# Permanent Magnet Synchronous Machine (PMSM)

## Description

Permanent Magnet Synchronous Machine (PMSM)

Model of a three-phase Permanent Magnet Synchronous Machine (PMSM) with the sinusoidal back Electro-Motive Force (EMF). In motor operation torque and speed have the same sign. Rotor Angle is defined as the angle between the a-phase and the d-axis. In this model, the saturation can be modeled using a piecewise linear approximation of direct and quadratic axis inductances.

### Electrical model and equations

v_a = R_s i_a + \frac{d}{dt} (L_{aa} i_a + L_{ab} i_b + L_{ac} i_c) + \frac{d }{dt} \phi_m \sin{(\omega_r t)}
v_b = R_s i_b + \frac{d}{dt} (L_{ba} i_a + L_{bb} i_b + L_{bc} i_c) + \frac{d }{dt}\phi_m \sin{(\omega_r t - \frac{2 \pi }{3})}
v_c = R_s i_c + \frac{d}{dt} (L_{ca} i_a + L_{cb} i_b + L_{cc} i_c) + \frac{d }{dt} \phi_m \sin{(\omega_r t + \frac{2 \pi }{3})}

where $\phi_m = \frac{K_e}{ N_{ pp} }$ is the permanent magnet flux linkage, and $\omega_r = N_{pp} \Omega$ is the electrical speed of the rotor field. Phase inductances ($L_{aa}$, $L_{ab}$, $L_{ac}$,..., $L_{cc}$) are calculated from the rotor reference frame inductance ($L_{d}$, $L_{q}$) defined in the property panel as the incremental inductance.
This component will use the lineary interpolated inductance when the current is between the two segmented points.

### Electromechanical equations

Electro-magnetic torque:

T_e = 1.5 * N_{ pp}*(i_q * \phi_d - i_d * \phi_q)

where $\phi_d = L_d i_d + \phi_m$ and $\phi_q = L_q i_q$

Mechanical rotational speed $\Omega$:

J \frac{d\Omega}{ dt} = T_e - B \Omega

## Library

Electrical > Motors

## Pins

Name Description
Pin_A Phase A (Electrical)
Pin_B Phase B (Electrical)
Pin_C Phase C (Electrical)
Pin_R Rotor (Rotational Mechanical)
Pin_Angle Rotor Angle in radians, electrical angle (Control)

## Parameters

Name Description
Rs Stator Winding Resistance [Ohm]
Ld Direct Axis Inductance [H]