The Reciprocity Theorem

THE RECIPROCITY THEOREM

Statement

In any linear network containing bilateral linear resistances and energy sources, the ratio of a voltage V introduced in one mesh to the current I in any second mesh is the same as the ratio obtained if the positions of V and I are interchanged, other voltages sources are being removed.

The reciprocity theorem leads to a definition of the transfer resistance RT where V is the voltage in mesh 1 and I is the current in any second mesh.

R_T12 = V₁ / I₂ = R_T21 = V₂ / I₁

Proof

Let us consider the network which consists of resistors R₁, R₂ and R₃

and the theorem is proved. A network which satisfies equation (3) is said to be bilateral.

Limitations

Networks containing ordinary resistors including capacitors as circuit elements are bilateral. When electron tubes or other semiconductor devices are introduced, the circuits may not be bilateral and the reciprocity theorem may not apply.

EXAMPLE 37: In the network shown in figure, find the current in the 82 resistor. Verify the reciprocity theorem.

Solution:

Current through the 8 Ω resistor = 40 / 8 = 5 A

Current through 10 Ω resistor = 40 / 10 = 4 A

Current through 20 Ω resistor = 40 / 20 = 2 A

On changing the voltage source to the 8 Ω branch,

The current through the short = 40 / 8 = 5 A

Thus, the reciprocity theorem is verified.

I₁₀ = I₂₀ = 0

EXAMPLE 38: In the circuit of figure, find I₃ and verify the reciprocity theorem.

Solution:

The current I₃ is found by the mesh current method.

Now, transferring the battery to the branch with 2.75 Ω resistor, we get the circuit of figure.

We shall again find I₃ by mesh method.

EXAMPLE 39: Verify the reciprocity theorem for the network shown in figure.

Solution :

The current I is found by the mesh current method.

Applying the reciprocity theorem, by inter changing the source and response, we get

By applying mesh method, we find the I₃.

which verifies reciprocity theorem.