# Amplitude Distribution of Partial Discharge Signals on Tunnel-Installed High-Voltage Cables
## 3 Amplitude Distribution Characteristics of Signals on Cross-bonding Wires
### Amplitude Distribution of PD Signal  
PD usually occurs only in one phase of the three-phase cables. Pulse currents with the same amplitude and opposite polarity will be induced in the inner conductor and metal sheath of the cable [22]. The currents propagate in a coaxial mode, so the distribution of the current amplitude on each conductor is not affected by attenuation. Assuming that the PD is in phase A, the incident current at its adjacent cross-bonding joint is \(\mathbf{I_{\text{inc}}}=[I_{\text{P}}\ 0\ 0-I_{\text{P}}\ 0\ 0]^{\text{T}}\), where \(I_{\text{P}}\) represents the amplitude of the current when it propagates to the cross-bonding joint. The unit of the current variables is A. The current amplitude \(\mathbf{I_{\text{cb}}}\) at the cross-bonding joint is the product of the transmission coefficient \(\mathbf{T_{\text{bond}}}\) and the incident current \(\mathbf{I_{\text{inc}}}\):  
\[\mathbf{I_{\text{cb}}}=\mathbf{T_{\text{bond}}}\mathbf{I_{\text{inc}}}=\frac{I_{p}}{4 \alpha+3}\left[\begin{array}{c}4\alpha+1\\ 1\\ 1\\ -2\alpha-1\\ -1\\ -2\alpha-1\end{array}\right] \tag{16}\]  
The 4 th, 5 th, and 6 th elements in \(\mathbf{I_{\text{cb}}}\) are the currents of the cross-bonding wires in phase A, B, and C (the phase sequence is based on the phase of the discharge side, as shown in Figure 1). When the PD occurs inside the cable, the amplitude distributions of the signals on its adjacent cross-bonding wires have the following characteristics:  
1. The current signals on the two cross-bonding wires connected to the defective phase are equal and denoted as \(I_{\text{major}}\). In the case of A-B, B-C, and C-A, as shown in Figure 1, if PD occurs in phase A, the signals on wire A and wire C are the same.  
The current signal on the remaining cross-bonding wire is denoted as \(I_{\text{minor}}\). The current \(I_{\text{minor}}\) has the same polarity with \(I_{\text{major}}\), but the amplitude is smaller. The proportional relationship between the two currents is determined by the characteristic admittance ratio \(\alpha\):  
\[\frac{I_{major}}{I_{minor}}=1+2\alpha \tag{17}\]  
When the PD current signal passes through the first cross-bonding joint, it will no longer be composed of only the coaxial mode. It comprises the coaxial mode, the inters heath mode, and the earth mode [23]. The propagation velocity and attenuation rates of the three modes are different to each other, so a single pulse will gradually attenuate and be separated into multiple pulses as the propagation distance increases. Unlike the directly buried cables, the propagation velocity of the inters heath mode and the earth mode is often greater than or close to the propagation velocity of the coaxial mode for tunnel-installed cables, and thus cannot be neglected [19]. At the same time, the attenuation rate of the inters heath mode and the earth mode will vary significantly with frequency. Therefore, the amplitude distribution of the PD signal after passing through the first cross-bonding joint will become very complicated and may exhibit different distribution characteristics, which limits its guidance effect on on-site detection. For that reason, this paper focuses on the amplitude distribution at the adjacent joints to the PD source.