This paper gives an overview of the formulation and solution of network equations, with emphasis on the historical development of this area. Networks are mathematical models. The three ingredients of network descriptions are discussed. It is shown how the network equations of one-dimensional multi-port networks can be formulated and solved symbolically. If necessary, the network graph is modified so as to obtain an admittance representation for all kinds of multi-ports. N-dimensional networks are defined as graphs with the algebraic structure of N-dimensional vectors. In civil engineering, framed structures in two and three spatial dimensions can be modeled as 3-dimensional or 6-dimensional networks. The separation of geometry from topology is a characteristic feature of such networks.
@article{bwmeta1.element.bwnjournal-article-amcv11i1p237bwm,
author = {Reinschke, Kurt},
title = {On network models and the symbolic solution of network equations},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {11},
year = {2001},
pages = {237-269},
zbl = {1033.93004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv11i1p237bwm}
}
Reinschke, Kurt. On network models and the symbolic solution of network equations. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) pp. 237-269. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv11i1p237bwm/
[000] Bellert S. (1962): Topological analysis and synthesis of linear systems. - J. Franklin Inst., Vol.274, No.6, pp.425-443. | Zbl 0142.12703
[001] Branin F.H.jr. (1966): The algebraic-topological basis for network analysis and the vector calculus. - Proc. Symp. Generalized Networks, Brooklyn, pp.453-491. | Zbl 0163.39604
[002] Bott R. and Duffin R.J. (1953): On the algebra of networks. - Trans. AMS, Vol.74, pp.99-109. | Zbl 0050.25104
[003] Cauchy A.L. (1815): Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs. - J. l'Ecole Polytechnique, Vol.10, pp.29-97, reprinted in Oeuvres Complètes, Serie II, Tome I, pp.91-169.
[004] Cayley A. (1857): On the theory of the analytical forms called trees. - Philos. Mag., Vol.4, No.13, pp.172-176.
[005] Chen W.K. (1967): Topological network analysis by algebraic methods. - Proc. IEE, Vol.114, No.1, pp.86-88.
[006] Chua L.O. and Lin P.-M. (1975): Computer Aided Analysis of Electronic Circuits-Algorithms and Computational Techniques. - New Jersey: Prentice-Hall. | Zbl 0358.94002
[007] Coates C.L. (1959): Flow-graph solutions of algebraic equations. - IEEE Trans. Circuit Theory, Vol.6, No.2, pp.170-187.
[008] Dmitrischin R.W. (1969): Rasčet linejnych cepej s metodom rasširennych strukturnych čisel. - Isv. Vuzov SSSR, Radioelektreonnika, Vol.12, No.8, pp.806-813 (in Russian).
[009] Duffin R.J. (1959): An analysis of the Wang algebra of networks. - Trans. AMS, Vol.93, pp.114-131. | Zbl 0114.06503
[010] Firestone B.F. (1933): A new analogy between mechanical and electrical systems. - J. Acoust. Soc. Am., Vol.4, pp.249-267.
[011] Gieben G. and Sansen W. (1991): Symbolic Analysis for Automated Design of Analog Integrated Circuits. - Boston: Kluwer.
[012] Gnther M. and Feldmann U. (1999): CAD-based electric-circuit modeling in industry, Part I and II. - Surv. Math. Ind., Vol.8, pp.97-129 and 131-157.
[013] Hhnle W. (1932): Die Darstellung elektromechanischer Gebilde durch rein elektrische Schaltbilder. - Wiss. Verff. a. d. SIEMENS-Konzern, Vol.11, No.1, pp.1-23.
[014] Jacobi C.G.J. (1841): De formatione et proprietatibus determinantium. - Crelle's J. f. reine und angew. Mathem., Vol.22, pp.285-318.
[015] Kirchhoff G. (1845): Uber den Durchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisfrmige. - Poggendorfs Annalen der Physik und Chemie, Vol.64, pp.497-514.
[016] Kirchhoff G. (1847): Uber die Auflsung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strme gefhrt wird. - Poggendorfs Annalen der Physik und Chemie, Vol.72, pp.497-508.
[017] Knig D. (1936): Theorie der Endlichen und Unendlichen Graphen. - Leipzig: Akademische Verlagsgesellschaft.
[018] Krtzig W.B. (1998): Tragwerke 2. 3. Aufl. - Berlin: Springer,
[019] Kron G. (1939): Tensor Analysis of Networks. - New York: Wiley. | Zbl 0022.16801
[020] Le Corbeiller P. (1950): Matrix Analysis of Networks. - New York: Wiley. | Zbl 0041.55304
[021] Lin P.-M. (1991): Symbolic Network Analysis. - Amsterdam: Elsevier.
[022] Mason S.J. (1953): Feedback theory; some properties of signal-flow-graphs. - Proc. IRE, Vol.41, No.9, pp.1144-1156.
[023] Maxwell J.C. (1870): On reciprocal figures, frames and diagrams of forces. - Trans. Roy. Soc. Edinb., Vol.26, pp.1-40.
[024] Maxwell J.C. (1882): Electricity and magnetism I, 3rd Ed, (Ch. VI and Appendix). - Oxford: Clarendon Press.
[025] Mayeda W. and Seshu S. (1965): Generation of trees without duplication. - IEEE Trans. Circuit Theory, Vol.12, No.2, pp.181-185.
[026] Poincar'e H. (1895): Analysis situs. -J. de l'Ecole Polytech., Vol.2, No.1, pp.1-121.
[027] Reibiger A. and Elst G. (1983): Network theoretic interpretation of multibody systems. - Acta Polytechnica Prague, Vol.8, pp.109-129.
[028] Reinschke K.J. (1988): Multivariable Control-A Graph-Theoretic Approach. - Berlin: Springer-Verlag. | Zbl 0682.93006
[029] Reinschke K. and Schwarz P. (1976): Verfahren zur Rechnergesttzten Analyse Linearer Netzwerke. - Berlin: Akademie-Verlag. | Zbl 0325.94015
[030] Roth J.P. (1959): An application of algebraic topology: Kron's method of tearing. - Quart. Appl. Math., Vol.17, No.1, pp.1-24.
[031] Seshu S. and Reed M.B. (1961): Linear Graphs and Electrical Networks. - Reading: Addison-Wesley. | Zbl 0102.34001
[032] Synge J.L. (1951): The fundamental theorem of electrical networks. - Quart. Appl. Math., Vol.9, No.2, pp.113-127. | Zbl 0043.20003
[033] Tellegen B.D.H. (1948): The gyrator, a new electric network element. - Philips Res. Rep., Vol.3, pp.81-101.
[034] Tellegen B.D.H. (1953): A general network theorem with applications. - Proc. Inst. Radio Engrs. Australia, Vol.14, pp.265-270.
[035] Ting S.L. (1935): On the general properties of electrical network determinants. - Chinese J. Phys., No.1, pp.18-40.
[036] Trent H.M. (1955): Isomorphismus between oriented linear graphs and lumped physical systems. - J. Acoust. Soc. Am., Vol.27, No.3, pp.500-527.
[037] Trochimenko J.K. (1972): Metod obobščennych čisel i analiz linejnych cepej. - Moscow: Izd. Sov. Radio (in Russian).
[038] Tsai C.T. (1939): Short cut methods for expanding determinants involved in network problems. - Chinese J. Phys., No.3, pp.148-181. | Zbl 0022.27704
[039] Veblen O. (1916): Analysis situs. - (The Cambridge Colloquium, Part II, Lectures 1916); New York: AMS, 1922.
[040] Wang K.T. (1934): On a new method for the analysis of electrical networks. - Nat. Res. Inst. Eng. Acad. Sinica, Memoir No.2, pp.1-11.
[041] Weyl H. (1923): Repartici'on de corriente en una red conductora (Introducci'on al an'alisis combinatorio). - Revista Matematica Hispano-Americana, Vol.5, pp.153-164. | Zbl 49.0412.03
[042] Weyl H. (1924): An'alisis situs combinatorio (continuaci'on). - Revista Matematica Hispano-Americana, Vol.6, pp.1-9 and pp.33-41.