',m(= kr ia ), represented by a dependent current
source necessary to rotate the mechanical load at a speed terminaIs, the rotation at the speed of
úJm
úJm '
Across the annature
induces a voltage, called the backemf
ea (= kEúJm ) , represented by a dependent voltagesource. On the electrical side, the applied voltage v a overcomes the backemf ea and causes the current ia to flow.
Recognizing that there is a voltage drop across both the annature
winding resistance Ra (which includes the voltage drop across the carbon brushes) and the annature winding inductance La' we can write the equation of the electrical side as (1115) On the mechanical side, the electromagnetic torque produced by the motor overcomes the mechanicalIoad torque TL to produce acceleration: dúJm
dt
=_I_(T T) J em L
(1116)
'q
where J ,q is the total etfective value of the combined inertia of the dc machine and the mechanical load.
Figure IIII
De motor equivalent circuito
1144 TorqueSpeed Characteristics Note tbat the electric and the mechanical systems are coupled. The torque 1',m in the mechanical system (Eq. 1112) depends on the electrical current ia' The backemf ea in the electrical system (Eq. 1113) depends on the mechanical spced úJm • The electrical
1110
power absorbed from the electrical source by the motor is converted into mechanical power and vice versa. ln a dc steady state, with a voltage Va applied to the armature terminais, and a loadtorque TL be ing supplied to the load, the equivalent circuit is as shown in Fig. 11l2a, where the inductance La (not shown) in the electrical portion and the capacitance (shown dotted) representing J ,q in the mechanical portion of the circuit do not play a role in dc steady state. Hence, in Fig. 1112 (1117)
(1118) Based on Eqs. 1117 and 1118, the steady state torquespeed characteristics for various values of Va are plotted in Fig. 1112b. J,,
+ PPU
+
1
W.
I I
'..,J ..
v.
"'m
: :
+,
Va 3
I I
Vai> Va2 > Va3 > Va4
Wm,rated Va/=rated Va 2
T,
Va4
1'_,1.:. ___ _ ,
E=kwT =~ 1 .. Em~m"TQ
(a)
(b)
Figure 1112 DC motor in the dc steady state.
115
PERMANENTMAGNET AC MACHlNES
We will now study an important class of ac drives, namely sinusoidalwaveform, permanentmagnet ac (PMAC) drives, which are used in applications where a high efficiency and a high power density are required in controlling the speed and position. ln trade literature, they are also called "brushless dc" drives. We will examine these machines for speed and position control applications, usually in small ÂŤ 1O kW) power ratings, where these drives have threephase ac stator windings and the rotor has dc excitation in the form of permanent magnets. ln such drives, the stator windings of the machine are supplied by controlled currents, which require a closedIoop operation, as shown in the block diagram of Fig. 1113. The discussion of PMAC drives also lends itself to the analysis of lineconnected synchronous machines, which are used in very large ratings in the central power plants ofutilities to generate electricity.
1111
Power Processing
@u tility
Unit
ControU input l
ia h ic
Sinuso idal
Load
W
motor
(1m
Controller
fm={ Position I sensor
~
PMAC
(I)
I
Figure 1113 Block diagram ofthe c losed loop operation ofa PMAC drive.
1151 The Basic Structure ofPermanentMagnet AC Machines We will consider 2pole machines, like the one shown schematically in Fig. 1114a. This analysis can be generalized to ppole machines where p > 2. The stator contains threephase, wyeconnected windings shown in the crosssection ofFig. 1114a, each ofwhich produce a sinusoidallydistributed f1uxdensity distribution in the air gap, when supplied by a current. 1152 PrincipIe of Operation The permanentmagnet pole pieces mounted on the rotor surface are shaped to ideally produce a sinusoidallydistributed f1ux density in the air gap. The rotor f1uxdensity distribution (represented by a vector ĂŠ,) peaks at an axis that is at an ang le
em(I)
with
respect to the a axis of phase winding a, as shown in Fig. 1114b. As the rotor tums, the entire rotorproduced f1ux density distribution in the air gap rotates with it and "cuts" the statorwinding conductors and produces emf in phase windings that are sinusoidal functions of time. ln the ac steady state, these voltages can be represented by phasors. Considering phasea as the reference in Fig. 1114b, the induced voltage in it due to the rotor f1ux cutting it can be expressed as
.~ b axlS '. ,
,, ,
Br(l)

aaxis
:CaxIS ,
Cb)
Ca) Figure 1114 Twopole PMAC mach ine.
1112
(1119) It should be noted that the induced voltage in phasea peaks when the rotorflux peak in
Fig. 1114 is pointing downward, 90° before it reaches the phasea magnetic axis. Since the flux produced by the permanent magnets on the rotor is constant, the rms magnitude of the induced voltage in each stator phase is linearly related to the rotational speed mm by a perphase voltageconstant
k E . pha"
(1120) An important characteristic of the machines under consideration is that they are supplied through a currentregulated powerprocessing unit shown in Fig. 1113, which controls the currents ia ' ib , and ic supplied to the stator at any instant oftime. To optimize such that the maximum torqueperampere is produced, eaeh phase eurrent in ae steady state is eontrolled in phase with the indueed voltage. Therefore, with the voltage expressed in Eq. 1119, the eurrent in ae steady state is (1121) ln ae steady state, aeeounting ali threephases, the input eleetrie power, supplied by the eurrent in opposition to the induced baekemf, equals the meehanieal output power. Using Eqs. 1119 through 1121
T.mmm
=
3(kE .pha" mm)/,m,
(1122)
'v'
E_
The torque eontribution of eaeh phase ean be written as
Tem ,lphase _T.m_k 3  E, phose / rms
(1123)
ln Eq. 1123, the eonstant that relates the rms eurrent to the perphase torque is the perphase torqueconstant
k". ph~•
kT , phase = k E, phase
. Therefore, similar to that in de maehines, in MKS units
(1124)
At this point, we should note that PMAC drives eonstitute a elass, whieh we will eall selfsynchronous motor drives, where the term "self' is added to distinguish these maehines from the eonventional synehronous maehines. The reason for this is as follows: in PMAC drives, the stator phase eurrents are synehronized to the meehanieal position of the rotor sueh that, for example, the eurrent into phasea, in order to be in phase with the inducedemf, peaks when the rotorflux peak is pointing downward, 90° before it reaehes
1113
the phasea magnetic axlS. This explains the necessity for the rotorposition sensor, unless the rotor position is mathematically computed by sensed voltages and currents. 1153 Mechanical System ofPMAC Drives
The electromagnetic torque acts on the mechanical system connected to the rotor, and the resulting speed mm can be obtained from the equation below: _d_m_ m = _T,, 'm,::_T"L dt J ,q
(1125)
where J ,q is the combined motorIoad inertia and TL is the load torque, which may include friction . The rotor position Om (t)
IS
(1126) where 0m(O) is the rotor position at time t=O. 1154 PMAC Machine Equivalent Circuit
Similar to a dc machine, it is convenient to discuss a PMAC machine in terms of its equivalent circuit of Fig. I115 a, which shows conversion between electrical and mechanical power. ln the ac steady state, using phasors, the current I . is ensured to be in phase with the phasea induced voltage
Em.
currents produce the total electromagnetic torque
by the feedback control.
1'.m ( = 3kT.PhMJ~, ) ,
The phase
represented by a
dependent currentsource, necessary to rotate the mechanical load at a speed mm' The induced backemf
Em.
= E~LOo, whose rms magnitude is linearly proportional to the
speed of rotation mm' is represented by a dependent voltagesource.
7;. = +
L.
• +
PPU
I...u L Oo .,,,
+1
lV", :
,,
..
'J , 1
(b)
f",DCj T_ = 31<,..,••...1~
Figure 1115 Equivalent circuit diagram and the phasor diagram ofPMAC (2 pole). On the electrical s ide, the applied voltage and causes the current 1114
7.
V.
in Fig. I 115a overcomes the backemf
Em.
to f1ow. The frequency of the phasors in Hz equals mm /(2n)
in a 2pole PMAC machines. There is a voltage drop across both the perphase stator winding resistance R, (neglected here) and a perphase inductance L" which is the sum of the leakage inductance caused by the leakage f1ux of the stator winding, and the effect of the combined flux produced by the currents f10wing in the stator phases. On the mechanical side in Fig. ll15a in steady state, the capacitance representing J ,q in the mechanical portion ofthe circuit is of no effect, and
T.m
= TL
•
Note that the electric and the mechanical systems are coupled. ln the electrical system, the backemf
Emo
magnitude (Eq. 1120) depends on the mechanical speed
mechanical system, the torque current
T.m
aJm •
ln the
(Eq. 1123) depends on the magnitude ofthe electrical
lo, which depends on the load torque being demanded of the machine. The
electrical power absorbed from the electrical source by the motor is converted into mechanical power and vice versa. The phasor diagram, neglecting R" is shown in Fig. 1115b. 1155 PMAC TorqueSpeed Characteristics
ln PMAC machines, the speed is independent ofthe electromagnetic torque developed by the machine, as shown in Fig. 1116a, and depends on the frequency of voltages and currents applied to the stator phases of the machine.
J;>fz>J, > f. f         J;
Vo
ffz f         J,
11. /
o ''l~ T.m
0''1f=
(a)
(b)
aJm
2n Figure I I  I 6 Torquespeed characteristics and the voltage versus frequency in PMAC. ln the perphase equivalent circuit of Fig. 1 115a and the phasor diagram of Fig. I I  I 5b, the backemf is proportional to the speed linearly related to
aJm
,
aJm •
Similarly, since the electrical frequency is
the voltage drop across L, is also proportional to
aJm
•
Therefore,
neglecting the perphase stator winding resistance R" in the phasor diagram of Fig. 1 115b, the voltage phasors are ali proportional to
aJm ,
requiring that the perphase voltage
magnitude that the powerprocessing unit needs to supply is proportional to the speed aJm , as plotted in Fig. 1116b. This relationship between the voltage and speed shown in Fig. 1 1 I 6b is approximate.
A substantially higher voltage, called the voltage boost, 1115
above that indicated by the dotted line, is needed at very low speeds, where the voltage drop across the stator winding resistance R, becomes substantial at higher torque loading, and hence cannot be neglected. 1156 The Controller and the PowerProcessing Unit (pPU)
As shown in the block diagram ofFig. 1113, the task ofthe controller is to dictate the switching in the powerprocessing unit, such that the desired currents are supplied to the PMAC motors. The reference torque signal is generated /Tom the outer speed and position loops discussed in Chapter 13.
The rotor position Bm is measured by the
position sensor connected to the shaft. Knowing the torque constant
kT . pha"
allows us to
calculate the rms value ofthe reference current /Tom Eq. 1123. Knowing the current rms vai ue and Bm allows the reference currents for the three phases to be calculated, which the PPU delivers at any instant oftime. 116
INDUCTION MACIDNES
lnduction motors with squirrelcage rotors are the workhorses of industry because of their low cost and rugged construction. When operated directly from line voltages (a 50 or 60Hz utility input at essentially a constant voltage), induction motors operate at a nearly constant speed. However, by means of power electronic converters, it is possible to vary their speed efficiently. 1161 Strnctnre of Induction Machines
The stator of an induction motor cons ists of threephase windings, distributed in the stator slots. These three windings are displaced by 120 0 in space with respect to each other, as shown by their axes in Fig. 111 7a. The rotor, consisting of a stack of insulated laminations, has electrically conducting bars of copper or aluminum inserted (molded) through it, close to the periphery in the axial bms
(b) (a) Figure 1117 (a) Threephase stator; (b) squirrelcage rolor,
1116
direction. These bars are electrically shorted at each end of the rotor by electrically conducting endrings, thus producing a conducting cagelike structure, as shown in Fig. 1117b. Such a rotor, called a squirrelcage rotor, has a low cost, and rugged nature. 1162 Principies oflnduction M o tor Operation
Figure 1118a shows the stator windings whose voltages are shown in the phasor diagrarn ofFig. 1118b, where the frequency ofthe applied linevoltages to the motor is j in Hz, and
V~,
is the magnitude in rms v;,=V~, LO째 ,
V.=V~,L 120째, and
v  'V,}'+'

~ = V~,L 240째
 
n
(1127)
1<
v.
v;.
I ma
(b)
(a)
Figure 1118 Induction mach ine: applied voltages and magnetizing currents. By Faraday' s law, the applied stator voltages given in Eq. 1127 establish a rotating fluxdensity distribution in the air gap by drawing magnetizing currents of rms value 1m' which are shown in Fig. ll18b: (1128) As these currents vary sinusoidaUy with time, the combined fluxdensity distribution in the air gap produced by these currents rotates at a synchronous speed
CO",.
= 2ffj
(
CO 'yn 
2ff j p l2
for appole maChine)
CO,yn'
where (1129)
The rotor in an induction motor tums (due to the electromagnetic torque developed, as will be discussed shortly) at a speed
com
gap fluxdensity distribution, such that
in the sarne direction as the rotation of the air com
< CO" n. Therefore, there is a relative speed
between the fluxdensity distribution rotating at
CO,yn
and the rotor conductors at
COm .
This relative speed, that is the speed at which the rotor is "slipping" with respect to the rotating fluxdensity distribution, is called the slip speed: slip speed
OJslip = l체.syn 
mm
(1130)
1117
By Faraday's Law (e = B l u) , voltages are induced in the rotor bars at the slip frequency due to the relative motion between the fluxdensity distribution and the rotor, where the slip frequency
hUp
in terms ofthe frequency
f
ofthe stator voltages and currents is
slip frequency
(1131)
Since the rotor bars are shorted at both ends, these induced bar voltages cause slipfrequency currents to flow in the rotor bars. The rotorbar currents interact with the fluxdensity distribution established by the statorapplied voltages, and the result is a net electromagnetic torque
T.m
in the sarne direction as the rotor's rotation.
The sequence of events in an induction machine to meet the load torque demand IS as follows: At essentially no load, an induction machine operates nearly at the synchronous speed that depends on the frequency of applied stator voltages (Eq. 1129). As the load toque increases, the motor slows down, resulting in a higher value of slip speed. Higher slip speed results in higher voltages induced in the rotor bars and hence higher rotorbar currents. Higher rotorbar currents result in a higher electromagnetic torque to satisfy the increased loadtorque demando Neglecting secondorder effects, the air gap flux is totally determined by the applied stator voltages. Hence, the air gap flux produced by the rotorbar currents is nullified by the additional currents drawn by the stator windings, which are in addition to the magnetizing currents in Eq. 1128. 1163 PerPhase Equivalent Circuit ofInduction Machines ln this balanced threephase sinusoidal steady state analysis, we will neglect secondorder effects such as the stator winding resistance and leakage inductance, and the rotor circuit leakage inductance. As shown in Fig. 1119a for phasea in this perphase circuit, and so on are applied at a frequency
ti;,
f, which results in magnetizing currents to
establish the air gap fluxdensity distribution in the air gap, represented as flowing through a magnetizing inductance Lm' The voltage magnitude and frequency are such that the magnitude of the fluxdensity distribution in the air gap is at its rated value and this distribution rotates counterclockwise at the desired
Ăš},yn
(Eq. 1129) to induce a
backemf in the stationary stator phase windings such that (1132) where,
1118
k E.p/w"
is the machine perphase voltageconstant.
Next, we will consider the effect ofthe rotorbar currents. The rotorbar currents result in additional stator currents (in addition to the magnetizing current), which can be represented on a perphase basis by a current 7;." inphase with the applied voltage (since the rotorcircuit leakage inductance is ignored), as shown in Fig. 1119a
l'=
= l' LOo =
(J 133)
+ á!.
t
PPU
T..
I I
'.,I
Jq
+ T, (b)
v:. =
I' Ema
=
kE . ,.,.,.,~(j}syr> LO°
'E; =
DCj kE.pIttuJomLO° r:", = 3kr .phsuJ: (a)
Figure 1 119 Induction motor equivalent circuit and phasor diagram. The fluxdensity distribution (although produced differently than in PMAC machines) interacts with these stator currents, just like in PMAC machines, and hence the perphase torqueconstant equals the perphase voltage constant
k T,
pha.~e
(1134)
=kE, phase
The perphase torque can be expressed as (1135) This torque at a mechanical speed wm results in perphase electromagnetic power that gets converted to mechanical power, where using Eqs. 1 [34 and 1135
Pem,phase =wT. 1'ra m em,p ase =wk m E., p • a~'e
(1136)
ln Eq. 1136, (WmkE.p.~, ) can be considered the backemf E~ ,just like in the DC and the PMAC machines, as shown in Fig. 1 [[9a (1 [37) ln the rotorcircuit, the voltages induced depend on the slipspeed W' lip' and overcome the
IR voltage drop in the rotor bar resistances. The rotorbar resistances, and the voltage drop and the power losses in them, are represented in the perphase equivalent circuit of
1119
Fig. l119a by a voltage drop across an equivalent resistance
R;.
Using the Kirchhoff's
voltage law in Fig. 1119a, (1138) Hence, (1139) Using Eqs. 1134, 1136 and 1139, the combined torque ofall three phases is (1140)
where k r ", . ~"P in the above equation is a machine constant that shows that the torque produced is linear1y proportional to the slip speed. Using Eqs. 1130 and 1140, lVm = (j}syn
(1141)
k T.dI.sI"
8ased on Eq. 1141, the torquespeed characteristics are shown in Fig. 1120a for various applied frequencies to the stator. The stator voltage as function of frequency is shown in Fig. 1120b to maintain the peak of the fluxdensity distribution at its rated vaI ue. The relationship between the voltage and the synchronous speed or f shown in Fig. 1120b is approximate. It needs a substantially higher voltage than indicated (shown dotted) at very low frequencies where the voltage drop across the stator winding resistance R, becomes substantial at higher torque loading, and hence cannot be neglected.
m
OJ
f..>/,>/,>f.
f..
Va
f, /, f. O (a)
I;m
O (b)
f
=
msyn
2"
Figure 1120 lnduction motors: Torquespeed characteristics and voltage vs. frequency.
1120
117
SUMMARY
The discussion above shows the requirements of three common types of machines supplied through powerelectronics based power processing units in steady state.
REFERENCE I.
2.
N. Mohan, Electric Drives: An Integrative Approach, year 2003 Edition, MNPERE, Minneapolis, www.MNPERE.com. N. Mohan, Advanced Electric Drives: Analysis, Modeling and Simulation using Simulink, year 2001 Edition, MNPERE, Minneapolis, www.MNPERE.com.
PROBLEMS Mechanical Systems I lI
112
A constant torque of 5 Nm is applied to an unloaded motor at rest at time t = O. The motor reaches a speed of 1800 rpm in 4 s. Assuming the damping to be negligible, ca1culate the motor inertia. ln an electricmotor drive similar to that shown in Fig. 112a, the combined inertia is J eq= 5xIO3kg.m2. The load torque is T =0.05kg·m 2 . Draw the L
113
electrical equivalent circuit and plot the electromagnetic torque required /Tom the motor to bring the system linearly /Tom rest to a speed of 100 rad/s in 4 s, and then to maintain that speed. Plot the power required in Problem I 12.
DCMotors 114
A permanentmagnet dc motor has the following parameters: Ra = 0.35Q and kE
= k T = 0.5
in MKS units. For a torque of up to 8 Nm, plot its steady state
torquespeed characteristics for the following values of Va 115
Consider
the
J m = 0.02 kg· m
dc 2
•
motor
of
Problem
114
:
100 V, 75 V, and 50 V.
whose
momentofinertia
lts armature inductance La can be neglected for slow changes.
The motor is driving a load of inertia J L = 0.04 kg· m 2 •
The steady state
operating speed is 300 rad/s. Calculate and plot the terminal voltage vael) that is required to bring this motor to a halt as quickly as possible, without exceeding the armature current of 12 A.
PermanentMagnet AC Motors 116
ln a threephase, 2pole, brushlessdc motor, the torque constant k T = 0.5Nm / A. Ca1culate the phase currents ifthe motor is to produce a counterclockwise torque of 5Nm. 1121
117
ln a 2pole, threephase (PMAC) brushlessdc motor drive, the torque constant kr
and the voltage constant k E are 0.5 in MKS units.
The synchronous
inductance is 15 mH (neglect the winding resistance). This motor is supplying a torque of 3 Nm at a speed of 3,000 rpm in a balanced sinusoidal steady state. Calculate the perphase voltage across the powerprocessing unit as it supplies controlled currents to this motor.
Induction Motors 118
119
Consider an induction machine that has 2 poles and is supplied by a rated voltage of 208 V (linetoline, rms) at the frequency of 60 Hz. It is operating in steady state and is loaded to its rated torque. Neglect the stator leakage impedance and the rotor leakage fluxo The perphase magnetizing current is 4.0 A (rms). The current drawn perphase is 10 A (rms) and is at an angle of 23.56 degrees (lagging). Calculate the perphase current if the mechanical load decreases 50 that the slip speed is onehalfthat ofthe rated case. ln Problem 118, the rated speed (while the motor supplies its rated torque) is 3475 rpm.
IIlO
IIII
Calculate the slip speed OJ'''p' and the slip frequency f."p of the
currents and voltages in the rotor circuito ln Problem 119, the rated torque supplied by the motor is 8 Nm. Calculate the torque constant, which linearly relates the torque developed by the motor to the slip speed. A threephase, 60Hz, 4pole, 440 V (Iineline, rms) inductionmotor drive has a fullIoad (rated) speed of 1746 rpm. The rated torque is 40 Nm. Keeping the air gap fluxdensity peak constant at its rated value, (a) plot the torquespeed characteristics (the linear portion) for the following values of the frequency f:
60 Hz, 45 Hz, 30 Hz, and 15 Hz. (h) This motor is supplying a load whose torque demand increases linearly with speed, such that it equals the rated torque of the motor at the rated motor speed. Calculate the speeds of operation at the four values offrequency in part (a). 1112 ln the motor drive of Problem IIII , the induction motor is such that while applied the rated voltages and loaded to the rated torque, it draws 10.39 A (rms) perphase at a power factor of 0.866 (Iagging). R, = 1.50. Calculate the voltages corresponding to the four values ofthe frequency
1122
f to maintain Em'
=
E~.ro"d Âˇ
Chapter 12 SYNTHESIS OF DC AND LOWFREQUENCY SINUSOIDAL AC VOLTAGES FOR MOTOR DRIVES ANDUPS 121
INTRODUCTION
Motor drives (ac and dc) are important application areas ofpower electronics with market value of tens of billions dollars annually, as described in Chapter 1. Uninterruptible power supplies (UPS) are a special case of ac motor drives in discussing the role of power electronics, and will be briefly discussed in section 126 in this chapter. ln motor drive applications, the voltagelink structure of Fig. 116, repeated in Fig. 121 ,
is used, where our emphasis will be to discuss how the loadside converter with the dc voltage as an input synthesizes dc or lowfrequency sinusoidal voltage outputs. Functionally, this converter operates as a linear amplifier, amplifYing a control s ignal, dc in case of dcmotor drives, and ac in case of acmotor drives. The power flow through this converter should be reversible. + conv2
'V }l conv I
utility
Load
controJler
Figure 121 Voltagelink system. These converters consist ofbidirectional switching powerpoles discussed earlier, two in case of dcmotor drives and three in ac motor drives, as shown in Figs. 122a and b. ac motor
!
.l!..... A vA
d'
:::
B'
I, I
VB
C
....., ~
~
, ,I n
==__ c
cL__ (h)
Figure 122 Converters for dc and ac motor drives.
121
122
SWITCIllNG POWERPOLE AS THE BUILDING BLOCK
To synthesize de or lowfrequeney s inusoidal ae outputs, these eonverters eonsist of the switehing powerpole shown in Fig. 123a with the bidireetional eurrent eapability, whieh was introdueed in Chapter 3. [t eonsists of a parallel eombination of Buek and Boost eonverters, as shown in Fig. 123b, where the two transistors are eontrolled by eomplimentary gate signals. This implementation and the eomplimentary gating of transistors allow a eontinuous bidireetional power and eurrent eapability as diseussed below, and henee ideally a diseontinuous eonduetion mode does not existo The average representation by an ideal transforrner is shown in Fig. 123e. Buck .. , Boas! ., . .,. ~
+
+
(a)
(b)
(c)
Figure 123 Bidireetional switehing powerpole. 1221 PulseWidthModulation (pWM) ofthe BiDirectional Switching PowerPole Tt is elear from the diseussion of the switehing powerpoles that the output voltages of
these poles at the eurrentport are always of a positive polarity. However, the output voltages of eonverters for motor drives must be reversible in polarity. This is aehieved as shown in Fig. 122 by the differential output between two switehing powerpoles in demotor drives, and between any two of the three powerpoles in ae motor drives . The average representation of a switehing powerpole is shown in Fig. 124a. ln eaeh switehing pole, the output voltage at the eurrentport includes a de offset of Vd / 2 (that eorresponds to the dutyratio dA equal to 0.5). To synthesize a lowfrequeney sine wave, for example, varying the dutyratio sinusoidally around 0.5 , the average output voltage varies sinusoidaIly around Vd / 2, as shown in Fig. 124b. The average output voltage ean range from O and Vd
â€˘
ln the differential output between two powerpoles
ln
eonverters for de drives, and between any two of the three poles for ae drives, shown in Fig. 122a and b, the de offset is neutralized as will be diseussed in the foIlowing seetions in this ehapter.
122
,idA
+ Vd
,<
!

IA
+
r
T
T
VAN
rI v c, A
~
VAN
1: dA
~
(b)
Figure 124 Varying the dutyratio around 0.5 varies
V AN
around Vd / 2 .
ln motor drives, to generate the switching function qA' a triangularwaveform signal v,,; is used for comparison in the PWMIC, as shown in Fig. 125a, where the peak value V,ri is kept constant and the frequency of the triangular wavefonn establishes the switching frequency
J, .
i"
,
o
,,, , ,
I
~
q , (t) :
o
I
IT I
," ',
,
o
I
"
f ' j ± 115'. 0 , __ 1 0 o '
I .",,(1)1: v_ (1)
I
~
"
t
•I
r:
f  i f  =Jfi
i~~ 00
d J')
w
•
v_ t
• t
Figure 125 Switching powerpole and its voltage and current waveforms. The resulting switchingsignal waveforms are shown in Fig. 125b. The dutyratio d A' obtained by comparison of these linear waveforms, can be derived by considering two values of the control signal Vc A: V,ri results in d A to equal I, and (V,ri) results in d A to equal O. Linearly interpolating between these two values, the expression for d A can be written as 123
(121 ) which shows that it consists of two terms  a dc offset of 0.5 and a term that is linearly proportional to the control voltage V'.A(t). Therefore, the average output voltage of the powerpole can be written as (122)
where O.5Vd is the dc offset and kpo,< is the gain by which the control signal is amplified to produce the output voltage. The average representation of the pulsewidthmodulated switching powerpole is shown Fig. 126.

idA
+
r   ,
+1 vdT
1 i
iA
~
I
Vd
A i>AN
I
L_
_ ...J
q A(I)
Vc A(I)
Vc. A(I)
(a )
(b)
Figure 126 Average representation of the pulsewidthmodulated powerpole. 1223 Harmonics in the PWM Waveforms vA and idA
Although only their average values are of interest, we must recognize that the voltage vA and the current idA contain harmonics in addition to their intended average values. Fourier analysis of these waveform s shows that they contain components at frequencies at the multiples ofthe switching frequency /" with sidebands located at the multiples of the frequency !, being synthesized by the switching powerpole.
These harmonic
frequencies can be expressed as foll ows, where k, and k, are constants, which can take on values 1,2, 3, and
50
on:
fh = k,/, Âą k,!,

sidebands
124
(123)
This harmonic analysis is graphically shown in Fig. 127, which highlights the importance of selecting a high switching frequency. Clearly, the harmonic components in v A at frequencies
f.
switching powerpole.
given by Eq. l23 are undesirable but also unavoidable in a However, the minimum value in
.f"
depends on
J"
and by
selecting a high switching frequency, ali the harmonic frequencies are pushed to the correspondingly high values. At these high values, the series inductance at the currentport becomes very effective in ensuring that the current iA has a small ripple in spite of the pulsating vA '
Similarly, the highfrequency components in
i dA
are filtered by a
relatively small capacitor across the voltageport. For this reason in modem power electronic systems, it is typical to use switching frequencies a few hundred kHz in dcdc converters at low power, and a few tens of kHz in converters for motor drives up to a few hundred kW.
Vh
3fs
f,
4 fs
Figure 127 Harmonics in the switching powerpole. 123
DeMOTOR DRlVES
ln dcmotor drives as shown by Fig. l28a, the dc motor is controlled by the dc voltage Vo
applied to its terminals. The converter consists oftwo bidirectional switching power
poles, which are pulsewidthmodu lated. Allowing the power (and hence the current) through this converter to be bidirectional facilitates the machine to operate both in motor and as well in its generator mode, in either direction of rotation , as shown in Fig. 128b in terms of average terminal quantities. Motoringmode in the forward direction rotation is represented by quadrant 1 with positive voltage and current. While rotating in the forward direction, this machine can be slowed down by making the machine go into its generator mode in which the power flow reverses, as shown by quadrant 2 with a positive voltage and a negative current. Motoring in the reverse direction of rotation requires voltage and current both negative, corresponding to quadrant 3 . While rotating in the reverse direction, the generatormode in quadrant 4 slows down the machine .
125
I
+
Vd
I T A
I
B
I
x
X
I
+
Vo
Vo
ea l
l______________ ~_J
( a)
x
io
(b)
Figure 128 Demotor fourquadrant operation. ln dcmotor drives, as shown in Fig. 129a, the control voltage vc(t) produced by the feedback controller is applied as vc.A (t), equal to vc(t), to produce the switching function
q A (t) for poleA. The control voltage for poleB is negative of the control voltage for poleA: vc.B (t) = vc(t). The average representation ofthe two poles is shown in Fig. 12
9b and the switching waveforms are shown in Fig. 129c. Using Eq. 12l (124)
(125)
â€˘
( 126)
where kpwm is the gain by which the control voltage is amplified by this switchmode converter to apply the voltage at the terminais of this dc machine. This output voltage can be controlled in a range from (Vd
)
to Vd
.
Thus, a fourquadrant converter
IS
rea1ized by using twopoles, each of which is capable of a twoquadrant operation. The total average dcside current is the sum of the average dcside currents of each pole in Fig. 129b: (127)
Using the values of the dutyratios of the two poles, and recognizing the directions with which the currents are defined, (128)
126
Tbus, using Eq. 121 to aehieve d A and d B in terms of vc(t) and substituting them in
Eq. 127 along with the currents from Eq. 128: "'(I) ~ vcY) T,,(t)
(129)
V,ri
+ A
B N
vJt)
vo(t)
(a)
i
~ ;0
IX Motor
~
== :
v...ft): ,
::
idO
+
I
â€˘
BW
io(t)
I , ,
tt
,
:, ,
 
~
n~~_+_:~~f___L___ t
.
Vc . A (I)
I dB
()~ , 7,i+~,f+, () ~T
t
:: I I
I I
""
id (t)o;1 (h)
! ,,
Or>++~f4_i__+_ t
," idA(t)  ~N~)
t
,, ,
,
,,,
t;j
(c)
~ ~
,:: '" , I ,.
t
~
n f1t= ~ I ' , , ' ,
Figure 129 Converter for demotor drives. The instantaneous quantities assoeiated with this switehmode dede converter are pulsating, as shown in Fig. 12ge. Based on the diseussion of harmonic eomponents in the voltage at the eurrentport and in the eurrent at the voltageport of eaeh powerpole, we ean eaIculate eaneellation of some harrnonies in the eombination of the two powerpoles in a dedrive converter. A detailed diseussion ofthis ean be found in referenee [1].
127
+
t.
124
ACMOTOR DRlVES
Acmotor drives are the workhorse of industry. These threephase motors are controlled in speed and position by applying adjustablemagnitude, and adjustablefrequency ac voltages by three bidirectional switching powerpoles, as shown in Fig. 12 I Oa. Their average representation is shown in Fig. 12 I Ob.
A '=II~""{ +
.
acmotor
N
(b)
Figure 12 I O Converter for threephase motor drive and UPS.
ln order to synthesize balanced threephase voltages, the PWM signals for the switches in Fig. 12IOa are obtained by comparing three control signals, which establish the amplitude and the frequency of the output voltages, with a triangular waveform v"" as shown in Fig. 1211
Vc sin cqt vc.B (t) = Vc sin(úV 120°) vcc (t) = Vc sin(av  240°) VC.A (t) =
(1210)
The resulting voltage waveforms are shown in Fig. 121 I. Using Eq. 1210 into equations similar to Eq. 12 I for ali three phases produces the following dutyratios in the average representation shown in Fig. 1210b, where the control voltages and the dutyratios are plotted in Fig. 1212:
d B(t) = 0.5 +
~c
sin(lV,t  120°)
~c
sin(cqt  240°)
V,ri d c (t) = 0.5 +
128
V;rt
(121 I)
38bl_ D DDDDD DDn~ ~ ~ ~~n ~ v". i~ I,    ~,,   ~''   ~~ ' ~ D~ D   ~' '  ~ 0,,0 'O 'D"' L D'0,,0
'''o
Figure 121 1 Switching waveforms in threephase converter. Applying the dutyratios in Eq. 1211 in Fig. 12IOb results in the following average output voltages:
vAN (t)=0 .5Vd
+0.5
~d
'v"
~d
V;,sin(w,t1200)
'v'
k,...
VBN (t)=0.5Vd +0.5
Vc sin Ăş,/
~,.;
v c ,A
V,ri '
'v'
.
'
(1212)
Vc .B
k,...
where the pole gain kpo'c is the sarne as in Eq. 122, and it amplifies the control voltage for each switching powerpole. These voltages are plotted in Fig. 1212. The average output voltages of the switching powerpoles have the sarne offset voltage, equal to 0.5 Vd
â€˘
These equal offset voltages in each phase are considered zerosequence voltages,
which cannot cause any current to flow when applied to a balanced threephase motor load shown in Fig. 12IOa or b.
129
0.5
OL__ t
Vd
2 t
OL_ Figure 1212 Dutyratios and the average output voltages ofthe powerpole. Henee the de eomponent in eaeh phase voltage (terminal to neutral n) of the motor load equals zero. As a eonsequenee in Fig. 1210b, the average phaseneutral voltages aeross the motor phases, with respeet to the motor neutral n, are as intended
(1213)
where the amplitude of these phase voltages are (1214)
1210
We should note that at high switching rrequency
Is
relative to the fundamenta l
rrequency j, being synthesized, average voltages in Eq. 1213 equal the fundamentalfrequency components (v Anl' VBnl ' VCnl ) in the voltages applied to the motor phases (at the terminais with respect to the motorneutral n). To obtain the average currents drawn rrom the voltageport of each switching powerpole, we will assume the average currents drawn by the motor load in Fig. 1210b to be sinusoidal but lagging with respect to the average voltages in each phase by an angle ~(t) = j sin(w,t 
tA)
T,;(t)=jsin(w,ttA 120°) I.;(t) = j sin(w,t 
tA ,
tA 
(1215)
240°)
Therefore in Fig. 1210b, the average currents drawn from the voltageport are
(I216)
The total average current drawn by the converter from the dcside voltage source is the sum ofthe three currents above: (121 7) ln Eq . 1215, applying Kirchoff's law to average currents at the motorneutral, the sum of
three average currents equals zero, (1218) Therefore, substituting Eqs. 1216 and making use ofEq. 1218 in Eq. 1217 I:t(t) =
~5 [V<.A(t)~(t) + Vc.B(t)T,;(t) + Vc.C(t)1.; (t) ]
(I2J9)
'n Substituting for the control voltages from Eq. 1210 and for the output currents from Eq. 1215 into Eq. 1219
_id(t) = 0.5,!iV I,[Sin(w,t)Sin(liVtA)+sin(llJlt 1200)Sin(llJlttA 1200)] V;ri
(1220)
+sin(w,t2400)sin(w,ttA 240° )
1211
that simplifies to a dc current

3
V _
iA/) = I d = "EI costA
4 V,ri
(1221)
Substituting Eq. 1214 into Eq. 122 1 (1222) which shows that the average power input at the voltageport equals the total threephase power out ofthe currentport in this converter that is assumed lossIess. ln Fig. 12IOa, the instantaneous quantities associated with this switchmode converter are pulsating, as shown in Fig. 1211. Based on the discussion of harmonic components in the voltage at the currentport and the current at the voltageport of each powerpole, we can calculate cancellation of some harmonics in the combination of the three powerpoles in this acdrive converter. A detailed discussion of this can be found in reference [ 1]. 1241 Space Vector Pulse Width Modulatioo The above method of synthesizing the threephase sinusoidal output is called the sinusoidalPWM. As discussed in the appendix to this chapter on the accompanying CD, the sinusoidalPWM is near ideal, except it does not utilize the available dclink voltage to its fullest. A modified PWM technique called the space vector PWM (SVPWM) is described in the Appendix on the accompanying CD, which can produce from the sarne dclink voltage a threephase output voltage that is higher by approximately fifteen percent. 125
VOLTAGELINK STRUCTURE WITH BIDIRECTIONAL POWER FLOW
ln many applications of acmotor dr ives, the voltagelink structure in Fig. 121 is such that the power tlow through it is bidirectiooal. Normally, power tlows from the utility to the motor, and while slowing down, the energy stored in the inertia of machineIoad combination is recovered by operating the machine as a generator and feeding power back into the utility grid. This can be accomplished by using threephase converters discussed in section 124 at both ends, as shown in Fig. 1213a, recognizing that the power tlow through these converters is bidirectional. ln the normal mode, converter at the utilityend operates as a rectifier and the converter at the machineend as an inverter. The roles of these two are opposite when the power tlows in the reverse direction during energy recovery.
1212
The average representation of these converters by means of ideal transformers is shown in Fig. 1213b. ln this simplified representation, where los ses are ignored, the utility source is represented by a source L, .
v ,a '
etc. in series with the internal system inductance
The machine is represented by its steady state equivalent circuit, the machine
equivalent inductance L â€˘â€˘ in series with the backemf e A
,
etc . Under balanced three
phase operation at both ends, the role of each converter can be analyzed by means of the perphase equivalent circuits shown in Fig. 1213c. ln these perphase equivalent circuits, the fundamentalfrequen cy voltages produced at the acside by the two converters are
v aI
and
v An l ,
and the acside currents at the two sides can be expressed in
the phasor form as (1223)
(1224) where, the phasors in Eq. 1223 represent voltages and current at the utilitysystem frequency
IV,
typically 60 (or 50 Hz), and the phasors in Eq. 1224 at the fundamental
frequency cq synthesized by the m achineside converter.
Tal
(c)
Figure 1213 Voltagel ink structure for bidirectional power flow .
1213
ln Fig. 1213a, for a given utility voltage, it is possible to control the current drawn from
the utility by controlling the voltage synthesized by the utilityside converter in magnitude and phase. ln these circuits, if the losses are ignored, the average power drawn from the utility source equals the power supplied to thc motor. However, the reactive power at the utilitys ide converter can be controlled, independent1y of the reactive power at the machineside converter that depends on the motor load. 126
UNINTERRUPTffiLE POWER SUPPLlES (UPS)
ln considering the synthesis of a lowfrequency ac from a dc voltage, the uninterruptible power supplies (UPS) can be considered as a special case of acmotor drives. UPS are used to provi de power to criticaI loads in industry, business and medical facilities to which power should be available continuously, even during momentary utility power outages. The Computer Business Equipment Manufactures Association (CBEMA) has specifications that show the voltage tolerance envelope as a function of time. This CBEMA curve shows that the UPS for criticaI loads are needed, not just for complete power outages but also for "swells" and " sags" in the equipment voltage due to disturbances on the utility grid. ln a UPS, a voltagelink structure shown in Fig. 121 is used where the dclink consists of batteries to shield criticaI loads from voltage disturbances, as shown in the block diagram of Fig. 1214. Normally, the power to the load is provided through the two converters, where the utilityside converter also keeps the batteries charged. ln the event of powerline outages, energy stored in the batteries allows load power to be supplied continuously. The function ofthe loadside inverter is to produce, from dc source, ac voltages similar to that in acmotor drives, except it is a special case where the output voltage has constant specified magnitude and frequency (e.g., 115 V rms, and 60 Hz). The load may be threephase, where the analysis of section 125 applies. ln the following section, the case of a singlephase load is considered, since the threephase case is similar to that in motor drives.
Energy Storage
Figure 12 J 4 Bock diagram of UPS. 1261 SinglePhase UPS The loadside converter of 1phase UPS is similar in power topology to that in dcmotor drives, as shown in Fig. J 2 J 5a. The average representation is shown in Fig. 1215b. It
1214
consists of two switching powerpoles, where as shown, the inductance of the lowpass filer establishes the currentports of the two power poles. The control voltage is at the desired output frequency to produce the desired output voltage amplitude:
voU) = V, sin av + V,O "
.J
"
~   '
y
.. :v
,
b '
+
Gj~ f_ .!: '

(1225)
,
o
,
T
Y
,
Critical Load
+
v,
Âˇ0
Criticai Load
I
q, q.
(a)
(b)
Figure 1215 Singlephase UPS. This control signal equals
V ,.A.
For the switching powerpole B, a phase shift of 1800
IS
introduced in Eq. 1225, that is by a factor of (I) , to produce its control signal (1226) As shown in Fig. 1216a, these control signals are compared with a triangular waveform at the switching frequency, typically above 20 kHz to produce the switching functions for the transistors in the switching powerpoles. The switching waveforms are shown in Fig. 1216a. v,
v",..
o. o
(t)
/
\
o r+~~
\
/
\
v. (t)
/ ''
(b)
Figure 1216 UPS waveforms. Using the expression derived for the dutyratio in phase A in Eq. 121 , and substituting the control voltage for the other pol e: 1215
dA(I) = 0.5 + 0.5
d.(t) = 0.5  0.5
~c sin li'"
V,ri
~c
(1227) sin w,t
V,ri With these dutyratios, the average outputs of the two powerpoles and the average output voltage vo(t) are as follows, plotted in Fig. 1216b v AN (t) = 0.5Vd
+0.5~
Vd sinw,t
VBN (I) = 0.5Vd

~c
Vd sin w,t
V,ri
0.5
V,ri
(1228)
where, (1229) ln order to calculate the average current drawn from the dc source, we will assume that the average acside current is sinusoidal and lagging behind the output ac voltage by an angle
tA, as sbown
in Fig. 1217: (1230)
.0
Figure 121 7 Output voltage and current. Therefore, equating average input power to the output power, the average input current is
(1231)
1216
which shows that the average current drawn from the dc source has a dc component Id that is responsible for the average power transfer to the ac side of the converter, and a second harmonic component id 2 (at twice the frequency of the ac output), which is undesirable. The dclink storage in a 1phase inverter must be sized to accommodate the f10w of this large ac current at twice the output frequency, similar to that in PFCs discussed in Chapter 6. Of course, we should not forget that the above discussion is in terms of average representation of the switching powerpoles. Therefore, the dclink storage must also accommodate the flow of switchingfrequency ripple in id discussed below. The switching waveforms in the singlephase UPS are shown in Fig. 1216a, which confirm that the voltage vo(t) and id(t) are pulsating at the switching frequency. A lowpass filter is necessary to remove the output voltage harmonic frequencies, which were discussed earlier in a generic manner for each switching powerpole. The pulsating current ripple in iAt) can be bypassed from being supplied by the batteries by placing a small high quality capacitor with a very low equivalent series inductance.
REFERENCES I.
N. Mohan, T . M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and Design, 3 m Edition, Wiley & Sons, New York, 2003.
PROBLEMS Switching PowerPole 121
ln a switchmode converter poleA, Vd = 150V,
V;rl
= 5V, and
J,
= 20kHz.
Calculate the values of the control signal vc â€˘ A and the pole dutyratio dA during which the switch is in its top position, for the following values of the average output voltage: 122
V AN
= 125V and
V AN
= 50V .
ln a converter pole, the iA(t) waveform is as shown in Fig. 125b. lncluding the ripple, show that the current relationship of an ideal transformer is valid.
DCMotor Drives 123
A switchmode dcdc converter uses a PWMcontroller IC which has a triangular waveform signal at 25 kHz with Vd
=
V;,j
=
3 V.
If the input dc source voltage
150V , calculate the gain k pWM in Eq. 126 ofthis switchmode amplifier.
1217
124
ln a switchmode dcdc converter,
is
= 20kHz and Vd = IS0 V.
V o (I)
12S
VC
I V",
= 0.8
with a switching frequency
Calculate and plot the ripple in the output voltage
.
A switchmode dcdc converter is operating at a switching frequency of 20 kHz, and Vd
=
ISO V. The average current being drawn by the dc motor is 8.0 A. ln
the equivalent circuit of the dc motor, Eo = I OOV , Ro = 0.2S0 ,and Lo = 4mH .
126
(a) Plot the output current and calculate the peaktopeak ripple, and (b) plot the dcside current. ln Problem 12S, the motor goes into regenerative braking mode. The average current being supplied by the motor to the converter during braking is 8.0 A. Plot the voltage and current waveforms on both sides of this converter at that this instant. Calculate the average power f10w into the converter.
i,;,.,
],;0' and
],;(= l d) Âˇ
127
ln Problem 12S, calculate
128
Repeat Problem 12S if the motor is rotating in the reverse direction, with the same current draw and the sarne induced emf Eu value of the opposite polarity.
129
Repeat Problem 128 if the motor is braking while it has been rotating in the reverse direction. It supplies the same current and produces the sarne induced emf Eo value ofthe opposite polarity.
1210
Repeat problem 12S if a bipolar voltage switching is used in the dcdc converter. ln such a switching scheme, the two bipositional switches are operated in such a manner that when switchA is in the top position, switchB is in its bottom position, and vice versa. The switching signal for poleA is derived by comparing the control voltage (as in Problem 12S) with the triangular waveform.
ThreePhase ACMotor Drives 1211
Plot
d A(t)
if
the
output
voltage
of
the
converter
poleA
is
VAN(t) = Vd +0.8S Vd sin(/XI,t),where a>, =2trx60rad ls .
2
12 I 2
2
ln a threephase dcac
.f,
= 4S Hz.
inverter,
Vd = 300 V,
v,,, = I V, V = 0.7S V, C
and
Calculate and plot dA(I), dB(t), d c (I) , VA N(t), VBN(t), VCN(I) , and
vAn(t), VBn(t) , and VCn(t)Âˇ 12 I 3
ln a balanced threephase dcac inverter, the phaseA average output voltage is
VAn(t) = ;
0.7Ssin(/XI,t) , where
Vd =3 00V
and
/XI,
= 2tr x4S rad ls.
The
inductance L in each phase is S mH. The acmotor internal voltage in phase A 1218
ean be represented as eA(t) = 106.14sin(w,t6.6Â° )V. (a) Caleulate and plat d A(t), dB(t) , and d e (t), and (b) sketeh 7;.(t) and t.;,,(t).
1214
ln Problem 1213, ealeulate and plat 7;(t), whieh is the average de eurrent drawn
fram the deside. SinglePhase UPS
1215
ln
a
lphase
UPS,
!.(t) = 10sin(2Jr x 60t30") A. VBN(t), I d
1216
,
Vd = 300V,
vo(t) = 170sin(2Jr x 60t) V ,
Caleulate and plat d A(t) , d B(t),
and
VAN (t),
id2 (t), and i,;(t ).
ln Prablem 1215, ealeulate q A(t) and q B(t) at a.>t = 90Â°.
1219
Chapter 13
DESIGNING FEEDBACK CONTROLLERS FOR MOTOR DRIVES 131
INTRODUCTION
Many applications, such as robotics and factory automation, require precise control of speed and position. ln such applications, a feedback control, as illustrated by Fig. 131, is used. This feedback control system consists of a powerprocessing unit (PPU), a motor, and a mechanical load. The output variables such as torque and speed are sensed and are fed back, to be compared with the desired (reference) values. The error between the reference and the actual values are amplified to control the powerprocess ing unit to minimize or eliminate this error. A properly designed feedback controller makes the system insensitive to disturbances and changes in the system parameters. error des ired
(reference) sigo ai
'0
error amplifier
p p U
I I I I
Electri c Machine
I I I I
Mec h Load
output
~
ElectricaJ Systern
Mechanical System
measured o utout s ignal
Fig 13 1 Feedback controlled drive. The objective of this chapter is to discuss the design of motordrive controllers. A dcmotor drive is used as an example, although the same design concepts can be applied in controlling brushlessdc motor drives and vectorcontrolled inductionmotor drives. ln the following discussion, it is assumed that the powerprocessing unit is of a switchmode type and has a very fast response time. A perrnanentmagnet dc machine with a constant field flux r/lf is assumed. 132
CONTROL OBJECTIVES
The control system in Fig. 131 is shown simplified in Fig. 132, where Gp(s) is the Laplacedomain transfer function of the plant consisting of the powerprocessing unit, the motor, and the mechanical load. G Js ) is the controller transfer function. ln response to a desired (reference) input X'(s) , the output of the system is X(s) , which (ideally) equals the reference input.
The controller Gc(s) is designed with the following
objectives in mind:
131
•
a zero steady state error.
•
a good dynamic response (which implies both a fast transient response, for example to a stepchange in the input, and a small settling time with very little overshoot). Controller X'(s)
, ""I
Ge(s)
Plant
,I
Gp (s)
I
,
X(s)
I
Fig 132 Simplified control system representation. To keep the discussion simple, a unity feedback will be assumed. The openIoop transfer function (incIuding the forward path and the unity feedback path) G OL (s) is (131 ) The cIosedloop transfer function
X( s) in a unity feedback system is X'(s)
(132) ln order to define a few necessary control terms, we will consider a generic Bode plot of the openIoop transfer function G OL (s ) in terms of its magnitude and phase angle, shown in Fig. 133a as a function of frequency. The frequency at which the gain equals unity (that is
IG
OL (s)1
= Odb) is defined as the crossover frequency
fc
(angular frequency me)·
At the crossover frequency, the phase delay introduced by the openIoop transfer function must be less than 180° in order for the closedIoop feedback system to be stable. Therefore, at fe' the phase angle ~oL IJ; ofthe openIoop transfer function, measured with respect to 180° , is delmed as the Phase Margin (PM): (133) Note that ~oLlf. has a negative value.
For a satisfactory dynamic response without
oscillations, the phase margin should be greater than 45°, preferably cIose to 60° . The magnitude of the closedIoop transfer function is plotted in Fig. 133b (idealized by the asymptotes), in which the bandwidth is defined as the frequency at which the gain drops to ( 3 dB). As a Irrstorder approximation in many practical systems, ClosedIoop bandwidth '"
132
fc
(134)
100
o 路50 路50
3dB
OdB1:====~==:s~3.il~'@ I
路 100
BW
; 0 1
路 150
, ""''''
""" !
(b)
"""
200 '.'2 ''''.......ll....,, ''''''=;; O'........"''""t, ''...............'J 2
10
10
10
10
(a)
10
..
Figure 133 (a) Phase Margin; (b) bandwidth. For a fast transient response by the control system, for example a response to a stepchange in the input, the bandwidth of the closedIoop should be high. From Eq. 134, this requirement implies that the crossover frequency f c (of the openIoop transfer function shown in Fig. 133a) should be designed to be high.
133
CASCADE CONTROL STRUCTURE
ln the following discussion, a cascade control structure such as that shown in Fig. 134 is used. The cascade control structure is commonly used for motor drives because of its f1exibility. It consists of distinct control loops; the innennost current (torque) loop is followed by the speed loop. If position needs to be controlled accurately, the outennost position loop is superimposed on the speed loop. Cascade control requires that the bandwidth (speed of response) increase towards the inner loop, with the torque loop being the fastest and the position loop being the slowest. The cascade control structure is widely used in industry. torque
pos; l ; o n 0 Pos ition + I: controller
.

~~ +
Speed
controller
~~ +


Torque controller

Electrical
System
speed
hiJ
E~
H
Mech System
POS ;I; on
torque(current)
speed position
Figure 134 Cascade control of a motor drive.
133
134
STEPS lN DESIGNING THE FEEDBACK CONTROLLER
Motion control systems ofien must respond to large changes in the desired (reference) values of the torque, speed, and position. They must reject large, unexpected load disturbances. For large changes, the overall system is ofien nonlinear. This nonlinearity comes about because the mechanical load is ofien highly nonlinear. Additional nonlinearity is introduced by voltage and current limits imposed by the powerprocessing unit and the motor. ln view of the above, the following steps for designing the contro!ler are suggested: I.
2.
135
The first step is to assume that, around the steadystate operating point, the input reFerence changes and the load disturbances are ali small. ln such a smallsignal analysis, the overall system can be assumed to be linear around the steadystate operating poin!, thus allowing the basic concepts of linear control theory to be app lied. Based on the linear control theory, once the controller has been designed, the entire system can be simulated on a computer under largesignal conditions to evaluate the adequacy of the controller. The controller must be "adjusted" as appropriate.
SYSTEM REPRESENTATlON FOR SMALLSIGNAL ANALYSIS
For ease of the analysis described below, lhe system in Fig. 134 is assumed linear and the steadystate operating point is assumed to be zero for ali ofthe system variables. This linear analysis can be then extended to nonlinear systems and to steadystate operating conditions other than zero. The control system in Fig. 134 is designed with the highest bandwidth (associated with the torque loop), which is one or two orders of magnitude smaller than the switching frequency .f,. As a result, in designing the controller, the switchingfrcqucncy components in variou s quantities are of no consequence. Therefore, we will use the average variables discussed in Chapter 3, where the switchingfrequency components were eliminated.
1351 The Average Representation of the PowerProcessing Unit (PPU) For the purposes of designing the feedback controller, we will assume that the debus voltage Vdc within the PPU shown in Fig. 135a is constant. Following the averaging analysis in Chapter 3, the average representation of the switchmode converter is shown in Fig. 135b. ln terms ofthe dcbus voltage V'c and the triangularfrequency waveforrn peak
v" p the average output voltage vaU)
oFthe converter is linearly proportional to the
control voltage: ) (k PIPA., _ V'c .
v,,,
134
( 135)
where
k l'WM
is the gain constant of the PWM converter. Therefore, in Laplace domain,
the PWM controller and the dcdc switchmode converter can be represented s imply by a gainconstant k pWM , as shown in F ig. 135c: (1 36) where Va( s) is the Laplace transfo rm of Va(t) , and v,,(s) is the Laplace transform of vc(t)Âˇ The above representation is valid in the linear range, where
+
v,,, ,;:;
Vc
,;:;
v,,, .
+ A
B
Va
= iB
(a) ;;; (1) +
â€˘ â€˘ 1~~ d( l )
,,r  ,,, ,,, + ,: ,, ,,, ,, ,,, ,
Va
(11
ed
, ,,
,, ,
,, ,, ,,
Vc (S)
Va(S)
~           j
(c) Figure 135 (a) Switchmode conve rter fo r dc motor drives; (b) average model of the switchmode converter; (c) linearized representation.
135
1352 Modeling of the De Machine and the Mechanical Load The dc motor and the mechanical load are modeled as shown by the equivalent circuit in Fig. 136a, in which the speed (J)m(t) and the backemf ea(t) are assumed not to contain switchingfrequency components. corresponding to Fig. 136a are
The electrical
and the mechanical equations
(137) and (138) where the equivalent load inertia J ,q (= J M + J L ) is the sum of the motor inertia and the load inertia, and the damping is neglected (it could be combined with the load torque TL
).
ln the simplified procedure presented here, the controller is designed to follow the changes in the torque, speed, and position reference values (and hence the load torque in Eq. 138 is assumed to be absent). Eqs. 137 and 138 can be expressed in the Laplace domain as (139) or (1310) We can define the Electrical Time Constant T
,
T,
as
La
=
R
(1311)
a
Therefore, Eq. 1310 can be written in terms of <, as (1312)
From Eq. 138, assuming the load torque to be absent in the design procedure, (1313) Eqs. 1310 and 1313 can be combined and represented in blockdiagram form, as shown in Fig. 136b. 136
7 â€˘
=
1'em kr
TL(s)
+ R.
1.(5~
VQ(s)
+ v.
Ior
~
(UM
s./,
6.(5) eQ _
k~
kE
IV..
(b)
(a)
Fig 136 De motor and mechanicalload (a) equivalent circuit; (b) block diagram. 136
CONTROLLER DESIGN
The controller in the cascade control structure shown in Fig. 134 is designed with the objectives discussed in section 132 in mind. ln the following section, a simplified design procedure is described. 1361 PI ControIlers Motion control systems often utilize a proportionalintegral (PI) controller, as shown in Fig. 137.
The input to the contro ller is the error E(s) = XÂˇ (s)  X(s), which is the
difference between the reference input and the measured output. ln Fig. 137, the proportional controller produces an output proportional to the error input: (1314) where kp is the proportionalcontro ller gain.
ln torque and speed loops, proportional
controllers, if used alone, result in a steadystate error in response to stepchange in the input reference. Therefore, they are used in combination with the integral controller described below. ln the integral controller shown in F ig. 137, the o utput is proportional to the integral of the error E(s), expressed in the Laplace domain as (1315) where k, is the integralcontroller ga in. Such a controller responds slowly because its action is proportional to the time integral of the error. The steadystate error goes to zero
137
5
for a stepchange in input because the integrator action continues for as long as the error is not zero.
Vc, p(s)
kp G p (.1")
x '(s) + '
!L E(s)
s
V<.I (s~~
X(s)
Vc(s)
Figure 137 PI controller. ln motioncontrol systems, the P controllers in the position loop and the PI controllers in the speed and torque loop are ofien adequate. Therefore, we will not consider differential (O) controllers. As shown in Fig. 137, VÂŤ s) = V<. p(s) + v".i (s). Therefore, using Eqs. 1314 and 1315, the transfer functi on ofa PI controller is (1316) 137
EXAMPLE OF A CONTROLLER DESIGN
ln the following discussion, we will consider the example of a permanentmagnet dcmotor supplied by a switchmode PWM dcdc converter. The system parameters are given as follows in Table 131 : Table 131 DCMotor Drive System System Parameter Value 2.0n Ra
138
La
5.2mH
J ,q
152 X 10 6 kgÂˇ m '
B
KE
O O.IV /( rad /s)
kT
O.I N m / A
Vd
60V
V,ri
5V
/,
33kHz
We will design the torque, speed, and position feedback controllers (assuming a unity feedback) based on the smallsignal analysis, in which the load nonlinearity and the effects ofthe limiters can be ignored. 1371 The Design ofthe Torque (Current) Control Loop As mentioned earlier, we will begin with the innermost loop in Fig. I38a (utilizing the transfer function block diagram of Fig. 136b to represent the motorIoad combination, Fig. 135c to represent the PPU, and Fig. 137 to represent the PI controller).
PI
(a) v, (S) 1;(S~ + '~
PI
f
k pWM

~ 
Ea(s)
k EkT
sJ,q
la(s)
10 (s)
11 R" l +sTe
f..L
kT
I
sJ,q
~
I
(b) PI J;(s)
+ :::~
kil (1+killk J
V,(s )
S
S
k
II R"
Va(s)
pWM
la(s)
I+STe
pJ
l a(s)
(c) Fig. 138 Design ofthe torque controlloop. ln permanentmagnet dc motors in which
,pj
is constant, the current and the torque are
proportional to each other, related by the torque constant k T
â€˘
Therefore, we will consider
the current to be the control variable because it is more convenient to use. Notice that there is a feedback in the current loop from the output speed. This feedback dictates the induced backemf. Neglecting 7;., and considering the current to be the output, can be calculated in terms of
Ia(s)
in Fig. 138a as
Ea(s) =
;;E
Ia(s).
E a (s)
Therefore, Fig.
'q
138a can be redrawn as shown in Fig. 138b. Notice that the feedback term depends 139
inversely on the inertia J ,q'
Assuming that the inertia is sufficient1y large to justifY
neglecting the feedback effect, we can simplifY the block diagram, as shown in Fig. 138c. The currentcontroller in Fig. 138c is a proport.i onalintegral (PI) error amplifier with the proportional gain k p , and the integral gain ki/' lts transfer function is given by Eq. 1316.
The subscript "I" refers to the current loop.
The openIoop transfer function
G, OL(S) ofthe simplified current loop in Fig. 138c is G' OL (S)=k"[I+ S ] k ~ S ki// k p , . PPU
PIco~lrolru
1/ R. s
(1317)
1+  I/T, 'v' mo/Or
To select the gain constants of the PI controller m the current loop, a simple design procedure, which results in a phase margin of 90 degrees, is suggested as follows: â€˘
Select the zero (kil / k p , ) ofthe PI controller to cancel the motor pole at (I Ir, ) due to the electrical timeconstant
T,
of the motor. Under these conditions, (1318)
or
Cancellation of the pole m the motor transfer function renders the openIoop transfer function to be =k,.OL G IOL () S .
(1319a)
S
where (1319b) â€˘
ln the openIoop transfer function of Eq. 1319a, the crossover frequency m" = k , OL . We will select the crossover frequency
h ,(= m"
/ 27t) ofthe current openIoop to be
approximately one to !WO orders of magnitude smaller than the switching frequency of the powerprocessing unit in order to avoid interference in the control loop from the switchingfrequency noise. T herefore, at the selected crossover frequency, from Eq.1319b, k  me/ R.
i/ k
PWM
1310
(1320)
This completes the design of the torque (current) loop, as illustrated by the example below, where the gain constants k p, and k" can be calculated from Eqs. 1318 and 1320. Design the current loop for the example system of Table 131, Example 131 assuming that the crossover frequency is selected to be 1 kHz. ~
From Eq. 1320, for
Solution
OJ"
= 2" x 10
3
rad / s ,
= 1050.0
and, from Eq. 1318,
M,,~~~~~~~~~~~~ I 1 1 11 11 11 1 11 1 11111 I 1 11 1 111
~40
§2O '2
f
o
1 11111 111
1 1 1111111
.., TI I
nur
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III
 111111111 11 flfill! '  I T in i'
1 1 1 1 11111
1 1 11 1 111
' ,"'T 'l lIii  1l llTl _'_ I .!I~I I I
20
I I I 11 111
o <õ
~ 10
~
~ ·20
:>
4O ~LU~~~Wlli~~~~Lll~~UW~
10f)
102
10'
103
~ r__________~ F~ ~ ~~Y~~ ~)~________~
f ;
I 11 111111
89.5
90
~ 90. 5 a.
I
1111 111 1
111 1 1111
I 1 1 11 111
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"""i
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111 1 1 1111
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I ' 1 11 11' 1
I 1 11 1 1111
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i i illi ,ii
i i
i'
i i
r " ,TI
I
I I 1 1 1111
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ri HItT
I I 111111
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I 1 I 111 11 1
 I"'"
"i ' H II1"
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 I I "i lti ltt 
I I 111111
O
I
r
I 1 1 111
1 11 !TI
f•
50
Ii
10'
10'
, , ,
·100 10'
,, ,, ,,
,
,,
1 1 111111 1 1 111111 1 1 111111
I 111 1111 11 111111 I 11 11 111
10'
' " '"
10' 10' Frequency (Hz)
10'
,
~
I I I 11111
,, ,, ,,
, , , ,
1 1 111111 I I II . I I 11 111 1 1 1 11 111 11 111111 1 1 11 1111 11 111111 1 11 1111 11 111 111 1 1 111111 I..J. 101 U.III _I l .11 UIIL _ 11 .J.IU III. _ I ..J.1.1J1ll_1 1l1.J.1 1 1 11 1111 1 11 1111 11111 1 1 1 11111 1 11 11111 1 11 11111 1 11 11111 1 11 1111 1 1 111 111 1.1 LJ U.III _I...J .11 U IIL _ 11 .J. I U III. _ 1_ 1..I.1.1J _Li...JUI 11 1111 11 11 111111 1 11 1111 1 1 11 11 11 1 11 11111 11 111111 1 11 1111 1 1 111 111 1 1 111111 11 1111i1
30 10'
10~
10·
,,
I I 11111 I I 111 111 1 11 11111
,, ,
I I 111 11 1 I 11 11 11 1 I 111111 1
10'
,
10'
I I 11 1111 I I 11 1111 I I ' 11 111 '1 11111
, 11 1 '111 , 11 11111
10' 10' Frequencv (HZ)
10'
10'
(b) (a) Figure 139 Frequency response ofthe current loop (a) open loop; (b) closed loop. The openIoop transfer function is plotted in Fig. 139a, which shows that the crossover frequency is 1 kHz, as assumed previously. The closedIoop transfer function is plotted ~ in Fig. 139b. 1372 The Design ofthe Speed Loop
We will select the bandwidth ofthe speed loop to be one order ofmagnitude smaller than that of the current (torque) loop. Therefore, the closedcurrent loop can be assumed ideal for design purposes and represented by unity, as shown in Fig. 1310. The speed controller is of the proportionalintegral (PI) type. The resulting openIoop transfer function Gn.oL(s) of the speed loop in the block diagram of Fig. 1310 is as follows, where the subscript "Q" refers to the speed loop:
1311
, S
,
PI
l:.L
!
GO.OL (S) = k,o [1 + s I(k,o I kpn )]
(1321)
sJ'"I
currenf loop

co~u'()lJel'
lorque +mu1ia
Eq. 1321 can be rearranged as G O ,OL
(s) = (k,okT ) 1+ s l (kiO I kpn ) J 2
'(s)
O
+ "
(1322)
S
.q
PI
I';(S
,
I'·(S)
kT
i

,
ilIm{S)
sJ. q
com(s)
Fig 1310 Block diagram ofthe speed loop. This shows that the openIoop transfer function consists of a double pole at the origino At low frequencies in the Bode plot, this double pole at the origin causes the magnitude to decline at the rate of  40 db per decade while the phase angle is at 180° . We can select the crossover frequency w'" to be one order of magnitude smaller than that of the current loop.
Similarly, we can choose a reasonable value of the phase margin I/Jpm.o '
Therefore, Eq. 1322 yields two equations at the crossover frequency: (1323)
=1 and = 180°
+ I/Jpm.o
(1324)
The two gain constants of the PI controller can be calculated by solving these two equations, as illustrated by the following example. '" Example 132 Design the speed loop controller, assuming the speed loop crossover frequency to be one order of magnitude smaller than that of the current loop in Example 132; that is, f cn = 100Hz, and thus w'" = 628rad I s. The phase margin is selected to be 60° .
Solution and I/JPM.o
1312
ln Eqs. 1323 and 1324, substituting k T = 0.1 Nm I A , J ,q = 152 x 106 kg· m 2
= 60·
at the crossover frequency, where s
,
= )w", = )628 , we can calculate that
kpn =0.827 and k,n =299.7. '" r~,~,~,,~,,",,',~,~,,~,,",,~,~,~"~,~,,.'~,~,~"~"",,,',~,~,~,,~,,',,:,~,~"~,,::l,,
150,;,',:c",::::",,';,'"C:"::::,,,,';,:C"C:";;',,,,',CC"",,"::::,,,',:C,,"'"::::,,,:, , :C"",,,eJ"
~ 100 : ~I!.!':* :~ H:~:: ~:~ :*::~ ~ i:H:m ::H:*:: TH:~:: '§ I t 11111
:a ~
S'J
111111111
o
1 11 111111
HHII+!  11 H~II 
1 111 11 11
+ HI~I
" "",~'~"~'~"'~";:'~'i'~"~'''~'~'~'~"~"~''':;J':'~'::::;~'±''~''~'''~'~''~'~''']' 5 10.'
,0'
10°
lcJ
,rl
104
10
~ o
I t IIIIU IIIIIU
{l20
'"11111
l..,
1.,00 j ·1~ o.
I 11111111 1 1 1111111
I IIIII!II IIIIIIUI
I 11111111 , t 111111 111111111 11111111 _' _LUWI 1 11111 11 I IIIIUII I 111 1111
t 11111111 IIIIIIIU
' I 1111 11
I 111111
.ooL:::~::::::~"~~~~~"~'~"'~"~'~~~~'~"~"~" ~
~
~
~
~
~
I I I I I'IA 11111111
11 1111111
'
1 1111111
1 1 1111 11 11i'i1l1
1
1 11111111
11111111
tllllii,íllli'illl I "11 11
,. r i ",n  ,, ntnl, r n mi" ..., T rrl 11I1I1i1
""11111
11111 11.
 rITITl1!r , TIni ,
111 111111
I
'1111"
+ HHIH  I .... H ,''',, +HI+I I .... III+IHIIiI+I
'"11111
11111111 I +1 ... ,
11111111
1111111
"1111'"
"'"111
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2i
~ e ~
I I 1111111
1 1111 111 I IIII II! "111 11
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"
BD
Tnnln " 11111 111 I I I " ' II "IIIUI
......
11111111 '11111" 11111101
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nI1Tl1rnm
11 1111111 I I 1111111 1111 11111
~
1 1 11111.111111111 111111111 11111111 1111' 11 11 11111111
,TrTUmrITlnllri'Tlnl
1111111 I I 11111' 1111 1 111
111 111111 I 1111111 I
11111111111111111 " I 111111 I I 111111 • • 11111111
10'
10' F~(Hz1
FA3QJI3IlCY (Hz)
(a)
(b)
Figure 1311 Speed loop response (a) open loop; (b) closed loop. The open and the closedIoop transfer functions are plotted in Figs. 13 I I a and 13 I I b.
À
1373 The Design of the Position Control Loop We will select the bandwidth of the position loop to be one order of magnitude smaller than that of the speed loop. Therefore, the speed loop can be ideal ized and represented by unity, as shown in Fig. 1312. For the position controller, it is adequate to have only a proportional gain kpe because ofthe presence ofa true integrator
(J/s) in Fig. 1312 in
the openIoop transfer function. This integrator will reduce the steady state error to zero for a stepchange in the reference position. With this choice of the controller, and with the closedIoop response of the speed loop assumed to be ideal, the openIoop transfer function G. OL (s) is (1325)
O:(S)
o
+\.. 
aJ~(s)
k.
aJ".(s)
J
J 
°m ( s)
S
Fig 1312 Block diagram of position loop. Therefore, selecting the crossover frequency wc. of the openIoop allows k. to be calculated as ( 1326)
1313
.. Example 133 For the example system of Table 131 , design the positionIoop controller, assuming the positionIoop crossover frequency to be one order of magnitude smaller than that of the speed loop in Example 133 (that is,
f di
=
10Hz and
llJdI = 62.8 rad / s).
Solution
From Eq. 1326, k. =llJ,. =62.8rad/s.
50 ,
I I I II HI
I I I 111m
iii
I I 1 111111
"
t
1 1 1 111111
I I I ti· til I 1 1 1 1111 ' I "IIIU,
o r ;,OC":C,,;;,,,C,C 7," ,",f',;;:::;;~::;;cc;;c;,:,:,:7";;;],,,
' I I 11111 1111111 I 111111
'111lil!
I
111111>,
, ,
1 1111111
I 'I
1111
...J J. Ul1l1' I ' II IHI' I 1 11 11'
o I 1 1 111111
,.
I 1 1 1 Hill
I 1 , 11 1111
1111 1111
I 11 111111
1 11 11'11 11 1 ' 1,11
1 ' 1' 1"
,
I 111111
,
11111111
,
'11 11 " 11 1111
.5Q
10.
1
10'
10'
10'
10'
.ao
f ...·5 "Q;'"
90
2a.. 90.5
J I ' 111111
'
I 111U1I
I I' 111111
I "
11111
I 1111 11
... 1. I.!I!JII_ 1. 1.11111)'_ !.. LI UI!!.! _ L 1.l1.!1I!;. _ U l I!! " I • 1111111 1 I I IIIm
I
I ; 1111111 1 11111111
' ii "'"
i I
1 1 111 1111 111111111
, ii lll
'iI'ii
"
I I I IIII U I I I I I IUI I t HlttU t .. I I I t 11111 I I 1 111111
1 II I II UI H HIIIl I I IUlII
,+11'' ' ;
1 1 1111 11 1 1 1 111 11"
1 1111111 I I I IIIU
, , ' 01, "
,,
",n,
I 1111111 1 1 1 1111 1 11 tt l lll  I Itlttl I I 1 111111 I I 111111
~1 ~~wm~~~~~rua~~~~LW~
1~
1~
~
~
~
~
r
!
,
111I I 'II 1111 I 11 11I 1
10' 10' Frequency (tu )
,
,,
11111111 , 11 1'11
1 111 1111
1 11111<
.5Q
r I 1111·'1
.....
11 1 11011
·100
I
10'
10'
,,, ,
I I 11111 I I IIU' 111111,1 I I IUlII
,,
, "1111 I 11 1111 111 11 11 1111
"
I I 11 1111
I 11 1111
1111111
I 1'1 1111
L;.;.'.:.":;."::;"'~:..'.:.';;",:;";;:"',;'~'~'~'"::':;~::;:::;;~'..:'..:';;'';;:J'''
~
FreQuet'lCV (Hz)
~
1~
1~
~
1~
Freouencv (Hz)
(b
(a)
Figure 1313 Position loop response (a) open loop; (b) c10sed loop. The open and the c1osedloop transfer functions are plotted in Figs. 13l3a and 1313b . .. REFERENCE 1.
M. Kazmierkowski and H. Tunia, "Automatic Control of ConverterFed Drives," Elsevier, 1994, 559 pages.
PROBLEMS 131
ln a unity feedback system, the openIoop transfer function is of the form GOL(s ) =
k l+s / llJp
. Calculate the bandwidth ofthe closedIoop transfer function.
How does the bandwidth depend on k and llJp ? 132
ln a feedback system, the forward path has a transfer function of the form G(s) =
k l+s /llJp
, and the feedback path has a gain of kfb which is less than
unity . Calcu late the bandwidth of the closedIoop transfer function. How does the bandwidth depend on k fb ? 133
1314
ln designing the torque loop of Examp le 131 , include the effect ofthe backemf, shown in Fig. 138a. Desig n a PT controller for the sam e openIoop crossover frequency and for a phase margin of 60 degrees. Compare your results with those in Example 131.
134
\35
\36
137 138 139
1310
ln designing the speed loop of Exarnple 132, include the torque loop by a first
order transfer function based on the design ln Exarnple 131. Design a PI controller for the sarne openIoop crossover frequency and the same phase margin as in Exarnple 132 and compare results. ln designing the position loop ofExample 133, include the speed loop by a firstorder transfer function based on the design ln Example 132. Design a Ptype controller for the sarne openIoop crossover frequency as in Example 133 and for a phase margin of 60 degrees. Compare your results with those in Exarnple 133. ln an actual system in which there are Iimits on the voltage and current that can be supplied, why and how does the initial steadystate operating point make a difference for largesignal d isturbances? Obtain the time response of the system designed in Example 132, in terms of the change in speed, for a stepchange ofthe loadtorque disturbance. Obtain the time response of the system designed in Example 133, in terms of the change in position, for a stepchange of the loadtorque disturbance. ln the example system of Table \31, the maximum output voltage of the dede converter is limited to 60 V. Assume that the current is limited to 8 A in magnitude. How do these two limits impact the response ofthe system to a large stepchange in the reference vai ue? ln Exarnple 132, design the speedIoop controller, without the inner current loop, for the same crossover frequency and phase margin as in Exarnple 132. Compare results with the system of Example 132.
1315
Chapter 14
THYRISTOR CONVERTERS 141
INTRODUCTION
Historically, thyristor converters were used to perform tasks that are now performed by switchmode converters discussed in previous chapters. Thyristor converters are now typically used in utility applications at very high power leveIs. ln this chapter, we will examine the operating principIes of thyristorbased converters. 142
Thyristors (SCRs)
Thyristors are a device that can be considered as a controlled diode. Like diodes, they are available in very large voltage and current ratings, making them atlractive for use in applications at very high power leveIs. Thyristors, shown by their symbol in Fig. 14la are sometimes referred to by their trade name of Silicon Controlled Rectifiers (SCRs). These are 4layer (pnpn) devices as shown in Fig. 141 b. When a reverse (v AK < O) voltage is applied, the flow of current is blocked by the junctions pn I and pn3.
When a forward (v AK > O) polarity voltage is
applied and the gate terminal is open, the flow of current is blocked by the junction pn2, and the thyristor is considered to be in a forwardblocking state. ln this forwardblocking state, applying a small positive voltage to the gate with respect to the cathode for a short interval supplies a pulse of gate current iG that latches the thyristor in its on state, and subsequently the gatecurrent pulse can be removed. A
p
A
(a)
G~ K
1l....... p.~..I... (b)
N p
G
....._.p..~~.
1l ........P.V.2. N K
Figure 141 Thyristors. The operation of thyristors is iIlustrated by means of a simple RL circuit in Fig. 142a. At mI = O, the positivehalfcycle of the input voltage begins, beyond which a forward
141
voltage appears across the thyristor (anode A is positive with respect to cathode K), and if the thyristor were a diode, a current would begin to flow in this circuito This instant wl = O in this circuit we will refer as the instant ofnatural conduction. With the thyristor forward blocking, the start of conduction can be controlled (delayed) with respect to wt = O by a delay angle a at which instant the gatecurrent pulse is applied. Once in the conducting state, the thyristor behaves Iike a diode with a very small voltage drop of the order of 1 to 2 volts across it (we wi\l idealize it as zero), and the RL load voltage vd equals v" as shown in Fig. 142b. i,
(a)
(b)
Figure 142 A simple thyristor circuito The current waveforrn in Fig. 142b show that due to the inductor, the current comes to zero sometime in the negative halfcycle of the input voltage. The current through the thyristor cannot reverse and remains zero for the remainder of the input voltage cycle. ln the next voltage cycle, the current conduction again depends on the instant during the positive halfcycle at which the gale pulse is applied. By controlling the delay angle (or the phase control as it is ofien referred), we can control the average voltage vd across lhe
RL load. This principie can be extended to the practical circuits discussed below. 143
SinglePhase, PhaseControlIed Thyristor Converters
Fig. 143a show> a commonly used FullBridge phasecontrolled converter for controlled rectification of the singlephase ulility voltage. To undersland the operating principie, it is redrawn as in Fig. 143b, where the acside inductance L, is ignored and the dcside load is represented as drawing a constant current ld. The wavefonns are shown in Fig. 144.
142
+ id
is
1
1
3
3
Ls
is +
Vs
rv
vd
vd
â€˘
Id
2
4 '4
+
2
(a)
(b)
Figure 143 FullBridge, singlephase thyristor converter.
mt mt 3, 4
I , 2
â€˘I
Figure 144 Singlephase thyristor converter waveforrns. Thyristors 1 and 2, and thyristors 3 and 4 are treated as two pairs, where each thyristor pair is supplied gate pulses delayed by an angle a with respect to its instant of natural conduction; mt = 0 0 for tbyristors 1 and 2, and 0Jt = 1800 for tbyristors 3 and 4, as shown in Fig. 144. ln tbe positive halfcycle of the input voltage, thyristors 1 and 2 are forwardblocking until they are gated at 0Jt = a when tbey immediately begin to conduct, and thyristors 3 and 4 become reverseblocking. ln tbis state, vAI) = v,(t)
and
i,(t) = 1d
a<mt";a+1r
(141)
These relationships hold true until a + 1r in the negative halfcycle of the input voltage, when thyristors 3 and 4 are gated and begin conducting. ln this state, Vd(1) = v,(t)
and
i,(t) = 1d
a+1r < mt"; a+21r
(142)
which holds true for one halfcyc1e, until the next halfcycle begins with the gating of thyristors 1 and 2.
143
Assuming the input ac voltage to be sinusoidal, the average value Vd of the voltage across the dcside of lhe converter can be obtained by averaging lhe vAI) waveform in Fig. 144 ove r one halfcycle during a < wr ,., a + 1[ :
2 " f V, sinwr·d(lüt) =v, cosa
1 a+.>r V, = 1[
"
a
(143)
1[
On the acside, lhe input current i, waveform is shifted by an angle a wilh respect to lhe input voltage as shown in Fig. 144, and the fundamentalfrequency component i' l (I) has a peak value of 4 1' 1 = 1,
(144)
1[
ln terms of voltage and current peak values, the power drawn from the acside is p =
I  v, / ' 1cosa 2
(145)
Assuming no power loss in the thyristor converter, the input power equals lhe power to the dcside ofthe converter. Using Eqs. 143 and 144, we can reconfirm the following relationship: (146) Power flow can be controlled by the delay angle a; increasing it toward 90° reduces lhe average dcside voltage Vd
while simultaneously shifting lhe input eurrent i, (/)
waveform farther away wilh respect to lhe input voltage waveform. The de voltage as a function of a is plotted in Fig. 145a and the eorresponding power direction is shown in Fig. 145b. The waveforms in Fig. 144 show Ihat for the delay angle a in a range of 0° to 90°, Vd has a positive value as plotted in Figs. 145a and b, and the converter operates as a reetifier, wilh power flowing from the acside to lhe dcside. Delaying the gating pulse sueh that a is greater Ihan 90° in Fig. 144 makes V, in Eq. 143 negative, and the converter operates as an inverter as shown in Fig. 145b, with power flowing from the dcside to the acside. ln practiee, the upper limit on a in lhe inverter is below 180°, for example 160° lo avoid a phenomenon known as the commutation failure, where the current fails to commulate fully from the conducting Ihyristor pair to lhe next pair, prior to the instant beyond which the conducting pair keeps
144
on conducting for another halfcycle. described in [I].
This commutationfailure phenomena is fully
Rectifier P=VdId = + ,j+:rr:.,,~ 1500., 1/800
,, ,, ,
a
,
Inverter
" (a)
(b)
P=VdId =
Fig. 145 E ffect ofthe delay angle a. 1431 The Effect of L, on Current Commutation
Previously, our assumption was that the acside inductance L, equals zero.
ln the
presence of this inductance, the input current takes a finite amount of time to reverse its direction, as the current "commutates" from one thyristor pair to the next.
1
3
, iI isL
vL +
v
O t i3
+
+ '\.t
vd
j
" t _ ~~s /
ld
iG
!
.
n
b . . j
1:7: '
4
1;4
2
ti2
~~)
\
tId
! i
a: u :
\
t
I1d
t
(b)
(a)
Figure 146 Effect of L, on Current Commutation. From basic principies, we know that changing the current through the inductor L, in the circuit of Fig. 146a requires a finite amount of voltseconds. represented by a dc current Id'
The dcside is stiH
The waveforms are shown in Fig. 146b, where
thyristors 3 and 4 are conducting prior to mI = a, and i, = Id
â€˘
a , thyristors 1 and 2, which have been forward blocking, are gated and hence they immediately begin to conduct. However, the current through them doesn ' t jump instantaneously as in the case of L , = O where i, instantaneously changed from CId ) to At
ĂşJ{
=
145
(+ I d
).
W ith a tinite L" during a short interval called the commutation interval u, ali
thyristors conduct, applying v, across L, in Fig. 146a. The voltradians needed to change the inductor current /Tom ( I d ) to (+ I d ) can be calcu lated by integrating the inductor voltage v L (= L, . di, / dI) from a to (a + u), as follows: a+/I
a+u
a
a
d"
IJ
f v"d(รกJI)=L, f ;; d(รกJI)=รกJL, f di, =รกJL,(2Id )
(147)
I"
The above voltradians are "Iost" from the integral of the dcside voltage waveform in Fig. 146b every halfcycle, as shown by the shaded area in Fig. 146b. Therefore, dividing the voltradians in Eq. 147 by the 1C radians each halfcycle, the voltage drop in the dcside voltage is (148)
This voltage is lost from the dcside average voltage in the presence of L, . Therefore, the average voltage is smaller than that in Eq. 143: 2
2
1C
1C
(149)
Vd =V,cosaรกJL,Id
We should note that the voltage drop in the presence of L, doesn't mean a power loss in L, ; it simply means a reduction in the voltage available on the dcside.
144
THREEPHASE, FULLBRIDGE THYRISTOR CONVERTERS
Threephase FullBridge converters use six thyristors, as shown in Fig. 147a. A simplified converter for initial analysis is shown in Fig. 147b, where the acside inductance L, is assumed zero, thyristors are divided into a top group and a bottom group, similar to the threephase diode rectitiers, and the dcside is represented by a current source Id' The converter waveforms, where the delay angle a (measured with respect the instant at which phase voltage waveforms cross each other) is zero are similar to that in diode rectifiers discussed in Chapter 5 (see Figs. 511 and 512). The average dc voltage is as calculated in Eq. 523, where
VL"
is the peak value ofthe ac input voltage: (1410)
146
id
ia
va
J
3
5
6
2
J P
L,
4
(b)
(a)
Figure 147 Threephase FullBridge thyristor converter. Oelaying the gate pulses to the thyristors by an angle a measured with respect to their instants of natural conductions, the waveforms are shown in Fig. 148, where L, = O .
wt = o
Figure 148 Waveforms with L, = O . ln the deside output voItage waveforms, the area Aa corresponds to " voltradians loss" due to delaying the gate pulses by a every
7r / 3
radiano Assuming the timeorigin as
shown in Fig. 148 at the instant at whieh the phase voltagc waveforms cross, the lineline voltage v
Qe
waveform ean be expressed as
Vu
sin rol. Therefore from Fig. 148, the
drop LI. Va in the average dcside voltage can be calculated as (1411)
1441 Effect of L,
As shown in the waveforms of Fig. 149, the average de output voltage due to the presence of L, is reduced by the area
A"
every
7r / 3
radiano Ouring the eommutation
interval u , from a to a + u , the instantaneous de voItage is redueed due to the vo1tage drop across the inductanee in series with the thyristor to whieh the current is
147
o
commutating from
to (+Id
Using the procedure
).
ln
Eq. 147 for singlephase
converters, a+u
Au
=
J
VI.
Id
d(ml)
= mL,
Jdi, = mL,Id
(1412)
O
a
and therefore the voltage drop due to the presence of L, is I1V u
= ~ = ~mL I d ,,/3
"
(1413)
'
Figure 149 Waveforms with L,. Therefore, the dcside output voltage can be written as (1414)
Vd = Vdo I1Va I1Vu
Substituting results from Eqs. 1410, 1411, and 1413 into Eq. 1414 (1415) Similar to s inglephase converters, threephase thyristor converters go into inverter mode with the delay angle a exceeding 90 0
â€˘
ln the inverter mode, the upper limit on a is
less than 1800 to avoid commutation failure, just like in singlephase converters. Further details on thyristor converters can be found in Reference [I J.
148
145
CurrentLink Systems
Thyristors are available in very large current and voltage ratings of several kAs and several kVs that can be connected in series. ln addition. thyristor converters can block voltages of both polarities but conduct current only in the forward direction. This capability has led to the interface realized by thyristorconverters with a dccurrent link in the middle, as shown in Fig. 1410. Unlike in voltagelink systems, the transfer of power in currentlink systems can be reversed in direction by reversing the voltage polarity of the dclink. This structure is used at very high power leveIs, at tens of hundreds ofMWs, for example in H igh Voltage DC Transmission (HVDC).
+
+
Figure 1410 Block diagram of currentlink systems. Thyristors in these two converters shown in Fig. 1410 are connected to allow the flow of current in the dclink by thyristors in converter I pointing up, and the thyristors in converter 2 connected to point downward. ln Fig. 1410, subscripts 1 and 2 refer to systems I and 2 and Rd is the res istance of the dclink inductance.
Assuming each
converter a sixpulse thyristor converter as discussed previously, (1416)
(1417) 0
0
By controlling the delay angles a I and a 2 in a range of 0 to 180 (practically, this value is limited to approximately 150
0 )
,
the average voltage and the average current in
the system ofFig. 1410 can be controlled, where current flow through the dclink can be expressed as (1418)
149
where the dcIinl< resistance Rd is generally very small. ln such a system, for the power f10w from system I to system 2, Vd2 is made negative by controlling a, sucb that it operates as an inverter and establishes the voltage of the deI ink.
The converter 1 is
operated as a rectifier at a delay angle aI such that it controls the current in the dclink. The converse is true for tbese two converters if the power is to f10w from system 2 to system 1. The above discussion of currentlink systems shows the operating principie behind HVDC transmission systems discussed in the next chapter, where using transformers, sixpulse thyristor converters are connected in series on the dcside and in parallel on the acside to yield higher effective pulse number. REFERENCE 1.
N. Mohan, T. M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and Design, 3,d Edition, Wiley & Sons, New York, 2003.
PROBLEMS SinglePhase Thyristor Converters ln a singlephase thyristor converter, V,
= 120V(rms)
at 60Hz , and L,
= 3mH.
The
delay angle a = 30Â° . This converter is supplying 1kW of power. The dcside current id can be assumed purely de. 141 142 143
Calculate the commutation angle u. Draw the waveforms for the converter variables. Calculate the DPF, %THD in the input current, and the PF of operation of this converter.
ThreePhase Thyristor Converters ln a threephase thyristor converter, VLL = 460V(rms) at 60Hz, and L, = 5mH. The delayangle a = 30Â° . This converter is supplying 5kW of power. The dcside current id can be assumed purely de. 144
Calculate the commutation angle u.
145 146
Draw the waveforms for the converter variables. Calculate the DPF, %THD in the input current, and the PF of operation of this converter.
1410
Chapter 15
UTILITY APPLICATIONS OF POWER ELECTRONICS 151
INTRODUCTION
Power electronics applications in utility systems are growing very rapidly, which promise to change the landscape of future power systems in terms of generation, operation and control. The goal of this chapter is to briefly present an overview of these applications to prepare students for new challenges in the deregulated utility environrnent and to motivate them to take further courses in this field, where a separate advancedIevel graduate course is designed, for example at the University of Minnesota, to discuss these topics in detail. ln discussing these app lications, we will observe that the power electronic converters are
the same or modifications of those that we have already discussed in earlier chapters. Therefore, within the scope of this book, it will suffice to discuss these applications in terms of the block diagrams of various converters. These utility applications of power electronics can be categorized as follows: •
Distributed Generation (DG) Renewable Resources (Wind, Photovoltaic, etc.) Fuel Cells and MicroTurbines Storage  Batteries, Superconducting Magnetic Storage, Flywheels
•
Power Electronic Loads  Adjustable Speed Drives
•
Power Quality Solutions Dual Feeders Uninterruptible Power Supplies Dynamic Voltage Restorers
•
Transmission and Distribution (T&D) High Voltage DC (HVDC) and HVDCLight Flexible AC Transmission (FACTS) • Shunt Compensation • Series Compensation • Static Phase Angle Control and Unified Power Flow Controllers
151
152
POWER SEMICONDUCTOR DEVICES AND THEm CAPABILITlES [IJ
Fig. 151 a shows the commonly used symbols of power devices. The power handling capabilities and switching speeds of these devices are indicated in Fig. 151 b. Ali these devices allow current flow only in their forward direction (the intrins ic antiparallel diode of MOSFETs can be eXplained separately). Transistors (intrinsically or by design) can block only the forward polarity voltage, whereas thyristors can block both forward and reverse polarity voltages. Diodes are uncontrolled devi ces, which conduct current in the forward direction and block a reverse voltage. At very high power leveIs, integratedgate controlled thyristors (IGCTs), which have evolved from the gatetumoff thyri stors, are used. Thyristors are semicontrolled devices that can switchon at the desired instant in their forwardblocking state, but cannot be switched off from their gate and hence rely on the circuit in which they are connected to switch them off. However, thy ristors are available in very large voltage and current ratings.
10' ~
e ~ o
"Thyristor
IGCT
IGBT
MOS FET
106 IGBT
lO'
~OSF0
lO'
(a)
103 lO' lO' 10' Switching Frequency (Hz) (b)
Figure 151 Power semiconductor devices. 153
CATEGORlZING POWER ELECTRONIC SYSTEMS
ln a very broad sense, the role of power electronics in these power system applications can be categorized as follows : 1531 SolidState Switches
By connecting two thyristors in antiparallel (backtoback) as shown in Fig. 152, it is possible to realize a solidstate switch which can conduct current in both directions, and tumon or tumoff in an ac circuit with a delay of no more than onehalf the linefrequency cycle. Such switches are needed for applications such as dual feeders , shuntcompensation for injecting reactive power at a bus for voltage control, and seriescompensation oftransmission lines.
152
Figure 152 Backtoback thyristors to act as a solidstate switch. 1532 Converters as an Interface Power electronic converters provide the needed interface between the electrical source, often the utility, and the load, as shown in Fig. 153. The electrical source and the electrical load can, and often do, differ in frequency, voltage amplitudes and the number of phases. The power electronics interface allows the transfer of power from the source to the load in the most energy efficient manner, in which it is possible for the source and load to reverse roles. These interfaces can be classified as below.
,,, ,,, , ,
}~
Converter
f+{ Load
Source
, :_ L.__..:_::.:_:.:_:.:_:.:_:.:_:.:_"__..J_ J
Figure 153 Power electronics interface. 15321 VoltageLink Systems The semiconductor devices such as transistors ofvarious types and diodes can only block forwardpolarity voltages. These devices with only unipolar voltageblocking capability have led to the structure with two converters, where the dc ports of these two converters are connected to each other with a parallel capacitor forming a dclink in Fig. 154, across which the voltage polarity does not reverse, thus allowing unipolar voltagehandling transistors to be used within these converters. The transfer of power can be reversed in direction by reversing the direction of currents associated with the dclink system. Voltagelink converters consist of switching powerpoles as the building blocks, which can synthesize the desired output by means of Pulse Width Modulation (PWM) and have the bidirectional power flow capability. Such switching powerpoles can be modeled by means of an ideal transformer with a controllable tumsratio, as discussed in Chapter 12.
153
6>i ACl
Fig. 154 Block diagrarn ofthe voltagelink systems. 15322 CurrentLink Systems Thyristors can block voltages of both polarities but conduct current only in the forward direction. This capability has led to the interface realized by thyristorconverters with a dccurrent link in the middle, as shown in Fig. 155. The transfer of power can be reversed in direction by reversing the voltage polarity of the dclink, but the currents in the link remain in the sarne direction. This structure is used at very high power leveis, at tens ofhundreds of MWs, for example in High Voltage DC Transmission (HVDC).
ACl
Figure 155 810ck diagram ofthe currentlink systems. ln the following sections, various utility applications and the role of power electronics in them are exam ined further. 154
DISTRIBUTED GENERATION (DG) APPLICATIONS
Distributed generation, shown in Fig. 156 from the control perspective, promises to change the landscape ofhow power systems ofthe future will be operated and controlled. These generators may have decentralized (local) control, in addition to a central supervisory control.
I
I Control Center I Central Power $tation
I. I
Combi ned Heat and Power Plant (eHP)
>Factory
>Commercial Bui lding > House >Apanrnent Building
Fuel Cell :> Smart House ,. Perfonnance Building
_I Solar Power Plants );;o
~
eHP House
Wind Power Plants ). Village );o
Commerc ial Building
MicroTurbine ). Hospital :> Commercial Building
Figure 156 Distributed Generation (DG) systems. 154
There is a move away from large central power plants towards distributed generation due to environrnental and economic reasons. Renewable resources such as wind and photovoltaic systems are growing in their popularity. There are proposals to place highly efficient smallscale power plants, bascd on fuel cells and microturbines, near load centers to sirnultaneously avoid transmission congestion and line losses. Many of distributed generation systems are discussed below. 1541 WindElectric Systems Wind energy is the fastest growing energy resource in the world. ln 2002, the amount of installed wind energy in the world reached nearly 30,000 MW, and the installed wind energy in the U.S. alone increased by 66% in 2001. This remarkable growth is spurred by a combination of environrnental concems, decreasing costs of wind electric systems, and financial incentives for wind park developers. There are many schemes for hamess ing wind energy, out of which those using doublyfed induction generators are enjoying increased popularity. As shown in Fig. 157, in this system, the stator which produces most of the power is connected directly to the grid. The rotor windings are accessible througb sliprings and brushes to a power electronic converter system, which supplies the rotor current. By injecting currents into the rotor, the machine can be made to act like a generator at both subsynchronous and supersynchronous speeds. The converter rating is about 15% of the rated power, for example, in the 750 kW doublyfed induction machines used in the southwestem Minnesota for windelectric systems, only 150 kW flows through the rotor, and the rest through the stator. Injecting current into the rotor provides control over the real and the reactive power (Ieading or lagging). Wound rotor
lnduction Generator
r~~~
w,~\.:~t:"rl /~,..~_...~=DC==; /i1~rtr A=CC= ]
Turbine
V
DC
Generators lde Converter
/
AC Gndslde Converter
Figure 157 Doublyfed induction generators for windelectric systems. 1542 PbotovoItaic (PV) Systems Photovoltaic systems are the ultimate in distributed generation and have even a greater potential than the windelectric systems. ln 1997, PV systems surpassed the 500 MW
155
threshold in worldwide installation, and annual global sales of $1 billion. A few years back, the Clinton administration announced the 'million solar roof initiative', emphasizing the growing importance of PV systems. ln PV systems, the PV arrays (typically four of them connected in series) provide a voltage of 52V to 90V DC, which the power e1ectronic system, as shown in Fig. 158, converts to 120V160Hz sinusoidal voltage suitable for interfacing with the singlephase utility. PV modules are still very expensive, as much as $1,500 for only a 300 W peak output. Such PV modules can be connected in parallel for higher output capacity. Presently there is not much economy of scale in the PV modules, making it prudent to invest in steps. Isolated DCDC Converter
PWM Converter
Uti lity Icp
Figure 158 Photovoltaic systems. 1543 Fuel Cell Systems Lately, there has been a great deal of interest in, and effort being devoted to fuel cell systems. The reason has to do with their efficiency, which can be as high as 60 percent. The input to the fuel cell can be natural gas or gasoline and the output is a dc voltage. The need for power electronics converters to interface with the utility is the sarne in the fuelcell systems as that for the photovo ltaic systems. 1544 MicroTurbines These use highly efficient aircraft engines, for example to produce peak power when it is needed, with the natural gas as the input fue\. To improve efficiency, the turbine speed should be al\owed to vary based on loading. This will cause the frequency of the generated output to vary, requiring a powerelectronic interface as that in adjustablespeed drives. 1545 Energy Storage Systems Although not a primary source of energy, storage plants offer the benefit of loadIeveling and peakshaving in power systems, because of the diurnal nature of electricity usage. Energy is stored, usually at night when the load demand is low, and supplied back during the peakIoad periods. It is possible to store energy in leadacid batteries (other exotic 156
hightemperature batteries are being developed), in Superconducting Magnetic Energy Storage (SMES) coils and in the inertia of f1ywheels. Ali these systems need power electronics interface, where the interface for the f1ywheel storage is shown in Fig. 159.
GridÂˇside Converter
Machineside Converter Flywheel
Figure 159 Flywheel storage system.
155
POWER ELECTRONIC LOADS
As discussed in Chapter 1, power electronics is playing a significant role in energy conservation, for example, as users discover the benefits of reduced energy consumption and better process control by operating electric drives at adjustable speeds. An adjustablespeed drive (ASD) is shown in Fig. 1510 in a block diagram formo Power electronic loads of this type often use an interface with the utility that results in distorted line currents. These currents result in distorted voltage waveforms, affecting the neighboring loads. However, it is possible to design the utility interface (often called the powerfactorcorrected frontend) that allows sinusoidal currents to be drawn from the grid. With the proliferation of power electronics loads, standards are being enforced that limit the amount of distortion in currents drawn . SwitchÂˇmode Converter
Utility
~
I
Rectifier
Figure 15] O Adjustablespeed drive.
156
POWER QUALITY SOLUTIONS
Poor quality of power can imply any of the following: distorted voltage waveforms, unbalances, swells and sags in voltage and power outages, etc. This problem is exacerbated in a deregulated env ironment where utilities are forced to operate at marginal profits, resulting in inadequate maintenance of equipment. ln this section, we will also see that power electronics can solve many ofthe power quality problems.
157
1561 Dual }'eeders The continuity of service can be enhanced by dual power feeders to the load, where one acts as a backup to the other that is supplying the load as shown in Fig. 1511 . Usi ng backtoback connected thyristors, acting as a solidstate switch, it is poss ible to switch the load quickly, without interruptio n, from the main feeder to the backup feeder and back. Feeder I
)11
Feeder2
I
Load
L_ _ _ _ '
Figure 1511 Dualfeeders.
1562 Uninterruptible Power Supplies Power outages, even for a few cyc1es, can be very disruptive to criticaI loads. To provide immunity from such outages, powerelectronicsbased uninterruptible power supplies (UPS) shown in Fig. 1512 can be used, where the energy storage can be by means of batteries, SMES, f1ywheels or supercapacitors.
Energy Storage
Figure 1512 U ninterruptible power supplies.
1563 Dynamic Voltage Restorers Dynamic Voltage Restorers (DVR), shown in Fig. 1513, can compensate for voltage sags or swells by injecting a voltage
V ;nj
in series with that supplied by the utility.
+ v,
Power Electronic Interface
Load
Figure 15 I3 Dynamic vo ltage restorers .
157
TRANSMISSION AND DISTRIBUTION (T&D) APPLICATIONS
ln the past, utilities operated as monopolies, where users had little or no option ln selecting whom they can purchase power from. The recent trend , which seems 158
irreversible in spite of recent setbacks, is to deregulate where utilities must compete to sell power on an open market and the customers have a choice of selecting their power provider. ln such an environment, utilities that were vertically integrated are now forced to split into generation companies that produce power, and transmissionline operators that maintain the transmission and distribution network for a fee. [n this deregulated environment, it is highly desirable to have the capability to dictate the flow ofpower on designated power lines, avoiding overloading oftransmission lines and excessive power losses in them. ln this section, we will look at some such options. 1571 High VoItage DC (HVDC) and Medium Voltage DC (MVDC) Transmission
Direct current (DC) transmission represents the ultimate in flexibiI ity, isolating two interconnected ac systems from the requirement of operating in synchronism or even at the same frequency. High voltage DC (HVDC) systems using thyristorbased converters have now been in operation for several decades. Lately, such systems at medium voltages have been proposed. Both ofthese types of systems are discussed below. [5711 High Voltage DC (HVDC) Transmission Fig. 1514 shows the block diagram of a HVDC transmission system, where power is transmitted over dc lines at high voltages in excess of 500 kV. First, the voltages in ac system I at the sending end are stepped up by means of a transformer. These voltages are rectified into dc by means of a thyristorbased converter, where the ac line voltages provide the commutation of current within the converter, such that ac currents drawn from system I tum into dc current on the other side of the converter. These currents are transmitted over the dc line where additional series inductance is added to ensure that the dc current is smooth, free of ripple as much as possible. At the receiving end, there are thyristorbased converters, which convert the dc current into ac currents and injects them into the ac system 2, through a stepdown transformer. The roles of the sending and the receiving ends can be reversed by reversing the voltage polarity in the dc system.
ACl
AC2
Figure 1514 HVDC system block diagram (transformers are not shown). At very high power leveis in excess of 1,000 MW, use of thyristors, at least for now, represents the only reasonable choice.
159
15712 Medium Voltage DC (MVDC) Transmission At lower power leveis, there are proposals to use medium voltages, in which case the block diagram ofthe system is as shown in Fig. 1515. ln such a system, the direction of power f10w is reversed by changing the direction of current in the dc line.
6>1 ACI
Figure 1515 Block diagram for mediumvoltage dc transmission systems. 1572 Flexible AC Transmission Systems (FACTS) [2]
DC transmission systems discussed earlier are an excellent choice where large amount of power needs to be transmitted over long distances, or if the system stability is a serious issue. ln existing ac transmission networks, limitation on constructing new power lines and their cost has led to other ways to increase power transmission capability without sacrificing the stability margino These techniques may also help in directing the power f10w to designated lines. Power f10w on a transmission line connecting two ac systems in Fig. 1516 is given as
P
E,E, .
<"
(I 51)
= SInu
X
where E, and E, are the magnitudes of voltages at the two ends of the transmission line,
X is the line reactance and <5 is the angle between the two bus voltages. E,LO
E,L<5 jX
~~~__nnn__ +_ _~ ACI
AC2
Figure 1516 Power flow on a transmission line . Eq. 151 shows that the power flow on a transmission line depends on three quantities: (1) the voltage magnitude E, (2) the line reactance X, and (3) the power angle <5. Various devices which are based on rapidly controlling one or more of the above three quantities are discussed in the follow ing sections:
1510
15721 ShuntConnected Devices to Control the Bus Voltage Magnitude E The reactive power compensation is very important and may even be necessary at high loadings to avoid voltage collapse. Shuntconnected devices can draw or supply reactive power from a bus, thus controlling the bus voltage, albeit in a limited range, based on the internal system reactance. Various forms of such devices are being used in different combinations. These include ThyristorControlled Reactors (TCR) shown in Fig. 1517a, and ThyristorSwitched Capacitors (TSC) shown in Fig. 1517b for Static Var Compensation (SVC). The advanced Static Var Compensator (STATCOM) shown in Fig. 1517c can draw or supply reactive power.
1
= (a)
& (b)
L
Utility
I
STATCOM
l...l
(c)
Figure 1517 Shuntconnected devices for voltage control. Shuntcompensation devices have the following limitations for controlling the flow of active power: I.
2.
A large amount of reactive power compensation, depending on the system internal reactance, may be required to change the voltage magnitude. Of course, the voltage can only be changed in a limited range (utilities try to maintain bus voltages at their nominal values), which has a limited effect on the power transfer given by Eq. ISI. Most transmission systems consist of parallel paths or loops. Therefore, changing the vo ltage magnitude at a given bus changes the loading of ali the lines connected to that bus, and there is no way to dictate the desired change of power flow on a given line.
15722 SeriesConnected Devices to Control the Effective Series Reactance X These devices, connected in series with a transmission line, partially neutralize (or add to) the transm ission line reactance. Therefore, they change the effective value of X in Eq. 151, thus allowing the active power flow P to be controlled. Various forms ofsuch devices have been used. These include the ThyristorControlled Series Capacitor (TCSC) shown in Fig. 1518.
1511
Figure 1518 Thyristorcontrolled series capacitor. 15723 Static Phase Angle Control and Unified Power Flow Controller (UPFC) Based on Eq. 151 , a device connected at a bus in a substation, as shown in Fig. 1519a, can influence power flow in three ways by: I.
2. 3.
controlling the voltage magnitude E changing the line reactance X, and/or changing the power angle <5.
Such a devi ce called the Unified Power Flow Controller (UPFC) can affect power flow in any combination of the ways listed above. The block diagram of a UPFC is shown in Fig. 1519a at one side of the transmission line. lt consists of two voltagesource switchmode converters. The first converter injects a voltage E, in series with the phase voltage such that
E, +E, =E2
(152)
Therefore, by controlling the magnitude and the phase of the injected voltage E, within the circle shown in Fig. 1519b, the magnitude and the phase of the bus voltage
E2
can
be controlled. Ifa component ofthe injected voltage E, is made to be 90 degrees outofphase, for example leading with respect to the current phasor J, then the transmission line reactance X is partially compensated.
E,
substation   , I~
.E
j
+
E,
LI
E,
I converter
(b) Series
Shunt ~
~
converter
P2.Q2 ~ .QI ~  
(a) Figure 1519 UPFC.
1512
E2
The second converter in a UPFC is needed for the following reason: since converter 1
E" line (where p,
and the reactive power Q, to the
injects a series voltage
it delivers real power p"
transmission
and Q, can be either positive or negative):
p,
= 3 Re(E,IÂ°)
(153) (154)
Since there is no steady state energy storage capability within UPFC, the power P2 into converter 2 must equal
p,
if the los ses are ignored: (155)
However, the reactive power Q2 bears no relation to Q"
and can be independently
controlled within the voltage and current ratings ofthe converter 2: (156)
By controlling Q2 to control the magnitude of the bus voltage
E"
UPFC provides the
same functionality as that of an advanced static var compensator STATCOM. REFERENCES 1.
2. 3.
N. Mohan, T. M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and Design, 3"' Edition, Wiley & Sons, New York, 2003. N. G. Hingorani, and L. Gyugyi, Understanding FACTS, New York, IEEE Press, 2000. N. Mohan, A. Jain, P. Jose and R. Ayyanar, "Teaching Utility Applications ofPower Electronics in a First Course on Power Systems," submitted to the IEEE Transactions on Power Systems, Special Section on Power Engineering Education, 2003.
PROBLEMS 151 152 153 154 155
Show the details and the average representation of converters in Fig. 157 for a windelectric system. Show the details and the average representation of converters in Fig. 158 for a photovoltaic system. Show the details and the average representation of converters in Fig. 159 for a tlywheel storage system. Show the details and the average representation of converters in Fig. 1510 for an adjustablespeed drive. Show the details and the average representation of converters in Fig. 1512 for a UPS. 1513
156
Show the details and the average representation of converters in Fig. 15 \3 for a DVR. 157 Show the details and the average representation of converters in Fig. 1514 for an HVDC transmission system. 158 Show the details and the average representation ofconverters in Fig. ISIS for an MVDC transmission system . 159 Show the details and the average representation of converter in Fig. 1517c for a STATCOM. 1510 Show the details and the average representation of converters in Fig. I 519a for a UPFC system.
1514
DETAILED CHAPTER CONTENTS Chapter 1 Power Electronics and Drives: Enabling Technologies lI Introduction to Power Electronics and Drives, lI 12 Applications and the Role of Power Electronics and Drives, lI 121 Powering the Information Technology, lI 122 Robotics and Flexible Production, 12 13 Energy and the Environrnent, 14 131 Energy Conservation, 14 1311 ElectricMotor Driven Systems, 14 1312 Lighting, 15 1313 Transportation, 15 132 Renewable Energy, 16 133 Utility Applications of Power Electronics, 17 134 Strategic Space and Defense Applications, 17 14 Need for High Efficiency and High Power Density, 18 15 Structure ofPowerElectronics Interface, 19 16 SwitchMode LoadSide Converter, IlO 161 SwitchMode Conversion: Switching PowerPole as the Building Block, lII 162 PulseWidth Modulation (PWM) ofthe Switching Power
Pole (Constant j,), 112
163 Switching PowerPole in A Buck DCDC Converter: An Example, 113 17 Recent and Potential Advancements, I  14 References, II 5 Problems, 116 Chapter 2 Design ofSwitching PowerPoles 21 Power Transistors and Power Diodes, 21 22 Selection ofPower Transistors, 21 221 MOSFETs, 22 222 IGBTs, 23 223 PowerIntegrated Modules and lntelligentPower Modules, 24 224 Costs ofMOSFETs and IGBTs, 24 23 Selection ofPower Diodes, 24 24 Switching Characteristics and Power Losses in PowerPoles, 25 24 I Tumon Characteristic, 26 242 Tumoff Characteristic, 27 243 Calculating Power Losses within the MOSFET (assuming an ideal diode),28 2431 Conduction Loss, 28 2432 Switching Losses, 29 244 Gate Driver Integrated Circuits (ICs) with builtIn Fault Protection, 210 25 JustifYing Switches and Diodes as Ideal, 211 26 Design Considerations, 211 DCCl
261 Switching Frequency
is, 211
262 Selection of Transistors and Diodes, 212 263 Magnetic Components, 212 264 Capacitor Selection, 212 265 Thermal Design, 212 266 Design Tradeoffs, 214 27 The PWM ControJler IC, 214 References, 21 5 Problems, 21 5 APPENDIX 2A Diode Reverse Recovery and Power Losses, 216 2A I Diode F orward Loss, 216 2A2 Diode Reverse Recovery Characteristic, 216 2A3 Diode Switching Losses, 217 Chapter 3 SwitchMode DCDC Converters: Switching Analysis, Topology Selection and Design 31 DCDC Converters, 31 32 Switching PowerPole in DC Steady State, 31 33 Simplifying Assumptions, 34 34 Common Operating Principies, 34 35 Buck Converter Switching Analysis in DC Steady State, 34 36 Boost Converter Switching Analysis in DC Steady State, 37 37 BuckBoost Converter Analysi s in DC Steady State, 310 371 Other BuckBoost Topologies, 313 3712 Cuk Converter, 314 38 Topology Selection [2], 315
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39 WorstCase Design, 316 310 SynchronousRectified Buck Converter for Very Low Output Voltages [3],316 311 lnterleaving ofConverters [4], 317 312 Regulation of DCDC Converters by PWM, 317 313 Dynamic Average Representation of Converters in CCM, 318 314 BiDirectional Switching PowerPole, 321 315 DiscontinuousConduction Mode (DCM),322 References, 324 Problems, 324 Appendix,324 3A DiscontinuousConduction Mode (DCM) in DCDC Converters, 324 3AI Output Voltages in DCM, 324 3AII Buck Converters in DCM, 324 3AI2 Boost and BuckBoost Converters in DCM, 325 3A2 Average Representation ofDCDC Converters in DCM, 326 3A21 v. and i. for Buck and BuckBoost Converters in DCM, 326 3A22 v. and i. for Boost Converters in DCM, 328 Chapter 4 Designing Feedback Controllers in SwitchMode DC Power Snpplies 41 Introduction and Objectives of Feedback Control, 41 42 Review ofL inear Control Theory, 42
421 OpenLoop Transfer Function
G,, (s),43 422 Crossover Frequency .Ic(s) of
G L (s), 43 423 Phase and Gain Margins, 43 43 Linearization of Various Transfer Function Blocks, 45 431 Linearizing the PWM Controller IC,45 432 Linearizing the Power Stage of DCDC Converters in CCM, 46 4321 Using Computer Simulation to Obtain
"o1d , 49
44 Feedback Controller Design ln VoltageMode Control, 411 45 PeakCurrent Mode Control, 415 References, 420 Problems, 420
Chapter 5 Rectification ofUtility Input Using Diode Rectifiers 51 Introduction, 51 52 Distortion and Power Factor, 51 521 Rms Value ofDistorted Current and the Total Harrnonic Distortion (THD), 52 522 The Displacement Power Factor (DPF) and Power Factor (PF), 55 523 Deleterious Effects ofHarrnonic Distortion and a Poor Power Factor, 57 5231 Harmonic Guidelines, 57 53 Classifying the "FrontEnd" Of Power Electronic Systems, 59
54 DiodeRectifier Bridge "FrontEnds", 510 541 The SinglePhase DiodeRectifier Bridge, 513 54 lI Effects of L, and Cd on the Waveforrns and the THD, 511 542 ThreePhase DiodeRectifier Bridge, 5 13 543 Comparison of SinglePhase and ThreePhase Rectifiers, 5 I 5 55 Means to Avoid Transient lnrush Currents at Starting, 516 56 FrontEnds with BiDirectional Power Flow, 516 References, 516 Problems, 516
Chapter 6 PowerFactorCorrection (PFC) Circuits and Designing the Feedback Controller 61 I ntroduction, 61 62 Operating Principie of SinglePhase PFCs, 61 63 Control of PFCs, 63 64 Designing the Inner AverageCurrentControl Loop, 64 641 d(s)/Vc(s) for the PWM Controller, 65
642 IL(s) 1 d(s) for lhe Boost Converter in the Power Stage, 65 643 Designing the Current Controller
G,(s),65 65 Designing the Outer Voltage Loop, 66 66 Example ofSinglePhase PFC Systems, 68
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661 Design of the Current Loop, 68 662 Design ofthe Voltage Loop, 68 67 Simulation Results, 69 68 Feedforward ofthe Input Voltage, 69 References, 610 Problems, 611 Chapter 7 Magnetic Circuit Concepts 71 AmpereTums and Flux, 71 72 Inductance L, 72 721 Energy Storage due to Magnetic Fields, 73 73 Faraday ' s Law: Induced Voltage in a Coil due to TimeRate ofChange ofFlux Linkage, 74 74 Leakage and Magnetizing Inductances, 75 741 Mutual Inductances, 77 75 Transformers, 77 References, 79 Problems, 79
8 7 Practical Considerations ln TransformerIsolated SwitchMode Power Supplies, 811 References, 8 II Problems, 811 Chapter 9 Design of HighFrequency Inductors and Transformers 91 lntroduction, 91 92 Basics of Magnetic Design, 91 93 Inductor and Transformer Construction, 91 94 AreaProduct Method, 92 94 1 Core Window Area A,.mdow' 92 942 Core CrossSectional Area
A=~,
93 943 Core AreaProduct A p (= A=re A,.ind~ ) , 94
944 Design Procedure Based On AreaProduct A p' 95 95 Design Example of An Inductor, 96
Chapter8 SwitchMode DC Power Supplies 81 Application of SwitchMode D C Power Supplies, 81 82 Need for Electrical Isolation, 8 1 of Transformer83 Classification Isolated DCDC Converters, 82 84 Flyback Converters, 82 85 Forward Converters, 85 851 TwoSwitch Forward Converters, 86 86 FullBridge Converters, 87 861 HalfBridge and PushPull Converters, 810
DCC4
96 Design Example of A Transformer For A Forward Converter, 97 References, 98 Problem s, 98 Chapter 10 SoftSwitching in DCDC Converters and Converters for Induction Heating and Compact Fluorescent Lamps 101 Introduction, 101 102 HardSwitching in Switching PowerPoles, 101 103 SoftSwitching in Switching PowerPoles, 101
1031 Zero Voltage Switching (ZVS), 103 1032 Synchronous Buck Converter with ZVS, 103 1033 PhaseShift Modulated (PSM) DCDC Converter, 106 104 Inverters for Induction Heating and Compact Fluorescent Lamps, 107 References, 107 Problems, 107 Chapter 11 Electric Motor Drives 111 Introduction, 1 lI 112 Mechanical System Requirements, 112 1121 Electrical Analogy, 114 113 Introduction to Electric Machines and the Basic Principies of Operation, I 14 1131 Electromagnetic Force, 115 1132 lnduced Emf, 117 1133 Basic Structure, 117 I 14 DC Motors, I 18 1141 Structure ofDC Machines, 118 1142 Operating Principies of DC Machines, 119 1143 DCMachine Equivalent Circuit, 1110 1144 TorqueSpeed Characteristics, 1111 115 PermanentMagnet AC Machines, 1112 1151 The Basic Structure of PermanentMagnet Ac Machines, 1113 1152 Principie of Operation, 1113
1153 Mechanical System ofPMAC Drives, 1115 1154 PMAC Machine Equivalent Circuit, 1115 1155 PMAC TorqueSpeed Characteristics, 1116 1156 The Controller and the PowerProcessingUnit(PPU),1117 116 lnduction Machines, 1117 116 I Structure of Induction Machines, 118 1162 The Principies ofInduction Motor Operation, 1 lI 8 1163 PerPhase Equivalent Circuit of Induction Machines, 1120 References, 1122 Problems, 1122 Chapter 12 Synthesis ofDC and LowFrequency Sinusoidal AC Voltages for Motor Drives and UPS 121 Introduction, 121 122 Switching PowerPole as the Building Block, 122 1221 PulseWidthModulation (PWM) ofthe BiDirectional Switching PowerPole, 122 1223 Harmonics in The PWM Waveforms v A and idA' 125 123 DCMotor Drives, 125 124 ACMotor Drives, 129 1241 Space Vector Pulse Width Modulation, 1213 125 VoltageLink Structure with BiDirectional Power Flow, 1213 126 Uninterruptible Power Supplies (UPS), 1215
DCCS
1261 SinglePhase UPS, 1216 References, 1219 Problems, 1219 Cbapter 13 Designing Feedback Controllers for Motor Drives \31 lntroduction, 131 \32 Control Objectives, 131 133 Cascade Control Structure, 133 134 Steps in Designing the Feedback ControII er, 134 135 System Representation for SmallSignal Analysis, 134 \351 The Average Representation of the PowerProcessing Unit (PPU), 134 1352 Modeling ofthe DC Machine and the Mechanical Load, 136 136 Controller Design, 137 1361 PI Controllers, 137 137 Example of A Controller Design, 138 \371 The Design ofthe Torque (Current) Control Loop, 139 1372 The Design ofthe Speed Loop, 1311 1373 The Design of the Position Control Loop, 1313 References, 1314 Problems, 1314 Cbapter 14 Tbyristor Converters 141 lntroduction, 141 142 Thyristors (SCRs), 141 143 Sing1ePhase, PhaseControlled Thyristor Converters, 142
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1431 The Effect of Ls on Current Commutation, 145 144 ThreePhase, FullBridge Thyristor Converters, 146 145 CurrentLink Systems, 148 Reference, 1410 Problems, 1410 Cbapter 15 Utility Applications of Power Electronics 151 Introduction, 151 152 Power Semiconductor Devices and their Capabi lities [1], 152 153 Categorizing Power E1ectronic Systems, 152 1531 SolidState Switches, 152 1532 Converters as an Interface, 153 15321 VoltageLink Systems, 153 15322 CurrentLink Systems, 154 154 Distributed Generation (DG) Applications, 154 1541 WindElectric Systems, 155 1542 Photovoltaic (PV) Systems, 156 1543 Fue1 Cell Systems, 156 1544 MicroTurbines, 156 1545 Energy Storage Systems, 156 155 Power Electronic Loads, 157 156 Power Quality Solutions, 158 1561 Dual Feeders, 158 1562 Uninterruptible Power Supplies, 158 1563 Dynamic Voltage Restorers, 15
8 157 Transmission and Distribution (T &D) Applications, 159
1571 High Voltage DC (HVDC) And Medium Voltage DC (MVDC) Transmission, 159 15711 High Voltage DC (HVDC) Transmission, 159 15712 Medium Voltage DC (MVDC) Transmission, 1510 1572 Flexible AC Transmission Systems (Facts), 1510 15721 ShuntConnected Devices to Control the Bus Voltage Magnitude E, 1512
15722 SeriesConnected Devices to Control the Effective Series Reactance X, 1512 15723 Static Phase Angle Control and Unified Power Flow Controller (UPFC), 1513 References, 1514 Problems, 1514
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Published on Oct 20, 2016