A Necessary Addition to Kirchhoff’s Current Law of Circuits

Abstract

Kirchhoff’s current law for circuits is about the flux ? of charge, usually electrons, and is derived for systems without significant time dependence of total charge. Yet Kirchhoff’s current law is used to design circuits on nanosecond time scales where time dependence is important. Maxwell’s equations are general and deal with time dependence. They imply conservation of the source term for the magnetic field. ?_????? = ? + ?₀ ??⁄?? is conserved. It does not accumulate. ? accumulates. ? accumulates as charge ? as the electric field changes. ? is not conserved. It is necessary to add a term ?₀ ??⁄?? to reconcile Kirchhoff’s law with the Maxwell Ampere equation. But this term is not sufficient to allow a derivation of Kirchhoff’s law for general circuits. Additional constraints are needed that depend on details of the real circuit not usually included in simplified circuit diagrams, like the skin effect of current flow in wires and the locations and amount of stray capacitance. Kirchhoff’s current law can made general for the simplified circuits used by circuit designers, if the displacement term ?₀ ??⁄?? is included in a revised definition of current ?_????? in Kirchhoff’s current law ?_????? = ? + ?₀ ??⁄??. Including stray capacitance everywhere automatically adds the needed term ?₀ ??⁄??. Perhaps that is why the need for the added terms has not been widely recognized.