# An Analog Circuit Fault Diagnosis Method Based on Circle Model and Extreme Learning Machine
## 3 Test Verification  
### Simulation Test 1  
The Tow-Thomas filter (Figure 7) was selected for simulation test 1. The tolerances of the elements fall within [-10%,10%]. The fault states of elements fall outside the tolerance interval: [10%,50%]U[-50%,-10%]. The ELM is a classification algorithm based on the activation function. The classification effect of the algorithm depends on the test accuracy of the activation function relative to sample features. Hence, three popular activation functions, namely, Multiquadrics, Gaussian and Sigmoid, were selected for training and testing. The results of the three functions are recorded in Table 3 below. Figure 7: The Tow–Thomas filter. As shown in Table 3, the Gaussian function achieved the highest training and test accuracies in fault diagnosis. Therefore, the Gaussian function was adopted as the activation function of our ELM classification algorithm. Then, the fuzzy groups and the indistinguishable points at the current frequency were identified quickly by the circle model. The identification results are listed in Table 4.  
As shown in Table 4, there were a total of five fuzzy groups, in which (0,0) stand for the voltage values that are indistinguishable at the current frequency. Hence, this group was subjected to frequency adjustment, while the other four groups were classified. Through the identification of fuzzy groups, four groups that may affect the classification decision are screened out, reducing the influence of fuzzy groups on classification accuracy. During feature construction, three different features were created, including the 3D feature (\(x_{f_{1}},y_{f_{1}},x_{f_{2}}\)) designed by our method, the uncompressed 4D feature (\(x_{f_{1}},y_{f_{1}},x_{f_{2}},y_{f_{2}}\)) and the 2D feature (\(x_{f_{1}},y_{f_{1}}\)) before frequency adjustment. The three types of features were all normalized and compared with the original 3D feature. The training and test accuracies of the three normalized features and the original 3D feature are compared in Table 5 below. As shown in Table 4, normalized 3D feature and normalized 4D feature had comparable accuracies, both of which are higher than those of the original 3D feature. This means the normalization method in our feature engineering can effectively improve the diagnosis accuracy. In addition, the 2D feature, which had not undergone frequency adjustment, were much poorer than the original 3D feature, which was designed by our feature engineering approach, in test accuracy. The comparison shows the frequency adjustment can make features that are indistinguishable in 2D space distinguishable in 3D space. The addition of the 3D feature can effectively enhance the test accuracy, proving the validity of our feature engineering method. \begin{table}
\begin{tabular}{c c} \hline
**Fuzzy Groups** & **Fault Elements** \\ \hline
1 & R1 \\ \hline
2 & R2 \\ \hline
3 & C1, R6 \\ \hline
4 & C2, R3, R4, R5 \\ \hline
5 & (0,0) \\ \hline \end{tabular}
\end{table} 
Table 4: Identification of fuzzy groups for simulation test 1. \begin{table}
\begin{tabular}{c c c} \hline
**Activation Function** & **Training Accuracy** & **Test Accuracy** \\ \hline Multiquadrics & 92.4 & 91.2 \\ \hline Gaussian & 94.7 & 93.8 \\ \hline Sigmoid & 93.9 & 92.6 \\ \hline \end{tabular}
\end{table} 
Table 3: The training and test results of different activation functions. \begin{table}
\begin{tabular}{c c c} \hline
**Feature Engineering** & **Training Accuracy** & **Test Accuracy** \\ \hline Normalized 3D feature & 94.7 & 93.8 \\ \hline Normalized 4D feature & 94.5 & 93.2 \\ \hline Original 3D feature & 93.9 & 92.8 \\ \hline Normalized 2D feature & 90.1 & 88.4 \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison between different feature engineering approaches. As shown in Table 6, [PERSON]'s method and [PERSON]'s method exhibited low training and test accuracies, because neither of them considers fuzzy groups. By contrast, our method reclassifies faults based on fuzzy groups, and thus achieved much better accuracy in fault diagnosis.