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# Single-Phase Induction Motor Drive

Implement single-phase induction motor drive

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• Simscape / Electrical / Specialized Power Systems / Electric Drives / AC Drives ## Description

The Single-Phase Induction Motor Drive block models a vector-controlled single-phase machine drive. The drive configuration consists of a half-bridge rectifier, a divided DC bus with two filter capacitors, and a two-leg inverter that supplies the motor windings.

The single-phase induction machine (SPIM), without its startup and running capacitors, is treated as an asymmetric two-phase machine. The auxiliary and main windings are accessible and are in quadrature. This configuration provides good performances and operation in regenerating mode. ### Equations

The single-phase induction motor is asymmetrical due to the unequal resistances and inductances of the main and auxiliary windings. To obtain the mathematical model of the motor with constant parameters (voltage, current, and flux), it is necessary to transform all the variables to the stationary reference frame (d-q) fixed to the stator.

Mathematical Model

This diagram shows the mathematical model of the machine. Na and Nm represent the number of auxiliary and main stator windings, respectively.

#### Voltage

The equations that define the voltage for the model (in the stationary reference frame d-q) are:

`${V}_{qs}={R}_{s}{i}_{qs}+\frac{d{\varphi }_{qs}}{dt},$`

`${V}_{ds}={R}_{a}{i}_{ds}+\frac{d{\varphi }_{ds}}{dt},$`

`$0=R{\text{'}}_{r}i{\text{'}}_{qr}+\frac{d{\varphi }_{qr}^{\text{'}}}{dt}-\frac{1}{k}\cdot {\omega }_{r}{\varphi }_{qr}^{\text{'}},$`

`$0={k}^{2}R{\text{'}}_{r}i{\text{'}}_{dr}+\frac{d{\varphi }_{dr}^{\text{'}}}{dt}+k\cdot {\omega }_{r}{\varphi }_{dr}^{\text{'}},$`

and

`$k=\frac{{N}_{a}}{{N}_{m}},$`

where:

• Vqs is the q-axis stator voltage.

• Rs is the main stator resistance.

• iqs is the q-axis stator current.

• ϕqs is the q-axis stator flux linkage.

• Vds is the d-axis stator voltage.

• Ra is the auxiliary stator resistance.

• ids is the d-axis stator current.

• ϕds is the d-axis stator flux linkage.

• R'r is the rotor winding resistance referred to the main stator winding.

• i'qr is the q-axis rotor current referred to the main stator winding.

• ϕ'qr is the q-axis rotor flux linkage referred to the main stator winding.

• k is the turn ratio of Na to Nm.

• ωr is the rotor electrical angular velocity.

• i'dr is the d-axis rotor current referred to the main stator winding.

• ϕ'dr is the d-axis rotor flux linkage referred to the main stator winding.

• Na is the number of auxiliary stator windings.

• Nm is the number of main stator windings.

#### Flux

The equations that define the flux for the model (in the stationary reference frame d-q) are:

`${\varphi }_{qs}={L}_{ls}{i}_{qs}+{L}_{ms}\left({i}_{qs}+{i}_{qr}^{\text{'}}\right)$`

`${\varphi }_{ds}={L}_{la}{i}_{ds}+{k}^{2}{L}_{ms}\left({i}_{ds}+{i}_{dr}^{\text{'}}\right)$`

`${\varphi }_{qr}^{\text{'}}={L}_{lr}^{\text{'}}{i}_{qr}^{\text{'}}+{L}_{ms}\left({i}_{qs}+{i}_{qr}^{\text{'}}\right)$`

`${\varphi }_{dr}^{\text{'}}={k}^{2}{L}_{lr}^{\text{'}}{i}_{dr}^{\text{'}}+{k}^{2}{L}_{ms}\left({i}_{ds}+{i}_{dr}^{\text{'}}\right)$`

where:

• Lls is the leakage inductance of the main stator winding.

• Lla is the leakage inductance of the auxiliary stator winding.

• Lms is the magnetizing inductance of the main stator winding.

• L'lr is the leakage inductance of the rotor winding referred to the main stator winding.

The electromagnetic torque expressed as a function of the rotor flux linkages and currents is

`${T}_{e}=P\left(k\cdot {\varphi }_{qr}^{\text{'}}{i}_{dr}^{\text{'}}-\frac{1}{k}\cdot {\varphi }_{dr}^{\text{'}}{i}_{qr}^{\text{'}}\right)$`

where:

• Te is the electromagnetic torque.

• P is the number of pole pairs.

#### Field-Oriented Control

Using the stator currents and rotor flux linkages as state-space variables for the SPIM model, the electromagnetic torque equation is

`${T}_{e}=P\cdot \frac{{L}_{ms}}{{L}_{r}^{\text{'}}}\left(\frac{1}{k}\cdot {i}_{qs}{\varphi }_{dr}^{\text{'}}-k\cdot {i}_{ds}{\varphi }_{qr}^{\text{'}}\right).$`

Using the next change of variable,

`${i}_{qs}={k}^{2}\cdot {i}_{qs1}$`

and

`${i}_{ds}={i}_{ds1}.$`

Therefore, the electromagnetic torque equation can be rewritten as

`${T}_{e}=P\cdot \frac{{L}_{ms}}{{L}_{r}^{\text{'}}}k\left({\varphi }_{dr}^{\text{'}}{i}_{qs1}-{\varphi }_{qr}^{\text{'}}{i}_{ds1}\right).$`

In the indirect rotor flux-oriented control, the d-axis of the reference frame is oriented along the rotor flux linkage vector ϕ'r, then

`${\varphi }_{dr}^{\text{'}}={\varphi }_{r}^{\text{'}}$`

and

`${\varphi }_{qr}^{\text{'}}=0.$`

The electromagnetic torque results in

`${T}_{e}=P\cdot \frac{{L}_{ms}}{{L}_{r}^{\text{'}}}k\cdot {\varphi }_{r}^{\text{'}}{i}_{qs1}^{e}.$`

From here, the q-axis current component is

`${i}_{qs1}^{e}=\cdot \frac{{T}_{e}\cdot {L}_{r}^{\text{'}}}{k\cdot {L}_{ms}\cdot P\cdot {\varphi }_{r}^{\text{'}}}.$`

The resulting slip speed, ωs, is

`${\omega }_{s}=\frac{{k}^{3}\cdot {L}_{ms}\cdot {i}_{qs1}^{e}}{\left(\frac{{L}_{r}^{\text{'}}}{{R}_{r}^{\text{'}}}\right)\text{ }\text{ }\cdot {\varphi }_{r}^{\text{'}}}.$`

From here, the d-axis current component is

`${i}_{ds1}^{e}=\frac{{\varphi }_{r}^{\text{'}}}{{k}^{2}\cdot {L}_{ms}\text{ }}.$`

where the e superscript indicates that the variable is referred to the synchronous reference frame.

This block diagram shows field-oriented control. #### Direct-Torque Control

This type of control selects the voltage vector from a switching table to control the power switches in the inverter to obtain the required stator flux and corresponding motor torque. From the motor equations in the stationary reference frame d-q, estimate the stator flux and the torque:

`${\varphi }_{qs}=\int \left({V}_{qs}-{R}_{s}{i}_{qs}\right)\text{ }\text{ }\text{ }dt$`

and

`${\varphi }_{ds}=\int \left({V}_{ds}-{R}_{a}{i}_{ds}\right)\text{ }\text{ }\text{ }dt.$`

Assuming the approximation

`${L}_{la}={k}^{2}\cdot {L}_{ls},$`

and using the stator variables (flux linkages and currents) as state-space variables of the SPIM model, the electromagnetic torque is given by

`${T}_{e}=P\left(\frac{1}{k}\cdot {i}_{qs}{\varphi }_{ds}-k\cdot {i}_{ds}{\varphi }_{qs}\right).$`

Using the cross-product, the torque is

`${T}_{e}=P\left({\varphi }_{s}×{i}_{s}\right),$`

that is

`${T}_{e}=P|{\varphi }_{s}|\cdot |{\varphi }_{s}|\mathrm{sin}\delta ,$`

where:

• |ϕs| and |ϕr| are the magnitudes of the stator and rotor flux linkage space vectors, respectively.

• δ is the angle between the space vectors.

A change in the relative movement of ϕs and ϕr (defined by the angle, δ) affects the motor instantaneous torque. If the voltage drop on the stator resistance is omitted, the stator flux linkage directly depends on the inverter output voltage.

The next diagram shows the available voltage vectors, which correspond to possible inverter states, and the four distinct sectors in the d-q plane for a two-leg inverter. Selecting the appropriate inverter voltage vectors can directly change the magnitude of ϕs (flux control) and the rotating speed of ϕs (torque control) as shown in this diagram of sector 1. This block diagram shows direct-torque control. The estimated flux and torque are compared with the references using hysteresis control. The digitalized output variables and the stator flux position sector are used to select the appropriated voltage vector from the switching table. This table shows the appropriate voltage vector for the inverter where Hϕ and HTe are the output of the flux and torque hysteresis blocks

HϕHTeSector 1Sector 2Sector 3Sector 4
`1` (Flux is up)`1` (Torque is up)V1V2V3V4
`0` (Torque is down)V4V1V2V3
`0` (Flux is down)`1` (Torque is up)V2V3V4V1
`0` (Torque is down)V3V4V1V2

## Ports

### Input

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Speed set point, in rpm. The speed set point can be a step function, but the speed change rate follows the acceleration and deceleration ramps.

Data Types: `single` | `double`

Motor mechanical torque.

#### Dependencies

To enable this port, set the parameter to ```Torque Tm```.

Data Types: `single` | `double`

### Output

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The motor measurement bus. This bus allows you to use the Bus Selector block to observe variables for the motor. When the parameter is set to ```Single output busses```, this bus contains the three-phase converters measurement and the controller measurement signals.

Data Types: `bus`

Measurement signals of the DC bus voltage, the rectifier output current, and the inverter input current.

#### Dependencies

To enable this port, set the to ```Multiple output busses```.

Data Types: `bus`

This output allows you to observe the torque and the flux reference using the Bus Selector block.

#### Dependencies

To enable this port, set the to ```Multiple output busses```.

Data Types: `bus`

Mechanical motor speed.

#### Dependencies

To enable this port, set the parameter to ```Torque (Tm)```.

Data Types: `single` | `double`

The mechanical rotational port of the motor.

#### Dependencies

To enable this port, set the parameter to ```Mechanical rotational port```.

Data Types: `single` | `double`

### Conserving

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Phase A terminal of the motor drive.

Phase B terminal of the motor drive.

## Parameters

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Motor sample time.

Configure the bus ports.

#### Dependencies

Selecting `Multiple outport buses` enables the Conv and Ctrl outports.

Option to label bus signals.

Configure the mechanical input and output ports. If you select `Torque TM`, the output is the motor speed according to this differential equation, which describes the mechanical system dynamics:

`${T}_{e}=J\frac{d}{dt}{\omega }_{r}+F{\omega }_{r}+{T}_{m}.$`

where:

• Te is the electromagnetic torque.

• J is the inertia.

• ωr is the angular velocity of the rotor.

• F is the applied force.

• Tm is the mechanical torque.

If you select `Mechanical rotational port`, the connection port S acts as both the mechanical input and output port. It allows a direct connection to the Simscape™ environment. The mechanical system of the motor is also included in the drive and is based on the same differential equation.

#### Dependencies

Selecting `Torque TM` enables the Tm and Wm ports. Selecting `Mechanical rotational port` enables the S port.

### Motor

Electrical parameters > Nominal values

Motor nominal electrical power.

Motor nominal electrical voltage.

Motor nominal electrical frequency.

Electrical parameters > Equivalent circuit values
To see the equivalent circuit for the unsymmetrical SPIM, click .

Electrical parameters > Equivalent circuit values > Main winding stator

Resistance of the stator main winding.

Motor main winding stator equivalent circuit leakage inductance.

Mutual inductance of the stator main winding.

Electrical parameters > Equivalent circuit values > Auxiliary winding stator

Resistance of the stator auxiliary winding.

Leakage inductance of the stator auxiliary winding.

Mutual inductance of the stator auxiliary winding.

Electrical parameters > Equivalent circuit values > Auxiliary winding rotor

Resistance of the rotor main winding.

Leakage inductance of the rotor main winding.

Mechanical parameters

Motor inertia.

Motor friction factor.

Number of motor pole pairs.

Motor Initial speed.

### Converters and DC Bus

Rectifier > Snubbers

Resistance of the rectifier snubbers. To eliminate the snubbers from the model, specify `inf`.

Capacitance of the rectifier snubbers. To eliminate the snubbers from the model, specify `0`. For resistive snubbers, specify `inf`.

Rectifier > Diodes

Internal resistance of the rectifier diodes when they are conducting.

Voltage across the rectifier diodes when they are conducting.

DC Bus

Capacitance of the two capacitors in the DC bus

Braking Chopper

The braking chopper resistance is used to avoid bus overvoltage during motor deceleration or when the load torque accelerates the motor.

Braking chopper frequency.

The dynamic braking is activated when the bus voltage reaches the activation voltage value.

The dynamic braking is shut down when the bus voltage reaches the shutdown value.

Inverter > Switches

Internal resistance of the inverter IGBTs when they are conducting.

Inverter > Switches > Forward voltages (V)

Forward voltage of the inverter IGBTs

Forward voltage of the inverter diodes.

Inverter > Snubbers

Resistance of the inverter snubbers. To eliminate the snubbers from the inverter, specify `inf`.

Capacitance of the inverter snubbers. To eliminate the snubbers from the model, specify `0`. For resistive snubbers, specify `inf`.

### Controller

Type of regulation the controller is performing.

#### Dependencies

Selecting `Speed regulation` enables the Speed cutoff frequency (Hz), Speed ramps, and PI regulator parameters.

Speed controller

The sampling time must be a multiple of the simulation time step.

Cut-off frequency of the first-order low-pass filter of the speed controller.

#### Dependencies

Selecting `Speed regulation` for the Regulation type parameter enables this parameter.

Speed controller > Speed ramps (rpm/s)

Maximum change of speed allowed during motor acceleration in rpm/s. An excessively large positive value can cause DC bus undervoltage.

#### Dependencies

Selecting `Speed regulation` for the Regulation type parameter enables this parameter.

Maximum change of speed allowed during motor deceleration in rpm/s. An excessively large negative value can cause DC bus overvoltage.

#### Dependencies

Selecting `Speed regulation` for the Regulation type parameter enables this parameter.

Speed controller > PI regulator

Speed controller proportional gain.

#### Dependencies

Selecting `Speed regulation` for the Regulation type parameter enables this parameter.

Speed controller integral gain.

#### Dependencies

Selecting `Speed regulation` for the Regulation type parameter enables this parameter.

Speed controller > Torque output limits (N.m)

Maximum negative demanded torque applied to the motor by the current controller in N.m.

#### Dependencies

Selecting `Speed regulation` for the Regulation type parameter enables this parameter.

Maximum positive demanded torque applied to the motor by the current controller in N.m.

#### Dependencies

Selecting `Speed regulation` for the Regulation type parameter enables this parameter.

Machine flux

Initial motor flux.

Nominal motor flux.

Vector controller
To see schematics for field-oriented control and direct-torque control, click .

Type of control.

#### Dependencies

Selecting `FOC` enables the Current controller hysteresis bandwidth (A) parameter. Selecting ```DTC (two-level hysteresis)``` or ```DTC (three-level hysteresis)``` enables the Torque controller hysteresis bandwidth (N-m) and Flux controller hysteresis bandwidth (Wb) parameters.

Total bandwidth in Current control mode, distributed symmetrically around the current set point. This figure illustrates a case where the current set point is Is* and the current hysteresis bandwidth is set to dx. #### Dependencies

Selecting `FOC` for the Controller type parameter enables this parameter.

Total bandwidth distributed symmetrically around the torque set point. This figure shows a case where the torque set point is Te* and the torque hysteresis bandwidth is set to dTe. #### Dependencies

Selecting `DTC (two-level hysteresis)` or `DTC (three-level hysteresis)` for the Controller type parameter enables this parameter.

Total bandwidth distributed symmetrically around the flux set point. This figure shows a case where the flux set point is Ψ* and the torque hysteresis bandwidth is set to . #### Dependencies

Selecting `DTC (two-level hysteresis)` or `DTC (three-level hysteresis)` for the Controller type parameter enables this parameter.

Total bandwidth distributed symmetrically around the torque set point. This figure shows a case where the torque set point is Te* and the torque hysteresis bandwidth is set to .

#### Dependencies

Selecting `DTC (two-level hysteresis)` or `DTC (three-level hysteresis)` for the Controller type parameter enables this parameter.

Maximum inverter switching frequency.

The sampling time must be a multiple of the simulation time step.