# Synchronous Machine Winding Modeling Method Based on Broadband Characteristics
## 2 Theory Introduction
### High-Frequency Lumped Parameter Circuit Model
\right)V_{s}= \tag{2}\] \[\left( \begin{array}{cccccccccccc}C_{g1}R_{g1}&0&0&0&-C_{g1}R_{g 1}&2L_{zu}&0&0&0&0\\ -C_{g1}R_{g1}&C_{g2}R_{g2}&0&0&0&L_{d}&0&0&0\\ 0&0&0&0&0&-L_{d}&L_{1}&0&0\\ 0&0&C_{1}&0&0&0&0&0&0\\ 0&0&C_{1}&0&0&0&0&0&0\\ 0&C_{g2}R_{g2}&0&0&-C_{g1}R_{g1}&0&0&L_{d}&0\\ 0&0&0&0&0&0&L_{d}&-L_{1}\\ C_{g1}&0&0&0&0&\frac{L_{d}}{R_{e}}&0&0&0\\ 0&3C_{g2}&0&0&0&-\frac{L_{d}}{R_{e}}&0&-2\frac{L_{d}}{R_{e}}&0\\ 0&0&0&-2C_{g1}&0&0&-2\frac{L_{d}}{R_{e}}&0&0\end{array} \right)\left( \begin{array} []{c}\dot{V}_{1}\\ \dot{V}_{2}\\ \dot{V}_{3}\\ \dot{V}_{4}\\ \dot{V}_{5}\\ \dot{I}_{s}\\ \dot{I}_{1}\\ \dot{I}_{2}\\ \dot{I}_{3}\\ \dot{I}_{4}\end{array} \right) \tag{3}\]  
where \(V_{s}\), \(V_{1}\), \(V_{2}\), \(V_{3}\), \(V_{4}\), and \(V_{5}\) are the power supply voltage, voltage on U-phase \(C_{g1}\), voltage on each phase \(C_{g2}\), voltage on U-phase \(C_{1}\), voltage on V and W-phase \(C_{1}\), and the voltage on V and W-phase \(C_{g1}\), respectively; \(I_{s}\), \(I_{1}\), \(I_{2}\), \(I_{3}\), and \(I_{4}\) are current on \(L_{zu}\), current on U-phase \(L_{d}\), current on U-phase \(L_{1}\), current on V and W-phase \(L_{d}\), and current on V and W-phase \(L_{1}\), respectively. Then the Equations of (1) and (2) are written as follows:  
\[HI=C\dot{V}+GV \tag{4}\]  
\[TV=RI+L\dot{I} \tag{5}\]  
where \(H\) is the current coefficient matrix, \(T\) is the voltage coefficient matrix, \(R\) is the resistance matrix, \(L\) is the inductance matrix, \(C\) is the capacitance matrix, and \(G\) is the conductivity matrix. The input signal is \(V_{s}\), and the measurement signal is \(I_{s}\). \(V_{s}\) needs to be extracted to calculate the transfer function. \[T_{0}V_{s}+T^{\prime}V^{\prime}=RI+L\dot{I} \tag{6}\]  
\[HI=G_{0}V_{s}+G_{0}\dot{V}_{s}+G^{\prime}V^{\prime}+C^{\prime}\dot{V^{\prime}} \tag{7}\]  
where \(T_{0}\) is the first column matrix of \(T\), \(T^{\prime}\) is the matrix after \(T\) removes \(V_{s}\), \(G_{0}\) is the first column of \(G\), \(C_{0}\) is the first column of \(C\), \(G\) removes \(G_{0}\) to get \(G^{\prime}\), and \(C\) removes \(C_{0}\) to get \(C^{\prime}\). The matrix is written as follows:  
\[\left( \begin{array}{cc}C^{\prime}&0\\ 0&L\end{array} \right)\left( \begin{array}{c}\dot{V}^{\prime}\\ \dot{I}\end{array} \right)=\left( \begin{array}{cc}-{G^{\prime}}^{-1}&H\\ T^{\prime}&-R\end{array} \right)\left( \begin{array}{c}V^{\prime}\\ I\end{array} \right)+\left( \begin{array}{c}-G_{0}\\ T_{0}\end{array} \right)V_{s}+\left( \begin{array}{c}C_{0}\\ 0\end{array} \right)\dot{V}_{s} \tag{8}\]  
\[A\dot{X}=BX+PV_{s}+Q\dot{V}_{s} \tag{9}\]\[\text{where}\;A=\left( \begin{array}{cc}C^{\prime}&0\\ 0&L\end{array} \right)\!\!,X=\left( \begin{array}{cc}\dot{V}^{\prime}\\ \dot{1}\end{array} \right)\!\!,B=\left( \begin{array}{cc}-{G^{\prime}}^{-1}&H\\ T^{\prime}&-R\end{array} \right)\!\!,P=\left( \begin{array}{cc}-G_{0}\\ T_{0}\end{array} \right)\!\!,Q=\left( \begin{array}{cc}C_{0}\\ 0\end{array} \right)\!\!,\]  
\[I_{\text{s}}=MX \tag{9}\]  
Laplace transformations are used for Equations (8) and (9), and the transfer function can be obtained by eliminating state variable \(X\):  
\[Z(s)=\frac{V_{\text{s}}}{I_{\text{s}}}=\frac{1}{M{(sA-B)}^{-1}(P+sQ)} \tag{10}\]