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Induction Motor

Three-phase induction motor

Since R2020b

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).

$\begin{array}{l} ω_{e m} = P ω_{m} \\ ω_{s l i p} = ω_{s y n} - ω_{e m} \end{array}$

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:

$ω_{d A} = ω_{d a} - ω_{e m}$

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

$\begin{array}{l} ω_{d a} = 0 \\ ω_{d A} = - ω_{e m} \end{array}$

Calculation	Equation
Flux	$\begin{array}{l} \frac{d}{d t} [\begin{matrix} λ_{s d} \\ λ_{s q} \end{matrix}] = [\begin{matrix} v_{s d} \\ v_{s q} \end{matrix}] - R_{s} [\begin{matrix} i_{s d} \\ i_{s q} \end{matrix}] - ω_{d a} [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} λ_{s d} \\ λ_{s q} \end{matrix}] \\ \frac{d}{d t} [\begin{matrix} λ_{r d} \\ λ_{r q} \end{matrix}] = [\begin{matrix} v_{r d} \\ v_{r q} \end{matrix}] - R_{r} [\begin{matrix} i_{r d} \\ i_{r q} \end{matrix}] - ω_{d A} [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} λ_{r d} \\ λ_{r q} \end{matrix}] \end{array}$ $[\begin{matrix} \begin{matrix} λ_{s d} \\ λ_{s q} \end{matrix} \\ \begin{matrix} λ_{r d} \\ λ_{r q} \end{matrix} \end{matrix}] = [\begin{matrix} \begin{matrix} L_{s} & 0 \\ 0 & L_{s} \end{matrix} & \begin{matrix} L_{m} & 0 \\ 0 & L_{m} \end{matrix} \\ \begin{matrix} L_{m} & 0 \\ 0 & L_{m} \end{matrix} & \begin{matrix} L_{r} & 0 \\ 0 & L_{r} \end{matrix} \end{matrix}] [\begin{matrix} \begin{matrix} i_{s d} \\ i_{s q} \end{matrix} \\ \begin{matrix} i_{r d} \\ i_{r q} \end{matrix} \end{matrix}]$
Current	$[\begin{matrix} \begin{matrix} i_{s d} \\ i_{s q} \end{matrix} \\ \begin{matrix} i_{r d} \\ i_{r q} \end{matrix} \end{matrix}] = (\frac{1}{L_{m}^{2} - L_{r} L_{s}}) [\begin{matrix} \begin{matrix} - L_{r} & 0 \\ 0 & - L_{r} \end{matrix} & \begin{matrix} L_{m} & 0 \\ 0 & L_{m} \end{matrix} \\ \begin{matrix} L_{m} & 0 \\ 0 & L_{m} \end{matrix} & \begin{matrix} - L_{s} & 0 \\ 0 & - L_{s} \end{matrix} \end{matrix}] [\begin{matrix} \begin{matrix} λ_{s d} \\ λ_{s q} \end{matrix} \\ \begin{matrix} λ_{r d} \\ λ_{r q} \end{matrix} \end{matrix}]$
Inductance	$\begin{array}{l} L_{s} = L_{l s} + L_{m} \\ L_{r} = L_{l r} + L_{m} \end{array}$
Electromagnetic torque	$T_{e} = P L_{m} (i_{s q} i_{r d} - i_{s d} i_{r q})$
Power invariant dq transformation to ensure that the dq and three phase powers are equal	$[\begin{matrix} v_{s d} \\ v_{s q} \end{matrix}] = \sqrt{\frac{2}{3}} [\begin{matrix} cos (Θ_{d a}) & cos (Θ_{d a} - \frac{2 π}{3}) & cos (Θ_{d a} + \frac{2 π}{3}) \\ - sin (Θ_{d a}) & - sin (Θ_{d a} - \frac{2 π}{3}) & - sin (Θ_{d a} + \frac{2 π}{3}) \end{matrix}] [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}]$ $[\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] = \sqrt{\frac{2}{3}} [\begin{matrix} cos (Θ_{d a}) & - sin (Θ_{d a}) \\ \begin{matrix} cos (Θ_{d a} - \frac{2 π}{3}) \\ cos (Θ_{d a} + \frac{2 π}{3}) \end{matrix} & \begin{matrix} - sin (Θ_{d a} - \frac{2 π}{3}) \\ - sin (Θ_{d a} + \frac{2 π}{3}) \end{matrix} \end{matrix}] [\begin{matrix} i_{s d} \\ i_{s q} \end{matrix}]$

The equations use these variables.

ω_m	Angular velocity of the rotor (rad/s)
ω_em	Electrical rotor speed (rad/s)
ω_slip	Electrical rotor slip speed (rad/s)
ω_syn	Synchronous rotor speed (rad/s)
ω_da	dq stator electrical speed with respect to the rotor a-axis (rad/s)
ω_dA	dq stator electrical speed with respect to the rotor A-axis (rad/s)
Θ_da	dq stator electrical angle with respect to the rotor a-axis (rad)
Θ_dA	dq stator electrical angle with respect to the rotor A-axis (rad)
L_q, L_d	q- and d-axis inductances (H)
L_s	Stator inductance (H)
L_r	Rotor inductance (H)
L_m	Magnetizing inductance (H)
L_ls	Stator leakage inductance (H)
L_lr	Rotor leakage inductance (H)
v_sq, v_sd	Stator q- and d-axis voltages (V)
i_sq, i_sd	Stator q- and d-axis currents (A)
λ_sq, λ_sd	Stator q- and d-axis flux (Wb)
i_rq, i_rd	Rotor q- and d-axis currents (A)
λ_rq, λ_rd	Rotor q- and d-axis flux (Wb)
v_a, v_b, v_c	Stator voltage phases a, b, c (V)
i_a, i_b, i_c	Stator currents phases a, b, c (A)
R_s	Resistance of the stator windings (Ohm)
R_r	Resistance of the rotor windings (Ohm)
P	Number of pole pairs
T_e	Electromagnetic torque (Nm)

Mechanical System

The motor angular velocity is given by:

$\begin{matrix} \frac{d}{d t} ω_{m} = \frac{1}{J} (T_{e} - T_{f} - F ω_{m} - T_{m}) \\ \frac{d θ_{m}}{d t} = ω_{m} \end{matrix}$

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))
θ_m	Motor mechanical angular position (rad)
T_m	Motor shaft torque (Nm)
T_e	Electromagnetic torque (Nm)
T_f	Motor shaft static friction torque (Nm)
ω_m	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	P_mot	$P_{m o t} = - ω_{m} T_{e}$
		`PwrBus`	Electrical power	P_bus	$P_{b u s} = v_{a n} i_{a} + v_{b n} i_{b} + v_{c n} i_{c}$
	`PwrNotTrnsfrd` — Power crossing the block boundary, but not transferred Positive signals indicate an input Negative signals indicate a loss	`PwrElecLoss`	Resistive power loss	P_elec	$P_{e l e c} = - (R_{s} i_{s d}^{2} + R_{s} i_{s q}^{2} + - R_{r} i_{r d}^{2} + R_{r} i_{r q}^{2})$
		`PwrMechLoss`	Mechanical power loss	P_mech	When Port Configuration is set to `Torque`: $P_{m e c h} = - (ω_{m}^{2} F + \| ω_{m} \| T_{f})$ When Port Configuration is set to `Speed`: $P_{m e c h} = 0$
	`PwrStored` — Stored energy rate of change Positive signals indicate an increase Negative signals indicate a decrease	`PwrMtrStored`	Stored motor power	P_str	$P_{s t r} = P_{b u s} + P_{m o t} + P_{e l e c} + P_{m e c h}$

The equations use these variables.

R_s	Stator resistance (Ohm)
R_r	Rotor resistance (Ohm)
i_a, i_b, i_c	Stator phase a, b, and c current (A)
i_sq, i_sd	Stator q- and d-axis currents (A)
v_an, v_bn, v_cn	Stator phase a, b, and c voltage (V)
ω_m	Angular mechanical velocity of the rotor (rad/s)
F	Combined viscous damping of motor and load (N·m/(rad/s))
T_e	Electromagnetic torque (Nm)
T_f	Combined friction torque of motor and load (Nm)

Ports

Input

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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

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Info — Bus signal
bus

The bus signal contains these block calculations.

Signal			Description	Variable	Units
`IaStator`			Stator phase current A	i_a	A
`IbStator`			Stator phase current B	i_b	A
`IcStator`			Stator phase current C	i_c	A
`IdSync`			Direct axis current	i_d	A
`IqSync`			Quadrature axis current	i_q	A
`VdSync`			Direct axis voltage	v_d	V
`VqSync`			Quadrature axis voltage	v_q	V
`MtrSpd`			Angular mechanical velocity of the rotor	ω_m	rad/s
`MtrMechPos`			Rotor mechanical angular position	θ_m	rad
`MtrPos`			Rotor electrical angular position	θ_e	rad
`MtrTrq`			Electromagnetic torque	T_e	N·m
`PwrInfo`	`PwrTrnsfrd`	`PwrMtr`	Mechanical power	P_mot	W
	`PwrTrnsfrd`	`PwrBus`	Electrical power	P_bus	W
	`PwrNotTrnsfrd`	`PwrElecLoss`	Resistive power loss	P_elec	W
	`PwrNotTrnsfrd`	`PwrMechLoss`	Mechanical power loss	P_mech	W
	`PwrStored`	`PwrMtrStored`	Stored motor power	P_str	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

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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

Induction Motor

Description

Three-Phase Sinusoidal Model Electrical System

Mechanical System

Power Accounting

Ports

Input

LdTrq — Load torque on motor scalar

Dependencies

Spd — Rotor shaft speed scalar

Dependencies

PhaseVolt — Stator terminal voltages 1-by-3 array

Output

Info — Bus signal bus

PhaseCurr — Phase a, b, c current 1-by-3 array

MtrTrq — Motor torque scalar

Dependencies

MtrSpd — Motor speed scalar

Dependencies

Parameters

Simulation type — Select simulation type Continuous (default) | Discrete

Dependencies

Sample time, Ts — Sample time for discrete integration 0.001 (default) | scalar

Dependencies

Port configuration — Select port configuration Torque (default) | Speed

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

Number of pole pairs, P — Pole pairs 2 (default) | scalar

Stator resistance and leakage inductance, Zs — Resistance and inductance [1.77 0.0139] (default) | vector

Rotor resistance and leakage inductance, Zr — Resistance and inductance [1.34 0.0121] (default) | vector

Magnetizing inductance, Lm — Inductance 0.3687 (default) | scalar

Physical inertia, viscous damping, static friction, mechanical — Inertia, damping, friction [0.001 0 0] (default) | vector

Dependencies

Initial mechanical position, theta_init — Angular position 0 (default) | scalar

Initial mechanical speed, omega_init — Angular speed 0 (default) | scalar

Dependencies

References

Extended Capabilities

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

Version History

See Also

Topics

LdTrq — Load torque on motor
`scalar`

Spd — Rotor shaft speed
`scalar`

PhaseVolt — Stator terminal voltages
`1`-by-`3` array

Info — Bus signal
bus

PhaseCurr — Phase a, b, c current
`1`-by-`3` array

MtrTrq — Motor torque
`scalar`

MtrSpd — Motor speed
`scalar`

Simulation type — Select simulation type
`Continuous` (default) | `Discrete`

Sample time, Ts — Sample time for discrete integration
`0.001` (default) | `scalar`

Port configuration — Select port configuration
`Torque` (default) | `Speed`

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

Number of pole pairs, P — Pole pairs
`2` (default) | `scalar`

Stator resistance and leakage inductance, Zs — Resistance and inductance
`[1.77 0.0139]` (default) | `vector`

Rotor resistance and leakage inductance, Zr — Resistance and inductance
`[1.34 0.0121]` (default) | `vector`

Magnetizing inductance, Lm — Inductance
`0.3687` (default) | `scalar`

Physical inertia, viscous damping, static friction, mechanical — Inertia, damping, friction
`[0.001 0 0]` (default) | `vector`

Initial mechanical position, theta_init — Angular position
`0` (default) | `scalar`

Initial mechanical speed, omega_init — Angular speed
`0` (default) | `scalar`

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