IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 18,NO. 4,JULY 2003 985 Cyclic-Averaging for High-Speed Analysis of Resonant Converters Martin P. Foster, H. Isaac Sewell, Chris M. Bingham, Member, IEEE, David A. Stone, Dirk Hente, and Dave Howe Abstract—The paper describes the development and appli- cation of a cyclic-averaging technique for the rapid analysis of high-order resonant power converters. To provide a focus to the paper, particular emphasis is given to a 3rd-order LCC voltage output converter topology. The proposed methodology predicts steady-state voltages and currents throughout the circuit, and provides estimates of the stresses on the resonant circuit compo- nents. State-space simulations and experimental results from a 350 V-input/150 V-output converter are used to demonstrate a pre- diction accuracy comparable with time-domain integration-based techniques is achievable, while requiring only 1/10,000th of the computation time. In addition, a comparison with Spice simula- tion results shows that cyclic averaging provides commensurate predictions of voltage and current stresses on the resonant circuit components. Issues arising from the stray capacitance associated with the resonant inductor, and the corresponding sensitivity of the predicted output voltage, are also considered. Index Terms—Modelling, resonant power conversion, simula- tion. I. INTRODUCTION R ESONANT power converters offer a higher efficiency and reduced size compared with traditional switched-mode counterparts. Thus, there is significant interest in high-order resonant systems, such as that shown in Fig. 1, particularly in the consumer product industry, to satisfy the requirement for smaller power supplies for compact electronic equipment. However, the increased circuit complexity makes it more difficult to accurately predict the performance of the converters during an iterative design process, due to the protracted simulation times required for component-based simulation packages, such as Spice. Computationally efficient algorithms are, therefore, sought to accurately predict current and voltage waveforms throughout such converters, so as to facilitate their design. Utilizing state-variable models, and exploiting the periodic behavior of resonant converters in the steady-state, cyclic-aver- aging techniques will be shown to provide an attractive alter- native to traditional ‘integration-based’ and harmonic analysis methods for converter performance evaluation, and component stress prediction. Manuscript received February 8, 2002; revised March 3, 2003. Recom- mended by Associate Editor K. Smedley. M. P. Foster, H. I. Sewell, C. M. Bringham, D. A. Stone, and D. Howe are with the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, S1 3JD U.K. (e-mail: c.bingham@sheffield.ac.uk). D. Hente is with the Forschlungslaboratorien, Philips GmbH, Aachen, Germany. Digital Object Identifier 10.1109/TPEL.2003.813763 II. MODELLING CYCLIC BEHAVIOUR The proposed approach is considered to be a variant of Floquet-based techniques, [1], whereby averaged, steady-state values of state-variables are determined. Due to the multiple operating modes of resonant converters, Fig. 2, “classical” steady-state operating points cannot be determined due to the continuous switching nature of the input voltage. However, by considering converter operation to be periodic, then (1) where the state vector, , describing converter operation at time is equal to the state vector at time , being the pe- riod of the applied input voltage and an integer. The resulting performance of the converter can then be obtained from ana- lytical equations without the need for integration. The method can, thereby, provide steady-state performance predictions ex- tremely rapidly. In the steady-state, each period of a cycle can be decom- posed into multiple operating modes, each dependent on the state of the input voltage and the rectifier input/output volt- ages and currents. Fig. 2 shows typical steady-state voltage and current waveforms for the resonant converter circuit shown in Fig. 1, together with the sub-division of the cycle into operating modes, . In general, power converters operating in a cyclic-mode can be modeled by a system of piecewise linear (state-space) equations which describe the converter in each operating mode during the cycle; i.e. (2) where is the state vector, is the dynamical matrix and is the input vector, in the operating mode of the converter. Thus, they consist of a linear combination of circuit voltages and currents. For the mode, (2) can be solved analytically (3) where , , and are the initial conditions for the mode. If the time during which the circuit operates in the mode is , where is the associated duty-cycle, the complete solution for the dynamics of the converter can be obtained by employing the state vector at time as the initial condition for the subsequent dynamics of the mode. Nevertheless, the solution of (3) is com- plicated by the need to evaluate the integral, which is a major contribution to the computational overhead when analyzing the 0885-8993/03$17.00 © 2003 IEEE