The ac-dc difference of single-junction thermal converters

Abstract

A nonlinear parabolic partial differential equation of heat conduction subject to the Robin boundary conditions is considered. The equation describes energy conservation in the heater wire of a single-junction thermal converter for both ac and dc currents. In the audio-frequency range the temperature distribution along the heater is calculated and the ac-dc difference deduced by means of the Picard iterative technique. The previously neglected effects of radiation and the thermal properties of the heater wire are included. An expression for the ac-dc difference is also derived in the low-frequency regime. The thermal conductance of the thermocouple, which has previously been neglected, is taken into account. The calculated increase in the ac-dc difference is consistent with recent measurements. The solution in this limit is found by both an eigenfunction-expansion method and the Laplace-transform method. Comparison of the solutions obtained by the two methods gives some useful formulae for the summation of numerical series.