# CU-Boulder ECEN 5807 - Time-Varying Effects and Averaging Issues (9 pages)

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## Time-Varying Effects and Averaging Issues

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- University of Colorado at Boulder
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- Ecen 5807 - Modeling and Control of Power Electronic Systems

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IEEE TRANSACTIONS ON POWER ELECTRONICS VOL 12 NO 3 MAY 1997 453 Time Varying Effects and Averaging Issues in Models for Current Mode Control David J Perreault Student IEEE and George C Verghese Senior Member IEEE Abstract This paper investigates issues in modeling of currentmode control The effects of the current sampling intrinsic to current mode control are analyzed and inadequately recognized limitations of linear time invariant LTI models at high frequencies where the system behavior is time varying are exposed The paper also examines the geometric methods used to derive dutyratio constraints in averaged models of current mode control and points out the sources of discrepancies among various models The conclusions are supported by simulation and experimental results Index Terms Current mode control state space averaging Fig 1 Example boost converter I INTRODUCTION M ODELING of current mode controlled converters has been a topic of interest to the power electronics community for well over a decade Recently much effort has been focused on extending the traditional averaged models to capture high frequency behavior 1 4 Other research has been aimed at improving modeling accuracy by eliminating subtle flaws in the derivation of duty ratio constraints for current mode control 5 6 This paper which appeared in a preliminary form as 7 investigates these recent modeling approaches and in the process exposes some serious limitations that have not been adequately accounted for previously Section II of this paper investigates the impact of sampled data effects on small signal modeling of current mode controlled converters Section III examines the geometric methods used to derive duty ratio constraints used for averaged models of current mode control The appendix outlines the approach used in our simulations Throughout this paper comparisons between models are made using the boost converter example from 2 shown in Fig 1 Under normal operating conditions the switch is turned on every s and is turned off when the inductor current reaches a peak value of minus a compensating ramp II SAMPLED DATA EFFECTS Efforts to extend small signal linear time invariant LTI models of current mode controlled converters to high frequencies have been motivated by the desire to improve control design while retaining simplicity Typically low frequency Manuscript received May 21 1996 revised February 4 1997 This work was supported by the Bose Foundation and IEEE Convergence Fellowship in Transportation Electronics D J Perreault and G C Verghese are with the Laboratory for Electromagnetic and Electronic Systems Massachusetts Institute of Technology Cambridge MA 02139 USA Publisher Item Identifier S 0885 8993 97 03287 0 Fig 2 The approximate sample and hold relationship between perturbations in control and peturbations in instantaneous and average inductor current averaged models are used for feedback control design while a separate high frequency model is used for slope compensation of the well known ripple instability This is done because low frequency averaged models cannot predict the ripple instability even under open loop conditions On the other hand models used for predicting subharmonic oscillation do not always capture the behavior of converters operating under closed loop voltage control Thus many researchers have sought to develop LTI transfer functions that fully capture the small signal behavior of current mode controlled converters 1 4 8 Unfortunately these works have not sufficiently addressed the limitations imposed by the current sampling intrinsic to current mode control leading to results that are subject to misinterpretation This section of the paper investigates the effects of current sampling and assesses their impact on control design As 0885 8993 97 10 00 1997 IEEE 454 IEEE TRANSACTIONS ON POWER ELECTRONICS VOL 12 NO 3 MAY 1997 Fig 3 System for modeling the relation between perturbations in control and perturbations in current This model relies on the assumptions used in forming Hz described in 1 and illustrated in Fig 2 an approximate sample and hold relation exists between a perturbation in the control signal and the resulting perturbation in inductor current for a current mode controlled converter The corresponding perturbation in the one cycle average inductor current is also to first order These facts form the basis for the derivations in 1 3 of high frequency extensions to low frequency models A similar approach expressed in terms of duty ratio perturbations is used in 4 A more exact numerical approach to generating a transfer function is described in 8 but the limitations imposed by current sampling apply equally there as well What is not made clear in all these works is that because of the sampling and reconstruction the system becomes significantly time varying in fact periodically varying to perturbations in that approach half the switching frequency This leads to the injection of additional frequencies in and thereby causes significant deviations from the results suggested by existing treatments A Modeling Approach Consider the effect of a perturbation in the control signal of a current mode controlled converter With the assumption that the input and output voltages do not vary significantly the relation between the perturbation in control and the resulting current perturbation can be approximated by a sample and hold system Fig 2 That is the exact current perturbation which is the difference between the transient and steady state currents is well approximated by the zero order hold ZOH of its samples taken at the turnoff instants As discussed in 1 the main effects not modeled by the sample and hold approximation are the variation in sampling time and the finite slope of the current perturbation transition The samples of the instantaneous current perturbation can also be seen as approximate samples of the average current perturbation over the ensuing interval of length Discrete time relations can now be formed between the samples of the control perturbation and samples of 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 and are the slope magnitudes of the where rising inductor current falling inductor current and slopecompensation ramp respectively in the nominal steady state To keep notation streamlined we employ the same symbol for time domain and transform domain

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