Kirchhoff's Circuit Laws: Conservation Laws for Every Circuit

Two conservation laws, one circuit

Every circuit simulator, from a hand calculation on paper to full SPICE software, ultimately rests on two rules Gustav Kirchhoff wrote down in 1845 while still a student. They are not approximations - they are exact statements of conservation of charge and conservation of energy applied to an electrical network, and together they are enough to solve for every voltage and current in a circuit of arbitrary complexity.

Kirchhoff's Current Law

Kirchhoff's Current Law (KCL) says that the sum of currents flowing into any node equals the sum flowing out - equivalently, the signed sum of all currents at a node is zero. This is charge conservation: charge cannot pile up or vanish at a junction, so whatever flows in this instant must flow out.

sum(I_in) = sum(I_out)        at every node
equivalently:  sum(I_k) = 0    (currents leaving the node taken as positive)

Kirchhoff's Voltage Law

Kirchhoff's Voltage Law (KVL) says that the sum of voltage rises and drops around any closed loop is zero. This is energy conservation: if you walk all the way around a loop and add up every gain and loss in electric potential, you must return to exactly where you started, since potential is a well-defined function of position.

sum(V_k) = 0    around any closed loop (rises positive, drops negative, consistent direction)

KCL and KVL are not independent tricks - they follow from Maxwell's equations in the quasi-static limit (frequencies low enough, or wavelengths long enough compared with the circuit, that propagation delay and radiation can be ignored). KCL is Gauss's law applied to a small volume around a node with no charge accumulation; KVL is Faraday's law applied to a loop with negligible time-varying magnetic flux through it. At radio frequencies, when circuit dimensions become comparable to a wavelength, this quasi-static assumption breaks down and you need full transmission-line or electromagnetic analysis instead.

Turning laws into a solvable matrix

By hand, KCL and KVL let you write down enough independent equations to solve any circuit, but modern circuit solvers formalize the process as Modified Nodal Analysis (MNA), the technique used inside SPICE and every derivative of it. MNA treats every node's voltage (relative to a reference ground node) as an unknown, writes one KCL equation per node - the sum of currents leaving through every resistor, capacitor and source connected to it must be zero - and assembles all of those equations into one linear system G·V = I, where G is a conductance matrix, V is the vector of unknown node voltages, and I is a vector of known current sources.

for each two-terminal element between nodes i and j with conductance g:
    G[i][i] += g;   G[j][j] += g
    G[i][j] -= g;   G[j][i] -= g     // this is the "stamp"
# independent current sources add directly to the I vector;
# voltage sources need an extra unknown (their own current) and an extra KVL row - hence "Modified" nodal analysis
solve  G * V = I   for V

This element-by-element "stamping" procedure is what makes MNA practical for automated tools: each component only ever touches the rows and columns of the nodes it is connected to, so building the system for a circuit with thousands of components is just thousands of small, local, four-entry additions to a sparse matrix, followed by one linear solve. Reactive elements (capacitors, inductors) are handled by discretizing their differential current-voltage relationship in time and folding the result into the same conductance-matrix form at each simulated time step; AC analysis instead replaces every element's real conductance with its complex impedance and solves the same linear system with complex numbers.

A worked example: two resistors and a source

For a simple series circuit - a 12V source driving two resistors R1 and R2 in series - KVL around the single loop gives 12 = I*R1 + I*R2, so I = 12/(R1+R2) directly. For a circuit with a branch point, say R1 from the source to a middle node and then R2 and R3 in parallel from that node to ground, KCL at the middle node states that the current entering through R1 equals the sum of the currents leaving through R2 and R3, giving one equation in one unknown node voltage that, once solved, hands you every current in the circuit via Ohm's law.

Frequently asked questions

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