they are still low. And again, good agreement between the neural model predictions and the rigorous analysis with the SFE method is observed. 6. CONCLUSIONS A new CAD tool for the design of microwave circuits, which combines segmentation, FEM, and ANNs, was presented. First, a fast and accurate neural model of an arbitrary mi- crowave device is developed, starting from a set of training points computed using the SFE method. Afterwards, this neural model, whose evaluation is very fast, is used together with optimization algorithms for the microwave design. The first step in the neural model development is the division of the microwave device into small regions connected by waveguide segments. Then, the GSM of each region is modeled separately by MLPs. To achieve this, training points are needed, which are obtained in a very efficient way using the SFE method. The use of this analysis method allows us to deal with arbitrarily shaped 3-D regions connected by wave- guide segments, also with arbitrarily shaped cross sections, and to easily include a very high number of modes in the connection ports into the computations. Finally, the complete GSM system is computed connecting all of the GSMs mod- eled in each region. By using segmentation techniques, the number of training points needed for a correct modeling is drastically reduced. Therefore, the development of a neural model is simplified considerably because, in the neural model development for microwave devices, the generation of the training set is usually the largest time-consuming step. This CAD tool is used for the design of a transition between coaxial and rectangular waveguides with good re- sults. The neural model developed in this example shows very good agreement with the SFE method used to generate the training set, but it computes the response of the transition in a much shorter time. REFERENCES 1. Q.J. Zhang and K.C. Gupta, Neural networks for RF and mi- crowave design, Artech House, Norwood, MA, 2000. 2. J.M. Cid and J. Zapata, CAD of rectangular-waveguide H-plane circuits by segmentation, finite elements and artificial neural net- Ž . works, Electron Lett 37 2001 , 9899. 3. J. Rubio, J. Arroyo, and J. Zapata, Analysis of passive microwave circuits by using a hybrid 2-D and 3-D finite-element mode-match- Ž . ing method, IEEE Trans Microwave Theory Tech 47 1999 , 17461749. 4. K. Hornik, M. Stinchcombe, and H. White, Multilayer feedforward Ž . networks are universal approximators, Neural Networks 2 1989 , 359366. 5. G. Cybenko, Approximation by superpositions of sigmoidal func- Ž . tion, Math Contr Signals Syst 2 1989 , 303314. 6. P.J. Werbos, Backpropagation through time: What it does and Ž . how to do it, Proc IEEE 78 1990 , 15501560. 7. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical recipes in FORTRAN: The art of scientific computing, Cambridge University Press, Cambridge, England, 1992, 2nd ed.  2002 John Wiley & Sons, Inc. A PRACTICAL TECHNIQUE FOR OPTIMUM MATCHING OF TWO-ELEMENT ANTENNA ARRAYS Caner Ozdemir 1 1 Department of Electrical and Electronics Engineering Engineering Faculty Mersin University 33342 Mersin, Turkey Recei  ed 9 August 2001 ABSTRACT: A fast and effecti  e method for the matching of two- element symmetric antenna arrays is presented. The full geometry is subdi  ided into two half-space problems with the help of an e en- and ( ) odd-mode EOM analysis scheme. An interpretation based on the EOM representation is made for any radiation pattern mode that is constructed by arbitrarily exciting the antenna ports. Also, a relationship between the radiation efficiency for the full geometry and the e en- and odd-mode S-parameters is deri  ed. A unique technique for matching the two- element antenna arrays is de eloped by exploiting the EOM representa- tion. Numerical examples that illustrate the effecti  eness of our matching technique for different radiation pattern modes are pro ided.  2002 John Wiley & Sons, Inc. Microwave Opt Technol Lett 32: 224229, 2002. Key words: e enodd mode analysis; antenna matching; two-element array; radiation pattern modes DOI 10.1002  mop.10138 I. INTRODUCTION Matching of antenna arrays has always been a challenge because of the inevitable electromagnetic interaction be-   tween the antenna elements 1  2 . This mutual coupling between the elements makes the matching process very com- plex in such a way that, when one uses a matching network to match the original array, the coupling between the elements is altered, and the original matching condition is no longer valid. As a result, the matching circuit does not work since the radiation impedance condition is changed. For this rea- son, antenna designers usually employ an iterative design- Ž . and-measure or design-and-simulate procedure to minimize the coupling between the antenna elements, while maximiz- ing the radiated power from the array for a particular radia- tion mode. However, this type of iterative search is quite expensive in the lab since it requires numerous prototypes to be built, and is also very time consuming when using an- tenna-design simulators since the simulations have to be performed again and again until finding an optimum point. In this paper, we present a unique method for the opti- mum matching of two-element antenna arrays for a particu- lar radiation mode with a scheme especially tailored to the Ž .  even- and odd-mode EOM analysis technique 3 . This method uses the symmetric feature of the antenna geometry to decompose the problem into two one-element even and one-element odd symmetry half problems. Until now, rota- tionally symmetric array antennas have been studied by some   researchers 4, 5 ; however, their works were mainly focused on the analysis of the array, not on the matching of them. In this work, our main focus is the matching issue of two- element symmetric arrays. With the help of our method that utilizes the EOM scheme, we can easily match any two-ele- ment symmetric array without resorting to a design-and- simulate cycle that may take a long time for completion of Contract grant sponsor: Mersin University Research Fund Ž . Contract grant number: MUH F EEM CO 2001-3 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 32, No. 3, February 5 2002 224