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Modelling and controlof DC-DC convertersThis tutorial a rticle shows how the widely used analysis techniques of averagingan d linearisation a re applied to the buck or step-down DC-DC converter to obtainsimple equations which may then be usedfor control design. Three common controlmethods are described. Their principal characteristics are illustrated using Matlaband the Simulink block diagram system along with exp erimental results. Theanalysis procedures described may be applied directly to other DC-DC convertersand the principles may be extended to more complex power electronic systems.

by A. J. Forsyth and S. V. M o l l o vC-DC converters are some of thesimplest power electronic circuits.They are widely used in the powersupply equipment for most

electronic instruments and also in specialisedhigh-power applications such as batterycharging, plating and welding. In addition to acontrollable and theoretically lossless DCvoltage transformation, DC-DC convertercircuits may also provide voltage isolationthrough the incorporation of a small high-frequency transformer. The wide variety ofcircuit topologies ranges from the single-transistor buck, boost and buclzhoostconverters to complex configurationscomprising two or four devices and employingsoft-switching or resonant techniques tocontrol the switching losses.',2 However,similar methods of analysis and control areapplied to many of these converters.Buck converterFig. 1 shows a circuit diagram of a buckconverter along with idealised waveforms forthe inductor voltage and current. The transistoroperates a t a fixed frequency, period T, and withan on-time to period ratio or duty-ratio d. Theinductor current is assumed to be continuous,the circuit components are lossless and theoutput capacitor ripple voltage is considerednegligible. The relationship between steady-state output voltage and duty-ratio is obtainedby equating the positive and negative inductor

volt-seconds in a switching cycle. The volt-seconds must balance in steady-state operation:

Since the value of d lies between 0 and 1, theconverter output voltage must be less than orequal to the input voltage.

, Buc:kDC-DCconved,er and ideali sedwavefoirms

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L

+ -1 1 1 AON EONtransistor on

Ly x $ j ( n l -r4 T T I

+ -AOFF BOFF

transistor off

2 Circuit The usual requirement of a control systemconfigurations of the for the converter is to maintain the outuutbuck converterassuming continuousinductor current

voltage constant irrespective of variations in theDC source voltage V, , and the load current.According to the steady-state equation for theoutput voltage (eqn. l ) , V is independent ofload conditions. However, as we will see, loadchanges affect the output voltage transiently,possibly causing significant deviations from thesteady-state level. Furthermore, in a practicalsystem circuit losses introduce an outputvoltage dependency on steady-state loadcurrent which must be compensated for by thecontrol system.Modelling power electronic converters byaveragingThe inherent switching operation of powerelectronic converters results in the circuitcomponents being connected together inperiodically changing configurations, eachconfiguration being described by a separate setof equat ions. The transient analysis and controldesign for converters is therefore difficult sincea number of equations must be solved insequence. The technique of averaging providesa solution to this problem. A single equationmay be formed to described the converterapproximately over a number of switchingcycles by simply taking a linearly weightedaverage of the separate equations for eachswitched configuration of the converter. State-space a~eragingl-~s the most commonaveraging technique and is used here to modelthe buck converter.

Fig. 2 shows the two circuit configurations ofthe buck converter corresponding to the twostates of the transistor. A third configurationoccurs if the inductor current becomesdiscontinuous but is not considered here. Inorder to provide a facility in the model forexamining the response of the converter to loadchanges, a current generator I, is added inparallel with the load resistor in Fig. 2. TheFigure also shows the equations for each circuitconfiguration expressed n standard state-spaceform. The inductor current I and outputcapacitor voltage V are the two elements of thestate vector x, whilst the input vector u haselements V,, and I,.

The state-space averaged model of theconverter is formed by taking a weightedaverage of the equations in Fig. 2, and may beexpressed as:

The averaged matrices for the buck converterare then

Eqn. 2 approximates the behaviour of theconverter over many cycles, but the averagingprocess has removed all information abut theswitching frequency ripple component of thevariables. For the averaging approximation tobe valid two main conditions must be~atisfied:~irst the state variables must evolvein an approximately linear manner in the twocircuit configurations, and second theswitching frequency ripple component of thestate variables must be small in comparisonwith the average component. Both theseconditions are usually satisfied in simple DC-D C converters.

The control input to the converter, the duty-ratio, appears within the B matrix of theaveraged model (eqn. 2) rather than as anelement in the input vector. The averagedmodel is therefore time varying and difficult tosolve. To simplify he model, eqn. 2 is linearisedby considering small variations in the variables.Each variable is written as the sum of a steady-

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x = AX t BU t Edconvertercontroller H(s) modulator

state or DC component and a small-signal orAC component, denoted by -, i.e.

4 Siniulink model ofconverier with single-loop control

Ix = x + 2 , u = u + ii and d = d + dwhere 2


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TutorialTable 1 Parameter valuesparameter valueVi"VRLC7

24 V12v11n335 pH10pF21 ps

values listed in Table 1. The parameters aretaken from a prototype system which is usedbelow to provide supporting experimentalresults. The small-signal source voltage andload current are shown as external inputswhilst the duty-ratio is determined by thecontrol loop. The mux block simply combinesthe three signals V,,, I, and d into vector formfor the state-space equations. There is a singleoutput from the state space equation block, theoutput voltage V. The PW M modulator isrepresented by a small-signal gain k, which isdetermined below.

Assuming that the peak and valley levels ofthe ramp waveform are denoted by V, and V,,respectively, then the duty ratio d is given interms of the control voltage V, by:

(4)

5 Open-loop ii/iicfrequency response plot The small-signal gain of the modulator is then(5)- prediction, k - - = -d 1---measurement) ,- v, V,-V,

30

I 03 10 4frequency,Hz

-200 I I I103 104frequency, Hz

the value of k, in the prototype was 0.5.The controller transfer function H(s) is

chosen to have an integral characteristic at lowfrequency in order to ensure zero steady-stateerror. A compensation term is added at higherfrequency to provide a satisfactory crossoverfrequency and stability margin. The crossoverfrequency of the control loop is typicallyrestricted to around one-tenth of the switchingfrequency since this usually provides anacceptable compromise between speed ofresponse and avoiding switching frequencyrelated instabilities.

Fig. 5 shows the open-loop control-to-output frequency response ?/Vc for theconverter. The solid line represents datagenerated from Simulink by removing thecontrol loop shown in Fig. 4. The broken lineshows measured data from the prototypesystem. The measurement was made using anetwork analyser; the oscillator output fromthe analyser was superimposed onto themodulator control signal V, using a widebandsignal transformer. The resulting small-signaldisturbance in the converter output voltage wasmeasured by the analyser and the frequencyresponse plotted. There is closecorrespondence between the two sets of data upto the maximum frequency plotted, 20 kHz,illustrating the accuracy of the model. Thesmall discrepancy is due to the absence ofcircuit losses from the model. Averaged modelsare typically accurate up to around one-third ofthe switching frequency. The frequencyresponse has the familiar form of a second-order transfer function with under-dampedpoles at 2.5 kHz.

In order to achieve a crossover frequency ofaround 5kHz (one-tenth the switchingfrequency) the controller transfer function H ( s )was designed to be

0.24 ( S + 10k)'H ( s ) = s(s + 60k)The two zeros are placed at lokrads', justbelow the converter pole frequency and a highfrequency pole is placed at 60krads-l,approximately twice the target crossoverfrequency. The magnitude of H(s) was chosento give a crossover frequency of 4kHz, the phasemargin being 65".

Fig. 6 shows the closed-loop performance ofthe system. The magnitude of the DC sourcevoltage to output voltage transfer functionF/V, , ,is plotted as a frequency response; the solid line

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mux

is from the Simulink model and the broken linerepresents measured data. There is at least 7dBatte.nuation of input voltage disturbancesacross the entire frequency range. Fig. 6 alsoshows a predicted and measured time domainresponse of the output voltage to a 0.12 A stepincrease in load current. The prediction fromthe model has no switching frequency ripple orsteady-state component. In addition to beingunderdamped, the response also has a slowlydecaying component which arises due to thetwo zeros in H(5 ) and the relatively low loopgain at low frequency.

~

--cx':Ax+Bu-EdV

Input voltage feedforwardInput voltage feedforward increases theimmunity of the converter output voltage todisturbances in the DC input voltage. This isaccomplished by making the peak value of thePW M modulator ramp waveform V,proportional to the DC input voltage Vln. Anincrease in the DC source voltage will thenincrease the slope of the PW M ramp and lead toan almost instantaneous reduction in transistorduty-ratio, thereby compensating for theincrease in source voltage and reducing theresultant disturbance in the converter outputvoltage.

A linearised small-signal representation forthe PW M modulator is again determined fromthe duty-ratio equation (eqn. 4), but with thepeak value of the ramp voltage V, expressed asa linear function of VI,, V, = kfv,,:

(7)

A linear expression relating small changes induty-ratio 2 to small changes in input voltageand control voltage, c,,, nd qc, espectively, isobtained by taking the first-order terms in theTaylor series for2

- dd -, dd -d = -Vc + -V,,, = k,i Vc km2G,,, ( 8 )av, av,,

Expressions for the constants k,l and k d areobtained by undertaking the partialdifferentiation of eqn. 7:

A simulink model for the system is shown inFig. 7. The converter is represented as before bythe small-signal state-space averaged model(eqn.3) ; he modulator is represented by eqn. 8;an output voltage feedback loop is also shown.

The addition of the feedfonvard does notaffect the control-to-output transfer function ofthe converter; therefore the control loop designis identical to that described in the previoussection, the controller transfer function H(s )being given by eqn. 6. As a result the outputvoltage transient due to a step load change isidentical to that shown in Fig. 6.However, animprovement is seen in the source-to-output

6 Claised-loopperformance- sinylz-loop control . Top: V/Kn(- prediction, ---measurement); Bottom:output response to0-12A Step increase nload (pirediction on left,measuirement on right)

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103frequency, Hz

104

8 i?/krom Simulink voltage frequency response, the Simulink- ~edfOnnrard ontrol prediction is shown in Fig. 8 where hiwas taken

to be 0.1 and V,= 0.5 V. The magnitude isreduced by over 10 dB across the full frequencyrange compared with the results in Fig. 6.Current-mode controlIn this control m e t h ~ d l , ~ , ~Fig. 9) , the PWMmodulator is replaced by a transistor currentfeedback loop. The sketched waveform of thefeedback current signal If. (Fig. 9) illustrates theoperation of the control loop. The transistor is

9 Current-modecontrol

switched on at the start of each cycle by a clockpulse which sets the output of the latch. Thetransistor current rises linearly while the deviceis conducting. The current is fed back as signalrf. and is compared with a reference signal.When If. is equal to the reference thecomparator output switches low, resetting thelatch and turn ing the transistor off.

The reference signal for the comparator isformed by an output voltage feedback oop, butthe signal produced by the control transferfunction Ixf is modified by subtracting a rampwaveform of slope m which is synchronisedwith the clock. The ramp is necessary toprevent switching frequency relatedinstabilities 4, 5

By using the transistor current to determinethe turn-off instant, this control methodinherently provides a current limit andprotection function for the power circuit. Also,the technique provides an inherent sourcevoltage feedforward function; an increase in thesource voltage causes the transistor current to

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Tutorialrise more rapidly, reaching the reference levelearlier and therefore reducing the duty-ratioalmost immediately. Furthermore, the controlmethod simplifies the design of the controllersince the control-to-output transfer functionfor the converter becomes first-orderdominated; this is demonstrated below.

A small-signal representation of the currentcontrol loop is formed by considering theinductor current waveform under transientconditions (Fig. 10). The average inductorcurrent across the switching cycle may beexpressed as

(9 )gi d2 T2 g2 (1- d)2 T2Iavg = I p --where lp s the peak value of the current,gl andg2 are the rising and falling slopes of thewaveform.

The current control loop gives the relation

where Ks is the gain of the current feedbacktransducer.

Eliminating gl and g2 from eqn. 9 using theexpressions in Fig. 10, and eliminating lpusingeqn. 10:

Iavg =1,f mdT d2T V - V (1-d)2T VK, R, 2 [ InL ] 2 [c] (I1)~ _ _ _ _ _

A linearised small-signal relation between thevariables is obtained by taking the first-orderterms in a Taylor series expansion of lavg:

carrying out the partial differentiation and re- 10 lniductor currentarranging to give an expression for 2 under transientconditions

The negative term h~ represents the sourcevoltage feedforward effect of current-modecontrol. Fig. 11 shows a Simulink blockdiagram of the system. The controller ismodelled as before using the sn-tall-signal tate-space averaged equations (eqn. 3) and thecurrent control loop is represented by eqn. 13.The parameters listed in Table IL are again usedwith current feedback gain R, = 1.5Q and m =3.8 x l o 4 V/s. The output from the converterblock is defined as the state vector, and thedemux block separates the state vector into itselements.

Fig. 12 shows the predicted and measuredcontrol-to-output transfer function of theconverter and current control loop, v/?,+ heprediction in solid lines and the measurement

11 Siimulink model ofc~rent-tno de ontrol

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103 104frequency, Hz

-200 I I I103 104frequency,Hz

10 3 104frequency, Hz,.III , - 2v-0.1-0 2-0 3-0.4 t0 0.2 0 4 0 0.8ms

12 Open-loop i /Lfrequency response plotfo r current-modecontrol (-prediction,--- measurement)13 Closed-loopperformance- cu ry r t -mode control. Top: VIV,(-prediction, ---measurement); Bottom:output response to0.12A step increase inload [prediction on left,measurement on right)

in broken lines. As a result of the currentcontrol loop the response is first-orderdominated rather than the second-ordercharacteristic of the converter alone (Fig. 5).

The controller in the voltage feedback loopH(s1 is designed to have an integralcharacteristic at low frequency and a zero at 20krads-I to cancel the converter pole. The low-frequency magnitude of H(s) was chosen togive a cross-over frequency of 5 kHz, he phasemargin being 65. Eqn. 14 gives the transferfunction for H(s):

( s + 20k)H(s) = 0.45 ____________The upper plot in Fig. 13shows the closed-loopsource to output voltage response c&t,,or thesystem, the solid line shows the prediction andthe broken line the measurement. Comparedwith the single-loop control, the attenuation ofsource voltage disturbances is increased by a tleast 30 dB, confirming the inherent sourcevoltage feed forward characteristics of currentmode control. The correspondence betweenmeasurement and prediction is not as close asbefore; this is because small errors in themeasurement of the current control loopparameters have a large influence on the sourceto output response. Fig. 13 also shows thepredicted and measured output response due toa 0.12A step increase in load. A rapid and well-damped response is seen.ConclusionThe tutorial has shown how averaging andlinearisation techniques may be used to obtainlinear transfer functions for power electronicsystems, specifically DC-DC converters . Linearsystem design techniques may then be used toundertake controller design and examineclosed-loop performance. Three commonlyused control methods for the buck DC-DCconverter are described and compared.Current-mode control is seen to offer superiorperformance in both the rejection of sourcevoltage disturbances and the response to loadtransients.Furtherreading1 MOHAN, N ., UNDELAND, T.M., and ROBBINS, W.

P: Power electronics: converters, applications anddesign (Wiley, 1995, 2nd Edn.)2 KASSAKIAN, J. G. , SCHLECHT, M. E, andVERGHESE, G . C.: Principles of power electronics(Addison Wesley, 1991)

3 MIDDLEBROOK, R. D., and CUK, S.: A generalunified approach to modelling switching converterpower stages, Intemational Joumal of Electronics,1977,42, (61, pp.521-5504 MITCHELL, D. M.: DC-DC switching regulatoranalysis (McGraw Hill, 1988)

5 VERGHESE, G. C., BRUZOS,C. A. and MAHABIR,K.N.: Averaged and sampled-data models for currentmode control: a re-examination, IEEE PowerElectronics Specialists Conference, 1989, pp.484-491

0 EE: 1998The authors are with the School of Electronic 6sElectrical Engineering, University of Birmingham,Edgbaston, Birmingham B1 5 2TT, UK

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