# Amplitude Distribution of Partial Discharge Signals on Tunnel-Installed High-Voltage Cables
## 2 Cable Model  
### Distributed Parameters of Tunnel-Installed Cables  
The three-phase cable can be regarded as a transmission line system consisting of 6 conductors. The propagation characteristics of the voltage and current on the conductors are determined by the distributed impedance matrix \(\mathbf{Z}\) (unit: \(\Omega\)) and the distributed admittance matrix \(\mathbf{Y}\) (unit: S). The skin effect in the metal sheath is enhanced at high frequencies and the current is bound to the inner surface and outer surface of the sheath, which makes the transfer impedance decrease rapidly and it can be ignored [20]. In this case, the distributed parameter matrix \(\mathbf{Z}\) and \(\mathbf{Y}\) of the three-phase cable are as follows:  
\[\mathbf{Z}=\left[ \begin{array}{ccccc}z_{cc}&z_{mu}&z_{mu}&z_{mu}&z_{ss}&z_{mu}&z _{mu}\\ z_{mu}&z_{cc}&z_{mu}&z_{mu}&z_{mu}&z_{ss}&z_{mu}\\ z_{mu}&z_{mu}&z_{cc}&z_{mu}&z_{mu}&z_{mu}&z_{ss}\\ z_{ss}&z_{mu}&z_{mu}&z_{s}&z_{mu}&z_{mu}&z_{mu}\\ z_{mu}&z_{ss}&z_{mu}&z_{mu}&z_{s}&z_{mu}&z_{mu}\\ z_{mu}&z_{mu}&z_{ss}&z_{mu}&z_{mu}&z_{s}&z_{mu}\\ \end{array} \right] \tag{1}\]  
Figure 1: High-frequency current method applied to a three-phase cross-bonded cable unit. \[\mathbf{Y}=\left[ \begin{array}{cccccc}y_{cs}&0&0&-y_{cs}&0&0\\ 0&y_{cs}&0&0&-y_{cs}&0\\ 0&0&y_{cs}&0&0&-y_{cs}\\ -y_{cs}&0&0&y_{ss}&-y_{mu}&-y_{mu}\\ 0&-y_{cs}&0&-y_{mu}&y_{ss}&-y_{mu}\\ 0&0&-y_{cs}&-y_{mu}&-y_{mu}&y_{ss}\end{array} \right] \tag{2}\]  
where \(z_{cc}\), \(z_{ss}\), \(z_{ss}\), and \(z_{mu}\) represent the self-impedance of the inner conductor, self-impedance of the metal sheath, and mutual impedance between the conductors, respectively. \(y_{ss}=y_{se}+y_{cs}+2y_{mu}\), where \(y_{cs}\), \(y_{se}\), and \(y_{mu}\) are the admittance of the inner conductor, admittance of the metal sheath to the earth, and mutual admittance between the metal sheaths, respectively. The formulas to calculate the impedance and admittance of directly buried cables have been given in [14; 15]. The calculation of the admittance of tunnel-installed cables is the same, but the self-impedance \(z_{s}\) and mutual impedance \(z_{m}\) used to obtain the impedances \(z_{cc}\), \(z_{ss}\)\(z_{ss}\), and \(z_{mu}\) need to be calculated by [PERSON]'s high-frequency formulas. Figure 2 shows the structure of the tunnel-installed cables, where \(R\) represents the radius of tunnel, \(b\) represents the distance from cable to the tunnel center, \(d\) represents the center spacing of the cables, \(\theta\) represents the included angle between two cables, and \(r_{1}\), \(r_{2}\), and \(r_{3}\) represent the radius of the inner conductor, insulating layer, and metal sheath respectively. In this paper, the unit of all the length variables is meter and that of the angle is rad. Then the self-impedance \(z_{s}\) and mutual impedance \(z_{m}\) of the cable-ground loop are [18]:  
\[ \begin{array}{l}z_{s}=\frac{|z_{0}\mu_{0}}{2\pi}\ln\frac{R}{d}+\frac{m\rho}{ 2\pi R}\frac{K_{0}(mR)}{K_{1}(mR)}+\frac{m\rho}{2\pi R}\sum\limits_{n=1}^{ \infty}\frac{\cos n\rho}{n}\Big{(}\frac{R^{2}}{R^{2}}\Big{)}^{n}\\ \times\left[\frac{\frac{mR}{d}K_{n}^{2}(mR)+nK_{n}(mR)}{\frac{mR}{mR}K_{n}(mR) -K_{n}^{\prime}(mR)}\right]\\ \\ z_{m}=\frac{|z_{0}\mu_{0}}{2\pi}\ln\frac{R}{d}+\frac{m\rho}{2\pi R}\frac{K_{0 }(mR)}{K_{1}(mR)}+\frac{m\rho}{2\pi R}\sum\limits_{n=1}^{\infty}\frac{\cos n \rho}{n}\Big{(}\frac{R^{2}}{R^{2}}\Big{)}^{n}\\ \times\left[\frac{\frac{mR}{d}K_{n}^{2}(mR)+nK_{n}(mR)}{\frac{mR}{mR}K_{n}(mR) -K_{n}^{\prime}(mR)}\right]\end{array}  \tag{4}\]  
where \(\rho\) (unit: \(\Omega\)-m) and \(\mu_{e}\) (no units) are the resistivity and the relative permeability of the earth respectively, \(\omega\) (unit: rad/s) is the angular frequency, \(m=(j\omega\mu_{e}\mu_{0}/\rho)^{1/2}\) and \(K_{n}\) represents the \(n\)-th order second modified Bessel function. To simplify the calculation, assume that the three cables are completely symmetrical, that is, the structure parameters of each cable are the same, and the distance and angle between any two cables are equal.