## 3 Model Calculation Based on Actual Parameters of True Transformer
### Displacement and Acceleration Variation Law of Disks at Power Frequency  
Assuming that the initial displacement and initial speed of each disk coil are zero, Period T = 2 \(\times\) pi/\(\omega\), when the short-circuit current frequency is equal to the power frequency current, it can be seen from Equation (4) that the short-circuit electromotive force frequency is equal to twice the frequency of the power frequency current, that is, 100 Hz; the external short-circuit current frequency is 50 Hz, that is, at this time, the short-circuit electrodynamic frequency of the disk on the winding is 100 Hz. In the initial stage of system vibration, transient vibration and steady vibration coexist, and the total vibration of the system is the synthetic vibration of two simple harmonic vibrations with different frequencies. In the damped system, transient vibration is quickly  
Figure 6: Acceleration of each disk at 0–250 Hz. (**a**) Acceleration of Each disk at 0–40 Hz; (**b**) 40–100 Hz; and (**c**) 100–250 Hz. attenuated due to damping, and finally, only the steady vibration is retained. Therefore, the steady motion of damped forced vibration in steady vibration reflects the motion law of a damped forced vibration system. This paper focuses on the transformation and law of displacement and acceleration of the windings when they vibrate axially under short-circuit conditions, so only the steady vibration is studied with the transient vibration omitted in the simulation analysis. Figure 7 shows the change of the displacement of the top layer of the wire cake with time during a short circuit. When the time is at 0.15, the transient amplitude of the short-circuit current tends to zero, and the steady-state vibration displacement amplitude is at the maximum moment. It can better observe the displacement change of each disk in the axial direction, so the time t = 0.15 s is taken. When t = 0.15 s and the damping ratio takes different values, the displacement change at this time is shown in Figure 8. The axial vibration displacement waveform of the winding is in a 'valley' shape, which is roughly consistent with the vibration mode law of the eighth order vibration mode, which is just around the frequency of 100 Hz, so it is believed the radial displacement distribution of the winding conforms to the theoretical calculation. The displacement amplitude reaches the peak at one-seventh of both ends, the displacements at the top and the end are close to zero, and the middle disk displacement is distributed in an \"M\" shape. Therefore, the deformation near both ends of the winding is large, and small in the middle part. As the damping ratio increases, the displacement oscillation amplitude of each disk gradually decreases. Figure 8: Displacement of each disk under different damping ratios. Figure 7: Displacement response of the top disk. When t = 0.15 s, the acceleration change of each winding disk at this time is shown in Figure 9. Similar to the displacement waveform of each winding disk, the radial vibration acceleration waveform of the winding is in the shape of a 'valley', which is also roughly consistent with the vibration mode law of the eighth order mode. The acceleration near both ends of the windings reaches a maximum value, about twice the acceleration of the middle part. With the increase of the damping ratio, the acceleration oscillation amplitude of each disk gradually decreases. It can be concluded that when the power transformer is in a short circuit, the winding acts as a damped multi-DOF system and the short-circuit electrodynamic force of the winding is 100 Hz. The displacement distribution law of each disk is consistent with the displacement distribution law of the eighth vibration mode. The displacement is the largest near both ends of the winding, and the middle part is relatively gentle. The displacements of the top and end disks are close to zero.