# Voltage Distribution on Transformer Windings Subjected to Lightning Strike Using State-Space Method
## 1 Introduction  
Increasing electrical energy consumption makes a continuous and high-quality energy supply essential for both individual and commercial consumers. Accordingly, various measures are taken to prevent failures that may occur in electricity networks during the production, transmission, and distribution of energy. Maintenance and repair of network elements, such as power lines, protective devices, transformers, and others, are performed at regular intervals to minimize failures. In addition, when designing a power line or electrical equipment, both steady-state and transient analyses are necessary. The main reason for this is that the peak current-voltage values of the system in a steady state may be lower than those in the transient state. If the design relies solely on steady-state conditions, protection and insulation failures may occur in the line and other network elements. Events such as lightning strikes, short circuits, switching, and sudden load changes can induce transients. When transformer windings are subjected to very high voltage during transients, malfunctions such as insulation breakdown, insulation deterioration, overheating of the windings, and arcing between the windings can occur. Therefore, when designing a power system, it is crucial to ensure that it can withstand the worst conditions to maintain energy continuity, durability, and long-term performance. Transient analysis in transformers has been a subject of research in recent years. In [1], transformer windings were divided into a finite number of segments to represent an electrical network, and a mathematical model for this network was created. In [2], a 345 kV transmission system was energized from a limited-capacity diesel generator, and the inrush currents of the transformer were analyzed using the Electromagnetic Transients Program (EMTP). Current analyses were performed under different load conditions, and inrush current values were examined. In [3], a power transformer was modeled using the ATP-EMTP. To analyze the performance of the transformer relay modeled in MATLAB, transient waveforms were generated under various operating conditions. In [4], voltage calculations were performed in the frequency domain using a hybrid model comprising a Multi-Conductor Transmission Line (MTL) and a Single Transmission Line (STL). In [5], the voltage distribution across the windings was investigated employing the input terminal impedance method. In [6], a method was proposed that utilized the state variables technique, with inductance and capacitance matrices, as well as resistance matrices, obtained from Finite Element Method (FEM) simulations. In [7], voltage distributions in windings were studied using black-box and vector-fitting techniques, alongside a voltage distribution factor. In [8], the resistance, inductance, and capacitance values of the windings were calculated, and the circuit equations were solved in the frequency domain using Wolfram Mathematics. In [9], a study on a 245 kV voltage transformer was conducted to investigate voltage distribution within the windings when subjected to a 2.5/50 us lightning strike. The Finite Element Method and analytical formulas were employed to determine the model parameters, and a simulation program was used for analysis. The data obtained were compared with experimental results, revealing that the results were almost consistent and that voltage levels decreased from the first disc to the last disc. In [10], transient analysis of a transformer was performed using a Hybrid Analytical-FEM approach. A high-frequency model was used for the transformer windings, which were modeled as lumped RLC parameters and solved in the time domain using [PERSON]'s method. The data from the proposed method were compared with measured data, concluding that the hybrid model approach is suitable for transient analysis of transformer windings. In [11], lightning impulse testing was conducted using full-wave (FW) and chopped-wave (CW) impulses in the time domain.