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Permanent Magnet Synchronous Machine (PMSM)



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. Stator winding is star connected in this model. To access the neutral point pin, check the "Neutral Point" checkbox in the properties menu.

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


Electrical > Motors


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)
Pin_N Neutral Point (Electrical)


Name Description
Rs Stator Winding Resistance [Ohm]
Ld Direct Axis Inductance [H]
Lq Quadratic Axis Inductance [H]
Ke Back-EMF Constant
J Rotor Inertia [kg.m²]
B Rotor Friction Coefficient [N.m/(rad/s)]
InitialSpeed Rotor initial speed [ rad/s]
InitialCurrent Initial Current Vector [IA IB IC]
NPP Number of pole pairs
ExposeNeutralPoint Expose Neutral Point (Boolean)