# Amplitude Distribution of Partial Discharge Signals on Tunnel-Installed High-Voltage Cables
## 3 Amplitude Distribution Characteristics of Signals on Cross-bonding Wires  
### Current Reflection at Cross-Bonding Links  
The characteristic admittance matrix \(\mathbf{Y_{c}}\) of a three-phase cable system changes after the cross-bonding link, causing current reflection. If the cross-bonded line length is ignored, the transmission coefficient \(\mathbf{T_{bond}}\) is as follows [21]:  
\[\mathbf{T_{bond}}=E+\Big{(}\mathbf{M}\mathbf{Y_{c}}\mathbf{M^{T}}-\mathbf{Y_{c}}\Big{)}\Big{(}\mathbf{ M}\mathbf{Y_{c}}\mathbf{M^{T}}+\mathbf{Y_{c}}\Big{)}^{-1} \tag{11}\]  
where \(E\) is a \(6\times 6\) identity matrix. \(\mathbf{M}\) represents the rearrangement of the characteristic admittance matrix \(\mathbf{Y_{c}}\) after cross-bonding:  
\[\mathbf{M}=\left[\begin{array}{cccccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 0&0&0&1&0&0\end{array}\right] \tag{12}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Laying Parameters**} \\ \hline Tunnel radius \(R\)/m & 2 & Earth resistivity \(\rho\)/(\(\Omega\cdot\)m) & 100 \\ Cable center spacing \(d\)/m & 0.2 & Cable distance from the tunnel wall \(R\)-\(b\)/m & 0.5 \\ \hline \multicolumn{3}{c}{**Cable parameters (YJLW03 127/220 1 \(\times\) 1000)**} \\ \hline Inner conductor radius \(r_{1}\)/mm & 19 & Relative dielectric constant of insulation \(\varepsilon_{1}\) & 2.3 \\ Insulation radius \(r_{2}\)/mm & 46 & Relative dielectric constant of sheath \(\varepsilon_{2}\) & 3 \\ Sheath radius \(r_{3}\)/mm & 48.4 & Inner conductor resistivity \(\rho_{1}\)/(\(\Omega\cdot\)m) & \(1.7\times 10^{-8}\) \\ Cable radius \(r\)/mm & 53.4 & Sheath resistivity \(\rho_{2}\)/(\(\Omega\cdot\)m) & \(2.1\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of layout and cable.  
Figure 4: Relationship between the real parts of the characteristic admittances and the frequency.  
Equation (7) is then substituted into Equation (11) and the matrix form of the transmission coefficient \(\mathbf{T_{\text{bond}}}\) is obtained:  
\[\mathbf{T_{\text{bond}}}=\left[\begin{array}{cccccc}1&T_{cc}&-T_{cc}&T_{sc}&0&-T_{ sc}\\ -T_{cc}&1&T_{cc}&-T_{sc}&T_{sc}&0\\ T_{cc}&-T_{cc}&1&0&-T_{sc}&T_{sc}\\ T_{cs}&-T_{cs}&0&1&-T_{ss}&T_{ss}\\ 0&T_{cs}&-T_{cs}&T_{ss}&1&-T_{ss}\\ -T_{cs}&0&T_{cs}&-T_{ss}&T_{ss}&1\end{array}\right] \tag{13}\]  
where,1  
\[\left\{\begin{array}{c}T_{cc}=T_{ss}=\frac{1}{3+4\alpha}\\ T_{cs}=\frac{2+2\alpha}{3+4\alpha}\\ T_{sc}=\frac{3+4\alpha}{3+4\alpha}\end{array}\right. \tag{14}\]  
\[\alpha=\frac{3Y_{ss}+Y_{se}}{Y_{cs}} \tag{15}\]  
where \(T_{cc}\), \(T_{cs}\), \(T_{ss}\) and \(T_{sc}\) are the reflection coefficients from inner conductor to inner conductor, from inner conductor to metal sheath, from metal sheath to inner conductor, and from metal sheath to metal sheath, respectively. According to the characteristic admittance model, \(Y_{ss}\) and \(Y_{se}\) are the characteristic admittances between the cable and the external environment, while \(Y_{cs}\) is the characteristic admittance inside the cable. Therefore, \(\alpha\) can be regarded as the ratio of the external characteristic admittance to the internal characteristic admittance. From Equation (14), it can be seen that the reflection of the current at the cross-bonding is only related to the characteristic admittance ratio \(\alpha\).