Interior PMSM

Three-phase interior permanent magnet synchronous motor with sinusoidal back electromotive force

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Libraries:
Powertrain Blockset / Propulsion / Electric Motors and Inverters
Motor Control Blockset / Electrical Systems / Motors

Description

The Interior PMSM block implements a three-phase interior permanent magnet synchronous motor (PMSM) with sinusoidal back electromotive force. The block uses the three-phase input voltages to regulate the individual phase currents, allowing control of the motor torque or speed.

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.

On the Parameters tab, if you select Back-emf, the block implements this equation to calculate the permanent flux linkage constant.

$${\lambda }_{pm}=\frac{1}{\sqrt{3}}\cdot \frac{K_e}{1000P}\cdot \frac{60}{2\pi }$$

Motor Construction

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

The motor magnetic field due to the permanent magnets creates a sinusoidal rate of change of flux with motor angle.

For the axes convention, the a-phase and permanent magnet fluxes are aligned when motor angle θ_r is zero.

Three-Phase Sinusoidal Model Electrical System

The block implements these equations, expressed in the motor flux reference frame (dq frame). All quantities in the motor reference frame are referred to the stator.

$$\begin{array}{l}{\omega }_e=P{\omega }_m\\ \frac{d}{dt}{i}_d=\frac{1}{L_d}{v}_d-\frac{R}{L_d}{i}_d+\frac{L_q}{L_d}P{\omega }_m{i}_q\end{array}$$

$$\frac{d}{dt}{i}_q=\frac{1}{L_q}{v}_q-\frac{R}{L_q}{i}_q-\frac{L_d}{L_q}P{\omega }_m{i}_d-\frac{{\lambda }_{pm}P{\omega }_m}{L_q}$$

$$T_e=1.5P[{\lambda }_{pm}{i}_q+(L_d-L_q)i_d{i}_q]$$

The L_q and L_d inductances represent the relation between the phase inductance and the motor position due to the saliency of the motor.

The equations use these variables.

Lq, Ld	q- and d-axis inductances (H)
R	Resistance of the stator windings (ohm)
iq, id	q- and d-axis currents (A)
vq, vd	q- and d-axis voltages (V)
ωm	Angular mechanical velocity of the motor (rad/s)
ωe	Angular electrical velocity of the motor (rad/s)
λpm	Permanent flux linkage constant (Wb)
Ke	Back electromotive force (EMF) (Vpk_LL/krpm, where Vpk_LL is the peak voltage line-to-line measurement)
P	Number of pole pairs
Te	Electromagnetic torque (Nm)
Θe	Electrical angle (rad)

Mechanical System

The motor 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 motor and load (kgm^2)
F	Combined viscous friction of motor and load (N·m/(rad/s))
θm	Motor mechanical angular position (rad)
Tm	Motor shaft torque (Nm)
Te	Electromagnetic torque (Nm)
Tf	Motor shaft static friction torque (Nm)
ωm	Angular mechanical velocity of the motor (rad/s)

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 (ohm)
ia, ib, ic	Stator phase a, b, and c current (A)
isq, isd	Stator q- and d-axis currents (A)
van, vbn, vcn	Stator phase a, b, and c voltage (V)
ωm	Angular mechanical velocity of the rotor (rad/s)
F	Combined motor and load viscous damping (N·m/(rad/s))
Te	Electromagnetic torque (Nm)
Tf	Combined motor and load friction torque (Nm)

Amplitude invariant dq transformation

The block uses these equations to implement amplitude invariant dq transformation to ensure that the dq and three phase amplitudes are equal.

$\left[\begin{array}{c}{v}_{sd}\\ v_{sq}\end{array}\right]= \frac{2}{3} \left[\begin{array}{ccc}\text{cos}({\Theta }_{da})& \text{cos}({\Theta }_{da}-\frac{2\pi }{3})& \text{cos}({\Theta }_{da}+\frac{2\pi }{3})\\ -\text{sin}({\Theta }_{da})& -\text{sin}({\Theta }_{da}-\frac{2\pi }{3})& -\text{sin}({\Theta }_{da}+\frac{2\pi }{3})\end{array}\right]\left[\begin{array}{c}{v}_a\\ v_b\\ v_c\end{array}\right]$

$\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]= \left[\begin{array}{cc}\text{cos}({\Theta }_{da})& -\text{sin}({\Theta }_{da})\\ \begin{array}{c}\text{cos}({\Theta }_{da}-\frac{2\pi }{3})\\ \text{cos}({\Theta }_{da}+\frac{2\pi }{3})\end{array}& \begin{array}{c}-\text{sin}({\Theta }_{da}-\frac{2\pi }{3})\\ -\text{sin}({\Theta }_{da}+\frac{2\pi }{3})\end{array}\end{array}\right]\left[\begin{array}{c}{i}_{sd}\\ i_{sq}\end{array}\right]$

The equations use these variables.

Θda	dq stator electrical angle with respect to the rotor a-axis (rad)
vsq, vsd	Stator q- and d-axis voltages (V)
isq, isd	Stator q- and d-axis currents (A)
va, vb, vc	Stator voltage phases a, b, c (V)
ia, ib, ic	Stator currents phases a, b, c (A)

Ports

Input

LdTrq — Load torque on motor

scalar

Load torque on the motor shaft, T_m, in N·m.

Dependencies

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

Spd — Motor shaft speed

scalar

Angular velocity of the motor, 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 motor	ωm	rad/s
MtrPos			Motor 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 Mechanical input 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 Mechanical input configuration parameter.

Parameters

Block Options

Mechanical input 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

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.

Parameters

Number of pole pairs (P) — Pole pairs

4 (default) | scalar

Motor pole pairs, P.

Stator phase resistance per phase (Rs) — Resistance

.2 (default) | scalar

Stator phase resistance per phase, R_s, in ohm.

Stator d-axis and q-axis inductance (Ldq) — Inductance

[3.752e-4 4.148e-4] (default) | vector

Stator d-axis and q-axis inductance, L_d, L_q, in H.

Permanent flux linkage constant (lambda_pm) — Flux

0.1194 (default) | scalar

Permanent flux linkage constant, λ_pm, in Wb.

Back-emf constant (Ke) — Back electromotive force

scalar

Back electromotive force, EMF, K_e, in Vpk_LL/krpm. Vpk_LL is the peak voltage line-to-line measurement.

To calculate the permanent flux linkage constant, the block implements this equation.

$${\lambda }_{pm}=\frac{1}{\sqrt{3}}\cdot \frac{K_e}{1000P}\cdot \frac{60}{2\pi }$$

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

[0.002700,4.924e-4,0] (default) | vector

Mechanical properties of the motor:

• Inertia, J, in kg.m^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.

Initial Values

Initial d-axis and q-axis current (idq0) — Current

[0 0] (default) | vector

Initial q- and d-axis currents, i_q, i_d, in A.

Initial mechanical position (theta_init) — Angle

0 (default) | scalar

Initial motor angular position, θ_m0, in rad.

Initial mechanical speed (omega_init) — Speed

0 (default) | scalar

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

Dependencies

To enable this parameter, select the Torque configuration parameter.

References

[1] Kundur, P. Power System Stability and Control. New York, NY: McGraw Hill, 1993.

[2] Anderson, P. M. Analysis of Faulted Power Systems. Hoboken, NJ: Wiley-IEEE Press, 1995.

Extended Capabilities

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

Version History

Introduced in R2017a