# An Analog Circuit Fault Diagnosis Method Based on Circle Model and Extreme Learning Machine
## 2 Feature Engineering Based on Circle Model  
### Principle of Circle Modeling  
Figure 1 illustrates the procedure of circle modeling. In Figure 1a, N is a linear time-invariant circuit, \(U_{\text{s}}\) is the voltage phasor of the independent voltage source that stimulates N, \(\dot{U}_{\text{o}}\) is the output  
voltage phasor, \(x\) is a passive element, and \(\dot{U}_{\rm x}\) is the voltage phasor across \(x\). The theory is described below [6; 9; 10]:  
According to the substitution theorem, the passive element \(x\) can be replaced with an independent voltage source with voltage phasor \(\dot{U}_{\rm x}\) (Figure 1b). The voltage phasor \(\dot{U}_{\rm x}\) can be derived by the [PERSON]'s theorem:  
\[\dot{U}_{x}=\dot{U}_{\rm\infty}\frac{z_{x}}{z_{\rm o}+z_{x}} \tag{1}\]  
where, \(\dot{U}_{\rm\infty}\) is the open circuit voltage phasor across terminals a and b in the equivalent circuit (Figure 1b); \(Z_{\rm o}\) is the Thevenin impedance between terminals a and b. According to the [PERSON]'s theorem, \(\dot{U}_{\rm\infty}\) and \(Z_{\rm o}\) are independent from the value of x and uniquely determined by its location and the fault-free elements in N.  
According to the substitution theorem, the output voltage phasor \(\dot{U}_{\rm o}\) in Figure 1a is equal to \(\dot{U}_{\rm o}\) in Figure 1b, where N is stimulated by two independent voltage sources \(\dot{U}_{\rm s}\) and \(\dot{U}_{\rm x}\). According to the superposition theorem, the \(\dot{U}_{\rm o}\) in Figure 1b equals the algebraic sum of the responses to \(\dot{U}_{\rm s}\) or \(\dot{U}_{\rm x}\). Figure 1c,d illustrate the circuit under the operation of \(\dot{U}_{\rm s}\) and that of \(\dot{U}_{\rm x}\) respectively. The following equation holds under the operation of \(\dot{U}_{\rm s}\):  
\[\dot{U}_{\rm o}^{\prime}=\dot{U}_{\rm s}H^{\prime}\left(j\omega\right) \tag{2}\]  
where, \(H^{\prime}(j\omega)\) is independent from the value of x. The following equation holds under the operation of \(\dot{U}_{\rm x}\):  
\[\dot{U}_{\rm o}^{\prime\prime}=\dot{U}_{\rm x}H^{\prime\prime}\left(j\omega\right) \tag{3}\]  
where, \(H^{\prime\prime}(j\omega)\) is independent from the value of x.  
Figure 1: The procedure of circle modeling. According to the superposition theorem, we have:  
\[\dot{U}_{o}=\dot{U}_{o}^{\prime}+\dot{U}_{o}^{\prime\prime}=\dot{U}_{s}H^{\prime} (j\omega)+\dot{U}_{x}\dot{H^{\prime\prime}}(j\omega) \tag{4}\]  
Replacing \(\dot{U}_{x}\) with Formula (1), we have:  
\[\dot{U}_{o}=\dot{U}_{s}H^{\prime}(j\omega)+\dot{U}_{oc}\frac{z_{x}}{z_{0}+z_{x}} H^{\prime\prime}(j\omega) \tag{5}\]  
\[\Rightarrow\frac{z_{o}}{z_{x}}=\frac{\dot{u}_{oc}H^{\prime\prime}(j\omega)- \left(\dot{U}_{0}-\dot{U}_{s}H^{\prime}(j\omega)\right)}{\dot{U}_{0}-\dot{U}_{ s}H^{\prime}(j\omega)} \tag{6}\]  
\[=\frac{\dot{u}_{oc}H^{\prime\prime}(j\omega)}{\dot{U}_{os}}-1\]  
where,  
\[\dot{U}_{os}=\dot{U}o-\dot{U}_{s}H^{\prime}(j\omega) \tag{7}\]  
Without loss of generality, it is assumed that:  
\[\left\{ \begin{array}{l}Z_{o}=R_{o}+jX_{o}\\ \dot{U}_{os}=U_{osr}+jU_{osj}\\ \dot{U}_{oc}H^{\prime\prime}(j\omega)=m+jn\end{array} \right. \tag{8}\]  
where, \(j\) is the imaginary unit of the imaginary part and denotes the imaginary part. Since \(\dot{U}_{s}\), \(H^{\prime}(j\omega)\), \(\dot{U}_{oc}\), \(H^{\prime\prime}(j\omega)\), and \(Z_{o}\) are independent from \(Z_{x}\), \(R_{0}\), \(X_{0}\), m and n must be independent from \(Z_{x}\), too. Then, Formula (6) can be rewritten as:  
\[\frac{R_{o}+jX_{o}}{Z_{x}}=\frac{mU_{osr}+uU_{osj}}{U_{osr}^{2}+U_{osj}^{2}}-1 +j\frac{uU_{osr}-mU_{osj}}{U_{osr}^{2}+U_{osj}^{2}} \tag{9}\]  
If fault component \(x\) is a resistor and \(Z_{x}=R_{x}\), the following equations hold:  
\[\left\{ \begin{array}{l}\frac{R_{o}}{R_{x}}=\frac{mU_{osr}+uU_{osj}}{U_{osr}^ {2}+U_{osj}^{2}}-1\\ \frac{X_{o}}{R_{x}}=\frac{uU_{osr}-mU_{osj}}{U_{osr}^{2}+U_{osj}^{2}}\end{array} \right.