Calculation of the temperature development in electronic systems by convolution integrals

The temperature evolution at selected locations in an electronic set-up can be determined for arbitrary pulses of dissipated power once the thermal impedance of these locations to the ambient is known. The method used is the evaluation of the convolution integral of power evolution and thermal impedance in the time domain. A precise definition of the thermal impedance `junction to ambient' ZthJA for power semiconductor devices and its relation to ZthJC (junction to case) is given. By using a quantum mechanical analog to the heat conduction equation, a representation of the impedances in terms of eigenfunctions and eigenvalues of the underlying differential operator is established. It is shown that generally, for systems of finite extension, the time-constant spectrum is discrete, positive and approaching zero with increasing number index. The model thus obtained can be fitted to the Zth functions, which are received either by measurement or by simulation of the complete set-up. Analytical closed-form expressions for the temperature evolution in case of simple power wave forms are derived. Using this for the interpolation of arbitrary power pulses, a considerable gain in computation speed for the convolution integrals by a factor 50 is obtained. The method described is far superior to fast Fourier transformation with respect to both accuracy and speed, and it can be adapted to the nonlinear case, where the power evolution depends not only on time but also on the temperature itself. The results are elucidated by application to a numerically challenging problem. Also multichip-modules under load cycle stress conditions are treated. The resulting temperature functions of the different layers of the set-up are a valuable basis for discussing reliability issues.