# Voltage Distribution on Transformer Windings Subjected to Lightning Strike Using State-Space Method
## 4 Application and Results  
The correct operation of the model used in the study first of all requires the correct modeling of the source. The Heidler lightning model used as the source was simulated with codes written in MATLAB and drawn in the ATP-EMPT program. The graphs of Heidler current (I\({}_{\rm H}\)) given in Figures 10 and 11 are obtained from the MATLAB model and the ATP-EMPT model, respectively. When the current-time graphs in both figures are examined, it is seen that they are compatible with each other. The voltage-time graphs of the transmission line sending end (V\({}_{\rm s}\)) and receiving end (V\({}_{\rm r}\)) obtained using the state-space method are given in Figure 12. The voltage-time graph of the ATP-EMPT model is given in Figure 13.  
At t = 0 ms, the lightning strike generates a sharp voltage peak at the sending end, reaching approximately 1.2 MV, followed by a significant negative peak (\(-1.8\) MV) at t = 2 ms due to wave reflections. The delay between V\({}_{\rm s}\) and V\({}_{\rm r}\) reveals the wave propagation time along the line, with V\({}_{\rm r}\) exhibiting lower-amplitude responses due to attenuation and  
Figure 9: Flowchart for state-space analysis. energy dispersion. The lightning wave reaches the receiving end of the transmission line with a delay of \(\tau=1\) ms. The delay in the line is calculated by dividing the length of the line by the characteristic impedance \(\left(\tau=d/\sqrt{lc}\right)\). The receiving-end voltage demonstrates significant attenuation compared to the sending-end voltage, highlighting the lossy nature of the transmission line and its distributed parameters. A secondary high-amplitude transient at t = 4 ms in \(\mathrm{V_{s}}\) indicates potential resonance or multi-reflection effects within the system. It is seen that the waveforms and voltage values given in Figures 12 and 13 are compatible with each other. Figure 11: Lightning strike current by using ATP-EMTP. Figure 12: Sending-end and receiving-end voltage waveforms of transmission line using state-space method. Figure 10: Lightning strike current using the ATP-EMTP. The transmission line sending-end (I\({}_{\text{s}}\)) and receiving-end (I\({}_{\text{r}}\)) current-time graphs are given in Figure 14. The current-time graph of the ATP-EMTP model is given in Figure 15. The application of a lightning strike at the sending end of the transmission line results in a sudden rise in I\({}_{\text{s}}\). The current at the sending end sharply increases following the lightning strike, reaching an initial peak of approximately 3.5 kA. In contrast, the current at the receiving end exhibits significantly higher amplitudes, with the first peak reaching approximately 6.6 kA. This difference represents the transient effects caused by the high-frequency energy introduced by the lightning impulse. The initial peak reflects the rapid rise time of the impulse and the system's response to this high-energy disturbance. Multiple peaks and troughs are observed in both I\({}_{\text{s}}\) and I\({}_{\text{r}}\), indicating the presence of reflection and damping effects along the transmission line. The difference in impedances between the transmission line and transformer causes some of the waves to reflect back, leading to the transient oscillations observed at both ends. High-frequency transients dominate during the first few milliseconds. Over time, the amplitudes of these transients decay, and the system approaches a more stable state, demonstrating the effectiveness of the line's damping mechanisms. Lightning impulses typically have a broad-frequency spectrum, and the sharp rises and damped oscillations observed in the graph reflect the transmission line's response to these high-frequency components. This behavior aligns with the distributed-parameter model of the line. Figure 14: Sending-end and receiving-end current waveforms of transmission line by using state-space method. Figure 13: Sending-end and receiving-end voltage waveforms of transmission line using the ATP-EMTP.