Wound-rotor asynchronous machine with fundamental parameterization

Machines / Asynchronous Machine (Wound Rotor)

The Asynchronous Machine Wound Rotor (fundamental) block models a wound-rotor asynchronous machine using fundamental parameters. A wound-rotor asynchronous machine is a type of induction machine. All stator and rotor connections are accessible on the block. Therefore, you can model soft-start regimes using a switch between wye and delta configurations or by increasing rotor resistance. If you do not need access to the rotor windings, use the Asynchronous Machine Squirrel Cage (fundamental) 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. If you do not need to vary rotor resistance, connect rotor port ~1r to a Floating Neutral block and rotor port ~2r to a Grounded Neutral block.

The rotor circuit is referred to the stator. Therefore, when you use the block in a circuit, refer any additional circuit parameters to the stator.

The asynchronous 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

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

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

where

The rotor equations are mapped to another reference frame, defined
by the difference between the electrical angle and the product of
rotor angle θ_{r} and number of pole pairs
N:

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

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:

*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 ~1 and ~2._{c}*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 ~1 to port ~2._{c}

The rotor voltage equations are defined by

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

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

and

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

where:

*v*,_{dr}*v*, and_{qr}*v*are the_{0r}*d*-axis,*q*-axis, and zero-sequence rotor voltages, defined by$$\left[\begin{array}{c}{v}_{dr}\\ {v}_{qr}\\ {v}_{0r}\end{array}\right]={P}_{r}\left[\begin{array}{c}{v}_{ar}\\ {v}_{br}\\ {v}_{cr}\end{array}\right].$$

*v*,_{ar}*v*, and_{br}*v*are the rotor voltages across ports ~1r and ~2r._{cr}*ψ*,_{dr}*ψ*, and_{qr}*ψ*are the_{0r}*d*-axis,*q*-axis, and zero-sequence 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*,_{dr}*i*, and_{qr}*i*are the_{0r}*d*-axis,*q*-axis, and zero-sequence rotor currents, defined by$$\left[\begin{array}{c}{i}_{dr}\\ {i}_{qr}\\ {i}_{0r}\end{array}\right]={P}_{r}\left[\begin{array}{c}{i}_{ar}\\ {i}_{br}\\ {i}_{cr}\end{array}\right].$$

*i*,_{ar}*i*, and_{br}*i*are the rotor currents flowing from port ~1r to port ~2r._{cr}

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

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

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

and

${\psi}_{0r}={L}_{rrd}{i}_{0r},$

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

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

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.

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

**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`

.

**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 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.**Initial rotor zero-sequence magnetic flux linkage**Initial rotor zero-sequence flux linkage. The default value is

`0`

pu.

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.

`~1r`

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

`~2r`

Expandable three-phase port associated with the rotor 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

[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.

Asynchronous Machine Measurement | Asynchronous Machine Squirrel Cage (fundamental) | Asynchronous Machine Squirrel Cage (fundamental, SI) | Asynchronous Machine Wound Rotor (fundamental, SI)

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