Permanent Magnet Synchronous Machine

This block implements a permanent magnet synchronous machine.

The PMSM block implements a three-phase Permanent Magnet Synchronous Motors (PMSM) model with resolvers and encoders.

Equations & Characteristics

General PMSM Solver Equation

The equation of the PMSM model can be expressed as follows:

where L_abc is the time-varying inductance matrix (global inductance for DQ and VDQ models), I_abc is the stator current inside the winding, R is the stator resistance and V_abc is the voltage across the stator windings. As for ψ_abc, it defines the magnet flux linked into the stator windings (for DQ and VDQ models), or the total flux (for the SH model),

Standard DQ Motor Characteristics

In normal conditions, the ideal sinusoidal stator voltages of the PMSM, back-EMFs, and inductances all have sinusoidal shapes. One can transform the equation using the Park transformation with a referential locked on the rotor position θ using (2a) and (2b).

The Park transform (also called ‘DQ’ transform) reduces sinusoidal varying quantities of inductances, flux, current, and voltage to constant values in the D-Q frame thus greatly facilitating the analysis and control of the device under study.

It is important to note that there are many different types of Park transforms and this often leads to confusion when interpreting the motor states inside the D-Q frame. The one used here presents the advantage of being orthonormal (notice the √3/2 factor). This particular Park orthonormal transform is power-invariant which means that the power computed in the D-Q frame by performing a dot product of currents and voltages will be numerically equal to the one computed in the phase domain. With this transform (and only this transform) the PMSM torque can be expressed by (3), where pp is the number of pole pairs.

One may notice the absence of the 3/2 factor in (3), which is usually present in the PMSM torque equation when using non-orthonormal transforms. This is, again, because this model uses the orthonormal Park transform. Figure 1 explains the principle of the Park transform. Considering fixed ABC referential with all quantities ( V_bemf, motor current I) rotating at the electric frequency ω, if we observe these quantities in a D-Q frame turning at the same speed we can see that the motor quantities will be constant.

This is easy to see for the Back-EMF voltage V_bemf that directly follows the Q-axis (because the magnet flux is on the D-axis by definition). In Figure 3, I leads and the Q-axis by an angle called β (beta). The modulus of the vector I is called I_amp. In the figure below, θ is the rotor angle, aligned with the D-axis.

Park transform with definitions for theta and beta

The Vd and Vq are not calculated directly inside the blackbox. But using P and Q could estimate Vd and Vq using the following equations since Instantaneous calculated P and Q in the phase domain are equivalent to instantaneous calculated P and Q in the DQ domain.

FPGA Implementation

Since the simulation is performed on FPGA hardware, the block implements a low pass filter on V_abc of stator set at 1000 Hz to help visualize the voltage. Because the original traces are square-like, it is harder to figure out the trace of V_abc without the filter.

The FPGA block also implements a low-pass filter that is used to remove high harmonics of the DQ currents from accessing the table, which could make the model stiffer. Its default cut-off frequency is set at 200 Hz, but it can be adjusted by the parameter “TableInputCutoffFrequency“ in the json parameter file (see PMSM Variable DQ parameters file).

Parameters & Measurements

The PMSM's parameters and measurements are separated in 4 different tabs, Electrical, Mechanical, Resolver and Encoder.

Electrical Parameters & Measurements

Symbol	Name	Description	Unit	Type

Symbol	Name	Description	Unit	Type
M_conf	Motor configuration type	Type of motor : Constant DQ, Variable DQ or Spatial Harmonics. In Variable DQ and Spatial Harmonics mode, electrical input parameters are hidden and a resource file must be provided.		Input
M_file	Motor configuration	Field only visible in Variable DQ and Spatial Harmonics modes. Allows to import a Variable DQ or Spatial Harmonics motor configuration resource file.		Input
R_s	Stator resistances	Resistances of the stator windings specified for every phase, A, B and C.	Ω	Input
L_ls	Stator inductances	d and q axis inductances	H	Input
δΦ/δθ	Back EMF profile	Profile of the back EMF, either Sinusoidal or User defined		Input
EMF_file	Back EMF profile table	Field only visible in User defined mode. Allows to import a back EMF resource file.		Input
λ_m	Permanent magnet flux linkage	Amplitude of the rotor permanent magnet flux	Wb	Input
pp	Number of pole pairs	Number of pole pairs		Input
i_s	Stator currents	Currents measured at phases A, B and C of the stator	A	Measurement
i_sdq	Stator currents (DQ)	Currents measured of axis d and q	A	Measurement
V_s	Stator voltages	Voltages measured at phases A, B and C of the stator	V	Measurement
B_emf	Back EMF voltages	Phase to neutral voltage generated from the permanent magnet flux linkage	V	Measurement
P	Active power (3ph, instantaneous)	Instantaneous electrical active power	W	Measurement
Q	Reactive power (3ph, instantaneous)	Instantaneous electrical reactive power	var	Measurement
θ_e	Electrical rotor position	Position of the rotor from 0 to 360 degrees	°	Measurement
R_s	Snubber resistance	Resistances of the snubber on phase A, B and C	Ω	Input
C_s	Snubber capacitance	Capacitance of the snubber on phase A, B and C	F	Input

Mechanical Parameters & Measurements

Symbol	Name	Description	Unit	Type

Symbol	Name	Description	Unit	Type
J	Rotor inertia	Moment of inertia of the rotor	kg*m²	Input
F_v	Viscous friction coefficient	Viscous friction	Nms/rad	Input
F_s	Static friction torque	Static friction	N*m	Input
ctrl	Mechanical control mode	Control mode of the mechanical model. Has two possible values: speed or torque. In speed mode, the mechanical model is bypassed and the speed command is sent directly. In torque mode, the torque command is used to measure the speed using the mechanical parameters of the machine.		Input
T	Torque command	Torque command sent to the mechanical model	N*m	Input
ω_rc	Rotor speed command	Speed command sent to the mechanical model	rpm	Input
ω_r	Rotor speed	Speed of the rotor	rpm	Measurement
T_e	Electromagnetic torque	Torque measured at the rotor	N*m	Measurement
θ₀	Initial rotor angle	Rotor position at time t = 0	°	Input
θ	Rotor angle	Rotor position from 0 to 360 degrees	°	Measurement

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Resolver Parameters & Measurements

Symbol	Name	Description	Unit	Type

Symbol	Name	Description	Unit	Type
R_en	Enable resolver	Whether or not to enable the resolver	N/A	Input
R_sc	Resolver feedback signals	The two two-phase windings producing a sine and cosine feedback current proportional to the sine and cosine of the angle of the motor	N/A	Measurement
R_pp	Number of resolver pole pairs	Number of pole pairs of the resolver	N/A	Input
R_dir	Direction of the sensor rotation	Direction in which the sensor is turning, either clockwise or counterclockwise	N/A	Input
R_θ	Angle offset Δθ ( Sensor- Rotor )	Angle offset between the resolver and the rotor position from 0 to 360 degrees	°	Input
E_mod	Excitation mode	Number of resolver excitation windings. Can either be single (just one excitation signal) or double (two excitation signals)	N/A	Input
R_k1	Resolver sine cosine gains	The sine/cosine modulation output sine/cosine component amplitude. Default value are 1, 0, 0 and 1. When the double excitation is selected, the default values shall be set 0, 1, 1, and 0.	N/A	Input
R_k2	Resolver sine cosine gains for the second excitation	The sine/cosine modulation output sine/cosine component amplitude for the second excitation. Default value are -1, 0, 0 and 1	N/A	Input
E_type	Excitation source type	The source from which the excitation of the resolver is generated. Can either be AC, which is generated inside the FPGA with the specified frequency, DC (only for single excitation), and External, which is generated from outside the model	N/A	Input
E_f	Excitation frequency	Frequency of the excitation when in AC mode. When in Double excitation, the second excitation signal has the same frequency and is 90º leading.	Hz	Input
E_src	Excitation source	Source of the external excitation source when in External mode	N/A	Input
E_src2	Second excitation source	Source of the second external excitation source when in External mode and Double excitation	N/A	Input
E_ts	Excitation time shift	This parameter is used to compensate the time offset between the carrier generation's input in the system and modulated signals' output	s	Input

Encoder Parameters & Measurements

Symbol	Name	Description	Unit	Type

Symbol	Name	Description	Unit	Type
Enc_en	Enable encoder	Whether or not to enable the encoder	N/A	Input
Enc_type	Encoder type	Encoder type, either Quadrature or Hall Effect	N/A	Input
Q_ABZ	A B Z encoder signals	A B and Z signals of the encoder	N/A	Measurement
Q_ppr	Number of pulses per revolution	Number of pulses in one full revolution of the encoder	N/A	Input
Q_dir	Direction of the sensor rotation	Direction in which the sensor is turning, either A leads B or B leads A	N/A	Input
Q_θ	Angle offset Δθ ( Sensor - Rotor )	Angle offset between the encoder and the rotor position from 0 to 360 degrees	°	Input
Q_rat	Encoder speed ratio ( sensor to mechanical position )	Mechanical to encoder ratio. Angle of Encoder = Qrat * machine mechanical angle.	N/A	Input
H_θ	Hall effect sensor position	Position of sensor phases A, B and C in Hall effect mode	°	Input
H_rat	Hall effect sensor speed ratio ( sensor to mechanical position )	Mechanical to encoder ratio. Angle of Encoder = Hrat * machine mechanical angle in Hall effect mode	N/A	Input

Visualization of Resolver Encoder Parameters Effects

Number of resolver pole pairs affects the number of electrical turns per mechanical turns. On the left figure, the number of resolver pole pairs is 2, on the right figure, the number of resolver pole pairs is 4.

Resolver sine cosine gains affect the sine (first axe) and cosine (second axe) modulation output. Default values set to 1, 0, 0, 1 make it so the sine modulation has a sine form and the cosine modulation has a cosine form. If set to 0, 1, 1, 0, the sine modulation would have a cosine and the cosine modulation would have a sine form.

Excitation frequency, in AC excitation source type, affects the frequency of the carrier signal. We can see the time step highlighted in red. ( Figure 3 is an enlarged view of Figure 2 )

Number of pulses per revolution (Q_ppr) defines how many times signals A and B pulse between two Z pulses ( one full rotation ).

Direction of the sensor rotation describes if A leads B ( Clockwise ) or if B leads A ( Counterclockwise )

Electrical Ports

This block has three electrical ports, the three terminals of the stator.