Single-phase induction machine with SI or pu fundamental parameterization

**Library:**Simscape / Electrical / Electromechanical / Asynchronous

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_{qs}v_{ds} | Stator voltages in the d-q
representation |

i_{qs}i_{ds} | Stator currents in the d-q
representation |

Ψ_{qr}Ψ_{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 |

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:

${L}_{ms}={L}_{mas}=\frac{1}{{a}^{2}}{L}_{mbs}$

${R}_{ar}^{\text{'}}=\frac{1}{{a}^{2}}{R}_{br}^{\text{'}}$

${L}_{lar}^{\text{'}}=\frac{1}{{a}^{2}}{L}_{lbr}^{\text{'}}$

The voltage equations for the stator and rotor are:

${v}_{qs}={R}_{as}{i}_{qs}+\frac{d{\psi}_{qs}}{dt}$

${v}_{ds}={R}_{bs}{i}_{ds}+\frac{d{\psi}_{ds}}{dt}$

$0={R}_{ar}^{\text{'}}{i}_{qr}^{\text{'}}+\frac{d{\psi}_{qr}^{\text{'}}}{dt}-\frac{1}{a}{\omega}_{r}{\psi}_{dr}^{\text{'}}$

$0={a}^{2}{R}_{ar}^{\text{'}}{i}_{dr}^{\text{'}}+\frac{d{\psi}_{dr}^{\text{'}}}{dt}+a{\omega}_{r}{\psi}_{qr}^{\text{'}}$

${\psi}_{qs}=\left({L}_{las}+{L}_{ms}\right){i}_{qs}+{L}_{ms}{i}_{qr}^{\text{'}}$

${\psi}_{ds}=\left({L}_{lbs}+{a}^{2}{L}_{ms}\right){i}_{ds}+{a}^{2}{L}_{ms}{i}_{dr}^{\text{'}}$

${\psi}_{qr}^{\text{'}}=\left({L}_{lar}^{\text{'}}+{L}_{ms}\right){i}_{qr}^{\text{'}}+{L}_{ms}{i}_{qs}$

${\psi}_{dr}^{\text{'}}={a}^{2}\left({L}_{lar}^{\text{'}}+{L}_{ms}\right){i}_{dr}^{\text{'}}+{a}^{2}{L}_{ms}{i}_{ds}$

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

$T=p\left(a{\psi}_{qr}^{\text{'}}{i}_{dr}^{\text{'}}-\frac{1}{a}{\psi}_{dr}^{\text{'}}{i}_{qr}^{\text{'}}\right)$

The mechanical equation is

$J\frac{d{\omega}_{m}}{dt}=T-{T}_{L}-{B}_{m}{\omega}_{m},$

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:

${v}_{c}={R}_{s}{i}_{ds}+{{\displaystyle \int}}^{\text{}}\frac{1}{{C}_{s}}{\omega}_{r}{i}_{ds}.$

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

${v}_{ds}={R}_{bs}{i}_{ds}+\frac{d{\psi}_{ds}}{dt}-{R}_{s}{i}_{ds}-{{\displaystyle \int}}^{\text{}}\frac{1}{{C}_{s}}{\omega}_{r}{i}_{ds}.$

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

${v}_{ds}={R}_{bs}{i}_{ds}+\frac{d{\psi}_{ds}}{dt}-{R}_{r}{i}_{ds}-{{\displaystyle \int}}^{\text{}}\frac{1}{{C}_{r}}{\omega}_{r}{i}_{ds}.$

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**.

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 (Simscape).

[1] Krause, P. C. "Simulation of
Unsymmetrical 2-Phase Induction Machines." *IEEE Transactions on
Power Apparatus and Systems*. Vol 84, Number 11, 1965, pp.
1025-1037.

Induction Machine Direct Torque Control (Single-Phase) | Induction Machine Field-Oriented Control (Single-Phase)