*2.1. Motor Equations*

The simplified design of the interior permanent magne<sup>t</sup> synchronous motor (IPMSM) is shown in Figure 1. It has three pole pairs formed by the interior permanent magnets in the rotor and three-phase winding at the stator. The interaction of the flux formed by permanent magnets and the flux formed by the three-phase stator winding produces the torque.

**Figure 1.** Simplified design of IPMSM.

The basic electrical equations of IPMSM in the synchronous reference frame *dq* under assumption that hysteresis loss, eddy currents, etc., can be neglected, are:

$$\begin{aligned} u\_d &= i\_d r\_s + \frac{d\psi\_d}{dt} - \omega \psi\_{q'} \\ u\_q &= i\_q r\_s + \frac{d\psi\_q}{dt} + \omega \psi\_d; \end{aligned} \tag{1}$$

where:

> *ud*, *uq*—*d*- and *q*-axis voltage components, respectively, *id*, *iq*—*d*- and *q*-axis currents components, respectively, ψ*<sup>d</sup>*, ψ*q*—*d*- and *q*-axis flux linkages respectively,

*rs*—stator resistance,

ω—electrical angular velocity.

The flux linkages cab be expressed as:

$$\begin{aligned} \Psi\_d &= L\_d \dot{\imath}\_d + \Psi\_{m\prime} \\ \Psi\_q &= L\_q \dot{\imath}\_q \end{aligned} \quad \begin{aligned} \Psi\_{m\prime} \\ \to \tag{2}$$

where:

> *Ld*, *Lq*—full *d*- and *q*-axis inductances, respectively,

Ψ*m*—permanen<sup>t</sup> magne<sup>t</sup> flux linkage.

Combining (1) and (2) the motor equations in synchronous reference frame *dq* can be derived:

$$\begin{aligned} \mu\_d &= i\_d r\_s + L\_d^{diff} \frac{\text{di}\_d}{\text{d}t} - \omega L\_q i\_{q\_f} \\ \mu\_q &= i\_q r\_s + L\_q^{diff} \frac{\text{di}\_q}{\text{d}t} + \omega L\_d i\_d + \omega \Psi\_m; \end{aligned} \tag{3}$$

where *Ldi f f d* and *Ldi f f q* are differential inductances of *d*- and *q*-axis, respectively. Thetorqueproducedbythemotor is definedasandcontainsfullinductances:

$$T = \frac{3}{2} p i\_q \left( (L\_d - L\_q) i\_d + \Psi\_m \right) \tag{4}$$

where:

*p*—number of pole pairs.

Equations (3) and (4) are conventional equations of IPMSM, which are used for the design of an overwhelming majority of control systems.
