# Passive Intermodulation at Contacts of Rough Conductors
## 3 Thermal Nonlinearities at Contacts of Good Conductors
### Electro-Thermal Effect in Contact Joints and PIM Generation  
Electrothermal PIM (ET-PIM) is caused by self-heating due to conductor and dielectric losses [37]. ET-PIM has a distinct signature of the nonlinear coupling of electrical and thermal domains as illustrated by Figure 5. The basic mechanism of this nonlinearity is the thermal modulation of resistivity. In essence, the heat due to RF losses alters the resistance, which is heated by the high-power carriers, and this results in the generation of PIM products. The effects of ET-PIM have been studied in the termination resistors [37; 47], printed TLs [36; 70] and thin-film coplanar waveguides with spatially inhomogeneous current distributions [35]. The developed theory of the ET-PIM and supporting experiments [36; 37; 70] have revealed that the baseband resistivity of conductors is modulated by the heat oscillations. The effect of resistivity variation on the skin depth due to modulation of the RF carriers was analysed in [42; 43]. It enables both the development of the qualitative analytical model, which sheds light on the principal mechanisms of ET-PIM generation, and provides an accurate assessment of signal distortion by ET-PIM in the full-wave EM simulations [33; 34; 35; 36; 70]. Evaluation of the dissipative losses and the rate of self-heating are the critical steps in the ET-PIM analysis. While the skin effect is routinely modelled in the EM simulators when calculating the losses of imperfect conductors, surface roughness is often ignored despite its proven major impact on the performance of printed circuit boards [48; 49] and MEMS switches [50; 51; 52]. The conductors with rougher surfaces exhibit not only higher RF losses but also worse thermal performance and a higher level of PIM [32]. The ET-PIM phenomenon was described first in [37], where a thermal modulation of the load resistance by two high-power RF signals was examined. A load with a size much smaller than the carrier wavelength has the resistance proportional to its temperature-dependent resistivity \(\rho_{L}(\delta T)\) where \(\delta T\) is the temperature increment. Since \(\rho_{L}(\delta T)\) has the same temperature dependence as \(\rho_{c}(\mathbf{x})\) in (5), its modulation by the high-power carriers is described by (9). This basic model, illustrated by Figure 5, directly relates an instantaneous power of the electrical current in the load with resistance \(R_{cm}\) to the heat flow at a thermal node with the thermal resistance \(R_{th}\) and thermal capacitance \(C_{th}\). Such a thermal system acts as a low pass filter as the thermal response of the resistive load is much slower than variations of the high-power RF signals of frequencies \(\omega_{1}\) and \(\omega_{2}\). However, when the frequency of thermal oscillations is comparable with an intermediate frequency \(\Delta\omega=1\,\omega_{2}\)-\(\omega_{1}\,|\) generated by the electro-thermal nonlinearity, the resulting temperature oscillations modulate the load resistance. Then the PIM products fall in the operational frequency band [35, 36, 37, 42, 46, 47] as illustrated by Figure 6. When the difference of the carrier frequencies exceeds several MHz, the thermal oscillations become too slow to follow the frequency \(\Delta\omega\) and the magnitude of the ET-PIM products decreases [36, 37, 70]. The distinctive feature of the ET-PIM is that the power slope of the PIM products versus a difference of carrier frequencies is only 10 dB per frequency decade [37]. Such a rate of the ET-PIM decay with the carrier frequency offset implies that ET-PIM products may have a significantly stronger impact on the signal distortion in broadband communications systems with a dense spectrum of closely spaced high-power carriers. The ET-PIM in contact joints is strongly affected by the surface roughness as the asperity height and size of a contact spot determine its resistance and reactance. The current constriction and contact heating at rough surfaces increase the spot temperature and asperity resistivity.