Nonlinear Problems
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
- PDF / 846,645 Bytes
- 16 Pages / 439 x 666 pts Page_size
- 35 Downloads / 235 Views
2.1
INTRODUCTION
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, p, = ~, is nonhomogeneous 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 p,. 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. The most popular method of dealing with nonlinear problems in magnetics is the Newton-Raphson method described in section 2.3. Before we describe the Newton-Raphson method however, we first look at the saturation characteristic and how it is represented.
17 S. J. Salon, Finite Element Analysis of Electrical Machines © Kluwer Academic Publishers 1995
18
CHAPTER
IInitialize Problem I
! ! !
ISet fk in iron to unsaturated value I ISolve Problem I no
IUpdate values of fk I Is
I
Figure 2.1
f!new -f!old /-Lold
I< E ?
yes
rc;:::::::-,
---)o~~
Simple Iteration For Nonlinear Problems
2
19
Nonlinear Problems B
H
Figure 2.2
2.2
Family of Hysteresis Loops and Magnetization Curve
REPRESENTATION OF THE B-H CURVE
If we take a piece of magnetic steel and slowly magnetize it to a value HI then slowly reduce the field until it reaches -HI and keep repeating this process, (i.e. varying the field slowly between HI and -HI) until the characteristic repeats itself, we obtain a hysteresis loop such as the one shown in Figure 2.2. If we now increase the field to H2 and repeat the process, and then to H 3 , etc. we obtain a family of nested hysteresis loops as shown in Figure 2.2. If we now connect the tips of the hysteresis curves we obtain the normal magnetization curve. It is this curve which is most commonly used in finite element analysis to represent the steel's magnetic characteristics. For soft magnetic materials, the hysteresis loop is narrow and this is a good approximation. When we consider the fact that the loops found in Figure 2.2 can vary from batch to batch of steel from the same supplier and that these curves are also affected by mechanical pressure and
Data Loading...
