Superposition Theorem — Circuits

The superposition theorem finds one voltage or current in a linear circuit with several independent sources by solving the circuit one source at a time and adding the signed results. The method itself is short; the part that decides whether you get the right answer is turning the other sources off correctly and tracking signs.

The Method And Its Rules

To find the current through one resistor or the voltage across one branch in a circuit with several independent sources:

1. keep one independent source active
2. deactivate the other independent sources
3. solve for that source's contribution
4. repeat for each remaining source
5. add the signed results

Deactivating a source has a precise meaning. An ideal voltage source set to zero volts becomes a short circuit. An ideal current source set to zero current becomes an open circuit. A dependent source is never turned off, because its value is tied to circuit variables, so it stays active throughout. The sum equals the same total voltage or current you would get by solving the whole linear circuit at once.

Why Superposition Works

The theorem holds only because the circuit model is linear. In ordinary introductory problems that means resistors and linear source models, not nonlinear device behavior. Linearity is exactly the property that lets a combined response equal the sum of individual responses: each source drives the circuit independently, and because every relationship between voltage and current is proportional, the partial effects simply add. Remove linearity, with a diode or a transistor in a nonlinear regime, and the add-up-the-responses logic breaks, which is why the very first check before using superposition is whether the circuit is linear.

Worked Example: Two Opposing Sources In One Loop

A single loop has two resistors in series, $R_1 = 2\ \Omega$ and $R_2 = 4\ \Omega$, and two ideal voltage sources, $V_1 = 12\ \mathrm{V}$ and $V_2 = 6\ \mathrm{V}$, that oppose each other. Define clockwise current as positive. The total resistance is

Contribution from $V_1$ alone. Deactivate $V_2$ by replacing the ideal voltage source with a short circuit:

This drives current clockwise, so the partial result is $+2\ \mathrm{A}$.

Contribution from $V_2$ alone. Deactivate $V_1$ with a short circuit:

Because this source pushes current opposite to the chosen positive direction, keep the sign:

Add the signed currents.

Each source creates part of the response, and the total current is the algebraic sum of those parts.

Try It Yourself

Rework the same loop with $V_2 = 9\ \mathrm{V}$ instead of $6\ \mathrm{V}$, keeping the resistors unchanged. Solve the two single-source currents first, then add the signed results. As a check, $V_1$ alone still gives $+2\ \mathrm{A}$, while $V_2$ now gives $9/6 = 1.5\ \mathrm{A}$ in the negative direction, so the total is $2 - 1.5 = 0.5\ \mathrm{A}$. If the opposing source ever drives the total negative, that simply means the net current reversed, which the signs handle automatically.

Pitfalls In Superposition Problems

• Using it in a nonlinear circuit. With diodes or transistors in nonlinear conditions, the theorem does not apply in this simple form.
• Turning off dependent sources. Only independent sources are deactivated one at a time; dependent sources remain.
• Adding power contributions directly. Superposition applies to voltages and currents, but power depends on products such as $P = VI$ or $P = I^2R$, so find the total voltage or current first, then compute power from that total.
• Losing the sign convention. Each partial contribution keeps its sign; a source driving current opposite to your chosen positive direction contributes a negative value.

Where The Theorem Is Used

Superposition appears in linear DC and AC circuit analysis, especially when a circuit has multiple sources and you want one branch current or voltage. In AC work the same idea applies within a linear phasor model. It is less useful when one direct equation is faster, and most valuable when source-by-source reasoning makes a messy circuit easier to organize, understand, or check.