# Amplitude Distribution of Partial Discharge Signals on Tunnel-Installed High-Voltage Cables
## 2 Cable Model
### Cable Characteristic Admittance Model  
The conductors are coupled to each other in a three-phase cable system, and therefore the voltage and current on each conductor is affected by the voltage and current on other conductors. The three-phase cable system can be transformed into a number of independent two-conductor  
Figure 2: Cross section of tunnel-installed cable system and structure of cable.  
transmission lines by using a modal transformation, then the characteristic admittance matrix \(\mathbf{Y_{c}}\) can be obtained from the distributed parameter matrix \(\mathbf{Z}\) and \(\mathbf{Y}\)[21]:  
\[\mathbf{Y_{c}}=\mathbf{Z}^{-1}\mathbf{T_{V}}\mathbf{\Lambda}\mathbf{T_{V}}^{-1} \tag{5}\]  
where \(\mathbf{\Lambda}\) is the propagation constant matrix of each mode after decoupling and \(\mathbf{T_{V}}\) is the voltage transformation matrix. \(\mathbf{\Lambda}\) and \(\mathbf{T_{V}}\) are the eigenvalues and eigenvector matrix derived from the diagonalization of the matrix product \(\mathbf{Z}\mathbf{Y}\), that is [21]:  
\[\mathbf{T_{V}}^{-1}\mathbf{Z}\mathbf{Y}\mathbf{T_{V}}=\mathbf{\Lambda}^{2} \tag{6}\]  
Similar to the two-conductor transmission line, the characteristic admittance matrix \(\mathbf{Y_{c}}\) characterizes the relationship between the voltage and the current on the multi-conductor transmission line. Based on the assumption that the cable is completely symmetrical, the extended matrix of the Clarke transform matrix can be used as the voltage transform matrix \(\mathbf{T_{V}}\)[20]. Equations (1), (2), and (6) are substituted into Equation (5) and simplified to obtain \(\mathbf{Y_{c}}\):  
\[\mathbf{Y_{c}}=\left[\begin{array}{cccccc}Ycs&0&0&-Ycs&0&0\\ 0&Ycs&0&0&-Ycs&0\\ 0&0&Ycs&0&0&-Ycs\\ -Ycs&0&0&Yse+Ycs+2 Yss&-Yss&-Yss\\ 0&-Ycs&0&-Yss&Yse+Ycs+2 Yss&-Yss\\ 0&0&-Ycs&-Yss&-Yss&Yse+Ycs+2 Yss\end{array}\right] \tag{7}\]  
where:  
\[Y_{cs}=\sqrt{\frac{y_{cs}}{z_{cc}-z_{ss}}} \tag{8}\]  
\[Y_{se}=\sqrt{\frac{y_{ss}-2y_{mu}-y_{cs}}{2z_{mu}+z_{ss}}} \tag{9}\]  
\[Y_{ss}=\frac{1}{3}\sqrt{\frac{y_{ss}+y_{mu}-y_{cs}}{z_{ss}-z_{mu}}}-\frac{1}{3 }\sqrt{\frac{y_{ss}-2y_{mu}-y_{cs}}{2z_{mu}+z_{ss}}} \tag{10}\]  
where \(Y_{cs}\) is the coaxial characteristic admittance, namely, the reciprocal of the characteristic impedance of a single coaxial cable. \(Y_{cs}\) is only related to the structure of the cable, usually between 0. 025 S and 0.1 S. \(Y_{se}\) is the characteristic admittance between the cable sheath and the earth, and \(Y_{ss}\) is the characteristic admittance between the sheaths of the cable. \(Y_{se}\) and \(Y_{ss}\) are affected by the external environment of the cable system. If \(\mathbf{Y_{c}}\) is regarded as a nodal admittance matrix, it can be expressed in the form of self-admittance and mutual admittance, as shown in Figure 3.  
Figure 3: Three-phase cable characteristic admittance model.  
Figure 4 shows the frequency variation curves of the real parts of the characteristic admittances \(Y_{cs}\), \(Y_{sc}\), and \(Y_{ss}\). The laying parameters are shown in Table 1 and the relative permeability value is 1. The real parts of the characteristic admittances changed very slowly at high frequencies. At the same time, the imaginary parts are 1 to 4 orders of magnitude smaller than their real parts, and thus can be ignored [20]. Considering that the detection band of the high-frequency current method is usually between 3 to 30 MHz, the characteristic admittances \(Y_{cs}\), \(Y_{se}\) and \(Y_{ss}\) can be regarded as fixed values in this range. In this paper, the admittances are calculated at 10 MHz.