Superposition theorem with example

Statement

Superposition theorem: in a linear network containing more than one independent source, the current through (or voltage across) any element is the algebraic sum of the currents (or voltages) produced by each source acting alone, with all the other independent sources replaced by their internal resistances.

Rules

• Replace each unused voltage source by a short circuit.
• Replace each unused current source by an open circuit.
• Add the individual results with their signs (direction matters).
• It applies only to linear circuits and to currents and voltages, not to power (power is not linear).

Steps

1. Keep one source, set the others to zero.
2. Find the required current or voltage.
3. Repeat for every source.
4. Add the results algebraically.

Example

A 10 V source and a 5 V source feed a 2 Ω resistor R through resistors R₁ = 2 Ω (10 V side) and R₂ = 4 Ω (5 V side), all joined at the same node.

 (10V)--[2]--+--[4]--(5V)
             |
            [R=2]
             |
 -----------------------

• 10 V alone (5 V shorted): total R = 2 + (4 ∥ 2) = 3.333 Ω, I = 3 A; current in R = 3 × 4/(4 + 2) = 2 A.
• 5 V alone (10 V shorted): total R = 4 + (2 ∥ 2) = 5 Ω, I = 1 A; current in R = 0.5 A.
• Total current in R = 2 + 0.5 = 2.5 A.

Check by nodal analysis: V_R(1/2 + 1/4 + 1/2) = 10/2 + 5/4, so V_R = 6.25/1.25 = 5 V and I_R = 2.5 A.