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

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

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.

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. For an example that shows how initialize an induction machine using data from a load flow analysis, see Induction Motor Initialization with Loadflow.

For the SI implementation, the block converts the SI values that you enter to per-unit values for simulation. The converted values are based on the machine being connected in a delta-winding configuration.

For the per-unit implementation, you must specify the resistances and inductances in the impedances tab based on the machine being connected in a delta-winding configuration.

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

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

where *f _{rated}* is the
value of the

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

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

where *θ _{e}* 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:

*v*,_{ds}*v*, and_{qs}*v*are the_{0s}*d*-axis,*q*-axis, and zero-sequence stator voltages, defined by$$\left[\begin{array}{c}{v}_{ds}\\ {v}_{qs}\\ {v}_{0s}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$$

*v*,_{a}*v*, and_{b}*v*are the stator voltages across ports_{c}**~1**and**~2**.*ω*is the per-unit base electrical speed._{base}*ψ*,_{ds}*ψ*, and_{qs}*ψ*are the_{0s}*d*-axis,*q*-axis, and zero-sequence stator flux linkages.*R*is the stator resistance._{s}*i*,_{ds}*i*, and_{qs}*i*are the_{0s}*d*-axis,*q*-axis, and zero-sequence stator currents defined by$$\left[\begin{array}{c}{i}_{ds}\\ {i}_{qs}\\ {i}_{0s}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$$

*i*,_{a}*i*, and_{b}*i*are the stator currents flowing from port_{c}**~1**to port**~2**.

The rotor voltage equations are defined by

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

and

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

where:

*v*and_{dr}*v*are the_{qr}*d*-axis and*q*-axis rotor voltages.*ψ*and_{dr}*ψ*are the_{qr}*d*-axis and*q*-axis rotor flux linkages.*ω*is the per-unit synchronous speed. For a synchronous reference frame, the value is 1.*ω*is the per-unit mechanical rotational speed._{r}*R*is the rotor resistance referred to the stator._{rd}*i*and_{dr}*i*are the_{qr}*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 *L _{ss}* is the stator
self-inductance and

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 *L _{rrd}* 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 *L _{ss}*, stator
leakage inductance

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

The rotor self-inductance *L _{rrd}*, rotor
leakage inductance

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

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

${L}_{m\_sat}=f({\psi}_{m})$

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

For no saturation, the equation reduces to

${L}_{m\_sat}={L}_{m}$

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 line 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,*L*, and the saturation factor,_{m_sat}*K*, versus magnetic flux linkage,_{s}*ψ*, characteristics._{m}

**Plot Saturation Factor**— Plots saturation factor,*K*, versus magnetic flux linkage,_{s}*ψ*, in a MATLAB figure window using the machine parameters. This parameter is derived from other parameters that you specify:_{m}No-load line current saturation data,

*i*Terminal voltage saturation data,

*v*Leakage inductance,

*L*_{ls}

**Plot Saturated Inductance**— Plots magnetizing inductance,*L*, versus magnetic flux linkage,_{m_sat}*ψ*, in a MATLAB figure window using the machine parameters. This parameter is derived from other parameters that you specify:_{m}No-load line current saturation data,

*i*Terminal voltage saturation data,

*v*Leakage inductance,

*L*_{ls}

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

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.

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`

.

[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.

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