The Lorentz reciprocity theorem enables us to establish that the transmitting and receiving patterns of any antenna are identical, provided some hypotheses on this antenna and the surrounding medium are satisfied. But reciprocity does not mean that the antenna behaves the same in the transmitting and the receiving modes. In this paper, array antennas fed by multiple beam forming networks are discussed, highlighting the possibility to have different values of internal losses in the beam forming network depending on the operation mode. In particular, a mathematical condition is derived for the specific case of a multiple beam forming network with lossless transmitting mode and lossy receiving mode, such a behavior being fully consistent with the reciprocity theorem. A theoretical discussion is provided, starting from a simple 2-element array to a general

The Lorentz reciprocity theorem enables us to establish that the transmitting and receiving patterns of any antenna are identical, provided some hypotheses on this antenna, the probe antenna used to evaluate the patterns and the medium in between them are satisfied. This is quite a simple and natural property. Still, its interpretation is not always straightforward. This difficulty is well illustrated by recent discussions on the modeling of receiving antennas and the distinction between its absorbed and scattered powers [

Before starting with a specific simple case to illustrate the difference in behavior between transmitting and receiving array antennas, let us first introduce the hypotheses. An important application of the Lorentz reciprocity theorem is the comparison between the receiving and transmitting patterns of an antenna. Let us consider two distinct antennas, labeled 1 and 2. First, antenna 1 transmits while antenna 2 receives, and second, antenna 2 transmits while antenna 1 receives [

To introduce the property discussed in the introduction, we consider a 2-element linear array with the notations defined in Figure

2-element linear array parameters definition.

This matrix is unitary because the following relation is verified:

This property is a necessary and sufficient condition to have a so-called lossless network, that is, a network that has no internal power dissipation whatever the input power distribution applied to any combination of its ports [

All the power supplied at the antenna port is delivered to the array element ports because

In Figure

Radiation pattern of a 2-element linear array.

Now, we consider the receiving mode assuming an incident plane wave. The signals captured by the array elements depending on the incidence angle

Interestingly, the power received at each array element does not depend on the incidence angle (only the relative phase between the two received voltages varies with the incidence angle) and is equivalent to a gain of 3 dB, which is the maximum gain observed in the transmitting mode, reached in the broadside direction. Using (

One can note that

Let us consider now a Wilkinson power divider. Its scattering matrix is defined as follows:

This component requires internal loads to enable matching of all ports

If we consider the transfer function only, the two beam forming networks produce exactly the same result in the transmitting and in the receiving mode. Interestingly, the gain variation with the considered angular direction in the transmitting mode is compensated in the receiving mode by either reradiated power (case of a “lossless” beam forming network) or dissipated power (case of a “lossy” beam forming network). In the next section, this property is generalized to any multiple beam forming network.

Multiple beam forming networks are circuits that generate several simultaneous beams from a same array antenna. Figure

Multiple beam

Reciprocity at circuit level imposes the matrix

A typical requirement for antenna design is that all beam ports are matched and decoupled; that is,

This particular case is of high interest because it corresponds to a well-known practical implementation: the Butler matrix [

In the transmitting mode, a beam forming network characterized by a unitary transfer matrix is lossless because the excitations produced by feeding any beam port are unit vectors (the power supplied at the input port is fully transferred to the output ports). From an antenna point of view, the transfer function in the transmitting mode is defined by the radiation pattern produced by feeding a given beam port, which naturally varies with the observation angle (location of antenna 2). In the receiving mode, as already discussed, the power captured by the array elements does not depend on the plane wave incidence angle. How does reciprocity hold? This can be understood having a look at all the patterns produced by a unitary transfer matrix. As an illustration, we consider a

Radiation patterns of a

In the general case, the number of beam ports and array element ports are not necessarily equal, resulting in a transfer matrix that is not a square matrix. Consequently, the transfer matrix is no longer unitary and (

The right side of (

Let us assume that

Let us illustrate this with a practical example. We consider the

Layout of a

Radiation patterns of a

Losses as a function of the plane wave incidence angle in a

Radiation patterns produced by the loaded ports of the considered

This discussion could be extended to another case of multiple beam forming networks: those that are lossy in the transmitting and the receiving modes. A typical example is the Blass matrix [

This paper discussed reciprocity and multiple beam forming network properties from an antenna point of view. Differences in behavior were highlighted comparing the transmitting and receiving modes of linear array antennas fed by such networks. Reciprocity is obviously confirmed at antenna level, but this does not imply that the antenna operation is the same in the transmitting and receiving modes. As demonstrated, the beam forming network can be lossless in the transmitting mode and lossy in the receiving mode without infringing the reciprocity theorem. Mathematical condition to observe such a phenomenon was discussed in the general case of a

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank Dr. Piero Angeletti, from the European Space Agency, for interesting discussions on this topic.