# Effect of Tree Quantity and Distribution on the Electric Field under Transmission Lines
## 2 Charge Simulation Method Solving Electric Field near Transmission Lines
### Field Equation  
Due to the fact that the span of transmission lines is much smaller than their operating wavelength, the induced electric field generated by time-varying magnetic fields is much smaller than the Coulomb electric field generated by their own charges. Therefore, the line voltage and line current of the transmission line remain almost constant along the span direction. Meanwhile, because the radius of the transmission line is much smaller than the installation height and parallel to the ground, and the voltage and current vary in a three-phase symmetrical sine law, it can be considered that the transmission line is uniformly distributed by line charges, and its power frequency electric field strength value is often calculated using the charge simulation method (CSM). The field equation of the simulated charge method is shown below. \[\
abla^{2}\varphi=-\frac{\rho}{\varepsilon}\text{ or }\
abla^{2}\varphi=0 \tag{1}\]  
\[\varphi|_{L_{1}}=f_{1}(P) \tag{2}\]  
\[\varphi_{1}=\varphi_{2},\ \varepsilon_{1}\frac{\partial\varphi_{1}}{\partial n}- \varepsilon_{2}\frac{\partial\varphi_{2}}{\partial n}=0 \tag{3}\]  
where \(\varphi\) is the potential, \(\rho\) is the density of the charged body, and \(\varepsilon\) is the dielectric constant of the medium. The meanings of \(\varphi_{1}\) and \(\varphi_{2}\) are the potential at the interface between two media, and \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the dielectric constants of the two media, respectively. \(L_{1}\) is the boundary of a given potential, \(p\) is the point on the boundary, and \(f_{1}\) is the Dirichlet boundary condition. However, in practical engineering applications, the free charges inside the conductor and the bound charges at the interface are unknown and cannot be obtained through boundary conditions. Therefore, the simulated charge method uses continuous point charges, line charges, or surface charges to replace unknown charges while ensuring that the boundary conditions remain unchanged, in order to obtain simulated charges. Then, select matching points on the surface of the wire with a known potential that are equal in number to the simulated charges and use them to calculate the coefficient matrix based on the relative position between the matching points and the simulated charges [21]. At the same time, combine the given boundary conditions to write the potential equation for the transmission line wire:  
\[[P][\tau]=[\varphi] \tag{4}\]  
where \([P]\) is the \(l\)-order potential coefficient matrix, \([\tau]\) is the column vector of line charge density, and \([\varphi]\) is a potential column vector. The calculation accuracy is verified through several verification points on the surface of the wire. If it does not meet the accuracy requirements, the position, number, and shape of the simulated charge will be re-selected. Finally, by satisfying the precision requirements of discrete line charge density, the potential and electric field values at any point can be calculated based on the superposition principle. The flow chart of the simulated charge method is shown in Figure 1, where \(\Delta\Phi\) is the relative error. For transmission lines or bus bars, line charges are generally selected to be arranged at the center of the conductor for equivalent substitution. For some complex-shaped power equipment, more general point charges can be used for the corresponding arrangement. In order to facilitate the analysis of three-dimensional transmission lines, simulated line charges are selected to calculate the voltage distribution around the wires. In the three-dimensional coordinate system, the wire is divided into basic line charge units of length L, and the amount of charge carried by each line charge unit also follows a linear distribution [22; 23; 24].