Dynamic model of a PMSM

The two phase equivalent circuit is widely used to analyse the permanent magnet synchronous machine. In this blog I will present the two phase equivalent circuit of the PMSM dynamic model. 

The three phase electrical dynamic equations can be written as:
$$U_a^S=R_sI_a^S+\frac{d\psi_a^S}{dt}\\U_b^S=R_sI_b^S+\frac{d\psi_b^S}{dt}\\U_c^S=R_sI_c^S+\frac{d\psi_c^S}{dt}\tag{1}$$
where the index $S$ denotes the stator coordinate system.
Also the motor model can be express in the rotating coordinate system
$$U^R=R_sI^R+\frac{d\psi^R}{dt}+j\omega\psi^R\tag{2}$$

Now we can write in the two phase equivalent circuit which is rotating with the same frequency as the rotate magnetic field.

$$u_d=R_sI_d+\frac{d\psi_d}{dt}-\omega\psi_q\tag{3}$$
$$u_q=R_sI_q+\frac{d\psi_q}{dt}+\omega\psi_d\tag{4}$$
where
$$\psi_d=L_dI_d+L_m\tag{5}$$
$$\psi_q=L_qI_q\tag{6}$$
The produced torque can be expressed as:
$$T_e=\frac{3P}{2}\left(L_mI_q+(L_d-L_q)I_d\right)I_q\tag{7}$$
and the motor dynamics can be represented by:
$$T_e=J\frac{d\omega_r}{dt}+B\omega_r+T_L\tag{8}$$  

Coordinate transformation 

Figure 1. PMSM

As you can see in figure 1, the three phase stationary reference frame can be transformed directly into a two phase reference frame using Park's transformation. Let X represent any of the variables (current, voltage, fluxe), the transformation matrix is given by[4]

$$\left[ \begin{array}{c} X_d \\ X_q \\ X_0 \end{array} \right] =\frac{2}{3} \begin{bmatrix} sin(\theta) & sin(\theta-\frac{2\pi}{3}) & sin(\theta+\frac{2\pi}{3}) \\ cos(\theta) & cos(\theta-\frac{2\pi}{3}) & cos(\theta+\frac{2\pi}{3}) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix}\left[ \begin{array}{c} X_a \\ X_b \\ X_c \end{array} \right]\tag{9}$$

The transformation can be a combination of two transformations. First the three phase reference frame can be transformed into a two phase reference frame($abc$ to $\alpha\beta$) by replacing $\theta$ with 0

$$\left[ \begin{array}{c} X_\alpha \\ X_\beta \\ X_0 \end{array} \right] =\frac{2}{3} \begin{bmatrix} 0 & -{\sqrt3\over2} & {\sqrt3\over2} \\ 1 & -{1\over2} & -{1\over2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix}\left[ \begin{array}{c} X_a \\ X_b \\ X_c \end{array} \right]\tag{10}$$

The second transformation is a conversion from stationary to rotating reference frame ($\alpha\beta$ to $dq$)

$$\left[ \begin{array}{c} X_d \\ X_q \\ X_0 \end{array} \right] = \begin{bmatrix}  cos(\theta) & sin(\theta) & 0 \\ -sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix}\left[ \begin{array}{c} X_\alpha \\ X_\beta \\ X_0 \end{array} \right]\tag{9}$$

Appendix:

$P$ - Pole pairs
$R$ - Stator phase resistance
$L_m$ - Permanent magnets flux

$B$ - Viscous friction coefficient
$J$ - Inertia
$L_d$ - Direct axis inductance
$L_q$ - Quadrature axis inductance

$\omega$ - Angular velocity

Sources:

1. Dal Y. Ohm: DYNAMIC MODEL OF PM SYNCHRONOUS MOTORS
2. Wikipedia: dqo transformation
3. Mohamed S. Zaky: Adaptive and robust speed control of interior permanent magnet synchronous motor drives
4. Lecture Set 6.pdf