What is Reciprocity Theorem & Its Statement

In electromagnetics, there are many important network theorems available, in that the reciprocity theorem is one of the most important ones. By using this theorem, we can build up physical intuition to ascertain if a specific design otherwise experiment is correct or incorrect. This theorem also states that what is achievable or unfeasible within the design of numerous systems. However using mathematics in electromagnetics, this relationship is expressed exactly and briefly. This article discusses an overview of the reciprocity theorem.

What is Reciprocity Theorem?

This theorem states that, in any circuit or network, the flow of current is because of a single voltage source within any specific branch of circuit is equivalent to the current value within the original branch wherever the voltage source was arranged once the source is moved to that specific branch of the circuit. This theorem is used in a reciprocal network that is linear & bilateral circuit that has simply one independent source and this theorem can also be used for both AC & DC circuits.

In a simple way, this theorem can be defined as; once the voltage and current source in any circuit is interchange the amount otherwise magnitude of current then the voltage supplying within the circuit remains the same. This theorem is applicable for solving several AC & DC networks which are used in electromagnetism electronics. These circuits do not have any time-varying element.

Statement

The statement of reciprocity theorem can be explained through the following circuit diagram. In the following circuit, the positions of two sources like voltage & the current may be exchanged without changing current. But, the voltage source’s polarity must be the same through the way of the branch current in every location.

The resistors in the circuit like R1, R2 & R3 are connected through a voltage & a current source. It is very clear from the above circuit that these sources are exchanged for solving the circuit using Reciprocity Theorem.

The drawback of this theorem is that it is used in single-source networks only but not applicable for the multi-source network. So, to apply this theorem to the network, the network must be linear& include resistors, capacitors, inductors & coupled circuits. That circuit should not include elements that change with time

Steps to Solve a Reciprocity Theorem

• At first, choose the branches in the circuit among which reciprocity has to be created.
• The flow of current within the branch can be obtained through any conventional network analysis technique.
• The voltage source can be interchanged between the branches which are selected.
• The flow of current within the branch wherever the voltage source existed before is measured.
• At the present, it is observed that the current attained within the earlier connection that is in the second step & the flow of current which is measured once the source is exchanged, that is in step 4 are equal to each other.

In this theorem, when the ammeter and voltage source is used then that must ideal. So, the internal resistance of voltage sources should be zero. The reciprocal theorem applied circuit may be simple otherwise complex. However, each complex reciprocal network can be simplified into an easy circuit. In a linear passive network, based on the reciprocity theorem, the voltage supply, as well as o/p current, is moveable equally. The relation between voltage and current is known as the transfer resistance.

Reciprocity Theorem Circuit Theory

The statement of reciprocity theorem can be explained through the following circuit. First, we need to check whether the circuit is a bilateral network or not so that we can apply this theorem. After that, check the validity of the theorem within the branch of a network like x-y & a-b. So for that first, we have to find the current within a-b branch like the following.

Equivalent resistance among x-y branch

R = [(2+1)||3] + 2

So this equation will be

= (9/6) +2 = ((12+9)/6) = 21/6 = 3.5 ohms

So, current like ‘I1’

10/3.5 A => 2.86A

Similarly, I2 = 2.86 X (3/3+3) =1.43A

I3 = I1-I2 = 2.86- 1.43= 1.43A

In the above circuit, change the voltage source position & place it into a-b branch, then the circuit will be like the following. Calculate the equivalent resistance across a-b terminals first, and then the resistance can be calculated like the following.

Equivalent resistance across terminals x-y

R = (2||3/5) + 1+2

= (6/5) + 3 = 21/5= 4.2 ohms

I2 = 10/4.2 => 2.34A

I1 = I2 X (3/3+2)

= 2.384 x (3/5) = 1.43A

Therefore we notice that in the first circuit, the sources within branch x-y & the a-b branch current can be 1.43A. Similarly, once the source in the a-b branch, the current in an x-y branch will become 1.43A. So, this proofs the reciprocity theorem.

Reciprocity Theorem in Antenna

The most useful property in an antenna is Reciprocity that states that the properties of transmitting and of an antenna are equal. Therefore antennas don’t include different transmit & receive radiation patterns. If we know the radiation pattern within the transmit mode that we can also recognize the pattern within the receive mode.

The main function of an antenna is for transmitting and receiving. Once the antenna’s operating mode is changed then properties can also be changed. The antenna’s properties being unchanged is called the reciprocity property. Under reciprocity, the properties of an antenna include the following.

• Directional patterns Equality
• Directivities Equality
• Effective lengths Equality
• Antenna impedances Equality

Lorentz Reciprocity Theorem

Reciprocity theorem is a significant notion within antennas as it produced implications once we turn over the position of transmitting &receiving antennas. The definition of the Lorentz Reciprocity Theorem starts by considering a volume that includes two sets of sources like J1 & J2, where each source generates fields like E1, H1 & E2, H2.

Experiment

The steps involved in the reciprocity theorem are explained below.

Aim:

The main objective of this experiment is the Verification of the Reciprocity Theorem

Required Apparatus

The required apparatus of this theorem includes a training system for the DC circuit, connecting wires, a DC power supply, and a digital AVO meter.

Theory

In any bilateral linear n/w that includes one otherwise more generator then the percentage of a voltage initiated in on mesh network to the current within any next mesh network is similar to the ratio attained if the voltage position & current are exchanged other e.m.f being detached.

Procedure

• With the help of a DC circuit trainer, the circuit can be connected and take voltage = 5V, resistors R1, R2 & R3 values are 10kΩ, 100Ω & 1kΩ
• Calculate the voltage as well as current for resistors, after that record the values within the observation table.
• This table contains voltage and current values for these resistors
• Detach the power supply of DC to measure the equal resistance by using the AVO meter simply.
• Modify the location of the voltage supply as shown in the circuit.
• Calculate the current & voltage values for these resistors and note them down in the observation table
• Again, detach the DC power supply to gauge the equal resistance with the help of the AVO meter simply.

Result

Evaluate the results among theoretical and practical between the theoretical and practical results. Find the values of current and resistor in two cases like when the battery and ammeter are at the same place and when the battery and ammeter are interchanged. So that the values of current and resistor can be measured.

Thus, this is all about an overview of the reciprocity theorem. The application of the reciprocity theorem in the antenna is, there are different forms of the reciprocity theorems which are utilized in several applications of electromagnetic like antenna systems and electrical networks analysis. For instance, the reciprocity theorem implies that antennas function uniformly like transmitters otherwise receivers & particularly those radiations of an antenna as well as receiving patterns are the same. Here is a question for you, please mention the different types of theorems available in electromagnetics?