# Research on the Mechanism and Dynamic Characteristics of Intermitt Failure of Electrical Connectors
## 3 Analysis of Intermittent Failure Dynamic Characteristics
### Dynamic Model of Intermittent Failure  
To analyze the dynamic characteristics of intermittent failure, it is necessary to determine the initial state of the contact surface first. The initial state of the contact consists of the mean radius of the \"a-spot\", the relative position between the asperity and the rigid plate, and so on. The mean radius of the conductive spots is denoted by \(\overline{a}\). Before calculating the value of \(\overline{a}\) the ratio of the \"a-spot\" area to the total contact area and the ratio of the number of conductive spots to the total number of asperities or contact spots are proposed in this paper, denoted by \(\varepsilon\) and \(\eta\), respectively. \(\varepsilon\) and \(\eta\) are given by  
\[\varepsilon=A_{\mathrm{c}}/A \tag{7}\]  
\[\eta=N_{\mathrm{c}}/N \tag{8}\]  
where \(A_{\mathrm{c}}\) and \(N_{\mathrm{c}}\) denote the total conductive area and the total number of conductive spots, respectively, and \(A\) and \(N\) denote the total contact area and the total number of actual contact asperities, respectively. Let the total number of asperities be \(N_{0}\); thus, \(N\) and A can be deduced from the G-W model. \[N=\int_{h_{0}}^{\infty}N_{0}\varphi(z)dz \tag{9}\]  
\[A=\int_{h_{0}}^{\infty}N_{0}\varphi(z)\pi r(z-h_{0}) \tag{10}\]  
where \(\varphi(z)\) represents the probability density function (PDF) of the asperity height, \(h_{0}\) represents the average height of the rough surface, and \(r\) represents the radius of asperities. Combining Equations (9) and (10), \(\overline{a}\) is derived as follows. \[\overline{a}=\sqrt{\frac{A_{c}}{\pi\cdot N_{\mathrm{c}}}}=\sqrt{\frac{\int_{h _{0}}^{\infty}N\varphi(z)\pi r(z-h_{0})dz}{\int_{h_{0}}^{\infty}N\varphi(z)dz} \frac{\varepsilon}{\eta}} \tag{11}\]  
The position of each \"a-spot\" is related to its corresponding plowing region and is represented using polar coordinates, denoted by (\(l_{\mathrm{i}}\), \(\theta_{\mathrm{i}}\)), \(i=1\), \(2\ldots N_{\mathrm{c}}\). where \(l_{\mathrm{i}}\) denotes the distance between the \(i\)th center of conductive spot and its corresponding center of the plowing area. \(\theta_{\mathrm{i}}\) denotes the relative angle between the \(i\)th center of conductive spot and its corresponding center of the plowing area. It is assumed that \(l_{\mathrm{i}}\) and \(\theta_{\mathrm{i}}\) both follow a uniform distribution. \[f_{\theta}(y)=\left\{ \begin{array}{cl}1/2\pi&y\in[0,2\pi]\\ 0&\mbox{otherwise}\end{array} \right. \tag{12}\]  
\[f_{l}(x)=\left\{ \begin{array}{cl}1/L&x\in[0,L]\\ 0&\mbox{otherwise}\end{array} \right. \tag{13}\]  
where \(L\) denotes the radius of the plowing region. The initial contact resistance could be calculated using the initial parameters mentioned above. Then, the contact resistance can be calculated via an iterative algorithm. Ignoring the transition effect resulting from the size of the spot, each conductive spot resistance only has two states: one state is the low-resistance state \(R_{\mathrm{I}}\), which representsthe conductive state; the other state is the high-resistance state \(R_{\rm h}\), which denotes the nonconductive state. \(R_{\rm l}\) and \(R_{\rm h}\) both can be calculated according to Equation (2); however, when calculating \(R_{\rm h}\), \(\rho_{1}\), or \(\rho_{2}\) represent the resistivity of the oxidized film material. Considering the parallel structure of each conductive spot, it is necessary to calculate the contact resistance \(R_{\rm ci}\) (i = 1, 2... \(N_{c}\)) of each conductive spot at the current time. Using the relative motion equation \(z_{\rm c}(t)\), \(R_{\rm ci}\) is given as follows:  
\[R_{\rm ci}(t)=\left\{ \begin{array}{ll}R_{l}&-L<l_{\rm i}\cos\theta_{i}+z_{c }(t)<L\\ R_{h}&\mbox{otherwise}\end{array} \right. \tag{14}\]  
The contact resistance \(R_{\rm c}\) at time \(t\) is derived as follows. \[R_{c}(t)=\frac{1}{\sum\limits_{i=1}^{N_{c}}1/R_{ci}(t)} \tag{15}\]  
Let \(R\)th be the predetermined fault threshold.