## 2 Traveling Wave Principle and Phase-Mode Transformation  
The generation of a traveling wave can be analyzed by the superposition principle, as shown in Figure 1, assuming that the voltage of a certain point F of the power supply cable is normal when the voltage is \(\dot{U_{F}}\). After the ground fault occurs at this point, point F has the same potential as the Earth, and the switch \(K\) is closed. Point F is equivalent to superimposing an additional power supply \(-\dot{U_{F}}\) with the same voltage and opposite direction when the point is normal. Under the action of additional power supply \(-\dot{U_{F}}\), point F will produce traveling wave signals transmitted to both ends.  
The specific traveling wave transmission process is expressed by the wave equation, as shown in Equation (1).  
\[\begin{cases}\frac{\dot{\partial}^{2}U}{\partial x^{2}}=LC\frac{\dot{\partial }^{2}U}{\partial t^{2}}\\ \frac{\dot{\partial}^{2}I}{\partial x^{2}}=CL\frac{\dot{\partial}^{2}I}{ \partial t^{2}}\end{cases} \tag{1}\]  
Within the equations, \(U\) and \(I\) represent the phase-to-ground voltage and conductor line current, respectively, while \(x\) denotes the line length and \(t\) signifies time. The per-unit-length parameters \(L\) and \(C\) correspond to the inductance and capacitance parameter matrices along the transmission line. The electromagnetic coupling between three-phase conductors results in non-diagonal \(L\) and \(C\) matrices, making direct equation solving highly complex. To simplify the analytical and computational process, phase-mode transformation is applied to diagonalize \(L\), \(C\), \(LC\), and \(CL\) matrices. This study employs Karenbauer trans  
Figure 1: Principal diagram of traveling wave generation.  
formation to accomplish the phase-mode conversion of the traveling wave, as expressed in Equation (2).  
\[\begin{bmatrix}u_{0}\\ u_{1}\\ u_{2}\end{bmatrix}=\frac{1}{3}\begin{bmatrix}1&-1&0\\ 1&0&-1\\ 1&1&1\end{bmatrix}\begin{bmatrix}u_{A}\\ u_{B}\\ u_{C}\end{bmatrix} \tag{2}\]  
In the formula, \(u_{0}\) is a zero-mode traveling wave, which forms a loop through the Earth. \(u_{1}\) and \(u_{2}\) are line-mode traveling waves, which form a loop through two non-fault phases. \(u_{A}\), \(u_{B}\), and \(u_{C}\) are three-phase voltages.  
After phase-mode transformation, the lines are decoupled, and the wave equation is shown in Formula (3).  
\[\left\{\begin{array}{l}\frac{\partial^{2}U_{m}}{\partial x^{2}}=\Lambda_{u} \frac{\partial^{2}U_{m}}{\partial x^{2}}\\ \frac{\partial^{2}U_{m}}{\partial x^{2}}=\Lambda_{u}\frac{\partial^{2}U_{m}} {\partial x^{2}}\end{array}\right. \tag{3}\]  
In the formula, \(\Lambda_{u}\) is the modulus diagonal matrix after matrix transformation.  
Thus, the D'Alembert form general solution of the modulus voltage and current at any time and place is shown in Formula (4).  
\[\left\{\begin{array}{l}u_{m}(x,t)=u_{m}^{+}(t-x/c_{m}+\alpha_{m}^{+})+u_{m}^ {-}(t+x/c_{m}+\alpha_{m}^{-})\\ i_{m}(x,t)=i_{m}^{+}(t-x/c_{m}+\alpha_{m}^{+})+i_{m}^{-}(t+x/c_{m}+\alpha_{m}^ {-})\end{array}\right. \tag{4}\]  
In the formula, \(\alpha_{m}^{+}\) and \(\alpha_{m}^{-}\) are arbitrary constants, where both the forward traveling wave \(u_{m}^{+}\) and the reverse traveling wave \(u_{m}^{-}\) can be arbitrary functions. The modulus wave velocity formula of each modulus is shown in Formula (5).  
\[v_{m}=\frac{1}{\sqrt{L_{m}C_{m}}} \tag{5}\]  
Equation (5) demonstrates that line-mode and zero-mode traveling waves propagate at distinct velocities due to their different transmission channel parameters, resulting in varying arrival times at the measurement point. This velocity differential between modes enables fault location determination. A key advantage of this approach is that it only requires detection of the initial wave heads for both line-mode and zero-mode traveling waves, thus circumventing the challenges associated with identifying reflected wave signatures.