Direct Eigen Control for Induction Machines and Synchronous Motors

Appendix D

State-Space Eigenvector Calculation

The third and fourth coordinates of the state-space eigenvector of IPM-SM, was calculated in section 3.3.2.1.

We now have to calculate the first two coordinates of the state-space equation system:

(D.1)    

This system is calculated in section 3.2.5 (cf. equation (3.57)).

The state-space eigenvectors (cf. equation (3.76)) will be calculated from coefficients of the inverse of the matrix P (Appendix C):

(D.2)    

The input matrix of state-space equations was calculated (cf. equation (3.72)) in section 3.3.2:

(D.3)    

as well as the exponential function of the diagonal matrix D, multiplied by the sampling period T (cf. equation (3.69)):

(D.4)    

Let us calculate the first coordinate of the state-space vector (D.2) in equation (D.1), written in the (d₀, q₀) reference frame:

(D.5)    

For the second coordinate, we obtain, in the same way:

(D.6)    

For the current ...