## Models for phase

As demonstrated in the VHM Chapter, there are two alternative methods for predicting an electrical machine's behaviour. An equivalent circuit method, whereby the machine is split into contributing parts of magnet induced open circuit emf and a series inductance, or a look up table method, whereby exact flux-current-position reference tables are used at every time step. As the inductance is independent of current in the air cored tubular machine, the simple equivalent circuit model will suffice. When considering just one phase, this model is simple to implement as both the flux linking the coil and its inductance may be accurately modelled as sinusoidal functions.

The position dependant flux linkage is differentiated to make the emf a product of the velocity and a cosine function of position.

5.4.2.2 Simple Equivalent Circuit Model for 3 phases

5.4.2.2 Simple Equivalent Circuit Model for 3 phases

Figure 5.26: Equivalent circuit of red phase

Figure 5.26 shows the equivalent circuit for the red phase, including the mutual inductances (M), self inductance (L), the internal resistance (r) and the load (R). The voltage equation for this loop is given below (5.12).

Er — L, di dL diy dMry di, dM u , n r • r " y ' ' 17 + Mrb —+ ib-dM^ + ir (r + R) (5.12)

■ + Mr dt 1 dt Liy dt +iy dt iU dt u dt If the machine were perfectly balanced then two simplifications could be employed, both given in (5.13).

In an unbalanced machine, however, neither of these assumptions are valid and so the equations for the three currents must be re-stated as the differential equations of (5.14).

"dT

dib dt dir dt

y ry dt r dMry - Mrb —-b - ibdMrb y dt dMrb dt dM

dt di dt

by y ry dt

dt di

When considering the entire three phase machine and utilising the results from the larger FEA model, the inductances and magnet induced flux flow can no longer be simply approximated to a single sinusoidal function. The inductance waveforms are no longer pure sinusoids due to the imbalance caused by using coil support spacers only every third coil. Furthermore, the importance of maintaining the correct phase differences encourages the use of Fourier series approximations, which allows periodic functions to be approximated to the form given in equation ( 5.15).

r, N ^f f 2nnx"] 7 . f 2nnx f (x) = a0 + an cosl —— | + bn sin|

Where f(x) = periodic function a0 = average value an & bn = constants l = period of function The MATLAB [80] fast Fourier transform function (FFT) takes a discretely sampled set of data and returns its Fourier series.

Figure 5.27: Fourier series approximation

Figure 5.27 shows the inductance relationship between the blue and yellow phases plotted on the same axis as the first two summations from ( 5.15). The close correlation between the sum of these terms and the original results makes further summation redundant. A similar effect was found for all the inductance relationships, whilst the magnet flux linkage required only the first terms, the second being 105 times smaller.

Each phase current must be solved simultaneously and numerically integrated over time. A SIMULINK [80] model was constructed within MATLAB to achieve this.

Equations (5.14) to (5.16) give the flux, emf, and self inductance for the red phase, with similar relationships existing for the other two phases. Coefficients of the equations governing all the phases are given in Appendix C.

. ( 2nx i 7 ( 2nx a1 sinl —— I - b1 cosl ——

Lv = a0 + a1 cosl ——— I + b1 sinl-I + a2 cosl-I + b2 sin

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