Reciprocity Theorem states that, the value of current due to a single source in any particular branch of circuit is equal to the value of current in the original branch where the source was placed when the source is shifted to that particular branch of circuit. This theorem is only applicable for a reciprocal network i.e. linear and bilateral circuit having only one independent source. This theorem can, however, be used for both AC and DC circuit.
Explanation of Reciprocity Theorem:
The statement of reciprocity theorem seems a bit confusing but hold on, I will explain this theorem here in detail to make it crystal clear to you. Let us consider the circuit below.
Let us first check if we can apply the theorem of not. Since, the circuit is a linear bilateral network and hence reciprocal. Therefore, we can apply reciprocity theorem in this circuit.
Let us assume that we want to check the validity of reciprocity theorem in branch x-y and a-b. For this, first we find the current though the branch a-b as below.
Equivalent Resistance between x-y
= (3×3)/6 + 2
= 3.5 Ω
Therefore, current I1
= 10/3.5 A
= 2.86 A
Now the current through branch a-b may be calculated using current division rule as shown below.
I2 = (3x I1)/(3+3)
Let us now change the position of voltage source and put it into branch a-b. This shown in figure below.
Our aim is to find the value of current in branch x-y i.e. the branch where the source was originally placed when it is shifted to the particular branch a-b where we calculated the current.
Let us first find the equivalent resistance across terminals a-b first. This resistance is calculated as below.
Equivalent resistance across terminals x-y
= 4.2 Ω
Current I2 = 10/4.2
= 2.38 A
Using current division rule, the current in branch x-y may be find as shown below.
I1 = (2.38×3)/5
= 1.43 A
Carefully notice the value of currents I1 and I2. Aren’t they same? They are same.
Thus, Reciprocity Theorem can be understood in another way. In electrical, cause is voltage and effect is current. This simply means, voltage causes current to flow. Let us call voltage as excitation and current as response. It is clear from the above workout that, interchanging the position of excitation and response does not alter the value of response.
Reciprocity theorem may be stated as: the ratio of excitation and response remains constant even if we interchange the position of excitation and response in a reciprocal network. This definition matches with the literal meaning of word “reciprocity”.
Steps for using Reciprocity Theorem:
Step-1: As a very first step, check if the given circuit is a linear bilateral network (reciprocal circuit) having only one source or not. If the circuit is having more than one source, this theorem can not be applied. Hence, you are all done with your job. If, the circuit is reciprocal, then follow next steps.
Step-2: Select the branch between which reciprocity is to be checked.
Step-3: Find the value of current in branch using conventional network analysis.
Step-4: The voltage source is interchanged between the branches concerned.
Step-5: Calculate the current in the branch where the voltage source was existing earlier.
Step-6: Check that current calculated in step-3 & step-5 are same. If they are same, this means reciprocity is validated between the concerned branches.
Hope you got the concept. In case of doubt, kindly write in comment box. Your voice will be heard and responded.