Midya Thesis

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NONLINEAR CONTROL AND OPERATION OFDC TO DC SWITCHING POWER CONVERTERS

BYPALLAB MIDYA

B. Tech., Indian Institute of Technology Kharagpur, 1988M. S., Syracuse University, 1990

THESISSubm itted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Electrical Engineeringin the Graduate College of theUniversity of Illinois at Urbana-Champaign, 1995

Urbana, Illinois


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U N I V E R S I T Y O F I L L I N O I S A T U R B A N A - C H A M P A I G N

T H E G R A D U A T E C O L L E G E

MAY 1995

W E H E R E B Y R E C O M M E N D T H A T T H E T H E S I S B YPALLAB MIDYA

E N T I T L E D . NONLINEAR CONTROL AND OPERATION OF

DC TO DC SWITCHING POWER CONVERTERS

B E A C C E P T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R

T H E D E G R E E O F . DOCTOR OF PHILOSOPHY

ALA=^ Director of Thesis Researchr\J -HVf9U\*vv \"@\ Max)

Head of Department

C o m m i t t e e o n F i n a l E x a m i n a t i o nLk-T^/^S. Chairperson

P - K . VC/ywvJ>t Required for doctor's degree but not for master's.


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NONLINEAR CONTROL AND OPERATION OFDC TO DC SWITCHING POWER CONVERTERS

Pallab Midya, Ph.D.Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-Champaign, 1995

Philip T. Krein, AdvisorABSTRACT

Conventional control of switching power conversion is based on a linear model of thesystem, which limits its scope and performance. This thesis addresses a family of control andoperation issues that arise from the nonlinear nature of the switching power conversion. Thefocus is on dc to dc power converters with a small number of states. Nonlinear noise analysisof conventional feedback has been performed to evaluate this important aspect of converterperformance. Alternative schemes are developed that reduce noise susceptibility, reduceeffects of source and load disturbance, and improve system operation. A nonlinear controlscheme called sensorless current mode (SCM) control has been developed that emulatescurrent mode control without current sensing. SCM has very significant advantages indynamic range and noise susceptibility. Optimal control approaches to switching powerconversion has been explored. O ptimization ofanonmonotonic tuning problem has also beenexplored. A generalized tuning scheme has been developed and implemented for practicalconverters with excellent results.

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ACKNOWLEDGEMENTSThis work was supported in part by the University of Illinois Power Affiliates Program

and in part by a contract with the Sorensen Company, Paxton. IL. The latter also provided avery challenging research problem that contributed to the thesis.

I would like to thank my advisor Professor P. T. Krein for his expert guidance. I ammost thankful to him for also providing me with the opportunity to work independently. Hisinsight into theoretical issues contributed greatly to the success of the thesis. I would like tothank him for the long hours we spent in the laboratory; they were most educational.

I would also like to thank the Ph.D. committee, Professor P. R. Kumar,Professor P. W. Sauer, and Professor B. S. Song for their comments that helped broaden theperspective of the thesis.

I would like to thank Dr. S. Dharanipragada, Dr. G. W. Wright, Dr. S. Ponnapalli, andMr. J. Locker for numerous technical discussions that were very helpful. Thanks are due toMr. A. Kulkarni for maintaining the computer system and for software support. The staff inthe Electrical and Computer Engineering Store were very helpful in providing parts for theexperimental work.

I would like to especially thank my wife Lipika for bearing with me while I waswriting the thesis and for helping out with the numerous figures in the thesis.

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DEDICATIONThis thesis is dedicated to my parents, Mr. Gunakar Midya and M rs. Runu M idya, who

gave me the most important ideas in life.

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TABLE OF CONTENTSCHAPTER PAGE

1 INTRODUCTION 11.1 Control Issues in Power Con verter 21.2 Description of a Switched Power Converter 41.3 Noise Ana lysis of Switching Control and its Mitigation 81.4 Optimal, Lyapunov and Energy Based Control 101.5 Nonm onotonic Track ing 111.6 Conclusions 122 LITERATURE SURVEY 132.1 Small Signal Ana lysis 132.2 Large Signal Ana lysis 142.3 Linear Feedback Control Schemes 152.4 Nonlinear Control Schemes 163 NOISE ANALYSIS O F TH E PW M PROCESS 203.1 Model of the PWM Process 213.2 Description of the Noise Process 233.3 Duty Ratio Analysis 263.4 Interpretation of the Results 283.5 Conclusions 324 NOISE ANALYSIS O F PW M IN A CONTRO L LOO P 334.1 Description of the PWM Control Loop 344.2 Time Domain Noise Analysis 374.3 Frequency Domain Analysis 404.4 Internal Noise Effects 464.5 Conclusions 475 SENSORLESS CURRENT MO DE CONTROL 495.1 Sensorless Curren t Mode Control Theory 505.2 Discontinuous Mo de 545.3 Flux Balance Control of Transformer Coupled Converter 565.4 Experimental Verification 58

5.5 SCM Performance Com parison 656 OPTIMAL CONTROL APPROACHES 736.1 Introduction 736.2 Approximate Optimal Control Methods 786.3 Lyapunov Function Control 82

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6.4 Gradient Approximation to Optimal Control 876.5 Fast Switching Optimal Control 916.6 Experimental Results 966.6 Conclusions 98

7 AUTO TUNING SCH EM ES 1007.1 Tuning Theory 1017.2 Maximum Power Point Tracker 1057.3 Experimental Results 1077.4 Active Filter 1107.5 Conclusions 1128 CON CLUSION S 1138.1 Important Analytical Results 1138.2 Performance Improvements 116

8.3 Future Work 117REFERENCES 119APPENDIX A: CIR CU IT DIAGRAMS 123APPENDIX B: DER IVATION OF NOISE IN PW M 135APPENDIX C: DERIVATION OF NOISE IN CLOSED LOOP PWM ...140

VITA 146

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CHAPTER 1INTRODUCTION

Power electronic systems control energy transfer between a source and a load byswitch action. Switch action m akes these systems discontinuous and fundamentally nonlinear.Conventional control is based on a linear model of a system, which limits the scope andperformance of conventional methods for nonlinear applications. The usual advantages oflinear state feedback control are not obtained because of the inherent nonlinearity of powerconverters. Secondary nonlinear effects such as saturation in magnetics, uncontrolled switchaction in diodes and nonlinear input sources further constrain the scope of linear control inswitching power converters.

This thesis addresses a family of control and operation issues that arise from thenonlinear nature of switching power conversion. The focus is on dc to dc power converterswith a small number of states. Three types of disturbances are considered here: disturbancesat the source, disturbances at the load and disturbances due to noise. Noise, for example,affects converter control in unusual ways that have not been analyzed in previous work.Nonlinear noise analysis of conventional feedback has been performed to evaluate thisimportant aspect of converter performance. Alternative schemes are developed that reducenoise susceptibility, reduce effects of source and load disturbance and improve systemoperation.

The performance criteria for a power converter, as defined by user specifications, donot correspond to linear performance bounds. Thus a linear feedback scheme that satisfies theperformance requirements around a given operating point does not satisfy the same

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requirements around another operating point. Nonlinear controllers can be developed to meetgiven performance requirements over the entire range of operation.1.1 Control Issues in Power Converters

A power co nverter ideally transfers energy between a power source and a load, withoutloss, such that the output is unperturbed by transients in the source or the load. Usually thesource and the load are at different current and voltage levels, and there is a range of valuesover which each is allowed to vary. The power converter output is required to stay within aspecified tolerance level in response to these changes. The ability to handle large high speeddisturbances is one measure of the performance of a power converter.

In the absence of source and load transients, a disturbance due to noise makes thesteady-state operation of a power converter nontrivial. The ability to maintain a constantoperation and output in the presence of noise is another measure of the performance of apower converter.

The sw itching frequency, hence the cycle time, of the switch is usually fixed. The on-time of the switch is a variable and is the principal control input to the system. Duty ratio isdefined as the ratio oftheon time to the switching cycle time. Duty ratio is a number betweenzero and one. The digital signal to the switch is called the switching function. It determinesthe switch action and is obtained by pulse width modulation (PWM) of an analog duty ratiosignal. It varies the on-time keeping the cycle time a constant.

The PWM process is a highly nonlinear process which is susceptible to noise. Theanalysis of this process allows one to predict performance degradation due to noise. It also

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enables us to compare noise susceptibility of different systems and to synthesize controlschemes that are robust in their resistance to noise.

States that are used in feedback in power converters are capacitor voltage and inductorcurrent. Voltage sensing is simp le and relatively robust to noise. Current sensing is often doneby introducing a small resistor in the path of the current. The voltage drop in this resistor isthen sensed. To m aintain high efficiency, this drop is a very small fraction of the voltages inthe system. This makes the current signal prone to noise. To improve noise robustness, controlschemes are presented here that do not require current sensing. Instead current estimators arebuilt that do not introduce noise.

Optimal control that minimizes a cost function is a reasonable target for nonlinearcontrol. For example, some cost functions to be optimized are recovery time following atransient, overshoot in the output or the energy stored, or error during a transient. Owing tothe switching action, a converte r's output does not converge to an equilibrium point but movesaround it. This produces the output ripple, which makes unbounded the traditional costfunction based on the integral of the output error squared. An optimal control approach,modified to take these factors into account, is presented.

Power and energy are fundamentally nonlinear quantities. A power converter can becontrolled by monitoring the energy or the power in the system. This would address thenonlinear aspect of power conversion in the most fundamental way. Based on this idea , controlschemes involving direct computation of a Lyapunov function or of energy stored have beendeveloped.

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A tracking problem focused on power is described as an example of a staticoptimization problem. A nonlinear tuning scheme has been developed that handles suchtracking problems. Examples include output ripple cancellation and m aximum pow er trackingof a solar panel. These nonmonotonic tuning problems originate in nonlinear ties such asinductor saturation and sources with nonlinearities. The nonlinear tuning scheme developedhere is quite general and could be used in a wide variety of static optimization problems.

Chapter 2 is a survey of existing control techniques, both linear and nonlinear, usedin switching power converters. Some of these methods are widely used in commercial powerconverters while others are relatively new but show considerable merit.

The relationship between the methods developed here to those in the survey will beexamined. The similarities and the differences between the methods will be brought out. Inparticular, the relative advantages and disadvantages of these methods will be analyzed, andthe scope for performance improvement will be outlined.1.2 Description of a Switching Power Converter

Physically, a power converter is a network of switches and passive storage elements-inductors, capacitors and transformers. The passive elements store energy in order to smoothpower transfer without loss. Figure 1.1 shows a boost converter, which is a typical exampleof a switching power converter for interfacing a dc source to a dc load. When the switch ison, the inductor stores energy from the source. When the switch is turned off, this energy isfed to the load and the output capacitor through the diode. The energy stored in the capacitorprovides the load power when the switch is on. The output voltage obtained is higher than theinput voltage by a factor of 1/(1-D).

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/ S M ILV.

D >SWITCH /

DIODEC R

Figure 1.1 Boost converter1.2.1 Analytical model of converter

There are idealized models of power converters with ideal passive components andswitches. Reference [1] provides analytical models for typical pow er converters. Controlaction of a power converter must be performed solely through choice of switch position. Twoor more switch configurations are possible for any power converter. In a particular switchconfiguration, the converter is a network of passive components connecting the input sourceto the load. Neglecting component nonlinearity, each configuration can be characterized bya set of linear differential equations. The system behavior in the i* configuration is determinedby the differential equation:

X(r) =AX{t) *fl (1.1)

The overall system can be characterized by a combination of the linear differential equationsand the switching functions Hs. For the ^configuration H; is unity and the other terms arezero.


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X(t)= / / ( 0 ) ( A * ( / ) + B) (1.2)

Often the switching action is set to a predetermined frequency. Then the control inputin each cycle is in effect the duty ratio D. In conventional power converter control analysisand design, duty ratio is taken as a linear input to the system . Reference [2] is a represen tativeof the feedback design procedures that are used in the industry.

Note that in general Equation (1.2) is a time-varying sys tem. In general, this systemdoes not satisfy a Lipsch itz condition, since A(t,X) is neither con tinuous in time nor in state.However, the system is passive and lossy, and the system states (X) are continuous.

rosinL

V.inC R

Voutf from

LiT c 1

v.inR

Vout

CONTINU OUS MODE (1) CONTINUO US MODE (2)

roi%L

1C r

V.inR

v out

DISCONTINUOUS MODE (3)Figure 1.2 Different configurations of boost converter

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Consider again the boost converter. It has two switches: a transistor, which is activelycontrolled, and a diode. Figure 1.2 shows the three possible configurations for this converter.In normal operation when the transistor is off the diode is on, and vice versa, and only twopossible switch configurations occur. If the transistor is on, H, is 1 and H2 is 0. If thetransistor is off, H, is 0 and H2 is 1. The A and B matrices are as follows.

\ =1 0RC0 0

A, =1 1

~RC ~ C-1 0L

a, = s2=0v.L

(1.3)

For the purpose of analysis, the system is normalized such that the desired operatingpoint is [1,1]. The desired operating point before normalization for a boost converter is[V ref,V0Ut2/(RV ref)].The normalization is performed by dividing the state by the reference state.1.2.2 Discontinuous mode of operation

Under certain conditions, usually light loads, the diode current can go to zero and thediode can switch off prior to transistor turn-on. In the boost converter, this third configurationis described by the following:

^ 3 =

1 0RC0 0

* 3 = (1.4)

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The different configurations are numbered from 1 to 3 and the circuits correspondingto each configuration are shown in Figure 1.2 It is possible to take the system fromconfiguration 2 to configuration 1 and back by turning the switch on or off. However, thetransition between configurations 2 and 3 is determined by the load. If a converter operatesin configuration 3, it is said to be in discontinuous mode. This is often avoided because of thecontrol problems it introduces through state reduction. However, the nonlinear controltechniques developed here handle this mode without degradation in performance. Thus, the

nonlinear control techniques offer new alternatives for operation of power converters.1.3 Noise Analysis of Switching Control and Noise Mitigation

PWM is a convenient technique for the generation of switching signals in commercialpower converters. This process modulates any noise in the analog input into a timing error inthe switching signal. Linear small signal analysis does not adequately capture thisphenomenon. Detailed nonlinear analysis of the phenomenon is presented here to elucidate thetime domain and frequency domain effects of this noise modulation.

Various conventional techniques are analyzed and compared for noise susceptibility.Perhaps the most effective way to mitigate noise effects is to feed back only those states thatcan be sensed with low noise. In many dc-dc converters, this translates to sensing voltages butnot currents. Information about the currents can be reconstructed from the voltages in thesystem. Power loss associated with the current sensing is avoided by this method.

Chapter 3 deals with the noise analysis of the PWM process. This is a time domainanalysis of the PWM process is presented. The statistics of the duty ratio variation areexplored. An unusual result is uncovered in the analysis: it is found that the average duty ratio

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is modified in the presence ofnoiseeven when the noise has an average of zero. The variationoftheduty ratio is determined by the rms values of the noise and its derivative. The nonlinearnature of the noise modulation is also reflected in the statistics of the duty ratio. Theprobability density and distribution functions of the duty ratio are obtained in closed form.These theoretical results match Monte Carlo simulation data and experiments in whichcommercial PWM ICs were subjected to a noise generator.

Chapter 4 deals with noise analysis in closed-loop PWM systems. Here conventionaltechniques such as voltage m ode control, peak current mode control and current mode controlwith ramp are tested for noise susceptibility. The standard deviation oftheduty ratio in noisyconditions is obtained in terms of ramp slopes and noise power. Frequency domain analysisof this process predicts that the noise is aliased to subsynchronous frequencies (frequencieslower than the switching frequency). This quantifies for the first time a commonly observedphenomenon in power electronics, namely, that a noisy converter has subsynchronousoscillations. The theoretical results are compared to Monte Carlo simulation data and toexperimental results in which a PWM loop is subjected to random noise from a noisegenerator. There is very good agreement both in the time domain and frequency domainresults.

Chapter 5 describes a sensorless approach to current mode feedback control. Thissensorless current m ode (SCM ) control method minimizes noise problems by avoiding currentfeedback. The current is estimated from the intermediate voltages in the system. The methodretains most of the advantages of conventional current feedback approaches but has superiornoise rejection properties. For low voltage applications where current sensing is harder this

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method is particularly useful. A family of 2Voutput converters has been built using the SCMcontrol method.

In certain transformer coupled converters, there is a possibility of transformer failuredue to dc offset which sa turates the transformer core. The flux in the transformer has positiveand negative excursions, and the transformer flux must be balanced to keep the excursionscentered about zero. Magnetic flux in a transformer is not readily measurable, andconventional control estimates the flux by the transformer current. SCM control estimates theflux and enforces flux balance using the transformer voltages. A flux balance approach ispresented for push-pull forward converters.1.4 Optimal, Lyapunov and Energy Based Control

In converters where all states, including currents, are measurable, the dynam ic optimalcontrol problem of disturbance can be approached. Here the integral of the output errorsquared is defined to be the cost function. Minimizing this cost function is useful in convertersthat must respond to rapid load and source transients. The finite horizon optimal controlproblem and the infinite horizon optimal control problem are considered. The control lawobtained is complex and difficult to implement directly.

Boundary control for power electronics [3] is a large signal control method. Inboundary control, a switching boundary is defined, which is a surface in state space. Theswitching boundary divides the state-space into regions. The switch action is determined bycomparing the system state to the boundary. A switching boundary is denied whichcorresponds to the optimal control action. This transforms the optimal control law to aboundary control problem and simplifies the implementation.

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Instead of minimizing an integral of the output error, it is simpler to explicitlyminimize a Lyapunov function. This would no t necessarily minimize the integral oftheoutputerror. However, this ensures large signal stability and a w ell-defined response to large signaltransients. Control based on the quantity of stored energy is another way to address theproblem of optimal power transfer in a power converter. These techniques exploit thefundamental nature of power converters as opposed to attempting to fit a linear feedbackscheme to a nonlinear circuit.

Chapter 6 approaches the optimal control problem for switching power converters.Switching boundaries are developed that correspond to the finite horizon and the infinitehorizon optimal control problem. Implementation of some simple optimal control problemshas been attempted. The issue of system complexity and other limitations for optimal controlare discussed.1.5 Nonmonotonic Tracking

In the steady-state optimization problem, the output being optimized could be anonmonotonic function of the controlled variable. Often the output is dependent on thetemperature, the load impedance, and other unknown and variable quantities. Thus, it is veryuseful to have an automatic tuning scheme that would take the system to the desired operatingpoint using only input and output information.

A generalized tuning scheme has been developed that uses the correlation betweenchanges in the input and corresponding changes in the output to tune the operating point.When the system reaches the desired operating point, the correlation goes to zero and thesystem converges. This corresponds to the point at which the derivative of the output with

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respect to the input is zero. This tuning scheme is appropriate for any tuning problem whichhas a single maximum or minimum. A variety of tuning problems in power electronics fallinto this category. A tuning scheme based on correlation usually requires an excitation input.The switching action perturbs all the states and provides this excitation. Thus, this tuningscheme is appropriate for switching power applications.

Chapter 7 introduces the tuning theory and establishes criteria for stability andconvergence. Two tuning problems, adaptive ripple cancellation and maximum power pointtracking, are implemented using this method.

Adaptive ripple cancellation [4] uses a feedforward scheme to estimate the currentripple shape and the tuning scheme to determine the amplitude. This method is used toestablish a new form of an efficient active filter for high performance applications.1.6 Conclusions

This thesis analyzes a family of steady-state and dynamic nonlinear operation andcontrol problems of dc to dc switching converters and develops appropriate nonlinear controltechniques. A family of converters has been designed to test these techniques.

Chapter 9 sum marizes the important theoretical results, the performance improvementsand areas for future work. Novel analytical results have been obtained in the area of noiseanalysis of PWM, feedforward control and large signal optimal control. Significantimprovements have been attained in the areas of noise immunity, transient response, ripplereduction and automatic tracking.

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CHAPTER 2LITERATURE SURVEY

Switched dc-dc power converters are nonlinear, discontinuous and time-varyingsystems. The only control input is the switch duty ratio. Switch action is a bounded anddiscrete input. This makes the analysis, simulation and control of switched power convertersa complex and challenging problem. Much of the prior analysis of power converters is smallsignal analysis, which is obtained by linearizing the nonlinear system.

The design of the feedback structure for switched power converters is often limited tolinear feedback. This is an artificial constraint and often a restricting one. Some nonlinearcontrol schemes have been developed that exhibit significant performance improvements overthose for linear control schemes. In addition there are nonlinear tracking or optimizationproblems that are beyond the scope of linear feedback control.2.1 Small Signal Analysis

The circuit level model of switched power converters, including switch characteristics,is complex and offers limited scope for analysis. The model is simplified by assuming idealswitching, and this analysis provides insight into converter action in open loop [1]. There areexcellent small signal analysis techniques for converter action that include the action of thefeedback network [5,6,7]. Reference [5] combines sampled data modelling with a pole-zeromodel of the control loop. It provides a very accurate small signal model of current modecontrol. Reference [6] provides an accurate small signal model for average current modecontrol. Reference [7] analyzes existing m odels for switched power converters, including bothlinear and nonlinear m odels.

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The interaction between the converter and the feedback network introducesnonlinearities in addition to the nonlinearities due to switch action. Furthermore, thedisturbances are often large. Thus, the accuracy and validity of small signal analysis arelimited.2.2 Large Signal Analysis

A large signal analysis technique that provides the most intuitive understanding ofconverter action is geometric control. In this technique the converter action is observed asthe movement of the system state in the state space. The control law is expressed as aboundary in state space. The switch is toggled when the system state crosses the boundary.Reference [8] presents geometric control and does a large signal analysis of linear andnonlinear control laws in state space. This provides a pictorial representation of theoperation of a switched power converter.

Novel control laws have been developed as boundaries in state space, and eventraditional control laws can be interpreted using the idea of boundary control [8]. Theoptimal control approaches developed in this thesis have been interpreted in terms ofboundary control.

Averaging theory has been developed as an alternative to small signal approximation[9]. Averaging analysis allows one to predict some converter characteristics that are lost insmall signal analysis. The added complexity of averaging analysis is justified because o f thisfeature. Reference [9] develops a framework for applying averaging theory to the analysisof switched power converters and shows that this analysis recovers an importantphenomenon such as output ripple that is not captured by linear analysis.

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PWM (pulse width modulation) is a process used to convert from a duty ratio signalto the switch function. The characteristics of its noise properties and noise interaction withthe modulation process are well-understood from a communication point of view. Reference[10] provides a detailed analysis of band limited Gaussian noise. References [11,12] presentdetailed analyses of the noise behavior of the PWM process and other pulse modulationschemes. The analyses are linear and the quantity of interest are signal to noise ratio. Thereis no comparable existing analysis from the point of view of power electronics in which theduty ratio and switching function are the noise issues rather than the transmission channelnoise.2.3 Linear Feedback Control Schemes

Linear feedback considers duty ratio to be a continuous and unbounded input to alinear system. The duty ratio signal is obtained by a linear combination of the output and thestates of the system. Perhaps the simplest of feedback schemes is direct output feedback. Forvoltage output converters, this is called voltage mode control and is used extensively. Currentmode control is a relatively recent control technique that has greatly improved transientperformance [13].

A ramp signal is introduced in current mode control to maintain a fixed switchingfrequency and avoid subharmonic instability. The design of feedback networks for currentmode control has been optimized for performance. Current mode control, however, is proneto noise and difficult to implement for low voltage converters. To improve noise performance,average current mode control has been introduced [1]. Further, some of the noise associatedwith current sensing has been avoided using alternative techniques [14]. System complexity

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has been increased for reduced noise in the system. In this thesis, sensorless current modecontrol has been developed that shares certain advantages of current mode control but hasimproved noise performance and dynamic range and does not require current sensing.2.4 Nonlinear Control Schemes

Stability criteria and performance requirements for switched power converters for m ostelectronic loads are different for sm all signal and large signal disturbances. These requiremen tsdo not readily translate to linear criteria, as does pole placement. This provides an incentive

to introduce nonlinear control schemes.Nonlinear control techniques have been developed that address the nonlinear control

problem of switched power conversion. Some of these techniques attack the problem usingnonlinear feedforward and feedback techniques [15,16,17]. Reference [16] implements anonlinear control law by sensing the inductor current and integrating it in a capacitor. Thecharge in the capacitor is used to determine the switch position. This provides a simple controllaw similar to current mode control. Reference [17] analyzes a nonlinear difference equationand uses the results to construct control laws for dc to dc converters. The control laws areshown to have good small signal behavior in theory and simulations.

Others attempt the nonlinear control problem using formal Lyapunov theory. Reference[18] develops a formal Lyapunov function based on the energy in a converter and constructscontrol laws that minimize the Lyapunov function. The optimal control problem has also beenaddressed for some converters [19]. Reference [19] computes an error based on linear statefeedback in addition to the output feedback . This scheme has been simulated for a buck

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converter and was shown to improve performance. These nonlinear techniques have muchpotential. However, m uch of commercial power converter control relies on linear feedback [2].

One important class of nonlinear optimization problems attempts to maximize orminimize a nonmonotonic function of the states. These problems require nonlinear controlschemes. An autotuning technique that has been used traditionally for online tuning is thecross-correlation technique. It has been implemented in the digital domain for slow tuning ofcontrollers [20]. Reference [20] considers a general linear system and tunes the gain of thePID feedback for optimum performance. The scheme imposes a pseudo-random binarysequence no ise signal to probe the system. The algorithm is implemented in the digital domainand is computationaly intensive.

In the analog domain, tuning of a scalar quantity is feasible. This has been attemptedfor the maximum power transfer problem [21]. In reference [21]. a simple analog tuningscheme has been developed and implemented. A small sinusoidal signal is injected into theduty ratio and the response is used to tune the system. The scheme maximizes the chargingcurrent to the battery and thus achieves maximum power point tracking.

It is difficult to guarantee the stability of such a nonlinear control algorithm . Even so,stability has been achieved by constructing a very slow tuning process whose dynamics do notinteract with the other dynamics of the system. These systems track the maximum powertransfer problem with a delay of about a second. A fast tracking scheme has been developedin Chapter 7 that tracks with a delay of only 2 milliseconds.

References [22,23] describe a high efficiency inverter being driven from a solar arraythrough a maximum power tracking inverter. The maximum power point is obtained by

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introducing a small step and computing the power digitally. Response times on the order ofa few seconds are obtained. Reference [24] describes a system that computes the maximumpower point by calculating a small signal model ofthesystem. The maximum power point isagain obtained by using the response to a small perturbance to the system.

Another control problem of considerable interest is output ripple cancellation whichis implemented by injection of a cancellation current, which is an inverse oftheripple current.There are a number of active filters that cancel the harmonics introduced into the powersystem by inverters and rectifiers [25,26,27,28]. The cancellation current is obtained usingfeedback, feedforward, or a com bination of the two. These controllers have to react to thepower line frequency, and this allows for the computation and cancellation of individualharmonics [28].

By contrast, the calculation and cancellation ofahigh frequency ripple require a circuitwith small delay in computation. Reference [29] describes a converter in which coupledinductors are used to obtain and cancel current waveforms. The cancellation is achievedalmost perfectly for a high performance converter. Since the cancellation is based on coupledinductors, the precision requirements on the magnetics are extreme.

Reference [30] describes a simple feedback active filter that can cancel ripple for avariety of power con verters. It obtains the cancellation current signal by monitoring the output.The scope of this filter is limited because the ripple at the output of a high performanceconverter is hard to monitor accurately. As the cancellation progresses, the signal beingmonitored disappears, causing signal to noise problems.

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Reference [31] describes an active filter that cancels line harmonics at the input of aninverter. Since it has to respond to the line frequency and its higher harmon ics, it is very slowand is implemented digitally. The active filter developed here suppresses ripple frequencieson the order of 100 kHz. The gain of the cancellation current is optimized by minimizing thecross-correlation between the cancellation current and the output ripple. There is a significantreduction in hardware by implementing this in the analog domain.

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CHAPTER 3NOISE ANALYSIS OF THE PWM PROCESS

In steady state, disturbances due to noise perturb converter operation and distort theoutput. The converter output is a function of the input voltage (or current) and the duty ratio.Noise can affect the converter output through either of these two variables. The propagationof noise from the input to the output has been studied extensively . Quantitatively, the effectis represented by audio susceptibility, and converters are designed to have low or zero audiosusceptibility. Audio suscep tibility is a small signal analysis that does not consider the dutyratio variation and nonlinear ty of the PWM process. Therefore, it has a limited ability topredict the noise performance of a power converter under operating conditions.

The variation in duty ratio due to noise is very significant to the converter output. Thisis a common phenomenon that has not been quantified in previous work. The variation in dutyratio often degrades performance, and it is important to be able to quantify this phenomenon.Understanding of this process would help in the comparison of noise performance of feedbackschemes and the synthesis of robust schemes.

The PWM process uses a comparison between an analog signal and a ramp time baseto generate a switching function. This process has been analyzed from noise performancecriteria in the communication literature for large SNR (signal to noise ratio) [32] as well asfor small SNR [33,34]. This analysis of PWM assumes uniform sampling, which is anoversimplification in the context of many power converters. In a power converter, using aconventional PWM IC, the sample time is the same as the switch on-time. Thus variation of

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the sample time is of special interest. A rigorous noise analysis of PWM in the context ofpower electronics is required.

In previous work, noise analysis of PWM-based dc-dc converters almost alwaysaddressed small-signal performance measures such as audio susceptibility [35] and control-to-output response [36]. These give an indication oftherobustness of a given control circuit butdo no t give any insight into the important nonlinear noise effects. Con sider, for instance, thataudio susceptibility and other measures do not show dramatic distinctions between voltage-mode and current-mode control loops. This contrasts with qualitative observations made bymany designers, who report clear distinctions in the noise behavior of different controlmethods [37,38]. Noise effects would be expected to depend on switching frequency, thenominal duty ratio, and control loop properties; this is not captured in small-signal analysis.

In this chapter a nonlinear model of the PWM process is constructed and the PWMprocess is analyzed. Bandlimited G aussian noise is imposed and the variation in duty ratiois estimated. The cumulative probability distribution function and the probability densityfunction are obtained analytically.3.1 Model of the PW M Process

A schematic of the PWM process, based on a commercial converter control IC, isillustrated in Figure 3 .1. The analog input is compared to an internally generated ramp. Alatch prevents multiple transitions per cycle. The standard practice is to use the comparatorto reset the latch and a clock signal to set the latch. The set and reset operations could beexchanged, and the results obtained here apply with minor modifications.

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A VRA M PA N A L O GSIGNAL

+ v COMPARATOR

CLOCK

LATCHR

SQ

SWITCHINGFUNCTION

i

Figure 3.1 Schematic of PWM ICThe switch is set by a synchronous clock signal and reset when the analog input signal

crosses the ram p signal. The presence of a noise pulse affects the reset timing and, hence, theduty ratio. Figure 3.2 shows the switching function and associated waveforms obtained withand without noise in the analog input.

R A M PA N A L O G S I G N A L

1

SWITCH0

INGF U N C T I O N

:: CLOCK

-r>rZ-vJAi

0

I

[

R A M PA N A L O G S I G N A L

11 1 o

S W I T C H I N GF U N C T I O N

:: CLOCK

1I 0

(a) Waveforms withou t noise (b) Waveforms with noise in analog inputFigure 3.2 PWM waveforms

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3.2 Description of the Noise ProcessLet us assume that noise interferes with the analog signal at the PWM input. As a

reasonable model, let us further assume that the noise has a Gaussian probability densityfunction and limited bandwidth. The bandwidth limitation is realistic given that low passfiltering is normally applied explicitly or implicitly by the comparator speed limitation.Further, the noise process is assumed to be stationary. This means that the statistics of thenoise process (mean, standard deviation, etc.) are not a function of time. The noise and itsderivative are assumed to have zero mean and a defined standard deviation. Further, the noiseand its derivative are assumed to be uncorrected.3.2.1 DefinitionsRandom variables:

Duty ratio = XNoise = YNoise derivative = ZDuty ratio in absence of noise = D 0Noise standard deviation = c YNoise derivative standard deviations a zRamp amplitude = V^,Switch cycle time period = T sThe ramp is linear and has zero reset time. In the absence of noise, the duty ratio is

a function of the signal and ramp level. After noise is added to the process, the duty ratio isstochastic, and it is possible to find only its statistical nature. Noise is added to the analog

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xx)=__L_f2tr (3.1)Tv/2T

Figure 3.4 is a plot of cumulative and probability distribution functions for the standardGaussian distribution (mean=0, standard deviation =1). Note that pdf is the density functionand PDF is the distribution function.

0 . 8 -0 . 6 -0 . 4 -

0 . 2 -0 - 4 - 3

' p d f' P D F '

- 2 - 1 0Figure 3.4 Gaussian distribution: cumulative and density functions

Let us define two separate Gaussian distributions. The noise variable is denoted Y,while its derivative is denoted Z. The corresponding pdf s for Y and Z are as follows.

-r :p,m= 1 .TV (3.2)o-/2rT

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-z 1p2(Z)= l e^ (3.3)

3.3 Duty R atio AnalysisThe duty ratio D is a random variable. Consider the probability that D is between X

and (X+AX). The probability of this event can be expressed as [P(X+AX)-P(X)]. This eventoccurs if the analog signal is larger than the ramp in the interval (0 to X) and the analogsignal is smaller than the ramp voltage in the interval (X to X+AX).

For the analysis of duty ratio, we have to introduce conditional probability. Given twoevents A and B, the conditional probability of A given B is defined as [P(AIB)], which isdefined as the ratio of the p robability of both A and B occurring relative to the probability ofB occurring. This definition is valid for discrete probability as well as for continuousprobability functions.

P(Af]B)=P(B)P(A\B) OR P(A\B)=P(Af]B) (3.4)P(B)Define A to be the event in which the analog signal is less than the ramp voltage in

the interval (X to X+AX). Define the eventBin which the analog signal has not become lessthan the ramp in the interval (0 to X). The switch is reset in the interval (X, AX) if and onlyif both events occur. Thus P (A nB ) is by definition [P(X+AX)-P(X )]. Let Q denote the valueof the conditional probability P(AIB) of the comparator toggling in the interval (X to X+AX)given that there has not been a transition between 0 and X.

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P(X^6X) -P(X)=(1-P(X))Q (3.5)

The event Q is the event that the switch is reset in the interval (X to X+AX). Since theswitch is reset only once in a cycle, the switch has not been reset in the interval (0 to X ). Thisconstrains the noise (Y) to certain values only. Starting from these values of noise (Y), thereare further constraints on the noise (Y) and its derivative (Z), such that the switch is rest inthe interval (X to X+AX).

The value of Q is obtained by calculating the probability of all noise values (Y) andnoise derivative values (Z) that will result in a switch transition in the interval (X to X+AX).This is an area integral in no ise and its derivative in the Y-Z plane. It is given by the integralin the Y-Z plane of the density functions p,(Y) and p2(Z) over values of Y and Z that resultin a transition in the interval(X to X+AX).

The noise (Y) determines the difference between the ramp and the sum of the signaland the noise. The noise derivative (Z) determines the rate at which the ramp and the sum ofthe signal and the noise approach each other. The area integral computes the probability ofthe switch being reset in the interval(X to X+AX).

g - ' W * - M, V AX (3.6)

The value of Q is substituted in Equation (3.6) and the differential equation obtainedis solved analytically. It turns out that results can be expressed in closed form as follows. Thedetailed derivation is provided in Appendix B.

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magnetics and prevent successful operation. In a system with an outer integral loop, a dcoffset in the outer loop would compensate for this effect.3.4.2 Comparison with simulation and experimental data

Figures 3.5(a) and 3.5(b) show the pdf of the duty ratio in the presence of noise withdq=0.5 . The analytical formula is compared to Monte Carlo simulation. Figure 3.5(a)corresponds to small Gz (standard deviation of noise derivative) and consequently, the noiseprocess is fairly linear. Figure 3.5(b) corresponds to large o~z, and the mean duty ratio hasshifted significantly. The analytical formulae predict this shift accurately.

1 -

'THEORY' SIMULATION' o

0 'eoof l f l 48oi o 'oo oe cr a

THEORY S I M U L A T I O N '

J ^ o o o o o o o o o oo o e0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8

(a) Small Noise Bandwidth (b) Large Noise BandwidthX-axis: Duty ratio, Y-axis: pdf of duty ratioFigure 3.5 Probability density function: theoretical and Monte Carlo simulation results

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Figures 3.6(a) and 3.6(b) show the PD F oftheduty ratio in the presence of noise. Thetheoretical results are compared to the experimental values. The nominal duty ratio in theabsence of noise is 50% . Figure 3.6(a) corresponds to a small noise d erivative (M p l) ;consequently the mean duty ratio is unchanged due to the presence of noise. Figure 3.6(b)corresponds to the large noise derivative ( M , l ) and the mean duty ratio has shiftedsignificantly. The analytical formula predicts this shift accurately. Appendix A describes thecircuit used to obtain the experimental data used.

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In current mode control, the inductor current is used for feedback. The current has awaveform like a ramp with one slope when the switch is on and another when the switch isoff. The switch is reset when the current becomes more than the reference current. Currentmode control can be divided into two categories depending on whether an external ramp isused to generate the duty ratio. In peak current mode control, the reference current is a dcquantity. In current mode control with an external ramp, the reference current is a sawtooth.Figure 4.2 shows the waveforms in current mode control with an external ramp.

Figure 4.2 Current mode control waveforms4.1.1 Noise propagation effect

If noise alters the switching time, the duty ratio perturbation will alter the startingsignal value during subsequent cycles as control action corrects the operation. This effectpropagates into subsequent cycles. The basic process, based on an idealized current-modecontrol ramp, is illustrated in Figure 4.3.In the figure, a brief noise spike changes the timingof the comparator trip point. The duty ratio is disturbed, but in this particular case recoversto its nominal value after about three cycles.

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In voltage mode feedback the output voltage is used for feedback. The output voltageis approximately a constant dc quantity because of the action of the output filter. By settingthe current ramp rate to zero, the current mode control results can represent the results forvoltage mode control.

NO NOISERAMP ANALOG SIGNAL WtTHNOISE

I , NO NOISE | 1 | 1m I i i ,SWITCHINGFUNCTION, ' 1 WITH NOISE | 1 | 1

I o I I I IFigure 4.3 Noise propagation waveforms

4.1.2 DefinitionsRamp slope = SRRamp amplitude = VRPRate of change of current feedback signal with switch on = S NRate of change of current feedback signal with switch off = SFNominal duty ratio = D 0Noise value at the trip point of cycle N = E NStandard deviation of EN= aEStandard deviation of injected noise = o \Difference between duty ratio in cycle N and nominal duty ratio = FN

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Standard deviation of FN= o>Standard deviation of duty ratio = CT DSwitching frequency = F sSwitching cycle time period = T s4.2 Time Domain Noise Analysis

The PWM ramp starts from a value of zero at the beginning of each cycle and riseslinearly until it reaches the end of the cycle, when it is reset to zero. The current signal is thenegative of the inductor current and ramps down when the switch is on and ramps up whenthe switch is off. The switch turns on at the beginning of each cycle and is reset when thecurrent equals the ramp value. The expression for the current at the switch transition is givenby the following:

/ ( ( A N % ) = % S * ( 4 . 1 )

Taking the difference between the equations at cycle N and cycle N +l, the following recursiveequation is obtained.

/ W V ^ M F + o ^ - f f ) (4-2)Noise is introduced into the system and the expression for the current is modified.

W+DJTJ+EN-DJfK (4.3)

The duty ratio is obtained in recursive form as follows:

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the case of peak current mode with no ramp, the quantity 'K* is a function of the duty ratio.The result for peak current mode control is function of the average duty ratio is given by:

~ 0 = 2 d - D J% + S J \ 1-2A, (4.7)

Figure 4.4 shows the standard deviation of the duty ratio as ob tained from Equations (4.6) and(4.7) compared with Monte Carlo simulation results. Note that for voltage mode control andcurrent mode control with optimal ramp the results are not a function of the nominal dutyratio. For peak current mode control the performance degrades as the nominal duty ratioincreases, and the system becomes unstable at D 0=l/2.

X axis: Nominal duty ratioYaxis:Normalized aD -

P E C U R M X T KO O E ,

/

u o

V OL TA GE HOOe

T H E O R Y -S I M U L A T I O N '

CURReKT HOOE WITH O PTI MA L (UUCP

0 0 .2 0 4 0 6 0 8 1

Figure 4.4 Standard deviation of duty ratio: theoretical and experimental results4.2.2 Effect of noise on ripple

The r ipple value represents variation oftheload voltage in steady state. In the absenceof any significant noise effects, the ripple is obtained from the switching waveform and the

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output filter transfer function. The PWM process aliases high frequency components of noisein the control signal to subsynchronous frequencies. The additional ripple due to noise can beobtained from the noise spectrum of the switching function and the output filter transferfunction.

The output filter is designed for rejecting the switching components and higherharmonics. It normally does not provide significant attenuation at subharmonics of theswitching frequencies. Thus a small duty ratio variation, due to noise , results in adisproportionate increase in the ripple.4.3 Frequency Domain Analysis

The energy spectral density of a random process can be obtained from theautocorrelation function of the process. From Equation (4.5), we know that the duty ratiovariation in any cycle is correlated to the duty ratio variation of previous cycles. Anapproximate expression can be obtained for the spectrum by treating the duty ratio as adiscrete variable and by obtaining the autocorrelation in the discrete time domain and thentaking its Fourier transform to obtain the energy spectral density in the frequency domain.While these results are not exact, they provide useful approximate closed-form expressions forthe spectrum.

The autocorrelation function and the power spectral density are a Fourier transformpair. The autocorrelation function is the expected value of the product of the signal with adelayed version ofitself. The function is defined as follows.

Rx(x)= (4.8)

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The autocorrelation of the error (FN) at the origin is simply the power of the signal:

RJ0)=-1 5 : (4.9)

For a time difference of 'm' cycles the correlation can be obtained from Equation (4.4).

R^mT^K^lLLoy 2 _ J _ (4.10)^1 r/( ):

The power spectral density is obtained by taking a Fourier transform of the discrete timeautocorrelation function. The detailed mathematical derivation is provided in Appendix C.

2av2(l-cos0)SJf) = Y - where0 =2rc//F, (4.11)(l+K2-2KcosQ)Ts(S R+SN)2

The maximum of this spectrum occurs at halftheswitching frequency, which corresponds tothe first subharmonic. The energy spectrum at this frequency is

4 a 2SJFS IT) = _ _ I (4.12)( + # % + . ? /

Figures 4.5(a), 4.5(b) and 4.5(c) show the noise spectrum oftheswitching function fordifferent values of K. The simulation results are compared to the analytical result given byEquation (4.11). The curve for K=0 corresponds to current mode control with optimal ramp.

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The Figures 4.6(a) and 4.6(b) show the noise spectrum obtained experimentally frompeak current mode control with nominal duty ratios of 0.25 and 0.45, respectively. Thespectrum is measured in the subsynchronous frequency range. The switching frequency is 1.0kHz and the spectrum is measured from 250 Hz to 750 Hz.

ii " T M F O P V

-20-40-60-80

(a) D 0=0.25 - 2 5 0 .5 0 .75X axis: f/Fs,Y axis:Noise spectrum(dB)0

-20-40-60- 8 0

(b) D 0=0.45X axis: f/Fs. 0 .25 0 . 5 0 .75Y axis:Noise spectrum(dB)Figu re 4 .6 Noise spec t rum for peak curr ent m od e: theory and exper ime nta l da ta

1

1 '1-

Ii i H i 1_ | T ' i I 1 i'

-j*tpr Vum

ii__ J

T XEQEY:

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Figures 4.7(a) and 4.7(b) show the spectrum for current mode control with optimalramp.Thenominal duty ratiosare 0.25 and0.75, respectively. Current mo de control withoutinherent noise is realized by using a current estimator, then Gaussian noise with knownstatistics isadded.

-20

-40

-60

-80

(a)D 0=0.25 -2 5 -5 0-75X axis:f/Fs,Y axis:Noise spectrum(dB)o

-20

-40

-60

-80

(b) D 0 =0 .7 5 0. 25 0. 3 0 .3 5 0.4 0 . 45 0. 5 0 . 55 0.6 0 . 65 0. 7 0 75X axis:f/Fs,Y axis:Noise spectrum(dB)Figure 4.7 Noise spectrum current mode(optimaI ramp) :

theory and experimental data

44

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1,

i

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i iOiiy-l

-.iaijl

111

nim u i>\wiimjwp

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nMJili.iJii I L . Ifmmm-

iAktlUljfofflffttutet(pl XdeORY.'

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Figures 4.8(a) and 4.8(b) show the noise spectrum obtained for voltage mode control.The nominal duty ratios are 0.25 and 0.75, respectively. In all cases, the results are comparedto theoretical values obtained from Equation (4.11), and agreement is good. Circuit diagramsof the experimental setup are shown in Appendix A.

o- 2 0

- 4 0

- 6 0

- 8 0

1H rfkttkrtftilk

1 _ . . . .TMFnRV - .

;^;'^% i^y^j^^^trki ^fU^iVii.^.'iitfcr ff|r"r J1-r i F | f f f / , , ' [ l | f ' r iV l v ' i n , r i7 ' ' " i i

41

ii l : 'I i 1 .ii

(a)D 0=0.25 - 2 5X axis: f/Fs,Y axis:Noise spectrum(dB)0 . 5

0

- 2 0

- 4 0

- 6 0

- 8 0

it i kkrw_LI

fflOTW inMli

4iVWnHrIffr ILlULir n1 ""'1 "i i ]11i1

jiltJJL/W Wf

r-

\\kf\ifrflf

' T -

W r W ,W'1

-

i1

.

' F . O R Y '

tihu LkAh1 i "irmri

-(b) D0=0.75X axis: f/Fs, 0.25Y axis:Noise spectrum(dB) 0 . 5 0 . 7 5

Figure 4.8 Noise spectrum for voltage mode: theory and experimental data

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4.4 Internal Noise EffectsSwitching action in the converter is itself a significant source of noise. This internal

noise is distinct from thermal noise, which is nondeterministic, and external noise, whichoriginates outside the converter. Internal noise affects the control signal by electromagneticcoupling between the switched current and the control circuit. In addition to this couplednoise, there is also a spike in the current signal ( defined If ) due to stray capacitance in theinductors [37]. This noise is present only at the switch turn on-time. However, it could belarge enough to cause the switch to reset. The following analysis shows the noise margin ofcurrent mode control and the range of possible duty ratios.

Noise (nTs)= / ^ Z I(nTs) (4.13)

'Z ' is the stray coupling coefficient from the power circuit to the control circuit. There is nofalse trigger due to commutation noise only if the commutation noise is less than thedifference between the signal and the ramp. For a nominal duty ratio of D 0 , ramp slope ofSR, and current ramp rate of SN, this difference is D0(SR+SN)TS. Thus the condition forpreventing a false trigger at the beginning of the cycle is given by

The current at the beginning of the cycle has been approximated by the nominal dccurrent 1^. Iftheduty ratio required is less than this m inimum, either in steady state or duringa transient, the duty ratio will be set to zero. Ifthenominal duty ratio of the converter is lessthan this value, the converter will not operate properly. This result also illustrates that current

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mode control has an internal noise problem at large current levels. This is also compoundedby dynamic range problems as observed in [37].

This analysis and reference [37] point to the fact that current mode control, inparticular, peak current mode control, is not an appropriate choice if the current variation issmall compared to the dc current level. To reduce loss in the inductor, small current variationis considered to be good design practice.4.5 Conclusions

To obtain good response in steady state, PWM conversion is embedded in an outerintegral loop. This modifies the noise properties of the overall system. The noise enhancement/suppression properties of this outer loop were examined. In particular, voltage mode, peakcurrent mode, and current mode with optimal ramp were considered. For the sam e noise level,the three systems were compared at various nominal duty ratios. It was observed that thevoltage mode has the least noise enhancement and the peak current mode has the most noiseenhancement. The results confirm qualitative observations that current mode controls are"noisier" than voltage mode controls. Analytical formulae for these schemes have beenobtained and compared with simulation results.

In the absence of noise, the spectrum of a switching function is identically zero belowthe switching frequency. The PWM process aliases any noise in the system to subharmonicfrequencies. Analytical expressions have been obtained for these subharmonics. It is observedthat the spectra for voltage mode control, peak current mode control and current mode controlwith optimal ramp are quite different both in amplitude and distribution. There is goodcorrelation between the analytical results, simulation and experimental data.

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The variation in duty ratio leads to an increase in output ripple; this phenomenon wasalso explored. The spectrum of a switching function in the presence of noise was obtained forvoltage and current mode controls. By multiplying this spectrum by the transfer function ofthe output filter, the spectrum at the load is easily obtained.

Apart from random Gaussian noise, there is commutation noise in a power converter.A brief analysis was done to explore this issue. This analysis showed that commutation noiseis a critical issue, especially in current mode control. This confirms various observations inthe literature.

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CHAPTERSENSORLESS CURRENT MODE CONTROL

It is desirable that the control schemes of power converters be robust to noise, havegood dynamic performance and be simple to design. A control scheme has further merit ifit requires minimal sensing of states and can be implemented using commercial PWM ICs.With these objectives in mind sensorless current mode (SCM) control was developed as partof this project.

Current mode control is based on sensing of an inductor current with the objective ofmaintaining it at a desired level. This scheme has good dynamic response to disturbances atthe input and output oftheconverter. It relies on current sensing, which can be prone to noiseand can be hard to implement for low voltage power converters.

It is possible to obtain information about the currents in the converter using the voltageinformation. For exam ple, the current through an inductor is the integral of the voltage acrossit. In principle, it is possible to reconstruct the current entirely from the voltage. In practice,it is simple to construct the ac part of the current from the inductor voltage. M ore importantly,it is possible to formulate control laws in terms of the internal voltages in the converter. It isnot necessary to explicitly reconstruct the current signal and then implement current modecontrol. Sensorless current mode control is a scheme to control the output voltage and currentin a power converter using the intermediate voltages in the converter.

In periodic steady-state operation, the average voltage across an inductor m ust be zero.In a buck converter, the SCM control law sets the voltage at the switch side of the inductorto a desired reference value. This indirectly sets the output voltage to the reference voltage.

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SWITCHINGFUNCTION

V L I JTRRTL V L2

L 2

L I

Ho

/ -

CLO CK

S 0

COMPARATOR LATCH

LIV R

Figure 5.2 Schematic of SCM with PWMThe duty ratio at cycle N is defined to be D N and the cycle time is defined to be T.

The magnitude of the ramp is equal to V%p. It is assumed to be zero at the beginning of thecycle and equal to V RPat the end of the cycle. The ramp slopeSR=V%p/T. At the reset time,the signals into the com parator are equal. Thus V, is

y / ( w + D , ) n = % r (5.2)Define the derivative of V, to be SNwhen the switch is on and SFwhen the switch is

off. A recu rsive equation can be obtained for the duty ratio by taking the difference ofEquation (5.2) at cycle N from cycle N+l.

VNDm Vji DTi l DS JD DT 5.3)Figure 5.3 shows the waveforms of the voltages and the switching functions. A differenceequation for the duty ratio can be obtained as

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SR~ SN ^ - ^(5.4)

The pole of this difference equation is at (SR-SF)/(SR-SN).The slope SNis always negative andSp is always positive. The minimum ramp slope for a stable system is therefore (SN+S F)/2.A good choice is to select SR= SF. This sets the pole of the difference equation to zero. Apole at the origin of the Z-plane corresponds to the most stable system . This is analogous tothe optimal ramp rate in current mode control.

111

Slope =S y ^

\ v v ^ % o p e . ^

/ S l o p e = ^ON

| OFFSWITCH POSITIOf

RAMP

ANALOG SIGNAL V,

4

Figure 5.3 Waveforms in SCMFor a stable system, the steady-state duty ratio is independent of the ramp slope. It is

obtained by taking the limit N tending to infinity in Equation (5.4).

a,=sFs (5.5)

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The minimum ramp slope for stability can be interpreted in terms of the steady stateduty ratio using Equation (5.5). For steady-state duty ratio equal to half the minimum rampslope for stability is zero. For smaller duty ratios the minimum ramp slope is negative. Thismeans that the system can work without a ramp for duty ratio less than half. This result isanalogous to the stability criterion of peak current mode control.

The ramp stability criterion for SCM does not depend on values of the storagecomponents of the power converter. This makes the design robust to variations in storagecomponents. The slopes are determined by the input and the reference voltage. The latter isa constant and the variations in the former are known a priori. Thus designing a stable oroptimal loop is simple.

Any real inductor has some parasitic series resistance. Thus there is a steady-state errorin the output voltage, which is equal to the resistive drop in the inductor. For a powerconverter with high efficiency, this drop is of the order of about 1% oftheoutput voltage. Forsome applications, this is acceptable. In other cases, a Proportional-Integral (PI) loop cancorrect for this error. Figure 5.4 shows the schematic of the scheme with the outer PI loop.5.2 Discontinuous mode

In a conventional switched dc to dc pow er converter, passive diodes are used for manyof the switch elements. It is possible to have diode turn-off prior to closing of the activeswitch because the diode current goes to zero. The behavior of a dc to dc converter in thisdiscontinuous conduction mode is dramatically different from normal operation. Conventionaldesign of dc to dc converters tends to avoid this mode ofoperation.The SCM control scheme,in contrast, handles the discontinuous mode of operation in a manner very similar to the

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continuous operation. Since the inductor voltage is sensed and fed back into the control law,the control reacts immediately if the converter makes a transition between the continuous anddiscontinuous modes of operation.

VLI i " 0 SWITCHINGFUNCTION"L2

VLI +

Ho

RAMP

CLOCK

S Q

COMPARATOR LATCH

"L iK%VRv O

Figure 5.4 Schematic of SCM with outer PI loopThe stability analysis is similar except that now we have three circuit configurations:

switch on, switch off and a third mode in which both the switch and the diode are off. In thelast case the inductor is open circuited. This corresponds to zero current and a slope of V,equal to zero. Figure 5.5 shows the waveforms.

Equation (5.2) is still valid. However, the ramp rates are affected and Equation (5.3)is modified since there is an additional interval of zero slope. We define a time weightedaverage slope S ^ for the total off-period of the active switch. For all positive SF, the averageS r oisless than SF. The original criterion for stability is SR> (SN+ SF)/2 implies that SR> (SN+ Sn))/2 is also true. Thus stability in discontinuous mode is ensured if the system is stablein continuous mode.

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R AMP

AN AL O G S IG N A L V

j OFF I I I |SWITCH POSITION

Figure 5.5 Waveforms in SCM (discontinuous mode)5.3 Flux Balance of Transformer Coupled Converter

Magnetic flux in a transformer can be treated as a state variable. It is necessary toestimate this flux to avoid transformer saturation. Direct measurement of flux in thetransformer of a power converter is rarely attempted. In conventional design, an estimator isused to estimate the flux. This estimate is used to control the excursion of the flux. The goalis to set the duty ratios such that the flux excursion is balanced and centered around zero.

In current mode control, the transformer current is used to estimate the flux, which isproportional to the current. A dual of this scheme is to use voltage to estimate the flux. Thiseliminates the problem of current sensing. The flux is proportional to the integral of thevoltage. The integration process provides a degree of high frequency noise suppression whichmakes it preferable to current mode control.

Figure 5.6 shows the schematic for flux balancing in a transformer coupled converter.It is possible to b alance flux and control the output voltage of the converter by modifying theflux boundaries. The following schematic shows such a control strategy.

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21

PITRANSFORM

P2

"22

CLOCKI I COMPARATORFLUXESTIMATE

INTEGRATOR

1 ^H|2

S . Q .

L AT C HC L O C K

COMPARATOR

%

12

"r>

LATCH

Figure 5.6 Schematic for flux balancingUsing this strategy the flux estimate O is maintained between 3>mmand Om ax . The time

base is obtained from two phase clock pulses applied to the set pins of the SR(set-reset)latches.

The average duty ratio is set by the difference between 4>max and min.

To ensure that there is no direct current path across the input voltage source duringswitch transition a small dead time of about 200 ns has been introduced between thecomplementary switch functions H,,,H 2I and H12,H22.

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5.4 Experimental verification5.4.1 Buck converter

A buck converter operating from 5 V to 2 V with load current from 0 to 15 A isconsidered here. It is built with an inner SCM loop and an outer PI loop. The SCM schemeis particularly simple for a buck converter because the output inductor is always connectedto the load. When the control law sets the voltage on the input side of the inductor equal tothe reference voltage the output becomes insensitive to the converter supply. Figure 5.7 is aschematic of the SCM con trol without the outer loop. A complete circuit diagram with theouter PI loop is given in Appendix A.

i i VsHJolTL v ViNrTy;z LOAD

BUCK CONVERTER

VS s INTEGRATOR^ U ^ l

CLOCK

S Q

R

COMPARATOR LATCHRAMP

Figure 5.7 SCM schematic for buck converterTable 5.1 compares steady-state operation for the two cases: with and without

the outer PIloop.The source regulation is less than0.3%without the outer loop over a 20%input range. Load regulation and source regulation improve with addition of the outer loop.

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Withtheouter loop, combined loadandsource regulationis less than0.02%overtheentirerange.

Table5.1Buck converter d ata

Input Voltage(V)

4.04.04.04.05.05.05.05.06.06.06.06.0

Output Voltage(V)(No Output Loop)1.99581.95381.91331.87201.99581.95261.91211.86751.99581.95101.91071.8680

Output Voltage(V)(With Outer Loop)1.99741.99731.99731.99721.99741.99731.99721.99731.99721.99711.99711.9971

Output Current(A)0.04510150.04510150.0451015

Figures5.8 and 5.9 show steady-state waveforms withthe SCMscheme. Waveformsfor the switch voltage,theoutput ripple, inductor currentand its estimateV, areshowninFigure 5.8. The switch voltage and the inductor current estimate V, for continuous anddiscontinuous modesare showninFigure5.9.Note thatthe current estimate V, has asigninversion from the inductor current becauseit is the negative integralof VA.

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Switch Voltage (2 V/div) (upper), Inductor Current (1 A/div) (upper),Output Voltage Ripple(20 mV /div)(lower) V, (1 V/div)(lower)X axis: Time (2 us/div) X axis: Time (2 us/div)Figure 5.8 SCM control for buck converter: steady state behavior

(continuous mode) (discontinuous mode)X axis: Time (2 us/div) Y axis :V, (1 V/div) (upper),Switch Voltage (2 V/div) (lower)

Figure 5.9 SCM control for buck converter: current estimate waveforms

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5.4.2 Boost converterA boost converter for 70 V to 210 V step-up with a load current of 0 to 0.3 A is

considered here. It is built with a single loop SCM. The converter worked reliably for aninput voltage range of 60 V to 100 V. It can operate with no more than a 5% change in outputvoltage over its entire load range.

Figure 5.10 provides a simplified schematic for the control scheme. Appendix Aprovides a complete circuit diagram of the converter.

f o W Lv,

- W - D2XHi LOAD

V|

BOOST CONVERTER

H2

T _%/_H2

-(I.

INTEGRATOR

CLOCK

COMPARATORRAMP LATCH

Figure 5.10 SCM schematic for boost converterThe source regulation was found to be better than 5%. Table 5.2 shows the variation

in output voltage with input voltage and load impedance. Note the regulation is better than +/-4% over the entire range. This performance is obtained without using an outer PI loop.

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Table 5.2 Boost converter dataInput Voltage (V)

60708090100

Output Voltage(Load=710Q.)217213210211212

Output Voltage(Load=5 k#)219217215214212

Output Voltage(Load=200 kfi)226223220218216

Figures 5.11 and 5.12 show characteristic steady-state waveforms for the boostconverter both in continuous and discontinuous modes of operation. Figure 5.11 shows theswitch voltage and the voltage integrator output V,. Figure 5.12 shows the current waveformand compares it to the voltage integrator output V,.

N - i7

T+,

' V i

i 1 1 . 1n r H/N

(continuous mode) (discontinuous mode)X axis:Time(5 us/div) Y axis:Inductor Current (0.5 A/div)(upper), Switch Voltage(50V/div)(lower)Figure 5.11 SCM control for boost converter: steady state behavior

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(continuous mode)Inductor current (1 A/div) (upper)V, (1 V/div)(lower)X axis: Time (5 us/div)

(discontinuous mode)Inductor current (0.5 A/div) (upper)V, (0.5 V/div)(lower)X axis: Time (5 us/div)

Figure 5.12 SCM control for boost converter: comparison of current and its estimateSCM control is very effective in handling load and source transients. For the boost

converter, which is designed for an automotive application, it is necessary for the converterto maintain reliable output voltage during large signal transients. Figure 5.13 shows theresponse of the converter to load and source transients. During the load transient the converterload is changed from full load to no load (710 Q to 100kQ.) and the converter responds withno more than 10% overshoot in the output voltage. Note that this also takes the converter fromthe continuous mode of operation to the discontinuous mode of operation. During the sourcetransient the input voltage is changed from 55 V to 80 V and there is no more than 3%change in the output voltage.

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ensures flux balance and prevents saturation. The only analog component required for thecontrol circuit is an opamp. The bandwidth requirement on the opamp is modest even thoughthe switching frequency is 300 kHz.

V VPI2 N 2 2 2 2 A 2 6 6 C T I 2 5 - 3 E 2 A c o r e L&60J

(W5g2

2N2907

^620pF

2

6 T urns

36 Turns

luF

? i I N 5 8 2 I10k

vp2 -vWrv p l - A WI Ok

620pF

V47k

- v W V -

^I Ok

l 5 0 0 p F-Cx> H>o

V \ , v *MC34071Figure 5.14 Circuit for pushpull forward converter with SCM control

5.5 SCM Performance ComparisonHere, SCM control is compared to current mode control and voltage mode control for

a 5 V to 2 V buck conv erter. Dynam ic performance is compared for the three control schem es.The circuits for each control schemes are shown in Appendix A.5.5.1 Noise and dynamic range issues in SCM

In current m ode control, it is necessary to sense the inductor current. Th is is often doneby a small sense resistor. To maintain efficiency, the sense voltage is small compared to theoutput voltage. To ensure that the loss in the sense resistor is at most 1%, the sense resistor

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voltage has to be 1 of the output voltage. By contrast, in SCM the feedback signal is aswitch voltage which is of the order of the output voltage. Thus, the signals in the two casesare different by a factor of 100, or 40 d B . If the converter is running at 10% load , then thedifference is a factor of 1000, or 60 dB. Thus SCM has a very significant advantage overcurrent mode control in terms of signal magnitude. For low voltage converters, this is moresignificant since the current mode signal could be less than 50 mV at full load.

Transform er voltage (2 V/div) (upper) V, (2 V/div) (upper)Transform er Current (0.1 A/div)(low er) Transformer Current (0.1 A/div )(lowe r)X axi s: Tim e (1 us/div) X axis: Time (1 us/div)Figure 5 .15 SCM control for pushpull forward converter : s teady s ta te behavior

For current mode control with an external ramp, the current ramp rate is usuallymatched to the PWM ramp. The latter is fixed by design in most commercial PWM ICs. Thegain of the current signal is fixed to create this match. For large load currents, the currentsignal gain has to be limited to maintain the signals within the I C s dyn am ic ran ge. Thisproblem is accentuated when the ratio of current ripple to its peak value is small, a condition

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that is actually quite a common design choice since it allows for low loss in the magnetics.SCM on the other hand does not have these problems. The inductor voltages being fed backare almost independent of the load current. The outer loop only corrects for the dc drop in themagnetics and needs a very small dynamic range of operation.5.5.2 Implementation without current sensing

Current sensing usually increases losses and system complexity. The construction ofprecision noninductive current sense resistors can present serious problems, especially for lowvoltage power converters. Hall effect sensors and other nondissipative methods of sensingcurrent offer a relatively expensive alternative. SCM bypasses these problem s and matches theperformance of current mode control using only voltage measurement.5.5.3 Source regulation

SCM has exce llent steady-state source regulation as illustrated experimentally. Tables5.1 and 5.2 show the steady-state performance of test buck and boost converters in the absenceof outer loop action for source and load regulation tests.

Good regulation of transients in the source voltage is expected for SCM because theinput voltage is used in the feedback (Equation (5.2)). The input voltage is essentially beingmonitored on a cycle by cycle basis. Since there is only one control input per cycle (the dutyratio), this is the fastest control action feasible.

Figures 5.16, 5.17 and 5.18 compare performance of SCM control, voltage modecontrol and current mode control during a source transient. Both continuous mode anddiscontinuous mode of operation are considered. SCM control and current mode control showexcellent response, while voltage mode control does not respond effectively.

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^ j y .*.i'.?-rr"'i1*if^wyW *i .

X axis: Time (50 us/div)Y axis :Input VoItage(l V/div) (upper), Output Voltage(20 mV/div) (lower)Figure 5.16 SCM control for buck converter: source transient

X axis: Time (50 us/div)Y axis:Input Voltage(l V/div)(upper), Output Voltage(20 mV/div)(lower)Figure 5.17 Voltage mode control for buck converter: source transient

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""ft

#uFIFTH*

s**

# U jlp T 'F rl l

i

1 ,Vjvjjj,PffW

(^ j*M

H'*rri

> 1JJ*U*H l*f \

StjiL." T 1

, t ^ T ' t *

j l l\ 1

1*>*tsT< _ * . A.;

WfM w ' t ' . - i f M *4. .*-*l

1

1^V/L/kAJ1= T P "

- iX axis: Time (50 us/div)Y axis : Input Voltage( 1 V/div)(upper), Output Voltage(20 mV/div)(lower)

Figure 5.18 Current mode control for buck converter: source transient5.5.4 Load Regulation

Steady-state load regulation is good for SCM even without the action of the outer loop.In contrast, current mode control requires the outer loop to set the current level and cannothandle load variation without the outer loop. Tables 5.1 and 5.2 show the steady-state loadregulation data for a buck converter and a boost converter using SCM control.

Dynamic load regulation is improved by the action oftheouter control loop . The outerloop for the SCM control and current mode control are the same. Figures 5.19, 5.20 and 5.21show response to a 5 A load transient. In continuous m ode, the performances of three schemesare comparable. However, the results in discontinuous mode are quite different.5.5.5 Discontinuous mode of operation

In discontinuous mode, the action of the converter changes. For example, a voltagemode control that performs well in continuous mode performs very poorly during a load

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transient that takes the converter to the discontinuous m ode of operatio n. In contrast, the SCMcontrol scheme can handle the discontinuous mode of operation very well. Figures 5.18 and5.19 highlight the difference.

The regime of discontinuous mode of operation is usually avoided due to problems ofconventional control. However, SCM facilitates the design of converters that allowdiscontinuous mode of operation. There are many advantages of operating a converter in thediscontinuous mode for at least a part of the load range. These include operating the converterat a lower switching frequency, sm aller inductor values and elimin ation of internal loads toforce continuous m ode. This results in an overall gain in power conv ersion efficiency.

(continuous mode) (discontinuous m od e)X axis:Time(50 us/div)Y axis:Output Voltage (50 mV/div) (upper), Output Current(2 A/div)(lower)

Figure 5 .19 SCM control for buck converter : load t ransient

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(continuous mode) (discontinuous mode)Xaxis: Time (100 us/div)Y axis:Output Voltage (50 mV/div) (upper), Output Current(2 A/div)(lower)Figure 5.20 Voltage mode control for buck converter: load transient

v V v v

ij1

7 " -

^

//

^r^

i

W A A ; / ^A A W * V * ^

.

> w \ A\i

" 1J W W

(continuous mode) (discontinuous mode)X axis:Time(50 us/div)Y axis:Output Voltage (50 mV/div) (upper;, Output Current(2 A/div)(lower)

Figure 5.21 Current mode control for buck converter: load transient

" ~ w

/

\ uysf

-

W j ^ Vviv

.\VAA/

=.

V W V A ,

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Note that the dynamic response to a load transient is comparable for the three controlschemes. The dynamic response to a source transient is superior for SCM and conventionalcurrent mode control.

SCM is a very attractive option for the control of dc to dc converters. It offers theperformance merits of current mode control without the need for current sensing. Thecomplexity level is almost the same as in voltage mode control except for an additionalopamp. There is a dramatic performance improvement in source regulation and load regulationduring discontinuous mode compared to conventional converter control methods. Eliminatingthe need for current sensing also saves the power loss associated with current sensing whichmay be as high as 2.5% for a 2 V converter.

The difference in signal level between SCM and current mode controls is large. Forexample, in current mode control the current sense resistor may have a drop of 50 mVcompared to more than 1 V across the inductor. A difference in signal level of 40 dB to 60dB is not unusual. Numerous low voltage converters have been built by the author (48 V to2 V; 12 V to 2 V and 5 V to 2 V) with output currents up to 60 A. In such circuits there isa problem of noise from the power stage coupling with the control signals. This increase insignal level was critical for reliable operation of these converters.

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CHAPTER 6OPTIMAL CONTROL APPROACHES

Minimization of a specified cost function is a standard approach to optimal control.Control of this type unifies large-signal and small-signal performances. In this thesis, quadraticoutput-error cost functions are considered for power converters. The conventional form ofquadratic function is intractable for many converters. Severa l cost functions whichapproximate it, but are easier to apply, are introduced. Exam ples include a cycle-by-cyclequadratic function, a cost function based on the stored energy in a converter, andfast-switching approximations to quadratic functions. These functions are studied for dynamicperformance and s

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