A Rigorous Series Solution for Diamond Heat Spreaders with Temperature-Dependent Thermal Conductivity Used in Microwave Power Devices

Abstract

In this paper we derive a rigorous series solution of the nonlinear heat transfer equation using the modified Kirchhoff transformation for a cylindrical diamond heat spreader with temperature-dependent thermal conductivity, situated on the top of a copper heat sink. We point out what we believe to be errors in two analytical approaches applied to the same problem which were published recently [Proc. R. Soc. Lond. A 441 (1993) 181; Proc. R. Soc. Lond. A 445 (1995) 375; and IEEE Trans. Microwave Theor. & Technol. 42 (1994) 573]. The basic difference between our work and these previous studies lies in the boundary condition (b.c.) of the transformed heat potential over the spreader-sink interface–we use nonlinear b.c. whereas the boundary conditions in the previous studies have been assumed to be linear. Calculations of change in the thermal resistance with the geometrical dimensions of the spreader, the input power, and the ambient temperature show up the profound influence which the assumed b.c. have on the problem. Our study has also revealed errors in numerical results presented in another paper [IEEE Semi-Therm Proc. (4th Annu. Semiconductor Thermal and Temperature Measurement Symp.), p. 113].

On the effect of nonlinear boundary conditions for heat conduction in diamond heat spreaders with temperature-dependent thermal conductivity

“…From our derivation in an earlier paper [8], we have the rigorous series solution of the transformed temperature in the spreader and in the sink as (5) (6) where and (7) and is the th positive zero of the first order Bessel function of the first kind . For convenience, we have introduced two dimensionless quantities:…”

“…For the chosen thermal conductivities of diamond IIa spreader and copper sink, the explicit form of the nonlinear boundary condition over the interface (4) is taken as (8) The expansion coefficients in solution (5) and (6) can be determined by solving (8) numerically using an iterative scheme [8]. An initial approximate solution is needed to start the iterative process.…”

“…This yields the approximate boundary condition (4). Then, the expansion coefficients will be further refined through the iterative process developed from (8). It is seen that the approximate solution with the linear boundary condition (4) in fact is the first approximation to the solution with the nonlinear boundary condition (8).…”

“…Then, the expansion coefficients will be further refined through the iterative process developed from (8). It is seen that the approximate solution with the linear boundary condition (4) in fact is the first approximation to the solution with the nonlinear boundary condition (8). The temperature is then obtained from through the inverse Kirchhoff transformation.…”

“…In fact, for region , is defined only by and as in (1). We have compared and examined Luy and Schmidl's results in detail against our rigorous solution [8].…”