### review of magnetic circuit equations for synchronous machine model


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parent 2d361a71
 ... @@ -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: % % \begin{equation} \begin{equation} ... ...
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