# Modeling scalar control of PMSM

I have met with a scalar control algorithm for a permanent magnet synchronous motor. I haven´t heard about it before so I decided to develop a dynamic model to further analyze this control algorithm. Unfortunatelly I haven´t got the Matlab with the SimPowerSystems library so I decided to create this model in Scilab.

I have been using below given equations for PMSM simulation:

$$\\frac{\mathrm{d}i_d}{\mathrm{d}t} = \frac{1}{L_d}\cdot(u_d-R\cdot i_q+L_q\cdot\omega_e\cdot i_q), \\ \frac{\mathrm{d}i_q}{\mathrm{d}t} = \frac{1}{L_q}\cdot(u_q-R\cdot i_q-L_d\cdot\omega_e\cdot i_d - \omega_e\cdot\psi_m), \\ \frac{\mathrm{d}\omega_m}{\mathrm{d}t} = \frac{1}{J}\cdot(1.5\cdot p\cdot\psi_m\cdot i_q - T_l),\\ \omega_e = p\cdot\omega_m\$$

where $$\p\$$ is number of pole pairs

The parameters of the model:

$$\L_d = 1.365\cdot10^{-3}\,H \\ L_q = 1.365\cdot10^{-3}\,H \\ R = 0.416\,\Omega \\ \psi_m = 0.166\,Wb \\ p = 2 \\ J = 3.4\cdot10^{-4}\,kg\cdot m^2 \\ kF= \frac{3.2}{(2\cdot\pi\cdot\frac{1200}{60})^2}\,\frac{N\cdot m}{(rad\cdot s^{-1})^2}\$$

Here is my overall simulation model

The motor is loaded by below given torque

$$\T_l = kF\cdot\omega^2\$$ which is removed in step wise manner at $$\t=0.4\,s\$$

The U/f block

The PMSM Model block

The Inverse Park Transform block

The Inverse Clarke Transform block

Ther results are following

Stator currents

Motor torque

Reference and actual mechanical speed

Stator voltage in q axis

I have doubts about correctness of my model due to the fact that the actual motor speed differs from the reference speed. I expected that they will equal because of the modeled machine is a synchronous motor. Does anybody know where I did a mistake? Thanks in advance for any suggestions.

• Have you tried with the power invariant Clarke transform? – a concerned citizen Feb 12 at 7:59
• @a concerned citizen Thank you for your reaction. Do you mean changing the gains in the Inverse Clarke Transform block? – Steve Feb 12 at 21:02
• Yes, but now I realize the matrix is only for displaying the currents. Maybe there are similar scaling coefficients that need to be added when using the dq equations? – a concerned citizen Feb 13 at 7:39
• As far as I know the Park transformation only realizes rotation of coordinate system via multiplying by $e^{j\phi}$ and does not change module of the space vector. – Steve Feb 13 at 10:42
• Yes, but you don't only apply the Park transform to get to the dq-reference frame. That is, it's not a simple matter of multiplying with a sin/cos and adding/subtracting in a 3phase to a 2phase, the Clarke matrix, or, at least, the coefficients are also there. That $e^{j\phi}$ has meaning in an orthogonal system, not a 3phase, so you first apply 3->2 and then sin/cos, unless you complicate with doing it directly, but even then you have scaling coefficients. That's what I referred to when I mentioned the scaling coefficients. But I am also not sure, that's why I'm stuck with comments. – a concerned citizen Feb 13 at 13:24