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This block implements a permanent magnet synchronous machine.

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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:

I_{a b c}=\left[L_{a b c}(\theta)\right]^{-1}\left[\int\left(V_{a b c}-R I_{a b c}\right) d t-\psi_{a b c}\right]

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).

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This block implements a permanent magnet synchronous machine.

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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:

I_{a b c}=\left[L_{a b c}(\theta)\right]^{-1}\left[\int\left(V_{a b c}-R I_{a b c}\right) d t-\psi_{a b c}\right]

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).

\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)} & {\sin (\theta)} \\
{-\sin (\theta)} & {\cos (\theta)}
\end{array}\right]=\mathrm{T}\left[\begin{array}{lccc}
{V_{a s}1} \\& {V_{b s}-0.5} & {-0.5} \\
{V_{c s}}
\end{array}\right]

\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 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.

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

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.

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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.

P = Vd*Id + Vq*Iq

Q = Vq*Id - Vd*Iq

Vd = (P*Id - Q*Iq)/(Id^2 + Iq^2)

Vq = (P*Iq + Q*Id)/(Id^2 + Iq^2)

Mechanical Model Characteristics

The equation of the mechanical model in torque mode can be expressed as follows:

\omega_{r_{r}}=a \omega_{r z-1}+b\left(T_{e}-T_{L}\right)

a=\left(1-T_{S}\right) \frac{F_{v}}{J}

b=60 \frac{T_{r}}{2 \pi J}

where ω_r is the rotor speed, T_e is the electromagnetic torque, T_L is the torque command, F_vis the viscous friction coefficient, J is the inertia and T_s the sample time. There is a dead-zone implementation with the static friction torque if the electromagnetic doesn't exceed the static friction torque, the speed remains zero.

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0} & {\sqrt{3} / 2} & {-\sqrt{3} / 2}
\end{array}\right]

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.

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

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 vectorI is called I_amp. In the figure below, θ is the rotor angle, aligned with the D-axis.

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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.

P = Vd*Id + Vq*Iq

Q = Vq*Id - Vd*Iq

Vd = (P*Id - Q*Iq)/(Id^2 + Iq^2)

Vq = (P*Iq + Q*Id)/(Id^2 + Iq^2)

Resolver Characteristics

The equations of the resolver sensors can be expressed as follows:

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\text { sine signal }=\text { Excitation } *\left(\cos \left(\theta_{\text {res }}\right) * R_{k_{\text {ce }}}+\sin \left(\theta_{\text {res }}\right) * R_{k_{\text {ge }}}\right)

where θ_res is the resolver angle, θ_mec is the mechanical angle of the machine, θ_offset is the angle offset, R_pp is the Number of pole pairs of the resolver and R_k are the resolver sine cosine gains.

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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.

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Electrical Parameters & Measurements

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Mechanical Parameters & Measurements

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Resolver Parameters & Measurements

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

Sine \; Output = [Sin.Sin*sin(pp(\theta_m - \theta_{Offset})) + Sin.Cos * cos(pp * \theta_m - \theta_{Offset}))] * Excitation

Cosine \; Output = [Cos.Sin*sin(pp(\theta_m - \theta_{Offset})) + Cos.Cos * cos(pp * \theta_m - \theta_{Offset}))] * Excitation

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  to 0.  This results in the following equations:

Sine \; Output = sin(\theta_m) * Excitation

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Encoder Parameters & Measurements

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Visualization of Resolver Encoder Parameters Effects

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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.Image Removed

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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 )Image Removed

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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 )Image Removed

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Electrical Ports

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