## 2 EMTR and Correlation Estimator Methods  
### EMTR-Based Fault Location Method in Transmission Lines  
To understand the application of EMTR to diagnose the fault location in transmission lines, Figure 1 is taken as a reference, where \(Z_{T}\) and \(Z_{C}\) represent the equivalent terminal impedance and the characteristic impedance of the line, respectively, while \(V_{f}(t)\) is the fault voltage of the source, which can be a step-like function [7], or the exponential charge [8], and the fault impedance was assumed to be zero. This method can be summed up in three steps as follows [7]:  
(I) Forward propagation stage: In this stage, fault-generated transient signals are measured at one of the line extremities, such as \(V_{m}(t)\) at x = 0 in Figure 1. (II) Back-propagation stage: The measured transients in Step 1 are time-reversed and back-injected into the simulated system model from the same observation point, e.g., \(V_{m}(T-t)\). Then, on each pre-defined GFL (\(x_{g}\)), the fault signals are observed. (III) Finally, the fault location is computed by employing different proposed metrics. Figure 1: Simplified representation of post-fault line. The recorded transients in the first step along with the necessary details are expressed in the frequency domain as follows:  
\[ \begin{split}& V_{m}(\omega)=\frac{(1+\rho)e^{-\gamma z_{f}}}{1+ \rho e^{-2\gamma z_{f}}}V_{f}\\ &\rho=\frac{Z_{T}-Z_{\text{C}}}{Z_{T}+Z_{\text{C}}}\\ &\gamma=\sqrt{Z^{\prime}Y^{\prime}}\\ & Z_{\text{C}}=\sqrt{\frac{Z^{\prime}}{Y}}\end{split}  \tag{1}\]  
where \(x_{f}\), \(\rho\), \(\gamma\), \(Z^{\prime}\), and \(Y^{\prime}\) are the fault location, reflection coefficient, propagation constant, total per-unit length impedance, and per-unit admittance, respectively. Impedance and admittance are defined as follows:  
\[ \begin{split}& Z^{\prime}=j\omega L^{\prime}+Z^{\prime}_{w}+Z^{ \prime}_{g}\\ & Y^{\prime}=\frac{(G^{\prime}+j\omega C^{\prime})Y^{\prime}_{g}}{ G^{\prime}+j\omega C^{\prime}+Y^{\prime}_{g}}\end{split}  \tag{2}\]  
where \(Z_{w}\), \(L^{\prime}\), \(C^{\prime}\), \(G^{\prime}\), \(Z_{g}\), and \(Y_{g}\) are the wire impedance, inductance, capacitance, conductance, ground impedance, and ground admittance, respectively, further defined as:  
\[ \begin{split}& Z^{\prime}_{w}=\frac{1}{2\pi r_{w}}\sqrt{\left( \frac{i\omega\mu_{w}}{2\sigma_{w}}\right)}\\ & L^{\prime}=\frac{R_{0}}{2\pi}\ln\left(\frac{2\mu}{\sigma_{w}} \right)\\ & C^{\prime}=\frac{2\pi r_{0}}{1+\left(\frac{R_{0}}{2\pi}\right)} \\ & G^{\prime}=\frac{\sigma_{w}}{\epsilon_{0}}C^{\epsilon}\\ & Z^{\prime}_{g}=\frac{|\omega\mu_{0}}{2\pi}\ln\left(\frac{1+ \gamma_{g}h}{\gamma_{g}h}\right)\\ & Y^{\prime}_{g}=\frac{\gamma g^{2}}{Z^{\prime}_{g}}\\ &\gamma_{g}=\sqrt{j\omega\mu_{0}(\sigma_{g}+j\omega\epsilon_{0} \epsilon_{g})}\end{split}  \tag{3}\]  
where \(h\), \(r_{w}\) are the height and radius of the conductor, \(\sigma_{air}\), \(\sigma_{g}\), and \(\sigma_{w}\) are the conductivity constants of the air, ground, and wire respectively, while \(\mu_{0}\), \(\epsilon_{0}\), and \(\gamma_{g}\) are the permeability, permittivity, and propagation constant in ground respectively. It is worth noticing that the line parameter expressions defined above, namely (2) and (3), are based on the line configuration with its equivalent circuit shown in Figure 2. The following is a mathematical representation of Step 2, in which the conjugated recorded transients, i.e., \(V_{m}\)*, are sent back to the system [7; 15]:  
\[ \begin{split}& V_{m}{}^{*}=conj(V_{m})\\ & V(x_{g},\omega)=\frac{(1+\rho)e^{-\gamma z_{g}}}{1+\rho e^{-2 \gamma z_{g}}}V_{m}{}^{*}\\ & V(x_{g},\omega)=\frac{(1+\rho)^{2}e^{-\gamma\left(x_{g}-x_{f} \right)}}{\left(1+\rho e^{-2\gamma z_{g}}\right)\left(1+\rho e^{+2\gamma z_{f} }\right)}V_{f}{}^{*}\end{split}