Flux-Based PMSM

Flux-based permanent magnet synchronous motor

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  • Powertrain Blockset / Propulsion / Electric Motors

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.

CalculationEquation
q- and d-axis voltagevd=dψddt+Rsidωeψqvq=dψqdt+Rsiq+ωeψd
q- and d-axis currentid=f(ψd,ψq)iq=g(ψd,ψq)
Electromechanical torqueTe=1.5P[ψdiqψqid]

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.

ωe=Pωmdθedt= ωe

TransformDescriptionEquations

Clarke

Converts balanced three-phase quantities (a, b) into balanced two-phase quadrature quantities (α, β).

xα= 23xa 13xb 13xcxβ= 32xb 32xc

Park

Converts balanced two-phase orthogonal stationary quantities (α, β) into an orthogonal rotating reference frame (d, q).

xd= xαcosθe+ xβsinθexq= xαsinθe+ xβcosθe

Inverse Clarke

Converts balanced two-phase quadrature quantities (α, β) into balanced three-phase quantities (a, b).

xa= xaxb= 12xα+ 32xβxc= 12xα 32xβ

Inverse Park

Converts an orthogonal rotating reference frame (d, q) into balanced two-phase orthogonal stationary quantities (α, β).

xα= xdcosθe xqsinθexβ= xdsinθe+ xqcosθe

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:

ddtωm=1J(TeTfFωmTm)dθmdt=ωm

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 DescriptionVariableEquations

PwrInfo

PwrTrnsfrd — Power transferred between blocks

  • Positive signals indicate flow into block

  • Negative signals indicate flow out of block

PwrMtr

Mechanical power

Pmot

Pmot= ωmTe
PwrBus

Electrical power

Pbus

Pbus= vania+ vbnib+vcnic

PwrNotTrnsfrd — Power crossing the block boundary, but not transferred

  • Positive signals indicate an input

  • Negative signals indicate a loss

PwrElecLoss

Resistive power loss

Pelec

Pelec= 32(Rsisd2+Rsisq2)
PwrMechLoss

Mechanical power loss

Pmech

When Port Configuration is set to Torque:

Pmech= (ωm2F+ |ωm|Tf)

When Port Configuration is set to Speed:

Pmech= 0 

PwrStored — Stored energy rate of change

  • Positive signals indicate an increase

  • Negative signals indicate a decrease

PwrMtrStored

Stored motor power

Pstr

Pstr= Pbus+ Pmot+ Pelec + Pmech

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 CaseImplementation

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 InputExtrapolation

Ports

Input

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Rotor shaft input torque, Tm, in N·m.

Dependencies

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

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

Dependencies

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

Stator terminal voltages, Va, Vb, and Vc, in V.

Dependencies

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

Output

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The bus signal contains these block calculations.

Signal DescriptionVariableUnits

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
PwrStoredPwrMtrStored

Stored motor power

Pstr

W

Parameters

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Block Options

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.

Integration sample time for discrete simulation, in s.

Dependencies

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

This table summarizes the port configurations.

Port ConfigurationCreates Ports

Torque

LdTrq

PhaseVolt

Info

Speed

Spd

PhaseVolt

Info

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

d-axis flux, Ψd, breakpoints, in Wb.

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.

q-axis flux, Ψq, breakpoints, in Wb.

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.

Array of values for d-axis current, id, as a function of M d-fluxes, Ψd, and N q-fluxes, Ψq, 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.

Array of values for q-axis current, id, as a function of M d-fluxes, Ψd, and N q-fluxes, Ψq, 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 breakpoint maximum extrapolation endpoint, u1max, in Wb.

Dependencies

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

Flux breakpoint minimum extrapolation endpoint, u1min, in Wb.

Dependencies

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

Flux breakpoint maximum extrapolation endpoint, u2max, in Wb.

Dependencies

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

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, in ohm.

Motor pole pairs, P.

Initial d- and q-axis flux, Ψq0 and Ψd0, in Wb.

Initial rotor angular position, θm0, in rad.

Initial angular velocity of the rotor, ωm0, in rad/s.

Dependencies

To enable this parameter, select the Torque configuration parameter.

Mechanical properties of the rotor:

  • Inertia, J, in kgm^2

  • Viscous damping, F, in N·m/(rad/s)

  • Static friction, Tf, 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, Vol. 51, No. 4, July/August 2015.

[2] Chen, Xiao, Jiabin Wang, Bhaskar Sen, Panagiotis Lasari, 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, Vol. 62, No. 7, July 2015.

Extended Capabilities

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

Introduced in R2017b