Lumped elements and Kirchhoff laws

Let the largest relevant interconnect dimension be d and the propagation speed along its environment be v. A sinusoidal signal accumulates a phase of order angular frequency times travel time across that dimension. Lumped modeling requires this phase to be small wherever one voltage or current is assigned to an extended object. For a pulse, use the fastest significant rise time rather than its repetition period; a slow clock can still have electrically long connections.

Electrical smallness is necessary but does not remove every parasitic effect. A tightly wound coil can have appreciable capacitance between turns even while its outer dimensions are small. A narrow wire can contribute inductance that matters relative to a small intended inductance. The useful test is whether ignored storage, radiation, delay, and loss are small compared with the quantities retained at the accuracy required. There is no universal dimension in centimetres that guarantees success.

Integrating charge continuity over a small junction says that inward conduction currents increase the charge stored there. The usual Kirchhoff current law replaces this charge accumulation by explicit capacitive branches, leaving an ideal node with negligible unmodeled storage. Thus current through a capacitor branch is included in circuit accounting even though conduction electrons do not cross its dielectric. The capacitor terminal current records charging of its plates, not a violation of conservation.

A circuit voltage is a consistently defined difference between terminals. In an electroquasistatic region, it can be associated with a nearly path independent potential difference. An inductive loop requires more care because Faraday induction creates a circulating electric field. Assign induced electromotive force to an inductor, transformer, or explicit source; then use a loop balance with that contribution included. A bare statement that every physical electric-field loop integral vanishes would contradict Maxwell's equations.

For linear ideal elements, a capacitor stores electric energy and an inductor stores magnetic energy. Their constitutive laws introduce time derivatives, so lumped does not mean static. A resonant circuit can alternate rapidly between electric and magnetic storage while remaining electrically small. Its resonance wavelength can greatly exceed its dimensions because the element geometry concentrates fields. A resistor instead converts electrical energy into other forms and need not carry a storage variable in the ideal model.

Choose current entering the positive terminal so that voltage times current is absorbed power. This passive sign convention makes a resistor's power positive and gives storage rates directly for constant capacitance and inductance. If a supposed ideal wire has a measurable phase difference between its ends, promote it into an element or transmission line. Improving a circuit model usually means assigning omitted field effects explicitly, rather than deciding that charge conservation or induction has stopped working.