• word_count: 200

Kirchhoff's two laws are the foundation of circuit analysis:

Junction Rule (KCL): At any node, $\sum I_{\text{in}} = \sum I_{\text{out}}$. Based on charge conservation — charge cannot accumulate at a junction in steady state.

Loop Rule (KVL): Around any closed loop, $\sum V = 0$. Based on energy conservation — the electric field is conservative. Convention: voltage rises (battery $-$ to $+$) are positive; voltage drops (resistor in current direction) are negative.

Systematic approach: (1) Assign current directions (arbitrary — if wrong, the answer comes out negative). (2) Apply KCL at each junction to reduce the number of unknowns. (3) Write KVL equations for independent loops (number of independent loops = branches $-$ nodes $+$ 1). (4) Solve the system of equations.

For complex circuits, additional techniques help: symmetry arguments (identify equipotential points to simplify), star-delta transformations (convert between Y and $\Delta$ configurations), superposition theorem (analyze one source at a time), and Thevenin/Norton equivalents (replace complex networks with a single source and resistance). The Wheatstone bridge balance condition $P/Q = R/S$ is a direct consequence of KVL applied to the bridge loops.