164 © IEEE / Therminic 2013 2013 (( 19th INTERNATIONAL WORKSHOP on Thermal Investigations of ICs and Systems )) 1 I ntroduction In the last decades many efforts, reported in literature, have led to effective procedures for constructing compact models of dynamic heat diffusion problems for electronic components. A particularly efficient method is the projection-based approach used by the present author in the multi-point moment matching technique. Almost all the proposed approaches, for constructing compact thermal models, apply only to linear thermal prob- lems, in which material properties are assumed not to depend on temperature [3]-[28]. However for heat diffusion in electronics components, the temperature dependencies of thermal conductivities on temperature can be neglected only for small temperature rises [1]. Moreover, with the advent of nanotechnology, the scaling of electronics devices implies increasing current densities that result in greater Joule heating and in greater temperature rises [2]. In all the cases in which the dependence of material prop- erties on temperature have to be taken into account, the most robust approach is still based on Kirchhoff’s transformation [29], by which a nonlinear thermal problem is transformed into an equivalent linear thermal problem. However this method is exact only in very particular situations. In the general case it introduces large inaccuracies which cannot be removed [30]. In this paper a novel approach for generating compact models of dynamic heat diffusion problem is proposed, in the nonlinear case. It takes into account the dependencies of material properties on temperature relevant in electronics applications. It considers a novel way of writing the non- linear heat diffusion problem. A projection is applied, in such a way that the nonlinear structure of the equations is preserved. The projection space is obtained from the Volterra series expansion of the solution of the nonlinear problem. In this way a much more effective approach is achieved with respect to previous attempts reported in literature. The details of the derivation are provided for the analytical heat diffu- sion problem. However the method can be straighforwardly applied to any discretization of the heat diffusion problem, for instance by means of finite difference or finite elements methods. An in-depth investigation of a simple example problem, shows that the proposed technique is very efficient and accurate for variations of temperatures larger than needed in applications. 2 Reformulation of the nonlinear heat diffusion problem A dynamic heat diffusion problem in the spacial domain Ω is ruled by the well known equation ∇ ¨ p´kpr,upr,tqq∇upr,tqq ` cprq Bu Bt pr,tq“ gpr,tq (1) in which the unknown upr,tq, function of the position vector r and of the time instant t, is the temperature rise with respect to ambient temperature, due to the power density gpr,tq. In order to take into account nonlinear effects, the thermal conductivity kpr,upr,tqq is assumed to depend on upr,tq, in the form [1] kpr,upr,tqq “ kpr, 0q e μprqupr,tq . (2) in which μprq is the sensitivity of the thermal conductivity with respect to temperature. For the sake of simplicity, the volumetric heat capacity cprq is assumed to be independent on upr,tq, as it is usually the case. Equations (1), (2) are completed by conditions on the spacial boundary BΩ of Ω, having outward normal unit vector nprq, assumed of Robin’s type ´ kpr,upr,tqq Bu Bn pr,tq“ hprqupr,tq, (3) in which hprq is the heat exchange coefficient. For the sake of simplicity homogeneous initial conditions are considered upr, 0q“ 0. (4) When introducing compact thermal models, the power density is assumed in the form [7] gpr,tq“ n ÿ i“1 g i prqP i ptq, Novel Approach to Compact Modeling for Nonlinear Thermal Conduction Problems Lorenzo Codecasa* 1 1 Politecnico di Milano, Dipartimento di Elettronica, Informazione e Bioingegneria, Milan, Italy * Corresponding Author: Codecasa@elet.polimi.it, +39 02 2399 3534 Abstract In this paper a novel approach is proposed for generating dynamic compact models of nonlinear heat diffusion problems for electronics components. The method is very efficient and leads to accurate approximations of the space-time distribution of temperature rise within the component for all waveforms of the injected powers.