Multirate modeling and simulation of pulse-width modulated power converters

Abstract

Pulse-width modulated power converters are devices which transform voltages and currents between different levels to meet the requirements of the appliances. In contrast to conventional transformers, they use transistors and other semiconductor components to abruptly switch on and off the input voltage source to generate an output voltage or current which, averaged in time, equals the desired values. The so-generated pulsed signal is usually filtered by an active or passive filter to suppress the high-frequency components and thus smoothen the output. The numerical simulation of these devices is computationally expensive since with conventional time discretization very small time steps are necessary to properly represent the steep transients induced by the abrupt switching of the semiconductor devices. If the semiconductor behavior is idealized, a switch event detection is often necessary to prevent a failure of the time integration algorithm. In this dissertation a multirate approach is developed to efficiently tackle these problems. The idea is to split the solution of the ordinary differential or differential-algebraic equations describing the power converters into slowly varying parts and fast varying parts. The output of the converter is represented as a sum of fast periodically varying ripples and a slowly varying envelope. The differential equations are, in a first step, reformulated into so-called multirate partial differential equations (MPDEs), which allow to explicitly split the solution by associating the different components to different artificial time scales. The MPDEs are solved using a combination of two methods. First, a Galerkin approach is applied to solve along the fast time scale. Three different types of basis functions, namely PWM basis functions, PWM eigenfunctions and B-spline basis functions, are employed. Second, a conventional time integration algorithm is used on the remaining differential equation system. It is assumed that the semiconductor switching behavior can be idealized as such that it can be represented by an ideal pulsed voltage source. The solution components along the fast varying time scale, i.e. the ripples, are represented by basis functions, which are specifically designed for this purpose. Since in some of the solution components the ripples are only continuous and not smooth, the basis functions take these points of C0 continuity at the proper position into account by construction. The MPDE approach is applied to different examples to demonstrate its accuracy and efficiency. It is applicable to single-phase DC-DC and DC-AC power converters. If the power converter or the application consists of nonlinear elements the computational effort increases. For nonlinear elements, e.g. nonlinear inductors, a simplification is proposed which keeps the simulation efficient. To generate the pulsed excitation, a pulse-width modulation (PWM) with constant switching frequency and varying duty cycle is applicable. It is generated by either sawtooth or triangle carriers. Both natural or regular sampling are supported. Besides circuit simulation the method is also applied to a field-circuit coupled model.