# Induction Motor

## Library

Rotational Actuators

## Description

The Induction Motor block represents the electrical and torque characteristics of an induction motor powered by an ideal AC supply. The following figure shows the equivalent circuit model of the Induction Motor block.

In the figure:

• R1 is the stator resistance.

• R2 is the rotor resistance with respect to the stator.

• L1 is the stator inductance.

• L2 is the rotor inductance with respect to the stator.

• Lm is magnetizing inductance.

• s is the rotor slip.

• $\overline{V}$ and $\overline{I}$ are the sinusoidal supply voltage and current phasors.

Rotor slip s is defined in terms of the mechanical rotational speed ${\omega }_{m}$, the number of pole pairs p, and the electrical supply frequency ω by

`$s=1-\frac{p{\omega }_{m}}{\omega }$`

This means that the slip is one when starting, and zero when running synchronously with the supply frequency.

For an n-phase induction motor the torque-speed relationship is given by:

`$T=\frac{np{R}_{2}}{s\omega }\frac{{V}_{rms}{}^{2}}{{\left({R}_{1}+{R}_{2}+\frac{1-s}{s}{R}_{2}\right)}^{2}+{\left({X}_{1}+{X}_{2}\right)}^{2}}$`

where:

• Vrms is the line-neutral supply voltage for a star-configuration induction motor, and the line-to-line voltage for a delta-configuration induction motor.

• n is the number of phases.

You can parameterize this block in terms of the preceding equivalent circuit model parameters or in terms of the motor ratings the block uses to derive these parameters.

This block produces a positive torque acting from the mechanical C to R ports.

### Thermal Ports

The block has two optional thermal ports, one per winding, hidden by default. To expose the thermal ports, right-click the block in your model, and then from the context menu select Simscape > Block choices > Show thermal port. This action displays the thermal ports on the block icon, and adds the Temperature Dependence and Thermal Port tabs to the block dialog box. These tabs are described further on this reference page.

Use the thermal ports to simulate the effects of copper resistance losses that convert electrical power to heat. For more information on using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

## Basic Assumptions and Limitations

The model is based on the following assumptions:

• The block does not model the starting mechanism for a single-phase induction motor.

• When you parameterize the block by motor ratings, the block derives the equivalent circuit model parameters by assuming that the effect of the magnetizing inductance Lm is negligible, and the magnetizing inductance is not included in the simulated component.

## Dialog Box and Parameters

### Electrical Torque Tab

Model parameterization

Select one of the following methods for block parameterization:

• `By motor ratings` — Provide electrical torque parameters that the block converts to an equivalent circuit model of the motor assuming that the effect of the magnetizing inductance Lm is negligible. This is the default method.

• `By equivalent circuit parameters` — Provide electrical parameters for an equivalent circuit model of the motor.

Stator resistance R1

Resistance of the stator winding. The default value is `1` Ω. This parameter is only visible when you select ```By equivalent circuit parameters``` for the Model parameterization parameter.

Rotor resistance R2

Resistance of the rotor, specified with respect to the stator. The default value is `1` Ω. This parameter is only visible when you select ```By equivalent circuit parameters``` for the Model parameterization parameter.

Stator inductance L1

Inductance of the stator winding. The default value is `0.02` H. This parameter is only visible when you select ```By equivalent circuit parameters``` for the Model parameterization parameter.

Rotor inductance L2

Inductance of the rotor, specified with respect to the stator. The default value is `0.02` H. This parameter is only visible when you select `By equivalent circuit parameters` for the Model parameterization parameter.

Magnetizing inductance Lm

Magnetizing inductance of the stator. This parameter is only visible when you select `By equivalent circuit parameters` for the Model parameterization parameter. Its value is hard to estimate from motor parameters, but the effect is usually small. If you do not know its value, use a typical value of 25 times the Stator inductance L1 value. The default value is `0.5` H.

Rated mechanical power

Mechanical power the motor delivers when running at the rated speed. The default value is `825` W. This parameter is only visible when you select `By motor ratings` for the Model parameterization parameter.

Rated speed

Speed at which the motor delivers the specified Rated mechanical power value. The default value is `3.5e+03` rpm. This parameter is only visible when you select ```By motor ratings``` for the Model parameterization parameter.

Rated RMS line-to-line voltage

Line-to-line voltage at which the motor ratings are specified. The default value is `200` V. This parameter is only visible when you select `By motor ratings` for the Model parameterization parameter.

Rated supply frequency

Frequency of the AC supply voltage at which the motor ratings are specified. The default value is `60` hertz. This parameter is only visible when you select `By motor ratings` for the Model parameterization parameter.

Rated RMS line current

Line current at which the motor delivers the specified Rated mechanical power value. The default value is `2.7` A. This parameter is only visible when you select ```By motor ratings``` for the Model parameterization parameter.

R1 parameterization

Select one of the following parameterizations for the equivalent circuit resistance, R1, of the motor:

• `From motor efficiency` — Calculate R1 from the motor efficiency. This is the default method.

• `From power factor` — Calculate R1 from the motor power factor.

• `Use measured stator resistance R1` — Measure R1 directly.

This parameter is only visible when you select ```By motor ratings``` for the Model parameterization parameter.

Motor efficiency (percent)

the percentage of input power to the motor that gets delivered to the mechanical load when running at the Rated speed value. The default value is `95`. This parameter is only visible when you select `By motor ratings` for the Model parameterization parameter and ```From motor efficiency``` for the R1 parameterization parameter.

Motor power factor

The cosine of the angle by which the supply current lags the supply voltage when running at the Rated mechanical power value. The default value is `0.93`. This parameter is only visible when you select `By motor ratings` for the Model parameterization parameter and ```From power factor``` for the R1 parameterization parameter.

Measured stator resistance R1

the measured stator resistance. The default value is `1` Ω. This parameter is only visible when you select ```By motor ratings``` for the Model parameterization parameter and `Use measured stator resistance R1` for the R1 parameterization parameter.

Number of pole pairs

Total number of pole pairs for the motor. The default value is `1`.

Number of phases

Number of supply phases. The default value is `3`.

Stator connections

Select one of the following motor configurations:

• `Delta configuration` — Connect the motor stator windings in delta configuration. This is the default method.

• `Star configuration` — Connect the motor stator windings in star configuration.

### Power Supply Tab

Supply RMS line-to-line voltage

The line-to-line voltage that supplies the motor. The default value is `200` V.

Supply frequency

Frequency of the AC supply voltage. The default value is `60` hertz.

### Mechanical Tab

Rotor inertia

Rotor inertia. The default value is `0.1` kg*m2. The value can be zero.

Rotor damping

Rotor damping. The default value is `2e-06` N*m/(rad/s). The value can be zero.

Initial rotor speed

Speed of the rotor at the start of the simulation. The default value is `0` rpm.

### Temperature Dependence Tab

This tab appears only for blocks with exposed thermal ports. For more information, see Thermal Ports.

Resistance temperature coefficients, [alpha_1 alpha_2]

A 1 by 2 row vector defining the coefficient α in the equation relating resistance to temperature, as described in Thermal Model for Actuator Blocks. The first element corresponds to the stator, and the second to rotor. The default value is for copper, and is ```[ 0.00393 0.00393 ]``` 1/K.

Measurement temperature

The temperature for which motor parameters are defined. The default value is `25` °C.

### Thermal Port Tab

This tab appears only for blocks with exposed thermal ports. For more information, see Thermal Ports.

Thermal masses, [M_1 M_2]

A 1 by 2 row vector defining the thermal mass for the stator and rotor windings. The thermal mass is the energy required to raise the temperature by one degree. The default value is ```[ 100 100 ]``` J/K.

Initial temperatures, [T_A T_B]

A 1 by 2 row vector defining the temperature of the stator and rotor thermal ports at the start of simulation. The default value is `[ 25 25 ]` °C.

## Ports

The block has the following ports:

`W`

Real power.

`wm`

Mechanical speed.

`VAR`

Imaginary power.

`s`

Motor slip.

`C`

Mechanical rotational conserving port.

`R`

Mechanical rotational conserving port.

`H1`

`H2`

## References

[1] S.E. Lyshevski. Electromechanical Systems, Electric Machines, and Applied Mechatronics, CRC, 1999.