Multiplicity Analysis of a Thermistor Problem—A Possible Explanation of Delamination Fracture

Abstract

In the present study, a numerical bifurcation analysis of a PTC thermistor problem is carried out, considering a realistic heat dissipation mechanism due to conduction, nonlinear temperature-dependent natural convection, and radiation. The electric conductivity is modeled as a strongly nonlinear and smooth function of the temperature between two limiting values, based on measurements. The temperature field has been resolved for both cases were either the current or the voltage (nonlocal problem) is the controlling parameter. With the aid of an efficient continuation algorithm, multiple steady-state solutions that do not depend on the external circuit have been identified as a result of the inherent nonlinearities. The analysis reveals that the conduction–convection parameter and the type of the imposed boundary conditions have a profound effect on the solution structure and the temperature profiles. For the case of current control, depending on the boundary conditions, a complex and interesting multiplicity pattern appears either as a series of nested cusp points or as enclosed branches emanating from pitchfork bifurcation points. The stability analysis reveals that when the device edges are insulated, only the uniform solutions are stable, namely, one “cold” and one “hot”. A key feature of the “hot” state is that the corresponding temperature is proportional to the input power and its magnitude could be one or even two orders of magnitude higher than the “cold” one. Therefore, the change over from the “cold” to the “hot” state induces a thermal shock and could perhaps be the reason for the mechanical failure (delamination fracture) of PTC thermistors.

Keywords

• PTC thermistor
• delamination fracture
• Joule heating
• nonlocal problem
• bifurcation analysis
• BaTiO₃ and titanates