Superposition Theorem: Electrical Circuit Analysis

Superposition theorem state that:
The voltage across (or current through) an element in a linear electric circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.

The above superposition theorem statement means that each independent source has its effect on each circuit passive elements. And the final effect can be found using the algebraic sum of all the above effects.

If an electrical circuit has multiple independent sources and the circuit is a linear circuit, so, the circuit can be analyzed using superposition theorem.

superposition theorem steps:

To use superposition theorem for electrical circuit analysis, follow the steps below.

  1. Replace all current sources with an open circuit and voltage sources with a closed circuit, expect, the source you are considering in next steps.
  2. Find the current in each branch and voltage at each node, whatever technique you like.
  3. Repeat the above two steps for each independent source.
  4. Finally, find the algebraic sum of each branch current and node voltage from the above individual currents and voltages.

Consider the following example for explanation.

Example:

Find the value of voltage V2 across R2 using superposition theorem in the following circuit.

SuperPosition Theorem

Solution:

Short Circuit Sources:

Replace all independent current and voltage sources with short circuit expect we are considering in the next step. And I am going to consider only the voltage source V=6v in the next step.

Solve for single source:

To find the effect of Voltage Source, V= 6v, first of all, we open the circuit source, I, and the circuit will become something like this.

SuperPosition Theorem step1 Solved example

Now applying the applying Voltage Divider Rule to the above diagram will give us:
$v_{12}= \frac{V \times R_{2}}{R_{1}+R_{2}}$
$v_{12}= \frac{6 \times 4}{8+4}$
$v_{12}= \frac{24}{12}
v_{12}= 2$

Solve for another source:

To find the effect of Current Source, I= 3A, I short circuit the voltage source V=6v and the circuit will become something like this:

SuperPosition Theorem solved example

Now, apply Current Divider Rule to the above circuit.
$i_{2}= \frac{I \times R_{2}||R_{1}}{R_{2}}$
$i_{2}= \frac{3 \times 4||8}{4}$
$i_{2}= \frac{3 \times 2.66}{4}$
$i_{2}=2 A$
Now finding the voltage across V$_{2}$, using Ohm’s law: 
$v_{22}=i_{2}R_{2}$
$v_{22}=2 \times 4$
$v_{22}=8$

Find Algebraic sum:

Finally, find the algebraic sum of the voltage from individual effects obtained in the above steps and we are done. \(\)

$v_{2}=v_{12}+v_{22}$
$v_{2}=2+8$
$v_{2}=10 v$

To wrap things up, superposition theorem is the circuit analysis tool considering the individual effect of sources. And the theorem can only be applied to the linear circuits. Where linear circuit is those which poses the property of homogeneity and additivity.

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  1. onesies June 27, 2018

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