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Simplified synchronous machine with electromotive force

**Library:**Simscape / Electrical / Electromechanical / Synchronous

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:

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

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

where:

*θ*is the electrical angle._{e}*N*is the number of pole pairs.*θ*is the rotor angle._{r}

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:

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

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:

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

*e*,_{a}*e*, and_{b}*e*are the per-unit internal voltage sources, defined by:_{c}$$\begin{array}{l}{e}_{a}={E}_{pu}sin{\theta}_{e}\\ {e}_{b}={E}_{pu}sin({\theta}_{e}-120\text{\xb0)}\\ {\text{e}}_{c}={E}_{pu}sin({\theta}_{e}+120\text{\xb0)}\end{array}$$

*e*is the per-unit amplitude of the internal generated voltage._{pu}*v*,_{d}*v*, and_{q}*v*are defined by:_{0}$$\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\\ {v}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$$

*v*,_{a}*v*, and_{b}*v*are the stator voltages measured from port_{c}**~**to neutral port**n**.*ω*is the per-unit base electrical speed._{base}*ψ*,_{d}*ψ*, and_{q}*ψ*are the_{0}*d*-axis,*q*-axis, and zero-sequence flux linkages.*ω*is the per-unit rotor rotational speed._{r}*R*is the stator resistance.*i*,_{d}*i*, and_{q}*i*are the_{0}*d*-axis,*q*-axis, and zero-sequence stator currents, defined by:$$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$$

*i*,_{a}*i*, and_{b}*i*are the stator currents flowing out of port_{c}**~**.

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:

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