# A Novel Excitation Approach for Power Transformer Simulation Based on Finite Element Analysis
## 3 Finite Element Analysis Simulation  
### Simulation Model  
The 2D model of a three-phase five-limb transformer is shown in Figure 2. The model is composed of the exciting coil, the iron core, and the test environment. The model of the iron core has a gap. In actual transformers, the iron core has a certain thickness. The 2D simulation model has no third dimension to establish the thickness of the iron core. Therefore, the 2D model depth is added to the 2D simulation model. The 2D model depth makes the equivalent total volume of the simulation iron core model the same as the total volume of the actual iron core model. According to the iron core of the experimental model, the cross-section area of the single limb is 4960.992 mm\({}^{2}\). The length of the iron core is 480 mm in the simulation model. The 2D model depth is 10.3354 mm, which is calculated by the Equation (30). The parameters of the simulation model as shown in Table 1.  
Figure 2: 2D simulation model diagram. The materials of the simulation model are defined before simulating iron loss and magnetic flux density. The materials of the simulation model are shown in Table 2. The material items are the excitation coil, iron core, gap, and environment. The non-linear factors of the iron core are considered. The \(B\)-\(H\) curve of the iron core is shown in Figure 3. The \(P\)-\(B\) curve of the iron core is shown in Figure 4. \begin{table}
\begin{tabular}{c c c} \hline
**Name** & **Material** & **Parameter** \\ \hline Excitation Coil & Copper & Resistivity: 1.68 \(\times\) 10\({}^{-8}\) [\(\Omega\cdot\) m] \\ \hline Iron Core & Electrical Grade Sheet & Conductivity: 2,083,300 [S/m] \\  & & Mass Density: 7650 [kg/m\({}^{3}\)] \\  & & Relative Permeability: \\  & & Non-linear \(B\)-\(H\) Curve \\ \hline Gap, Environment & Air & Magnetic of Relative Permeability: \\  & & 1.0000004 \\ \end{tabular}
\end{table} 
Table 2: Materials of the simulation model. Figure 4: The \(P\)-\(B\) curve of the iron core diagram. Figure 3: The \(B\)-\(H\) curve of the iron core diagram. The simulation results of the winding flux linkage without a slow start are shown in Figure 8, and its deviation is 75%. Moreover, the simulation results of the winding flux linkage under conventional external excitation equations are shown in Figure 9. When \(\beta=50\) and \(\beta=5\), the winding flux linkage deviation is 20.7% and 2.325%, respectively. The simulation results of winding flux linkage under the proposed external excitation equation are shown in Figure 10. When \(\gamma=25\), \(\gamma=50\), \(\gamma=75\), and \(\gamma=100\), the winding flux linkage deviations are 0.387%, 0.312%, 1.043%, and 5.87%, respectively. Figure 8: Simulation results of the winding flux linkage without a slow start. Figure 6: Simulation results of the winding voltage under the conventional equation. (**a**) \(\beta=50\); (**b**) \(\beta=5\). Figure 7: Simulation results of the winding voltage under the proposed equation. (**a**) \(\gamma=25\); (**b**) \(\gamma=50\); (**c**) \(\gamma=75\); (**d**) \(\gamma=100\). The simulation results of the core loss without a slow start are shown in Figure 11. The simulation results of the core loss under conventional external excitation equations are shown in Figure 12. The simulation results of the core loss under the proposed external excitation equation are shown in Figure 4:0 The average core losses under different slow start functions are shown in Figure 4:1 The total core loss is composed of two loss components, which are the eddy current loss component and the hysteresis loss component. The average interval time is 200 ms. According to Figure 6, when the \(\beta=50\) and \(\beta=5\), the time for the external excitation voltage to reach the steady-state is 0.08 s and 0.98 s, respectively. According to Figure 7, when the \(\gamma=25\), \(\gamma=50\), \(\gamma=75\), and \(\gamma=100\), the time for the external excitation voltage to reach steady-state is 0.04 s, 0.06 s, 0.08 s, and 0.14 s, respectively.