# Investigation of the Opposite-Electrode Effect on the Planar Solid-State Pulse-Forming Line
## 2 Theoretical Analysis of the Solid-State PFL
### Infinite far  
Let \(C\) represent the capacitance per unit length of the PFL, and let \(\varepsilon_{0}\) denote the permittivity of vacuum. Let \(\varepsilon_{e}\) stand for the equivalent relative permittivity of the ceramic plate and vacuum as a single dielectric medium. The capacitance \(C\) per unit length of the PFL can be determined through a conformal mapping transformation [31,32].  
\[C=\varepsilon_{0}\varepsilon_{e}\frac{w}{h}+\frac{\varepsilon_{0}\varepsilon_ {e}}{\pi}[1+\ln(1+\frac{2\pi w}{h}+\ln(1+\frac{2\pi w}{h}))+\ln(1+\frac{2\pi d }{4})^{2})]\quad,\quad w/h\geq 1 \tag{1}\]  
Figure 1: The components of the classic planar solid-state PFL.  
Figure 2: PFL’s cross-section.  
\[\epsilon_{\rm e}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}\frac{1}{\sqrt{1+ \frac{h}{w}}} \tag{2}\]  
Divide the high-voltage electrode into \(n\) segments that are mutually parallel. According to the generalized definition of capacitance, the relationship between the unit length charge distribution \(q_{i}\) of each segment and the voltage and the generalized capacitance matrix can be expressed as  
\[\begin{bmatrix}q_{1}\\ q_{2}\\ \vdots\\ q_{n}\end{bmatrix}=\begin{bmatrix}\sum\limits_{k=1}^{n}c_{1k}&-c_{12}&\cdots&- c_{1n}\\ -c_{12}&\sum\limits_{k=1}^{n}c_{2k}&\cdots&-c_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ -c_{1n}&-c_{2n}&\cdots&\sum\limits_{k=1}^{n}c_{nk}\end{bmatrix}\begin{bmatrix} V_{1}\\ V_{2}\\ \vdots\\ V_{n}\end{bmatrix} \tag{3}\]  
From the formula, it can be concluded that the charge distribution tends to peak at the edge of the high-voltage electrode and that the peak value is related to the formation of the edge capacitance of the PFL. It can be seen that as the width of the ground electrode increases, the stray capacitance at the edge of the high-voltage electrode also increases, thereby causing the charge density at the edge of the high-voltage electrode to increase.  
According to the integral form of [PERSON]'s law, the following relationship exists between the enclosed free charges and the displacement vector on the Gaussian surface with the normal direction outward:  
\[\oint\limits_{S}\overrightarrow{D}\cdot ds=\sum q \tag{4}\]  
Therefore, the maximum electric field strength near the PFL high-voltage electrode is synchronized with the edge charge density of the high-voltage electrode and is further synchronized with the edge capacitance of the PFL. Therefore, for a PFL with an unequal ground electrode width to the high-voltage electrode, as the width of the ground electrode increases, the electric field at the edge of the high-voltage electrode also increases. The full-electrode PFL uses a ground electrode that covers the entire plane, which will result in additional electric field enhancement at the edge of the high-voltage electrode.  
The loss in the PFL is mainly divided into conductor loss and dielectric loss. Of these, the loss due to the alternating polarization and grain collision of the dielectric molecules when an electric field passes through the dielectric is referred to as dielectric loss. The relationship between the transmitted power \(P_{0}\), the loss power per unit length of the forming line \(P_{d}\), and the dielectric loss attenuation constant \(\alpha_{d}\) is [33; 34]  
\[\alpha_{d}=\frac{p_{d}}{4P_{0}} \tag{5}\]  
Therefore, after continuous operation, the average heating power in the electric field enhancement area near the high-voltage electrode edge of the planar solid-state PFL is larger, which may cause uneven heating of the PFL and a higher steady-state temperature in the electric field enhancement area.