Induction Motor

Three-phase induction motor

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.

Three-phase induction motor currents and magnetic fields

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

expand all

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

expand all

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

expand all

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`

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 R2017a

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`

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

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

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`

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™.