@@ -377,6 +377,77 @@ In both cases, the function "init" is setting every initial value to zero.

...

@@ -377,6 +377,77 @@ In both cases, the function "init" is setting every initial value to zero.

The model is according to \cite{wang2010methods} and \cite{kundur1994power}.

The model is according to \cite{wang2010methods} and \cite{kundur1994power}.

\subsubsection{Prerequisites}

\subsubsection{Prerequisites}

\subsubsection{Magnetic Circuits}

Considering two magnetically coupled windings with number of turns $N_1$ and $N_2$, resistance $r_1$ and $r_2$ and winding currents $i_1$ and $i_2$ respectively, the flux linkage with the respective windings, produced by the total effect of both currents are given by equations \ref{eq:FluxLinkage1} and \ref{eq:FluxLinkage2}.

\begin{equation}\label{eq:FluxLinkage1}

\Psi_1 = N_1(\Phi_{l1} + \Phi_{m1} + \Phi_{m2})

\end{equation}

\begin{equation}\label{eq:FluxLinkage2}

\Psi_2 = N_2(\Phi_{l2} + \Phi_{m2} + \Phi_{m1})

\end{equation}

\\

where

$\Phi_{m1}$ is the mutual flux linking both windings due to current $i_1$;

$\Phi_{m2}$ is the mutual flux linking both windings due to current $i_2$;

$\Phi_{l1}$ is the leakage flux linking winding 1 only;

$\Phi_{l2}$ is the leakage flux linking winding 2 only.

The flux linkage can also be expressed in terms of self and mutual inductance as in equation \ref{eq:FluxLinkageInductances}

\begin{equation}\label{eq:FluxLinkageInductances}

\begin{bmatrix}

\Psi_1\\

\Psi_2

\end{bmatrix}

=

\begin{bmatrix}

L_{11}& L_{12}\\

L_{21}& L_{22}

\end{bmatrix}

\cdot

\begin{bmatrix}

i_1\\

i_2

\end{bmatrix}

\end{equation}

\\

where $L_{11}$ and $L_{22}$ are the self inductances of windings 1 and 2 respectively and defined as the flux linkage per unit current in the same winding. $L_{12}$ and $L_{21}$ are the mutual inductance between two windings, defined as the flux linkage with one winding per unit current in the other winding.

\begin{equation}

L_{11} = N_1(\Phi_{m1} + \Phi_{l1}) / i_1

\end{equation}

\begin{equation}

L_{22} = N_2(\Phi_{m2} + \Phi_{l2}) / i_2

\end{equation}

\begin{equation}

L_{12} = N_1 \Phi_{m2} / i_2

\end{equation}

\begin{equation}

L_{21} = N_2 \Phi_{m1} / i_1

\end{equation}

The terminal voltages $e_1$ and $e_2$ are defined as following:

\begin{equation}\label{eq:VoltageEquation1}

e_1 = \frac{d \Psi_1}{d t} + r_1 i_1

\end{equation}

\begin{equation}\label{eq:VoltageEquation2}

e_2 = \frac{d \Psi_2}{d t} + r_2 i_2

\end{equation}

Equations \ref{eq:FluxLinkageInductances}, \ref{eq:VoltageEquation1} and \ref{eq:VoltageEquation2} give the performance of the circuit.

\subsubsection{Park's transformation}

Park's transformation is commonly used to achieve a model with static parameters:

Park's transformation is commonly used to achieve a model with static parameters: