# Design of a Compact 1-MV-Class Radial Closing Switch for Tesla Type Generator
## 3 Switch Parameters  
The operating voltage of this switch is up to 1 MV. The gap separation and the gas pressure must be adjusted according to the Paschen Law [29] in order that the switch can work properly. Therefore, these two parameters need to be first determined. The breakdown characteristics of SF\({}_{6}\) under nanosecond impulses were thoroughly studied by [PERSON] and the conclusions were summarized in her Ph. D. thesis [28]. Two important characteristics are provided in ref. [28]: the breakdown field per atm of SF\({}_{6}\) in a uniform electric field, i.e., 88.5 kV/(cm-atm), and the surface flashover electric field per atm of SF\({}_{6}\), i.e., 73.5 kV/(cm-atm). Since a quasi-uniform electric field is expected to be generated between the switch electrodes, the above data are useful, and the operating parameters of the radial switch can therefore be established. In ref. [1], the following formula is suggested to calculate the breakdown voltage for common insulation gases, such as air, N\({}_{2}\), H\({}_{2}\), and SF\({}_{6}\):  
\[U_{b}=c_{0}\cdot pd+c(pd)^{0.5} \tag{1}\]  
where \(U_{b}\) is in kV, \(pd\) is in cm\(\cdot\)atm. Two sets of (\(U_{b}\), \(p\), \(d\)) are used for SF\({}_{6}\) from ref. [28]. One is (120 kV, 2.7 atm, 0.5 cm); another is (88.5 kV, 1 atm, 1 cm). With these data, \(c_{0}\) and \(c\) are calculated to be 88.54 kV/(cm-atm) and \(-0.04\) (kV/(cm-atm))\({}^{0.5}\), respectively. So, \(U_{b}\) of SF\({}_{6}\) in a uniform filed can be expressed as follows:  
\[U_{b}=88.54 pd-0.04(pd)^{0.5}\approx 88.54 pd(\text{kV})\text{ or }E_{b}\approx 88.54p(\text{kV}/\text{cm}) \tag{2}\]  
With (2), Table 1 lists few examples of gas pressure (\(p\)), gap distance (\(d\)), and breakdown field (\(E_{b}\)) as well as the flashover field (\(E_{f}\)) for a voltage of 1 MV applied between electrodes. It is worth mentioning that once the pressure is fixed, the electric field generated between the switch electrodes should be larger than \(E_{b}\), to ensure the switch closes (\"fires\") while the electric field produced along the two insulator surfaces inside the gas chamber should be smaller than \(E_{f}\), to prevent any surface flashover phenomena. Figure 2: 3-D model of the radial main switch. HV electrode means the electrode mounted as part of the inner conductor of the generator and LV electrode means the grounded electrode. Based on Table 1, the dependences of the switch gap distance and of the breakdown field on SF\({}_{6}\) pressure for a 1-MV voltage, are both presented in Figure 3a. Theoretically, based on the Paschen Law, all the combinations of the pressure and the corresponding gap distance can result in a breakdown at 1-MV voltage. However, a low pressure would correspond to a too long gap and would correspondingly increase the switch resistance and self-inductance. In addition, the jitter would also increase. Therefore, the gas chamber pressure should be high, but it cannot be too high because it would produce large forces acting on the two insulators. In such cases, the insulators must be thick enough to withstand the forces. This means that the total size and the overall weight of such a switch would be unpractical. For the present design, the theoretical pressure was chosen to be around 10 atm and the corresponding switch gap distance was set as 12 mm. In this case, the electric field on the electrode surface should be larger than 885 kV/cm to ensure a breakdown. It is worth mentioning that if the finally-tested voltage is smaller than 1 MV, the pressure in the switch chamber can be increased moderately to realize such a high voltage. The dependence of the electric field threshold on pressure for surface flashover is shown in Figure 3b. The electric field on the insulator surface should be smaller than the threshold value, and this is presented as the highlighted operational region. Once the operating pressure is fixed as 10 atm, the electric field on the insulator's inner surface should be no greater than 735 kV/cm.