Analytical modelling of steady-state temperature distribution in thermal microsensors using Fourier method

Abstract

An analytical method is presented that allows one to determine the steady-state temperature distribution in thermal microsensors based on thermally isolated structures with arbitrary rectangular edges. The structure of thermal microsensors is treated as a 2D structure with a number of rectangular regions which are classified into some types depending on the boundary conditions at their edges. For each type of the regions, the equivalent parameters and heat exchange conditions are determined and the expression for temperature distribution in the region is obtained by means of Fourier method. Heat flux densities between the regions are represented as sums of orthogonal functions with weighting coefficients. The expressions for temperature distribution in the regions contain unknown weighting coefficients whose values are determined from adjoint boundary conditions between all the adjacent regions. The system of equations for the weighting coefficients obtained with the help of the adjoint boundary conditions is that of linear equations. As an example, a system of linear equations for the weighting coefficients of thermal microsensors based on the membrane thermally isolated structure is presented.