Introduction

An electromagnetic motor is capable of generating a torque using different interaction principles:

• between stator and rotor current;
• between stator current and rotor permanent magnet of vice and versa;
• between stator current and iron structure.

In order to better understand and how to realize the different performances and limits of these possibilities, the different ways of force and torque calculation will be developed and applied to specific examples.

Energy Balance

Expressed in a variation or incremental form, the energy balance in an electromechanical transformation is:

dWel = dWmag + dWth + dWmech (1)

The electrical energy (Wel) can be transformed in a mechanical one (Wmech), but going through a magnetic intermediate form, mainly in the motor airgap, (dWmag) and with thermal losses (dWth).

This conversion can be represented by Figure 1. An electromechanical energy transformation can only occur if a magnetic energy exists.

Laplace's expressions

The most frequently used expression to determine an electromagnetic force is the Laplace's equation:

With  and  perpendicular:

F = il B = m0 il H

The force is perpendicular to the plane defined by the current and the flux density vector . This last can be created by another current or by a permanent magnet. However, this expression cannot be applied to an interaction between a current or a permanent magnet and an iron part.

Energy Derivative

It is possible to show that the force or the torque can be calculated directly from the magnetic energy stored in an electromagnetic system. This energy can be defined as:

Wmag = =                          (3)

If the system is linear (no saturation), this magnetic energy becomes:
Wmag = i y = Li2 =                        (4)
The force created by such an electromagnetic system in the x direction is:

Fx = y = cst                        (5)

For a linear system:

Fx = i2                for a system with only one coil                (6)

Fx = i12 + i1 i2 + i22                for a system with 2 coils having a mutual               (7)

As an example, for an electromagnet with an ideal iron magnetic circuit (Fig. 3):

L = N2 L = coil inductance
L = = = permeance
Fx = N2 i2
Fx = N2 i2                (8)

For a rotating system or motor, the expressions (5), (6) and (7) become, for the corresponding torque relative to a rotation angle a:

(5) ® Ma = y = cst                general case
(6) ® Ma = i2                linear reluctant system
(7) ® Ma = i12 + i1 i2 + i22                2 coils

Maxwell's Stress Tensor

The Maxwell's stress tensor is a methodology based on the flux density distribution on the external surface of the moving (or standstill) part of the electromechanical device.

Figure 4 shows a surface element on which a flux density vector arises. The resulting force can be described by two components relatively to the surface vector:

dFn = m0 (Hn2 - Ht2) dS

= (Bn2 - Bt2) dS                (9)
dFt = m0 Hn Ht dS = Bn Bt dS                (10)

If b is the flux density angle relative to the surface vector (perpendicular to the surface) and a is the corresponding angle between the force and the surface vectors, they are corresponding to the following relation:

b = 2 a                                            (11)

= Ni
H2d = Ni
H = Ni / (2d)                                        (12)

Using the normal force expression defined by (9), the force becomes:

Fn = m0 Hn2 (2ab)
= N2 i2                                            (13)

This expression is similar to the one obtained using the energy derivative expression.

Methodology Choice

These three force or torque determination methods offer different possibilities of calculation and phenomena understanding.

The Laplace's force calculation is the most simple and efficient to calculate interaction between a permanent magnet and a coil.

The energy derivative approach is the most general one and very convenient for variable reluctance effects and interaction coils or between coils and permanent magnets.

The Maxwell's stress tensor (or pressure) is the best approach for numerical calculation such as finite elements.