Direct Eigen Control for Induction Machines and Synchronous Motors

Appendix A

Calculation of Vector PWM

The average voltage vector making it possible to obtain, at the end of the T sampling period, the set-point state-space vector, was calculated starting from a known initial state, for three different types of three-phase motors:

• a non-salient pole induction machine, equation (1.133) in section 1.3.3, in the (α, β) fixed frame
• a surface-mounted permanent-magnet synchronous motor, equation (2.81) in section 2.3.3, in the (α, β) fixed frame
• an interior permanent magnet synchronous motor (IPM-SM), equations (3.100) and (3.104) in section 3.3.3, in the (d₀, q₀) initial rotating reference frame; in this last case, it is enough to change the vector coordinates to express them in the (α, β) complex plane.

In all cases, the average control vector is thus expressed in a complex fixed frame, by its two coordinates:

(A.1)    

We now will approach, using an example, the realization of this voltage vector by the voltage inverter. To minimize the conduction losses regulating the average voltage over one sample period, the voltage inverter switches the supply voltage from zero to the line voltage, as shown in section 1.2; it thus remains to calculate the pulse width modulation (PWM) which will allows the average control vector to be generated.

A.1 PWM Types

There are many types of PWM for a voltage inverter, which we can classify first of all, by their frequency characteristics: ...