# Passive Intermodulation at Contacts of Rough Conductors
## 4 Mechanical Nonlinearities at Conductor Contacts
### Effect of Asperity Deformations on Conductor Resistivity  
Contact resistances of rough surfaces calculated with the aid of (1) or (3) take into account the effects of an external pressure and deformations of the compressed asperities [50; 51; 52; 53; 54; 56; 571]. The factors affecting expansion of the contact areas and asperity compression include the mechanical stresses, elasticity and surface finish. At low pressure, the colliding asperities experience elastic deformations and their contact areas enlarge. When contact pressure increases, the compressed asperities experience plastic deformations and their contact regions are hardened. This results in the slowing down of the expansion of the contact spots and a change in contact resistivity \(\rho_{c}(\mathbf{x})\)[50; 60; 64]. Thus, the combined effect of the contact area enlargement and asperity hardening is governed by the relations between the contact pressure and contact resistance, cf. Figure 1. The nonlinear effects of strain hardening and softening on the resistivity \(\rho(\mathbf{x})\) of a contact spot is accounted in [72]  
\[\rho(\mathbf{x})=\rho_{c}(\mathbf{x})\left(1+\frac{\epsilon_{p}}{\epsilon_{ ref}}\right)^{q}\left[1-\exp\left(-\frac{Q}{kT}\right)\right] \tag{11}\]  
where \(\rho_{c}(\mathbf{x})\) is the temperature dependent resistivity of a contact, defined in (5), \(\epsilon_{p}\) is plastic strain, \(\epsilon_{ref}\) is a reference strain, \(q\) is a material dependent parameter, \(Q\) is the activation energy for the mechanism of relaxing the stored dislocations, \(T\) is the absolute temperature and Boltzmann constant \(k=1.38\times 10^{-23}\,\mathrm{J/K}\). In addition to altering the contact resistance, heat softens contact asperities [50; 51; 52; 53] and increases their plastic and creep deformations. Very slow creep deformations have been examined in micro-contacts with a low current in [50], where the strain-rate \(\dot{\epsilon}\) is described by a power law dependence on the stress \(\sigma\) as  
\[\dot{\epsilon}=A\sigma^{p}\exp\left(-\frac{Q_{c}}{kT}\right) \tag{12}\]  
where \(A\) is a parameter defined by the material properties and the creep mechanism, \(\sigma\) is the stress, \(Q_{c}\) is the activation energy for creep. The stress exponent \(p\) used in (12) usually varies between 3 and 10, depending on the material composition. When contact joints are exposed to the high RF power, the contact area of asperities increases further due to its thermal expansion. But the conductor resistivity \(\rho_{c}(\mathbf{x})\) also grows due to Joule heat as discussed in Section 3. These two effects counteract each other as the rise of contact temperature increases both \(\rho_{c}(\mathbf{x})\) and the dissipative losses, whilst enlargement of the contact area reduces the resistance and a current density. Joule heat, generated by the current flowing through contact asperities, increases the temperature \(T\) of the contact spots. The temperature dependence on the contact voltage \(V_{c}\) was examined in [73; 74] for conductor joints with the characteristic size \(a\) of the contact spot considerably larger than the electron mean free path \(\lambda\). Since the boundary scattering has a minor effect on the self-heating of the contact spots in this case, the contact temperature is evaluated with the asperity heating model [50]  
\[T=\sqrt{T_{0}^{2}+\frac{\gamma}{4L}\frac{V_{c}^{2}}{\xi(\lambda/a)}} \tag{13}\]  
where \(T_{0}\) is ambient temperature, \(\gamma\) is the scaling parameter of the asperity distribution, \(L=2.45\times 10^{-8}\,\mathrm{W\cdot\Omega/K^{2}}\) is Lorentz number, \(V_{c}\) is the contact voltage, \(\xi(x)=1+\frac{8}{3\pi}\frac{x}{f(x)}\) and \(f(x)\) is defined in (2). It is interesting to note in (13) that \(T\) depends on \(\rho(\mathbf{x})\) only implicitly, through contact voltage \(V_{c}\). The slow deformations of contact asperities in conductor joints also affect the temperature dependent resistivity \(\rho(\mathbf{x})\) which is related to the material stress and strain.