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• W2562183967 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 <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L left-parenthesis i right-parenthesis StartFraction d i Over d t EndFraction equals StartFraction partial-differential upper P left-parenthesis i comma v right-parenthesis Over partial-differential i EndFraction comma upper C left-parenthesis v right-parenthesis StartFraction d v Over d t EndFraction equals minus StartFraction partial-differential upper P left-parenthesis i comma v right-parenthesis Over partial-differential v EndFraction period> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mfrac> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>i</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mfrac> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mfrac> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>v</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Lleft ( i right )frac {{di}}{{dt}} = frac {{partial Pleft ( {i,v} right )}}{{partial i}},Cleft ( v right )frac {{dv}}{{dt}} = - frac {{partial Pleft ( {i,v} right )}}{{partial v}}.</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> The function, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P left-parenthesis i comma v right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>Pleft ( {i,v} right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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." @default.
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• W2562183967 date "1964-01-01" @default.
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• W2562183967 title "A theory of nonlinear networks. I" @default.
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• W2562183967 doi "https://doi.org/10.1090/qam/169746" @default.
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