# A Novel Excitation Approach for Power Transformer Simulation Based on Finite Element Analysis
## 2 2D Finite Element Analysis Model
### External Excitation  
The open circuit of the transformer is shown in Figure 1. The circuit is composed of external excitation voltage \(V_{in}\), leakage inductance \(L_{k}\), series resistance \(R_{s}\), magnetizing inductance \(L_{m}\), and parallel resistance \(R_{p}\). To simplify the analysis, some conditions are assumed, namely:  
1. The external exciting voltage \(V_{in}\) is the ideal voltage source;  
Figure 1: The open circuit of the transformer diagram. Parallel resistance \(R_{p}\) is bigger than the magnetizing reactance \(X_{Ln}\);
3. The turns ratio is defined as \(N=N_{1}/N_{2}\);
4. The secondary side circuit is open;
5. The external excitation voltages are balanced in three phases. Since the external excitation voltages are balanced in three phases, the external excitation voltage of each phase can be written as Equations (16)-(18), where \(V_{p}\)nk is the peak value of the external excitation voltage and \(\alpha\) is the initial angle of the source:  
\[V_{in_{-PhaseA}}=V_{peak}\cdot\text{Sin}(2\pi ft+\alpha) \tag{16}\]  
\[V_{in_{-PhaseB}}=V_{peak}\cdot\text{Sin}[2\pi ft+\alpha+(\frac{2}{3}\pi)] \tag{17}\]  
\[V_{in_{-PhaseC}}=V_{peak}\cdot\text{Sin}[2\pi ft+\alpha+(\frac{4}{3}\pi)] \tag{18}\]  
The excitation current of each phase is calculated by Figure 1 and Equations (16)-(18). The excitation current equation of each phase can be written as:  
\[i_{\phi_{-}A}(t)=\frac{V_{peak}}{Z}\bigg{[}\text{sin}(2\pi ft+\alpha-\theta)- \text{sin}(\alpha-\theta)\cdot e^{-\frac{R_{s}}{k_{s}+\alpha}t}t\bigg{]} \tag{19}\]  
\[i_{\phi_{-}B}(t)=\frac{V_{peak}}{Z}\bigg{[}\text{sin}(2\pi ft+\alpha+120- \theta)-\text{sin}(\alpha+120-\theta)\cdot e^{-\frac{R_{s}}{k_{s}+\alpha}t}t \bigg{]} \tag{20}\]  
\[i_{\phi_{-}C}(t)=\frac{V_{peak}}{Z}\bigg{[}\text{sin}(2\pi ft+\alpha+240- \theta)-\text{sin}(\alpha+240-\theta)\cdot e^{-\frac{R_{s}}{k_{s}+\alpha}t}t \bigg{]} \tag{21}\]  
The Equations (19)-(21) are defined as follows:  
\[\text{Impedance}:\ Z=\sqrt{{R_{s}}^{2}+[2\pi f(L_{k}+L_{m})]^{2}} \tag{22}\]  
\[\text{Impedance angle}:\ \theta=\text{tan}^{-1}(\frac{2\pi f(L_{k}+L_{m})}{R_{s}}) \tag{23}\]  
According to Equations (16)-(18), the excitation current of each phase has AC steady-state components and DC transient components. When the AC steady-state component and the DC transient component are in the same square phase, the peak value of the exciting current will increase in the half-cycle. The peak value of the magnetic flux density is increased by this phenomenon. This will cause the iron core to enter the saturation region when the phenomenon is serious. Therefore, the slow start function is added to the equation of the external excitation voltage to avoid magnetic flux density deviation in the iron core. Therefore, the equations have been proposed by scholars, and the equations can be written as Equations (24)-(26) [7,16], where \(\beta\) is the slow start parameter:  
Figure 1:0  
Figure 1:1  
Figure 1:2  
According to Equations (24)-(26), the slow start parameter is constant. The simulation time is short and the accuracy of the simulation is lower under a big slow start parameter. Conversely, the simulation time is longer and the accuracy of the simulation is higher under the small slow start parameter. However, the small slow start parameter increases the time of each iteration in the coupling field, which consists of the magnetic field and the temperature field based on the sequential method. Therefore, this paper proposes the new external excitation voltage equation, in which the slow start function is the double exponential function. Comparing the conventional equation, the double exponential function makes the slow start parameter change over time. In the beginning, the slow start parameter is small. The slow start parameter becomes larger as the simulation time progresses. The proposed external excitation voltage equation of each phase can be written as Equations (27)-(29), where \(\gamma\) is the slow start parameter:  
Figure 1:3  
Figure 1:4  
Figure 1:5