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 – (Rs*id)/Ld+(we*Lq*iq)/Ld = 0

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

J*(dw_m)/dt=T_e – B*wm

where T_e = 1.5*P*ƛ_PM*iq

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!

Thanks,

## 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