The Permanent Magnet Synchronous Machine block implements a three-phase or a five-phase permanent magnet synchronous machine. The stator windings are connected in wye to an internal neutral point.
The three-phase machine can have a sinusoidal or trapezoidal back EMF waveform. The rotor can be round or salient-pole for the sinusoidal machine. The rotor is round when the machine is trapezoidal. Preset models are available for the sinusoidal back EMF machine.
The five-phase machine has a sinusoidal back EMF waveform and round rotor.
The Permanent Magnet Synchronous Machine block operates in either generator or motor mode. The mode of operation is dictated by the sign of the mechanical torque (positive for motor mode, negative for generator mode). The electrical and mechanical parts of the machine are each represented by a second-order state-space model.
The sinusoidal model assumes that the flux established by the permanent magnets in the stator is sinusoidal, which implies that the electromotive forces are sinusoidal.
The trapezoidal model assumes that the winding distribution and flux established by the permanent magnets produce three trapezoidal back EMF waveforms.
These equations are expressed in the rotor reference frame (qd frame). All quantities in the rotor reference frame are referred to the stator.
d d t i d = 1 L d v d − R L d i d + L q L d p ω m i q
d d t i q = 1 L q v q − R L q i q − L d L q p ω m i d − λ p ω m L q
T e = 1.5 p [ λ i q + ( L d − L q ) i d i q ]
q-axis and d-axis inductances
Resistance of the stator windings
q-axis and d-axis currents
q-axis and d-axis voltages
Angular velocity of the rotor
Amplitude of the flux induced by the permanent magnets of the rotor in the stator phases
Number of pole pairs
The Lq and Ld inductances represent the relation between the phase inductance and the rotor position due to the saliency of the rotor. For example, the inductance measured between phase A and B (when phase C is left open) is given by:
L a b = L d + L q + ( L q − L d ) cos ( 2 θ e + π 3 ) ,
where Θe represents the electrical angle.
The next figure shows the variation of the phase-to-phase inductance in function of the electrical angle of the rotor.
These equations are expressed in the rotor reference frame using an extended Park transformation (q1d1 and q2d2 frame). All quantities in the rotor reference frame are referred to the stator.
d d t i d 1 = 1 L v d 1 − R L i d 1 + L q L p ω m i q 1
d d t i q 1 = 1 L v q 1 − R L i q 1 − L d L p ω m i d 1 − λ p ω m L
d d t i d 2 = 1 L v d 2 − R L i d 2
d d t i q 2 = 1 L v q 2 − R L i q 2
T e = 2.5 p λ i q 1
Resistance of the stator windings
q1-axis and d1-axis currents
q1-axis and d1-axis voltages
q2-axis and d2-axis currents
q2-axis and d2-axis voltages
Angular velocity of the rotor
Amplitude of the flux induced by the permanent magnets of the rotor in the stator phases
Number of pole pairs
These equations are expressed in the phase reference frame (abc frame). Note that the phase inductance Ls is assumed to be constant and does not vary with the rotor position.
d d t i a = 1 3 L s ( 2 v a b + v b c − 3 R s i a + λ p ω m ( − 2 Φ a ′ + Φ b ′ + Φ c ′ ) ) d d t i b = 1 3 L s ( − v a b + v b c − 3 R s i b + λ p ω m ( Φ a ′ − 2 Φ b ′ + Φ c ′ ) ) d d t i c = − ( d d t i a + d d t i b ) T e = p λ ( Φ a ′ ⋅ i a + Φ b ′ ⋅ i b + Φ c ′ ⋅ i c )
Inductance of the stator windings
Resistance of the stator windings
a, b and c phase currents
a, b and c phase electromotive forces, in per-unit value to the amplitude of the flux λ
ab and bc phase to phase voltages
Angular velocity of the rotor
Amplitude of the flux induced by the permanent magnets of the rotor in the stator phases
Number of pole pairs
The electromotive force Φ ' is represented by:
d d t ω m = 1 J ( T e − T f − F ω m − T m ) d θ d t = ω m
Combined inertia of rotor and load
Combined viscous friction of rotor and load
Rotor angular position
Shaft mechanical torque
Shaft static friction torque
Angular velocity of the rotor (mechanical speed)
The power_brushlessDCmotor example illustrates the use of the Permanent Magnet Synchronous Machine block.
When you use Permanent Magnet Synchronous Machine blocks in discrete systems, you might have to use a small parasitic resistive load, connected at the machine terminals, to avoid numerical oscillations. Large sample times require larger loads. The minimum resistive load is proportional to the sample time. Remember that with a 25 μs time step on a 60 Hz system, the minimum load is approximately 2.5% of the machine nominal power. For example, a 200 MVA permanent magnet synchronous machine in a power system discretized with a 50 μs sample time requires approximately 5% of resistive load, or 10 MW. If the sample time is reduced to 20 μs, a resistive load of 4 MW is sufficient.
The Permanent Magnet Synchronous Machine block assumes a linear magnetic circuit with no saturation of the stator and rotor iron. This assumption can be made because of the large air gap usually found in permanent magnet synchronous machines.
Mechanical torque at the machine shaft. This input port is normally positive because the Permanent Magnet Synchronous Machine block is usually used as a motor. If you choose to use the block in generator mode, you can apply a negative torque input.
To enable this port, set Mechanical input to Torque Tm .
Machine speed, in rad/s.
To enable this port, set Mechanical input to Speed w .
Vector containing measurement signals. The block returns a 13-element vector when Number of phases is set to 3 , and a 16-element vector when Number of phases is set to 5 . The available signals depend on the model you selected. You can demultiplex these signals by using the Bus Selector block provided in the Simulink ® library. The signals include:
Stator current is_a
Stator current is_b
Stator current is_c
Stator current is_d
Stator current is_e
Stator current is_q
Stator current is_d
Stator current is_q1
Stator current is_d1
Stator current is_q2
Stator current is_d2
Stator voltage Vs_q
Stator voltage Vs_d
Stator voltage Vs_q1
Stator voltage Vs_d1
Stator voltage Vs_d2
Phase back EMF e_a
Phase back EMF e_b
Phase back EMF e_c
Hall effect signal h_a *
Three-Phase, Sinusoidal and Trapezoidal
Hall effect signal h_b *
Three-Phase, Sinusoidal and Trapezoidal
Hall effect signal h_c *
Three-Phase, Sinusoidal and Trapezoidal
Rotor angle thetam
Electromagnetic torque Te
The Hall effect signal provides a logical indication of the back EMF positioning. This signal is very useful to directly control the power switches. There is a change of state at each zero crossing of the phase-to-phase voltage. These signals must be decoded before being applied to the switches.
Mechanical rotational port, that represents the rotational shaft of the machine.
To enable this port, set Mechanical input to Mechanical rotational port .
Specialized electrical conserving port associated with the electrical terminal of phase A.
Specialized electrical conserving port associated with the electrical terminal of phase B.
Specialized electrical conserving port associated with the electrical terminal of phase C.
Select between a three-phase machine model or a five-phase machine model.
Select between Sinusoidal and Trapezoidal electromotive force.
To enable this parameter, set Number of phases to 3 .
Select between Salient-pole and Round rotors.
To enable this parameter, set Number of phases to 3 and set Back EMF waveform to Sinusoidal .
Select whether input is supplied by torque applied to the shaft, rotor speed, or a machine shaft represented by a Simscape™ rotational mechanical port.
Select Torque Tm to specify a torque input in N.m and expose the Tm port. The machine speed is determined by the machine Inertia J and by the difference between the applied mechanical torque Tm and the internal electromagnetic torque Te. The sign convention for the mechanical torque is when the speed is positive. A positive torque signal indicates motor mode and a negative signal indicates generator mode.
Select Speed w to specify a speed input in rad/s and expose the w port. The machine speed is imposed and the mechanical part of the model (Inertia J) is ignored. Using the speed as the mechanical input allows modeling a mechanical coupling between two machines.
The next figure indicates how to model a stiff shaft interconnection in a motor-generator set when friction torque is ignored in machine 2. The speed output of machine 1 (the motor) is connected to the speed input of machine 2 (the generator), while the machine 2 electromagnetic torque output, Te, is applied to the mechanical torque input of machine 1, Tm. The Kw factor takes into account the speed units of both machines (pu or rad/s) and the gear box ratio w2/w1 . The KT factor takes into account the torque units of both machines (pu or N.m) and machine ratings. Also, because the inertia J2 is ignored in machine 2, J2 is added to the machine 1 inertia, J1.
Select Mechanical rotational port to expose a Simscape mechanical rotational port that allows you to connect the machine shaft to other Simscape blocks with mechanical rotational ports.
The next figure indicates how to connect an Ideal Torque Source block from the Simscape library to the machine shaft to represent the machine in motor mode or in generator mode.
