Flux-Based PMSM

Flux-based permanent magnet synchronous motor

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Libraries:
Powertrain Blockset / Propulsion / Electric Motors and Inverters

Description

The Flux-Based PMSM block implements a flux-based three-phase permanent magnet synchronous motor (PMSM) with a tabular-based electromotive force. The block uses the three-phase input voltages to regulate the individual phase currents, allowing control of the motor torque or speed.

Flux-based motor models take into account magnetic saturation and iron losses. To calculate the magnetic saturation and iron loss, the Flux-Based PMSM block uses the inverse of the flux linkages. To obtain the block parameters, you can use finite-element analysis (FEA) or measure phase voltages using a dynamometer.

By default, the block sets the Simulation Type parameter to Continuous to use a continuous sample time during simulation. If you want to generate code for fixed-step double- and single-precision targets, considering setting the parameter to Discrete. Then specify a Sample Time, Ts parameter.

To enable power loss calculations suitable for code generation targets that limit memory, select Enable memory optimized 2D LUT.

Three-Phase Sinusoidal Model Electrical System

The block implements equations that are expressed in a stationary rotor reference (dq) frame. The d-axis aligns with the a-axis. All quantities in the rotor reference frame are referred to the stator.

The block uses these equations.

Calculation	Equation
q- and d-axis voltage	$$\begin{array}{l}{v}_d=\frac{d{\psi }_d}{dt}+R_s{i}_d-{\omega }_e{\psi }_q\\ v_q=\frac{d{\psi }_q}{dt}+R_s{i}_q+{\omega }_e{\psi }_d\end{array}$$
q- and d-axis current	$$\begin{array}{l}{i}_d=f({\psi }_d,{\psi }_q)\\ i_q=g({\psi }_d,{\psi }_q)\end{array}$$
Electromechanical torque	$$T_e=1.5P[{\psi }_d{i}_q-{\psi }_q{i}_d]$$

The equations use these variables.

ωm	Rotor mechanical speed
ωe	Rotor electrical speed
Θda	dq stator electrical angle with respect to the rotor a-axis
Rs, Rr	Resistance of the stator and rotor windings, respectively
iq, id	q- and d-axis current, respectively
vq, vd	q- and d-axis voltage, respectively
Ψq, Ψd	q- and d-axis magnet flux, respectively
P	Number of pole pairs
Te	Electromagnetic torque

Transforms

To calculate the voltages and currents in balanced three-phase (a, b) quantities, quadrature two-phase (α, β) quantities, and rotating (d, q) reference frames, the block uses the Clarke and Park Transforms.

In the transform equations.

$\begin{array}{l}{\omega }_e=P{\omega }_m\\ \frac{d{\theta }_e}{dt}= {\omega }_e\end{array}$

Transform	Description	Equations
Clarke	Converts balanced three-phase quantities (a, b) into balanced two-phase quadrature quantities (α, β).	$\begin{array}{l}{x}_{\alpha }= \frac{2}{3}{x}_a- \frac{1}{3}{x}_b -\frac{1}{3}{x}_c\\ x_{\beta }= \frac{\sqrt{3}}{2}{x}_b- \frac{\sqrt{3}}{2}{x}_c\end{array}$
Park	Converts balanced two-phase orthogonal stationary quantities (α, β) into an orthogonal rotating reference frame (d, q).	$\begin{array}{l}{x}_d= x_{\alpha }\cos {\theta }_e+ x_{\beta }\sin {\theta }_e\\ x_q= -x_{\alpha }\sin {\theta }_e+ x_{\beta }\cos {\theta }_e\end{array}$
Inverse Clarke	Converts balanced two-phase quadrature quantities (α, β) into balanced three-phase quantities (a, b).	$\begin{array}{l}{x}_a= x_a\\ x_b= -\frac{1}{2}{x}_{\alpha }+ \frac{\sqrt{3}}{2}{x}_{\beta }\\ x_c= -\frac{1}{2}{x}_{\alpha }- \frac{\sqrt{3}}{2}{x}_{\beta }\end{array}$
Inverse Park	Converts an orthogonal rotating reference frame (d, q) into balanced two-phase orthogonal stationary quantities (α, β).	$\begin{array}{l}{x}_{\alpha }= x_d\cos {\theta }_e- x_q\sin {\theta }_e\\ x_{\beta }= x_d\sin {\theta }_e+ x_q\cos {\theta }_e\end{array}$

The transforms use these variables.

ωm	Rotor mechanical speed
P	Motor pole pairs
ωe	Rotor electrical speed
Θe	Rotor electrical angle
x	Phase current or voltage

Mechanical System

The rotor angular velocity is given by:

$$\begin{array}{c}\frac{d}{dt}{\omega }_m=\frac{1}{J}\left(T_e-T_f-F{\omega }_m-T_m\right)\\ \frac{d{\theta }_m}{dt}={\omega }_m\end{array}$$

The equations use these variables.

J	Combined inertia of rotor and load
F	Combined viscous friction of rotor and load
θm	Rotor mechanical angular position
Tm	Rotor shaft torque
Te	Electromagnetic torque
Tf	Combined rotor and load friction torque
ωm	Rotor mechanical speed

Power Accounting

For the power accounting, the block implements these equations.

Bus Signal			Description	Variable	Equations
PwrInfo	PwrTrnsfrd — Power transferred between blocks Positive signals indicate flow into block Negative signals indicate flow out of block	PwrMtr	Mechanical power	Pmot	$P_{mot}= -{\omega }_m{T}_e$
		PwrBus	Electrical power	Pbus	$P_{bus}= v_{an}{i}_a+ v_{bn}{i}_b+v_{cn}{i}_c$
	PwrNotTrnsfrd — Power crossing the block boundary, but not transferred Positive signals indicate an input Negative signals indicate a loss	PwrElecLoss	Resistive power loss	Pelec	$P_{elec}= -\frac{3}{2}(R_s{i}_{sd}^2+R_s{i}_{sq}^2)$
		PwrMechLoss	Mechanical power loss	Pmech	When Port Configuration is set to Torque: $P_{mech}= -\left({\omega }_m^{2}F+ \left|{\omega }_m\right|T_f\right)$ When Port Configuration is set to Speed: $P_{mech}= 0$
	PwrStored — Stored energy rate of change Positive signals indicate an increase Negative signals indicate a decrease	PwrMtrStored	Stored motor power	Pstr	$P_{str}= P_{bus}+ P_{mot}+ P_{elec} + P_{mech}$

The equations use these variables.

Rs	Stator resistance
ia, ib, ic	Stator phase a, b, and c current
isq, isd	Stator q- and d-axis currents
van, vbn, vcn	Stator phase a, b, and c voltage
ωm	Angular mechanical velocity of the rotor
F	Combined motor and load viscous damping
Te	Electromagnetic torque
Tf	Combined motor and load friction torque

Lookup Table Memory Optimization

The data for the Corresponding d-axis current, id and Corresponding q-axis current, iq lookup tables are functions of the d- and q-axis flux.

To enable current calculations suitable for code generation targets that limit memory, select Enable memory optimized 2D LUT. The block uses linear interpolation to optimize the current lookup table values for code generation. This table summarizes the optimization implementation.

Use Case		Implementation
d- and q-axis flux aligns with the lookup table breakpoint values.		Memory-optimized current is current lookup table value at intersection of flux values.
d- and q-axis flux does not align with the lookup table breakpoint values, but is within range.		Memory-optimized current is linear interpolation between corresponding flux values.
d- and q-axis flux does not align with the lookup table breakpoint values, and is out of range.		Cannot compute an memory-optimized current. Block uses extrapolated data.

Extrapolation

The lookup tables optimized for code generation do not support extrapolation for data that is out of range. However, you can include pre-calculated extrapolation values in the power loss lookup table by selecting Specify Extrapolation.

The block uses the endpoint parameters to resize the table data.

User Input	Extrapolation

Ports

Input

LdTrq — Rotor shaft torque

scalar

Rotor shaft input torque, T_m, in N·m.

Dependencies

To create this port, select Torque for the Port Configuration parameter.

Spd — Rotor shaft speed

scalar

Angular velocity of the rotor, ω_m, in rad/s.

Dependencies

To create this port, select Speed for the Port Configuration parameter.

PhaseVolt — Stator terminal voltages

1-by-3 array

Stator terminal voltages, V_a, V_b, and V_c, in V.

Dependencies

To create this port, select Speed or Torque for the Port Configuration parameter.

Output

Info — Bus signal

bus

The bus signal contains these block calculations.

Signal			Description	Variable	Units
IaStator			Stator phase current A	ia	A
IbStator			Stator phase current B	ib	A
IcStator			Stator phase current C	ic	A
IdSync			Direct axis current	id	A
IqSync			Quadrature axis current	iq	A
VdSync			Direct axis voltage	vd	V
VqSync			Quadrature axis voltage	vq	V
MtrSpd			Angular mechanical velocity of the rotor	ωm	rad/s
MtrPos			Rotor mechanical angular position	θm	rad
MtrTrq			Electromagnetic torque	Te	N·m
PwrInfo	PwrTrnsfrd	PwrMtr	Mechanical power	Pmot	W
		PwrBus	Electrical power	Pbus	W
	PwrNotTrnsfrd	PwrElecLoss	Resistive power loss	Pelec	W
		PwrMechLoss	Mechanical power loss	Pmech	W
	PwrStored	PwrMtrStored	Stored motor power	Pstr	W

PhaseCurr — Phase a, b, c current

1-by-3 array

Phase a, b, c current, i_a, i_b, and i_c, in A.

MtrTrq — Motor torque

scalar

Motor torque, T_mtr, in N·m.

Dependencies

To create this port, select Speed for the Port configuration parameter.

MtrSpd — Motor speed

scalar

Angular speed of the motor, ω_mtr, in rad/s.

Dependencies

To create this port, select Torque for the Port configuration parameter.

Parameters

Block Options

Simulation type — Select simulation type

Continuous (default) | Discrete

By default, the block uses a continuous sample time during simulation. If you want to generate code for single-precision targets, considering setting the parameter to Discrete.

Dependencies

Setting Simulation Type to Discrete creates the Sample Time, Ts parameter.

Sample time, Ts — Sample time for discrete integration

0.001 (default) | scalar

Integration sample time for discrete simulation, in s.

Dependencies

Setting Simulation Type to Discrete creates the Sample Time, Ts parameter.

Port Configuration — Select port configuration

Torque (default) | Speed

This table summarizes the port configurations.

Port Configuration	Creates Input Port	Creates Output Port
Torque	LdTrq	MtrSpd
Speed	Spd	MtrTrq

Enable memory optimized 2D LUT — Selection

off (default) | on

Enable generation of optimized lookup tables, suitable code generation targets that limit memory.

Vector of d-axis flux, flux_d — Flux breakpoints

1-by-M vector

d-axis flux, Ψ_d, breakpoints, in Wb.

Resample storage size for flux_d, n1 — Flux bit size

2 (default) | 4 | 8 | 16 | 32 | 64 | 128 | 256

Flux breakpoint storage size, n1, dimensionless. The block resamples the Corresponding d-axis current, id and Corresponding q-axis current, iq data based on the storage size.

Dependencies

To create this parameter, select Enable memory optimized 2D LUT.

Vector of q-axis flux, flux_q — Flux breakpoints

1-by-N vector

q-axis flux, Ψ_q, breakpoints, in Wb.

Resample storage size for flux_q, n2 — Flux bit size

2 (default) | 4 | 8 | 16 | 32 | 64 | 128 | 256

Flux breakpoint storage size, n2, dimensionless. The block resamples the Corresponding d-axis current, id and Corresponding q-axis current, iq data based on the storage size.

Dependencies

To create this parameter, select Enable memory optimized 2D LUT.

Corresponding d-axis current, id — 2D lookup table

M-by-N array

Array of values for d-axis current, i_d, as a function of M d-fluxes, Ψ_d, and N q-fluxes, Ψ_q, in A. Each value specifies the current for a specific combination of d- and q-axis flux. The array size must match the dimensions defined by the flux vectors.

If you set Enable memory optimized 2D LUT, the block converts the data to single precision.

Corresponding q-axis current, iq — 2D lookup table

M-by-N array

Array of values for q-axis current, i_d, as a function of M d-fluxes, Ψ_d, and N q-fluxes, Ψ_q, in A. Each value specifies the current for a specific combination of d- and q-axis flux. The array size must match the dimensions defined by the flux vectors.

If you set Enable memory optimized 2D LUT, the block converts the data to single precision.

flux_d max endpoint, u1max — Flux breakpoint

0.22457 (default) | scalar

Flux breakpoint maximum extrapolation endpoint, u1max, in Wb.

Dependencies

To create this parameter, select Enable memory optimized 2D LUT and Specify Extrapolation.

flux_d min endpoint, u1min — Flux breakpoint

-0.22607 (default) | scalar

Flux breakpoint minimum extrapolation endpoint, u1min, in Wb.

Dependencies

To create this parameter, select Enable memory optimized 2D LUT and Specify Extrapolation.

flux_q max endpoint, u2max — Flux breakpoint

0.42959 (default) | scalar

Flux breakpoint maximum extrapolation endpoint, u2max, in Wb.

Dependencies

To create this parameter, select Enable memory optimized 2D LUT and Specify Extrapolation.

flux_q min endpoint, u2min — Flux breakpoint

-0.4296 (default) | scalar

Flux breakpoint minimum extrapolation endpoint, u2min, in Wb.

Dependencies

To create this parameter, select Enable memory optimized 2D LUT and Specify Extrapolation.

Stator phase resistance, Rs — Resistance

0.02 (default) | scalar

Stator phase resistance, R_s, in ohm.

Number of pole pairs, P — Pole pairs

4 (default) | scalar

Motor pole pairs, P.

Initial flux, fluxdq0 — Flux

[0 0] (default) | vector

Initial d- and q-axis flux, Ψ_q0 and Ψ_d0, in Wb.

Initial mechanical position, theta_init — Angle

0 (default) | scalar

Initial rotor angular position, θ_m0, in rad.

Initial mechanical speed, omega_init — Speed

0 (default) | scalar

Initial angular velocity of the rotor, ω_m0, in rad/s.

Dependencies

To enable this parameter, select the Torque configuration parameter.

Physical inertia, viscous damping, and static friction, mechanical — Inertia, damping, friction

[0.0027, 4.924e-4, 0] (default) | vector

Mechanical properties of the rotor:

• Inertia, J, in kgm^2

• Viscous damping, F, in N·m/(rad/s)

• Static friction, T_f, in N·m

Dependencies

To enable this parameter, select the Torque configuration parameter.

References

[1] Hu, Dakai, Yazan Alsmadi, and Longya Xu. “High-Fidelity Nonlinear IPM Modeling Based on Measured Stator Winding Flux Linkage.” IEEE^® Transactions on Industry Applications 51, no. 4 (July/August 2015).

[2] Chen, Xiao, Jiabin Wang, Bhaskar Sen, Panagiotis Lasari, and Tianfu Sun. “A High-Fidelity and Computationally Efficient Model for Interior Permanent-Magnet Machines Considering the Magnetic Saturation, Spatial Harmonics, and Iron Loss Effect.” IEEE Transactions on Industrial Electronics 62, no. 7 (July 2015).

[3] Ottosson, J., and M. Alakula. “A Compact Field Awakening Controller Implementation.” International Symposium on Power Electronics, Electrical Drives, Automation and Motion, July 2006.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2017b