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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 3, JUNE 2004733

[8] Z. Wang, J. Lam, and K. J. Burnham, “Stability analysis and observer

designforneutraldelaysystems,”IEEE.Trans.Automat.Contr.,vol.47,

pp. 478–483, Mar. 2002.

[9] L. Dugard and E. Verriest, Stability and Control of Time-Delay Sys-

tems.Berlin, Germany: Springer, 1998.

[10] Y. J. Cao and Q. H. Wu, “A note on stability of analog neural networks

withtimedelays,”IEEETrans.NeuralNetworks,vol. 7,pp. 1533–1535,

Nov. 1996.

[11] Z. Shao and J. Rowland, “Robust stability of time-delay singularly per-

turbed systems,” Proc. Inst. Elect. Eng., pt. D, vol. 142, pp. 111–113,

1995.

[12] T. Mori and H. Kokame, “Stability of ? ???? ? ????? ? ???? ? ??,”

IEEE. Trans. Automat. Contr., vol. 34, pp. 460–462, Apr. 1989.

A Comprehensive Analysis of Current-Mode Control for

DCM Buck-Boost Converters

Keng C. Wu

Abstract—Comprehensive analyses for the buck–boost, pulsewidth-

modulation dc–dc converters applying peak current current-mode control

are given. The analysis provides closed-form solutions for steady-state

output, small-signal loop gain, and conducted susceptibility. It also proves

that the state-space averaged model developed for converter using a

single-loop voltage-mode control is valid for a current-mode-controlled

converter.

Index Terms—Conducted susceptibility, current-mode control, loop

gain, sensitivity.

I. INTRODUCTION

By nature, signals in current forms have advantages over voltage

form, since voltage is an accumulation of electron flux and therefore

is slow in time as far as control mechanism is concerned. This

understanding spawned in the late 1970s a new tide in switch-mode

power supply design, namely, the current-mode control [1]. However,

by adding a current loop, the conventional concept of loop gain is

blurred, since multiple loops exist and make it difficult to identify the

main loop [2].

The current-mode control techniques, in addition to introducing dif-

ficulties in loop identification, also create new territories for analysis.

[1] and [2] developed current programmed model using state-space av-

eragingandresultedinaverycomplicatedy-parameterbase.Reference

[3] considered current-mode control as a sort of conductance control

and suggestedsuch a converterasacurrentsource. Reference[4] intro-

duced the sampled-data control concept and additional gain factor that

was claimed to improve theoretical loop gain prediction. Reference [5]

attempted to improve the average-current current-mode model. How-

ever,givenstreamsofstudiesforcurrent-modeoperation,mostexisting

reports either did not provide a model that was easy to use or employed

a mathematical procedure thatproduced questionableresults. With this

in mind and using an actual design as the base, this paper presents a

complete analysis for the flyback converter in the discontinuous con-

duction mode (DCM) with peak-current current-mode control.

Manuscript received July 15, 2002; revised November 18, 2003. Abstract

published on the Internet January 14, 2004.

The author is with Lockheed Martin, Moorestown, NJ 08057 USA (e-mail:

keng.c.wu@lmco.com).

Digital Object Identifier 10.1109/TIE.2004.825204

II. CLOSED-LOOP STEADY STATE

A. Closed-Form Output Equation

Refer to Fig. 1(a), where the schematic of a flyback converter is

shown. At steady state, that is, under constant load and constant line

input, the closed-loop output is given as

?? ????? ???? ???????????????? ????????

?????? ???? ??? ?????? ??? ???

????

??

????

??

?????

??

????? ????????????????????

?? ?? ? ????

?? ?

? ?

??

?? ????

?? ? ? ??

?? ????

??????????

????? ????????????????????

??

????????????

?? ????????????

??????????

?? ??

????????

?? ? ??

??

???????

??? ?? ? ??? ??

??? ???

??? ???? ????

??

The power stage gain ???is given in [6] and [7].

?? ?

??

???

??? ? ? ??

???

?

? ? ??? ???

B. Output Sensitivity

The closed-form solution gives designers the ability to evaluate

many performance merits of a design analytically and numerically.

Among them, load regulation, line regulation, and component sensi-

tivities are the three most sought after figures. For instance, the load

regulation sensitivity can be expressed as

???

???

????

????

??

?

III. AC LOOP GAIN

In order to study the small-signal behavior of the converter, the

schematic of Fig. 1(a) is transformed into Fig. 1(b) in which the

transfer functions of individual blocks are identified and derived as

follows.

The first error amplifier gives

?????? ? ?

??

??? ???

????

??

?

????????

?

???

where ?? ? ?????? and

???? ?

?

?

?????? ?

?

The second-stage error amplifier gives

?????? ???

??

? ???? ? ?

????

?????? ?

??????

?

????

?

?

?

0278-0046/04$20.00 © 2004 IEEE

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734IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 3, JUNE 2004

Fig. 1.(a) Circuit schematic. (b) Small-signal block diagram.

The transistor stage gains are

??????

?

???

?

?

?

?

?

?????

??

?

?

?

??

?

??

?

??

?

?

???

?

???

??

?

?

??

?????? ?

???

?

?

?

?

?

??

??

???

?

?

?

?

??

?

?

???

?

??

?

??

???

?

?

?

??

?

?

?

?

??

??

???

?

?

?

?

? ???

???? ????

???? ???? ?????????? ? ??

??

???

????

?

?

? ??? ?

???

The PWM gain is

?? ?

??

????

???? ??? ?

??? ??

?? ?

??

???

????? ??? ? ? ???

??? ??

?

?

Byinvokingthecanonicalmodelgivenin[6]and[7],thepowerstage

duty-cycle-to-output transfer function can be shown to be

????? ? ? ? ????

??

??? ?????

??? ?????

??

??

? ????

?? ??? ??

The loop gain is finally given as

???? ??????? ? ???? ?????? ? ????? ? ??? ??????

? ? ?????? ? ????? ? ??? ??????

?

The theoretical loop gain as shown in Fig. 2(a) compares extremely

well against the actual measurement of Fig. 2(b).

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 3, JUNE 2004735

Fig. 2.Loop gain. (a) Theoretical. (b) Measurement 10 dB/div, 45 ?div.

Fig. 3. Conducted susceptibility. (a) Theoretical. (b) Measurement. ?10 dB/div.

IV. CONDUCTED SUSCEPTIBILITY

By including the feedforward effect, a different loop gain is given

?????? ? ??????? ? ?????? ? ?????? ? ???????

?????? ? ??? ??????

The conducted susceptibility is then expressed as

????? ?

?

?????

?

? ?????

?

??? ??? ??? ?????

? ? ??????

?

The theoretical conducted susceptibility Fig. 3(a) matches well the

actual measurement, Fig. 3(b).

V. CONCLUSION

A complete analysis was given for the flyback dc–dc converter

with peak-current current-mode control. The power stage transfer

functions are derived based on the voltage-mode canonical model

given in [6]. However, combinational effects of voltage-loop feedback

???? and current-loop feedforward ???? make the model valid for the

current-mode control. A superb match between the analytical results

and the actual measurements supports the viewpoint.

REFERENCES

[1] S.-P. Hsu, A. Brown, L. Rensink, and R. D. Middlebrook, “Modeling

and analysis of switching DC-to-DC converters in constant-frequency

current-programmedmode,”inAdvancesinSwitched-ModePowerCon-

version, S.Slobodan Cuk and R. D. Middlebrook, Eds.

TESLAco, 1983, pp. 284–301.

[2] R. D. Middlebrook, “Topics in multiple-loop regulators and cur-

rent-mode programming,” IEEE Trans. Power Electron., vol. PE-2, pp.

109–124, Apr. 1987.

[3] D. O‘Sullivan, H. Spruijt, and A. Crausaz, “Pulse-width-modulation

(PWM) conductance control,” ESA J., vol. 13, pp. 33–46, 1989.

[4] R.B.Ridley,“Anew,continuous-timemodelforcurrent-modecontrol,”

IEEE Trans. Power Electron., vol. 6, pp. 271–280, Apr. 1991.

[5] W. Tang, F. C. Lee, and R. B. Ridley, “Small-signal modeling of av-

erage current-mode control,” IEEE Trans. Power Electron., vol. 8, pp.

112–119, Apr. 1993.

[6] S.CukandR.D.Middlebrook,“Ageneralunifiedapproachtomodeling

switching DC-to-DC converters in discontinuous conduction mode,” in

Proc. IEEE PESC’77, 1977, pp. 36–56.

[7]

, “Modeling, analysis, and design of switching converters,” NASA,

Houston, TX, NASA CR-135174, 1980.

Pasadena, CA: