## 2 Theoretical Considerations about Electrical Contacts
### The Flattened Ellipsoid Model  
In the case of the flattened ellipsoid model, the half-spaces 1 and 2 of finite conductivity are touching by means of a flattened ellipsoid, as shown in Figure 4. The equipotential surfaces are confocal ellipsoids, with the flattened ellipsoid as their base ellipsoid. To calculate the electrical resistance between the equipotential surface of the base ellipsoid (as a place of contact) and the surface of another equipotential confocal ellipsoid, formal analogy is used, which exists between the formulae characterizing the stationary electric field of direct current in a conducting medium and the formulae characterizing the electric field in an uncharged dielectric.  
Figure 3: The model of the infinite conductivity sphere.  
In this case, the expression of striction conductance is identical to the expression of capacitance, in which permittivity \(\epsilon\) is replaced by conductivity \(\sigma\). Solving the problem in this way is possible assuming the same boundary conditions are allowed for both the conductive medium with conductivity \(\sigma\) and the dielectric medium with permittivity \(\epsilon\). Thus, both the base ellipsoid and the medium delimiting the surface of the confocal ellipsoid are assumed to have infinite conductivity, and the conductive medium between the surfaces of the ellipsoids has constant conductivity \(\sigma\). Only in this case are the surfaces of the two ellipsoids equipotential surfaces in the stationary electric field, and the current lines are orthogonal to the surfaces of the ellipsoids.  
If it is considered that the base ellipsoid is flattened, as in Figure 4, and that the contact surface is an ellipse, then the equipotential surfaces in the contact members are semi-ellipsoids, with the equation:  
\[\frac{x^{2}}{\alpha^{2}+\mu}+\frac{y^{2}}{\beta^{2}+\mu}+\frac{z^{2}}{\mu}=1 \tag{5}\]  
where \(\alpha\) and \(\beta\) are the semi-axes of the flattened base ellipsoid in the plane \(xy\). Confocal ellipsoids are defined by the parameter \(\mu\) (\(\mu\) = 0.1: \(a^{2}\), \(\mu\) = 0.5 \(\cdot\)\(a^{2}\), \(\mu\) = 2 \(\cdot\)\(a^{2}\), \(\mu\) = 5 \(\cdot\)\(a^{2}\),...). Thus, the semi-axes are: \(\sqrt{\alpha^{2}+\mu}\) on axis \(x\), \(\sqrt{\beta^{2}+\mu}\) on axis \(y\), and \(\sqrt{\mu}\) on axis \(z\).  
The constriction resistance between the base semi-ellipsoid and a random confocal semi-ellipsoid is:  
\[R_{s}=\frac{1}{4\pi\sigma}\int\limits_{0}^{\mu}\frac{\mathrm{d}\mu}{\sqrt{( \alpha^{2}+\mu)\Big{(}\beta^{2}+\mu\Big{)}\mu}} \tag{6}\]  
If it is admitted that the contact ellipse is actually a circle, meaning that \(\alpha=\beta\) = a, then the elliptical integral becomes a simple transcendent integral, and  
\[R_{s}=\frac{1}{4\pi\sigma}\int\limits_{0}^{\mu}\frac{\mathrm{d}\mu}{(a^{2}+ \mu)\sqrt{\mu}} \tag{7}\]  
Figure 4: The flattened ellipsoid model.  
With the variable changes \(\sqrt{\upmu}=z\), \(\upmu=z^{2}\), and \(\mathrm{d}\upmu=2z\,\mathrm{d}z\), we obtain  
\[R_{s}=\frac{1}{2\pi\upsigma}\int\limits_{0}^{\sqrt{\upmu}}\frac{\mathrm{d}z}{a^ {2}+z^{2}}=\frac{1}{2\pi\upsigma a^{2}}\int\limits_{0}^{\sqrt{\upmu}}\frac{ \mathrm{d}z}{1+\left(\frac{z}{a}\right)^{2}} \tag{8}\]  
With a new variable change \(\frac{z}{a}=u\), and \(\mathrm{d}z=a\cdot\mathrm{d}u\), we obtain  
\[R_{s}=\frac{1}{2\pi\upsigma a}\int\limits_{0}^{\sqrt{\upmu}}\frac{\mathrm{d}u }{1+u^{2}}=\frac{\uprho}{2\pi a}\mathrm{arctan}\left(\frac{\sqrt{\upmu}}{a}\right) \tag{9}\]  
If instead of an ellipsoid a semi-space is considered, then \(\upmu\rightarrow\infty\), and Equation (9) becomes:  
\[R_{s}=\frac{\uprho}{4a} \tag{10}\]  
The total constriction resistance is twice as high as \(R_{s}\), that is,  
\[R=\frac{\uprho}{2a} \tag{11}\]  
According to [PERSON] (1963), the current density across a flattened ellipsoid contact area can be approximated as  
\[J(r)=\frac{i}{2\pi a}\cdot\frac{1}{\sqrt{a^{2}-r^{2}}} \tag{12}\]