# Numerical Analysis of a Spiral Tube Damping Busbar to Suppress VFTO in 1000 kV GIS
## 2 The Suppression Mechanism of the New Method
### The Equivalent Circuit of the Damping Busbar  
The equivalent circuit of the damping busbar is resistance and inductance connected in series with the GIS bus, as shown in Figure 3. \(L_{i}\)(i = 0,1,2,... ) represents the inductance of each unit coil of the metal spiral tube conductor. \(R_{i}\) and \(L_{Ri}\)(i = 0,1,2,... ) represent the noninductive resistance and its residual inductance of the parallel connection of each turn, respectively, and the value of the resistance can be adjusted to achieve the best damping effect. \(g_{i}\)(i = 0,1,2,... ) is the following gap of the damping busbar and \(r_{i}\)(i = 0,1,2,... ) is the arcing resistance of the gap, which represents the losses formed by the discharge channel.  
Inductance Calculation of the Damping Busbar at High Frequency-Damping Busbar Parameters  
The purpose of this section is to calculate a fairly precise value of the inductance. Determining the inductance was carried out by studying the effect of high-frequency fields with all the geometrical parameters of the busbar [28]. To this end, a calculation method for the self-inductance of the damping busbar at high frequency was developed and analyzed by the finite element method (FEM). In order to design the damping busbar, we needed to use a simulation method to obtain the VFTO waveform and then calculate the VFTO distributions. Thus, the mathematical expression of the VFTO waveform was theoretically calculated using the curve fitting method to get the fitting data of the VFTO waveform [29]. Then, the curve fitting method was applied to get the fitting data of the associated very fast transient current (VFTC) waveform for a single SF\({}_{6}\) gas discharge. Due to VFTC being assigned as current excitation for the simulation procedures, the VFTC equation was obtained for Fourier 8, with goodness of fit parameters R-square = 0.9768 and adjusted R-square = 0.9759.  
Figure 3: The distributed parameters of the equivalent circuit of the damping busbar.  
Figure 2: The structure of the damping busbar.  
\[VFTC(t)=a_{0}+\sum_{i=1}^{8}a_{i}\cdot\cos(i\omega t)+\sum_{i=1}^{8}b_{i}.\sin(i \omega t) \tag{1}\]  
Table 1 illustrates the coefficients of the VFTC equation.  
The aim of the simulation study was to calculate the equivalent inductance of the damping busbar at high frequency, so some assumptions were proposed for ease of simulation. In this paper, the simple geometry model was imported into ANSYS Maxwell, and the eddy current solution type was chosen. However, this design depends on many variables according to ANSYS analysis: material properties (aluminum); the geometry of the busbar (busbar body); and operating frequency and input current (the VFTC frequency is 2 MHz and the VFTC peak value is 8256 A--see Equation (1)--for singular switching opening operation, as shown in Figure 4). All of these can be included in design sweeps to study how the impedance matrix--and, therefore, the inductance--is affected by changes in these properties (frequency sweep setup: set the frequency field from 1 to 2 MHz with 0.1 MHz step size).  
The simulation results show a decrease in the inductance value of the damping busbar taking place for different frequencies, as illustrated in Figure 5. Notably, however, Figure 5 shows that the inductance value of the damping busbar is 0.33656 mH at 2 MHz. As a consequence, the damping busbar parameters are illustrated in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**Values**} \\ \hline \(i\) & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \(a_{i}\) & 1025 & 403.3 & –7.241 & 809.5 & –28.8 & 46.59 & 23.15 & –1149 & 1881 \\ \(b_{i}\) & 0 & 642.6 & 2540 & –3334 & –808.4 & 161.9 & –859.5 & 1063 & –1489 \\ \(\omega\) & \(1.372\times 10^{7}\) & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Coefficients of the very fast transient current (VFTC) equation.  
Figure 4: VFTC for a single breakdown at 0.0195 s.