Computation of Losses, Resistance and Inductance
The finite element solution gives the potential at the nodes. This information by itself is usually of little interest in machinery analysis. We use this information to compute useful quantities such as flux density (see Chapter 1). Using the finite eleme
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5.1
INTRODUCTION
The finite element solution gives the potential at the nodes. This information by itself is usually of little interest in machinery analysis. We use this information to compute useful quantities such as flux density (see Chapter 1). Using the finite element solution to find useful quantities is called postprocessing. In this chapter we will see how to use the field solution to find eddy current loss, resistance and inductance. Slot leakage inductance will be found as an example.
5.2
COMPUTATION OF EDDY CURRENT LOSS
A question which arises frequently in magnetic design is how to reduce (or sometimes increase) loss in an eddy current region. When selecting a material for use in a time varying magnetic field, should we choose a high resistivity materialor a low resistivity material? The answer is not always obvious and finite element analysis is invaluable in the determination. To see this, consider the two limiting cases illustrated in Figure 5.1. In the case of very high resistivity materials (insulators), we have low loss due to the fact that we have practically no induced current. At the other extreme, approaching superconductivity, we again have low loss due to the fact that J2 R is low. In the first case (high resistance) we have resistance limited eddy currents. Here the losses behave as As the resistance increases the loss decreases. Losses are resistance limited when the flux produced by the eddy currents has a negligible influence on the total field. Resistance limited losses are found from a magnetostatic solution (see Chapter 1) or a time varying solution in which the eddy current region
i:.
75 S. J. Salon, Finite Element Analysis of Electrical Machines © Kluwer Academic Publishers 1995
76
CHAPTER
5
Loss
Resistivity, p Figure 5.1
Loss vs. Resistivity
was given zero conductivity. This region is sometimes called a fine wire region. An example in which a fine wire region is used is in the computation of the stator eddy loss in an ac machine (computed below). The strand dimensions are small compared to a skin depth and the strands are frequently transposed. To a good approximation then, the current density is uniform. In regions where the conductor dimensions are greater than the skin depth, we use the methods of Chapter 4 to find the eddy current distribution. The losses in these solid conductor regions are computed in the following section.
5.2.1
Eddy Current Loss In A Triangular Element
We will now compute the loss in a first order triangle. The finite element solution gives the complex vector potential at the three nodes. The eddy current density is J = jWCTA
(5.1)
The instantaneous eddy current loss is written in terms of the current density as
(5.2)
Computation of Losses, Resistance and Inductance
77
If A corresponds to the peak value of the vector potential then the loss in a first order triangle will be
p =
~e{ ~ 2CT
11
~
JJ*dx dy}
(5.3)
Since A varies linearly over the element, then by equation 5.1 so does J. Therefore
Substituting into equation 5.3
p
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