Superposition Theorem — Circuits

The superposition theorem in circuits lets you find one voltage or current in a linear circuit with multiple independent sources by solving one source at a time and adding the signed results. If you are searching for how superposition works, the key rule is simple: turn off the other independent sources correctly, solve the partial response, then add the contributions.

This only works when the circuit model is linear. In ordinary introductory problems, that usually means resistors and linear source models, not nonlinear device behavior.

What The Superposition Theorem Says

Suppose a circuit has several independent sources and you want the current through one resistor or the voltage across one branch. Instead of solving the full circuit in one step, you can:

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

The result is the same total voltage or current you would get by solving the whole linear circuit at once.

How To Turn Off Sources Correctly

This is the step that causes most mistakes.

For ideal voltage sources, set the source to zero volts. In the circuit model, that means replacing the source with a short circuit.

For ideal current sources, set the source to zero current. In the circuit model, that means replacing the source with an open circuit.

If the circuit contains a dependent source, do not turn it off just because you are using superposition. Dependent sources stay active because their values are tied to circuit variables.

Worked Example: Two Opposing Voltage Sources In One Loop

Consider a single loop with two resistors in series, $R_1 = 2\ \Omega$ and $R_2 = 4\ \Omega$. The same loop also contains two ideal voltage sources: $V_1 = 12\ \mathrm{V}$ and $V_2 = 6\ \mathrm{V}$. Assume the sources oppose each other, and define clockwise current as positive.

The total resistance is

$$R_{total} = 2 + 4 = 6\ \Omega$$

Now solve the same loop one source at a time.

Contribution From $V_1$ Alone

Deactivate $V_2$. Because it is an ideal voltage source, replace it with a short circuit.

Then the loop current caused by $V_1$ is

$$I_1 = \frac{12}{6} = 2\ \mathrm{A}$$

This contribution is positive because it drives current in the chosen clockwise direction. So the partial result is $+2\ \mathrm{A}$.

Contribution From $V_2$ Alone

Deactivate $V_1$ by replacing it with a short circuit.

Now $V_2$ alone drives current through the same $6\ \Omega$ total resistance:

$$I_2 = \frac{6}{6} = 1\ \mathrm{A}$$

But this source pushes current opposite to the chosen positive direction, so you must keep the sign:

$$I_2 = -1\ \mathrm{A}$$

Add The Signed Currents

The total loop current is

$$I = I_1 + I_2 = 2 + (-1) = 1\ \mathrm{A}$$

That is the whole idea behind superposition. Each source creates part of the response, and the total current is the algebraic sum of those parts.

Why Superposition Helps In Circuit Analysis

Superposition is useful when a circuit has several independent sources and you want to see what each source is doing separately. It often makes a messy circuit easier to organize, and it gives physical insight instead of only producing one final number.

It is especially useful in introductory network analysis, small-signal linear models, and any linear circuit where separate source effects are worth comparing.

Common Mistakes In Superposition Problems

Using It In A Nonlinear Circuit

If the circuit model is not linear, the theorem does not apply in this simple form. Components such as diodes or transistors in nonlinear operating conditions can break the add-up-the-responses logic.

Turning Off Dependent Sources

Only independent sources are deactivated one at a time. Dependent sources remain in the circuit.

Adding Power Contributions Directly

Superposition applies directly to voltages and currents. Power depends on products such as $P = VI$ or $P = I^2R$, so you should first find the total voltage or current and then compute power from that total result.

Losing The Sign Convention

Each partial contribution must keep its sign. If one source drives current opposite to your chosen positive direction, its contribution is negative.

When The Superposition Theorem Is Used

The superposition theorem is used in linear DC and AC circuit analysis, especially when a circuit contains multiple sources and the target is one branch current or voltage. In AC work, the same idea still applies if the circuit is analyzed in a linear phasor model.

It is less useful when one direct equation is faster. It becomes valuable when source-by-source reasoning makes the circuit easier to understand or check.

Try A Similar Circuit

Change the example so $V_2 = 9\ \mathrm{V}$ instead of $6\ \mathrm{V}$ and keep the same resistor values. Work out the two single-source currents first, then add the signed results. If you want a quick check after solving it yourself, compare your steps with the same circuit in GPAI Solver.