Induction Machine (Single-Phase)

Single-phase induction machine with SI or pu fundamental parameterization

Library:
Simscape / Electrical / Electromechanical / Asynchronous

Description

The Induction Machine (Single-Phase) block represents a single-phase induction machine with a squirrel cage rotor with fundamental parameters expressed in per-unit or in the International System of Units (SI).

Each of four model variants for the Induction Machine (Single-Phase) block corresponds to a Block choice option. To access the block choices, in the model window, right-click the block, and then use either of these methods:

From the context menu, select Simscape > Block choices.
In the Simulink^® Editor, on the Modeling tab, in the Design section, in the Data Management category, select Property Inspector. In the Property Inspector window, select a Block Choice.

The model variant options are:

Split-phase
Capacitor-start
Capacitor-start-capacitor-run
Main and auxiliary windings

The single phase induction machine consists of a squirrel cage rotor and two stator windings, the main and the auxiliary windings. The auxiliary winding is typically only active during startup. However, to improve performance, the auxiliary winding can be active during running for low-power applications. The figure shows the equivalent d- and q-axis circuits for the main and auxiliary windings.

The table defines the variables.

Variable	Definition
v_qsv_ds	Stator voltages in the d-q representation
i_qsi_ds	Stator currents in the d-q representation
Ψ_qrΨ_dr	Rotor fluxes in the d-q representation
⍵	Rotor electrical speed
a	Auxiliary/main windings turn ratio
R_as	Main winding stator resistance
L_as	Main winding stator leakage inductance
R'_ar	Main winding rotor resistance
L'_lar	Main winding rotor leakage inductance
L_mas	Main winding mutual inductance
R_bs	Auxiliary winding stator resistance
R_lbs	Auxiliary winding stator leakage inductance
R'_br	Auxiliary winding rotor resistance
L'_lbr	Auxiliary winding rotor leakage inductance
L_mbs	Auxiliary winding mutual inductance

Equations

The SI model converts the SI values that you enter in the dialog box to per-unit values for simulation. For information on the relationship between SI and per-unit machine parameters, see Per-Unit Conversion for Machine Parameters. For information on per-unit parameterization, see Per-Unit System of Units.

These transformations reduce the rotor resistance, rotor leakage inductance, and stator mutual inductance to a single set of values by defining the equivalent circuits with respect to the main winding:

$L_{m s} = L_{m a s} = \frac{1}{a^{2}} L_{m b s}$

$R_{a r}^{'} = \frac{1}{a^{2}} R_{b r}^{'}$

$L_{l a r}^{'} = \frac{1}{a^{2}} L_{l b r}^{'}$

The voltage equations for the stator and rotor are:

$v_{q s} = R_{a s} i_{q s} + \frac{d ψ_{q s}}{d t}$

$v_{d s} = R_{b s} i_{d s} + \frac{d ψ_{d s}}{d t}$

$0 = R_{a r}^{'} i_{q r}^{'} + \frac{d ψ_{q r}^{'}}{d t} - \frac{1}{a} ω_{r} ψ_{d r}^{'}$

$0 = a^{2} R_{a r}^{'} i_{d r}^{'} + \frac{d ψ_{d r}^{'}}{d t} + a ω_{r} ψ_{q r}^{'}$

$ψ_{q s} = (L_{l a s} + L_{m s}) i_{q s} + L_{m s} i_{q r}^{'}$

$ψ_{d s} = (L_{l b s} + a^{2} L_{m s}) i_{d s} + a^{2} L_{m s} i_{d r}^{'}$

$ψ_{q r}^{'} = (L_{l a r}^{'} + L_{m s}) i_{q r}^{'} + L_{m s} i_{q s}$

$ψ_{d r}^{'} = a^{2} (L_{l a r}^{'} + L_{m s}) i_{d r}^{'} + a^{2} L_{m s} i_{d s}$

The expression for the electromagnetic torque, T, is obtained by applying the principle of virtual displacement [1].

$T = p (a ψ_{q r}^{'} i_{d r}^{'} - \frac{1}{a} ψ_{d r}^{'} i_{q r}^{'})$

The mechanical equation is

$J \frac{d ω_{m}}{d t} = T - T_{L} - B_{m} ω_{m},$

where ω_m is the rotor angular velocity.

In split-phase machines, the auxiliary winding is displaced at 90 electrical degrees from the main winding and operates only until the speed reaches the disconnection speed, which is typically 70 to 80 percent of rated speed. In this configuration, the auxiliary winding has high resistance and small reactance compared to the main winding. The resulting phase difference makes the machine behave like a two-phase machine.

The capacitor-start machine is a type of split-phase machine that uses a capacitor in series with the auxiliary winding to start the machine. In this configuration, auxiliary and main windings have the same number of turns. The value of the capacitor ensures that the current in the auxiliary coil leads the current in the main winding by approximately 80 electrical degrees.

The capacitor-start voltage is:

$v_{c} = R_{s} i_{d s} + \int^{} \frac{1}{C_{s}} ω_{r} i_{d s} .$

When a capacitor is connected in series with the auxiliary winding, the voltage equation for the d-axis is

$v_{d s} = R_{b s} i_{d s} + \frac{d ψ_{d s}}{d t} - R_{s} i_{d s} - \int^{} \frac{1}{C_{s}} ω_{r} i_{d s} .$

The extension is obtained immediately for capacitor-start-capacitor-run machines. In this configuration, two capacitors are connected in parallel.

The d-axis voltage after disconnecting the capacitor-start is

$v_{d s} = R_{b s} i_{d s} + \frac{d ψ_{d s}}{d t} - R_{r} i_{d s} - \int^{} \frac{1}{C_{r}} ω_{r} i_{d s} .$

Display Option

To display the machine per-unit base values in the MATLAB^® Command Window, right-click the block and, from the Electrical menu, select Display Base Values.

Variables

Use the Variables settings to specify the priority and initial target values for the block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables.

Ports

Output

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`pu` — Machine per-unit measurements
physical signal | vector

Physical signal vector port associated with the machine per-unit measurements. The vector elements are:

pu_torque
pu_velocity
pu_vds
pu_vqs
pu_v0s = 0
pu_ids
pu_iqs
pu_i0s = 0

Conserving

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`a1` — Main winding positive terminal
electrical | scalar

Electrical conserving port associated with the main winding positive terminal.

`a2` — Main winding negative terminal
electrical | scalar

Electrical conserving port associated with the main winding negative terminal.

`b1` — Auxiliary winding positive terminal
electrical | scalar

Electrical conserving port associated with the auxiliary winding positive terminal.

Dependencies

This port is visible only if you set the Block choice to Main and auxiliary windings.

`b2` — Auxiliary winding negative terminal
electrical | scalar

Electrical conserving port associated with the auxiliary winding negative terminal.

Dependencies

This port is visible only if you set the Block choice to Main and auxiliary windings.

`R` — Machine rotor
mechanical rotational

Mechanical rotational conserving port associated with the machine rotor.

`C` — Machine case
mechanical rotational

Mechanical rotational conserving port associated with the machine case.

Parameters

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Main

`Rated apparent power` — Rated apparent power
`180` `V*A` (default) | positive scalar

Rated apparent power of the induction machine.

`Rated voltage` — Rated voltage
`220` `V` (default) | positive scalar

RMS voltage.

`Rated electrical frequency` — Rated electrical frequency
`60` `Hz` (default) | positive scalar

Nominal electrical frequency corresponding to the rated apparent power.

`Number of pole pairs` — Number of pole pairs
`2` (default) | positive integer

Number of machine pole pairs. The value is used as a back-electromotive force constant.

`Parameterization unit` — SI or per-unit parameterization option
`SI` (default) | `Per unit`

Unit system for block parameterization.

Dependencies

Selecting:

SI exposes SI Electrical parameters.
Per unit exposes per-unit Electrical parameters.

Electrical

For the Parameterization unit parameter in the Main settings, select SI to expose the SI Electrical parameters, or Per unit to expose the per-unit Electrical parameters.

`Main winding stator resistance, Ras` — Main winding stator resistance
`1.8180` `Ohm` (default) | positive scalar

SI stator resistance for the main winding.

Dependencies

Selecting SI for the Parameterization unit parameter in the Main settings exposes SI Electrical parameters.

`Main winding stator leakage inductance, Llas` — Main winding stator leakage inductance
`0.0067` `H` (default) | positive scalar

SI leakage inductance for the main winding.

Dependencies

Selecting SI for the Parameterization unit parameter in the Main settings exposes SI Electrical parameters.

`Main winding rotor resistance, Rar'` — Main winding rotor resistance
`3.7080` `Ohm` (default) | positive scalar

SI rotor resistance for the main winding.

Dependencies

Selecting SI for the Parameterization unit parameter in the Main settings exposes SI Electrical parameters.

`Main winding rotor leakage inductance, Llar'` — Main winding rotor leakage inductance
`0.0050` `H` (default) | positive scalar

SI rotor leakage inductance for the main winding.

Dependencies

Selecting SI for the Parameterization unit parameter in the Main settings exposes SI Electrical parameters.

`Main winding mutual inductance, Lms` — Main winding mutual inductance
`0.1595` `H` (default) | positive scalar

SI mutual inductance for the main winding.

Dependencies

Selecting SI for the Parameterization unit parameter in the Main settings exposes SI Electrical parameters.

`Auxiliary winding stator resistance, Rbs` — Auxiliary winding stator resistance
`6.4260` `Ohm` (default) | positive scalar

SI stator resistance for the auxiliary winding.

Dependencies

Selecting SI for the Parameterization unit parameter in the Main settings exposes SI Electrical parameters.

`Auxiliary winding stator leakage inductance, Llbs` — Auxiliary winding stator leakage inductance
`0.0077` `H` (default) | positive scalar

SI stator leakage inductance for the auxiliary winding.

Dependencies

Selecting SI for the Parameterization unit parameter in the Main settings exposes SI Electrical parameters.

`Auxiliary/main windings turn ratio` — Turn ratio
`1.1` (default) | positive scalar

Winding ratio between the main and auxiliary winding.