Electrical conduction in solids II. Theory of temperature-dependent conductors

Abstract

The experiments in part I on the behaviour of the contact between metals when large currents pass the interface have yielded results which cannot be explained by the classical theory of constriction resistances. In an attempt to provide an account of this anomalous behaviour a new mathematical treatment of the general problem of the electrical heating of conductors has been developed. This treatment gives, under the appropriate conditions, a concise derivation of all the main results of the accepted theory; in addition it leads to three new conclusions. First, it is shown that the general problem of finding the spatial distribution of the potential, the current, and the temperature within a medium whose thermal and electrical conductivities vary with temperature may be reduced to the corresponding problem with constant conductivities, and a simple numerical integration. Secondly, it is shown that the spatial distribution of the current within the conductor is independent of the thermal and electrical properties of the medium, and that it is unaltered by variation s of the total current passing. Finally, it is demonstrated that there are certain conditions under which a steady solution of the problem is impossible. For many conductors steady conditions are possible only when the current is below a certain critical value. If a current greater than this critical value is maintained the temperature will rise continuously, and eventually the process will be limited by some other phenomenon, for example, melting. The new treatment is applied to the calculation of the spatial distribution of current for a particular shape of conductor, one which is relevant to many resistance welding processes. The predictions agree accurately with experimental data obtained from the examination of a series of welds. The theory is then applied to the special case of the electrical contact between gold pieces and shown to offer an explanation of the anomalous behaviour mentioned above. Good agreement is demonstrated between the theoretical predictions and the experimental results.

A simple method for testing the electromigration resistance of solders

“…Regarding the thermal problem of a sample which is symmetric about the diagonal BB, the steady temperature T at any point in the sample can be expressed as follows (Greenwood and Williamson 1958):…”

Effect of nanoscale heating on electrical transport in RF MEMS switch contacts

“…Also, is a scaling function. While this equation is well-known, most of the existing work on contact heating considers only the contribution of the Maxwell spreading resistance [10], [32]. Even for the smallest contact resistance in Fig.…”

“…The principle difference between the theory presented here and existing theory is that we assume that heating in the contact is due only to the Maxwell term in (5), . Greenwood and Williamson have previously shown that equipotential surfaces are also isothermals in a contact [32]. For initial development of the model, we assume that the contact is symmetric, with maximum temperature at the contact asperity.…”

Effects of alloying and local order in AuNi contacts for Ohmic radio frequency micro electro mechanical systems switches via multi-scale simulation

“…All samples show a peak temperature in the vicinity of the contact consistent with predictions from the VoltageTemperature relations developed by Kohlrausch 47 and refined by Greenwood and Williamson. 48 Samples with higher concentrations of Ni have a lower maximum temperature as compared to the poly crystalline Au samples. In all cases, this maximum temperature occurs at the narrowest point of contact due to the increased current density caused by constriction, which is the dominant factor in determining the thermal profile.…”