IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 12, NO. 3, MAY 1997 453Time-Varying Effects and Averaging Issuesin Models for Current-Mode ControlDavid J. Perreault, Student, IEEE, and George C. Verghese, Senior Member, IEEEAbstract—This paper investigates issues in modeling of current-mode control. The effects of the current-sampling intrinsic tocurrent-mode control are analyzed, and inadequately recognizedlimitations of linear time-invariant (LTI) models at high frequen-cies (where the system behavior is time-varying) are exposed. Thepaper also examines the geometric methods used to derive duty-ratio constraints in averaged models of current-mode control andpoints out the sources of discrepancies among various models.The conclusions are supported by simulation and experimentalresults.Index Terms—Current-mode control, state-space averaging.I. INTRODUCTIONMODELING of current-mode-controlled converters hasbeen a topic of interest to the power electronics com-munity for well over a decade. Recently, much effort hasbeen focused on extending the traditional averaged models tocapture high-frequency behavior [1]–[4]. Other research hasbeen aimed at improving modeling accuracy by eliminatingsubtle flaws in the derivation of duty-ratio constraints forcurrent-mode control [5], [6]. This paper (which appeared ina preliminary form as [7]) investigates these recent modelingapproaches and, in the process, exposes some serious limita-tions that have not been adequately accounted for previously.Section II of this paper investigates the impact of sampled dataeffects on small-signal modeling of current-mode-controlledconverters. Section III examines the geometric methods usedto derive duty-ratio constraints used for averaged models ofcurrent-mode control. The appendix outlines the approach usedin our simulations.Throughout this paper, comparisons between models aremade using the boost converter example from [2], shown inFig. 1. Under normal operating conditions, the switch is turnedon everys and is turned off when the inductor currentreaches a peak value of minus a compensating ramp.II. SAMPLED DATA E FFECTSEfforts to extend small-signal linear time-invariant (LTI)models of current-mode-controlled converters to high frequen-cies have been motivated by the desire to improve controldesign, while retaining simplicity. Typically, low-frequencyManuscript received May 21, 1996; revised February 4, 1997. This workwas supported by the Bose Foundation and IEEE Convergence Fellowship inTransportation Electronics.D. J. Perreault and G. C. Verghese are with the Laboratory for Elec-tromagnetic and Electronic Systems, Massachusetts Institute of Technology,Cambridge, MA 02139 USA.Publisher Item Identifier S 0885-8993(97)03287-0.Fig. 1. Example boost converter.Fig. 2. The approximate sample-and-hold relationship between perturbationsin control and peturbations in (instantaneous and average) inductor current.averaged models are used for feedback control design, while aseparate high-frequency model is used for slope compensationof the well-known ripple instability. This is done becauselow-frequency averaged models cannot predict the ripple in-stability, even under open-loop conditions. On the other hand,models used for predicting subharmonic oscillation do notalways capture the behavior of converters operating underclosed-loop voltage control. Thus, many researchers havesought to develop LTI transfer functions that fully capture thesmall-signal behavior of current-mode-controlled converters[1]–[4], [8]. Unfortunately, these works have not sufficientlyaddressed the limitations imposed by the current samplingintrinsic to current-mode control, leading to results that aresubject to misinterpretation.This section of the paper investigates the effects of currentsampling and assesses their impact on control design. As0885–8993/97$10.00  1997 IEEE454 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 12, NO. 3, MAY 1997Fig. 3. System for modeling the relation between perturbations in control and perturbations in current. This model relies on the assumptions usedin formingH(z).described in [1] and illustrated in Fig. 2, an approximatesample-and-hold relation exists between a perturbationin the control signal and the resulting perturbationin inductor current for a current-mode-controlledconverter. The corresponding perturbation in the one-cycleaverage inductor currentis also , to first order.These facts form the basis for the derivations in [1]–[3] ofhigh-frequency extensions to low-frequency models. A similarapproach, expressed in terms of duty-ratio perturbations, isused in [4]. A more exact numerical approach to generatinga transfer function is described in [8], but the limitationsimposed by current sampling apply equally there as well. Whatis not made clear in all these works is that, because of thesampling and reconstruction, the system becomes significantlytime varying (in fact, periodically varying) to perturbations inthat approach half the switching frequency. This leads tothe injection of additional frequencies inand therebycauses significant deviations from the results suggested byexisting treatments.A. Modeling ApproachConsider the effect of a perturbation in the control sig-nalof a current-mode-controlled converter. With theassumption that the input and output voltages do not varysignificantly, the relation between the perturbation in controland the resulting current perturbation can be approximatedby a sample-and-hold system (Fig. 2). That is, the exactcurrent perturbation(which is the difference between thetransient and steady-state currents) is well-approximated by thezero-order hold (ZOH) of its samples, taken at the turn-off instants. As discussed in [1], the main effects not modeledby the sample-and-hold approximation are the variation insampling time and the finite slope of the current perturbationtransition. The samplesof the instantaneous currentperturbation can also be seen as approximate samples of theaverage current perturbation over the ensuing interval of length. Discrete-time relations can now be formed between thesamplesof the control perturbation and samplesof the average inductor-current perturbation, as described in[1]. In the small-signal limit, the LTI model of [1]–[3] results,with the-transform transfer function given by(1)where, , and are the slope magnitudes of therising inductor current, falling inductor current, and slope-compensation ramp, respectively, in the nominal steady