The A.C. Admittance of Temperature-Dependent Circuit Elements

Abstract

The relation between the isothermal and steady-state current-voltage characteristics of a device in which the current depends on both the voltage and the temperature is deduced quite generally without assuming electrical linearity or Newtonian cooling. The phenomena of voltage turnover in a device with a positive temperature coefficient of d.c. conductance, and of current turnover in a device with a negative coefficient are deduced. The a.c. admittance for a small a.c. voltage superimposed on the d.c. voltage determining the operating point is derived in terms of the thermal admittance of the device, i.e. the complex ratio of the alternating components of power and temperature. For the special case in which the thermal admittance consists of a shunt combination of a fixed thermal capacity and thermal conductance the electrical admittance has been evaluated in detail. Three-element equivalent circuits (of two resistors and a capacitor or inductor) can represent the electrical admittance of such a device and its locus is semicircular; the reactive element represents the effect of thermal inertia and its nature is determined by the sign of the temperature coefficient of d.c. conductance. Where heat loss from the device is by a path which is not `thermally short at the frequency under consideration the thermal admittance no longer has simple form but has to be determined from the diffusion equation. This is done for the case of one-dimensional heat conduction (e.g. along the supporting wires of a thermistor or a lamp) or for radial heat flow (e.g. from the contact in a point-contact rectifier). The general condition for the admittance locus still to be circular, but with the centre not on the conductance axis, is formulated.