Calculation of Force and Torque
Since the purpose of a motor is to produce force or torque it is natural to compute these quantities from the finite element results. There are also a number of other cases, such as end winding forces during short circuits (see Chapter 11), in which we se
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TORQUE
6.1
INTRODUCTION
Since the purpose of a motor is to produce force or torque it is natural to compute these quantities from the finite element results. There are also a number of other cases, such as end winding forces during short circuits (see Chapter 11), in which we seek the forces and force distributions. In theory, the forces can be computed directly from the local field solution in a number of ways. This chapter explains some of the most popular methods. We begin this chapter with a disclaimer. The author has not found any of these force and torque calculation methods completely reliable. As will be illustrated, the force converges much more slowly than the field solution. Good field solutions can lead to erroneous force and torque computation and the methods described below will give different results using the same solution. Further, an attempt to find which of the force methods was "most accurate" was inconclusive. More than any other post-processing quantity, force (and torque) require careful scrutiny. We will consider three categories of force computation: •
Ampere's Force Law
•
The Maxwell Stress Method
•
The Virtual Work Method
All three methods can be used to find the global or total force on an object. The virtual work or Maxwell stress method can not (at least in a straight-
97 S. J. Salon, Finite Element Analysis of Electrical Machines © Kluwer Academic Publishers 1995
98
CHAPTER
6
forward way) be used to find the force distribution l . Ampere's force law, which can only be applied to nonmagnetic conductors, can be used to find the force distribution. For a more thorough discussion of these possibilities the reader is referred to DeBortoli [28].
6.2
AMPERE'S FORCE LAW
Ampere's Force Law is straightforward and simple to apply. Given the local value of flux density, iJ and current density (either input and therefore known, or computed in an eddy current region as illustrated in Chapter 4) we find the local force vector as
(6.1) This equation is useful for finding the force on conductors and will be used in the computation of end winding forces in Chapter 11. It is also possible to replace a piece of magnetic material by an equivalent current distribution. In the most general case this would involve both a surface and a volume current distribution. These currents in free space could then be used to find the global force. It is also possible to replace the magnetic material by an equivalent distribution of magnetic charges, qm, and use the magnetic equivalent of Coulomb's Law
(6.2) to find the force.
6.3
THE MAXWELL STRESS METHOD
We present here a rather complete development of the Maxwell Stress formula adapted from the work of Reyne[29] and DeBortoli[28]. While the final result is well known, we show here that the stress tensor can be obtained using Ampere's Force Law (above) and fictitious equivalent current distributions to replace magnetic material. We will then proceed to show in the next section that the lCarpenter [27] has shown that several different Maxwell stre
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