Asynchronous Machine Squirrel Cage (fundamental)

Squirrel-cage-rotor asynchronous machine with fundamental parameterization

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Machines / Asynchronous Machine (Squirrel Cage)

Description

The Asynchronous Machine Squirrel Cage (fundamental) block models a squirrel-cage-rotor asynchronous machine using fundamental parameters. A squirrel-cage-rotor asynchronous 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 Asynchronous Machine Wound Rotor (fundamental) or the Asynchronous Machine Wound Rotor (fundamental, SI) 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 or a Floating Neutral block.

Electrical Defining Equations

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

`${\theta }_{e}\left(t\right)=\underset{0}{\overset{t}{\int }}2\pi {f}_{rated}dt,$`
wherefrated is the value of the Rated electrical frequency parameter.

Park’s transformation maps stator equations to a reference frame that is stationary with respect to the rated electrical frequency. Park’s 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.

Park’s transformation is used to define the per-unit asynchronous 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

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

Plotting and Display Options

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

Right-click the block and, from the Power Systems 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.

Parameters

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

Main Tab

Rated apparent power

Rated apparent power of the asynchronous machine. The default value is `15e3` `V*A`.

Rated voltage

RMS line-line voltage. The default value is `220` `V`.

Rated electrical frequency

Nominal electrical frequency corresponding to the rated apparent power. The default value is `60` `Hz`.

Number of pole pairs

Number of machine pole pairs. The default value is `1`.

Impedances Tab

Stator resistance, Rs (pu)

Stator resistance. The default value is `0.0258`.

Stator leakage inductance, Lls (pu)

Stator leakage inductance. The default value is `0.0930`.

Referred rotor resistance, Rr' (pu)

Rotor resistance referred to the stator. The default value is `0.0145`.

Referred rotor leakage inductance, Llr' (pu)

Rotor leakage inductance referred to the stator. The default value is `0.0424`.

Magnetizing inductance, Lm (pu)

Magnetizing inductance, that is, the peak value of stator-rotor mutual inductance. The default value is `1.7562`.

Stator zero-sequence inductance, L0 (pu)

Stator zero-sequence inductance. The default value is `0.0930`.

Initial Conditions Tab

Initial rotor angle

Initial rotor angle. The default value is `0` `deg`.

Initial stator d-axis magnetic flux linkage

Initial stator d-axis flux linkage. The default value is `0` pu.

Initial stator q-axis magnetic flux linkage

Initial stator q-axis flux linkage. The default value is `0` pu.

Initial stator zero-sequence magnetic flux linkage

Initial stator zero-sequence flux linkage. The default value is `0` pu.

Initial rotor d-axis magnetic flux linkage

Initial rotor d-axis flux linkage. The default value is `0` pu.

Initial rotor q-axis magnetic flux linkage

Initial stator q-axis flux linkage. The default value is `0` pu.

Ports

The block has the following ports:

`R`

Mechanical rotational conserving port associated with the machine rotor.

`C`

Mechanical rotational conserving port associated with the machine case.

`~1`

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

`~2`

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

`pu`

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

References

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

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