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
d-axis
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 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.
Electro-Mechanical System
Electromagnetic torque:
Mechanical System
Mechanical rotor speed :
Parameters
- Model
- Implementation in the rotor reference frame or as a voltage behind reactance.
- Stator resistance
- Armature or stator resistance
in
.
- Stator inductance
- A two-element vector containing the combined stator leakage
and magnetizing inductance.
is referred to the d-axis and
to the q-axis of the rotor. The values are in henries (H).
- Flux induced by magnets
- Constant flux linkage
in Vs induced by the magnets in the stator windings.
- Inertia
- Combined rotor and load inertia
in
.
- Friction coefficient
- Viscous friction
in Nms.
- Number of pole pairs
- Number of pole pairs
.
- Initial rotor speed
- Initial mechanical rotor speed
in radians per second (
).
- Initial rotor position
- Initial mechanical rotor angle
in radians.
- Initial stator currents
- A two-element vector containing the initial stator currents
and
of phase a and b in amperes (A).
Inputs and Outputs
- Mechanical torque
- The input signal
represents the mechanical torque at the rotor shaft, in Nm.
The output vector “m” contains the following 3 signals:
- (1) Rotor speed
- The rotational speed
of the rotor in radians per second (
).
- (2) Rotor position
- The mechanical rotor angle
in radians.
- (3) Electrical torque
- The electrical torque
of the machine in Nm.
Probe Signals
- Stator phase currents
- The three-phase stator winding currents
,
and
, in A. Currents flowing into the machine are considered positive.
- Stator flux (dq)
- The stator flux linkages
and
in the stationary reference frame in Vs:
- Rotational speed
- The rotational speed
of the rotor in radians per second (
).
- Rotor position
- The mechanical rotor angle
in radians.
- Electrical torque
- The electrical torque
of the machine in Nm.
See also
If the stator inductance is independent of the rotor angle, i.e. , it is
computational 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. The sophisticated BLDC machine can be configured as a PMSM with sinusoidal back EMF if the parameters are converted as follows: