PMSM (DQ0)

Direct-quadrature-zero representation of permanent magnet synchronous machine

Since R2024b

Libraries:
Simscape / Electrical / Electromechanical / Permanent Magnet

Description

The PMSM (DQ0) block models the direct-quadrature-zero (dq0) frame representation of a permanent magnet synchronous motor (PMSM) with a three-phase stator. Use this block to model an interior permanent magnet synchronous machine (IPMSM), surface permanent magnet synchronous machine (SPMSM), transverse flux motor, axial flux (pancake) motor, or PMSM servomotor.

PMSMs have a permanent magnet rotor and a wound stator. You use these motors in traction applications such as electric vehicles (EVs) and in actuation systems. In actuation systems, PMSMs are also called brushless servomotors. You typically design a PMSM to have a sinusoidal back electromotive force (EMF) profile. This design reduces undesirable high-frequency mechanical and electrical current harmonics compared to motors like the brushless DC (BLDC) motor that have a trapezoidal back EMF profile.

Use this block to model the two main types of PMSM:

SPSM or surface permanent magnet (SPM) motor — The "S" signifies that the rotor magnets are surface mounted. Ld and Lq are equal.
IPSM or interior permanent magnet (IPM) motor — The "I" signifies that the rotor magnets are interior to the rotor. Ld and Lq are not equal.

You can also categorize PMSMs based on the main flux paths. The main flux paths are commonly radial but can be axial, which causes higher torques for the same physical size of motor. You can use the PMSM (DQ0) block to model both categories of PMSM.

Note

Simscape™ Electrical™ includes several blocks that can model the same type of motor or actuator. You must choose a block that has sufficient modeling detail for the engineering design questions that you need to answer. However, do not use more detail than you need, because higher-fidelity models slow down simulation and are more complex to parameterize.

Blocks like the PMSM (DQ0) block model motors with fixed or parameter-dependent coefficients with a simple equivalent circuit. These models have an intermediate level of fidelity. Use this block to design controls or systems in actuation applications, such as robotics and mechatronics, and for efficiency predictions when saturation and harmonics only weakly impact losses. For more information about choosing the right block to model your motor at the right level of fidelity, see Choose Blocks to Model Motors or Actuators.

Equations

Applying Park’s transformation to the block electrical equations produces an expression for torque that is independent of the rotor angle.

This equation defines the Park’s transformation:

$P = 2 / 3 [\begin{matrix} \cos θ_{e} & \cos (θ_{e} - 2 π / 3) & \cos (θ_{e} + 2 π / 3) \\ - \sin θ_{e} & - \sin (θ_{e} - 2 π / 3) & - \sin (θ_{e} + 2 π / 3) \\ 0.5 & 0.5 & 0.5 \end{matrix}],$

where θ_e is the electrical angle and is equal to Nθ_r. N is the number of pole pairs.

The block uses the Park's transformation on the stator winding voltages and currents to transform them to the dq0 frame, which is independent of the rotor angle:

$[\begin{matrix} v_{d} \\ v_{q} \\ v_{0} \end{matrix}] = P [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}]$

$[\begin{matrix} i_{d} \\ i_{q} \\ i_{0} \end{matrix}] = P [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}]$

$v_{d} = R_{s} i_{d} + L_{d} \frac{d i_{d}}{d t} - N ω i_{q} L_{q}$

$v_{q} = R_{s} i_{q} + L_{q} \frac{d i_{q}}{d t} + N ω (i_{d} L_{d} + ψ_{m})$

$v_{0} = R_{s} i_{0} + L_{0} \frac{d i_{0}}{d t}$

$T = \frac{3}{2} N (i_{q} (i_{d} L_{d} + ψ_{m}) - i_{d} i_{q} L_{q})$

where:

L_d = L_s + M_s + 3/2 L_m. L_d is the stator d-axis inductance.
L_q = L_s + M_s − 3/2 L_m. L_q is the stator q-axis inductance.
L₀ = L_s – 2M_s. L₀ is the stator zero-sequence inductance.
ω is the rotor mechanical rotational speed.
N is the number of rotor permanent magnet pole pairs.
T is the rotor torque. Torque flows from the port C, which represents the motor case, to port R, which represents the motor rotor.

The PMSM (DQ0) block uses the original, non-orthogonal implementation of the Park transform. If you try to apply the alternative implementation, you get different results for the dq0 voltage and currents.

Alternative Flux Linkage Parameterization

You can parameterize the motor by using the back EMF or torque constants by using the Permanent magnet flux linkage parameter.

The back EMF constant is the peak voltage that the permanent magnet induces in the per-unit rotational speed of each of the phases. The relationship between the peak permanent magnet flux linkage and the back EMF is:

$k_{e} = N ψ_{m} .$

The back EMF, e_ph, for one phase is:

$e_{p h} = k_{e} ω .$

The torque constant is the peak torque that the per-unit current of each of the phases induces. The torque constant is numerically identical in value to the back EMF constant when both are expressed in SI units:

$k_{t} = N ψ_{m} .$

When L_d=L_q and the currents in all three phases are balanced, the combined torque T is:

$T = \frac{3}{2} k_{t} i_{q} = \frac{3}{2} k_{t} I_{p k},$

where I_pk is the peak current in any of the three windings.

The factor 3/2 follows from this being the steady-state sum of the torques from all phases. Therefore, the torque constant k_t is also equal to:

$k_{t} = \frac{2}{3} (\frac{T}{I_{p k}}),$

where T is the measured total torque when testing with a balanced three-phase current with peak line voltage I_pk. The RMS line current is:

$k_{t} = \frac{\sqrt{2}}{3} \frac{T}{i_{l i n e, r m s}} .$

Calculating Iron Losses

Iron losses consist of two terms, one representing the main magnetizing path and the other representing the cross-tooth tip path that becomes active during the field-weakening operation. The iron losses model is based on [3].

The term that represents the main magnetizing path depends on the induced RMS line-to-neutral stator voltage, $V_{m_{r m s}}^{}$ :

$P_{O C} (V_{m_{r m s}}^{}) = \frac{a_{h}}{k} V_{m_{r m s}}^{} + \frac{a_{j}}{k^{2}} V_{m_{r m s}}^{2} + \frac{a_{e x}}{k^{1.5}} V_{m_{r m s}}^{1.5} .$

This term is dominant during the no-load operation. k is the back EMF constant relating the RMS volts per Hz. This term is equal to $k = V_{m_{r m s}}^{} / f$ , where f is the electrical frequency. The first term on the right side is the magnetic hysteresis loss, the second is the eddy current loss, and the third is the excess loss. The block derives the three coefficients on the numerators from the values that you specify for the open-circuit hysteresis, eddy, and excess losses.

The term representing the cross-tooth tip path is relevant when a demagnetizing field is set up and can be determined from a finite element analysis short-circuit test. This term depends on the RMS EMF associated with the cross-tooth tip flux, $V_{d_{r m s}}^{*}$ :

$P_{S C} (V_{d_{r m s}}^{*}) = \frac{b_{h}}{k} V_{d_{r m s}}^{*} + \frac{b_{j}}{k^{2}} V_{d_{r m s}}^{* 2} + \frac{b_{e x}}{k^{1.5}} V_{d_{r m s}}^{* 1.5} .$

The block derives the three numerator terms from the values you specify for the short-circuit hysteresis, eddy, and excess losses.

Model Thermal Effects

You can expose the thermal ports to model the effects of losses that convert power to heat. To expose the thermal ports, set the Modeling option parameter to one of these values:

No thermal port — The block contains the electrical conserving ports associated with the stator dq0 quantities but does not contain thermal ports.
Show thermal port — The block contains the electrical conserving ports associated with the stator dq0 quantities and the thermal conserving ports for the stator and for the rotor.

For more information about using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

Variables

To set the priority and initial target values for the block variables before simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. You can specify nominal values using different sources, including the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Examples

New

Control and Simulate Torque of IPMSM in DQ Frame

Control the torque in an automotive electrical traction drive of an interior permanent magnet synchronous machine (IPMSM). The example controls and simulates the torque in the rotor direct-quadrature (DQ) reference frame. You can use the DQ reference frame to design the controller and to speed up the simulation. The IPMSM operates in both motoring and generating modes according to the load. An ideal angular velocity source provides the load. The Control subsystem uses an open-loop approach to control the IPMSM torque and a closed-loop approach to control the current. The control algorithm converts the torque request to the relevant current references. The reference DQ voltages feed the IPMSM. The simulation involves several torque steps in both motor and generator modes.

Since R2024b
Open Model

New

Simulate PMSM Drive with Thermal Model in DQ Frame

Simulate a permanent magnet synchronous machine (PMSM) in the direct-quadrature (DQ) reference frame. The PMSM contains a thermal model and empirical iron losses. To design the PMSM controller and achieve the desired performance, select the architecture and the gains for the model. The initial temperature of the stator and rotor is 25 degrees Celsius. The ambient temperature is 27 degrees Celsius. The Scopes subsystem contains scopes that allow you to see the simulation results.

Since R2024b
Open Model

Ports

Conserving

expand all

d — d-axis component
electrical

Electrical conserving port associated with the d-axis component.

q — q-axis component
electrical

Electrical conserving port associated with the q-axis component.

0 — Zero-sequence component
electrical

Electrical conserving port associated with the zero-sequence component.

Dependencies

To enable this port, set Zero sequence to Include.

R — Motor rotor
mechanical

Mechanical rotational conserving port associated with the motor rotor.

C — Motor case
mechanical

Mechanical rotational conserving port associated with the motor case.

HR — Rotor thermal port
thermal

Thermal conserving port associated with the rotor.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HS — Stator windings thermal port
thermal

Thermal conserving port associated with the stator windings.

Dependencies

To enable this port, set Modeling option to Show thermal port.

Parameters

expand all

Modeling option — Whether to enable thermal ports
`No thermal port` (default) | `Show thermal port`

Whether to enable the thermal ports of the block and model the effects of losses that convert power to heat.

Main

Modeling fidelity — Modeling fidelity
`Constant Ld, Lq and PM` (default) | `Tabulated Ld, Lq and PM`

Select the modeling fidelity of the block:

Constant Ld, Lq and PM — Ld, Lq, and PM values are constant. The respective parameters define their values.
Tabulated Ld, Lq and PM — The block computes Ld, Lq, and PM online from the DQ currents lookup tables by using these equations:

$L_{d} = f_{1} (i_{d}, i_{q})$

$L_{d} = f_{2} (i_{d}, i_{q})$

$λ_{P M} = f_{2} (i_{d}, i_{q})$

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0.

Number of pole pairs — Number of permanent magnet pole pairs on rotor
`6` (default) | positive scalar

Number of permanent magnet pole pairs on the rotor.

Permanent magnet flux linkage parameterization — Parameterization option for permanent magnet flux linkage
`Specify flux linkage` (default) | `Specify torque constant` | `Specify back EMF constant`

Parameterization option for the flux linkage of the permanent magnet.

Permanent magnet flux linkage — Permanent magnet flux linkage
`0.03` `Wb` (default)

Peak permanent magnet flux linkage with any of the stator windings.

Dependencies

To enable this parameter, set Permanent magnet flux linkage parameterization to Specify flux linkage and Modeling fidelity to Constant Ld, Lq and PM.

Torque constant — Torque constant
`0.18` `N*m/A` (default)

Torque constant with any of the stator windings.

Dependencies

To enable this parameter, set Permanent magnet flux linkage parameterization to Specify torque constant and Modeling fidelity to Constant Ld, Lq and PM.

Back EMF constant — Back EMF constant
`0.18` `V*s/rad` (default)

Back EMF constant with any of the stator windings.

Note

The back EMF constant is the peak voltage that the permanent magnet induces in the per-unit rotational speed of each of the phases.

Dependencies

To enable this parameter, set Permanent magnet flux linkage parameterization to Specify back EMF constant and Modeling fidelity to Constant Ld, Lq and PM.

Stator parameterization — Parameterization option for stator
`Specify Ld, Lq, and L0` (default) | `Specify Ls, Lm, and Ms`

Parameterization option for the stator.

Stator d-axis inductance, Ld — Stator d-axis inductance
`0.00022` `H` (default)

d-axis inductance of the stator.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Constant Ld, Lq and PM.

Stator q-axis inductance, Lq — Stator q-axis inductance
`0.00022` `H` (default)

q-axis inductance of the stator.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Constant Ld, Lq and PM.

Direct-axis current vector, iD — Direct-axis current vector
`[-200, 0, 200]` `A` (default) | vector of two or more values

Direct-axis current vector.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld, Lq and PM.

Quadrature-axis current vector, iQ — Quadrature-axis current vector
`[-200, 0, 200]` `A` (default) | vector of two or more values

Quadrature-axis current vector.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld, Lq and PM.

Ld matrix, Ld(id,iq) — Ld matrix
`0.0002 * ones(3, 3)` `H` (default) | matrix of scalars

Ld matrix. The number of rows of this matrix must be equal to the value of the Direct-axis current vector, iD parameter. The number of columns of this matrix must be equal to the value of the Quadrature-axis current vector, iQ parameter.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld, Lq and PM.

Lq matrix, Lq(id,iq) — Lq matrix
`0.0002 * ones(3, 3)` `H` (default) | matrix of scalars

Lq matrix. The number of rows of this matrix must be equal to the value of the Direct-axis current vector, iD parameter. The number of columns of this matrix must be equal to the value of the Quadrature-axis current vector, iQ parameter.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld, Lq and PM.

Permanent magnet flux linkage, PM(id,iq) — Flux linkage of permanent magnet
`0.1 * ones(3, 3)` `Wb` (default) | matrix of scalars

Flux linkage of the permanent magnet. The number of rows of this matrix must be equal to the value of the Direct-axis current vector, iD parameter. The number of columns of this matrix must be equal to the value of the Quadrature-axis current vector, iQ parameter.

Dependencies

To enable this parameter, set:

Stator parameterization to Specify Ld, Lq, and L0
Modeling fidelity to Tabulated Ld, Lq and PM
Permanent magnet flux linkage parameterization to Specify flux linkage

Torque constant matrix, kt(id,iq) — Torque constant matrix
`0.18 * ones(3, 3)` `N*m/A` (default) | matrix of scalars

Torque constant matrix. The number of rows of this matrix must be equal to the value of the Direct-axis current vector, iD parameter. The number of columns of this matrix must be equal to the value of the Quadrature-axis current vector, iQ parameter.

Dependencies

To enable this parameter, set:

Stator parameterization to Specify Ld, Lq, and L0
Modeling fidelity to Tabulated Ld, Lq and PM
Permanent magnet flux linkage parameterization to Specify torque constant

Back EMF constant matrix, ke(id,iq) — Permanent magnet flux linkage
`0.18 * ones(3, 3)` `V/(rad/s)` (default) | matrix of scalars

Back EMF constant matrix. The number of rows of this matrix must be equal to the value of the Direct-axis current vector, iD parameter. The number of columns of this matrix must be equal to the value of the Quadrature-axis current vector, iQ parameter.

Dependencies

To enable this parameter, set:

Stator parameterization to Specify Ld, Lq, and L0
Modeling fidelity to Tabulated Ld, Lq and PM
Permanent magnet flux linkage parameterization to Specify back EMF constant

Stator zero-sequence inductance, L0 — Stator zero-sequence inductance
`0.00016` `H` (default) | positive scalar

Zero-sequence inductance of the stator.

Dependencies

To enable this parameter, set Zero sequence to Include.

Stator self-inductance per phase, Ls — Stator self-inductance per phase
`0.0002` `H` (default) | positive scalar

Average self-inductance of each of the three stator windings.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

Stator inductance fluctuation, Lm — Stator inductance fluctuation
`-0.0002` `H` (default) | scalar

Amount that the self-inductance and mutual inductance of the stator windings fluctuate with the rotor angle.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

Stator mutual inductance, Ms — Average mutual inductance between stator windings
`0.00002` `H` (default) | scalar

Average mutual inductance between the stator windings.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

Stator resistance per phase, Rs — Resistance of each of stator windings
`0.013` `Ohm` (default) | positive scalar

Resistance of each of the stator windings.

Zero sequence — Option to include or exclude zero-sequence terms
`Include` (default) | `Exclude`

Option to include or exclude zero-sequence terms.

Include — Include zero-sequence terms. To prioritize model fidelity, use this default setting. Using this option:
- Results in an error for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver.
- Enables the Stator zero-sequence inductance, L0 parameter.
Exclude — Exclude zero-sequence terms. To prioritize simulation speed for desktop simulation or real-time deployment, select this option.

Iron Losses

Model — Option to enable iron losses computation
`None` (default) | `Empirical`

Option to enable the iron losses computational model.

Open-circuit iron losses, [P_hysteresis P_eddy P_excess] — Open-circuit iron losses due to hysteresis, eddy, and excess losses
`[0, 0, 0]` `W` (default) | row vector of three nonnegative elements

Open-circuit iron losses due to the hysteresis, eddy, and excess losses, respectively, at the frequency you specify in the Electrical frequency at which losses determined parameter.

Dependencies

To enable this parameter, set Model to Empirical.

Short-circuit iron losses, [P_hysteresis P_eddy P_excess] — Short-circuit iron losses
`[0, 0, 0]` `W` (default) | row vector of three nonnegative elements

Short-circuit iron losses due to the hysteresis, eddy, and excess losses, respectively, at the frequency you specify in the Electrical frequency at which losses determined parameter.

Dependencies

To enable this parameter, set Model to Empirical.

Electrical frequency at which losses determined — Electrical frequency at which losses are determined
`60` `Hz` (default) | positive scalar

Electrical frequency at which the block measures the open-circuit and short-circuit iron losses.

Dependencies

To enable this parameter, set Model to Empirical.

Short-circuit RMS current for short-circuit iron losses — Short-circuit RMS current for short-circuit iron losses
`95` `A` (default) | positive scalar

Resulting short-circuit RMS phase current when measuring the short-circuit losses.

Dependencies

To enable this parameter, set Model to Empirical.

Mechanical

Rotor inertia — Rotor inertia
`0.01` `kg*m^2` (default) | nonnegative scalar

Inertia of the rotor that attaches to the mechanical translational port R.

Rotor damping — Rotary damping
`0` `N*m/(rad/s)` (default) | nonnegative scalar

Rotary damping.