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Permanent Magnet Synchronous Machine

Purpose

Synchronous machine excited by permanent magnets

Library

Electrical / Machines

Description

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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

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Stator flux linkages:

φq = Lqiq

φd = Ld id + φ′m

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:

    3
Te = 2 p (φd iq - φqid)

Mechanical System

Mechanical rotor speed ω
 m  :

       1-
ωm  =  J (Te - Fωm - Tm)

θm  = ωm

Parameters

Model
Implementation in the rotor reference frame or as a voltage behind reactance.
Stator resistance
Armature or stator resistance Rs   in _O_  .
Stator inductance
A two-element vector containing the combined stator leakage and magnetizing inductance. Ld   is referred to the d-axis and Lq   to the q-axis of the rotor. The values are in henries (H).
Flux induced by magnets
Constant flux linkage φ ′m    in Vs induced by the magnets in the stator windings.
Inertia
Combined rotor and load inertia J   in Nms2  .
Friction coefficient
Viscous friction F   in Nms.
Number of pole pairs
Number of pole pairs p  .
Initial rotor speed
Initial mechanical rotor speed ωm,0    in radians per second (s-1  ).
Initial rotor position
Initial mechanical rotor angle θm,0   in radians.
Initial stator currents
A two-element vector containing the initial stator currents ia,0   and ib,0   of phase a and b in amperes (A).

Inputs and Outputs

Mechanical torque
The input signal Tm   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 ωm    of the rotor in radians per second (s-1  ).
(2) Rotor position
The mechanical rotor angle θm   in radians.
(3) Electrical torque
The electrical torque Te   of the machine in Nm.

Probe Signals

Stator phase currents
The three-phase stator winding currents ia  , ib   and ic  , in A. Currents flowing into the machine are considered positive.
Stator flux (dq)
The stator flux linkages φ
 d   and φ
  q   in the stationary reference frame in Vs:

φ  = L i
  q   q q

φd = Ld id + φ′m

Rotational speed
The rotational speed ωm   of the rotor in radians per second (s-1  ).
Rotor position
The mechanical rotor angle θm   in radians.
Electrical torque
The electrical torque Te   of the machine in Nm.

See also

If the stator inductance is independent of the rotor angle, i.e. L  = L
  d   q  , 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:

L - M = L  = L
         d    q

KE = - φ′m ⋅p

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:

Kc,n = [0]

Ks,n = [- φ′m ⋅p]

L0- M = Ld-+-Lq
           2

L   = [0 L - L ]
 c,n       d   q

Ls,n = [0 0]