# Functions of Several Variables in the Theory of Finite Linear Structures Part I: Analysis

• M. Bessmertnyĭ
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 157)

## 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.

## Keywords

System theory electrical networks impedance matrices transfer matrices functions of several complex variables

## References

1. [Be1]
Бессмертный, М.Ф: Функцuu несколькuх комnлексных nеременных в mеорuu консчных лuнсuных сmрукmур. Кандидацкая диссертация. Харьковский университет. Харьков, 1982. 143 cc (in Russian). [Bessmertnyy, M.F. Functions of several complex variables in t hetheory of finite linear structures. PhD thesis, Kharkov University. Kharkov, 1982. 143 pp.]Google Scholar
2. [Be2]
Bessmertnyĭ, M.F.On realizations of rational matrix functions of several complex variables. Translated from the Russian by D. Alpay and V. Katsnelson. Oper. Theory Adv. Appl., 134, Interpolation theory, systems theory and related topics (Tel Aviv/Rehovot, 1999), 157–185, Birkhäuser, Basel, 2002.Google Scholar
3. [Ef]
Ефимов. А.В.: Об оdиом nрuмененuu mеоремы Лaнжевенa в mеорuu цеnеu. ДАН Армянской ССР, 49:3 (1969), 118–123 (in Russian). [Efimov, A.V.: On one application of Langevin’s theorem in the theory of electrical networks. DAN Armyansk. SSR, 49:3 (1969), 118–123.]Google Scholar
4. [EfPo]
Ефимов, А.В. и В.П. Потапов: J-рaсmяsuвaюшuе мamрuuы-функцuu u uх роль aнaлumuческоu mеорuu eлекmрuческuх цеnеu. Успехи мaтем. нaук, 28:1 (1973), 65–130 (in Russian). English transl.: Efimov, A.V. and V.P. Potapov: J-expanding matrix functions and their role in the analytical theory of electrical circuits. Russ. Math. Surveys, 28:1 (1973), pp. 69–140.Google Scholar
5. [Liv]
Лившиц, М.С.: Оnерamоры, Колебaнuя, Волны (оmкрыmые сuсmемы). Наука. Москва, 1966. 298 cc. [In Russian]. English transl.: Livšič, M.S. (Livshits, M.S.): Operators. oscillations. waves (open systems). (Translations of Mathematical Monographs, Vol. 34.) American Mathematical Society, Providence, R.I., 1973. vi+274 pp.Google Scholar
6. [SeRe]
Seshu, S. and M.B. Reed: Linear Graphs and Electrical Networks. (Addison-Wesley series in the engineering sciences. Electrical and control systems). Addison-Wesley, Reading, MA, 1961. 315 p. Russian transl.: Сещу, С., и М.Б. Рид: Лuнеuные Грaфы u Элекmрuческuе Цеnu. Издaтсльство “Высшaя щколa”. Москвa, 1971. 448 c.Google Scholar
7. [Zy]
Зыков, А.А.: Теорuя Конечных Грaфов. Издaтельство “Наука”, Сибирское Отделение. Новосибирск, 1969. 543 c. [In Russian: Zykov, A.A.: Theory of Finite Graphs]Google Scholar