Brushless DC Machine

This block implements a brushless DC machine.

The BLDC block implements a three-phase Brushless DC motor (BLDC) model with resolvers and encoders.

Equations & Characteristics

The BLDC machine shares the same equations as the permanent magnet synchronous machine.

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. In the case of the BLDC, the back-EMFs are considered has trapezoidal. 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.

are the partial derivative of instantaneous permanent magnet flux.

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.

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Trapezoidal back-EMF Characteristics

The main difference between the PMSM and the BLDC lies in the shape of the Back EMF voltage. The BLDC has a trapezoidal back EMF shape that is parametrized with λ_m the permanent flux linkage and H the back EMF flat area in degree.

The electromotive force is constructed from a cosine table as described in the following equations:

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 V_abc cut-off filter set at 200 Hz. It is used to remove high harmonics of the DQ currents from accessing the table, which could make the model stiffer.

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
R	Stator resistances	Resistances of the stator windings specified for every phase, A, B and C.	Ω	Input
L_s	Stator inductance	Phase to neutral winding inductance	H	Input
λ_m	Permanent magnet flux linkage	Amplitude of the rotor permanent magnet flux	Wb	Input
δΦ/δθ	Back EMF profile	Profile of the back EMF, either Trapezoidal 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
H	Back EMF flat area	Field only visible in Trapezoidal mode. Flat section width in degree of the back EMF voltage	°	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
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
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
R_k	Resolver sine cosine gains	The sine/cosine modulation output sine/cosine component amplitude. 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, which is generated with a 90° from the rotor and External, which is generated from outside the model	N/A	Input
E_f	Excitation frequency	Frequency of the excitation when in AC mode	Hz	Input
E_src	Excitation source	Source of the external excitation source when in External mode	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

Resolver Model

A resolver is a rotary transformer where the magnitude of the signal through the resolver windings varies sinusoidally as the shaft rotates.

Single excitation resover

The equations of the resolver can be expressed as follows:

where is the resolver angle, is the mechanical angle of the machine, is the angle offset, is the number of pole pairs of the resolver, , , , and are the resolver sine cosine gains and is the direction of the sensor rotation (0 = clockwise, 1 = counterclockwise).

Note: does not have an effect to the resolver angle, but change the resolver sine sign to match the counterclockwise convention.

When the gains are set to its default values (, , , and ), the sensor rotation is clockwise, and the excitation is a sinusoidal wave (, where is the excitation frequency), the outputs are given by the following expressions:

which means the excitaition signals are modulated by the sinus and the cosinus of the rotor position angle.

Excitation source

The excitation signal can be selected among one of the following options:

DC: a continuos signal is used as excitation, so the outputs are and ;
AC: a constant-frequency sinusoidal, unitary amplitude waveform is used as excitation , so the outputs are those expressed in the previous section.
External: a external signal, coming from one of the analog inputs, is used as excitation. This incoming signal may be rescaled by the Analog Input Differential Rescaling module (AIR).

Resolver gains

The resolver gains can be used to simulate non-ideal conditions of the sensor, occuring due to manufacturing imperfections.

Amplitude imbalance

Output phases have unequal inductances. Assuming the coupling between the rotor excitation winding and the cosine winding as the reference:

where .

Imperfect quadrature

Output phases are not in perfect spatial quadrature. Assuming the angle when the rotor excitation winding is aligned with the cosine winding:

where is the angle between the cosine winding and the sine winding (for there is a perfect quadrature, so one retrives the default gain values).

Faults

One of the windings is not connected due to a fault. In this case, the corresponding gain is equal to zero.

Excitation time shift

When the excitation signal comes from an external source (one of the analog input channels) the signal can be shifted by a given amount of time.

OPAL-RT's resolver models are based off of the following sets of equations:

(1)

(2)

Where Sin.Sin, Sin.Cos, Cos.Sin, and Cos.Cos represent gains that are applied to simulate a non-ideal resolver. To simulate an ideal resolver, set the Sin.Sin and Cos.Cos gains to 1, set the Sin.Cos and Cos.Sin gains to 0, set the pp to 1, and set the θ_Offset to 0. This results in the following equations:

(3)

(4)

Encoder Parameters & Measurements

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

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 (A B C from top to bottom).