IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 10, NO. 4, JULY 1995 ~ 397 A Unified Model for Current-Programmed Converters F. Dong Tan, Member, IEEE, and R. D. Middlebrook, Life Fellow, IEEE Abstruct- A unified model is established for a current- programmed converter, which is both a modification and an extension of familiar models. Inclusion of the sampling effect allows the presence of an additional pole up in the current-loop gain to be derived. The resulting final double-slope asymptote is fixed in position, and the crossover frequency cannot exceed half the switching frequency. A stability parameter, Q3, determines the additional pole and describes the degree of peaking in the closed-loop transfer function. Experimental verification employs an analog signal injection technique. I. INTRODUCTION URRENT programming has become the regulating C scheme of choice in dc-to-dc converters owing to its advantages over duty-ratio programming such as better line- noise rejection, automatic overload protection, easy paralleling of multiple converters, and especially design flexibility in improving small-signal dynamics. Since its initial conception [ 11, 121, a large number of small- signal models have been proposed, for example, 131-1121. A brief nonexhaustive review of some of the previous efforts is presented in the following to show the evolution of models. A low-frequency circuit-oriented approach was proposed in 1.51. A modulator model (also referred to as “duty-ratio control law” by other authors) is derived by perturbation of an expression for the average inductor current in steady state. Then, the modulator model is interfaced with the state-space averaged model for the power stage to obtain a complete model for the entire converter system. Current-loop gains are identified and then absorbed. This approach has gained wide acceptance owing to its simplicity and the insight gained into the properties of current programming. One of the major insights gained in 1.51 is that the crossover frequency of the current loop can in general be considered wideband, implying possible degradation of performance of a low-frequency model. To cope with potential deficiencies, a separate earlier work [4] needed to be used. This model is capable of predicting the well-known subharmonic oscillation which occurs, for example, when the duty ratio is greater than 0.5 and no compensating ramp is used. A general expression for the sampled-data version of the current-loop gain is de- rived, from which it is seen that the crossover frequency of the current loop is limited to approximately one third of the Manuscript received March 18, 1994; revised March 13, 1995. F. Dong Tan is with QSC Audio Products, Costa Mesa, CA 92626 USA. R. D. Middlebrook is with the Power Electronics Group, California Institute IEEE Log Number 94 12 189. of Technology, Pasadena, CA 91 125 USA. switching frequency. It will be shown later analytically as well as experimentally that this estimate is too conservative. Another earlier work [ 121 suggested that the modulator gain factor F, can approach infinity at the limit of stability. A few authors have contested this result on the ground that Shannon’s sampling theorem limits the crossover frequency to half the switching frequency. This apparent contradiction is resolved by the unified model. The unified model inherits the value of the gain factor but reveals further that the crossover frequency of the current-loop gain can not exceed half the switching frequency, not by limiting the value of the modulator gain factor, but by the presence of an additional pole due to sampling. The modulator model in [5] was derived by perturbation of an expression for inductor average current in steady state. This procedure was disputed in [6], where it is argued that the small- signal modulator model needs to be derived by perturbing an expression for inductor average current in perturbed state. Experimental measurements, however, did not support its prediction for the case where no compensating ramp is used. An explanation to this discrepancy is found in 171, which provides a geometric interpretation of the unified modulator gain to be proposed in this paper. A continuous-time model was proposed in 181, where a mod- ified Pad6 approximation of the complex exponential, accurate up to half the switching frequency, is used to approximate the sampled-data version of the current-loop gain derived in [4], allowing interpretation of sampled-data results in the continuous-time domain. Consequently, the mechanism behind the peaking, occurring at half the switching frequency in various closed-loop transfer functions, is seen to be due to the presence of a double right-half-plane zero at half the switching frequency. The expression for the current-loop gain crossover frequency, however, is found to be the same as in 141, which lacks experimental support and is inconsistent with the pro- posed corresponding closed-loop control-to-inductor-current transfer function. This inconsistency lies in the fact that diffi- culties were met in many attempts to derive the peaking at half the switching frequency, using the expression for the crossover frequency given in 141, [8]. Possible peaking at half the switch- ing frequency in closed-loop transfer functions was confirmed in 141, [8], [9] as well as by the present authors in [21]. A model for switches in a current-programmed converter was proposed in [lo], which enjoys easy implementation in computer simulation applications. The current-loop gain is not identified. A Q-factor is found lo determine the degree of peaking in closed-loop transfer functions. 0885-8993/95$04.00 0 1995 IEEE398 lEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 10. NO. 4, JULY 1995 In the following sections, a unified model for current- programmed converters is developed, which represents a low- frequency modification and a high-frequency extension over previous models. A low-frequency modification leads to a unified modulator model, which results in improved predic- tions for several essential quantities of current-loop gain. An extension to include sampling effect allows the presence of an additional pole wp in