International Journal of Antennas and Propagation

Volume 2015, Article ID 946289, 9 pages

http://dx.doi.org/10.1155/2015/946289

## Discussion on Reciprocity, Unitary Matrix, and Lossless Multiple Beam Forming Networks

Moltek Consultants Ltd. for the European Space Agency (ESA), Antenna and Sub-Millimetre Wave Section, Keplerlaan 1, P.O. Box 299, 2200AG Noordwijk, Netherlands

Received 15 December 2014; Revised 19 March 2015; Accepted 20 March 2015

Academic Editor: Yuan Yao

Copyright © 2015 Nelson Jorge G. Fonseca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 multiple beam forming network. A more practical example is then given, discussing a specific Nolen matrix design and comparing theoretical aspects with simulation results.

#### 1. Introduction

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 [1–5]. Recent work presented by the author has also raised discussions on this matter in connection with multiple beam forming network properties [6, 7]. Indeed, it was found that some beam forming networks exhibit an interesting property: they are theoretically lossless in the transmitting mode but have losses in the receiving mode which depend on the incidence angle of the incoming plane wave. Without any additional explanation, one might think that this property violates the reciprocity theorem. Interestingly enough, this property can be related to the one described in [8] in the case of nonsymmetric propagation scenarios: reciprocity holds between two antennas separated by a nonsymmetric propagation medium even if the path loss variations as a function of range along a given propagation path are totally different for the forward and backward scenarios. Clearly, reciprocity does not mean that the field distribution has to be the same in the transmitting and the receiving modes but only mean that the ratio of transmitted to received field strengths between the two considered antennas, when treated as a two-port network, is the same when the roles of transmitter and receiver are interchanged, provided the considered antennas and medium in which they are placed are linear. Having this in mind, a beam forming network with insertion loss values different in the transmitting and receiving modes is possible as long as this loss difference is compensated for somewhere else in the link budget between the two considered antenna ports. In this paper, we discuss the theoretical bases required to understand this peculiar property. We introduce the problem with a simple example: a two-element linear array fed by two types of power dividers. Then, we address the general case of multiple beam forming networks, setting the link between the scattering matrix properties and a theoretically lossless operation mode. We conclude this paper discussing a practical example previously introduced in the literature and related to this theoretical aspect.

#### 2. Hypotheses and Problem Description

##### 2.1. Calculation Hypotheses

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 [9]. Treating these two-antenna system as a two-port network, each port representing one of the two antennas, it can be shown that the mutual impedances between these two ports are identical . As we focus in this paper on beam forming network properties, we prefer to use this result translated into scattering parameters; that is, [10]. The main consequence of this relation is that the transmitting and receiving patterns of an antenna are identical, provided the two considered antennas are in a linear and isotropic medium [9]. To simplify the calculations, we assume that the distance between the two antennas is much longer than the wavelength. Consequently, the behavior of the antenna investigated in the transmitting mode can be derived from its far-field radiation pattern, while in the receiving mode an incident plane wave is assumed [11, 12]. Furthermore, we assume all the elementary antennas to be isotropic. Consequently, the effective aperture in receive is assumed to be the same for all the elementary antennas and is independent on the incident angle, resulting in a common multiplicative coefficient neglected in the following results. Similarly, the elementary antenna gain is assumed to be equal to one in any angular direction in the transmitting mode. This is consistent with the fact that reciprocity lies on relative values in linear medium, and thus directivity normalization at array element level does not affect our conclusions. The elementary antennas are considered perfectly matched and lossless: in the transmitting mode, all the power provided in guided-wave at the antenna port is radiated as a free-space wave and reciprocally, all the incident power captured by the antenna as a free-space wave is converted into a guided-wave delivered to the antenna port in the receiving mode. Additionally, the beam forming networks considered are all based on passive components, contain no anisotropic materials, and are fully characterized using only the fundamental mode at each port. In these conditions, reciprocity applies at beam forming network level; that is, , with and being two integers comprised between and in the case of a -port device. With these hypotheses, the scattering matrix of the beam forming network is necessarily symmetric [10]. In this paper, the notation is used for the amplitude of the voltage wave incident on port while is the amplitude of the voltage wave reflected from port , related by the following equation [10]: where and are the two vectors containing the values of and for varying from to .

##### 2.2. Problem Description through a Simple Case

To introduce the property discussed in the introduction, we consider a 2-element linear array with the notations defined in Figure 1. The array antenna has one beam port, referred to as port 1, and two array element ports, referred to as ports 2 and 3. The beam forming network is a 3-port junction. It can be demonstrated that such a device cannot be simultaneously lossless and matched at all ports [13]. But what is meant by lossless? And how this property may depend on the operation mode of the antenna, either transmitting or receiving? Actually, do we really need all ports to be matched when such a device is used as a beam forming network? To answer these questions, let us consider two technical solutions for the proposed 3-port junction: a tee junction and a Wilkinson power divider [13]. The scattering matrix of a tee junction is the following: