Kirchhoff's Circuit Laws

Sign convention does the work

Both laws are trivially true once you fix a convention and stick to it. The workable one is: assign every element a current arrow first, then write a voltage drop as positive when you traverse that element in the direction of its arrow. If the solved value comes out negative, the real current runs the other way — that is information, not an error. Most mistakes blamed on Kirchhoff's laws are actually sign errors introduced between the diagram and the equation.

Two laws solve every linear network

Nodal analysis writes the current law at each node and solves for node voltages; mesh analysis writes the voltage law around each independent loop and solves for loop currents. Either method alone is sufficient, and the choice is a matter of which produces fewer unknowns.

• Nodal analysis: one equation per node except the reference; unknowns are voltages. Best when there are many parallel branches and few loops.
• Mesh analysis: one equation per independent loop; unknowns are currents. Best for planar networks with few loops.
• Both methods scale as O(n³) in the number of unknowns, which is why circuit simulators use sparse matrix techniques instead.

Superposition and its limits

In a linear network the response to several sources is the sum of the responses to each source alone, with the others replaced by their internal impedance — a short for an ideal voltage source, an open for an ideal current source. Superposition simplifies hand analysis enormously but it does not apply to power, because power depends on the square of the current and squaring is not linear.