Superposition Theorem

Definition

The superposition theorem states that in any linear network containing two or more independent sources, the current through or voltage across any element is the algebraic sum of the currents or voltages produced by each independent source acting alone, while all other independent sources are replaced by their internal resistances.

For ideal source replacement:

An independent voltage source is replaced by a short circuit when deactivated.
An independent current source is replaced by an open circuit when deactivated.

This theorem is valid only for linear circuits. It cannot be directly applied to power calculations because power is not a linear quantity.

Main Content

1. Linear Circuits and the Principle of Superposition

A circuit is called linear when its voltage-current relationship follows linear laws, meaning the output is directly proportional to the input and the circuit parameters remain constant.
The superposition theorem works because linear circuits obey the principle that the total response caused by multiple independent sources is equal to the sum of the responses caused by each source individually.

In practical terms, if a circuit has two voltage sources and one current source, you analyze the circuit three times:

Keep one source active.
Deactivate the other sources one at a time.
Add all the individual voltage or current responses.

Example:
If a resistor carries 2 A due to one source and 3 A due to another source, then the total current is 5 A, provided the circuit remains linear. If the directions are opposite, then algebraic addition is used, such as 4 A upward and 1 A downward giving a net of 3 A upward.

2. Deactivating Independent Sources

To apply the theorem correctly, all independent sources except one must be turned off in each stage.
Deactivating a source means replacing it with its internal resistance:
Ideal voltage source → short circuit
Ideal current source → open circuit

This does not mean removing dependent sources. Dependent sources remain active because they depend on circuit variables and are part of the network behavior.

Important note:
If a voltage source has internal resistance, that resistance remains in the circuit when the source is deactivated. Similarly, if a current source has internal resistance, that resistance is retained.

Example:
Suppose a circuit contains a 10 V source, a 2 A source, and resistors.

For the first case, keep the 10 V source active and open-circuit the 2 A source.
For the second case, keep the 2 A source active and short-circuit the 10 V source.
Then calculate the required branch current or voltage in each case and add them.

3. Algebraic Addition of Individual Responses

The final value of current or voltage is obtained by adding the individual contributions algebraically, taking direction and polarity into account.
The theorem applies to voltage and current, but not directly to power, because power depends on square of voltage or current and does not follow linear superposition.

If the responses are:

$V_1 = 6 \text{ V}$ due to source 1
$V_2 = -2 \text{ V}$ due to source 2

Then total voltage is: $V = V_1 + V_2 = 6 + (-2) = 4 \text{ V}$

Example of current superposition:
If a branch current is:

$I_1 = 1.5 \text{ A}$ due to source A
$I_2 = 0.5 \text{ A}$ due to source B in the same direction Then total current: $I = 1.5 + 0.5 = 2 \text{ A}$ If the second current flows opposite, then it is subtracted algebraically.

Working / Process

1. Identify the quantity to be found

Determine whether the problem asks for branch current, voltage across an element, or node voltage.
Select the exact element or branch where the response is required.

2. Activate one independent source at a time

Keep only one independent source active.
Replace all other independent voltage sources by short circuits and current sources by open circuits.
Keep dependent sources active if present.

3. Solve the simplified circuit and add responses

Find the required current or voltage for each source separately.
Repeat the process for all independent sources.
Add all results algebraically to obtain the final answer.

Illustrative example:
Consider a circuit with two voltage sources and one resistor branch.

Case 1: Only source $V_1$ is active, and source $V_2$ is shorted. Find current $I_1$ .
Case 2: Only source $V_2$ is active, and source $V_1$ is shorted. Find current $I_2$ .
Final current: $I = I_1 + I_2$ If one current is opposite in direction, use negative sign accordingly.

Advantages / Applications

Simplifies complex circuit analysis

by breaking a multi-source circuit into smaller single-source circuits.

Helps in finding branch currents and voltages

in linear circuits quickly and systematically.

Widely used in electrical and electronics engineering

for DC networks, AC circuits, signal analysis, and troubleshooting.

Summary

The superposition theorem is a method for analyzing linear circuits with multiple independent sources.
It states that the total voltage or current in any element is the sum of the effects of each source acting alone.
Independent voltage sources are replaced by short circuits and current sources by open circuits while analyzing one source at a time.
The theorem is very useful for simplifying circuit problems in Unit I and finding voltages and currents efficiently.
Important terms to remember: linear circuit, independent source, voltage source, current source, short circuit, open circuit, algebraic sum