Permanent Magnet Synchronous Machine
Purpose
Synchronous machine excited by permanent magnets
Library
Electrical / Machines
Description
This three-phase permanent magnet synchronous machine has a sinusoidal back EMF.
The machine operates as a motor or generator; if the mechanical torque has the same sign as the rotational speed, the machine is operating in motor mode, otherwise in generator mode. All electrical variables and parameters are viewed from the stator side. In the component icon, phase a is marked with a dot.
Electrical System
Fig. 232 d-axis
Fig. 233 q-axis
Stator flux linkages:
The machine model offers two different implementations of the electrical system: a traditional rotor reference frame and a voltage behind reactance (VBR) formulation.
- Rotor Reference Frame
Using Park’s transformation, the 3-phase circuit equations in physical variables are transformed to the dq rotor reference frame. This results in constant coefficients in the differential equations making the model numerically efficient. However, interfacing the dq model with the external 3-phase network may be difficult. Since the coordinate transformations are based on voltage-controlled current sources, inductors and naturally commutated devices such as diode rectifiers may not be directly connected to the stator terminals.
- Voltage Behind Reactance
This formulation allows for direct interfacing of arbitrary external networks with the 3-phase stator terminals. The electrical system is described in circuit form. Due to the resulting time-varying inductance matrices, this implementation is numerically less efficient than the traditional rotor reference frame.
PLECS does not support code generation for models with time-varying inductance matrices. When generating code for the VBR model the machine equations are reformulated to interface electrically with a constant inductance and emulate the variable portion of the inductance with a voltage source.
Electro-Mechanical System
Electromagnetic torque:
Mechanical System
Mechanical rotor speed \(\omega_\mathrm{m}\):
Parameters
- Model
Implementation in the rotor reference frame or as a voltage behind reactance.
- Stator resistance
Armature or stator resistance \(R_\mathrm{s}\) in ohms \((\Omega)\).
- Stator inductance
A two-element vector containing the combined stator leakage and magnetizing inductance. \(L_\mathrm{d}\) is referred to the d-axis and \(L_\mathrm{q}\) to the q-axis of the rotor. The values are in henries \((\mathrm{H})\).
- Flux induced by magnets
Constant flux linkage \(\varphi'_\mathrm{m}\) in \((\mathrm{Vs})\) induced by the magnets in the stator windings.
- Inertia
Combined rotor and load inertia \(J\) in \((\mathrm{Nms}^2)\).
- Friction coefficient
Viscous friction \(F\) in \((\mathrm{Nms})\).
- Number of pole pairs
Number of pole pairs \(p\).
- Initial rotor speed
Initial mechanical rotor speed \(\omega_\mathrm{m,0}\) in \((\frac{\mathrm{rad}}{\mathrm{s}})\).
- Initial rotor position
Initial mechanical rotor angle \(\theta_\mathrm{m,0}\) in radians.
- Initial stator currents
A two-element vector containing the initial stator currents \(i_\mathrm{a,0}\) and \(i_\mathrm{b,0}\) of phase a and b in amperes \((\mathrm{A})\).
Probe Signals
- Stator phase currents
The three-phase stator winding currents \(i_\mathrm{a}\), \(i_\mathrm{b}\) and \(i_\mathrm{c}\), in amperes \((\mathrm{A})\). Currents flowing into the machine are considered positive.
- Stator flux (dq)
The stator flux linkages \(\varphi_\mathrm{d}\) and \(\varphi_\mathrm{q}\) in the rotating reference frame in \((\mathrm{Vs})\):
\[\varphi_\mathrm{d} = L_\mathrm{q} \, i_\mathrm{q}\]\[\varphi_\mathrm{d} = L_\mathrm{d} \, i_\mathrm{d} + \varphi'_\mathrm{m}\]- Rotational speed
The rotational speed \(\omega_\mathrm{m}\) of the rotor in \((\frac{\mathrm{rad}}{\mathrm{s}})\).
- Rotor position
The mechanical rotor angle \(\theta_\mathrm{m}\) in radians.
- Electrical torque
The electrical torque \(T_\mathrm{em}\) of the machine in \((\mathrm{Nm})\).
See also
If the stator inductance is independent of the rotor angle, i.e. \(L_\mathrm{d} = L_\mathrm{q}\), it is computationally more efficient to use the simplified Brushless DC Machine with a sinusoidal back EMF. The parameters need to be converted as follows:
For back EMF shapes other than sinusoidal, and/or if the stator inductance has a complex angle dependency, please use the sophisticated model of the Brushless DC Machine. It can be configured as a PMSM with sinusoidal back EMF if the parameters are converted as follows: