small-signal modeling of average current-mode control

7/28/2019 Small-Signal Modeling of Average Current-mode Control

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I1 2 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. E NO . 2. # p m 1993

Small-Signal Modeling ofAverage Current-Mode Control

Wei Tang, Student Mem ber, IEEE, Fred C . Lee, Fellow, IEEE an d Raymond B. Ridley, Member, IEEE

Abstract-A recently proposed average current-mode control isanalyzed. A complete small-signal model for the control schemeis developed. The model is accurate up to half the switchingfrequency. This control scheme is suitable for applications wherethe average inductor current needs to be controlled, as in powerfactor correction circuits and battery charger/dischargers. Thesubharmonic oscillation, commonly found in peak current-modecontrol, also exists in this control. This subharmonic oscillationcan be eliminated by choosing a proper gain of the compensationnetwork in the current loop. Model predictions are confirmedexperimentally.

I. INTRODUCTIONHE advantages of average current-mode control, such asT he ability to control the average inductor current and theimprovement of noise immunity, have been presented [ I ] . Thecontrol scheme is illustrated in Fig. 1. The inductor current issensed and fed into a compensation network to obtain its dcinformation. The output of the compensator is compared witha sawtooth ramp to generate PW M control. Since the averagecurrent is used as a controlled quantity, average current-mode

control is particularly suitable for power-factor correctioncircuits and for applications where a constant current sourceis needed, such as a battery charger circuit.The main difference between average current-mode controland peak current-mode control is that in the former, the induc-tor current is averaged and compensated by a compensationnetwork. In peak current-mode control, however, only theswitch current is sensed, and no compensation exists in currentloop. The PW M conductance control proposed in [2] sensesthe inductor current and compares it with a triangular rampto generate PW M control. Also, the switching frequency ofPWM conductance control is fixed; both turn-on time andtum-off time vary according to the control. Adding an integral-lead network to the current loop [3] makes PW M conductatxecontrol similar to average current-mode control. The low-frequency small-signal analysis of PWM conductance controlwas presented in [3].Small-signal analysis, especially a continuous-time small-signal model, is very useful in the control loop design ofpower supplies. As mentioned in [4], [ 5 ] , urrent-mode controlexhibits certain properties of a sampling system. One problem

Manuscript received November 6, 1991 ; revised September 25, 1992. Thiswork was supported in part by Lambda Electronics, Inc.The authors are with the Virginia Power Electronics Center, Departmentof Electrical Engineering, Virginia Polytechnic Institute and State University,Blacksburg, VA 24061.IEEE Log Number 9206671.

Fig. 1. Average current-mode control scheme.

of concern is the subharmonic oscillation at half the switch-ing frequency. In this paper, a complete small-signal modelis developed for average current-mode control. It generatesall the transfer functions needed for design purposes. Thesampling nature of current-mode control is considered in thedevelopment of the model, so that the subharmonic oscillationcan be predicted. Based on the small-signal model, designguidelines are established, and the model is experimentallyverified.11. SMALL-SIGNALODELING

One major difference between peak current-mode controland average current-mode control is that the inductor currentof the latter is averaged by a current compensator. Dueto the similarity of the two control methods, the modelingtechnique similar to that of peak current-mode control [ 5 ]canbe applied to the modeling of average current-mode control.The existence of the current compensator makes the modelingof average current-mode control more complicated.A . Modulator Gain

Because of the presence of the current compensator inaverage current-mode control, as shown in Fig. 2 , the currentloop transfer characteristics are quite different from those ofpeak current-mode control. In peak current-mode control, theinductor current is summed with an external ramp and directlycompared with the control voltage. The modulator gain of peakcurrent-mode control is [ 5 ]

where S, is the slope of the external ramp, S, is the inductorcurrent on-time slope, and T, is the switching cycle.0885-8993/93$03.00 0 993 IEEE


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TIW6:ol ai.: SMALL-SIGNAL MODELING OF AVERAGE CURRENT-MODE CONTROL 113

1Fig. 2. Current compensator and modulator.

From Fig. 2 , it can be seen that the inductor currentwaveform has been changed by the compensator, and theequivalent turn-on ( t=DT,) slope of the modified waveformSk can be calculated as

where

The modulator gain of average current-mode control is deter-mined by the sum of the external ramp slope and the turn-ontime slope of the modified current waveform:

B . Sampling GainIn the small-signal sense, the current loop behaves as asampling system [ 5 ] .According to sampling theory [6], thephase shift of the system transfer function is always real at halfthe sampling frequency (the sampling frequency of a current-

mode control system is equal to the switching frequency).When modeling peak current-mode control, the samplingeffect is approximated by the sampling gain [3] H e ( s ) , adouble RH P zero at half the switching frequency:

'Tw, =-Ts . (9)In average current-mode control, an integrator and a lead-lagnetwork are employed in the current loop. The zero is usuallyplaced before the power stage filter frequency to ensure thestability of the current loop, so the phase shift of the integratoris canceled by the zero at half the switching frequency.The effect of the second pole on the current loop was

studied. Due to the nature of the sampling system, the current

Vr MALL SIGNALWs)-AcG P H

Fig. 3. Small-signal model of average current-mode control.

loop gain always exhibits a 180" phase shift at half theswitching frequency no matter where the second pole isplaced. If the second pole is placed after half of the switchingfrequency, it does not significantly affect the current loop gain.Since the purpose of the second pole is to eliminate high-frequency noise, it should be placed after half of the switchingfrequency.

If the second pole of the current compensator is placed afterhalf of the switching frequency, the sampling gain used in peakcurrent-mode control can be used in average current-modecontrol. At high frequency, the characteristics of the currentloop of average current-mode control are almost the same asthose of peak current-mode control. Due to the existence of thelow-pass filter in the converter power stage, the voltage loopdoes not behave as a sampling system, and the compensatorpole exists in the voltage loop.C . Feedback and Feedfonvard Gainshown in Fig. 3, whereThe small-signal model of average current-mode control is

Wi (I +9

Although a current compensator exists in the current loop,the switching ripple at the output of the compensator is stillcomparable to the external ramp size, as shown in Fig. 4.Furthermore, the compensator output is a function of the inputand output voltages and the pole-zero locations of the currentcompensator. The duty cycle is affected by the perturbationsof input and output voltages.The effect of the input and output voltage perturbations onthe duty cycle can be modeled by feedback and feedforwardgain terms k , and k f from input and output voltages 131.


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114 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. E, NO . 2, APRIL 1993

TABLE IFEEDFORWARDND FEEDBACKAINSOR AVERAGE URRENT-MODEONTROLBuck Boost BuckBoost

DD 'T DD'Te tY E - % ( - ELL< -% (k , -E& - EL( * *k f LL 1 L

Fig. 4. Steady-state waveforms of average current-mode control

Referring to Fig. 2 , the following relation can be obtained:(12), =U, - G, ( s ) (R i i~ ,)

where G,(s) is the current compensator transfer functionGc(s )=G,(s)G,(s) (13)

and om is the output of the current compensator. If the average

b PLquantity is considered, (12) becomes v c =o

(14) Fig. 5. Simplified small-signal model for deriving feedforward gain k f .(wm)=21, -K (& ( ~ L ) ,)where the quantities ( i ~ ) nd ( w m ) denote the average valueof z~ and w respectively. K is the dc gain of the currentcompensator; it is finite for a real op-amp.

andFrom Fig. 4, the following equation can be derived:

( o m ) =Seton++AV,.Substituting (15) into (14),

K(RZ(2L)- ,) =U , - Set,, - +Equation (16) gives the relationship between the averageinductor current and input and output voltages. By perturbing(16) with respect to and w,, the Gg to transferfunction can be obtained. The same transfer function can alsobe derived from Fig. 5 , with k f as an unknown:

By comparing these two transfer functions, k f is obtained. k ,can be obtained through the same procedure. Table I lists k fand k , for three basic PW M converters employing averagecurrent-mode control where

a =

111. SMALL-SIGNALHARACTERISTICSBy using the small-signal model derived in the previoussection, all of the small-signal characteristics can be generatedthrough simulations. Simplified analytical transfer functionsare also derived to gain physical insight and to facilitate thedesign.A buck converter was used as an example to show the small-signal characteristics of the average current-mode control. Allof the characteristics are generated from the model shown inFig. 3 and are compared with those of peak current-modecontrol. The power stage parameters of the buck converter

wereV = 14 Vv, = 5 vR = 1 RL = 37.5 p HC = 380 pF


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TANG et al.: SMALL-SIGNAL MODELING OF AVERAGE CURRENT-MODE CONTROL 115

gain (dB)

-lo I-20 I I1m m 5m 1.m 2.000 5. m l 0 .W 2om

Frequency (HZ)phase (degree)

I

gain (dB)30 ,

u+ - 1435

.m'O lmLa xc ,.m z m L m l0.m mmFrequency Hz)

phase(degree)

Ilm w xa 1m z m I" l0.m amFrequency(Hz)

Fig. 6. Current loop gain with U ; as a running parameter.

Im zm rxr l.m z m 5 . m 1o .m ar.mFrequency(Hz)

Fig. 7. Current loop gain with wT as a running parameter.

R, =20 m52F, =50 kHz.The current loop elements value wereR; = 0.152Ri =2.2 kRR f = 30.5 k52Cj, =5.8 nFCf,= 220 pF.By varying R1 and Cf,, different w; and w; were obtained.

A . Current Loop GainCurrent loop gain is defined as the loop gain measured at

the output of the duty cycle modulator with the current loopclosed. Its analytical expression is given in (21):FmRiV, (1+sRC)T;(s )=

[l+s(;+cRc) + a Z L C ]

It can be seen from (21) that the dc gain of the current loop isaffected by both the extemal ramp (appearing in F,) and theintegrator gain U ; ; its shape is affected by the compensatorzero w,. While in peak current-mode control, the shape ofthe current loop gain is fixed, and the gain is only affectedby the slope of the extemal ramp; hence, there is morefreedom in designing the current loop for average current-mode control. Fig. 6shows a set of current loop gains withwi as a running parameter. The current loop gain with w , asa running parameter is shown in Fig. 7.It can be seen fromFigs. 6and 7 that the low-frequency portion of the current gainis much higher than that of peak current-mode control [3]. Itmeans that there is less low-frequency error in the current loopfor average current-mode control.

B . Control-to-Output Voltage GainThe control-to-output voltage gain is defined as the control-to-output voltage transfer function of average current-mode

control with the current loop closed. Fig. 8 shows thesetransfer functions with w; as the running parameter. It can beseen from Fig. 8 that the control-to-output gains have a low-frequency pole and a pair of double poles at half the switchingfrequency, and the damping of the double poles is affectedby the integrator gain of the current compensator (it is alsoaffected by the extemal ramp slope, which is not shown in Fig.8) . The current compensator pole w p also exists in control-to-output gain. An approximated expression of control-to-outputgain is given as follows:6 , R (1+sRcC)- N -, . -" Ri [1+ sR(C+Cx)]

where1c -- F,V,w;R;

and

I t - t -WnQpwhere

From (25), it can be seen that the damping factor of thedouble pole at half the switching frequency is affected by


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11 6 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 2, APRIL 1993

W

2 0 .

0

0

0

20

gain dB )wb.-.-..=.- \ ,\'wi - 7.55E+'$.


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-,o

-20

gain (dB)-20r

Experiment-Prediction - - - -

gain (dB)"Prediction - - - -

-Yi

Itm am xo 1m zm 5m mam 20000Frequency(Hz)

Fig. 11. Audiosusceptibility with w, as a running parameter..-

Since kf has a negative value, it is possible to completelyselecting Se and w,, but kf and F, have nonlinear relationswith Se and wi ,and it is difficult to find the exact values to nullis easy to choose an external ramp slope which totally nullsthe input perturbation.

null the circuit response to the input voltage perturbation by 9345

the audiosusceptibility. While in peak current-mode control, it o w r i m e n t -Prediction - - - -

45rm a0 xo i.m zmo 5m i0 .m zo.rmFrequency(Hz)

IV . DESIGNGUIDELINESAfter studying the small-signal characteristics of averagecurrent-mode control, certain design guidelines have beendeveloped.The current compensator can be designed as follows. First,the second pole should be placed after half the switchingfrequency. The zero should be placed at lesat one decadebefore half the switching frequency. Second, the function ofthe external ramp is similar to that of the sawtooth ramp involtage-mode control; this means that any sufficient large rampcan be used as the external ramp. Third, for a given ramp,choose the integrator gain w i which makes Q p =1. It givesproper damping on the resonant peak at half the switchingfrequency.By setting Q p in (25) equal to one

Fig. 12. Measurement and prediction of the current loop gain for abuck converter.

gain (de)ww - 7.55E+4-20Exprimen-

im am PO 1m zm 5m 10.m =.oxThe voltage loop design of average current-mode control issimilar to peak current-mode control.

Frequency (Hz)Fig. 13. Measurement and prediction of control-to-output voltage gain for abuck converter.

V. EXPERIMENTALERIFICATIONA buck converter was built with the same componentvalues as those given in the previous section. To measurethe current loop gain, a digital modulator [7] was used toensure that the correct sampled-data loop gain was obtained.

All other measurements were performed with conventionalanalog measurement schemes.The measured and predicted current loop gains are shownin Fig. 12. Both the gain and phase measurement agree verywell with predictions up to half of the switching frequency.

The control-to-output voltage and control-to-inductor cur-rent gains, measured with the current loop closed, are shownin Figs. 13and 14,respectively. The measurements again showvery good correlation with the theoretical results. The peak ofthe gains at half the switching frequency clearly shows theexistence of two complex poles.The measurement and predicted result of the audiosuscep-tibility of the buck converter with the current loop closed areshown in Fig. 15.Again, the measurement and prediction agreevery well.

.-


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118 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO . 2, APRIL 1993

gain (dB)

w, - 7.558+4\Iim w YI) ?.m z m s aa ro.mo z0.mFrequency (HZ)

phase (degree)

,80 Experiment-Prediction - - -

lca w %a l.ao z m Ia a 10.m w .m-zn Frequency (Hz)Fig. 14. Measurement and prediction of the control-to-inductor current gainfor a buck converter.

The current loop gain of average current-mode controlalways exhibits a 180 o phase shift at half the switchingfrequency no matter where the second pole of the currentcompensator is placed. If the pole is placed after half theswitching frequency, the noise picked up by the sensingnetwork is attenuated, while the stability of the system is notaffected.

The control-to-output voltage gain has more phase delaythan that of peak current-mode control because of the exis tenceof the second pole of the current compensator. The resonantpeak caused by the complex poles at half the switchingfrequency can be controlled by selecting the gain of the currentcompensator. In peak current-mode control, the resonant peakcan only be damped by the extemal ramp. This allows moreflexibility in the design of average current-mode control.Based on the small-signal analysis, a design guideline isproposed. By properly selecting the gain of the current com-pensator, subharmonic oscillation can be avoided. An almostflat control-to-inductor current gain can be achieved when thecurrent compensator is properly designed. The accuracy of themodel is confirmed with the measurements of a buck converter.

REFERENCES

Prediction - - -QIm aa ya lm 2 m la 0 ? o m am)Frequency(Hz)Fig. 15. Measurement and prediction of the audiosusceptibility for abuck converter.

VI . CONCLUSIONSIn peak current-mode control, the peak inductor current issensed and compared with the control voltage derived fromthe voltage loop. While in average current-mode control, itis the average inductor which compares with the controlvoltage. Hence, average current-mode control controls thereal averaged inductor current. This is particularly true whenthe converter is operated in the discontinuous conductionmode. When the buck converter employs average current-mode control, the output current, which equals the average

inductor current, is also controlled. It means that an idealcurrent source is achieved. Likewise, in the case of a boostconverter, the average input current is controlled, which makesit suitable for a power factor correction circuit.Because of the existence of the current compensator inthe control loop, the small-signal characteristics of averagecurrent-mode control are quite different from those of thepeak current-mode control. Due to the usage of an op-amp,the current loop gain of the average current-mode controlpossesses very high gain at low frequency. While in peakcurrent-mode control, the low-frequency gain of the currentloop is rather small.

[l ] L. H. Dixon, Average current-mode control of switching power sup-plies, in Unitrode Power Supply Design Seminar Handbook, 1990.[2] D. OSullivan, H. Spruyt, and A. Crausaz, PWM conductance control ,in IEEE Power E lectron. Specialists Con5 Rec. , 1988, pp. 351-359.[3] A. S . Kislovski, Small-signal low-frequency analysis of a buck typePWM conductance controller, in IEEE Power Electron. SpecialistsCon$ Rec. , 1990, pp. 88-95.[4] A. R. Brown, Topics in the analysis, measurement, and design of high-performance switching regulator, Ph.D. dissertation, Califomia Inst.Technol., Pasadena, May 1981.[ 5 ] R. B. Ridley, A new small-signal model for current-mode control,Ph.D. dissertation, Virginia Polytechnic Inst. State Univ., Blacksburg,Nov. 1990.[6] A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. En-glewood Cliffs, NJ: Prentice-Hall, 1975.[7] B. H. Cho and F. C. Lee, Measurement of loop gain with the digitalmodulator, in IEEE Power E lectron. Specialists Con5 Rec. , 1984, pp.363-373.

Wei Tang (S90) received the B.S. and M.S. degreesfrom Tsinghua University, Beijing, P.R. China, in1982 and 1985, respectively, both in electrical en-gineering. Since 1988 he has been working towardsthe Ph.D. degree at the Bradley Department ofElectrical Engineering, Virginia Polytechnic Insti-tute and State University, Blacksburg.From 1985 to 1988 he worked as a Senior En-gineer at the Research Center of Computer Appli-cations, Chengdu, China. He is now a ResearchAssistant at the Vireinia Power Electronics Center.His research interests include modeling, analysis, and control of the switchingpower converter, and power factor correction.Mr. Tang is a member of Eta Kappa Nu.


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Fred C. Lee (S72-M74-SM87-F90) receivedthe B.S. degree in electrical engineering from theNational Cheng Kung University, Taiwan, in 1968,and the M.S. and Ph.D. degrees from Duke Univer-sity, Durham, NC, in 1971 and 1974, respectively.From 1974 to 1977 he was employed as a memberof the Technical Staff at TRW Systems, RedondoBeach, CA. He was responsible for designing powerconverters for spacecraft power systems. He joinedVirginia Tech in 1977. He is presently the James S .Tucker Professor in the Department of Electrical En-gineering and the Director of the Virginia Power Electronics Center (VPEC).In 1987 he was appointed as the direc tor of the Technology DevelopmentCenter for Power Electronics of the Virginia Center for Innovative Technology.He is the founder of the Centers Industry Partnership Program. To date,62 companies from all over the world have subscribed to the program.His research interests include high-frequency power conversion, distributedpower systems, space power systems, device characterization, and modelingand control of converters and design optimization. During his career, he haspublished over 80 refereed joumal papers, and more than 150 technical papers.Dr. Lee has received nine best paper awards from various technicalconferences. He has been awarded ten patents, with five additional patentspending. He has been an active consultant for over 20 companies in thepower electronics industries. He is currently a member of the AdCom ofthe IEEE Power Electronics Society, and an Associate Editor of the IEEETRANSACTIONS ON POWER ELECTRONICS . He was the Chairmanof the 1987 IEEE Power Electronics Specialists Conference. Dr. Lee is arecipient of the Society of Automotive Engineering 1985 Ralph R. TeeterEducational Award, the IEEE Power Electronics Society 1989 William E.Newel1 Power Electronics Award, the 1990 PCIM Award for Leadership inPower Electronics Education, and the Virginia Tech 1990 Alumni Award forResearch Excellence.

Raymond B. Ridley (S90-M90) received theB.S. degree from Boston University, Boston, MA,in 1981 and the M.S. and Ph.D. degrees fromVirginia Polytechnic Institute and State University,Blacksburg, in 1986 and 1990, respectively , all inelectrical engineering.From 1981 to 1984 he was employed as a SeniorEngineer in the Power Systems Group at PrimeComputer, where he worked on the design andanalysis of computer power supplies. His researchspecialties nclude power converter control and anal-ysis, high-frequency converters, and computer-aided design for power sys-tems. He served as the Assistant Director of Virginia Power Electronics Centerfrom 1987 to 1991. Presently, he is a consultant in power electronics.

small-signal modeling of average current-mode control

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