9.5: Classical, Lumped-Parameter Synchronous Machine

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James Kirtley
Massachusetts Institute of Technology via MIT OpenCourseWare

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Now we are in a position to examine the simplest model of a polyphase synchronous machine. Suppose we have a machine in which the rotor is the same as the one we were considering, but the stator has three separate windings, identical but with spatial orientation separated by an electrical angle of \(\ 120^{\circ}=2 \pi / 3\). The three stator windings will have the same self- inductance \(\ \left(L_{a}\right)\).

With a little bit of examination it can be seen that the three stator windings will have mutual inductance, and that inductance will be characterized by the cosine of \(\ 120^{\circ}\). Since the physical angle between any pair of stator windings is the same,

\(\ L_{a b}=L_{a c}=L_{b c}=-\frac{1}{2} L_{a}\)

There will also be a mutual inductance between the rotor and each phase of the stator. Using M to denote the magnitude of that inductance:

\(\ \begin{aligned}
M &=\mu_{0} \frac{\pi}{4} \frac{\ell R N_{a} N_{f}}{g p^{2}} \\
M_{a f} &=M \cos (p \phi) \\
M_{b f} &=M \cos \left(p \phi-\frac{2 \pi}{3}\right) \\
M_{c f} &=M \cos \left(p \phi+\frac{2 \pi}{3}\right)
\end{aligned}\)

We show in Chapter 1 of these notes that torque for this system is:

\(\ T=-p M i_{a} i_{f} \sin (p \phi)-p M i_{b} i_{f} \sin \left(p \phi-\frac{2 \pi}{3}\right)-p M i_{c} i_{f} \sin \left(p \phi+\frac{2 \pi}{3}\right)\)