## 3 Cushion Compression and Offset Test Based on Short-Circuit Force Simulation
### Simulation Analysis of Winding Unbalanced Force after Short Circuit  
Based on the SFSZ7-31500/110 prototype transformer (Liaoning Hengyi electric power Equipment Manufacturing Co., LTD., Shenyang, China), an electromagnetic force finite element simulation model was established for electromagnetic force calculation [31], as depicted in Figure 3. The main geometric parameters of the transformer are shown in Table 1. The material parameters of windings and the cushion blocks are detailed in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Low-Voltage Winding Inner Diameter/mm & 331 \\ Low-Voltage Winding Outer Diameter/mm & 396 \\ Medium-Voltage Winding Inner Diameter/mm & 435 \\ Medium-Voltage Winding Outer Diameter/mm & 500 \\ High-Voltage Winding Inner Diameter/mm & 547 \\ High-Voltage Winding Outer Diameter/mm & 640.5 \\ Low-Voltage Winding Turns & 107 \\ Medium-Voltage Winding Turns & 226 \\ High-Voltage Winding Turns & 647 \\ Window Height/mm & 1670 \\ Core Diameter/mm & 600 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main geometric parameters of the transformer. Figure 3: Schematic diagram of finite element simulation model of transformer electromagnetic force. The radial magnetic induction intensity in the middle of the transformer winding is nearly zero 0.01 s after a short circuit. The maximum axial electromagnetic force is generated at the ends, and its direction is opposite. Excessive electromagnetic force can lead to axial instability in the winding [32]. To study the structural stability at the location of maximum stress in the winding, the topmost layer of the medium-voltage winding was selected to observe the axial force distribution, as shown in Figure 4. It is evident that there is a localized unbalanced stress on the winding. The maximum axial electromagnetic forces are generated at the upper and lower ends of the winding, and they are in opposite directions; the maximum value is 1.88 \(\times\) 10\({}^{9}\) N/m\({}^{3}\) and it is in the negative direction. The copper material used in transformer windings is a typical elastic-plastic material, with the stress-strain relationship described by the yield strength [33]. Cushion blocks are typically composed of nonlinear materials such as wood fiber or epoxy resin. According to the previous research results [34], it is generally believed that, when the value of the stress is less than 103 MPa, the stress-strain relationship of the insulation pad can be approximated as follows [35]:  
\[\sigma\mathrm{=a}\mathrm{\varepsilon+b}\mathrm{\varepsilon}^{3} \tag{23}\]  
where the linear constant a = 105.8 MPa and the hardening coefficient b = 1750 MPa. The defining equation of the elastic modulus is shown in (24), from which the relationship between the elastic modulus \(E\) and the strain value is obtained as (25). And the elastic modulus-strain relationship of the cushion blocks is plotted, as shown in Figure 5.  
\[E=\frac{\mathrm{d}\sigma}{\mathrm{d}\mathrm{\varepsilon}} \tag{24}\]  
\[E=105.8+5250\mathrm{\varepsilon}^{2} \tag{25}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Physical Quantity** & **Young’s Modulus** & **Density** & **Poisson’s Ratio** \\  & **[MPa]** & **[g/cm\({}^{3}\)]** & **Poisson’s Ratio** \\ \hline Winding & 124,000 & 8.96 & 0.325 \\ Cushion Block & 7600 & 1.3 & 0.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters for winding and cushion block. Figure 4: Axial force distribution at different positions of the same winding line. From Figure 4, the Lorentz force density and the axial height can be used to calculate the equivalent surface stress distribution of insulating cushion blocks. Based on the axial force distribution at different positions, the maximum difference in stress between adjacent cushion blocks on the circumference can be determined. Using Equation (23) and the stress-strain relationship for cushion blocks, the corresponding maximum unbalanced strain range can be calculated.