## 3 Test Details
### Some Main Modulators of TH FM on Electrical Contact
#### 3.2.3 Asperity Spots  
The contacting interfaces of two conductors, no matter how well the surface was processed, were always uneven on the micro level. If the hardness of the material was infinite, the contact force does not cause any deformation. There will be, at most, three contact spots for surface-to-surface contact; two contact spots for line to surface contact; and one contact spot for sphere to surface contact. However, in  
Figure 14: Contacting width and contact tip radius. reality, the material will deform. Thus, the number of contact spots will increase with the increase of the deformation, until the summation of the forces provided by point contacts is balanced by the external force, and the increase of contact spots stops, as shown in Figure 15. These spots, which actually touch, are called asperity spots (hereinafter referred to as a-spots), and are the ones that can actually conduct electricity. Assumptions  
1. As illustrated in Figure 16, the tip along the contact length direction was viewed as a spherical ball with a radius of \(r_{\text{t}}\) and with all a-spots on top of it. 2. The pad was viewed as an ideal spherical surface with an infinite radius of curvature or an ideal plane. 3. The a-spots were treated as a bunch of small balls, all with a radius of \(r_{\text{a}}\) (hereinafter referred to as a-spot ball radius). 4. The contacting interface of an a-spot was a circle, with a radius of \(a\) (hereinafter referred to as a-spot contacting radius), as shown in Figure 17.
5. When the touch just began, there was only one a-spot touching pad. As the actual contacting area was small, the force on the contact point was large, resulting in deforming the 1 st a-spot. With the increasing deformation of the 1 st a-spot, the 2 nd a-spot started touching the pad. Assuming that, when the 2 nd a-spot came into play, the 1 sta-spot was still in elastic region. Thus, the classical Hertz formula could be used to calculate the elastic deformation. Additionally, this procedure was repeated for all other a-spots until the supporting forces from a-spots could balance the external force. 6. The number of a-spots was n, and the distance between a-spots was much greater than a-spot contacting radius \(a\), and therefore any interference from the adjacent a-spots could be ignored. As a result, the a-spot contacting radius could be derived with the classic Hertz Formula [9], which was expressed as:  
\[a=\sqrt[3]{\frac{3}{4}P_{a}r_{\text{a}}\bigg{(}\frac{1-v_{1}^{2}}{E_{1}}+ \frac{1-v_{2}^{2}}{E_{2}}\bigg{)}} \tag{4}\]  
\(P_{\text{a}}\) is the contact force on an a-spot; \(r_{\text{a}}\) is a-spot ball radius; and \(V_{1}\), \(V_{2}\), \(E_{1}\), \(E_{2}\), are [PERSON]'s ratio and Young's modulus of the two conductor materials, respectively. As both contacting surfaces were Au plated, we have:  
\[v_{1}=v_{2}=v_{Au}=0.42 \tag{5}\]  
Figure 15: Real contact state. (**a**) Nominal contact region and conduct electricity spots; (**b**) Contacting status under equilibrium. LLCR  
As the actual contact interface was a group of a-spots, resistance increased as the path of current flow was elongated when current passes through a-spots. Such resistance was defined as constriction resistance \(R_{c}\), as depicted in Figure 18.  
Figure 16: Schematic representation of a contact tip design. Figure 17: Schematic representation of contact state of an a-spot. Figure 18: Schematic representation of constriction resistance. For the round contacting interface, there was the following relationship:  
Figure 15.0  
\(R_{ci}\) is the constriction resistance of an a-spot (hereinafter referred to as a-spot constriction resistance); \(a\) is a-spot contacting radius; and \(\rho\) is resistivity of the contacting material, which is also the resistivity of Au. Contact resistance consists of bulk resistance, constriction resistance, and film resistance, as shown in Figure 15.1 A-spot constriction resistances \(R_{ci}\) are in parallel at the interface.