Superposition Theorem
The Superposition Theorem states that in any linear circuit with multiple independent sources, the response (voltage or current) at any element equals the algebraic sum of the responses caused by each independent source acting alone, with all other independent sources deactivated. Voltage sources are deactivated by replacing them with short circuits, while current sources are deactivated by replacing them with open circuits. This theorem greatly simplifies the analysis of circuits with multiple sources and applies only to linear systems.
Worked Example
Problem
A circuit has a 10 V source (V1) and a 5 A current source (I1) driving a 2 Ω resistor (R). Find the current through R using superposition.
Solution
Step 1: Due to V1 alone (deactivate I1 → open circuit): Current through R from V1: I_R1 = V1 / R = 10 / 2 = 5 A. Step 2: Due to I1 alone (deactivate V1 → short circuit): With V1 shorted, R is in parallel with the short → all current bypasses R. I_R2 = 0 A (short circuit takes all current). Step 3: Total current: I_R = I_R1 + I_R2 = 5 + 0 = 5 A.
Answer
Current through R = 5 A
Source Deactivation Rules in Superposition
| Source Type | Deactivation Method | Replacement Element | Reason |
|---|---|---|---|
| Ideal voltage source | Short circuit (V = 0) | Wire (0 Ω) | Zero voltage = no drop = short |
| Ideal current source | Open circuit (I = 0) | Break in wire (∞ Ω) | Zero current = no flow = open |
| Dependent voltage source | Do NOT deactivate | Remains in circuit | Depends on circuit variable |
| Dependent current source | Do NOT deactivate | Remains in circuit | Depends on circuit variable |
| Real voltage source | Replace with internal resistance | Series resistor only | Accounts for source resistance |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Thevenin's Theorem
Thevenin's Theorem states that any linear electrical network with voltage sources, current sources, and resistances can be replaced by an equivalent circuit consisting of a single voltage source (V_th) in series with a single resistance (R_th). This simplification makes it much easier to analyse the behaviour of a load connected to a complex network, as only the terminal behaviour matters. It is widely used in circuit design, power systems, and electronics to simplify analysis without solving the full network repeatedly.
Norton's Theorem
Norton's Theorem states that any linear electrical network can be replaced by an equivalent circuit consisting of a single current source (I_N) in parallel with a single resistance (R_N). It is the dual of Thevenin's Theorem and is particularly convenient when analysing circuits where current distribution is of primary interest. Norton and Thevenin equivalents are interconvertible, and choosing between them depends on whether the circuit is better suited to series or parallel analysis.
Kirchhoff's Voltage Law
Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all voltages around any closed loop in a circuit equals zero. This principle is a direct consequence of the conservation of energy — as a charge traverses a complete loop, the energy gained from sources equals the energy lost across resistances. KVL is fundamental for analysing series circuits, mesh analysis, and determining unknown voltages in complex networks.
The word "superposition" comes from Latin "super" (above, over) and "positio" (placement), meaning placing one on top of another. In mathematics and physics, it describes the combination of independent solutions. The theorem was formalised in the context of circuit analysis in the 19th century as a consequence of linearity.