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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 I_d and I_q), 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.

This page includes the following components, configurable at edit-time only:

Section Channels

This section includes the following custom device channels:

f_{c} = \frac{1}{2\pi \times 1e-3} = 159Hz 

Model Description

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 DQ frame.

I_{abc} = [L_{abc}(θ_e)]^ {-1} \{ \int (V_{abc} - R_{abc} I_{abc})dt - \psi_{abc} \}

The electrical angle is expressed as follows:

\theta_e= pp * \theta_m + \theta_{offset}

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DQtransform DQtransform

DQ Transform

In normal conditions, the ideal 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_r using 

\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]

\mathrm{T}=\sqrt{2 / 3}\left[\begin{array}{cc}
{\cos (\theta_r)} & {\sin (\theta_r)} \\
{-\sin (\theta_r)} & {\cos (\theta_r)}
\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 DQ Transform (also called Park-Clarke transform) reduces sinusoidal varying quantities of inductances, flux, current, and voltage to constant values in the D-Q DQ 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 DQ transforms and this often leads to confusion when interpreting the motor states inside the D-Q DQ 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 DQ orthonormal transform is power-invariant which means that the power computed in the D-Q DQ frame by performing a dot product of currents and voltages will be numerically equal to the one computed in the phase domain, namely:

V_{abc}I_{abc} = V_{dq}I_{dq}

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Torque Torque

Torque Equation

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

T_{e}=p p\left[ \sqrt{I_{abc}\cdot\frac{3}{2\partial\psi_{abc}} {\; partial\psitheta_{Mr} \; i_{q}+\left(L_{d}-L_{q}\right) i_{d} i_{q}\right]

In the

case of the PMSM Constant Ld/Lq and PMSM Variable Ld/Lq motor types, the back EMF shape is sinusoidal and the torque can be further simplified into

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.

Image Removed

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

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

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 V_a, V_b, and V_c 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:

f_{c} = \frac{1}{\SI{1e4} \times Ts \times 2\pi} = \frac{1}{\SI{1e4} \times \SI{120e-9} \times 2\pi} = 133Hz

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:

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

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

f_{c} = \frac{1}{\SI{1e4} \times Ts \times 2\pi} = \frac{1}{\SI{1e4} \times \SI{120e-9} \times 2\pi} = 133Hz

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PMSM Constant Ld/Lq PMSM Constant Ld/Lq

Motor 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.  They are modifiable at edit-time only:

Torque Equation

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

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

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 Motor Model File.

Configuration Parameters

The following parameters are available:

Allows certain parameters to be exposed as tunable VeriStand Channels. See the Advanced Channels section below for more details.

This checkbox is only available when a Motor Model File has been specified. Otherwise, the option is greyed out.

Initial Angle of the machine

This may be useful when simulating two separate 3-phase machines that require a phase shift between them.

The timestep at which the machine model executes

Every T_s, new outputs are computed by the FPGA machine model. By default, this is set to the minimum achievable timestep.

The following VeriStand channels are displayed under the Advanced section when the Enable Advanced Channels option is enabled on the PMSM Variable Ld/Lq configuration page.

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PMSM Variable Ld/Lq PMSM Variable Ld/Lq

Motor 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 Motor Model File.

Configuration Parameters

The following parameters are available.  They are modifiable at edit-time only:

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AdvancedChannels AdvancedChannels

Advanced Channels

The following VeriStand channels are displayed under the Advanced section when the Enable Advanced Channels option is enabled on the PMSM Variable Ld/Lq configuration page. Channel values can be modified dynamically at execution time.

Torque Equation

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

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BLDC Constant Ls BLDC Constant Ls

Motor Type: BLDC Constant Ls

When set to the BLDC Constant L_S 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.  They are modifiable at edit-time only:

Torque Equation

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BLDCgraph BLDCgraph

Trapezoidal Back-EMF Characteristics

The BLDC has a trapezoidal back EMF shape that is parametrized with calculated using λ_m the _, the permanent magnet flux linkage and H the back EMF flat area , and H,the length of the flat portion of the trapezoid in degrees.

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

\Large\frac{\partial \psi_{a}}{\partial \theta_r \psilambda_M}\normalsize= max(min(\Large\frac{cos(\theta_r)}{cos(\frac{H}{2})}\normalsize,1),-1)