## 2 Fault Diagnosis Method Based on System Characteristics
### The Model of RC Motor Current and Excitation Signal Selection  
The stator current signal of induction motor is the most important excitation signal of RC system, and it is closely related to the valve leakage. Moreover, the stator current signal can be directly measured from the motor supply wire. Therefore, the stator current signal of the motor is used as the excitation signal in this paper. This paper takes the A phase, one of the three phases of the power supply system, as an example to study the electromagnetic relationship of the current, ignoring the higher harmonics, in order to study the motor current signal characteristics. The phase A current and magnetic flux of the motor under normal equipment can be expressed [24; 51], as Equations (1) and (2). \[i_{A}=\sqrt{2}I\cos(2\pi f_{s}t-\alpha_{I}), \tag{1}\]  
\[\phi_{A}=\sqrt{2}\phi cos\big{(}2\pi f_{s}t-\alpha_{\phi}\big{)}, \tag{2}\]  
The electromagnetic torque that is generated by the interaction of current and magnetic flux is as shown in Equation (3). \[i_{A}=\sqrt{2}I\cos(2\pi f_{s}t-\alpha_{I}), \tag{3}\]  
\(I,~{}\phi\) represent the RMS amplitude of the supply current and the magnetic flux, respectively. \(\alpha_{I}\), \(\alpha_{\phi}\) is the phase of the current and the flux with respect to the power supply. \(f_{s}\) is the fundamental frequency of the power supply and P is the number of pole pairs of the motor. When the motor drive equipment fails, the system will generate an additional fluctuating torque. The motor will generate an electric torque to balance the fluctuating torque [24; 52]. Assume that the fluctuating torque is a sine wave, with a frequency of \(f_{F}\), the corresponding current amplitude and  
Figure 3: Schematic diagram of RC working process; (**a**) Positive stroke, and (**b**) Negative stroke. phase angle are \(I_{F}\) and \(\alpha_{F}\), respectively. Afterwards, the fluctuating torque expression can be expressed by Equation (4). \[\Delta T=3P\phi I_{F}sin\left[2\pi f_{F}t-\left(\alpha_{I}-\alpha_{\phi}\right)- \alpha_{F}\right], \tag{4}\]  
This oscillating torque causes speed fluctuations, according to the motor torque balance equation, and the fluctuation speed can be derived as Equation (5), so that the fluctuation of the angle can be expressed [24] as in Equation (6). \[\Delta\omega=\frac{P}{J}\int\Delta Tdt=\frac{-3p^{2}\phi I_{F}}{2\pi f_{F}J}cos \left[2\pi f_{F}t-\left(\alpha_{I}-\alpha_{\phi}\right)-\alpha_{F}\right], \tag{5}\]  
\[\Delta\alpha_{F}=\int\Delta\omega dt=\frac{3P^{2}\phi I_{F}}{2\pi^{2}f_{F}^{2} J}\sin\left[2\pi f_{F}t-\left(\alpha_{I}-\alpha_{\phi}\right)-\alpha_{F} \right], \tag{6}\]  
where, \(J\) is the inertia the rotor system. The change of angle leads to the phase modulation to the magnetic flux [24; 52], so Equation (2) can be expressed as:  
\[\phi_{A}^{F}=\sqrt{2}\phi cos\left\{2\pi f_{S}t-\alpha_{\phi}-\Delta\phi sin \left[2\pi f_{F}t-\left(\alpha_{I}-\alpha_{\phi}\right)-\alpha_{F}\right] \right\}, \tag{7}\]  
where \(\Delta\phi=3P^{2}\phi I_{F}/\left(2\pi^{2}f_{F}^{2}J\right)\). The flux produces a nonlinear effect that produces a corresponding electromagnetic force, thus generating a nonlinear current signal in the stator. The flux now can be explicitly simplified and examined in three components [24]:  
\[ \begin{array}{l}\phi_{A}^{F}\approx\sqrt{2}\phi cos\left(2\pi f_{S}t-\alpha_{ \phi}\right)+\sqrt{2}\phi\Delta\alpha_{F}\sin\left(2\pi f_{S}t-\alpha_{\phi} \right)\\ =\sqrt{2}\phi cos\left(2\pi f_{S}t-\alpha_{\phi}\right)+\sqrt{2}\phi\Delta \phi cos\left[2\pi(f_{s}-f_{F})t-\alpha_{I}-\alpha_{F}\right]\\ \qquad\qquad-\sqrt{2}\phi\Delta\phi cos\left[2\pi(f_{S}+f_{F})t-2\alpha_{\phi }+\alpha_{I}-\alpha_{F}\right],\end{array}  \tag{8}\]  
Equation (8) shows that the fluctuating magnetic flux contains not only the fundamental frequency component, but also the two sidebands on both sides of the fundamental frequency. The current of the motor can be obtained [24; 52] based on the motor equivalent circuit. \[