Combination of thermal subsystems by use of rapid circuit transformation and extended two-port theory

This paper deals with the modeling problem of combining thermal subsystems (e.g. a semiconductor module or package with a cooling radiator) making use of reduced models. The subsystem models consist of a set of Foster-type thermal equivalent circuits, which are only behavioral models. Fast algorithms are presented in concise form by use of recursive relations for transforming (in both directions) unphysical Foster-type circuits in Cauer-circuits, which have physical behavior and therefore allow for the construction of the thermal model of the complete system. The R, C elements in the circuit can have negative values contrary to conventional network theory. Therefore the transformation methods have to be investigated under relaxed conditions concerning positive realness and passivity of the impedances. The method is exemplified by modeling and measurements on a single-chip IGBT package mounted on a closed water-cooled radiator. The thermal impedance of the complete system is constructed from the impedances of the subsystems, IGBT-package and radiator, and also the impedance of the package can be inferred from the measured impedance of the complete system. For modules or packages with large thermal contact area of largely inhomogeneous temperature a multi-port description is presented, which can be viewed as an extended two-port theory. In case of real two-ports the presented recursive analytic calculation methods for the impedances/admittances give an easy-to-use description for high calculation speed for all boundary conditions of the Cauer-network.