Electrical – Linearized state space model of PM synchronous motor


I intend to implement field oriented control of PM synchronous and Induction motor in Simulink. I started with the synchronous motor and was trying to find its linearized state space model.

I read that I should equate the following three equations to zero and find an equilibrium point.

(di_d)/dt=Vd/Ld – (Rsid)/Ld+(weLq*iq)/Ld = 0

(di_q)/dt=Vq/Lq – (Rsiq)/Lq-w_e(Ld*id+ƛ_PM)/Lq = 0

J*(dw_m)/dt=T_e – B*wm

where T_e = 1.5*Pƛ_PMiq

Since, there 5 unknowns, namely Vd, Vq, id, iq, and wm (we=p*wm, p= pole pairs) and three equations, I am unable to solve the above equations.

I tried to read few papers online but they did not explain it properly.
If you have any tips or suggestions to proceed further, it would be great!


Best Answer

You do not need to solve the set of equations to linealize them. You can read more about it in here:

https://en.wikipedia.org/wiki/Linear_approximation https://en.wikipedia.org/wiki/Linearization