6132 Voltage and Torque in Reference Frames

The abc variables can be transformed into the rotor reference frame, which is also the synchronous reference frame, for synchronous machines in steady state. The stator dq equations of the PMSM in the dq or rotor reference frame are as follows:

where

Here, id and iq are the dq axis stator currents, vd and vq are the dq axis stator voltages, Rs is the stator phase resistance, Ld and Lq are the dq axis phase inductances, d and q are dq axis flux linkages, and is the rotor speed in electrical rad/s. The subscript s used in Chapter 5 to refer to stator quantities for induction machines has been r dropped here for simplicity. The dand q axis inductances are Ld=Lis+Lmdand Lq =Lis+Lmq. (Llsis the leakage inductance.) Note that the d- and q-axes mutual inductances can be different in the case of PM machines. The electromagnetic torque is

where P is the number of poles.

Rotor position information gives the position of d- and q-axes. The control objective is to regulate the voltages vd and vq or the currents id and iq by controlling the firing angles of the inverter switches. The rotor position is given by

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