FAD1022 L11 — Kirchhoff's Rules (Theory)

Overview

This lecture introduces the fundamental principles of electric circuits and Kirchhoff's Rules for analyzing multi-loop circuits. The lecture covers circuit symbols, series/parallel networks, and the two fundamental laws for circuit analysis.

11.1 Fundamentals of Electric Circuit

Standard Symbols for Electrical Components

Component Symbol
Battery 🔋 ⎓
Resistor ／//\
Switch ——o
Capacitor ｜｜
Variable Resistor ／//\↗
LED —▷｜

Series and Parallel Networks

Series Circuit Characteristics

  • Current: The same in all parts of the circuit
  • Voltage: Sum of voltages equals total applied voltage: $$V = V_1 + V_2 + V_3$$

Parallel Circuit Characteristics

  • Current: Sum of branch currents equals total current: $$I = I_1 + I_2 + I_3$$
  • Voltage: Same across all parallel branches

11.2 Kirchhoff's Junction Rule / Current Law (KCL)

Statement

Kirchhoff's first rule (the junction rule) applies to the charge entering and leaving a junction or node (connection/intersection of three or more wires).

Physical Principle

Current is the flow of charge, and charge is conserved; thus, whatever charge flows into the junction must flow out.

Formula

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

Or equivalently (using sign convention): $$\sum I = 0$$

Key Points

  • Applies at any junction/node where three or more wires meet
  • Based on conservation of charge
  • Current directions can be assigned arbitrarily; if calculated value is negative, actual direction is opposite

11.3 Kirchhoff's Loop Rule / Voltage Law (KVL)

Statement

Kirchhoff's second rule (the loop rule) applies to potential differences. The algebraic sum of potential differences, including voltage supplied by voltage sources and resistive elements, in any closed loop must equal zero.

Formula

$$\sum \Delta V = 0$$

Or written as: $$\sum \mathcal{E} = \sum IR$$

Key Points

  • Based on conservation of energy
  • Applies to any closed loop in the circuit
  • Must account for sign conventions

Sign Conventions

For EMF (Voltage Sources)

  • Positive (+): Travel from (−) to (+) terminal (potential rise)
  • Negative (−): Travel from (+) to (−) terminal (potential drop)

For Resistors

  • Negative (−IR): Travel in same direction as current
  • Positive (+IR): Travel opposite to current direction

Worked Examples

Problem #11.1: Resistors in Series

Given: Five resistors in series with $V = 9.0$ V

  • $R_1 = R_2 = R_3 = R_4 = 20\ \Omega$
  • $R_5 = 10\ \Omega$

Solutions:

  • (a) $R_{\text{eq}} = 20 + 20 + 20 + 20 + 10 = 90\ \Omega$
  • (b) $I = V/R_{\text{eq}} = 9.0/90 = 0.1$ A (same through all resistors)
  • (c) Voltage drops: $V_1 = V_2 = V_3 = V_4 = 2.0$ V, $V_5 = 1.0$ V
  • (d) $P_{\text{dissipated}} = I^2 R_{\text{eq}} = (0.1)^2(90) = 0.9$ W
  • (e) $P_{\text{battery}} = V_{\text{batt}} \times I = 9.0 \times 0.1 = 0.9$ W ✓

Problem #11.2: Resistors in Parallel

Given: Three resistors in parallel with $V = 3.0$ V

  • $R_1 = 1.0\ \Omega$, $R_2 = 2.0\ \Omega$, $R_3 = 3.0\ \Omega$

Solution: $$R_{\text{eq}} = \left(\frac{1}{1.0} + \frac{1}{2.0} + \frac{1}{3.0}\right)^{-1} = \left(\frac{6 + 3 + 2}{6}\right)^{-1} = \frac{6}{11} = 0.55\ \Omega$$

Mermaid Diagram: Kirchhoff's Rules Overview

graph TD
    A[Kirchhoff's Rules] --> B[Junction Rule<br/>KCL]
    A --> C[Loop Rule<br/>KVL]
    B --> D["ΣI_in = ΣI_out<br/>Conservation of Charge"]
    C --> E["ΣV = 0<br/>Conservation of Energy"]
    D --> F[Apply at Nodes<br/>3+ wires meet]
    E --> G[Apply to Loops<br/>Closed paths]
    F --> H[Label Currents]
    G --> I[Follow Sign<br/>Conventions]

Key Formulas Summary

Rule Formula Conservation Law
KCL (Junction) $\sum I_{\text{in}} = \sum I_{\text{out}}$ Charge
KVL (Loop) $\sum V = 0$ Energy
Series Resistors $R_{\text{eq}} = R_1 + R_2 + ...$ —
Parallel Resistors $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ —

Links