SUPERPOSITION THEOREM

In general, the superposition principle states that the net effect produced due to two or more stimuli is the sum of the effects or responses which would have been caused by each stimulus individually. That means, if input X₁ produces Y and X₂ produces Z. Then X₁+X₂ = Y+Z.

Or, if a body is applied with three simultaneous forces as shown in the figure below:

Then the net force F acting on the body is given by $\overrightarrow{F}=\overrightarrow{{{F}_{1}}}+\overrightarrow{{{F}_{2}}}+\overrightarrow{{{F}_{3}}}$. And if F₁=F₂=F₃=F, then F₁ and F₂ cancel out and the net movement of body is in the direction of F₃. This was a short explanation of what superposition principle is. But can this principle be applied to electrical circuits? The answer is Yes!

If a circuit has two or more independent sources, then the value of a specific variable (voltage or current) can be derived using superposition theorem.

STATEMENT:

The superposition theorem states the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages or currents through that element due to individual sources acting alone.

One should keep in mind that the superposition principle is based upon the property of linearity.

THINGS TO KEEP IN MIND WHILE APPLYING SUPERPOSITION PRINCIPLE:

1. Consider only one independent source at a time and turn off all other independent sources i.e. Replace every voltage source by short circuit (0V) and every current source by an open circuit (0A).
2. Leave the dependent sources as it is, in the circuit because they are controlled by circuit variables. 

APPLYING THE SUPERPOSITION PRINCIPLE: 

1. Turn off all the independent sources, except one and find the response (current or voltage) due to that source using nodal or mesh analysis.
2. Repeat step 1 for all independent sources individually.
3. Find the total contribution by adding algebraically the responses due to the independent sources.

The only disadvantage in using the superposition theorem is that it involves more work. For example, if we have 4 independent sources in a circuit, then we have to analyze the circuit four times, which is surely not time saving.

Nevertheless, superposition is a good tool for reducing complex circuits to simpler ones. If you haven’t understood this theorem properly, then this illustrative example might help you. Consider the circuit given below: 

• When the current source is turned off (open circuited) –

  

Voltage across $4\Omega $ resistor, ${{V}_{1}}=\left( \frac{4}{4+8} \right)\times 6=2V$ (Voltage divider rule)

• When the voltage source is turned off (short circuited) – 

  

${{i}_{3}}=\left( \frac{8}{4+8} \right)\times 3=2A$ (current division rule)

 Voltage across $4\Omega $ resistor ${{V}_{2}}=4{{i}_{3}}=8V$ .

Thus the net voltage across $4\Omega $ resistor due to the two independent sources is given by V.

                                         \[V={{V}_{1}}+{{V}_{2}}=2V+8V=10V\]

Hope this example better explains the steps mentioned above to apply this theorem.

Keep in mind that superposition is based on linearity. As such, it cannot be used to find the power absorbed by an element as power absorbed by resistor depends on the square of voltage or current.