Induction Motor

Three-phase induction motor

Since R2020b

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

Description

The Induction Motor block implements a three-phase induction motor. The block uses the three-phase input voltages to regulate the individual phase currents, allowing control of the motor torque or speed.

Note

The block parameters use per-phase values of a star-equivalent induction motor.

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.

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 to calculate the electrical speed (ω_em) and slip speed (ω_slip).

$\begin{array}{l}{\omega }_{em}=P{\omega }_m\\ {\omega }_{slip}= {\omega }_{syn}-{\omega }_{em}\end{array}$

To calculate the dq rotor electrical speed with respect to the rotor A-axis (dA), the block uses the difference between the stator a-axis (da) speed and slip speed:

${\omega }_{dA}= {\omega }_{da}- {\omega }_{em}$

To simplify the equations for the flux, voltage, and current transformations, the block uses a stationary reference frame:

$\begin{array}{l}{\omega }_{da}=0\\ {\omega }_{dA}=-{\omega }_{em}\end{array}$

Calculation	Equation
Flux	$\begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}{\lambda }_{sd}\\ {\lambda }_{sq}\end{array}\right]= \left[\begin{array}{c}{v}_{sd}\\ v_{sq}\end{array}\right]- R_s\left[\begin{array}{c}{i}_{sd}\\ i_{sq}\end{array}\right]- {\omega }_{da}\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}{\lambda }_{sd}\\ {\lambda }_{sq}\end{array}\right]\\ \frac{d}{dt}\left[\begin{array}{c}{\lambda }_{rd}\\ {\lambda }_{rq}\end{array}\right]= \left[\begin{array}{c}{v}_{rd}\\ v_{rq}\end{array}\right]- R_r\left[\begin{array}{c}{i}_{rd}\\ i_{rq}\end{array}\right]- {\omega }_{dA}\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}{\lambda }_{rd}\\ {\lambda }_{rq}\end{array}\right]\end{array}$ $\left[\begin{array}{c}\begin{array}{c}{\lambda }_{sd}\\ {\lambda }_{sq}\end{array}\\ \begin{array}{c}{\lambda }_{rd}\\ {\lambda }_{rq}\end{array}\end{array}\right]= \left[\begin{array}{cc}\begin{array}{cc}{L}_s& 0\\ 0& L_s\end{array}& \begin{array}{cc}{L}_m& 0\\ 0& L_m\end{array}\\ \begin{array}{cc}{L}_m& 0\\ 0& L_m\end{array}& \begin{array}{cc}{L}_r& 0\\ 0& L_r\end{array}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{i}_{sd}\\ i_{sq}\end{array}\\ \begin{array}{c}{i}_{rd}\\ i_{rq}\end{array}\end{array}\right]$
Current	$\left[\begin{array}{c}\begin{array}{c}{i}_{sd}\\ i_{sq}\end{array}\\ \begin{array}{c}{i}_{rd}\\ i_{rq}\end{array}\end{array}\right]= \left(\frac{1}{L_m^2- L_r{L}_s}\right)\left[\begin{array}{cc}\begin{array}{cc}-L_r& 0\\ 0& -L_r\end{array}& \begin{array}{cc}{L}_m& 0\\ 0& L_m\end{array}\\ \begin{array}{cc}{L}_m& 0\\ 0& L_m\end{array}& \begin{array}{cc}-L_s& 0\\ 0& -L_s\end{array}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{\lambda }_{sd}\\ {\lambda }_{sq}\end{array}\\ \begin{array}{c}{\lambda }_{rd}\\ {\lambda }_{rq}\end{array}\end{array}\right]$
Inductance	$\begin{array}{l}{L}_s= L_{ls}+L_m\\ L_r= L_{lr}+L_m\end{array}$
Electromagnetic torque	$T_e=P{L}_m(i_{sq}{i}_{rd}- i_{sd}{i}_{rq})$
Power invariant dq transformation to ensure that the dq and three phase powers are equal	$\left[\begin{array}{c}{v}_{sd}\\ v_{sq}\end{array}\right]= \sqrt{\frac{2}{3}} \left[\begin{array}{ccc}\text{cos}({\Theta }_{da})& \text{cos}({\Theta }_{da}-\frac{2\pi }{3})& \text{cos}({\Theta }_{da}+\frac{2\pi }{3})\\ -\text{sin}({\Theta }_{da})& -\text{sin}({\Theta }_{da}-\frac{2\pi }{3})& -\text{sin}({\Theta }_{da}+\frac{2\pi }{3})\end{array}\right]\left[\begin{array}{c}{v}_a\\ v_b\\ v_c\end{array}\right]$ $\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]= \sqrt{\frac{2}{3}} \left[\begin{array}{cc}\text{cos}({\Theta }_{da})& -\text{sin}({\Theta }_{da})\\ \begin{array}{c}\text{cos}({\Theta }_{da}-\frac{2\pi }{3})\\ \text{cos}({\Theta }_{da}+\frac{2\pi }{3})\end{array}& \begin{array}{c}-\text{sin}({\Theta }_{da}-\frac{2\pi }{3})\\ -\text{sin}({\Theta }_{da}+\frac{2\pi }{3})\end{array}\end{array}\right]\left[\begin{array}{c}{i}_{sd}\\ i_{sq}\end{array}\right]$

The equations use these variables.

ωm	Angular velocity of the rotor (rad/s)
ωem	Electrical rotor speed (rad/s)
ωslip	Electrical rotor slip speed (rad/s)
ωsyn	Synchronous rotor speed (rad/s)
ωda	dq stator electrical speed with respect to the rotor a-axis (rad/s)
ωdA	dq stator electrical speed with respect to the rotor A-axis (rad/s)
Θda	dq stator electrical angle with respect to the rotor a-axis (rad)
ΘdA	dq stator electrical angle with respect to the rotor A-axis (rad)
Lq, Ld	q- and d-axis inductances (H)
Ls	Stator inductance (H)
Lr	Rotor inductance (H)
Lm	Magnetizing inductance (H)
Lls	Stator leakage inductance (H)
Llr	Rotor leakage inductance (H)
vsq, vsd	Stator q- and d-axis voltages (V)
isq, isd	Stator q- and d-axis currents (A)
λsq, λsd	Stator q- and d-axis flux (Wb)
irq, ird	Rotor q- and d-axis currents (A)
λrq, λrd	Rotor q- and d-axis flux (Wb)
va, vb, vc	Stator voltage phases a, b, c (V)
ia, ib, ic	Stator currents phases a, b, c (A)
Rs	Resistance of the stator windings (Ohm)
Rr	Resistance of the rotor windings (Ohm)
P	Number of pole pairs
Te	Electromagnetic torque (Nm)

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	$P_{elec}= -(R_s{i}_{sd}^2+R_s{i}_{sq}^2+-R_r{i}_{rd}^2+R_r{i}_{rq}^2)$
		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)
Rr	Rotor 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 viscous damping of motor and load (N·m/(rad/s))
Te	Electromagnetic torque (Nm)
Tf	Combined friction torque of motor and load (Nm)

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 — 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.

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
MtrMechPos			Rotor mechanical angular position	θm	rad
MtrPos			Rotor electrical angular position	θe	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

Load Parameter Values

File — Path to motor parameter ".m" or ".mat" file

scalar

Enter the path to the motor parameter ".m" or ".mat" file that you saved using the Motor Control Blockset™ parameter estimation tool. You can also click the Browse button to navigate and select the ".m" or ".mat" file, and update File parameter with the file name and path. For details related to the motor parameter estimation process, see Estimate PMSM Parameters Using Recommended Hardware.

• Load from file - Click this button to read the estimated motor parameters from the ".m" or ".mat" file (indicated by the File parameter) and load them to the motor block.

• Save to file - Click this button to read the motor parameters from the motor block and save them into a ".m" or ".mat" file (with a file name and location that you specify in the File parameter).

Note

Before you click Save to file button, ensure that the target file name in the File parameter has either ".m" or ".mat" extension. If you use any other file extension, the block displays an error message.

Parameters

Number of pole pairs, P — Pole pairs

2 (default) | scalar

Motor pole pairs, P.

Stator resistance and leakage inductance, Zs — Resistance and inductance

[1.77 0.0139] (default) | vector

Stator resistance, R_S, in ohms and leakage inductance, L_ls, in H.

Rotor resistance and leakage inductance, Zr — Resistance and inductance

[1.34 0.0121] (default) | vector

Rotor resistance, R_r, in ohms and leakage inductance, L_lr, in H.

Magnetizing inductance, Lm — Inductance

0.3687 (default) | scalar

Magnetizing inductance, L_m, in H.

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

[0.001 0 0] (default) | vector

Mechanical properties of the rotor:

• 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 Torque for the Port configuration.

Initial Values

Initial mechanical position, theta_init — Angular position

0 (default) | scalar

Initial rotor angular position, θ_m0, in rad.

Initial mechanical speed, omega_init — Angular speed

0 (default) | scalar

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

Dependencies

To enable this parameter, select Torque for the Port configuration.

References

[1] Mohan, Ned. Advanced Electric Drives: Analysis, Control and Modeling Using Simulink. Minneapolis, MN: MNPERE, 2001.

Extended Capabilities

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

Version History

Introduced in R2020b