Superposition Theorem

Analyze one source at a time, then add the results.

Circuit AnalysisElectrical Engineering Year 1
⏱️ About 16 min

Multiple sources in a circuit don't have to mean multiple headaches — analyze them one at a time and add the answers.

💡
The big idea: In a linear circuit, each independent source contributes independently to every voltage and current, so you can solve for each source acting alone and sum the results.
🎯 By the end, you'll be able to
  • State the superposition principle and its linearity requirement
  • Turn off independent voltage and current sources correctly
  • Calculate a branch current by summing individual source contributions
  • Explain why power cannot be superposed directly
📎 Helpful to know first
  • Source Transformations

The Core Idea

When a linear circuit contains two or more independent sources, you can find any voltage or current in the circuit by considering each source one at a time. While evaluating one source, you turn off all other independent sources: voltage sources become short circuits (0 V) and current sources become open circuits (0 A). The total response is the algebraic sum of the individual contributions.

✨ Why Linearity Matters

Superposition works because Ohm's law and Kirchhoff's laws are linear. If every element satisfies a linear relationship, the circuit's response scales linearly with each source. This is why superposition applies only to linear elements like resistors, capacitors, and inductors (under AC phasor analysis) — not to diodes, transistors, or other nonlinear devices.

\[ I_{\text{total}} = \sum_{k=1}^{N} I_k \qquad V_{\text{total}} = \sum_{k=1}^{N} V_k \]
Total current or voltage equals the sum of contributions from each independent source acting alone.
⚠️ Power Is Nonlinear — Don't Superpose It

Power follows P = I²R, which is quadratic in current. If source A alone produces 2 A through a 4 Ω resistor, that's P_A = (2)² × 4 = 16 W. If source B alone also produces 2 A, that's P_B = 16 W. But both sources together produce 4 A, giving P = (4)² × 4 = 64 W — not 32 W. Always find the total current or voltage first via superposition, then compute power once.

📝 Worked example: A circuit has a 12 V voltage source in series with R1 = 2 Ω. This branch is in parallel with R2 = 4 Ω and a 6 A current source. Find the current through R2 using superposition.
  1. Turn off the current source (replace with an open circuit). The circuit is now a 12 V source in series with R1 = 2 Ω and R2 = 4 Ω. Total resistance = 2 + 4 = 6 Ω. Current through R2 from the voltage source: I_V = 12 V/6 Ω = 2 A (top-to-bottom).
  2. Turn off the voltage source (replace with a short circuit). Now R1 = 2 Ω and R2 = 4 Ω are in parallel. The 6 A current divides between them. Using the current divider rule: I_I = 6 A × (R1/(R1+R2)) = 6 × (2/6) = 2 A (top-to-bottom through R2).
  3. Sum the contributions: I_R2 = I_V + I_I = 2 A + 2 A = 4 A.
✓ The total current through R2 is 4 A.
✏️ Practice: A 10 V voltage source is in series with R1 = 4 Ω, and this branch is in parallel with R2 = 6 Ω and a 5 A current source. Using superposition, find the total current (in A) through R2.
A
Solution
  1. Turn off the current source (open). Series circuit: R_total = 4 + 6 = 10 Ω. I_V = 10/10 = 1 A through R2.
  2. Turn off the voltage source (short). Current divider: I_I = 5 × (4/(4+6)) = 2 A through R2.
  3. Total: I_R2 = 1 + 2 = 3 A.

Check your understanding

1. When applying superposition, how do you deactivate an independent current source?
A current source set to zero delivers no current, which is equivalent to an open circuit.
2. Why can't you directly superpose power contributions from individual sources?
Power depends on the square of current, making it nonlinear. You must find the total current or voltage first, then compute power once.
🔎 Related lessons
Thévenin's Theorem
✅ Key takeaways
  • Superposition applies only to linear circuits with independent sources.
  • Turn off voltage sources (short) and current sources (open) one at a time.
  • Sum individual voltage or current contributions for the total response.
  • Power is nonlinear — find total current or voltage first, then compute power once.
➡️ Next, we'll use this linearity foundation to reduce entire networks to a single source and resistor.
Want to test yourself on this? Try the Electrical Aptitude test →