## 2 Theoretical Considerations about Electrical Contacts
### The Thermal Regime of Electrical Contacts  
From the point of view of the thermal regime of electrical contacts, electric current passing through a contact causes the development of heat (Joule effect) because of contact resistance. In permanent thermal regime, the overtemperature due to contact resistance must be maintained within relatively low limits of 2-10 degrees, whereas in short-term thermal regimes such as a short circuit, the overheating must not lead to the melting of the contact elements. The study of thermal behavior is carried out for closed contacts, as if the contact elements are separated, additional physical processes such as arcing and material migration occur. Figure 5 shows a contact point in whose touching area a number of m flow lines (tubes) of current \(i\) converge, such that the total current in the touch area is \(I=\upmu\cdot i\). Figure 6 schematically details such a tube of current, which in the contact point covers the area \(\Delta A_{0}\). Figure 5: Contact point with converging flow lines (tubes) of current. Calculations are made with the following assumptions:  
* The thermal flux developed in a certain current tube is transmitted outwards only through that current tube. There is no heat transfer between adjacent points m and n in Figure 5, which are assumed to be at the same temperature;
* The highest overtemperature is at the \(\Delta A_{0}\) surface, which also defines an isothermal surface, as a result of which the greatest restriction is located in the contact zone;
* The contact elements are made of the same homogeneous and isotropic material. By the passing of current \(i\) through a current tube, equipotential and isothermal surfaces are defined as shown in Figure 6, considering the surface \(\Delta A_{2}\) with the reference parameters in terms of electric potential \(V=0\), overtemperature \(\theta=\Theta\), and absolute temperature \(T=T_{\Theta}\). A surface \(\Delta A_{n}\) located far away, theoretically at a point infinitely far, is characterized by \(V=U/2\), where \(U\) is the voltage on the contact, \(\theta=0\) and \(T=T_{0}\). One can see that the equipotential surfaces are also isothermal, due to the fact that equal thermal fluxes go through equal thermal resistances. Thus, the equations of electrical resistance \(R\) and thermal resistances \(R_{t}\), for a segment of the current tube between the base surface \(\Delta A_{0}\) and the equipotential surface \(\Delta A_{1}\) situated at a distance d \(n\) from \(\Delta A_{0}\), are:  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]  
\[\mathrm{d}R_{t}=\frac{\mathrm{d}n}{\lambda\cdot\Delta\overline{A}_{01}} \tag{14}\]  
where \(\Delta\overline{A}_{01}\) represents an average value of the area over the distance d\(n\); \(\sigma\) is the electrical conductivity; \(\lambda\) is the thermal conductivity. For the elementary temperature variation, the equation of Fourier is valid when written in the form  
\[\mathrm{d}T=-P_{t}\cdot\mathrm{d}R_{t} \tag{15}\]  
where \(P_{t}\) is the thermal power and \(\mathrm{d}R_{t}\) the thermal resistance overcome by the thermal power. To integrate the differential Equation (15) based on Equations (13) and (14), one can write  
\[\mathrm{d}R_{t}=\mathrm{d}R\frac{\sigma}{\lambda}=\frac{\mathrm{d}R}{\lambda \rho} \tag{16}\]  
which can also be expressed as  
\[\mathrm{d}R=\frac{\mathrm{d}V}{i} \tag{17}\]  
Under these conditions, Equation (16) becomes  
\[-\lambda\,\rho\,\mathrm{d}T=V\,\mathrm{d}V \tag{18}\]  
Figure 6: Detailed view of a tube of current. Equation (18) allows us to express the dependence between the absolute temperature of the contact and the electrical potential, in the form  
\[-\int_{T_{\Theta}}^{T}\lambda\rho\mathrm{d}T=\int_{0}^{V}V\mathrm{d}V \tag{19}\]  
To solve Equation (19), \(\lambda\) and \(\rho\) are considered as functions of temperature. Two cases can arise, namely, the following:  
(a) The overtemperature is moderate, i.e., \(\Theta=1,\,\ldots,\,5\) degrees.