# Nonlinear Problems

## Abstract

The formulation and example given in Chapter 1 were linear. In the analysis of electrical machines the problems are almost always nonlinear due to the presence of ferromagnetic materials. Good designs will typically operate at or near the saturation point. The magnetic permeability, \(\mu = \frac{B}{H}\), is nonhomoge-neous and will be a function of the local magnetic fields which are unknown at the start of the problem. Since the permeability appears in all of the element stiffness matrices, we must use an iterative process and keep correcting the permeability until it is consistent with the field solution. A simple method is illustrated in Figure 2.1. We begin by assuming a permeability for each element in the mesh. For the magnetic regions this is usually taken as the unsaturated value of *μ*. We solve the problem, compute the magnitude of the flux density in each element and correct the permeabilities so that they are consistent with agree with the computed values of flux density. The problem is then solved again. New flux densities are found, permeabilities are corrected and the process continues until the results stop changing, *i.e.* the change is smaller than a specified value. A relaxation factor can be applied to the change of permeability at each element. The author has found this to be a successful and robust strategy but somewhat slow.

## Keywords

Hysteresis Loop Flux Density Nonlinear Problem Magnetic Permeability Local Magnetic Field## Preview

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