# Reliability Modelling and Evaluation for LTD System Based on Load-Sharing Model
## 2 Reliability Model of a Linear Transformer Driver (LTD) System
### System Reliability Model  
The pulsed outputs generated by the LTD system are critical to evaluate whether the LTD experiment successes or not. Considering that the circuit subsystem is directly related to the output performance, so we establish the reliability model of the circuit subsystem firstly and then extend it by considering the trigger unit failure. For convenience of description, Table 1 gives some basic notations used in this section. According to the hierarchical and modular structure used, the reliability model of the circuit subsystem could be established. Switches self-discharge before the trigger command is a most common cause to influence reliability, which is the core failure mode we considered. [PERSON] [34] and [PERSON] [35] indicated that the discharge voltage distribution of gas spark switch under thousands of amps current \begin{table}
\begin{tabular}{c c c} \hline
**Order** & **Name** & **Abbreviation** \\ \hline
1 & circuit subsystem & c \\
2 & branch & \(b_{i}\) \\
3 & level & \(l_{ij}\) \\
4 & module & \(m_{ijs}\) \\
5 & brick/circuit unit & \(u_{ijst}\) \\
6 & switch & sw \\ \hline \end{tabular} Where \(i=1,2,\ldots,30\), \(j=1,2,3,4\), \(s=1,2,\ldots,30\), \(t=1,2,\ldots,20\) represent the indexes of the branch, level, module, brick respectively.  
\end{table} 
Table 1: Some basic notations. circumstance is subject to three-parameter Weibull distribution [36; 37], which is applicable to the circumstance in the LTD system, namely:  
\[F(v)= \begin{cases}0&v<\gamma\\ 1-exp\big{[}-((v-\gamma)/\eta)^{\beta}\big{]}&v\geq\gamma\end{cases}  \tag{1}\]  
where \(v\) is the working coefficient here, \(F(v)\) is the failure probability, \(\eta>0\) is the scale parameter, \(\beta>0\) is the shape parameter, and \(\gamma>0\) is the location parameter. If we only focus on the case \(v\geq\gamma\), then the switch reliability \(R(sw)\) can be calculated by:  
\[R(sw)=1-F(v)=exp\big{[}-((v-\gamma)/\eta)^{\beta}\big{]} \tag{2}\]  
Correspondingly, the initial module reliability \(R\big{(}m_{ijs}\big{)}\) is:  
\[R\big{(}m_{ijs}\big{)}=\prod_{1\leq t\leq 20}R\big{(}u_{ijst}\big{)}=exp\big{[}-2 0((v-\gamma)/\eta)^{\beta}\big{]} \tag{3}\]  
The fault criteria show that only are there more than \(k\) modules failed in a level, is the level judged to fail. Considering the load-sharing feature, different failure sequences will lead to different load redistribution results. We define the failure sequence \(S^{k}\) whose elements are \(k\) different integers ranging from 1 to 30 as:  
\[S^{k}=s_{1}s_{2}\ldots s_{k-1}s_{k},1\leq s_{1},\ldots,s_{k}\leq 30,s_{1}\
eq \ldots\
eq s_{k} \tag{4}\]  
Then we use \(RS^{k}\big{(}m_{ijs}\big{)}\) to reflect the modules reliability under the failure sequence \(S^{k}\). Specifically speaking, if \(s\
otin S^{k}\), \(RS^{k}\big{(}m_{ijs}\big{)}\) represents the reliability of the module \(m_{ijs}\) after all the modules in the sequence \(S^{k}\) have failed; if \(s\in S^{k}\), \(RS^{k}\big{(}m_{ijs}\big{)}\) represents the reliability of the module \(m_{ijs}\) after all the modules previous to \(s\) in the sequence \(S^{k}\) have failed. Now, considering the load-sharing k-out-of-n: F mechanism, the level reliability \(R\big{(}l_{ij}\big{)}\) can be obtained by:  
\[R\big{(}l_{ij}\big{)}=1-\sum_{\forall\ \mathcal{S}^{k}}\left(1-\prod_{s\
otin S ^{k}}RS^{k}\big{(}m_{ijs}\big{)}\right)\prod_{s\in S^{k}}\left(1-RS^{k}\big{(} m_{ijs}\big{)}\right) \tag{5}\]  
Subsequently, we can write the branch reliability \(R(b_{i})\) and the circuit subsystem reliability \(R(c)\) as  
\[R(b_{i})=\prod_{1\leq j\leq 4}R\big{(}l_{ij}\big{)} \tag{6}\]  
\[R(c)=\prod_{1\leq i\leq 30}R(b_{i}) \tag{7}\]  
When we consider the trigger unit failure, we use the function \(T(m)\) to represent a trigger strategy, that is \(T\Big{(}m_{ijs}\big{)}\) represents the corresponding trigger unit connected to the module \(m_{ijs}\).