# Passive Intermodulation at Contacts of Rough Conductors
## 3 Thermal Nonlinearities at Contacts of Good Conductors
### Self-Heating Effect and Nonlinearity of Contact Resistivity  
Heat generation is an inherent feature of electromagnetic (EM) wave interactions with conductors and their contact joints. This multiphysics process couples the electric and thermal domains as illustrated by Figure 5. Namely, the dissipative losses of high-power RF signals generate heat which, in turn, changes the resistance of conductors and causes their thermal expansion and mechanical deformations. These nonlinear processes are interlinked but the time scales of the thermal and mechanical processes notably differ.  
High-power RF signals are attenuated by imperfect conductors due to their resistive losses. The dissipated power causes conductor self-heating, and the changes of local temperature at point \(\mathbf{x}\) alter a conductor resistivity \(\rho_{c}(\mathbf{x})\). In a broad range of temperatures, \(\rho_{c}(\mathbf{x})\) is well approximated by the linear temperature dependence  
\[\rho_{c}(\mathbf{x})=\rho_{0}[1+\alpha\cdot\delta T(\mathbf{x})] \tag{5}\]  
where \(\rho_{0}\) is an average resistivity of the conductors at ambient temperature, \(\alpha\) is the temperature coefficient of resistivity and \(\delta T(\mathbf{x})\) is the local temperature increment due to RF heating of the contact  
\[\delta T(\mathbf{x})=\sum_{n}\delta q(\omega_{n},\mathbf{x})\cdot R_{th,eq}( \omega_{n}) \tag{6}\]  
In (6), \(\delta q(\omega_{n},\mathbf{x})\) is the heat generated by the high-power carrier of frequency \(\omega_{n}\) with current density \(J_{c}(\omega_{n},\mathbf{x})\) in an imperfect conductor [49]. Then  
\[\delta q(\omega_{n},\mathbf{x})=\frac{1}{2}\mathrm{Re}\{Z_{c}(\omega_{n}, \mathbf{x})\}|J_{c}(\omega_{n},\mathbf{x})|^{2}\delta S \tag{7}\]  
Figure 5: Compact model of a pair of the compressed asperities where the contact resistance \(R_{cm}\) is coupled to a thermal node with the thermal resistance \(R_{th}\) and thermal capacitance \(C_{th}\).  
where \(Z_{c}(\omega_{n},\,\mathbf{x})\) is a local impedance at carrier frequency \(\omega_{n}\), and \(\delta S\) is a differential area of the contact spot. An equivalent thermal resistance \(R_{th,eq}(\omega_{n})\) can be defined as [37]  
\[R_{th,eq}(\omega_{n})=\text{Re}\Bigg{\{}\frac{R_{th}}{1+R_{th}\sqrt{j\omega_{n} C_{th}}}\Bigg{\}} \tag{8}\]  
where \(R_{th}\) and \(C_{th}\) are the thermal resistance and thermal capacitance, respectively. They form a thermal node shown in Figure 5.  
Simultaneous solution of (5)-(7) gives an approximation of the local resistivity  
\[\rho_{c}(\mathbf{x})=\rho_{0}\Big{[}1+\alpha\Delta_{\text{T}}(\mathbf{x})+O \Big{(}\alpha^{2}\Big{)}\Big{]} \tag{9}\]  
where  
\[\Delta_{\text{T}}(\mathbf{x})=\delta S\underset{n}{\sum}\Big{[}R_{th,eq}( \omega_{n})\delta q(\omega_{n},\mathbf{x})\Big{]}=\frac{1}{2}\delta S\underset {n}{\sum}\Big{[}R_{th,eq}(\omega_{n})\text{Re}\{Z_{c0}(\omega_{n})\}\left|J_{ c}(\omega_{n},\mathbf{x})\right|^{2}\Big{]} \tag{10}\]  
is the resistivity increment proportional to the heat generated by the contact resistances \(\text{Re}\{Z_{c0}(\omega_{n})\}\) at _ambient temperature_. Thus, (9), (10) show that the contact resistivity \(\rho_{c}(\mathbf{x})\) depends on the squared current density _magnitudes_\(\left|J_{c}(\omega_{n},\mathbf{x})^{2}\right|\) of individual carriers and has the nonlinearity of Kerr type. An important feature of (9) is that \(\rho_{c}(\mathbf{x})\) represents a local nonlinear resistivity, which may vary across the contact area due to inhomogeneity of the local thickness of the insulator in the MIM junction.