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The PMSM BLDC model implements three motor types which provide parametrization for different machine types (Permanent Magnet Synchronous Machine and Brushless DC Motor) and allow for different levels of model fidelity (Constant or Variable parametrization) : PMSM Constant Ld/Lq, PMSM Variable Ld/Lq, and BLDC Constant Ls.  The PMSM Constant Ld/Lq and BLDC Constant Ls motor modes simulate a machine with constant inductance and magnetic flux parameters. The PMSM Variable Ld/Lq motor type simulates a PMSM whose inductance and magnetic flux parameters are variable based on the operating state of the simulation (in this case, based on Id and Iq), which allows for greater model fidelity.

### Configuration Page

In the System Explorer window configuration tree, expand the Power Electronics Add-On custom device and select Circuit Model >> PMSM BLDC to display this page.  Use this page to configure the PMSM BLDC machine model.

 Machine Model Settings Name Specifies the name of the machine model. Description Specifies a description for the machine model. Motor Configuration Motor Type Choose from one of the following types.  The motor configuration parameters automatically populate depending on the selected Motor Type. Input Mapping Configuration Use the Input Mapping Configuration to route signals to the Voltage Phase A, Voltage Phase B, and Voltage Phase C inputs of the machine model.  Available routing options may vary depending on the selected Hardware Configuration. Group Specifies the group that will be routed to the input voltages of the machine. The available routing options are defined by the selected Hardware Configuration, however it is typical to see the following options by default:Measurements - eHS circuit model measurements Element Specifies the index of the measurement in the group that has been selected as the input voltage of the machine.

### Section Channels

This section includes the following custom device channels:

Channel Name

Symbol

Type

Units

Default Value

Description

Current Phase A

IaOutputAmpere0 APhase A current measured at the stator

Current Phase B

IbOutputAmpere0 APhase B current measured at the stator

Current Phase C

IcOutputAmpere0 APhase C current measured at the stator
Average Voltage AVa,avgOutputVolts0 V

Averaged Phase A voltage measured at the stator. The voltage is processed through a low-pass filter with a cutoff frequency of 159 Hz

LaTeX Math Block
anchor VCutoffFrequency left
f_{c} = \frac{1}{2\pi \times 1e-3} = 159Hz

Average Voltage BVb,avgOutputVolts0 VAveraged Phase B voltage measured at the stator. The voltage is processed through a low-pass filter with a cutoff frequency of 159Hz.
Average Voltage CVc,avgOutputVolts0 VAveraged Phase C voltage measured at the stator. The voltage is processed through a low-pass filter with a cutoff frequency of 159Hz
Three-Phase Active PowerPOutputWatts0 W

Three-phase instantaneous active electrical power in Watts

Three Phase Reactive PowerQOutputVolt-ampere reactive0 var

Three-phase instantaneous reactive electrical power in vars

Direct Stator CurrentIdOutputAmpere0 A

Direct-axis stator current in the reference frame aligned with the rotor

For a description of the DQ-transform used to compute this value, see D-Q Transform

Quadrature-axis stator current in the reference frame aligned with the rotor

For a description of the DQ-transform used to compute this value, see

Back-EMF Phase AVbemf,aOutputVolts0 VPhase A to neutral voltage induced by the electromotive force
Back-EMF Phase BVbemf,bOutputVolts0 VPhase B to neutral voltage induced by the electromotive force
Back-EMF Phase CVbemf,cOutputVolts0 VPhase C to neutral voltage induced by the electromotive force

ψM

OutputWeber0 Wb

Latest-value measurement of the Permanent Magnet Flux Linkage used at the input of the electrical model

In Constant mode, this will return the constant value input by the user in the Motor Configuration settings

In Variable mode, this will be the value that is looked-up in the 2D Flux Linkage table.

Direct InductanceLdOutputHenry0 H

Direct-axis inductance. This value is fed back from the input of the electrical model and describes only the latest value.

In Constant mode, this will return the constant value input by the user in the Motor Configuration settings

In Variable mode, this will be the value that is looked-up in the 2D Ld table.

Quadrature-axis inductance. This value is fed back from the input of the electrical model and describes only the latest value.

In Constant mode, this will return the constant value input by the user in the Motor Configuration settings

In Variable mode, this will be the value that is looked-up in the 2D Lq table.

Direct Stator VoltageVdOutputVolts0 V

Direct-axis stator voltage in the reference frame aligned with the rotor

For a description of the DQ-transform used to compute this value, see Standard DQ Motor Characteristics

Direct-axis stator voltage in the reference frame aligned with the rotor

For a description of the DQ-transform used to compute this value, see Standard DQ Motor Characteristics

### Model Description

Permanent Magnet Synchronous Machines are common electrical machines in the the automotive and transportation industry. The PMSM is usually chosen because of its excellent power density (produced power over size or weight) or its capability to reach higher speed than others motor types. However, controlling a PMSM is usually more challenging when compared to other machine types. Since it is a synchronous machine, the controller must be aware of the rotor position at all times in order to properly control the torque. In addition, there is a chance of de-fluxing the magnet if the control is not stable, which would lead to a modification of the machine properties.

The following figures illustrate the equivalent circuits of the PMSM motor model in the abc-frame and in the D-Q frame.

Figure 1.  Electrical Model for PMSM

Figure 2.  Electrical Model for PMSM in the D-Q frame

Excerpt

#### General Equation

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

LaTeX Math Block
anchor ElectricalModel center
I_{abc} = [L_{abc}(θ_e)]^ {-1} \{ \int (V_{abc} - R_{abc} I_{abc})dt - \psi_{abc} \}

where Labc is the time-varying inductance matrix (global inductance for Constant Ld/Lq and Variable Ld/Lq models), Iabc is the stator current inside the winding, Rabc are the stator resistances and Vabc is the voltage across the stator windings. ψabc defines the magnet flux linked into the stator windings.

#### AnchorDQtransformDQtransformD-Q Transform

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 to be trapezoidal. One can transform the equation using the Park transformation with a referential locked on the rotor position θ using

LaTeX Math Block Reference
anchor TransformEquations
and
LaTeX Math Block Reference
anchor Transform
.

LaTeX Math Block
anchor TransformEquations center
\left[\begin{array}{l}
{V_{d s}} \\
{V_{q s}}
\end{array}\right]=\mathrm{T}\left[\begin{array}{l}
{V_{a s}} \\
{V_{b s}} \\
{V_{c s}}
\end{array}\right]

LaTeX Math Block
anchor Transform center
\mathrm{T}=\sqrt{2 / 3}\left[\begin{array}{cc}
{\cos (\theta)} & {\sin (\theta)} \\
{-\sin (\theta)} & {\cos (\theta)}
\end{array}\right]\left[\begin{array}{ccc}
{1} & {-0.5} & {-0.5} \\
{0} & {\sqrt{3} / 2} & {-\sqrt{3} / 2}
\end{array}\right]

The D-Q Transform (also called Park-Clarke 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 D-Q transforms and this often leads to confusion when interpreting the motor states inside the D-Q frame. The one used here (which is typically standard in Japan) presents the advantage of being orthonormal (notice the sqrt(3/2) factor). This particular D-Q 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, namely:

LaTeX Math Block
anchor CFKB2 center
V_{abc}I_{abc} = V_{dq}I_{dq}

#### Torque Equation

With this transform (and only this transform) the machine torque can be expressed by

LaTeX Math Block Reference
anchor Torque
, where pp is the number of pole pairs.

LaTeX Math Block
anchor Torque center
T_{e}=p p\left[\psi_{M} \; \sqrt{\frac{3}{2}} \; i_{q}+\left(L_{d}-L_{q}\right) i_{d} i_{q}\right]

One may notice the absence of the 3/2 factor in

LaTeX Math Block Reference
anchor Torque
, which is usually present in the PMSM torque equation when using non-orthonormal transforms. This is, again, because this model uses the orthonormal D-Q transform. Figure 3 explains the principle of the Park transform. Considering fixed ABC referential with all quantities ( Vbemf, 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 Vbemf  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 Iamp. In the figure below, θ is the rotor angle, aligned with the D-axis.

Figure 3. Park Transform with angle definitions for θ and β

#### AnchorPowerPowerPower Equations

The  instantaneous active and reactive power, P and Q are calculated as follows:

LaTeX Math Block
anchor PandQ center
P = V_a I_a + V_b I_b + V_c I_c = V_d I_d + V_q I_q\\
Q = \frac {1} {\sqrt 3} [(V_b - V_c) I_a + (V_c - V_a) I_b + (V_a - V_b) I_c ] = V_q I_d - V_d I_q

where Va, Vb, and Vc are the instantaneous stator voltages

The active and reactive power are processed through low-pass filters dependent on the timestep of the machine and are calculated as follows. When Ts is set to the minimum of 120ns, the cutoff frequencies are 133Hz:

LaTeX Math Block
anchor VCutoffFrequency center
f_{c} = \frac{1}{\SI{1e4} \times Ts \times 2\pi} = \frac{1}{\SI{1e4} \times \SI{120e-9} \times 2\pi} = 133Hz

### AnchorPMSM Constant Ld/LqPMSM Constant Ld/LqMotor Type: PMSM Constant Ld/Lq

When set to the PMSM Constant Ld/Lq motor type, the machine model uses constant values for Direct Inductance, Quadrature Inductance, Magnetic Flux, Phase [A, B, C] Resistance, and Pole Pairs.

#### Configuration Parameters

The following parameters are available:

 Symbol Units Default Value Description Motor Configuration Direct Inductance Ld Henry 0.002984 H Direct-axis inductance of the machine Quadrature Inductance Lq Henry 0.004576 H Quadrature-axis inductance of the machine Permanent Magnetic Flux ψM Weber 0.25366 Wb Peak permanent magnet flux linkage Phase A Resistance Ra Ohm 0.12 Ω Phase A Resistance of the machine Phase B Resistance Rb Ohm 0.12 Ω Phase B Resistance of the machine Phase C Resistance Rc Ohm 0.12 Ω Phase C Resistance of the machine Pole Pairs pp 3 Number of pole pairs Direct Quadrature Transform Angle Offset Aligned Aligned - Indicates that the D axis is aligned with Phase A when the rotor angle θ=090 Degrees behind Phase A - Indicates that the Q axis is aligned with Phase A when the rotor angle θ=0 Solver Timestep Ts Second 1.2E-7 s The timestep at which the machine model executesEvery Ts, new outputs are computed by the FPGA machine model. By default, this is set to the minimum achievable timestep.

#### Torque Equation

The machine torque for the PMSM Constant Ld/Lq motor type can be expressed by

LaTeX Math Block Reference
anchor I73ID
.

LaTeX Math Block
anchor I73ID center
T_{e}=p p\left[\psi_{M} \; \sqrt{\frac{3}{2}} \; i_{q}+\left(L_{d}-L_{q}\right) i_{d} i_{q}\right]

### AnchorPMSM Variable Ld/LqPMSM Variable Ld/LqMotor Type: PMSM Variable Ld/Lq

When set to the PMSM Variable Ld/Lq motor type, the inductance and magnetic flux parameters are variable based on the operating state of the simulation, as defined in the JSON configuration file.

#### Configuration Parameters

The following parameters are available:

 Symbol Units Default Value Description Motor Configuration Model File Specifies the path to the JSON Motor Model file on disk. Refer to Motor Model File [JSON] for details regarding the file format. Solver Timestep Ts Second 1.2E-7 s The timestep at which the machine model executesEvery Ts, new outputs are computed by the FPGA machine model. By default, this is set to the minimum achievable timestep.

#### Torque Equation

The machine torque for the PMSM Variable Ld/Lq motor type can be expressed by

LaTeX Math Block Reference
anchor 5OZ0X
.

LaTeX Math Block
anchor 5OZ0X center
T_{e}=p p\left[\psi_{M} \; \sqrt{\frac{3}{2}} \; i_{q}+\left(L_{d}-L_{q}\right) i_{d} i_{q}\right]

### AnchorBLDC Constant LsBLDC Constant LsMotor Type: BLDC Constant Ls

When set to the BLDC Constant LS motor type, the machine model uses constant values for Stator Inductance, Magnetic Flux, Phase [A, B, C] Resistance, and Pole Pairs. The main difference between the PMSM and the BLDC motor types lies in the shape of the back EMF voltage, which is trapezoidal in the case of the BLDC.

#### Configuration Parameters

The following parameters are available:

 Symbol Units Default Value Description Motor Configuration Stator Inductance Ls Henry 0.002984 H Stator inductance of the machine Back EMF Flat Area H Degrees 0 Describes the length of the flat area in degrees of the trapezoidal back-EMF wavePlease see Trapezoidal Back-EMF Characteristics for a description of the wave. Permanent Magnetic Flux ψM Weber 0.25366 Wb Peak permanent magnet flux linkage Phase A Resistance Ra Ohm 0.12 Ω Phase A Resistance of the machine Phase B Resistance Rb Ohm 0.12 Ω Phase B Resistance of the machine Phase C Resistance Rc Ohm 0.12 Ω Phase C Resistance of the machine Pole Pairs pp 3 Number of pole pairs Direct Quadrature Transform Angle Offset Aligned Aligned - Indicates that the D axis is aligned with Phase A when the rotor angle θ=090 Degrees behind Phase A - Indicates that the Q axis is aligned with Phase A when the rotor angle θ=0 Solver Timestep Ts Second 1.2E-7 s The timestep at which the machine model executesEvery Ts, new outputs are computed by the FPGA machine model. By default, this is set to the minimum achievable timestep.

#### Torque Equation

LaTeX Math Block
anchor J2Q5O center
T_{e}=p p\left[ I_{a b c}\cdot\frac{\partial \psi_{abc}}{\partial \theta_r}\right]

#### AnchorBLDCgraphBLDCgraphTrapezoidal Back-EMF Characteristics

The BLDC has a trapezoidal back EMF shape that is parametrized with λm the permanent flux linkage and H the back EMF flat area in degrees.

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

LaTeX Math Block
anchor EMForce
\Large\frac{\partial \psi_{a}}{\partial \theta_r \psi_M}\normalsize= max(min(\Large\frac{cos(\theta_r)}{cos(\frac{H}{2})}\normalsize,1),-1)