## Performance

4.3.2.1 Open circuit emf

Emf is induced in the stator coils by virtue of a change in flux linking them, the magnitude and direction of which is given by Faraday's law as expressed in (4.19).

Where Eopen = open circuit emf (V) N = number of turns on coil

In the flux plots of Figure 4.13 and equation (4.13) it was demonstrated that when the machine is run on load there is flux in the machine due to the PM excitation and the armature excitation. Substitution of (4.13) into (4.19) implies that there are two emfs to consider: the driving or open circuit emf, resulting from interaction of PMs and teeth, and an opposing emf, called inductance or back emf, representing the coil's resistance to a change in current. The latter term may be expressed in several ways, the 'purest form' of which is (4.20)

Where y = flux linkage due to current (Wb-turns)

The two emfs can be treated combined or as separate entities in an equivalent circuit and described by a variety of equations depending on assumptions used. A total of five alternative models are considered in the following sections, becoming progressively more complex and computer intensive. In each model the circuit equation is formulated and then solved in the time domain with translator displacement fed in from an experimental position transducer to allow direct comparison with prototype results.

4.3.2.2 Simple Equivalent Circuit Model - Model A

The traditional per phase equivalent circuit for a permanent magnet synchronous machine consists of an emf source equal to the no load emf, feeding into a series reactance and resistance, as shown in Figure 4.26. Using the definition of inductance given in (4.21) substituted into (4.20) and applying Kirchoff s voltage law to the equivalent circuit gives the relationship of (4.22).

Where

Li flux linkage due to current with no PMs present in FEA (Wb-turns)

constant value of inductance (H)

Where the inductance is the flux flow with no PM excitation and can be deduced from the gradient of Figure 4.9, 0.425 H.

4.3.2.3 Step Variable Inductance Equivalent Circuit Model - Model B

Due to the effect of the magnets on the flux pattern, assuming that the inductance may be calculated using purely electrical excitation may have limitations. In order to account for the variation of inductance with current and relative magnet and tooth position, it has been suggested [77] that the inductance may be calculated according to (4.23).

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