Pwm Modelling

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 10, NO. 6, NOVEMBER 1995 659

PWM-Switch Modeling of DC-DC ConvertersEdwin van Dijk, Herman J. N. Spruijt, Dermot M. OSullivan, and J. Ben Klaassens

Abstract-The introduced PWM-switch modeling method isa simple method for modeling pulse-width-modulated (PWM)dc-dc converters operating in the continuous conduction mode.The main advantage of this method is its versatility and simpleimplementation compared to other methods. The basic idea isthe replacement of the switches in the converter by their time-averaged models. These switch models have been developedin such a way that the converter model provides the sameresults as the state-space-averaging technique but now includingnonlinear effects. Simple rules for determination of the switchmodels are obtained. The resulting model is a time-averagedequivalent circuit model where all branch currents and nodevoltages correspond to their averaged values of the correspondingoriginal currents and voltages. The model also includes parasitics,second-order effects and nonlinearities, and can be implementedin any circuit-oriented simulation tool. The same model is usedfor the simulation of the steady-state and the transient behavior.

Buck - boostI. INTRODUCTIONUMEROUS papers have been written on modeling PWMN c-dc converters, proposing a variety of methods formodeling switch-mode converters. An accepted method isthe state-space-averaging method as introduced in [l] . Adisadvantage of this method is the necessity to accomplish anumber of calculations to obtain the averaged state equationsfrom which an equivalent circuit model is derived.

Other methods replace a part of the converter by an equiv-alent circuit model to obtain the converter model. The equiv-alent circuit model of a part of the converter is already givenand can be used directly. These methods are typically restrictedto a few different topologies only.

It is desirable to have a single method which makes it

C/uk

C

Fig. . FourA. Definition ofpWM-Swit Model

converters with PWM-switches.

possible to model all PWM dc-dc converters. The modelshould be found by a simple inspection of the converter circuit.It should be possible to implement this model in a general-purpose simulation tool like SPICE. It should also be possibleto realize various analyzes such as steady-state, transient, andsmall-signal analysis, on one and the same model. This paperpresents such a modeling method, called PWM-switch mod-eling, for converters operating in the continuous conductionmode.

11. PWM-SWITCHModels for the PWM-switch are introduced in [2] and [3].

The method presented in [2] is used here and slightly adaptedto preserve its general character and to expand it for use inmore complex converters that do not have a PWM-switch assuch.

Manuscript received March 1, 1994; revised June 15, 1995.E. van Dijk and J. B.Klaassens are with the Delft University of Technology,H. J N. Spruijt and D. M. OSullivan are with the Power and EnergyIEEE Log Number 9414904.

Faculty of Electrical Engineering, 2628 CD Delft, The Netherlands.Conversion Division, ESA/ESTEC, Noordwijk, The Netherlands.

Fig. 1 shows four classical PWM converters: buck, boost,buck-boost, and Cuk converter. These converters have oneactive switch and one passive switch performing the switchingaction in the converter. The active switch is directly controlledby an external control signal. It is usually implemented witha bipolar or a field-effect transistor. The passive switch isindirectly controlled by the state of the active switch and thecircuit condition. It is usually implemented by a diode. Asshown in Fig. 1, these two switches can be combined intoone network with three terminals a , p , and c , which stands foractive, passive, and common, respectively. This three-terminalnetwork is called the PWM-switch. Since all other elementsof the converters are supposed to be linear, the PWM-switchis the only nonlinear element and therefore responsible for thenonlinear behavior of the converters.

Fig. 2shows the generic presentation of the PWM-switchoperating in the continuous conduction mode with the terminalcurrents and voltages. The active and the passive switchoperate like a single-pole double-throw switch. During the timeinterval dT, the passive switch is off and the active switch

0885-8993/95 04.00 1995 IEEE


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Fig. 2. PWM-switch.

a-

d ,/z

P

cPFig.3. Averaged model of the PWM-switch.is on, and the active terminal is connected to the commonterminal. During time interval d T, the active switch is off andthe passive switch is on, and the passive terminal is connectedto the common terminal. T, is the switching period of theactive switch, d stands for the duty ratio as the ratio of theon-time of the active switch and the switching period andd' = 1- d for the continuous conduction mode.

The relations between the instantaneous terminal currentsare found as

Fig.5. Boo st converter with PW M-switch replaced by the m odel shown inFig.3.

instantaneous terminal waveforms are time-averaged over onecycle T. The relations between the averaged terminal currentsare found from (1) and (2) as

( i a ) = d ( i c )( i p )= d'( i , ) .The relations between the averaged terminal voltages are

found from (3) and 4)

With 5 ) through (8) it is simple to obtain a time-averagedmodel for the PWM-switch.

To simplify the presentation in the follo win g, the symbols (.)indicating the time-averaged values are omitted.Fig. 3shows a model applying a controlled voltage source

and a controlled current source using (5)and (7). It is clear that(6) and (8) are also satisfied. The model for the P W-s wi tc halways satisfies either 5) or (6) and (7) or (8). This meansthat one element should satisfy 5) or (6) and another elementshould satisfy (7) or (8). This explains the function of thecontrolled current source and the controlled voltage source.The application in the model of two voltage sources is notpossible because it will leave the terminal currents undefined.The same is valid for the use of two current sources whichwill keep the voltages undefined.B. Application of a PW M-Switch Model(1)i c ( t ) during dT,

To illustrate the use and to demonstrate the validity of theduring d T,0 during dT, (2) PW-switch model, a boost converter as shown in Fig. 4 isi c ( t ) during d T,. used for the presentation. Fig. 5presents the boost converterwhere the PWM-switch is replaced by the model shown inFig. 3.

ia( t )=

{p( t )=The same applies for the instantaneous terminal voltages

during dT,{ :ap(t) during d T,.ac(t)=(3) We recognize from Fig. 4 that

4)a = -aL

uap= -uc. 9)It is sufficient to inspect the averaged behavior of the PWM-

switch to analyze the averaged behavior of a converter. The This results in the expressions for the controlled sources inFig.5.


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VAN DUK et al.: PWM-SWITCH MODELING OF DC-DC CONVERTERS 661

The equivalent circuit shown in Fig. 5 is a continuous,averaged circuit model of the boost converter as shown inFig. 4. The action of the switches and the resulting ripple areremoved from the circuit. However, the behavior of the circuitmodel is still nonlinear as demonstrated by the products ofthe duty ratio and a state variables in the expressions for theswitch model.

The state equations are easily recovered from the circuit of

E

Fig. . Boost converter with = R s .Fig. 5 U

Another method to derive the state equations is the well-known method of state-space-averaging [11. This methodaverages the state equations valid for time interval dT, withthose for time interval d T,, resulting in the averaged stateequations. The same equations are obtained with the PWM-switch model. Thus the use of the PWM-switch model resultsin a valid averaged equivalent circuit model of the boostconverter.

nI. PWM-SWrrCH MODEL ND ESR SA boost converter with the inductor and capacitor equivalent

series resistances (ESR) is shown in Fig. 6. Applying thePWM-switch model results in the equivalent circuit shownin Fig. 7where the controlled current source is indicated byi,, and the controlled voltage source by U,,. The expressionsfor the controlled sources are:

i =dir,U,, = duo. (1 1)

Note that the voltage across the active and passive terminalThe state equations are determined from the equivalent

of the PWM switch equals the output voltage uo.circuit model of Fig. 7and applying 1 1)

L-=- R RORc ] i ~ 1- dRo RCdiLdt [ (1- 2

RORo Rc uc E ,

To see whether the model is correct or not, the state equa-tions are also calculated with state-space-averaging technique[l]. This produces the following set of state equations for theboost converter with ESR s

RORc ] i ~ 1- dRo -I-Rcuc E,RORo Rc

Fig. 7 Boost converter model with controlled sources and ESR s.There is now a difference recognized between both sets of

(12) and (13). The difference is found in the equation for theinductor current. In (12), the factor multiplied by i~ containsthe term 1- d 2 , whereas in (13) it is 1- d . It is clearthat when the capacitor ESR Rc equals zero, both equationsare the same.

An explanation for this discrepancy is provided in [2] and[3]. The capacitor C that is connected to the active and passiveterminal of the PWM-switch, absorbs a pulsating current.When the ESR of this capacitor is zero, the instantaneousterminal voltage ua p(t) uC(t) s continuous and has only avoltage ripple, due to the switching process, which is neglectedin the averaging process. When the capacitor has an ESR, thepulsating current through the ESR causes a small square wavevoltage superimposed on the terminal voltage. The voltagenow also depends on the value of the ESR.

We can calculate the necessary expressions for the con-trolled sources to obtain the same equations as obtained withState-Space-Averaging by deriving the state equation from thecircuit shown in Fig. 7and comparing them to the set of (13).This results in the expressions for i,, and U,,:

i,, = d i ~U C .RORo RcU,, = d

When (14) is used in the equivalent circuit model shownin Fig. 7, this model gives exactly the same results as state-space-averaging

IV. GENERALIZEDWM-SWITCHMODELIt is desirable to find the expressions directly by inspectionof the converter circuit. In the case of the boost converterwith ESR s, we can see from (14), that i,, equals the duty ratiotimes the current flowing through the active switch during timeinterval dT,, during which the active switch is closed and thepassive switch is open. This current equals the inductor currenti ~ . similar approach is valid for U . The voltage acrossthe opened passive switch during time interval dT, can bedetermined as a division of the capacitor voltage uc across Roand Rc. From (14) it follows that the expression for U,, equalsthe duty ratio times this voltage across the passive switch.


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662 IEEE TRANSACTIONS ON POWER ELECTRONICS VOL. 10, NO. 6, NOVEMBER 1995

d c t i ve

Fig.8 shows the PWM-switch model of Fig. 3 , where dicand du,, in the expressions for the controlled sources arereplaced with diac-ive and dupasive, respectively.

Fig. 9.

PFig. 8. Generalized PWM-switch model.

f--f-TL37- RO

3 ClR C I

Rc3

Topology with separated PWM-switch [4].Ll RL, - Rc, Lz RLZU2

I L - - . l - IFig. 10. Model of converter shown in Fig. 9with active and passive switchreplaced by a control led current and a controlled voltage source resp ectiv ely.

V. SEPARATED OR MULTIPLEWM-SWITCHES

variables can be voltage or current sources. Similarly, upassiveequals the voltage across the opened passive switch (acrossterminals c and p ) , during time interval dT, , written as afunction of the state variables and input variables.

The State-Space-Averaging method averages the entire con-verter. This is realized by presenting the state equations forboth time intervals dT, and d'T, and evaluating the averagedequations. The waveforms and duty ratio signal are dividedin two parts: A constant (dc) term and a varying (ac) term.The cross-products of two ac terms are neglected assumingsmall-signal variations which results in linearized small-signalequations. An equivalent circuit model is obtained from theseequations. The model is limited to small-signal variations fromwhich only input and output variables are usually obtained.

The PWM-Switch modeling method averages the PWM-switches of the converter. This is realized by calculating theexpressions for the controlled sources of the PWM-switchmodel from inspection of the circuit as explained before.The PWM-switch model simply replaces a PWM-switch inthe circuit avoiding small-signal assumptions. Therefore, theresulting equivalent circuit model is valid for large-signalvariations. All internal branch currents and node voltagesare available from the circuit model. Small-signal transferfunctions can be calculated from the model.

The method is only valid for the continuous conductionmode. When a mosfet is used for the passive switch, insteadof a diode, and it is controlled by the duty ratio opposite to theactive switch, the converter can operate in both directions andwill always operate in the continuous conduction mode. ThePWM-Switch modeling method can be applied to this type ofbidirectional converters.

A . Converters with Separated PWM-SwitchA classification of PWM converter topologies with one

active switch and one passive switch has been presented in[4]. The presented topologies are derived from five basictopologies leading to a number of new converter topologies.In a number of topologies the active and the passive switch arenot connected to each other. The PWM-switch as such cannoteasily be distinguished in these topologies. Such a topology isshown in Fig. 9.This topology is referred to as topology 5.1in [4]. The inductor and capacitor ESRs are also included inFig. 9.It is a buck type topology with an ideal voltage gain of( 2 d - ) / d , were 0.5< d < 1 . It is clear that the two switchesare separated from each other. These two switches will bereferred to as a separated PWM-switch.

The examination of the boost converter in Fig. 6 and itsmodel in Fig. 7reveals that when the PWM-switch is replacedby its model, in fact, the active switch is replaced by thecontrolled current source and the passive switch is replaced bythe controlled voltage source. This can also be accomplishedwith a separated PWM-switch. Fig. 10 presents the convertershown in Fig. 9with the active and passive switch of theseparated PWM-switch replaced by their respective controlledsources.

Comparison of the state equations obtained from Fig. 10,with the state equations that are calculated with the state-space-averaging technique, will result in the expressions forthe controlled sources


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VAN DIJK er al.: PWM-SWITCH MODELING OF DC-DC CONVERTERS 663

L1 R L l s

Fig. 1 1 . Converter topology with two PWM-switches [5]

stFig. 12.sources. Converter as sho wn in Fig. 11 with switches replaced by controlled

Equation (15) is also obtained when the method, describedin section Generalized PWM-Switch Model, is used todetermine the expressions for the controlled sources. ThePWM-Switch modeling method cannot only be used for con-verters with a PWM-switch, but also for any PWM dc-dc converter that has one active/passive switch pair, notnecessarily connected to each other and operating in thecontinuous conduction mode.B. Converters with Multiple PWM -Switches

A classification of converters with one and two PWM-switches has been presented in [5]. A converter with twoPWM-switches is illustrated in Fig. 11. The ESRs are alsoindicated. Both active switches are controlled with the sameduty ratio signal. The ideal voltage gain equals 1 d .As shown in Fig. 12 the PWM-switches are replaced bythe model given in Fig. 8.Comparison of the state equationsobtained from Fig. 12,with the state equations that are calcu-lated with the state-space-averaging technique, results in thefollowing set of expressions:

i c s l = d i l

ucs2 = d E s (16)To find the missing expression for i C s 2 we assume that

the PWM-switch S2 is ideal, which means that its switchmodel has no losses. The following expression is derived fromFig. 12:

(17)Using the expression for ucs from (16) in (17) results in

u c s 2 ( i2 - cs2) cs2(ucsZ - s = 0.an expression for i c s 2 :

i c s 2 = d i 2 . 18)

lo8I23 8 1OO 2 4 8 T23 4 6 8 1

72uFGzl6O 2 6 8 , 1(C)

1 86EO 2 4 6 8 1

(d)Fig. 13.0.75 + 0.50 , (b) 0.25 0 .50 , (c ) 0 .50 + 0.25, (d) 0.75Output voltage waveforms for different steps in the duty ratio: (a)0.25.

The expressions of (16) and (18) for the switch model ofboth PWM-switches are also obtained, when the method forderiving the PWM switch model, as it is described in thesection Generalized PWM-Switch Model in this paper, isused.

VI. DEMONSTRATIONF PWM-SWITCHMODELINGA boost converter with inductor and capacitor ESRs, as

illustrated in Fig. 6, is taken from [6] as an example. Thecircuit model shown in Fig. 7using (14) is implemented inthe simulation program Micro-Cap 111. The data for the circuitis

E , = 60 V R, = 60 0L = 6 m H R ~ = 3 0C = 1mFT, = 100 ps

Rc = 10

Results in the time domain results are presented in Fig. 13.The output voltage waveforms are shown for different stepsin the duty ratio applied at t = 1 ms. The effect of the pulsewidth modulator on the transients is very small.

The same model is used to predict the transfer functions inthe frequency domain. The shown function in Fig. 14is thesmall-signal transfer function from the duty ratio to the outputvoltage for different steady-state duty ratios. Here, the effectof the pulse width modulator creates an extra phase lag whichincreases linearly with the frequency.


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664 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 10 NO. 6, NOVEMBER 1995

Gaind

0 100 l k 1OkFrequency n Hz

Frequency n Hz(b)

Fig. 14.(a) gain, (b) phase.Control to output frequency responses for different duty ratios D:

The waveforms and functions shown in Figs. 13and 14 areidentical to the experimental results presented in [ 6 ] .

VII. CONCLUSIONSThe PWM-switch modeling method is a simple method

to model PWM dc-dc converters operating in the continuousconduction mode. The PWM-switch model has been developedsuch that the model gives the same results as the state-space-averaging technique [13. This resulted in a general method forderiving the PWM-switch model. The state-space-averagingmethod averages the complete converter while the PWM-switch modeling method only averages the switches. Theaveraged models of the switches are simply obtained byinspection of the converter circuit. It is a simple and fastmethod to obtain the model.

Another advantage of the PWM-switch modeling method isthat the model corresponds directly to the original convertercircuit. Only the switches are replaced by their models. Allbranch currents and node voltages of the converter are directlyavailable from the model as averaged quantities.

The model developed with the PWM-switch modelingmethod can be implemented in any circuit-oriented simulationtool. It is a nonlinear, large-signal, averaged model. No small-signal assumptions have been made. The nonlinearity dueto the switching process in a converter is included in themodel. Other nonlinearities, such as the duty ratio limitation,can easily be modeled. Parasitic elements and other second-order effects can also be added to the model. The samemodel can also be used for frequency domain simulations. Allimportant transfer functions such as input/output impedances,susceptibility, or loop gain can easily be predicted.

The PWM-switch modeling method has the same limita-tion as state-space-averaging method. It is assumed that theswitching period of the switches is small compared to the timeconstants of the converter, which is usually the case.

The PWM-switch modeling method is only valid for thecontinuous conduction mode. It can be applied to bidirectionalconverters which always operate in the continuous conductionmode.

An interesting feature is the possibility that the methodpresented can be used to model other types of converters thanPWM dc-dc converters. Equivalent circuit models for resonantswitches in quasi-resonant converters are presented in [7]. Theequivalent circuits for the resonant switches show similaritiesto those for the PWM-switches. This could imply that it ispossible to join them into one general model.

REFERENCES[l ] R. D. Middlebrook and S. Cuk, A general unified approach to modelingswitching-converter power stages, in Proc. ZEEE Power Electron.Specialists ConJ, 1976, pp. 18-34.[2] R. P. E. Tymerski et al., Nonlinear modeling of the PWM switch,IEEE Trans. Pow er Electron., vol. 4, no. 2, pp. 225-233, 1989.[3] V. Vo@rian, Simplified analysis of PWM converters using model ofPWM switch, part I: Continuous conduction mode, IEEE Trans. Aerosp.Electron. Syst., vol. 26, no. 3, pp. 49M96, 1990.[4] E. R. W. Meerman and H. J. N. Spruijt, PWM converter topologies,in Proc. European Space Power Con , vol. ESA SP-294, 1989, pp.[5] R. P. E. Tymerski and V. VorpBrian, Generation and classification ofPWM dc-to-dc converters, IEEE Trans. Aerosp. Electron. Syst., vol.24, no. 6, pp. 743-754, 1988.[6] G. W. Wester and R. D. Middlebrook, Low-frequency characteriza-tion of switched dc-to-dc converters, in Proc. IEEE Power Electron.Specialists C onj , 1972, pp. 9-20.[7] V. Vorp6nan et al., Equivalent circuit models for resonant and PWMswitches, IEEE Trans. Power Electron., vol. 4, no. 2, pp. 205-214,1989.

297-305.

for Power ElectronicsTechnology.

holder of two ESA Datc

Edwin van Dijk was born in Oud-Beijerland, TheNetherlands, in 1968. He received the M.Sc. degreein electrical engineering from the Delft Universityof Technology, Delft, The Netherlands, in 1992.During his study in 1991 and 1992, he worked onthe modeling and the application of dc-dc convertertopologies in space power systems at the Powerand Energy Conversion Division of the EuropeanSpace Research and Technology Center (ESTEC),Noordwijk, The Netherlands. Since 1992, he isworking as a Ph.D. student on opening-switch sys-xrrents in pulsed power applications at the Laboratoryand Electrical Machines of the Delft University of

Herman J. N. Spruijt was born in Wassenaar, TheNetherlands, in 1940.He has worked at the European Space Agencyin Noordwijk, The Netherlands, since 1966, thefirst six years as a satellite project engineer dealingwith the HEOS-1, HEOS-2, GEOS-1, and GEOS-2 scientific satellites, and the last 23 years as apower subsystem engineer in the Power Energyconversion division. He was also responsible forthe execution of several research and developmentcontracts with the European industry. He is the:nts. one on the batten, charge termination using the1TDT (temperature derivative termination), the other on 16 new dc/dc f W Mconverter topologies. He is presently involved in the support of the ESAmoon program.


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VAN DUK et al.: PWM-SWITCH MODELING OF DC-DC CONVERTERS

Dermot M. OSullivanwas born on May 13, 1943,in Dublin, Ireland. He received the B.E. degree inelectrical engineering from the University CollegeDublin in 1965.In 1965, he joined Hawker Siddeley Dynamicsas a Power System Engineer. From 1969 to thepresent, he has been a member of the EuropeanSpace Agency in the Technical Center (ESTEC) ofNoordwijk, The Netherlands. He is presently Headof the Power and Energy Conversion Division andhas applied for several patents in this area.

665

J. Ben Maassens was born in Assen, The Nether-lands, in 1942. He received the B.S., M.S., andPh.D. degrees in electrical engineering from theDelft University of Technology, Delft, The Nether-lands.He i s currently an associate professor at theDelft University of Technology. His work has beenconcemed with inverter circuits, pulse width modu-lation, and the control of electrical machinery. Hisresearch work and professional publications are inthe area of converter systems with high internal

pulse frequencies for sub-megawatt power levels employing thyristors, powertransistors, and IGBTs. His current interest is in the area of control ofconverters and electrical drives. He has published a variety of papers on series-resonant converters for low and high power applications. He has designed andbuilt prototypes of the early dc-dc to the recent ac-ac series-resonantconvertersfor a wide variety of applications such as electric motors and generators,communication power supplies,radar signal generators, arc welders, and spaceapplications.

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