SimPowerSystems™

Permanent Magnet Synchronous Machine - Model the dynamics of three-phase permanent magnet synchronous machine with sinusoidal or trapezoidal back electromotive force (back EMF)

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Machines

Description

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. For the trapezoidal machine, the model assumes that the winding distribution and flux established by the permanent magnets produce three trapezoidal back EMF waveforms.

The block implements the following equations.

Sinusoidal Model Electrical System

These equations are expressed in the rotor reference frame (qd frame).

where (all quantities in the rotor reference frame are referred to the stator)

The L_q and L_d 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 (phase c is left open) is given by:

where Θ_e represents the electrical angle.

The next figure shows the variation of the line-line inductance in function of the electrical angle of the rotor:

• For a round rotor, there is no variation in the phase inductance. Therefore,

  

• For a salient round rotor, the dq inductances are given by:

  

  and

  

Trapezoidal Model Electrical System

These equations are expressed in the phase reference frame (abc frame). Note that the phase inductance L_s is assumed constant and does not vary with the rotor position.

where the electromotive force Φ^' is represented by

and

Mechanical System (for both models)

where

Dialog Box and Parameters

Configuration Tab

Back EMF waveform

Allows to select between the sinusoidal and the trapezoidal electromotive force.

Mechanical input

Allows you to select either the torque applied to the shaft or the rotor speed as the Simulink signal applied to the block's input.

Select Torque Tm to specify a torque input, in N.m., and change labeling of the block's input to Tm. 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 the following: 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 change labeling of the block's input to w. 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 and interfacing with SimMechanics and SimDriveline models.

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 (motor) is connected to the speed input of machine 2 (generator), while machine 2 electromagnetic torque output Te is applied to the mechanical torque input Tm of machine 1. The Kw factor takes into account speed units of both machines (pu or rad/s) and gear box ratio w2/w1. The KT factor takes into account torque units of both machines (pu or N.m) and machine ratings. Also, as the inertia J2 is ignored in machine 2, J2 referred to machine 1 speed must be added to machine 1 inertia J1.

Preset Model

Provides a set of predetermined electrical and mechanical parameters for various permanent magnet synchronous motor ratings of torque (N.m), DC bus voltage (V), rated speed (rpm), and continuous stall torque (N.m) . This parameter is available only for sinusoidal machine.

Select one of the preset models to load the corresponding electrical and mechanical parameters in the entries of the dialog box. Select No if you do not want to use a preset model.

Parameters Tab

Stator phase resistance

The stator phase resistance Rs (Ω).

Stator inductances

• Sinusoidal model: The d-axis and q-axis stator inductances Ld (H) and Lq (H).

• Trapezoidal model: The stator phase inductance Ls (H).

Specify a machine constant

Lets you select the machine constant that you wish to specify for block parameterization:

• Flux linkage established by magnets

• Voltage Constant

• Torque Constant

Once you select a constant, you can enter its value in the appropriate parameter field, while the other two parameters become inaccessible and are only shown for information.

Flux linkage established by magnets

The constant flux λ (Wb) induced in the stator windings by the magnets.

Voltage Constant

The peak line to line voltage per 1000 rpm. This voltage represents the peak open circuit voltage when the machine is driven as a generator at 1000 rpm.

Torque Constant

The torque per ampere constant. This constant assumes that the machine is driven by an inverter which provides a perfect synchronization between the current and the Back-EMF.

• Sinusoidal model: A sine wave current is assumed (see ac6_example_simplified.mdl for more detail).

• Trapezoidal model: A square ware current is assumed (see ac7_example_simplified.mdl for more detail).

Back EMF flat top area

The width of the flat top for a half period of the electromotive force Φ^' (degrees) (only for trapezoidal machine).

Inertia, friction factor and pole pairs

The combined machine and load inertia coefficient J (kg.m²), combined viscous friction coefficient F (N.m.s), and pole pairs p.

Initial conditions

Specifies the mechanical speed (rad/s), mechanical angle Θ_m (degrees), and instantaneous stator current (A):

[w_m, Θ_m, i_a, i_b]

Note that since the stator is wye-connected, the current i_c is given by i_c = -i_a-i_b.

Advanced Tab

Sample time (-1 for inherited)

Specifies the sample time used by the block. To inherit the sample time specified in the Powergui block, set this parameter to -1.

Inputs and Outputs

Tm

The Simulink input is the mechanical torque at the machine's shaft. This input should normally be positive because the Permanent Magnet Synchronous Machine block is usually used as a motor. Nevertheless, you can apply a negative torque input if you choose to use the block in generator mode.

m

The Simulink output of the block is a vector containing 13 signals for the sinusoidal model and 12 signals for the trapezoidal model. You can demultiplex these signals by using the Bus Selector block provided in the Simulink library.

Definition	Units	Symbol	Signal number
			Sinusoidal model	Trapezoidal model
Stator current is_a	A	ia	1	1
Stator current is_b	A	ib	2	2
Stator current is_c	A	ic	3	3
Stator current is_q	A	iq	4	N/A
Stator current is_d	A	id	5	N/A
Stator voltage Vs_q	V	vq	6	N/A
Stator voltage Vs_d	V	vd	7	N/A
Phase back EMF e_a	V	ea	N/A	4
Phase back EMF e_b	V	eb	N/A	5
Phase back EMF e_c	V	ec	N/A	6
Hall effect signal h_a*	logic 0-1	ha	8	7
Hall effect signal h_b*	logic 0-1	hb	9	8
Hall effect signal h_c*	logic 0-1	hc	10	9
Rotor speed wm	rad/s	ωr	11	10
Rotor angle thetam	rad	Θr	12	11
Electromagnetic torque Te	N.m	Te	13	12

The Hall effect signal provides a logical indication of the back EMF positioning. This signal is very useful to control directly 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.

Assumption

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.

Limitations

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, in order to avoid numerical oscillations. Large sample times require larger loads. The minimum resistive load is proportional to the sample time. As a rule of thumb, 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 PM 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 should be sufficient.

Example

The power_brushlessDCmotor demo illustrates the use of the Permanent Magnet Synchronous Machine block in motoring mode with a closed-loop control system built entirely with Simulink blocks. The complete system includes a six step inverter block from the SimPowerSystems library. Two control loops are used; the inner loop synchronizes the pulses of the bridge with the electromotive forces, and the outer loop regulates the motor's speed, by varying the DC bus voltage. The mechanical torque applied at the motor's shaft is originally 0 N.m (no load) and steps to its nominal value (3 N.m) at t = 0.1 second. The parameters of the machine are found in the dialog box section.

Set the simulation parameters as follows:

• Type: Fixed-step

• Integrator type: Runge-Kutta, ode4

• Sample time: 5e-6 (set automatically by the Model properties)

• Stop time: 0.2

Set the Flux distribution parameter to Trapezoidal and run the simulation to observe the motor's torque, speed, and currents. Change the Back EMF flat top area parameter of the trapezoidal model from 120 to 0 and observe the waveform of the electromotive force e_a.

The torque climbs to nearly 28 N.m when the motor starts and stabilizes rapidly when the motor reaches the reference value. The nominal torque is applied at t = 0.1 second and the controller reacts rapidly and increases the DC bus voltage to produce the required electric torque. Observe the saw tooth shape of the currents waveforms. This is caused by the six step controller, which applies a constant voltage value during 120 electrical degrees to the motor. The initial current is high and decreases during the acceleration to the nominal speed. When the nominal torque is applied, the stator current increases to maintain the nominal speed. The saw tooth waveform is also observed in the electromotive torque signal Te. However, the motor's inertia prevents this noise from appearing in the motor's speed waveform.

When the Back EMF flat top area parameter of the trapezoidal model is changed from 120 to 0, the model reacts exactly like the sinusoidal model. The electromotive force e_a is purely sinusoidal and the torque ripple is less than the previous case. The sinusoidal model requires a larger current to produce the same torque. That's why the trapezoidal machine is used in high torque applications, and the sinusoidal machine in precision applications.

References

[1] Grenier, D., L.-A. Dessaint, O. Akhrif, Y. Bonnassieux, and B. LePioufle, "Experimental Nonlinear Torque Control of a Permanent Magnet Synchronous Motor Using Saliency," IEEE Transactions on Industrial Electronics, Vol. 44, No. 5, October 1997, pp.680-687.

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