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International Journal of Antennas and Propagation
Volume 2015, Article ID 946289, 9 pages
Research Article

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.


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.