A novel generalized Lagrange multiplier method for functional optimization with inclusion of subsidiary conditions is presented and applied to the optimization of material distributions in thermoelectric converters. Multistaged devices are considered within the same formalism by inclusion of position-dependent electric current in the legs leading to a modified thermoelectric equation. Previous analytical solutions for maximized efficiencies for generators and coolers obtained by Sherman [J. Appl. Phys. 31, 1 (1960)], Snyder [Phys. Rev. B 86, 045202 (2012)], and Seifert et al. [Phys. Status Solidi A 207, 760 (2010)] by a method of local optimization of reduced efficiencies are recovered by independent proof. The outstanding maximization problems for generated electric power and cooling power can be solved swiftly numerically by solution of a differential equation-system obtained within the new formalism. As far as suitable materials are available, the inhomogeneous TE converters can have increased performance by use of purely temperature-dependent material properties in the thermoelectric legs or by use of purely spatial variation of material properties or by a combination of both. It turns out that the optimization domain is larger for the second kind of device which can, thus, outperform the first kind of device.

This paper will deal 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. A fast algorithm is presented for transforming the Foster-type circuits in Cauer-circuits which have physical behavior and therefore allow for the construction of the thermal model of the complete system. Then the set of Cauer-circuits for the complete system is transformed back into Foster-circuits to give a simple mathematical representation and applicability. The transformation algorithms are derived in concise form by use of recursive relations. 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.

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