# Understanding the Circuit and Antenna Reciprocity Theorem

### Key Takeaways

• In electromagnetic field theory, the reciprocity theorem, also known as the Lorentz reciprocity theorem, is associated with the coupling energy between fields produced by one source on another and vice versa.

• Circuit theory reciprocity is applicable when sources are lumped elements and reactions are voltage or current measurements.

• According to antenna reciprocity, the ratio of transmitted power from the transmitting antenna to the received power of the receiving antenna will not change even when the modes of the antennas are interchanged. The reciprocity theorem is useful when using an antenna in both transmitting and receiving modes

In electrical circuits, the reciprocity theorem is not a generalized theorem—there are limits to its application. Some of the limitations of this theorem are the systems containing dependent sources, non-linear devices, and time-varying elements.

The circuit reciprocity theorem is about the relationship between excitation and response. Simply put, if a system is reciprocal, then the ratio of response to excitation remains unchanged even when the position of the excitation and response are interchanged. The circuit reciprocity theorem is a special case of reciprocity in linear electromagnetic field theory

Reciprocity in antenna communication is desirable, as it offers the opportunity to interchangeably use a single pair of antennas in both receiving and transmitting modes. Circuit and antenna reciprocity are special cases of the reciprocity theorem in electromagnetic fields. In this article, we will identify the reciprocity theorem and analyze its uses in connection to electrical networks and antennas.

## The Circuit and Antenna Reciprocity Formula

In electromagnetic field theory, the reciprocity theorem, also known as the Lorentz reciprocity theorem, is associated with coupling energy between fields produced by one source on another source and vice versa. This theorem is the most basic form of reciprocity in linear electromagnetic systems.

Let’s take a closer look at the Lorentz reciprocity theorem.

Consider two sources, (J1, M1) and (J2, M2), occupying a volume of V[s] in a bounded surface of S. Let the fields associated with J1, M1 be E1, H1 and let J2, M2 be E2, H2. There is coupling energy between the fields produced from one source on another; this creates another field, which is the reaction of the fields on another source.

In this case, the reaction of fields E1, H1  on source J2, M2 is given by the equation <1,2> (below) and the reaction of field E2, H2 on the source J1, M1 is given by the equation <2,1> (below). The reactions in equation (1) and equation (2) are equal.

<1,2>=<2,1> (3)

### Circuit Theory Reciprocity

Circuit theory reciprocity is a special case of the Lorentz reciprocity theorem. Circuit theory reciprocity applies when sources are lumped elements and the reactions are voltage or current measurements. In such a system, Lorentz reciprocity reduces to circuit theory reciprocity.

In a linear circuit, there are two pairs of nodes (1,1’) and (2,2’) and the circuit consists of a voltage source Va , where current Ia is measured as the reaction.

Voltage is applied across node 1,1’ and the current is measured from nodes 2,2’. The position of the source and the measurement can be interchanged—the voltage Va can be at 2,2’ and the current Ia can be measured at 1,1’.

In both cases, the ratio of voltage to current remains constant. Both equations below present the reciprocity theorem in circuit theory: ### Antenna Reciprocity

The reciprocity theorem is fundamental to antenna theory. The following are all true in the case of antenna reciprocity:

• Surface S is of infinite radius and represents the unbounded problem of reciprocity.

• The ratio of transmitted power from the transmitting antenna to the received power by the receiving antenna will not change even when the modes of the antennas are interchanged.

• The whole system is considered isotropic and linear.

Consider two antennas: antenna 1 is in transmitting mode and antenna 2 is in receiving mode. The two antennas are matched by their feed networks and power fed to the transmitting antenna 1 is given by the following equation, where RA1 is the resistance of antenna 1: Power delivered to load or antenna 2 is given by the following equation, where RA2 is the resistance of antenna 2, RL2 is the load resistance, and Y21 is the combined transfer admittance of antenna 1, 2, and free space: The equation can be written as the following, where Pt and Pr are transmitted and receive the power of antenna 1 and antenna 2, respectively: When the antenna modes are interchanged, antenna 1 is receiving and antenna 2 is transmitting. That means the ratio remains unchanged: The combined transfer admittance Y21 is equal to Y12. Therefore, the ratio of received power to transmitted power given in equations (8) and (9) remains the same. This can be summarized as the ratio of received power to transmitted power remains constant when the modes of antennas are interchanged from transmitting to receiving and vice versa.

The circuit and antenna reciprocity theorems are special cases of the reciprocity theorem in linear electromagnetic systems. The reciprocity theorem in antenna theory is useful when using an antenna in both transmitting and receiving modes. This theorem is an integral part of ensuring proper antenna communication.

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