PMSM

Permanent magnet synchronous motor with sinusoidal flux distribution

Libraries:
Simscape / Electrical / Electromechanical / Permanent Magnet

Description

The PMSM block models a permanent magnet synchronous motor (PMSM) with a three-phase wye-wound or delta-wound 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 like 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 EMF profile. This design reduces undesirable high-frequency mechanical and electrical current harmonics compared to motors like the brushless DC (BLDC) motor which have a trapezoidal back EMF profile.

The PMSM block models the motor with fixed d-axis and q-axis inductances Ld and Lq. You can use this block to model the two main types of PMSM:

SPMSM or surface permanent magnet (SPM) motor — The "S" signifies that the rotor magnets are surface mounted. Ld and Lq are equal.
IPMSM 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 they can be axial which causes much higher torques for the same physical size of motor. You can use the PMSM block to model both categories of PMSM.

This figure shows the equivalent electrical circuit for a wye-wound stator.

You can model the permanent magnet synchronous motor in a delta-wound or open-end configuration by setting Winding type to Delta-wound or Open-end, respectively.

You can choose different built-in parameterizations for this block. For more information, see the Predefined Parameterization section.

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 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.

Motor Construction

This figure shows the motor construction with a single pole-pair on the rotor.

Permanent magnets generate a rotor magnetic field that creates a sinusoidal rate of change of flux based on the rotor angle.

For the axes convention, when you set the Rotor angle definition parameter to Angle between the a-phase magnetic axis and the d-axis, the a-phase and permanent magnet fluxes align when the rotor mechanical angle, θ_r, is zero. When you set the Rotor angle definition parameter to Angle between the a-phase magnetic axis and the q-axis, the rotor mechanical angle is the angle between the a-phase magnetic axis and the rotor q-axis.

Equations

Voltages across the stator windings are

$[\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] = [\begin{matrix} R_{s} & 0 & 0 \\ 0 & R_{s} & 0 \\ 0 & 0 & R_{s} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] + [\begin{matrix} \frac{d ψ_{a}}{d t} \\ \frac{d ψ_{b}}{d t} \\ \frac{d ψ_{c}}{d t} \end{matrix}],$

where:

v_a, v_b, and v_c are the individual phase voltages across the stator windings.
R_s is the equivalent resistance of each stator winding.
i_a, i_b, and i_c are the currents flowing in the stator windings.
$\frac{d ψ_{a}}{d t},$ $\frac{d ψ_{b}}{d t},$ and $\frac{d ψ_{c}}{d t}$ are the rates of change for the magnetic flux in each stator winding.

The permanent magnet and the three windings contribute to the total flux linking each winding. The total flux is

$[\begin{matrix} ψ_{a} \\ ψ_{b} \\ ψ_{c} \end{matrix}] = [\begin{matrix} L_{a a} & L_{a b} & L_{a c} \\ L_{b a} & L_{b b} & L_{b c} \\ L_{c a} & L_{c b} & L_{c c} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] + [\begin{matrix} ψ_{a m} \\ ψ_{b m} \\ ψ_{c m} \end{matrix}],$

where:

ψ_a, ψ_b, and ψ_c are the total fluxes linking each stator winding.
L_aa, L_bb, and L_cc are the self-inductances of the stator windings.
L_ab, L_ac, L_ba, and so on, are the mutual inductances of the stator windings.
ψ_am, ψ_bm, and ψ_cm are the permanent magnet fluxes linking the stator windings.

The inductances in the stator windings are functions of the rotor electrical angle and defined by

$θ_{e} = N θ_{r} + r o t o r o f f s e t$

$L_{a a} = L_{s} + L_{m} cos (2 θ_{e})$

$L_{b b} = L_{s} + L_{m} cos (2 (θ_{e} - 2 π / 3))$

$L_{c c} = L_{s} + L_{m} cos (2 (θ_{e} + 2 π / 3))$

$L_{a b} = L_{b a} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6))$

$L_{b c} = L_{c b} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6 - 2 π / 3))$

$L_{c a} = L_{a c} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6 + 2 π / 3))$

where:

θ_r is the rotor mechanical angle.
θ_e is the rotor electrical angle.
rotor offset is 0 if you define the rotor electrical angle with respect to the d-axis, or -pi/2 if you define the rotor electrical angle with respect to the q-axis.
L_s is the stator self-inductance per phase. This value is the average self-inductance of each of the stator windings.
L_m is the stator inductance fluctuation. This value is the amount the self-inductance and mutual inductance fluctuate with the changing of the rotor angle.
M_s is the stator mutual inductance. This value is the average mutual inductance between the stator windings.

The permanent magnet flux linking winding a is a maximum when θ_e = 0° and zero when θ_e = 90°. Therefore, the linked motor flux is defined by:

$[\begin{matrix} ψ_{a m} \\ ψ_{b m} \\ ψ_{c m} \end{matrix}] = [\begin{matrix} ψ_{m} \cos θ_{e} \\ ψ_{m} \cos (θ_{e} - 2 π / 3) \\ ψ_{m} \cos (θ_{e} + 2 π / 3) \end{matrix}]$

where ψ_m is the permanent magnet flux linkage.

Simplified Electrical Equations

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

Park’s transformation is defined by:

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

where θ_e is the electrical angle defined as Nθ_r. N is the number of pole pairs.

Using Park's transformation on the stator winding voltages and currents transforms them to the dq0 frame, which is independent of the rotor angle:

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

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

Applying Park’s transformation to the first two electrical equations produces these equations that define the block behavior:

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

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

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

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

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 motor case (block physical port C) to the motor rotor (block physical port R).

The PMSM 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 using the back EMF or torque constants which are more commonly given on motor datasheets by using the Permanent magnet flux linkage parameter.

The back EMF constant is the peak voltage induced by the permanent magnet 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 induced by the per-unit current of each of the phases. It 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 can also be:

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

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

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

The term representing the cross-tooth tip path becomes important when a demagnetizing field is set up and can be determined from a finite element analysis short-circuit test. It 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 three numerator terms are derived from the values you provide for the short-circuit hysteresis, eddy, and excess losses.

Predefined Parameterization

There are multiple available built-in parameterizations for the PMSM block.

This pre-parameterization data allows you to set up the block to represent components by specific suppliers. The parameterizations of these permanent magnet synchronous motors match the manufacturer data sheets. To load a predefined parameterization, double-click the PMSM block, click the <click to select> hyperlink of the Selected part parameter and, in the Block Parameterization Manager window, select the part you want to use from the list of available components.

Note

The predefined parameterizations of Simscape components use available data sources for the parameter values. Engineering judgement and simplifying assumptions are used to fill in for missing data. As a result, expect deviations between simulated and actual physical behavior. To ensure accuracy, validate the simulated behavior against experimental data and refine component models as necessary.

For more information about pre-parameterization and for a list of the available components, see List of Pre-Parameterized Components.

Model Thermal Effects

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

No thermal port — The block contains expanded electrical conserving ports associated with the stator windings, but does not contain thermal ports.
Show thermal port — The block contains expanded electrical conserving ports associated with the stator windings and thermal conserving ports for each of the windings 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

Electric Power Assisted Steering

Use a Permanent Magnet Synchronous Motor (PMSM) to amplify driver-applied force in a power-assisted automobile steering system.

Open Model

Faulted PMSM

Model a faulted permanent magnet synchronous motor (PMSM) using Simscape™ Electrical™. Normally when modeling a PMSM, you can represent each winding as a single entity with associated inductance, induced back electromotive force (EMF), and mutual inductive coupling to adjacent windings. However, when a winding fault occurs, the single entity assumption breaks down. To correctly capture the resulting dynamics, you have to model the motor at a winding slot level. This requires modeling in the magnetic domain.

Open Model

Three-Phase PMSM Drive

A Permanent Magnet Synchronous Machine (PMSM) in wye-wound and delta-wound configuration and an inverter sized for use in a typical hybrid vehicle. The inverter is connected directly to the vehicle battery, but you can also implement a DC-DC converter stage in between. You can use this model to design the PMSM controller, by selecting the architecture and gains to achieve the desired performance. To check the timing of IGBT turn-on and turn-off, you can replace the IGBT devices with the more detailed N-Channel IGBT block. For complete vehicle modeling, you can use the Motor & Drive (System Level) block to abstract the PMSM, inverter, and controller with an energy-based model. The Gmin resistor provides a very small conductance to ground that improves the numerical properties of the model when using a variable-step solver.

Open Model

Three-Phase PMSM Drive with Thermal Model

A Permanent Magnet Synchronous Machine (PMSM) and an inverter sized for use in a typical hybrid vehicle. The PMSM includes a thermal model and empirical iron losses. The inverter is connected directly to the vehicle battery, but you can also implement a DC-DC converter stage in between. You can use this model to design the PMSM controller, by selecting the architecture and the gains to achieve the desired performance. The initial temperature of the stator windings and rotor is set to 25 degrees Celsius. Ambient temperature is 27 degrees Celsius. The Scopes subsystem contains scopes that allow you to see the simulation results.

Open Model

Parameterize a Permanent Magnet Synchronous Motor

Estimate the back EMF and torque constants of a blackbox permanent magnet synchronous motor (PMSM) with an unknown flux linkage. You can use either the back EMF or torque constant to describe the flux linkage and parameterize a Simscape™ Electrical™ PMSM block. This parameterization allows you to accurately replicate the blackbox motor's behavior in simulation.

Open Model

IPMSM Torque Control

Control the torque in an interior permanent magnet synchronous machine (IPMSM) based automotive electrical-traction drive. A high-voltage battery feeds the IPMSM through a controlled three- phase converter. 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. At each sample instant, the torque request is converted to relevant current references. The current control is PI-based and uses a sample rate that is faster than the rate that is used for torque control. The simulation uses several torque steps in both motor and generator modes. The task scheduling is designed in Stateflow®. The Scopes subsystem contains scopes that allow you to see the simulation results.

Open Model

IPMSM Torque-Based Load Control

Control the torque in an interior permanent magnet synchronous motor (IPMSM) based drive. A high-voltage battery feeds the IPMSM through a controlled three-phase inverter. A ramp of torque request is provided to the motor controller. The load torque is quadratically dependent on the rotor speed. The Control subsystem uses an open-loop approach to control the IPMSM torque and a closed-loop approach to control the current. At each sample instant, the torque request is converted to relevant current references. The current control is PI-based and uses a sample rate that is faster than the rate that is used for torque control. The task scheduling is designed in Stateflow®. The Scopes subsystem contains scopes that allow you to see the simulation results.

Open Model

IPMSM Velocity Control

Control the rotor angular velocity in an interior permanent magnet synchronous machine (IPMSM) based automotive electrical-traction drive. A high-voltage battery feeds the IPMSM through a controlled three-phase converter. The IPMSM operates in both motoring and generating modes according to the load. An ideal torque source provides the load. The Scopes subsystem contains scopes that allow you to see the simulation results. The Control subsystem includes a multi-rate PI-based cascade control structure which has an outer angular-velocity-control loop and two inner current-control loops. The task scheduling in the Control subsystem is implemented as a Stateflow® state machine. During the one-second simulation, the angular velocity demand is 0 rpm, 500 rpm, 2000 rpm, and then 3000 rpm.

Open Model

Ports

Conserving

expand all

~ — Three-phase port
electrical

Expandable three-phase port. For more information, see Three-phase port.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Wye-wound or Delta-wound.

a — a-phase
electrical

Electrical conserving port associated with a-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

b — b-phase
electrical

Electrical conserving port associated with b-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

c — c-phase
electrical

Electrical conserving port associated with c-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

~1 — Stator positive-end connections
electrical

Since R2024a

Expandable three-phase port associated with the stator positive-end connections.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Open-end.

~2 — Stator negative-end connections
electrical

Since R2024a

Expandable three-phase port associated with the stator negative-end connections.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Open-end.