Kirchhoff's Laws and Circuit Analysis

Kirchhoff’s two rules are conservation laws wearing electrical clothing: junctions conserve charge (what flows in flows out) and loops conserve energy (a charge returning to its start must break even). Every circuit, however tangled, yields to these two bookkeeping rules.

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

Junction rule (KCL — conservation of charge):

$$\sum I_{\text{in}} = \sum I_{\text{out}}$$

Loop rule (KVL — conservation of energy):

$$\sum \Delta V = 0 \quad\text{around any closed loop}$$

Sign conventions for the loop rule

Systematic procedure

1. Assign current variables and directions to each branch.
2. Write KCL equations at each junction (except one).
3. Write KVL equations around independent loops.
4. Solve the system of linear equations.

RC circuits — Charging a capacitor through a resistor:

$$V_C(t) = \varepsilon\left(1 - e^{-t/RC}\right), \qquad \tau = RC$$

The time constant $\tau = RC$ is the time to reach $\approx 63\%$ of the final voltage.

Tip: The number of independent equations always equals the number of unknown currents. If it doesn't, you missed a loop or junction.

Common pitfall: Sign errors kill more circuit solutions than concept errors. Fix a travel direction around each loop first, then let the algebra decide: a negative answer just means the true current flows opposite your guess — it is not a mistake.