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# Simplified Synchronous Machine

Simplified synchronous machine with electromotive force

• Library:
• Simscape / Electrical / Electromechanical / Synchronous

• ## Description

The Simplified Synchronous Machine block models a simplified synchronous machine with a voltage source that represents electromotive force (EMF). You can specify the internal resistance and inductance with per-unit or SI parameters.

The equivalent circuits of the simplified synchronous machine for the direct axis, the quadrature axis, and the zero sequence are:   ### Equations

The simplified synchronous machine equations are expressed with respect to a rotating reference frame, which is defined by:

`${\theta }_{e}\left(t\right)=N{\theta }_{r}\left(t\right),$`

where:

• θe is the electrical angle.

• N is the number of pole pairs.

• θr is the rotor angle.

The Park transformation maps the synchronous machine equations to the rotating reference frame with respect to the electrical angle. The Park transformation is defined by:

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

The Park transformation is used to define the per-unit simplified synchronous machine equations. The voltage equations are defined by:

`${e}_{d}=\frac{1}{{\omega }_{base}}\frac{\text{d}{\psi }_{d}}{\text{d}t}-{\Psi }_{q}{\omega }_{r}+R{i}_{d}+{v}_{d}$`

`${e}_{q}=\frac{1}{{\omega }_{base}}\frac{\text{d}{\psi }_{q}}{\text{d}t}+{\Psi }_{d}{\omega }_{r}+R{i}_{q}+{v}_{q}$`

`${e}_{0}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{0}}{dt}+R{i}_{0}+{v}_{0}$`

where:

• ed, eq, and e0 are the d-axis, q-axis, and zero-sequence voltages, defined by:

`$\left[\begin{array}{c}{e}_{d}\\ {e}_{q}\\ {e}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{e}_{a}\\ {e}_{b}\\ {e}_{c}\end{array}\right].$`

ea, eb, and ec are the per-unit internal voltage sources, defined by:

`$\begin{array}{l}{e}_{a}={E}_{pu}sin{\theta }_{e}\\ {e}_{b}={E}_{pu}sin\left({\theta }_{e}-120\text{°)}\\ {\text{e}}_{c}={E}_{pu}sin\left({\theta }_{e}+120\text{°)}\end{array}$`

epu is the per-unit amplitude of the internal generated voltage.

• vd, vq, and v0 are defined by:

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

va, vb, and vc are the stator voltages measured from port ~ to neutral port n.

• ωbase is the per-unit base electrical speed.

• ψd, ψq, and ψ0 are the d-axis, q-axis, and zero-sequence flux linkages.

• ωr is the per-unit rotor rotational speed.

• R is the stator resistance.

• id, iq, and i0 are the d-axis, q-axis, and zero-sequence stator currents, defined by:

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

ia, ib, and ic are the stator currents flowing out of port .

The stator flux linkage equations are defined by

`$\begin{array}{l}{\psi }_{d}=L\cdot {i}_{d}\\ {\psi }_{q}=L\cdot {i}_{q}\\ {\psi }_{0}=L\cdot {i}_{0}\end{array}$`

where L is the stator leakage inductance.

The power equation of the simplified synchronous machine in per-unit is defined by:

`$P={e}_{d}{i}_{d}+{e}_{q}{i}_{q}.$`

## Ports

### Input

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Physical signal port associated with the amplitude of the internal generated voltage (phase-to-neutral), specified as a vector of physical signals.

#### Dependencies

To enable this port, set Parameterization unit to `SI`.

Physical signal port associated with the per-unit amplitude of the internal generated voltage, specified as a vector of physical signals.

#### Dependencies

To enable this port, set Parameterization unit to `Per unit`.

### Output

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Physical signal port associated with the machine per-unit measurements, returned as a vector of physical signals. The vector elements are:

• Electromotive force associated with phase a, pu_EMF(1)

• Electromotive force associated with phase b, pu_EMF(2)

• Electromotive force associated with phase c, pu_EMF(3)

• Electrical torque, pu_torque

• Rotor velocity, pu_velocity

• Stator `d`-axis voltage, pu_vd

• Stator `q`-axis voltage, pu_vq

• Stator zero-sequence voltage, pu_e0

• Stator `d`-axis current, pu_id

• Stator `q`-axis current, pu_iq

• Stator zero-sequence current, pu_i0

• Rotor electrical angle, electrical_angle_out

To connect to this port, use the Synchronous Machine Measurement block.

### 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 electrical port associated with the stator windings.

Electrical conserving port associated with the neutral point of the wye winding configuration.

## Parameters

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### Main

Rated apparent power.

RMS rated line-to-line voltage.

Nominal electrical frequency at which rated apparent power is quoted.

Number of machine pole pairs.

Block parameterization method. Options are:

• `SI`

• `Per unit`

Internal per-unit resistance.

#### Dependencies

To enable this parameter, set Parameterization unit to `Per unit`.

Internal per-unit inductance in per-unit.

#### Dependencies

To enable this parameter, set Parameterization unit to `Per unit`.

Internal resistance.

#### Dependencies

To enable this parameter, set Parameterization unit to `SI`.

Internal inductance.

#### Dependencies

To enable this parameter, set Parameterization unit to `SI`.

## See Also

Introduced in R2020b

## Support

#### 10 Ways to Speed Up Power Conversion Control Design with Simulink

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