Abstract
The notion of a finite linear structure is introduced which generalizes the notion of a linear electrical network. A finite linear structure is a Kirchhoff graph to which several pairs of external vertices (“terminals”) are coupled. A Kirchhoff graph is a finite linear graph g to whose edges p the quantities U g , and I g , are related. The values of U g , and I g , are complex numbers. It is assumed that the circuit Kirchhoff law holds for U g , and the nodal Kirchhoff law holds for I g ,. It is also assumed that for the quantities U g (generalized voltages) and the quantities I g (generalized currents) corresponding to the edges of the Kirchhoff graph the generalized Ohm law U g = z g · I g holds. The generalized impedances z g are complex numbers which are considered as free variables. The Kirchhoff laws and the state equations U g = z g · I g lead to linear relations for the values U g and I g corresponding to the “external” edges of the finite linear structures, i.e., the edges incident to the terminals. These linear relations between external voltages and currents can be expressed either in terms of the impedance matrix Z if the considered finite linear structure is k-port or in terms of the transmission matrix A if the considered finite linear structure is 2k-port. The properties of the impedance and transmission matrices Z and A as functions of the complex variables z g are studied. The consideration of the present paper served as a natural motivation for the study of the class of matrix functions which was introduced in the previous paper MR.2002589 of the author.
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References
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Bessmertnyĭ, M. (2005). Functions of Several Variables in the Theory of Finite Linear Structures Part I: Analysis. In: Alpay, D., Vinnikov, V. (eds) Operator Theory, Systems Theory and Scattering Theory: Multidimensional Generalizations. Operator Theory: Advances and Applications, vol 157. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7303-2_3
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