Kirchhoff's Circuit Laws
The two conservation laws that govern every electrical circuit: current conservation at nodes and voltage conservation around loops.
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Kirchhoff's circuit laws are the two conservation rules on which all circuit analysis rests. Stated by the German physicist Gustav Kirchhoff in 1845 as an extension of Ohm's law, they apply to any lumped-element circuit: Kirchhoff's current law (KCL) says that the sum of currents entering a node equals the sum leaving it, which is charge conservation — charge cannot accumulate at a junction. Kirchhoff's voltage law (KVL) says that the sum of voltage rises around any closed loop equals the sum of voltage drops, which is energy conservation — a charge returning to its starting point has the same potential energy it started with.
Together with the component equations — for a resistor V = IR, for a capacitor I = C·dV/dt, for an inductor V = L·dI/dt — the two laws yield a complete system of equations for any network. The standard solution procedure is either node-voltage analysis, which writes KCL at every node and solves for node voltages, or mesh-current analysis, which writes KVL around every independent loop and solves for loop currents. Both reduce to linear algebra for resistive circuits, and both are exactly what circuit simulation programs such as SPICE do internally.
A simple two-loop example shows the pattern. With loop currents I1 and I2, a shared resistor R2 carries I1 − I2:
Loop 1: V1 = R1·I1 + R2·(I1 − I2)
Loop 2: −V2 = R3·I2 + R2·(I2 − I1)
Two equations, two unknowns; solving gives both loop currents, and any branch current follows by combination. The signs matter: a current entering a resistor in the direction of the assumed loop produces a voltage drop, and a battery whose polarity opposes the loop direction contributes a negative rise.
The laws assume the lumped-element approximation: components are small compared with the wavelength of the signals involved, so voltages and currents are well defined at each point. At radio frequencies, in long transmission lines, and inside fast digital circuits, this approximation breaks down and fields must be analyzed directly — which is why high-speed circuit design uses transmission-line theory rather than simple KVL. For ordinary audio, power, and low-frequency electronics, Kirchhoff's laws are exact in practice and are the foundation of every textbook circuit analysis.
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circuits electricity electronics kirchhoff