%0 Journal Article %1 Brayton1964Theory %A Brayton, R. K. %A Moser, J. K. %D 1964 %J Quarterly of Applied Mathematics %K 37j05-finite-dimensional-hamiltonian-general-theory 94c05-analytic-circuit-theory 94c15-circuits-networks-applications-of-graph-theory %N 1 %P 1--33 %R 10.1090/qam/169746 %T A Theory of Nonlinear Networks. I %U http://dx.doi.org/10.1090/qam/169746 %V 22 %X This report describes a new approach to nonlinear RLC-networks which is based on the fact that the system of differential equations for such networks has the special form \$\$ L(i)didt = P (i, v)i, C (v)dvdt = -P (i, v)v. \$\$ The function, \$Płeft(i, v\right)\$, called the mixed potential function, can be used to construct Liapounov-type functions to prove stability under certain conditions. Several theorems on the stability of circuits are derived and examples are given to illustrate the results. A procedure is given to construct the mixed potential function directly from the circuit. The concepts of a complete set of mixed variables and a complete circuit are defined.

@article{Brayton1964Theory, abstract = {This report describes a new approach to nonlinear RLC-networks which is based on the fact that the system of differential equations for such networks has the special form \$\$ L(i)\frac{di}{dt} = \frac{\partial P (i, v)}{\partial i}, C (v)\frac{dv}{dt} = -\frac{\partial P (i, v)}{\partial v}. \$\$ The function, \$P\left(i, v\right)\$, called the mixed potential function, can be used to construct Liapounov-type functions to prove stability under certain conditions. Several theorems on the stability of circuits are derived and examples are given to illustrate the results. A procedure is given to construct the mixed potential function directly from the circuit. The concepts of a complete set of mixed variables and a complete circuit are defined.}, added-at = {2019-03-01T00:11:50.000+0100}, author = {Brayton, R. K. and Moser, J. K.}, biburl = {https://www.bibsonomy.org/bibtex/20b62857cc4f211770b1ca9f8a581a228/gdmcbain}, citeulike-article-id = {14488226}, citeulike-attachment-1 = {brayton_64_theoryI.pdf; /pdf/user/gdmcbain/article/14488226/1124134/brayton_64_theoryI.pdf; 47354a6f9bd7b48f23f896f9ca1d195a737738ad}, citeulike-linkout-0 = {http://dx.doi.org/10.1090/qam/169746}, doi = {10.1090/qam/169746}, file = {brayton_64_theoryI.pdf}, interhash = {254cf1b79d842d4bb573b7d7c2ccad9f}, intrahash = {0b62857cc4f211770b1ca9f8a581a228}, issn = {0033-569X}, journal = {Quarterly of Applied Mathematics}, keywords = {37j05-finite-dimensional-hamiltonian-general-theory 94c05-analytic-circuit-theory 94c15-circuits-networks-applications-of-graph-theory}, number = 1, pages = {1--33}, posted-at = {2017-12-04 16:32:59}, priority = {5}, timestamp = {2019-03-01T00:11:50.000+0100}, title = {A Theory of Nonlinear Networks. {I}}, url = {http://dx.doi.org/10.1090/qam/169746}, volume = 22, year = 1964 }