Superposition Theorem Easy Steps for Electric Circuit

superposition theorem

Superposition theorem will help us greatly when analyzing a circuit with multiple sources.

If a circuit has two or more independent sources, one way to determine the value of a specific variable (voltage or current) is to use nodal or mesh analysis like has been discussed before.

Another way is to determine the contribution of each independent source to the variable and then add them up. The latter approach is known as superposition.

Make sure to read what is electric circuit first.

These circuit analysis theorems are classified as:

  1. Superposition theorem
  2. Source transformation
  3. Thevenin theorem
  4. Norton theorem
  5. Maximum power transfer

Superposition Theorem Theory

The idea of superposition rests on the linearity property.

The superposition principle states that the voltage across (or currents through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.

The principle of superposition helps us to analyze a linear circuit with more than one independent source by calculating the contribution of each independent source separately.

However, to apply the superposition principle, we must keep two things in mind:

  1. We consider one independent source at a time while all other independent sources are turned off. This implies that we replace every voltage source by 0 V (or a short circuit), and every current source by 0 A (or an open circuit). This way we obtain a simpler and more manageable circuit.

  2. Dependent sources are left intact because they are controlled by circuit variables.

With these in mind, we apply the superposition principle in three steps:

Steps to Apply Superposition Principle:

  1. Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using the techniques in previous explanation.

  2. Repeat step 1 for each of the other independent sources.

  3. Find the total contribution by adding algebraically all the contributions due to the independent sources. 

Analyzing a circuit using superposition has one major disadvantage: It may very likely involve more work.

If the circuit has three independent sources, we may have to analyze three simpler circuits each providing the contribution due to the respective individual source.

However, superposition does help reduce a complex circuit to simpler circuits through replacement of voltage sources by short circuits and of current sources by open circuits.

Superposition Theorem Examples

For better understanding let us review examples below:

1. Use the superposition theorem to find v in the circuit of Figure.(1)

superposition theorem electric circuit analysis
Figure 1

Solution :

Since there are two sources, let

superposition theorem electric circuit analysis

where v1 and v2 are the contributions due to the 6 V voltage source and the 3 A current source, respectively. To obtain v1, we set the current source to zero, as shown in Fig.2(a).

superposition theorem electric circuit analysis
Figure 2

Applying KVL to the loop in Figure.(2a) gives

superposition theorem electric circuit analysis

Thus,

superposition theorem electric circuit analysis

We may also use voltage division to get v1 by writing

superposition theorem electric circuit analysis

To get v2, we set the voltage source to zero, as in Figure.(2b). Using current division,

superposition theorem electric circuit analysis
Hence,
superposition theorem electric circuit analysis
And we find
superposition theorem electric circuit analysis
 
2. Find io in the circuit of Figure.(3) using superposition.
superposition theorem electric circuit analysis
Figure 3

Solution :

The circuit in Figure.(3) involves a dependent source, which intact. We let

superposition theorem electric circuit analysis
(1)

where i’o and ino are due to the 4 A current source and 20 V voltage source respectively. To obtain i’o, we turn off 20 V source so that we have the circuit Figure.(4a). We apply mesh analysis in order to obtain i’o. For loop 1,

superposition theorem electric circuit analysis
(2)

For loop 2,

superposition theorem electric circuit analysis
(3)
superposition theorem electric circuit analysis
Figure 4

For loop 3,

superposition theorem electric circuit analysis
(4)

But at node 0,

superposition theorem electric circuit analysis
(5)

Substituting (2) and (5) into (3) and (4) gives two simultaneous equations

superposition theorem electric circuit analysis
(6)
superposition theorem electric circuit analysis
(7)

which can be solved to get

superposition theorem electric circuit analysis
(8)

To obtain ino, we turn off the 4 A current source so that the circuit becomes that shown in Figure.(3b). For loop 4, KVL gives

superposition theorem electric circuit analysis
(9)

and for loop 5,

superposition theorem electric circuit analysis
(10)

But i5 = -ino. Substituting this in (9) and (10) gives

superposition theorem electric circuit analysis
(11)
superposition theorem electric circuit analysis
(12)

which we solve to get

superposition theorem electric circuit analysis
(13)

Now substituting (8) and (13) to (1) gives

superposition theorem electric circuit analysis