Reciprocity Theorem - Statement, Applications & Examples

Q: What is a Reciprocal network?

I f any electrical network satisfies the Reciprocity theorem, it is known as a Reciprocal network. All the electrical networks which contain passive elements are always Reciprocal networks because they satisfy the Reciprocity theorem.

Reciprocity Theorem Statement

The reciprocity theorem states that in any linear and bilateral network or circuit having only one independent source, the ratio of response to excitation is constant even though the source is interchanged from the input terminals to output terminals.

Since we have two basic electrical quantities like voltage & current, we will get 4 possible ways regarding excitation and response of the electrical network in terms of voltage & current.

The reciprocity theorem is applicable only to single-source networks.
The reciprocity theorem is applicable only to the circuits containing dependent sources.

Reciprocity Theorem Application

The reciprocity theorem is applied for network falling in the following 2 cases.

Reciprocity Theorem to an electric circuit
Reciprocity Theorem to a 2-port network

Reciprocity Theorem to an Electric Circuit

In the following circuit, V_S is the input voltage and the current, and I flowing through the resistor R₃ is the response or output current.

In the following circuit, we interchanged the position of input voltage and output current. In this circuit, V_S is the input voltage and the current, and I flowing through the resistor R₁ is the response or output current.

In the first circuit, whatever the amount of current that flows through resistor R3, the same amount of current that will flow through resistor R₁ in the second circuit because the magnitude and polarities of voltage, V_S are the same in both circuits & the output current direction is same in both circuits.

Reciprocity Theorem to a 2-Port Network

As the name implies, a 2-port network consists of 2 ports, namely port 1 and port 2. In the following 2 port network, the voltage source, V_S is connected to port 1. This is the input voltage of the 2-port network. The current I₂ flows through port 2, the response or output current of the 2-port network.

In the following 2-port network, the voltage source, V_S' is connected to port 2. This is the input voltage of the 2-port network. The current, I₁' flows through port 1, the response or output current of the 2-port network.

According to the Reciprocity theorem,

I₁'V_S'=I₂V_S

If V_S'=V_S, then we will get I₁'=I₂.

Reciprocity Theorem Examples

Q1. Find the current, I of the circuit shown in fig(b) by using the Reciprocity theorem.

Ans: 1.5 A

Q2. Find the current, I of the 2-port network shown in fig(b) by using the Reciprocity theorem.

Ans: 0.2 A

Important Topics for Gate Exam
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Mohr's Circle	Surface Tension
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FAQs on Reciprocity Theorem

What is Reciprocity Theorem?

According to the reciprocity theorem, for linear and bilateral networks or circuits with only one independent source, the response to the excitation ratio is constant even if the source is swapped from the input port to the output port.

Whether we can apply the Reciprocity theorem directly to the circuits which have multiple independent sources?

No, we can apply the Reciprocity theorem directly to the circuits which contain only one independent source. So, if the circuits contain multiple independent sources, then first we can apply the Superposition theorem & then Reciprocity Theorem.

Whether we can apply the Reciprocity theorem to the AC circuit?

Yes. We can apply the Reciprocity theorem to the AC circuit just like the way we applied the Reciprocity theorem to the DC circuit. If the AC circuit consists of linear and bilateral elements and only one independent source, then we can apply the Reciprocity theorem.

What are the limitations of the Reciprocity theorem?

We can’t apply the Reciprocity theorem if the network is having non-linear and/or unilateral elements. Similarly, we can’t apply the Reciprocity theorem if the network has a dependent source because the dependent source makes the network active.

Whether Antennas will satisfy the Reciprocity theorem?

Yes. Antennas will satisfy the Reciprocity theorem. Hence, we can use the same antenna as either a transmitting antenna or receiving antenna based on the requirement. For effective communication, there should be impedance matching between the transmitter and receiver.

What is a Reciprocal network?

If any electrical network satisfies the Reciprocity theorem, it is known as a Reciprocal network. All the electrical networks which contain passive elements are always Reciprocal networks because they satisfy the Reciprocity theorem.

What is Reciprocity Theorem?