Abstract—In this paper, we present an overview of previously published approaches to dynamic modeling of current programmed converters, including basic low-frequency averaged models, as well as treatments of sampling and aliasing effects. The modeling assumptions are examined and the differences among the approaches are highlighted, with the objectives of making it easier to present these topics in power electronics courses and applying the models in practice. I. INTRODUCTION Constant-frequency current programmed mode (CPM) control [1], also known as peak current-mode control, is a popular control technique for switched-mode power converters. Advantages of CPM control include built-in over-current protection, robust dynamic responses, simplified voltage-loop compensator design, rejection of input voltage disturbances, and relatively simple current sharing for power modules operating in parallel. Dynamic modeling of CPM controlled switched-mode power converters has been addressed in a number of publications, including [2-11]. The purpose of this paper is to review the representative models, point to the similarities and differences among the models, and summarize the model assumptions and limitations. Our objectives are to present this review in a form suitable for instructors and students in power electronics courses addressing CPM control, and for practicing engineers applying the models in the design and evaluation of CPM controlled converters. A textbook knowledge [5] of switched-mode power converters, and associated modeling and control techniques is assumed. Throughout the paper, we assume that converters operate in continuous conduction mode. Section II gives a brief introduction to CPM control. Large-signal averaged models are presented in Section III. Continuous-time small-signal averaged models are discussed in Section IV. Section V explains the approaches to incorporating sampling effects into the continuous-time models. Limitations of the models due to aliasing effects are briefly summarized in Section VI. A simulation example is presented in the Appendix. II. CURRENT PROGRAMMED CONTROL A detailed introduction to CPM controller operation can be found in [4]. Figure 1 shows typical steady state and perturbed inductor current waveforms. It is well known that undesirable subharmonic oscillations occur for duty cycle D > 0.5 as illustrated by the waveforms in Fig. 1(a). An compensation ramp with slope Mc > 0, is added to ensure that a current perturbation in a switching period diminishes in the next period as shown in Fig. 1 (b). A simple geometrical argument leads to the following condition to prevent the subharmonic instability, 11212<+−=∆∆ccMMMMII, (1) where M1 = V1/L > 0 is the inductor current slope in the dTs subinterval when the active switch is on, and M2 = V2/L > 0 is the negative of the inductor current slope in the d’Ts subinterval when the rectifier switch is on, d’ = 1−d. From (1), it follows that selecting 2/2MMc> ensures stable operation of the current control loop under all steady-state operating conditions. A disadvantage of CPM control is a relatively high sensitivity to noise related to sensing the instantaneous switching current and comparing the sensed signal to the current command. To reduce the sensitivity to noise, the compensation ramp is commonly added in practical CPM designs, even when operating the converter at duty cycles less than 0.5 [5]. III. LARGE-SIGNAL AVERAGED MODELS The first step in a derivation of an averaged dynamic model for a CPM controlled converter is to describe a relationship between the current command ic and the average inductor current 〈iL〉Ts . In the simplest, first-order model [2, 5], it is assumed that the average inductor tracks the current command, cTsLii ≈, (2) which gives an approximate reduced-order model. More accurate models [2-8] are based on large-signal descriptions that take into account the inductor current slopes and the slope of the compensation ramp. Referring to the waveforms of Fig. 2, the mid-point current values in the dTs and the d’Ts subintervals can be found as: ssccdTmdTMii1121−−=, (3) Approaches to modeling converters with current programmed control F. J. Azcondo, Ch. Brañas, R. Casanueva University of Cantabria Department of Electronics Technology System and Automation Engineering Ave. de los Castros s/n 39005 Santander, SPAIN Email: [email protected] [email protected], [email protected] Dragan Maksimovic Colorado Power Electronics Center Department of Electrical and Computing Engineering University of Colorado, Boulder, CO 80309-0425, USA Email: [email protected] '2122−−=, (4) respectively, where d’=1 − d, and d is the switch duty cycle. It is of interest to examine how various CPM modeling approaches differ in approximating the average inductor current 〈iL〉Ts using the expressions (3) and (4). For example, assuming steady-state conditions, the large-signal averaged inductor current in [3] is found using (3) only, ssccTLdTmdTMiiis1121−−=≈. (5) In [6], however, the derivation is based on (4) only, ssccTLTdmdTMiiis'2122−−=≈. (6) Assuming m1d = m2d’, reference [7] combined (3) and (4) into an alternative expression: ssccTLTddmmdTMiis')(2121+−−≈. (7) Finally, reference [4] pointed to the fact that the large-signal expression for the average inductor current should be based on averaging the transient waveform in Fig. 2 over the switching period, sssccTLTdmTdmdTMiiddiis222121'2121' −−−=+≈, (8) which was the approach adopted in [5]. It should be noted that (5)-(8) represent different ways to approximate large-signal averaged dynamics of the inductor current in the CPM controlled converter. At low frequencies, as pointed in [8], all of the proposed approximations are nearly the same, and result in very similar predictions for the low-frequency dynamics. Low-frequency small-signal averaged models are discussed further in the next section. IV. SMALL-SIGNAL AVERAGED MODELS AT LOW FREQUENCIES Starting from the large-signal models presented in Section III, the corresponding low-frequency small-signal averaged models are derived by small-signal linearization, i.e. by finding how a perturbation in the current command ic relates to perturbations of the inductor current iL, the switch duty ratio d, and the voltages v1 and v2 that contribute to