Documentation

# Induction Machine Squirrel Cage

Squirrel-cage-rotor induction machine with per-unit or SI parameterization

• Library:
• Simscape / Electrical / Electromechanical / Asynchronous

## Description

The Induction Machine Squirrel Cage block models a squirrel-cage-rotor induction machine with fundamental parameters expressed in per-unit or in the International System of Units (SI). A squirrel-cage-rotor induction machine is a type of induction machine. All stator connections are accessible on the block. Therefore, you can model soft-start regimes using a switch between wye and delta configurations. If you need access to the rotor windings, use the Induction Machine Wound Rotor block instead.

Connect port ~1 to a three-phase circuit. To connect the stator in delta configuration, connect a Phase Permute block between ports ~1 and ~2. To connect the stator in wye configuration, connect port ~2 to a Grounded Neutral (Three-Phase) or a Floating Neutral (Three-Phase) block.

### Induction Machine Initialization Using Load-Flow Target Values

If the block is in a network that is compatible with the frequency-time simulation mode, you can perform a load-flow analysis on the network. A load-flow analysis provides steady-state values that you can use to initialize the machine.

For more information, see Perform a Load-Flow Analysis Using Simscape Electrical and Frequency and Time Simulation Mode (Simscape). For an example that shows how initialize an induction machine using data from a load flow analysis, see Induction Motor Initialization with Loadflow.

### Equations

For the SI implementation, the block 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.

The induction machine equations are expressed with respect to a synchronous reference frame, defined by

`${\theta }_{e}\left(t\right)=\underset{0}{\overset{t}{\int }}2\pi {f}_{rated}dt,$`

where frated is the value of the Rated electrical frequency parameter.

The Park transformation maps stator equations to a reference frame that is stationary with respect to the rated electrical frequency. The Park transformation is defined by

`${P}_{s}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\theta }_{e}& \text{cos}\left({\theta }_{e}-\frac{2\pi }{3}\right)& \text{cos}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ -\mathrm{sin}{\theta }_{e}& -\text{sin}\left({\theta }_{e}-\frac{2\pi }{3}\right)& -\text{sin}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right],$`

where θe is the electrical angle.

The Park transformation is used to define the per-unit induction machine equations. The stator voltage equations are defined by

`${v}_{ds}=\frac{1}{{\omega }_{base}}\frac{d{\psi }_{ds}}{dt}-\omega {\psi }_{qs}+{R}_{s}{i}_{ds},$`

`${v}_{qs}=\frac{1}{{\omega }_{base}}\frac{d{\psi }_{qs}}{dt}+\omega {\psi }_{ds}+{R}_{s}{i}_{qs},$`

and

`${v}_{0s}=\frac{1}{{\omega }_{base}}\frac{d{\psi }_{0s}}{dt}+{R}_{s}{i}_{0s},$`

where:

• vds, vqs, and v0s are the d-axis, q-axis, and zero-sequence stator voltages, defined by

$\left[\begin{array}{c}{v}_{ds}\\ {v}_{qs}\\ {v}_{0s}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$

va, vb, and vc are the stator voltages across ports ~1 and ~2.

• ωbase is the per-unit base electrical speed.

• ψds, ψqs, and ψ0s are the d-axis, q-axis, and zero-sequence stator flux linkages.

• Rs is the stator resistance.

• ids, iqs, and i0s are the d-axis, q-axis, and zero-sequence stator currents defined by

$\left[\begin{array}{c}{i}_{ds}\\ {i}_{qs}\\ {i}_{0s}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$

ia, ib, and ic are the stator currents flowing from port ~1 to port ~2.

The rotor voltage equations are defined by

`${v}_{dr}=\frac{1}{{\omega }_{base}}\frac{d{\psi }_{dr}}{dt}-\left(\omega -{\omega }_{r}\right){\psi }_{qr}+{R}_{rd}{i}_{dr}=0$`

and

`${v}_{qr}=\frac{1}{{\omega }_{base}}\frac{d{\psi }_{qr}}{dt}+\left(\omega -{\omega }_{r}\right){\psi }_{dr}+{R}_{rd}{i}_{qr}=0,$`

where:

• vdr and vqr are the d-axis and q-axis rotor voltages.

• ψdr and ψqr are the d-axis and q-axis rotor flux linkages.

• ω is the per-unit synchronous speed. For a synchronous reference frame, the value is 1.

• ωr is the per-unit mechanical rotational speed.

• Rrd is the rotor resistance referred to the stator.

• idr and iqr are the d-axis and q-axis rotor currents.

The stator flux linkage equations are defined by

`${\psi }_{ds}={L}_{ss}{i}_{ds}+{L}_{m}{i}_{dr},$`

`${\psi }_{qs}={L}_{ss}{i}_{qs}+{L}_{m}{i}_{qr},$`

and

`${\psi }_{0s}={L}_{ss}{i}_{0s},$`

where Lss is the stator self-inductance and Lm is the magnetizing inductance.

The rotor flux linkage equations are defined by

`${\psi }_{dr}={L}_{rrd}{i}_{dr}+{L}_{m}{i}_{ds}$`

and

`${\psi }_{qr}={L}_{rrd}{i}_{qr}+{L}_{m}{i}_{qs},$`

where Lrrd is the rotor self-inductance referred to the stator.

The rotor torque is defined by

`$T={\psi }_{ds}{i}_{qs}-{\psi }_{qs}{i}_{ds}.$`

The stator self-inductance Lss, stator leakage inductance Lls, and magnetizing inductance Lm are related by

`${L}_{ss}={L}_{ls}+{L}_{m}.$`

The rotor self-inductance Lrrd, rotor leakage inductance Llrd, and magnetizing inductance Lm are related by

`${L}_{rrd}={L}_{lrd}+{L}_{m}.$`

When a saturation curve is provided, the equations to determine the saturated magnetizing inductance as a function of magnetizing flux are:

`${L}_{m_sat}=f\left({\psi }_{m}\right)$`

`${\psi }_{m}=\sqrt{{\psi }_{dm}^{2}+{\psi }_{qm}^{2}}$`

For no saturation, the equation reduces to

`${L}_{m_sat}={L}_{m}$`

### Plotting and Display Options

You can perform plotting and display actions using the Electrical menu on the block context menu.

Right-click the block and, from the Electrical menu, select an option:

• Display Base Values — Displays the machine per-unit base values in the MATLAB® Command Window.

• Plot Torque Speed (SI) — Plots torque versus speed, both measured in SI units, in a MATLAB figure window using the current machine parameters.

• Plot Torque Speed (pu) — Plots torque versus speed, both measured in per-unit, in a MATLAB figure window using the current machine parameters.

• Plot Open-Circuit Saturation — Plots terminal voltage versus no-load stator current, both in per-unit, in a MATLAB figure window. The plot contains three traces:

• Unsaturated — Stator magnetizing inductance (unsaturated).

• Saturated — Open-circuit lookup table (v versus i) you specify.

• Derived — Open-circuit lookup table derived from the per-unit open-circuit lookup table (v versus i) you specify. This data is used to calculate the saturated magnetizing inductance, Lm_sat, and the saturation factor, Ks, versus magnetic flux linkage, ψm, characteristics.

• Plot Saturation Factor — Plots saturation factor, Ks, versus magnetic flux linkage, ψm, in a MATLAB figure window using the machine parameters. This parameter is derived from other parameters that you specify:

• No-load stator current saturation data, i

• Terminal voltage saturation data, v

• Leakage inductance, Lls

• Plot Saturated Inductance — Plots magnetizing inductance, Lm_sat, versus magnetic flux linkage, ψm, in a MATLAB figure window using the machine parameters. This parameter is derived from other parameters that you specify:

• No-load stator current saturation data, i

• Terminal voltage saturation data, v

• Leakage inductance, Lls

For the SI implementation, v is in V (phase-phase RMS) and i is in A (rms).

### 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 (Simscape).

The type of variables that are visible in the Variables settings depends on the initialization method that you select, in the Main settings, for the Initialization option parameter. To specify target values using:

• Flux variables — Set the Initialization option parameter to `Set targets for flux variables`.

• Data from a load-flow analysis — Set the Initialization option parameter to ```Set targets for load flow variables```.

If you select `Set targets for load flow variables`, to fully specify the initial condition, you must include an initialization constraint in the form of a high-priority target value. For example, if your induction machine is connected to an Inertia block, the initial condition for the induction machine is completely specified if, in the Variables settings of the Inertia block, the Priority for Rotational velocity is set to `High`. Alternatively, you could set the Priority to `None` for the Inertia block Rotational velocity, and instead set the Priority for the induction machine block Slip, Real power generated, or Mechanical power consumed to `High`.

## Ports

### Input

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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

• pu_ids

• pu_iqs

• pu_i0s

### Conserving

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Mechanical rotational conserving port associated with the machine rotor.

Mechanical rotational conserving port associated with the machine case.

Expandable three-phase port associated with the stator positive-end connections.

Expandable three-phase port associated with the stator negative-end connections.

## Parameters

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All default parameter values are based on a machine delta-winding configuration.

### Main

Rated apparent power of the induction machine.

RMS line-line voltage.

Nominal electrical frequency corresponding to the rated apparent power.

Number of machine pole pairs.

Unit system for block parameterization. Choose between `SI`, the international system of units, and `Per unit`, the per-unit system.

#### Dependencies

Selecting:

• `SI` exposes SI parameters in the Impedances and Saturation settings.

• `Per unit` exposes per-unit parameters in the Impedances and Saturation settings.

Option to specify a single or double squirrel cage for the machine.

Setting this parameter to `Double squirrel cage` exposes Impedances parameters for the second cage.

Zero-sequence model:

• `Include` — Prioritize model fidelity. An error occurs if you Include zero-sequence terms for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver (Simscape).

• `Exclude` — Prioritize simulation speed for desktop simulation or real-time deployment.

#### Dependencies

If this parameter is set to:

• `Include` and Parameterization unit is set to `SI` — The Stator zero-sequence reactance, X0 parameter in the Impedances settings is visible.

• `Include` and Parameterization unit is set to `Per unit` — The Stator zero-sequence inductance, pu_L0 (pu) parameter in the Impedances settings is visible.

• `Exclude` — The stator zero-sequence parameter in the Impedances settings is not visible.

Initialization method. You can initialize a machine for steady-state simulation using either flux data or data from a load-flow analysis.

If you select `Set targets for load flow variables`, to fully specify the initial condition, you must include an initialization constraint in the form of a high-priority target value. For example, if your induction machine is connected to an Inertia block, the initial condition for the induction machine is completely specified if, in the Variables settings of the Inertia block, the Priority for Rotational velocity is set to `High`. Alternatively, you could set the Priority to `None` for the Inertia block Rotational velocity, and instead set the Priority for the induction machine block Slip, Real power generated, or Mechanical power consumed to `High`.

#### Dependencies

The type of variables that are visible in the Variables settings depends on the initialization method that you select, in the Main settings, for the Initialization option parameter. To specify target values using:

• Flux variables — Set the Initialization option parameter to `Set targets for flux variables`.

• Data from a load-flow analysis — Set the Initialization option parameter to ```Set targets for load flow variables```.

### Impedances

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

Stator resistance.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI`.

Stator leakage reactance.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI`.

Rotor resistance referred to the stator.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI` and the Squirrel cage parameter in the Main setting to `Single squirrel cage`.

Rotor leakage reactance referred to the stator.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI` and the Squirrel cage parameter in the Main setting to `Single squirrel cage`.

Rotor resistance in cage 1.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI` and the Squirrel cage parameter in the Main setting to `Double squirrel cage`.

Rotor leakage reactance in cage 1.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI` and the Squirrel cage parameter in the Main setting to `Double squirrel cage`.

Rotor resistance in cage 2.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI` and the Squirrel cage parameter in the Main setting to `Double squirrel cage`.

Rotor leakage reactance in cage 2.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI` and the Squirrel cage parameter in the Main setting to `Double squirrel cage`.

Magnetizing reactance.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI`.

Stator zero-sequence reactance.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to `SI` and the Zero sequence parameter in the Main setting to `Include`.

Per-unit stator resistance.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit```.

Per-unit stator leakage inductance.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit```.

Per-unit rotor resistance referred to the stator.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit``` and the Squirrel cage parameter in the Main setting to `Single squirrel cage`.

Per-unit rotor leakage inductance referred to the stator.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit``` and the Squirrel cage parameter in the Main setting to `Single squirrel cage`.

Per-unit rotor resistance in cage 1.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit``` and the Squirrel cage parameter in the Main setting to `Double squirrel cage`.

Per-unit rotor leakage inductance in cage 1.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit``` and the Squirrel cage parameter in the Main setting to `Double squirrel cage`.

Per-unit rotor resistance in cage 2.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit``` and the Squirrel cage parameter in the Main setting to `Double squirrel cage`.

Per-unit rotor leakage inductance in cage 2.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit``` and the Squirrel cage parameter in the Main setting to `Double squirrel cage`.

Per-unit magnetizing inductance, that is, the peak value of stator-rotor mutual inductance.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit```.

Per-unit stator zero-sequence inductance.

#### Dependencies

This parameter is visible only if you set the Parameterization unit parameter in the Main setting to ```Per unit``` and the Zero sequence parameter in the Main setting to `Include`.

### Saturation

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

Block magnetic saturation representation.

#### Dependencies

If this parameter is set to ```Open-circuit lookup table (v versus i)```, related parameters are visible.

Current i data populates the voltage v versus field current i lookup table. This parameter must contain a vector with at least 10 elements.

#### Dependencies

This parameter is visible only if the Magnetic saturation representation parameter is set to ```Open-circuit lookup table (v versus i)``` and, in the Main settings, the Parameterization unit parameter is set to `SI`.

Terminal voltage v data populates the voltage v versus current i lookup table. This parameter must contain a vector with at least 10 elements. The number of elements must match the number of elements in the vector for the No-load stator current saturation data, i (rms) parameter.

#### Dependencies

This parameter is visible only if the Magnetic saturation representation parameter is set to ```Open-circuit lookup table (v versus i)``` and, in the Main settings, the Parameterization unit parameter is set to `SI`.

Current i data populates the voltage v versus field current i lookup table. This parameter must contain a vector with at least 10 elements.

#### Dependencies

This parameter is visible only if the Magnetic saturation representation parameter is set to ```Open-circuit lookup table (v versus i)``` and, in the Main settings, the Parameterization unit parameter is set to ```Per unit```.

Terminal voltage v data populates the voltage v versus current i lookup table. This parameter must contain a vector with at least 10 elements. The number of elements must match the number of elements in the vector for the Per-unit no-load stator current saturation data, i parameter.

#### Dependencies

This parameter is visible only if the Magnetic saturation representation parameter is set to ```Open-circuit lookup table (v versus i)``` and, in the Main settings, the Parameterization unit parameter is set to ```Per unit```.

## References

[1] Kundur, P. Power System Stability and Control. New York: McGraw Hill, 1993.

[2] Lyshevski, S. E. Electromechanical Systems, Electric Machines and Applied Mechatronics. Boca Raton, FL: CRC Press, 1999.

[3] Ojo, J. O., Consoli, A.,and Lipo, T. A., "An improved model of saturated induction machines", IEEE Transactions on Industry Applications. Vol. 26, no. 2, pp. 212-221, 1990.