Vector Control of Permanent Magnet Synchronous Motor (1)————Mathematical Model and Steady State Characteristics of Permanent Magnet Synchronous Motor

1. Mathematical model of permanent magnet synchronous motor

In the permanent magnet synchronous motor vector control system, there are two commonly used coordinate systems: two-phase rotating coordinate system ( $\tiny d-q$ coordinate system) and two-phase stationary coordinate system ( $\tiny \alpha -\beta$ coordinate system). The relationship between the coordinate systems is shown in the figure. In the figure, the direction of the flux linkage vector generated by the permanent magnet is consistent with the direction of the rotor pole.

The permanent magnet synchronous motor is a non-linear system with the characteristics of multivariable and strong coupling. When we analyze it, we have the following assumptions:

Ignore core saturation, eddy current and hysteresis loss
Ignore the armature response during commutation
There is no damping winding on the rotor, and the permanent magnet has no damping effect
The stator winding current produces only sinusoidally distributed magnetic potential in the air gap without high-order harmonics

Modeling according to motor application

Under this ideal condition:

1.1 The stator voltage equation of a permanent magnet synchronous motor in a three-phase static coordinate system:

$\small \begin{bmatrix} u_{a}\\ u_{b}\\ u_{c} \end{bmatrix}=\begin{bmatrix} R_{s}&0&0\\ 0&R_{s}&0\\ 0&0&R_{s} \end{bmatrix}\begin{bmatrix} i_{a}\\ i_{b}\\ i_{c} \end{bmatrix}+\begin{bmatrix} \Psi _{a}^{'}\\ \Psi _{b}^{'}\\\Psi _{c}^{'} \end{bmatrix}$

Wherein $\tiny R_{s}$ the armature resistance, $\tiny \Psi _{a},\Psi _{b},\Psi _{c}$ respectively, $\tiny abc$ a three-phase flux, $\tiny i_{a},i_{b},i_{c}$ respectively, for $\tiny abc$ the phase currents of the three phases.

1.2 The flux linkage equation in the three-phase stationary coordinate system

$\small \begin{bmatrix} u_{a}\\ u_{b}\\ u_{c} \end{bmatrix}=\begin{bmatrix} L_{aa}&M_{ab}&M_{ac}\\ M_{ba}&L_{bb}&M_{bc}\\ M_{ca}&M_{cb}&L_{ac} \end{bmatrix}\begin{bmatrix} i_{a}\\ i_{b}\\ i_{c} \end{bmatrix}+\Psi _{f}\begin{bmatrix} cos\theta \\ cos\left ( \theta -\frac{2\pi }{3} \right )\\ cos\left ( \theta +\frac{2\pi }{3} \right ) \end{bmatrix}$

Wherein, $\tiny L_{aa},L_{bb},L_{cc}$ for the self-inductance of each phase winding, and $\tiny L_{aa}=L_{bb}=L_{cc}$ , where $\tiny M_{ab}$ other and are equal to the mutual inductance between the windings. $\tiny \Psi _{f}$ It is the permanent magnet flux linkage, which $\tiny \theta$ is the angle between the rotor $\tiny N$ pole and the $\tiny a$ phase axis.

After $\tiny CLARK$ and $\tiny PARK$ transformation, $\tiny dq$ the mathematical model in the coordinate system is obtained :

1.3 $\small dq$ Voltage equation in the coordinate system

$\small \begin{bmatrix} u _{d}\\ u _{q} \end{bmatrix}=\begin{bmatrix} R_{s} & -\omega _{e}L_{q}\\ \omega _{e}L_{d}& R_{s} \end{bmatrix}\begin{bmatrix} i_{d}\\ i_{q} \end{bmatrix}+\frac{\mathrm{d}}{\mathrm{d} t}\begin{bmatrix} \Psi _{d}\\ \Psi _{q} \end{bmatrix}+\begin{bmatrix} 0\\ \omega _{e}\Psi _{f} \end{bmatrix}$

Wherein, $\tiny u_{d},u_{q}$ for the $\tiny dq$ axial voltage, $\tiny i_{d},i_{q}$ of $\tiny dq$ axis current $\tiny \Psi _{d},\Psi _{q}$ to $\tiny dq$ the axis flux, $\tiny L_{d},L_{q}$ to $\tiny dq$ axis inductance, $\tiny \omega _{e}$ the rotation speed.

1.4 $\small dq$ Flux linkage equation in the coordinate system

$\small \begin{bmatrix} \Psi _{d}\\ \Psi _{q} \end{bmatrix}=\begin{bmatrix} L_{d} & 0\\ 0& L_{q} \end{bmatrix}\begin{bmatrix} i_{d}\\ i_{q} \end{bmatrix}+\begin{bmatrix} \Psi _{f}\\ 0 \end{bmatrix}$

1.5 Torque equation

$\small \dpi{150} \small T_{e}=\frac{3}{2}n_{p}\left ( \Psi _{d}i_{q}-\Psi _{q}i_{d} \right )=\frac{3}{2}n_{p}\left ( \Psi _{f}i_{q}+\left ( L_{d} -L_{q}\right )i_{d}i_{q} \right )$

It can be seen from the torque equation in 1.5 that the electromagnetic torque is composed of two parts. The first term is produced by the interaction between the permanent magnet and the stator winding flux, and the second term is produced by the change of magnetic resistance. Here we need to distinguish the difference between salient pole and hidden pole motors. For hidden pole motors $\tiny L_{q}=L_{d}$ , the reluctance change torque is unique to salient pole motors. We also need to pay attention to the type of motor when building the simulation.
Summary: The mathematical model of the permanent magnet synchronous motor explains its internal structure and helps us design control strategies. We need to analyze its mathematical model when we carry out coordinate transformation and PI parameter tuning.