Solving nonlinear equations for selective harmonic eliminated PWM using predicted initial values

Abstract

This paper reports new methods to determine switching angles for selective harmonics eliminated P WM (SHE P WM) inverters. Such switching angles are defined by a set of nonlinear equations, and to solve these equations, a predicting algorithm is used to calculate initial values which are firstorder approximations of the exact solutions. With these predicted initial values, Newton algorithm can be w e d to find the so~utions within usually only one or two iterations. We also suggest another new approach for solving SHE PWM problem-ordinary differential equations approach. Advantages of this approach will be discussed and its applications be demonstmted by some ezamples. InductionCurrently used PWM techniques can be broadly classified as carrier modulated PWM, precalculated programmed PWM,and on-line waveform tracking PWM schemes ([l]-[5]). Programmed PWM waveforms are so determined as to minimize a particular objective function, such as to obtain minimum losses, reduced torque pulsations, or selective harmonic elimination (SHE). For SHE PWM, the switching instants (angles) can be determined through solving a set of nonlinear equations. Due t o nonlinear and transcendental characteristics, such equations can only be solved numerically. To obtain fast convergence, the initial (starting) values must be chosen to be close enough to the exact solutions. This is one of the most difficult task associated with programmed PWM techniques.There are now broadly two different approaches for determining these initial values. The first one could be termed as point-wise initial values where the initial values for different value of M (modulation index or normalized output voltage) are given separately. One example is IS] where the author gave some explicit linear functions which can be used to calculate initial values for given M. These linear functions are obtained from solving the nonlinear equations with small degree of freedom N (number of switchings in one-quarter wave) and then being generalized t o equations with large degree of freedom. This approach has the following advantages:1. The equations can be solved for random value of M; 2. If the linear functions are close to the actual switching trajectories, the numerical algorithm used to solve the equations will convergent very fast.But determination of these linear functions is not an easy task. Besides, we have found that for some waveforms (for example, for SLL, which will be discussed below), the switching trajectories can not be well approximated by few simple linear functions. It becomes then difficult to apply this method. The second approach could be termed as incremental initial values. The nonlinear equations are solved for incremental value of modulation index M, and the current solutions are directly taken as initial values for new equations where M is increased by a small amount AM ([lo]). This approach is based on the assumption that the solutions to the nonlinear equations vary continuously with modulation index M, so that the switching trajectories are cont...

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