multifrequency small-signal model for buck and multiphase buck converters

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 5, SEPTEMBER 2006 1185

Multifrequency Small-Signal Model forBuck and Multiphase Buck Converters

Yang Qiu, Ming Xu, Senior Member, IEEE, Kaiwei Yao, Senior Member, IEEE, Juanjuan Sun, andFred C. Lee, Fellow, IEEE

Abstract—To investigate the high-frequency behaviors of buckand multiphase buck converters, this paper develops a modelingmethod based on the harmonic-balance approach. Because thenonlinear pulse-width modulator generates sideband components,the sideband effect occurs in a closed-loop converter. Taking thiseffect into account, the multifrequency small-signal model is pro-posed, which is applicable beyond half of the switching frequency.In a voltage-mode-controlled buck converter, the introducedmodel predicts the measured phase delay of the loop gain, whilethe conventional average model fails to explain this phenomenon.Furthermore, this model is extended to the case of the multiphasebuck converter. The influence from the interleaving technique isdiscussed and the frequency-domain characteristics are clearlyexplained. Simulation and experimental results are provided toverify the achievements of the proposed model.

Index Terms—Buck converter, high frequency, multifrequencymodel, multiphase, sideband effect.

I. INTRODUCTION

MOORE’S law has proven true for more than 40 years,and will continue to predict the future of computer pro-

cessors. With more transistors and higher processing speeds,the operating voltage of the next generation of processors willcontinue decreasing to avoid overheating the silicon [1]. Mean-while, a more dynamic load is expected with higher speeds.Consequently, many stringent challenges are imposed on thevoltage regulator (VR), which is the dedicated power supply ofthe processor. One matter of particular concern is how to meetthe requirement of fast transient responses with fewer output ca-pacitors. Besides reasons related to cost, this is also necessarybecause of the limited space for VRs in the computer system.

The multiphase synchronous buck converter, as shown inFig. 1, is widely adopted in the VR design [2], [3]. Many papershave discussed the multiphase VR’s transient response andhow to improve it [4]–[8]. It has been shown that the feedbackcontrol loop’s bandwidth plays a very important role in thetransient response. With a higher bandwidth, fewer output

Manuscript received May 18, 2004; revised June 22, 2005. This work was pre-sented at APEC’04, Anaheim, CA, February 22–26, 2004. This work was sup-ported by Artesyn, Delta Electronics, Hipro Electronics, Infineon, Intel, Interna-tional Rectifier, Intersil, Linear Technology, National Semiconductor, Renesas,Texas Instruments, the Engineering Research Center Shared Facilities, Transim,and the National Science Foundation under NSF Award EEC-9731677.

Y. Qiu, M. Xu, J. Sun, and F. C. Lee are with the Center for Power ElectronicsSystems (CPES), Virginia Polytechnic Institute and State University, Blacks-burg, VA 24061 USA (e-mail: [email protected]).

K. Yao is with Monolithic Power Systems, Los Gatos, CA 95032 USA.Digital Object Identifier 10.1109/TPEL.2006.880354

Fig. 1. Multiphase synchronous buck converter.

capacitors are needed for the required transient performance[7], [8]. Therefore, it is advantageous to push the bandwidth ashigh as possible with a limited switching frequency.

In the past, most feedback controller designs have been basedon the average model of buck converters. However, because thestate-space averaging process eliminates the inherent samplingnature of the switching converter, the accuracy of the averagemodel is questionable at frequencies approaching half of theswitching frequency [9]. As a result, the relationship betweenthe control-loop bandwidth and the switching frequency has notbeen clearly understood.

To predict the sub-harmonic oscillations at half of theswitching frequency in peak-current-controlled converters,sampled-data approaches, hybrid approaches, and harmonic-balance approaches have been proposed [9]–[14]. For thevoltage-mode control, it is found that the average model shouldalso be reexamined if a high bandwidth is desired.

As an example, Fig. 2 compares the loop gain for a 1-MHzsingle-phase buck converter with voltage-mode control cal-culated from the average model with the loop gain obtainedby using SIMPLIS software. SIMPLIS performs very fastswitching-model small-signal ac analysis based on a schemesimilar to that used in the measurements, except that theswitching ripples are not considered. Hence, the simulated

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1186 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 5, SEPTEMBER 2006

Fig. 2. Loop gains of a 1-MHz buck converter with voltage-mode control:(a) 100-kHz-bandwidth design and (b) 400-kHz-bandwidth design.

transfer function from SIMPLIS is almost the same as that fromthe measurement.

In Fig. 2, in the case with a 100-kHz bandwidth of the voltageloop, the average model agrees with the simulation up to half ofthe switching frequency. However, in the 400-kHz bandwidthdesign, the average model is good only up to 100 kHz, i.e., one-tenth of the switching frequency. The simulation result has 25less phase margin than the average model. This excessive phasedelay would result in an undesired transient or reduced stabilityif a high-bandwidth converter is designed based on the averagemodel, which cannot predict high-frequency behaviors.

Although the harmonic-balance technique predicts the phasedelays at high frequencies, it is not an efficient means ofextracting physical insights out of the complicated model.In order to simplify the modeling, as well as to investigate

Fig. 3. Open-loop buck converter with V perturbation.

the control-loop bandwidth limitations and to improve thecontrol designs, it is essential to have a clear picture of theconverter up to the switching frequency. Therefore, this paperinvestigates the high-frequency characteristics of buck andmultiphase buck converters, and introduces the multifrequencymodel for the high-frequency analysis. First, the nonlinearityof the pulse-width modulation (PWM) scheme is reviewed inSection II. The sideband effect, which results in the discrepancybetween the average model and measurement, is identified forclosed-loop converters. After that, Section III introduces themultifrequency model to predict the system behavior. Withthe Fourier analysis, the relationships among different fre-quency components are derived for the PWM comparator. InSection IV, the proposed model is applied to the multiphasebuck converter. Simulation and experimental results are pro-vided as verification.

II. SIDEBAND EFFECT OF PWM CONVERTERS

WITH FEEDBACK LOOP

Fig. 3 illustrates the structure when analyzing the responseof an open-loop single-phase buck converter with a perturba-tion at the control voltage, . In the small-signal analysis, itis assumed that the perturbation is small enough not to changethe operating point of the converter. When studying the per-formance at a certain frequency, the perturbation is assumedto be sinusoidal for simplicity. In this paper, the trailing-edgePWM comparator is used as an example for discussion, whilethe same methodology can be extended to the leading-edge anddouble-edge modulations. With a sinusoidal perturbation fre-quency at , the spectra of , and that of the output, , are illus-trated in Fig. 4 [14], [15]. Clearly, the spectrum of consists ofthe dc component, the components at the switching frequency,

, and its harmonic frequencies. The components at the pertur-bation frequency, and , appear at the comparator outputas well. Meanwhile, because the PWM comparator works like asample-data function, its output, , has infinite frequency com-ponents at , , , , etc. Thesefrequencies are called the sideband frequencies or the beat fre-quencies around , , etc., which do not exist at the input ofthe comparator. Hence, the PWM comparator is a typical non-linear function.

In this paper, for the sake of simplicity only the perturbationat is considered. With the assumption that the input voltage,

, is constant, the buck converter’s phase voltage, , has awaveform similar to . Hence, has the same number of fre-quency components as and there is no additional frequencygenerated. The only difference is that the magnitudes of ’s

QIU et al.: MULTIFREQUENCY SMALL-SIGNAL MODEL 1187

Fig. 4. Spectra of the PWM comparator: (a) input spectrum and (b) outputspectrum.

Fig. 5. Frequency-domain representation for the open-loop buck converter withsideband components.

frequency components are times as high as those of . There-fore, with the constant input voltage assumption, the function ofthe two switches in the buck converter is a typical linear func-tion. Meanwhile, the output filter topology of buck convertersdoes not change during the switch on time and switch off time,so it is also a linear function. All the components at ap-pear at the output voltage, , through the output low-pass filter.In summary, the PWM comparator is the only nonlinearity forthe open-loop buck converter with perturbations at the controlvoltage.

When analyzing the small-signal stability and the transientperformance of a converter, it is not necessary to include thecomponents at dc, the switching frequency or its harmonics.Only the consequence of the perturbation, i.e., the componentsat the perturbation frequency and the sideband frequencies,needs to be included in the models, which is shown in Fig. 5.When studying the case with below , the lowest sidebandfrequency is – . In the conventional average model, it isassumed that the sideband components can be well attenuatedby the low-pass filters in the converter; therefore, only thecomponents are included.

Clearly, when is higher than 2, this assumption is nolonger valid since – is lower than . As an example, Fig. 6shows the waveforms when is 990 kHz for a 1-MHz open-loop buck converter. In this converter, is 12 V, is 1.2 V,

is 200 nH, is 1 mF, is 80 m , the peak-to-peak valueof the sawtooth ramp, , is 1 V. The sinusoidal perturbation at

Fig. 6. Simulated V waveform with 990-kHz V perturbation for a 1-MHzopen-loop buck converter.

Fig. 7. Voltage-mode-controlled buck converter with V perturbation.

Fig. 8. Sideband effect in a voltage-mode-controlled buck converter.

has a magnitude of 5 mV. Because – 10 kHz is muchlower than , the output filter has more attenuation at thanat – . Hence, the 10-kHz sideband frequency component isthe dominant one in the simulated waveforms. Therefore, thesystem performance cannot be reflected by only considering theperturbation-frequency components.

Nevertheless, in an open-loop buck converter, because thereis only one perturbation frequency at , ’s component canstill be predicted by the average model. However, for a buck con-verter with closed-loop control, as in Fig. 7, the system becomesmuch more complicated. Fig. 8 illustrates the relationship be-tween the components of and – . The generated sidebandcomponent, – , is fed back through the voltage compen-sator, , and added to the perturbation sources. Consequently,the control voltage, , includes both the perturbation-frequencycomponent, , and the sideband component, – .As – is sent into the PWM comparator, it generates theperturbation-frequency component of at the output ofthe comparator as well. Consequently, the component at


Fig. 9. Space vector representation ofV ’s f component with sideband effect.

Fig. 10. Simulated V waveform of a 1-MHz voltage-mode-controlled buckconverter with a 10-kHz perturbation.

includes two parts, as shown in Fig. 9. The part of is from, and the part of comes from – .

Therefore, unlike the open-loop case, contains not onlythe components, but also the – components. Throughthe PWM comparator, the components and the –components are coupled. The sideband effect happens inthe closed-loop converters, i.e., the perturbation-frequencycomponents are influenced by the fed-back sideband compo-nents, which are generated by the PWM comparator. In theconventional average model, only the component comingfrom is included. If the influence from the sidebandcomponents, , is so small that it can be ignored, theaverage model might be sufficient. Otherwise, the sidebandfrequencies should be considered to accurately represent thesystem performance in the model.

To qualitatively investigate the validity of the averagemodel for a buck converter with the voltage-mode control,the switching-model simulation is performed with different

. A buck converter with the parameters above is used as anexample. The voltage-loop bandwidth is 100 kHz with a 67phase margin. Figs. 10 and 11 compare the simulated wave-forms for the 1-MHz buck converter with 40-mV perturbations.With a 10-kHz perturbation, it is observed that at there exista 1-mV 10-kHz component and a 6.5-mV switching ripple. The990-kHz sideband component can hardly be observed. There-fore, is the dominant input component of the PWMcomparator. In the case of a 990-kHz perturbation, includeslarge magnitude components at both 10 kHz and 990 kHz. The10-kHz and 990-kHz components have similar magnitudes of39 mV and 40 mV, respectively. Therefore, the influence from

– is significant.From these two cases, it can be concluded that the sideband

effect becomes more significant with a higher and hencelower – . The reason for this is that the feedback controlloop (including the power-stage output filter and the compen-sator) functions as a low-pass filter. Therefore, there is suffi-cient attenuation for the sideband components when is low.

Fig. 11. Simulation of a 1-MHz voltage-mode-controlled buck converter witha 990-kHz perturbation: (a) V waveform and (b) closed-up V waveform.

When is high, the attenuation is weak for – and the side-band effect cannot be ignored. Clearly, the average model is notsuitable for the high-frequency analysis by only considering theperturbation-frequency component and ignoring the sidebandcomponent.

III. MULTIFREQUENCY SMALL-SIGNAL MODEL

The harmonic balance method is promising for further studyto explain the sideband effect. However, it is not widely uti-lized by designers because of the large amount of calculations.In a buck converter with ideal components and a constant inputvoltage, the only nonlinear function is the PWM comparator.If the extended describing functions considering the sidebandcomponents can be derived, it is possible to obtain a simplehigh-frequency model including the sideband components.

Using Fourier analysis, the PWM describing function hasbeen derived considering the same input and output frequency[16]. It is a simple gain as

(1)

if 2. Here, is the peak-to-peak value of thePWM ramp. For the relationship between different frequencies,a similar approach is employed in this paper. Fig. 12 illustratesthe input and output waveforms of the modulator. The controlvoltage is

(2)


Fig. 12. Input and output waveforms of the PWM comparator with a sinusoidalV perturbation.

and the duty ratio for the th cycle is

(3)where is the th cycle on-time, is the switching period,and

(4)

With the small-signal approximation, the sinusoidal wave’samplitude, , is much smaller than the dc value, . Therefore

(5)

In this paper, it is assumed that

(6)

where and are positive integers. If the relationship be-tween and cannot be expressed by (6), double integralsare required. However, the result is the same and this aspect willnot be discussed here.

To obtain the coefficient for the sideband component, Fourieranalysis is performed in the complex frequency domain. Itshould be noted that a real signal could be represented by eitherthe positive frequency or the negative frequency, although thenegative frequency has no physical sense. For example, when

is lower than , the sideband component can be representedby the positive frequency, – , or the negative frequency at

– . In the following modeling and derivations, the negativefrequency is utilized, because it leads to a simpler mathematicalexpression of the relationship between the sideband and theperturbation frequency components.

With the assumption in (6) and the definition of the Fouriercoefficient for a periodical signal

(7)

Because

(8)

it is derived that

(9)

If 2, (9) is rewritten by applying the small-signalapproximation as

(10)

The Fourier coefficient for the control signal is

(11)

therefore

(12)

Using the same method, it is obtained that at 2

(13)

With (1), (12), and (13), the proposed multifrequency modelis illustrated in Fig. 13. In this model, there are two feedbackloops, which represent the perturbation frequency and the side-band frequency, respectively. Although each feedback loop rep-resents a certain frequency, these two loops are coupled throughthe PWM comparator. With a certain frequency , the loop gainis

(14)

where is the transfer function of the compensator, andis the phase-voltage-to-output transfer function of the

filter. Clearly, the loop gain is exactly the loopgain in the traditional average model. With the multifrequencymodel, it is calculated that

(15)


Fig. 13. Multifrequency model of a voltage-mode-controlled buck converter.

Fig. 14. Loop gain of a 1-MHz buck converter with voltage-mode control.

is the loop gain that determines the stability and transient per-formance. The influence from the sideband effect is shown inthe denominator.

Up to the switching frequency, Fig. 14 compares the loop gainin the average model, in the SIMPLIS simulation and in the mul-tifrequency model. Fig. 15 shows the measurement result. When

is located in the low-frequency region, – is out of thevoltage-loop bandwidth. The denominator of (15) is approxi-mately equal to one. Therefore, the average model is accurate.When becomes high and approaches the switching frequency,

– goes into the voltage-loop bandwidth. Under this condi-tion, the denominator of (15) is approximately the loop gain at

– , which is much higher than one. This high loop gain re-sults in a dip around the switching frequency. While the averagemodel fails to predict the dip, it does exist in both the experimentand in the switching model simulation.

According to (15) and the previous analysis, the significanceof the sideband effect is determined by the bandwidth of .When the bandwidth of is higher, there is wider frequencyrange where the influence from the sideband components is sig-nificant, as shown in Fig. 2. Therefore, the sideband effect limitsthe possibility of high-bandwidth designs. For the stability anal-ysis at these cases, it is necessary to use the proposed multifre-quency model instead of the conventional average model.

Fig. 15. Measured loop gain of a 1-MHz buck converter with voltage-modecontrol.

IV. MULTIFREQUENCY MODEL OF MULTIPHASE

BUCK CONVERTER

From the output voltage point of view, the ripple frequencyis effectively increased with the multiphase buck converter. Ithas been argued that the equivalent switching frequency for an

-phase buck converter is -times the switching frequency ofeach phase, so the control loop bandwidth can be increased withthe multiphase buck converter [17]. Most of the previous anal-ysis is predominantly based on intuition or simulation results.To theoretically investigate the influence of a multiphase buckconverter, the multifrequency model is applied.

For the th phase in an -phase interleaving buck converter

(16)

Through a process similar to that used in the case of a single-phase buck converter, (1) is still correct for the same input andoutput frequency. For different input and output frequencies, itcan be derived that if 2, then

(17)

Similarly

(18)

Based on (1), (17), and (18), the small-signal model at foran -phase buck converter is illustrated in Fig. 16. Equation (17)shows that if and the transfer function are the samefor all the phases, the influence from the – component on

is cancelled at . Therefore, the dip is no longer expectedto occur around the switching frequency, as demonstrated by


Fig. 16. Multifrequency model of the n-phase buck converter.

Fig. 17. Loop gain of a 1-MHz two-phase buck converter with voltage-modecontrol.

the two-phase example in Fig. 17. The lowest-frequency dip isaround twice the switching frequency because the – com-ponent as the sideband of cannot be cancelled.

Because of the cancellation of the sideband components at theoutput voltage in the multiphase buck converter, it is possible topush the bandwidth higher than that in the single-phase case.For example, with voltage-mode control, the bandwidth of thetwo-phase buck converter can be pushed to 400 kHz with a 60phase margin, as shown in Fig. 18.

However, in the loop gain measurement result, a dip existsaround the switching frequency. This is because in the real im-plementation, the asymmetry of the two phases results in only apartial cancellation of sideband effect. According to Fig. 16, ifthe two phases’ or are different, or if the two channelsdo not have a phase shift of exactly 180 , the influence from the

– frequency component cannot be totally canceled. Fig. 19compares the ideal case and a practical case with 20% in-ductor tolerance for a 400-kHz bandwidth design. Because thesideband effect cannot be fully canceled in practice, it results inan additional 8 phase delay at the crossover frequency whencompared with symmetrical phases. Furthermore, the phase re-sponse decreases rapidly after the crossover frequency. In terms

Fig. 18. Experimental loop gain of a high-bandwidth 1-MHz two-phase buckconverter with voltage-mode control.

Fig. 19. Simulated loop gain of 1-MHz two-phase voltage-mode-controlledbuck converter: ideal case vs. �20% inductor tolerance.

of this aspect, the remaining sideband effect’s influence cannotbe ignored in high-bandwidth designs. Therefore, the bandwidthlimitation with voltage-mode control for the multiphase buckconverter is limited in the real case, although it is theoreticallyincreased. There are risks for using asymmetric phases to pushthe voltage bandwidth higher.

V. CONCLUSION

For high-bandwidth voltage-mode-controlled buck con-verters, the average model fails to predict the phase delays eveninside the control-loop bandwidth. This is because the sidebandeffect happens with a closed-loop converter; i.e., the sidebandcomponents generated by the PWM comparator are fed back toits input and then generate the perturbation-frequency compo-nents at the output. To describe this phenomenon and improvethe design, this paper introduces the multifrequency model,


which is valid above half the switching frequency. With thismodel, it is revealed that there is a bandwidth limitation posedby the sideband effect. Furthermore, the model is extendedto the case with multiphase buck converters, where the inter-leaving technique is possible to cancel the sideband effect inthe voltage loop. Therefore, the control-loop bandwidth can betheoretically pushed higher; however, the asymmetric phasesthat exist in implementation result in design risks in pushingthe control-loop bandwidth.

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[17] P.-L. Wong, “Performance Improvements of Multi-Channel In-terleaving Voltage Regulator Modules With Integrated CouplingInductors,” Ph.D. dissertation, Virginia Tech, Blacksburg, 2001.

Yang Qiu received the B.S. and M.S. degrees inelectrical engineering from Tsinghua University,Beijing, China, in 1998 and 2000, respectively,and the Ph.D. degree from the Virginia PolytechnicInstitute and State University (Virginia Tech),Blacksburg, in 2005.

He is currently a Postdoctoral Researcher with theCenter for Power Electronics Systems, Virginia Tech.His research interests include high-frequency powerconversion, distributed power systems, low-voltagehigh-current conversion techniques, and modeling

and control of converters.

Ming Xu (SM’00) received the B.S. degree inelectrical engineering from the Nanjing Universityof Aeronautics and Astronautics, Nanjing, China, in1991, and the M.S. and Ph.D. degrees from ZhejiangUniversity, Hangzhou, China, in 1994 and 1997,respectively.

He is currently an Assistant Research Professor atthe Center for Power Electronics Systems, VirginiaPolytechnic Institute and State University, Blacks-burg. He holds four U.S. patents, seven Chinesepatents, and has eight U.S. patents pending. He has

published one book and approximately 80 technical papers in journals andconferences. His research interests include high-frequency power conversion,distributed power system, power factor correction techniques, low-voltagehigh-current conversion techniques, high-frequency magnetic integration, andmodeling and control of converters.

Kaiwei Yao (SM’04) received the B.S. degree fromXian Jiaotong University, Xi’an, China, in 1992, theM.S. degree from Zhejiang University, Hangzhou,China, in 1995, and the Ph.D. degree from theVirginia Polytechnic Institute and State University(Virginia Tech), Blacksburg, in 2004, all in electricalengineering.

From 1995 to 1998, he was an Engineer for UPSDesign, Hwadar Electronics, Shenzhen, China. FromMarch 2003 to September 2004, he was TechnicalCoordinator at the Center for Power Electronics

Systems (CPES), Virginia Tech. Since October 2004 he has worked as a SeniorSystems Engineer at Monolithic Power Systems, Los Gatos, CA. His researchinterests include power management, resonant converters and inverters, andsystem modeling and control.

Juanjuan Sun received the B.S. and M.S. degreesin electrical engineering from Tsinghua University,Beijing, China, in 1999 and 2001, respectively, andis currently pursuing the Ph.D. degree at the Centerfor Power Electronics Systems, Virginia PolytechnicInstitute and State University, Blacksburg.

Her research interests include high-frequencypower conversion, low-voltage high-current conver-sion techniques, modeling and control of converters,and distributed power systems.

Fred C. Lee (S’72–M’74–SM’87–F’90) receivedthe B.S. degree in electrical engineering from theNational Cheng Kung University, Taiwan, R.O.C.,in 1968 and the M.S. and Ph.D. degrees in electricalengineering from Duke University, Durham, NC, in1971 and 1974, respectively.

He is a University Distinguished Professor withVirginia Polytechnic Institute and State University(Virginia Tech), Blacksburg, and prior to that he wasthe Lewis A. Hester Chair of Engineering at VirginiaTech. He directs the Center for Power Electronics

Systems (CPES). He is also the Founder and Director of the Virginia PowerElectronics Center (VPEC).

Dr. Lee received the Society of Automotive Engineering’s Ralph R. TeeterEducation Award (1985), Virginia Tech’s Alumni Award for Research Excel-lence (1990), and its College of Engineering Dean’s Award for Excellence inResearch (1997), in 1989, the William E. Newell Power Electronics Award, thehighest award presented by the IEEE Power Electronics Society for outstandingachievement in the power electronics discipline, the Power Conversion andIntelligent Motion Award for Leadership in Power Electronics Education(1990), the Arthur E. Fury Award for Leadership and Innovation in AdvancingPower Electronic Systems Technology (1998), the IEEE Millennium Medal,and honorary professorships from Shanghai University of Technology,Shanghai Railroad and Technology Institute, Nanjing Aeronautical Institute,Zhejiang University, and Tsinghua University.

multifrequency small-signal model for buck and multiphase buck converters

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