# Synchronous Machine Winding Modeling Method Based on Broadband Characteristics
## 2 Theory Introduction
### High-Frequency Lumped Parameter Circuit Model  
According to References [16; 17; 18], a single-phase machine high-frequency lumped parameter circuit model derived based on common mode and differential mode impedance is constructed, and the equivalent model of single-phase winding is shown in Figure 2a. In the Figure 2a, according to Reference [17], \(R_{g1}\) and \(C_{g1}\) represent the parasitic resistance and capacitance between the stator winding and the motor frame; \(R_{g2}\) and \(C_{g2}\) represent the parasitic resistance and capacitance between the stator neutral and motor frame; and \(R_{e}\), \(R_{Cu}\), and \(L_{d}\) represent the stator iron loss, stator winding copper loss, and stator winding leakage inductance, respectively. The parameters \(R_{1}\), \(L_{1}\), and \(C_{1}\) are introduced to characterize the secondary resonance of the machine frequency characteristics, which is caused by the inter-turn stator winding capacitance and the skin effect. The circuit wiring diagrams for measuring the differential mode and common mode impedance of synchronous machines are shown in Figure 2b,c. Among them, \(L_{zu}\) is the test lead inductance. Figure 1: Flow chart of proposed method. Figure 2: \(Cont.\)In order to obtain the value of each circuit parameter described above, the first step is to find the circuit equations that contain each parameter in the circuit topology shown in Figure 2b,c. When the above circuits are analyzed using the state space model, the capacitance voltage and inductance current in the single-phase lumped parameter circuit model are selected as the state variables. The KCL and KVL equations are written for the whole circuit, and these equations are finally integrated into matrices. The state space equation of the common mode impedance wiring diagram shown in Figure 2b is shown in Equation (1),  
\[\left( \begin{array}{cccccc}-1&0&0&0&0&0\\ 1&0&-1&0&0&-R_{\text{Cu}}\\ 0&-1&0&0&R_{\text{Cu}}&0\\ 0&0&0&0&1&0\\ 0&0&0&\frac{1}{3}&-1&-\left(1+\frac{R_{\text{Cu}}}{R_{e}}\right)\\ 0&0&0&0&1&1+\frac{R_{\text{Cu}}}{R_{e}}\\ \end{array} \right)\left( \begin{array}{c}V_{1}\\ V_{2}\\ V_{3}\\ I_{s}\\ I_{1}\\ I_{2}\\ \end{array} \right)+\left( \begin{array}{c}1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{array} \right)V_{3}= \tag{1}\] \[\left( \begin{array}{cccccc}R_{g1}C_{g1}&0&0&2L_{zu}&0&0\\ -R_{g1}\bar{C}_{g1}&0&C_{g2}R_{g2}&0&0&L_{d}\\ 0&R_{1}C_{1}&0&0&L_{1}&-L_{d}\\ 0&C_{1}&0&0&0&0\\ C_{g1}&0&0&0&-\frac{L_{d}}{R_{e}}\\ 0&0&C_{g2}&0&0&-\frac{L_{d}}{R_{e}}\\ \end{array} \right)\left( \begin{array}{c}\dot{V}_{1}\\ \dot{V}_{2}\\ \dot{V}_{3}\\ \dot{I}_{s}\\ \dot{I}_{1}\\ \dot{I}_{2}\\ \end{array} \right) \tag{2}\]  
where \(V_{s}\), \(V_{1}\), \(V_{2}\), and \(V_{3}\) are power supply voltage, the voltage on \(C_{g1}\) of each phase, the voltage on \(C_{1}\) of each phase and voltage on \(C_{g2}\) of each phase respectively; \(I_{s}\), \(I_{1}\), and \(I_{2}\) are current on \(L_{zu}\), current on \(L_{1}\) of each phase and current on \(L_{d}\) of each phase, respectively. The state space equation of the differential mode impedance wiring diagram shown in Figure 2c is shown in Equation (2),  
Figure 2: High-frequency lumped parameter circuit model of single-phase machine based on common mode and differential mode measurement:(**a**) single-phase winding model; (**b**)common mode impedance wiring diagram; and (**c**) differential mode impedance wiring diagram. \[\left( \begin{array}{cccccccccccc}-1&0&0&0&1&0&0&0&0\\ 1&-1&0&0&0&0&-R_{Cu}&0&0&0\\ 0&0&-1&0&0&0&R_{Cu}&-R_{1}&0&0\\ 0&0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&0&1\\ 0&-1&0&0&1&0&0&0&-R_{Cu}&0\\ 0&0&0&1&0&0&0&-R_{Cu}&R_{1}\\ 0&0&0&0&0&1&-\frac{R_{Cu}+R_{Fe}}{R_{e}}&-1&0&0\\ 0&0&0&0&0&\frac{R_{Cu}+R_{Fe}}{R_{e}}&1&2\frac{R_{Cu}+R_{e}}{R_{e}}&2\\ 0&0&0&0&1&0&0&2\frac{R_{Cu}+R_{e}}{R_{e}}&2\end{array} \right)\left( \begin{array} []{c}V_{1}\\ V_{2}\\ V_{3}\\ V_{4}\\ V_{5}\\ I_{s}\\ I_{1}\\ I_{2}\\ I_{3}\\ I_{4}\end{array} \right)+\left( \begin{array}{c}1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{array}