# Voltage Distribution on Transformer Windings Subjected to Lightning Strike Using State-Space Method
## 2 Network Elements
### Modeling of Power Transformer Winding
#### 2.3.3 Calculation of Series Capacitance of the Winding  
To calculate the series capacitance value according to the transformer winding structure, the capacitance value between the conductors in the same disc (\(C_{t}\)) and the capacitance between the conductors in the adjacent disc, also known as the capacitance between the discs (\(C_{d}\)), are calculated. Structure of single-phase disc-type winding is shown in Figure 5.  
Figure 4: Dimensions of winding discs [28].  
The capacitance value between the conductors in the same disc (between two turns) is calculated in (10) [27; 31].  
\[C_{t}=\frac{2\pi\varepsilon_{r}\varepsilon_{o}d_{w}}{ln\frac{b_{1}}{b_{2}}} \tag{10}\]  
Here, \(b_{1}\) represents the inner radius of the outer conductor, \(b_{2}\) represents the outer radius of the inner conductor, and \(d_{w}\) represents the length of the conductor. \(\varepsilon_{o}\) is the permittivity of free space and \(\varepsilon_{r}\) is the relative permittivity of the insulation paper between the inner and outer conductors, where \(\varepsilon_{o}=8.854\times 10^{-12}\) and \(\varepsilon_{r}=3.85\). Equations (11)-(14) are used to calculate the capacitance (\(C_{d}\)) value between any two discs [29].  
\[C_{d}=\varepsilon\frac{S}{d} \tag{11}\]  
\[S=\pi\Bigg{(}\left(\frac{d_{o}}{2}\right)^{2}-\left(\frac{d_{i}}{2}\right)^{ 2}\Bigg{)} \tag{12}\]  
\[S=\pi\Bigg{(}\frac{d_{o}^{2}}{4}-\frac{d_{i}^{2}}{4}\Bigg{)} \tag{13}\]  
where \(\varepsilon=\varepsilon_{r}\varepsilon_{o}\).  
\[C_{d}=\varepsilon_{r}\varepsilon_{o}\pi\frac{{d^{\prime}}_{o}^{2}-{d^{ \prime}}_{i}^{2}}{4d} \tag{14}\]  
where \({d^{\prime}}_{i}=d_{i}-d\) and \({d^{\prime}}_{o}=d_{o}+d\).  
In the equations, \(d\), \(d_{o}\), and \(d_{i}\) represent the distance between the turns, the outer diameter of the turn, and the inner diameter of the winding, respectively. \(S\) is the surface area of the disc. According to the equivalent circuit of the transformer capacitance in Figure 6, (15) is used to calculate the total \(C_{t}\) value and (16) is used to calculate the total \(C_{d}\) value.  
\[C_{t\ total}=\frac{1}{\frac{1}{c_{1}}+\frac{1}{c_{2}}+\frac{1}{c_{3}}+\frac{1} {c_{nn}}} \tag{15}\]  
\[C_{d\ total}=C_{d1}+C_{d2}+C_{d3}+C_{d\eta} \tag{16}\]  
Figure 5: Structure of single-phase disc-type winding [30].  
Using Figure 6 and Equations (15) and (16), the capacitance value of the transformer winding for a disc can be calculated as \(C_{d}+C_{t}\). However, (17) is used to calculate the total series capacitance value of the winding.  
\[C_{s}=\frac{1}{N_{d}}\left(\frac{C_{t}}{mt}+4C_{dt}\right) \tag{17}\]  
\[C_{dt}=\frac{(mt-1)(2 mt-1)}{6 mt}C_{d} \tag{18}\]  
where \(mt\) is the number of turns per disc.  
\[C_{g}=(2\pi e_{r}e_{o}d_{w})/\ln(d_{i}/d_{o}) \tag{19}\]  
Here, \(C_{dt}\) is calculated as the result of the capacitance between the discs. Apart from \(C_{t}\) and \(C_{d}\), there is also a capacitance to ground called \(C_{g}\) in the disc model of the transformer winding. This capacitance value of \(C_{g}\) is calculated by (19) [29].  
The voltage distribution factor \((a)\) is an important factor when analyzing the voltage distributions in transformer windings. This factor is calculated with the formula \(a=\sqrt{C_{g}/C_{s}}\)[32]. It is desired that the value \(a\) is small. Otherwise, transformer insulation may be at risk. Therefore, this factor can be beneficial for winding insulation in transformer design.