# An Analog Circuit Fault Diagnosis Method Based on Circle Model and Extreme Learning Machine
## 2 Feature Engineering Based on Circle Model
### Circle Model-Based Feature Engineering
#### 2.2.1 Circle Model-Based Feature Extraction
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Fault Source**} & \multirow{2}{*}{**Sampled Value**} & \multirow{2}{*}{**Output Voltage**} & \multicolumn{2}{c}{**Model Parameter**} \\ \cline{3-5}  & & & **Circle Center (a, b)** & **Radius r** \\ \hline \multirow{2}{*}{R1} & \(1\,\Omega\) & \(2.383\times 10^{4},3.158\times 10^{4}\) & \multirow{2}{*}{–} & \multirow{2}{*}{–} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(2.394\times 10^{-2},3.176\times 10^{-2}\) & & \\ \hline \multirow{2}{*}{R2} & \(1\,\Omega\) & \(5.080\times 10^{-4},5.548\times 10^{-4}\) & \multirow{2}{*}{(–15.393, 14.093)} & \multirow{2}{*}{20.870} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(4.261,7.073\) & & \\ \hline \multirow{2}{*}{R3} & \(1\,\Omega\) & \(6.061\times 10^{-8},6.283\times 10^{-4}\) & \multirow{2}{*}{(3.3031, -0.0005)} & \multirow{2}{*}{3.303} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(6.108,1.744\) & & \\ \hline \multirow{2}{*}{R4} & \(1\,\Omega\) & \(6.104\times 10^{-4},6.283\) & \multirow{2}{*}{(–0.8641, 3.1416)} & \multirow{2}{*}{3.258} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(6.114\times 10^{-2},1.746\times 10^{-2}\) & & \\ \hline \multirow{2}{*}{R5} & \(1\,\Omega\) & \(3.585\times 10^{-4},-2.25\times 10^{-4}\) & \multirow{2}{*}{(1.0614, 1.6897)} & \multirow{2}{*}{1.995} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(-4.321\times 10^{-1},3.013\) & & \\ \hline \multirow{2}{*}{R6} & \(1\,\Omega\) & \(-4.225\times 10^{-5},3.000\times 10^{-4}\) & \multirow{2}{*}{(2.758, 0.4105)} & \multirow{2}{*}{2.788} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(3.642,-2.234\) & & \\ \hline \multirow{2}{*}{R7} & \(1\,\Omega\) & \(-1.573\times 10^{-4},5.557\times 10^{-4}\) & \multirow{2}{*}{(2.3839, 0.6925)} & \multirow{2}{*}{2.483} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(4.861,8.568\times 10^{-1}\) & & \\ \hline \multirow{2}{*}{R8} & \(1\,\Omega\) & \(4.863,8.145\times 10^{-1}\) & \multirow{2}{*}{(5.1950, 3.6332)} & \multirow{2}{*}{2.838} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(3.988,6.202\) & & \\ \hline \multirow{2}{*}{R9} & \(1\,\Omega\) & \(4.038,6.224\) & \multirow{2}{*}{(5.1950, 3.6327)} & \multirow{2}{*}{2.838} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(4.810,8.212\times 10^{-1}\) & & \\ \hline \multirow{2}{*}{R10} & \(1\,\Omega\) & \(-0.6481,3.870\) & \multirow{2}{*}{(5.1950, 3.6332)} & \multirow{2}{*}{2.838} \\ \cline{2-2} \cline{5-5}  & \(100\,\mathrm{M}\Omega\) & \(4.863,8.145\times 10^{-1}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fault model parameter of the Leapfrog circuit. The three points of the measured data are taken to determine the characteristics of the analog circuit curves in Figure 3. It should be pointed out that because the parameter value of an actual analog component cannot be negative, the characteristic curve may not be a whole circle but an arc. It can be seen from the Table 1 and Figure 3 that the component set [R6, C2] has the same failure model (2.758,0.4105) 2.788, so they can't be distinguished and form a fuzzy group. Similarly, [R7,R13] and [R8,R9,R10,R12] are also fuzzy groups. The result of grouping is shown in Table 2. Except for the components of fuzzy group, other components have unique models.