# Single Phase Asynchronous Machine

Model dynamics of single phase asynchronous machine with squirrel-cage rotor

## Library

Simscape / Electrical / Specialized Power Systems / Electrical Machines

## Description

This machine has two windings: main and auxiliary. With the model, you can simulate the split-phase, the capacitor-start, the capacitor-start-capacitor-run, and main & auxiliary windings operation modes.

For the split-phase mode, the main and auxiliary windings are internally connected as follows:

For the capacitor-start mode, the main and auxiliary windings are internally connected as follows:

For the capacitor-start-capacitor-run mode, the main and auxiliary windings are internally connected as follows:

The electrical part of the machine is represented by a fourth-order state-space model and the mechanical part by a second-order system. All electrical variables and parameters are referred to the stator, indicated by the following prime signs in the machine equations. All stator and rotor quantities are in the stator reference frame (dq frame). The subscripts are defined in the following table.

Subscript

Definition

d

d axis quantity

q

q axis quantity

r

Referred to the main winding rotor quantity

R

Referred to the auxiliary winding rotor quantity

s

Main winding stator quantity

S

Auxiliary winding stator quantity

l

Leakage inductance

m

Magnetizing inductance

### Electrical System

 Vqs = Rsiqs + dφqs/dt φqs = Lssiqs + Lmsi'qr Vds = RSids + dφds/dt φds = LSSids + LmSi'dr V'qr = R'ri'qr + dφ'qr/dt – (Ns/NS)ωrφ'dr φ'qr = L'ri'qr + Lmsiqs V'dr = R'Ri'dr + dφ'dr/dt + (NS/Ns)ωrφ'qr where φ'dr = L'RRi'dr + LmSids Te = p[(NS/Ns)φ'qri'dr – (Ns/NS)φ'dri'qr] Lss = Lls + Lms LSS = LlS + LmS L'rr = L'lr + Lms L'RR = L'lR + LmS

### Mechanical System

`$\begin{array}{c}\frac{d}{dt}{\omega }_{m}=\frac{{T}_{e}-F{\omega }_{m}-{T}_{m}}{2H}\\ \frac{d}{dt}{\theta }_{m}={\omega }_{m}.\end{array}$`

Reference frame

The reference frame fixed in the stator converts voltages and currents to the dq frame.

The following relationships describe the ab-to-dq frame transformations applied to the single phase asynchronous machine.

`$\begin{array}{c}\left[\begin{array}{c}{f}_{qs}\\ {f}_{ds}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}{f}_{as}\\ {f}_{bs}\end{array}\right]\\ \left[\begin{array}{c}{f}_{qr}\\ {f}_{dr}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\left(\theta r\right)& -\mathrm{sin}\left(\theta r\right)\right)\\ -\mathrm{sin}\left(\theta r\right)& -\mathrm{cos}\left(\theta r\right)\end{array}\right]\left[\begin{array}{c}{f}_{ar}\\ {f}_{br}\end{array}\right].\end{array}$`

The variable f can represent either voltage, currents, or flux linkage.

The single phase asynchronous machine block parameters are defined as follows (all quantities are referred to the stator).

Parameter

Definition

Rs, Lls

Main winding stator resistance and leakage inductance

RS, LlS

Auxiliary winding stator resistance and leakage inductance

R′r, L′lr

Main winding rotor resistance and leakage inductance

R′R, L′lR

Auxiliary winding rotor resistance and leakage inductance. The two values are equal to the main winding rotor resistance and leakage inductances values, respectively.

Lms

Main winding magnetizing inductance

LmS

Auxiliary winding magnetizing inductance

Lss, L′rr

Total main winding stator and rotor inductances

LSS, L′RR

Total auxiliary winding stator and rotor inductances

Vas ,ias

Vbs ,ibs

Vqs, iqs

Main winding stator voltage and current

Auxiliary winding stator voltage and current

q axis stator voltage and current

V′qr, i′qr

q axis rotor voltage and current

Vds, ids

d axis stator voltage and current

V′dr, i′dr

d axis rotor voltage and current

ϕqs, ϕds

Stator q and d axis fluxes

ϕ′qr, ϕ′dr

Rotor q and d axis fluxes

ωm

Angular velocity of the rotor

Θm

Rotor angular position

p

Number of pole pairs

ωr

Electrical angular velocity (ωm x p)

Θr

Electrical rotor angular position (Θm x p)

Te

Electromagnetic torque

Tm

Shaft mechanical torque

J

F

Combined rotor and load inertia coefficient in (kg.m2). Set to infinite to simulate locked rotor.

Combined rotor and load viscous friction coefficient.

H

Combined rotor and load inertia constant in (s). Set to infinite to simulate locked rotor.

Ns

NS

Rst

Cs

Rrun

Crun

Number of main winding's effective turns.

Number of auxiliary winding's effective turns.

Capacitor-Start resistance

Capacitor-Start

Capacitor-Run resistance

Capacitor-Run

N

Ratio of number of auxiliary winding's effective turns and number of main winding's effective turns.

## Parameters

You can choose between two types of units to specify the electrical and mechanical parameters of the model, the per unit dialog box, and the SI dialog box. Both blocks are modeling the same machine. Depending on the dialog box that you use, Simscape™ Electrical™ Specialized Power Systems automatically converts the parameters that you specify into per unit parameters. The Simulink® model of the Single Phase Asynchronous Machine block uses per unit parameters.

### Configuration Tab

Mechanical input

Select the torque applied to the shaft as a Simulink input of the block, or to represent the machine shaft by a Simscape rotational mechanical port.

Select `Torque Tm` (default) to specify a torque input, in N.m or in pu, and change labeling of the block input to Tm. The machine speed is determined by the machine Inertia J (or inertia constant H for the pu machine) and by the difference between the applied mechanical torque Tm and the internal electromagnetic torque Te. The sign convention for the mechanical torque is when the speed is positive, a positive torque signal indicates motor mode and a negative signal indicates generator mode.

Select `Mechanical rotational port` to add to the block a Simscape mechanical rotational port that allows connection of the machine shaft with other Simscape blocks that have mechanical rotational ports. The Simulink input representing the mechanical torque Tm of the machine is then removed from the block.

The next figure indicates how to connect an Ideal Torque Source block from the Simscape library to the machine shaft to represent the machine in motor mode, or in generator mode, when the rotor speed is positive.

Units

Specify the per unit dialog box or the SI dialog box. Default is `SI`.

Type of machine

Specify one of the four types of single phase asynchronous machines: ```Split Phase``` (default), `Capacitor-Start`, `Capacitor-Start-Run`, or ```Main & auxiliary windings```.

Use signal names to identify bus labels

When this check box is selected, the measurement output uses the signal names to identify the bus labels. Select this option for applications that require bus signal labels to have only alphanumeric characters.

When this check box is cleared (default), the measurement output uses the signal definition to identify the bus labels. The labels contain nonalphanumeric characters that are incompatible with some Simulink applications.

### Parameters Tab

Nominal power, voltage, and frequency

The nominal apparent power Pn (VA), RMS Vn (V), and frequency fn (Hz). Default is `[ .25*746 110 60]`.

Main winding stator

The stator resistance Rs (Ω or pu) and leakage inductance Lls (H or pu). Default is `[2.02 7.4e-3]` (SI) and `[0.031135 0.042999]` (pu).

Main winding rotor

The rotor resistance Rr' (Ω or pu) and leakage inductance Llr' (H or pu), both referred to the stator. Default is `[4.12 5.6e-3]` (SI) and `[0.063502 0.03254]` (pu).

Main winding mutual inductance

The magnetizing inductance Lms (H or pu). Default is `0.1772` (SI) and `1.0296` (pu).

Auxiliary winding stator

The stator resistance RS (Ω or pu) and leakage inductance LlS (H or pu). Note that the Auxiliary winding rotor parameters are assumed to be equal to the main winding rotor resistance and leakage inductances values. Therefore it is not required to specify them in the dialog box. Default is ```[7.14 8.5e-3]``` (SI) and `[0.11005 0.049391]` (pu).

Inertia, friction factor, pole pairs, turn ratio (aux/main)

For the SI units dialog box: the combined machine and load inertia coefficient J (kg.m2), the combined viscous friction coefficient F (N.m.s), the number of pole pairs p and ratio of number of auxiliary winding's effective turns, and the number of main winding's effective turns. pu units dialog box: the inertia constant H (s), the combined viscous friction coefficient F (pu), and the number of pole pairs p. Default is ```[0.0146 0 2 1.18]``` (SI) and `[1.3907 0 2 1.18]` (pu).

Capacitor-Start

The start capacitance Cs (farad or pu) and capacitor series resistance Rst(Ω or pu). Default is `[2 254.7e-6]` (SI) and `[0.030826 6.2297]` (pu).

Capacitor-Run

The run capacitance Crun (farad or pu) and series resistance Rrun (farad or pu). Default is `[18 21.1e-6]` (SI) and `[0.27744 0.51608]` (pu).

Disconnection speed

Specifies the speed (%) when the auxiliary winding may be disconnected. Default is `75`.

Initial speed

Specifies the initial speed (%). Default is `0`.

Sample time (−1 for inherited)

Specifies the sample time used by the block. To inherit the sample time specified in the Powergui block, set this parameter to `−1` (default).

## Inputs and Outputs

`Tm`

The Simulink input of the block is the mechanical torque at the machine shaft. When you use the SI parameters mask, the input is a signal in N.m; otherwise it is in pu.

`m`

The Simulink output of the block is a vector containing measurement signals. You can demultiplex these signals by using the Bus Selector block provided in the Simulink library. Depending on the type of mask you use, the units are in SI, or in pu.

Name

Definition

Units

iar

Rotor current ir_a

A or pu

ibr

Rotor current ir_b

A or pu

iqr

Rotor current iq

A or pu

idr

Rotor current id

A or pu

phiqr

Rotor flux phir_q

V.s or pu

phidr

Rotor flux phir_d

V.s or pu

ias

Main winding stator current ia

A or pu

ibs

Auxiliary winding stator current ib

A or pu

phiqs

Stator flux phis_q(V.s)

V.s or pu

phids

Stator flux phis_d(V.s)

V.s or pu

vc

Voltage capacitor Vc

V or pu

w

Rotor speed

Te

Electromagnetic torque Te

N.m or pu

theta

Rotor angle thetam

The `power_singlephaseASM` example shows the use of the Single Phase Asynchronous Machine block in two modes of operation.