Reciprocity theorem is a fundamental principle in electrical engineering that relates to the behaviour of linear, passive, and bilateral circuits. It states that the ratio of the voltage across a given element to the current through another element in a circuit remains unchanged when the positions of these elements are interchanged. In simpler terms, the reciprocity theorem states that if we have two points in a circuit, say A and B, and we apply a voltage at point A and measure the current at point B, the result will be the same as if we apply the same voltage at point B and measure the current at point A. The reciprocity theorem is a fundamental concept in electrical engineering that establishes a relationship between the electromagnetic fields produced by two sources.

Reciprocity Theorem

The reciprocity theorem says that in any linear and bilateral network or circuit with one independent source, the response to excitation stays the same even if you swap the source from the input to the output terminals. Since we deal with both voltage and current, this can be seen in four different ways, depending on whether you use voltage or current for excitation and response. This theorem only works for single-source networks and circuits that include dependent sources. It's valid for almost all passive networks and reflects a broader principle of reciprocity in electromagnetism.

This detailed blog will apprise readers of the reciprocity theorem and its applications. This topic is important for upcoming examinations pertaining to Electrical engineering, including SSC JE, GATE Electrical Exam, etc.

What is the Reciprocity Theorem? 

The reciprocity theorem is a fundamental concept in electrical engineering that relates the electromagnetic fields generated by two sources. It states that the ratio of the voltage across a given element to the current through another element in a circuit remains unchanged when the positions of these elements are interchanged.

In terms of electromagnetism it states that the ratio of the electric field produced by one source at a point in space to the current generated by another source at the same point is equal to the ratio of the electric field produced by the second source at the original location of the first source to the current generated by the first source at its own location. This theorem is based on the principle of electromagnetic wave superposition, which states that the total electric field at a given point is the sum of the electric fields produced by all sources in the vicinity. In simpler terms, it means that if we have two sources, the electric field produced by one source at a particular location is the same as the electric field produced by the other source at the same location, but with the sources swapped.

The reciprocity theorem has various practical applications in the field of electrical engineering. Some notable applications include:

• Antenna Design: The reciprocity theorem is widely used in antenna design and analysis. It allows engineers to determine the radiation pattern and characteristics of an antenna by measuring the fields at a specific location with a known source. This principle is essential for optimising antenna performance, evaluating antenna efficiency, and designing antenna arrays.
• Network Analysis: The reciprocity theorem plays an important role in network analysis and circuit design. It enables engineers to simplify complex circuits by exploiting symmetry and reciprocal relationships. By applying the reciprocity theorem, engineers can determine the characteristics of a network, such as impedance, admittance, and scattering parameters, without the need for extensive measurements or simulations.
• Electromagnetic Compatibility: The reciprocity theorem is utilised in electromagnetic compatibility (EMC) testing and analysis. It allows engineers to predict the electromagnetic interference (EMI) and electromagnetic susceptibility (EMS) of devices and systems. By evaluating the fields produced by one source on another device, engineers can assess the potential for interference and design appropriate mitigation measures.
• Wireless Communication Systems: The reciprocity theorem is employed in wireless communication systems, particularly in the field of MIMO (Multiple Input Multiple Output) technology. MIMO systems rely on multiple antennas to improve data rates and reliability. The reciprocity theorem helps engineers calibrate and optimise the performance of MIMO systems by relating the transmit and receive antenna characteristics.
• Acoustic and Seismic Imaging: The reciprocity theorem is not limited to electromagnetic fields but can also be applied to acoustic and seismic waves. It enables engineers to analyse and predict the propagation of sound and seismic waves in various media. This has applications in fields such as medical imaging, geophysical exploration, and non-destructive testing.

Reciprocity Theorem to an Electric Circuit

The reciprocity theorem can be applied to electric circuits, allowing engineers to analyse the behaviour and characteristics of the circuit. Here's how the reciprocity theorem applies to an electric circuit:

• According to the reciprocity theorem, if a current source is placed at one point in the circuit and the resulting voltage is measured at another point, the same voltage will be obtained when the roles of the current source and measurement point are reversed. In other words, the response of the circuit to a current source is the same as the response to a voltage source at the same locations, but with the sources swapped.
• This principle is particularly useful in circuit analysis and design. By applying the reciprocity theorem, engineers can simplify calculations and obtain valuable insights into the behaviour of the circuit. It allows for the determination of parameters such as impedance, admittance, and transfer functions by exploiting the symmetry and reciprocal nature of the circuit.
• For example, in a network analysis, the reciprocity theorem can be used to determine the transfer function between two points in a circuit. Instead of calculating the response to a current source at one point and measuring the resulting voltage at another point, engineers can reverse the roles and calculate the response to a voltage source at the original measurement point. This simplifies the analysis and helps in understanding the circuit's behaviour.
• Overall, the reciprocity theorem is a powerful tool in electric circuit analysis, enabling engineers to relate the effects of current and voltage sources in a circuit. It aids in simplifying calculations, understanding circuit behaviour, and optimising circuit performance.
• In the given circuit, VS represents the input voltage and current, while the response or output current is denoted by I, which flows through the resistor R3.

In this modified circuit, we have swapped the positions of the input voltage and output current. Now, VS represents the input voltage and current, and the response or output current is denoted by I, which flows through the resistor R1.

In the first circuit, the current flowing through resistor R3 is equal to the current flowing through resistor R1 in the second circuit. This is because the magnitude and polarities of the voltage, VS, are the same in both circuits, and the direction of the output current is also the same in both circuits.

Reciprocity Theorem to a 2-Port Network

The reciprocity theorem can also be applied to a 2-port network, which is a common component in electrical systems. Here's how the reciprocity theorem applies to a 2-port network:

• In a 2-port network, there are two input terminals and two output terminals. According to the reciprocity theorem, if a signal is applied to one of the input terminals, the response measured at one of the output terminals will be the same as if the signal was applied to that output terminal and measured at the input terminal.
• This means that the transfer characteristics of the 2-port network remain the same regardless of the direction of the signal flow. The gain, impedance, and other parameters of the network will be the same for signals propagating from input to output and from output to input, but with the input and output terminals interchanged.
• The reciprocity theorem is particularly useful in analysing and designing communication systems and components, such as amplifiers, filters, and transmission lines. It allows engineers to simplify calculations and predict the behaviour of the network without performing separate analyses for each direction of signal flow.
• For example, in the case of an amplifier, the reciprocity theorem implies that the gain obtained when the input signal is applied to one terminal and the output is measured at the other terminal will be the same as if the roles of input and output were reversed.
• By applying the reciprocity theorem, engineers can gain insights into the behaviour of the 2-port network, determine its characteristics, and optimise its performance. It provides a valuable tool for analysing and designing communication systems, ensuring efficient signal transmission and reception.

Limitation of Reciprocity Theorem

While the reciprocity theorem is a valuable concept in electrical engineering, it does have certain limitations. Here are a few notable limitations:

• Linearity: The reciprocity theorem assumes linearity in the system under consideration. In real-world scenarios, some components or systems may exhibit nonlinear behaviour, which can limit the applicability of the reciprocity theorem. Non-linear elements can introduce distortions or nonlinear interactions that violate the reciprocity principle.
• Time-Invariance: The reciprocity theorem assumes that the system under study is time-invariant, meaning its characteristics and behaviour do not change over time. However, in practical applications, systems may exhibit time-varying behaviour due to factors like temperature variations, component ageing, or dynamic changes in the operating conditions. In such cases, the reciprocity theorem may not hold true.
• Losses and Energy Dissipation: The reciprocity theorem assumes an idealised system without losses or energy dissipation. In reality, every component and system has some level of losses due to resistive elements, radiation, or other dissipative processes. These losses can affect the reciprocity and introduce deviations from the ideal reciprocity theorem.
• Magnetic Materials and Frequency Dependency: The reciprocity theorem assumes that the electromagnetic properties of the medium in which the system operates remain constant. However, when dealing with magnetic materials or systems operating at different frequencies, the reciprocity principle may not hold due to the frequency-dependent characteristics and non-uniform magnetic properties of materials.
• Mutual Coupling and Isolation: In systems where there is significant mutual coupling or isolation between elements, the reciprocity theorem may not fully capture the complex interactions between the sources and measurement points. The presence of coupling or isolation mechanisms can introduce additional effects that deviate from the ideal reciprocity assumption.

Uses of Reciprocity Theorem

The reciprocity theorem has several practical uses in electrical engineering. Here are some notable applications:

• Antenna Design: The reciprocity theorem is extensively used in antenna design and analysis. It allows engineers to determine the radiation pattern and characteristics of an antenna by measuring the fields at a specific location with a known source. This principle aids in optimising antenna performance, evaluating antenna efficiency, and designing antenna arrays.
• Network Analysis: The reciprocity theorem is applied in network analysis and circuit design. It enables engineers to simplify complex circuits by exploiting symmetry and reciprocal relationships. By utilising the reciprocity theorem, engineers can determine the characteristics of a network, such as impedance, admittance, and scattering parameters, without the need for extensive measurements or simulations.
• Microwave Engineering: In the field of microwave engineering, the reciprocity theorem plays a significant role. It allows engineers to analyse and predict the behaviour of microwave components and systems. By leveraging the reciprocity principle, engineers can design and optimise devices such as filters, couplers, and waveguides.
• Electromagnetic Compatibility (EMC): The reciprocity theorem is employed in electromagnetic compatibility testing and analysis. It aids in predicting the electromagnetic interference (EMI) and electromagnetic susceptibility (EMS) of devices and systems. By evaluating the fields produced by one source on another device, engineers can assess the potential for interference and design appropriate mitigation measures.
• Signal Integrity Analysis: The reciprocity theorem is useful in signal integrity analysis, particularly in high-speed digital systems. It helps in predicting the coupling and crosstalk between signal traces and transmission lines. Engineers can use the reciprocity principle to analyse the effects of electromagnetic interference and optimise the performance of the system.
• Acoustic and Seismic Imaging: The reciprocity theorem is not limited to electromagnetic fields but can also be applied to acoustic and seismic waves. It enables engineers to analyse and predict the propagation of sound and seismic waves in various media. This has applications in fields such as medical imaging, geophysical exploration, and non-destructive testing.

Reciprocity Theorem Solved Example

Consider two antennas, Antenna A and Antenna B, separated by a distance of \(d\) meters. When Antenna A transmits a power of \(P_1\) watts, the power received by Antenna B is given by the Reciprocity Theorem as the power received by Antenna A when Antenna B transmits the same power. The formula for power received \(P_{\text{received}}\) is given by:

\[P_{\text{received}} = \frac{P_1 \cdot G_{\text{A}} \cdot G_{\text{B}} \cdot \lambda^2}{(4 \pi)^2 \cdot d^2}\]

where \(G_{\text{A}}\) and \(G_{\text{B}}\) are the gains of Antenna A and Antenna B, respectively, and \(\lambda\) is the wavelength.

Given values:

\(P_1 = 100\) W

\(G_{\text{A}} = 4\) (in decibels)

\(G_{\text{B}} = 6\) (in decibels)

\(d = 10\) m

\(\lambda = 2\) m

To find \(P_{\text{received}}\):

\[P_{\text{received}} = \frac{P_1 \cdot G_{\text{A}} \cdot G_{\text{B}} \cdot \lambda^2}{(4 \pi)^2 \cdot d^2} = \frac{100 \cdot 10^{\left(\frac{4}{10}\right)} \cdot 10^{\left(\frac{6}{10}\right)} \cdot 2^2}{(4 \pi)^2 \cdot 10^2} = \frac{100 \cdot 2 \cdot 4}{(4 \pi)^2} \approx 0.509\) W

Hence, \(P_{\text{received}} \approx 0.509\) W.

Given values:

\[P_1 = 100 \, \text{W}\]

\[G_{\text{A}} = 4 \, \text{(in decibels)}\]

\[G_{\text{B}} = 6 \, \text{(in decibels)}\]

\[d = 10 \, \text{m}\]

\[\lambda = 2 \, \text{m}\]

To find:  \(P_{\text{received}}\):

\[P_{\text{received}} = \frac{P_1 \cdot G_{\text{A}} \cdot G_{\text{B}} \cdot \lambda^2}{(4 \pi)^2 \cdot d^2} = \frac{100 \cdot 10^{\left(\frac{4}{10}\right)} \cdot 10^{\left(\frac{6}{10}\right)} \cdot 2^2}{(4 \pi)^2 \cdot 10^2} = \frac{100 \cdot 2 \cdot 4}{(4 \pi)^2} \approx 0.509 \, \text{W}\]

Hence, \(P_{\text{received}} \approx 0.509 \, \text{W}\).

Within the purview of this discussion, we covered the exam-oriented details about Reciprocity theorem and its applications in Electrical Engineering. If you are preparing for State and Central level Electrical examinations, get enrolled in AE/JE Electrical Coaching on the Testbook App.