# Voltage Distribution on Transformer Windings Subjected to Lightning Strike Using State-Space Method
## 1 Introduction
The test circuit was developed using the state-space method. The equivalent circuit model of the transformer was obtained through frequency response analysis (FRA) measurements and vector-fitting (VF) methods. Full- and chopped-voltage waveforms were generated using the state-space technique. The signals produced from the simulation were compared with measurement data, demonstrating compatibility between them. It was concluded that the proposed method saves time and effort in lightning impulse testing. In [12], a comparison of initial voltage distributions in the windings of layer-type and disc-type power transformers was conducted. Experimentally, a peak voltage of 50 V and a 1.2/50 us lightning pulse were applied to the transformer, alongside electrostatic potential simulations. The study revealed that initial voltage distributions differ in transformers with layer-type and disc-type windings due to variations in the lumped capacitance networks of the windings. Transient analysis can be performed in either the time domain or the frequency domain. Today, various simulation programs are available for this purpose, including the ATP-EMTP (Alternative Transients Program-Electromagnetic Transients Program), PSCAD (Power Systems Computer-Aided Design), MATLAB/Simulink, RTDS (Real-Time Digital Simulator), and PSS(r)E (Power System Simulator for Engineering). One method utilized for transient-state analysis in the time domain is the state-space model, which allows for the modeling of time-dependent behaviors of dynamic systems. The ATP-EMTP models capacitances and inductances as current sources and resistances, employing the trapezoidal rule for the transformation of these parameters. However, in this study, the parameters were modeled analytically, while the trapezoidal rule was utilized for numerical integration in solving the equations. The study, conducted using the state-space method, exhibits certain differences compared to other simulation programs. In the ATP-EMTP, the step size is constrained, making it impossible to achieve arbitrarily small step sizes. This limitation poses a disadvantage in analyzing very-short-duration transient regimes, as smaller step sizes correspond to higher accuracy. In contrast, the state-space method imposes no restrictions on step size. Lightning impulse tests are an important consideration in the transient analysis of transformer windings. Based on their effects and physical characteristics, lightning strikes are generally divided into two categories: direct lightning strikes and indirect lightning strikes. These two main categories are further divided into various subcategories, including positive and negative lightning strikes, as well as single and multiple strikes. Lightning strikes can be modeled in different ways. The double-exponential model, Bruce-Golde model, CIGRE (Conseil International des Grands Reseaux Electriques) model, Diendorfer-Uman model, and Heidler model are some of the lightning strike models used in the literature [13]. In this study, the time-dependent voltage distributions created in the windings of the high-voltage side of the transformer, due to a lightning strike on the phase conductor of a transmission line, were modeled using the state-space method. The modeling involved a transmission line with distributed parameters, a Heidler lightning model, and the high-voltage side of a disc-structured power transformer. First-order differential equations were derived using capacitor voltages and inductor currents, which were then transformed into state-space form. The equations in state-space form were solved using the trapezoidal integration rule, yielding the voltage and current waves of the transmission line and each disc on the high-voltage side of the power transformer. Additionally, the same circuit was drawn and simulated in the ATP-EMTP, and the results were compared with the data obtained from the state-space method. When constructing the equivalent circuit of disc-type power transformers, a separate circuit must be drawn to represent each disc.