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:
| Mathblock | ||
|---|---|---|
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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 Labc is the time-varying inductance matrix (global inductance for DQ and VDQ models), Iabc is the stator current inside the winding, R is the stator resistance and Vabc is the voltage across the stator windings. As for ψabc, 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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| alignment | left |
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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:
| Mathblock | ||
|---|---|---|
| ||
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 Labc is the time-varying inductance matrix (global inductance for DQ and VDQ models), Iabc is the stator current inside the winding, R is the stator resistance and Vabc is the voltage across the stator windings. As for ψabc, 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).
| Mathblock | ||
|---|---|---|
| ||
\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] |
| Mathblock | ||
|---|---|---|
| ||
\mathrm{T}=\sqrt{2 / 3}\left[\begin{array}{cc} {\cos (\theta)} & {\sin (\theta)} \\ {-\sin (\theta)} & {\cos (\theta)} \end{array}\right]\left[\begin{array}{lccc} {V_{d s}1} \\& {V_{q s}} \end{array}\right]=\mathrm{T}\left[\begin{array}{l} {V_{a s}} \\ {V_{b s}} \\ {V_{c s}} \end{array}\right] |
| Mathblock | ||
|---|---|---|
| ||
\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.
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-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.
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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 ( 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.
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Resolver Characteristics
The equations of the resolver sensors can be expressed as follows:
| Mathblock | ||
|---|---|---|
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\theta_{\text {res}}=\left(\theta_{\text {mee}}+\theta_{\text {offset}}\right) \cdot R_{p p} |
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|---|---|---|
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\text { cosine signal }=\text { Excitation } *\left(\cos \left(\theta_{\text {res }}\right) * R_{k_{c a}}+\sin \left(\theta_{\text {res}}\right) * R_{k_{\text {st }}}\right) |
| Mathblock | ||
|---|---|---|
| ||
\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 θres is the resolver angle, θmec is the mechanical angle of the machine, θoffset is the angle offset, Rpp is the Number of pole pairs of the resolver and Rk are the resolver sine cosine gains.
FPGA Implementation
Since the simulation is performed on FPGA hardware, the block implements a low pass filter on Vabc 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 Vabc without the filter. The FPGA block also implements a Vabc 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
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Symbol
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Name
...
Description
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Unit
...
Type
...
Mconf
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Motor configuration type
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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.
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Input
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Mfile
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Motor configuration
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Field only visible in Variable DQ and Spatial Harmonics modes. Allows to import a Variable DQ or Spatial Harmonics motor configuration resource file.
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Input
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Rs
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Stator resistances
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Resistances of the stator windings specified for every phase, A, B and C.
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Ω
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Input
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Lls
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Stator inductances
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d and q axis inductances
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H
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Input
...
Back EMF profile
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Input
...
EMFfile
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Back EMF profile table
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Field only visible in User defined mode. Allows to import a back EMF resource file.
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Input
...
λm
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Permanent magnet flux linkage
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Amplitude of the rotor permanent magnet flux
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Wb
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Input
...
pp
...
Number of pole pairs
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Number of pole pairs
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Input
...
is
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Stator currents
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Currents measured at phases A, B and C of the stator
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A
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Measurement
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isdq
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Stator currents (DQ)
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Currents measured of axis d and q
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A
...
Measurement
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Vs
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Stator voltages
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Voltages measured at phases A, B and C of the stator
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V
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Measurement
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Bemf
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Back EMF voltages
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Phase to neutral voltage generated from the permanent magnet flux linkage
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V
...
Measurement
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P
...
Active power (3ph, instantaneous)
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Instantaneous electrical active power
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W
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Measurement
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Q
...
Reactive power (3ph, instantaneous)
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Instantaneous electrical reactive power
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var
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Measurement
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θe
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Electrical rotor position
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Position of the rotor from 0 to 360 degrees
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°
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Measurement
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Rs
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Snubber resistance
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Resistances of the snubber on phase A, B and C
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Ω
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Input
...
Cs
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Snubber capacitance
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Capacitance of the snubber on phase A, B and C
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F
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Input
...
Mechanical Parameters & Measurements
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Symbol
...
Name
...
Description
...
Unit
...
Type
...
J
...
Rotor inertia
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Moment of inertia of the rotor
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kg*m2
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Input
...
Fv
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Viscous friction coefficient
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Viscous friction
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N*m*s/rad
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Input
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Fs
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Static friction torque
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Static friction
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N*m
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Input
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ctrl
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Mechanical control mode
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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.
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Input
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T
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Torque command
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Torque command sent to the mechanical model
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N*m
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Input
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ωrc
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Rotor speed command
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Speed command sent to the mechanical model
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rpm
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Input
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ωr
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Rotor speed
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Speed of the rotor
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rpm
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Measurement
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Te
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Electromagnetic torque
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Torque measured at the rotor
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N*m
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Measurement
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θ0
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Initial rotor angle
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Rotor position at time t = 0
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°
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Input
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θ
...
Rotor angle
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Rotor position from 0 to 360 degrees
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°
...
Measurement
.
Resolver Parameters & Measurements
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Symbol
...
Name
...
Description
...
Unit
...
Type
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Ren
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Enable resolver
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Whether or not to enable the resolver
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N/A
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Input
...
Rsc
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Resolver feedback signals
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The two two-phase windings producing a sine and cosine feedback current proportional to the sine and cosine of the angle of the motor
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N/A
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Measurement
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Rpp
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Number of resolver pole pairs
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Number of pole pairs of the resolver
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N/A
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Input
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Rdir
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Direction of the sensor rotation
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Direction in which the sensor is turning, either clockwise or counterclockwise
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N/A
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Input
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Rθ
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Angle offset Δθ ( Sensor- Rotor )
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FPGA Implementation
Since the simulation is performed on FPGA hardware, the block implements a low pass filter on Vabc 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 Vabc 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 |
|---|---|---|---|---|
Mconf | 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 | |
Mfile | 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 | |
Rs | Stator resistances | Resistances of the stator windings specified for every phase, A, B and C. | Ω | Input |
Lls | 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 | |
EMFfile | Back EMF profile table | Field only visible in User defined mode. Allows to import a back EMF resource file. | Input | |
λm | Permanent magnet flux linkage | Amplitude of the rotor permanent magnet flux | Wb | Input |
pp | Number of pole pairs | Number of pole pairs | Input | |
is | Stator currents | Currents measured at phases A, B and C of the stator | A | Measurement |
isdq | Stator currents (DQ) | Currents measured of axis d and q | A | Measurement |
Vs | Stator voltages | Voltages measured at phases A, B and C of the stator | V | Measurement |
Bemf | 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 |
θe | Electrical rotor position | Position of the rotor from 0 to 360 degrees | ° | Measurement |
Rs | Snubber resistance | Resistances of the snubber on phase A, B and C | Ω | Input |
Cs | Snubber capacitance | Capacitance of the snubber on phase A, B and C | F | Input |
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Mechanical Parameters & Measurements
Symbol | Name | Description | Unit | Type |
|---|---|---|---|---|
J | Rotor inertia | Moment of inertia of the rotor | kg*m2 | Input |
Fv | Viscous friction coefficient | Viscous friction | N*m*s/rad | Input |
Fs | 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 |
ωrc | Rotor speed command | Speed command sent to the mechanical model | rpm | Input |
ωr | Rotor speed | Speed of the rotor | rpm | Measurement |
Te | 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 | ° |
Input
Rk
Resolver sine cosine gains
The sine/cosine modulation output sine/cosine component amplitude. Default value are 1, 0, 0 and 1
N/A
Input
Etype
Excitation source type
Measurement |
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Resolver Parameters & Measurements
Symbol | Name | Description | Unit | Type | ||
|---|---|---|---|---|---|---|
Ren | Enable resolver | Whether or not to enable the resolver | N/A | Input | ||
Ef | Excitation frequency | Frequency of the excitation when in AC mode | Hz | Input | ||
Esrc | Excitation source | Source of the external excitation source when in External modeRsc | 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 |
Rpp | Number of resolver pole pairs | Number of pole pairs of the resolver | N/A | Input | ||
Ets | 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 |
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OPAL-RT's resolver models are based off of the following sets of equations:
| Mathblock | ||||
|---|---|---|---|---|
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Sine \; Output = [Sin.Sin*sin(pp(\theta_m - \theta_{Offset})) + Sin.Cos * cos(pp * \theta_m - \theta_{Offset}))] * Excitation |
| Mathblock | ||||
|---|---|---|---|---|
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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 θOffset to 0. This results in the following equations:
| Mathblock | ||||
|---|---|---|---|---|
| ||||
Sine \; Output = sin(\theta_m) * Excitation |
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| anchor | simpleCosineOutput |
|---|---|
| alignment | center |
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Rdir | Direction of the sensor rotation | Direction in which the sensor is turning, either clockwise or counterclockwise | N/A | Input |
|---|---|---|---|---|
Rθ | Angle offset Δθ ( Sensor- Rotor ) | Angle offset between the resolver and the rotor position from 0 to 360 degrees | ° | Input |
Emod | Excitation mode | Number of resolver excitation windings. Can either be single (just one excitation signal) or double (two excitation signals) | N/A | Input |
Rk1 | 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 |
Rk2 | 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 |
Etype | 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 |
Ef | 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 |
Esrc | Excitation source | Source of the external excitation source when in External mode | N/A | Input |
Esrc2 | Second excitation source | Source of the second external excitation source when in External mode and Double excitation | N/A | Input |
Ets | 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 |
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| Include Page | ||||
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Encoder Parameters & Measurements
Symbol | Name | Description | Unit | Type |
|---|---|---|---|---|
Encen | Enable encoder | Whether or not to enable the encoder | N/A | Input |
Enctype | Encoder type | Encoder type, either Quadrature or Hall Effect | N/A | Input |
QABZ | A B Z encoder signals | A B and Z signals of the encoder | N/A | Measurement |
Qppr | Number of pulses per revolution | Number of pulses in one full revolution of the encoder | N/A | Input |
Qdir | Direction of the sensor rotation | Direction in which the sensor is turning, either A leads B or B leads A | N/A | Input |
Qθ | Angle offset Δθ ( Sensor - Rotor ) | Angle offset between the encoder and the rotor position from 0 to 360 degrees | ° | Input |
Qrat | Encoder speed ratio ( sensor to mechanical position ) | Mechanical to encoder ratio. Angle of Encoder = Qrat * machine mechanical angle. | N/A | Input |
Hθ | Hall effect sensor position | Position of sensor phases A, B and C in Hall effect mode | ° | Input |
Hrat | 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 |
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Visualization of Resolver Encoder Parameters Effects
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