A theory of nonlinear networks. I

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">L\left ( i \right )\frac {{di}}{{dt}} = \frac {{\partial P\left ( {i,v} \right )}}{{\partial i}},C\left ( v \right )\frac {{dv}}{{dt}} = - \frac {{\partial P\left ( {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">P\left ( {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.