Kirchhoff's Laws and Circuit Analysis

Kirchhoff's Laws

Kirchhoff's rules are conservation laws wearing electrical clothing: junctions conserve charge and loops conserve energy.

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

• Junction rule (KCL): $\sum I_{\text{in}} = \sum I_{\text{out}}$, conserving charge.
• Loop rule (KVL): $\sum \Delta V = 0$ around any closed loop, conserving energy.

Systematic procedure: Assign branch currents, write KCL at junctions, write KVL around loops, and solve the linear system.

Common pitfall: Fix a travel direction first and let algebra decide signs. A negative answer just means true current flows opposite your guess.

RC Circuits and Analysis

An RC circuit combines resistors and capacitors, governing how voltage builds over time.

Charging equation:

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

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

Sanity check: The number of independent equations always equals the number of unknown currents. If they do not match, you missed a loop or junction.

Tip: Sign errors kill more circuit solutions than concept errors. Always trust your established sign conventions.