International Journal of Antennas and Propagation

Volume 2015, Article ID 759439, 12 pages

http://dx.doi.org/10.1155/2015/759439

## Applications of Generalized Cascade Scattering Matrix on the Microwave Circuits and Antenna Arrays

National Key Laboratory of Science and Technology on Antennas and Microwaves, Xidian University, Xi’an, Shaanxi 710071, China

Received 9 December 2014; Revised 8 March 2015; Accepted 19 March 2015

Academic Editor: Angelo Liseno

Copyright © 2015 Shun-Feng Cao et al. 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 ideal lossless symmetrical reciprocal network (ILSRN) is constructed and introduced to resolve the complex interconnections of two arbitrary microwave networks. By inserting the ILSRNs, the complex interconnections can be converted into the standard one-by-one case without changing the characteristics of the previous microwave networks. Based on the algorithm of the generalized cascade scattering matrix, a useful derivation on the excitation coefficients of antenna arrays is firstly proposed with consideration of the coupling effects. And then, the proposed techniques are applied on the microwave circuits and antenna arrays. Firstly, an improved magic-T is optimized, fabricated, and measured. Compared with the existing results, the prototype has a wider bandwidth, lower insertion loss, better return loss, isolation, and imbalances. Secondly, two typical linear waveguide slotted arrays are designed. Both the radiation patterns and scattering parameters at the input ports agree well with the desired goals. Finally, the feeding network of a two-element microstrip antenna array is optimized to decrease the mismatch at the input port, and a good impedance matching is successfully achieved.

#### 1. Introduction

Microwave system consists of microwave elements and transmission lines, and the microwave network theory is one of the most important analytical approaches to design and synthesize the microwave equipment. A field analysis using Maxwell’s equations is complete and rigorous. However, usually we are only interested in the terminal characteristics and the power flow through a device. And, sometimes, it is convenient to combine the elements together and find the responses without having to reanalyze the behavior of each element in combination with its neighbors. As a result, the complex and sensitive parts can be analyzed with a rigorous field analysis approach, while the characteristics of the entire network are obtained by microwave network theory.

Generally, a microwave network with an arbitrary number of ports can be characterized by the impedance, admittance, and scattering matrices. The transmission (*ABCD*) matrix has been widely adopted in many applications, but it only works for two or more two-port networks. Recently, a generalized method for arbitrary networks was discussed by Hallbjörner [1]. It was based on the calculation of the sum of admittances by considering both the “internal” and “external” ports. At each intersection, the sum of admittances is obtained separately. And then, construct the scattering matrices of the elements and calculate the scattering matrix of the entire network according to the equations raised by the authors. In this paper, the principle is completely different, where the scattering matrices are cascaded one-by-one and based on different algorithms. The algorithm of generalized cascade scattering matrix has been presented in many researches and educational books [2–6]. However, it seems that the existing descriptions only present approaches to cope with the one-by-one interconnections. There is no distinct explanation on the complex interconnections. Besides, as far as we know, the algorithm is usually employed on microwave circuits, and it has never been applied on the excitation coefficients of antenna arrays.

In this paper, the algorithm of the generalized cascade scattering matrix is studied and expanded to deal with the complex interconnections and the excitation coefficients of antenna arrays. The ideal lossless symmetrical reciprocal network (ILSRN) is firstly presented to facilitate the generalized microwave cascade network. It can be treated as the “virtual joint” to combine the surrounding networks together. As a result, the generalized cascade scattering matrix can be expediently operated with an intuitive idea. The only thing one should to do is just to offer the interconnection relationships, and then the operations of the generalized cascade scattering matrices can be automatically performed on the computer. Moreover, a useful derivation on the excitation coefficients of antenna arrays is firstly proposed with consideration of the coupling effects. The antenna array as a whole is considered to be a network load of the feeding network. The excitation coefficients of each unit can be obtained through the interconnection of the feeding network and antenna array. To demonstrate the accuracy of the proposed techniques, applications both in the fields of microwave circuits and antenna arrays will be presented in the following.

The framework of this paper is as follows. In Section 2, the algorithm of the generalized cascade scattering matrix is derived, and the ILSRNs are illuminated in detail. Moreover, the derivation on the excitation coefficients of antenna arrays is presented. In Section 3, the proposed techniques are employed on several applications. Finally, we draw the conclusions in Section 4.

#### 2. Algorithms on the Generalized Cascade Scattering Matrix and Antenna Arrays

Considering two arbitrary microwave networks I and II, the scattering matrices can be written aswhere the scattering matrices are separated into four cells related to the remaining (R) and vanished (V) ports, respectively. If the vanished ports of two different networks are interconnected one-by-one, they are satisfied by

Combining (1) and (2) yields the renewed scattering parameters at both the remaining and vanished portsHere, , , , and are given bywhere is the unit matrix.

Note that the derivation is under a hypothesis that each pair of interconnection ports is interconnected one-by-one. Although similar conclusions have been drawn in many researches and educational books, none clearly indicate how to deal with the complex situations, for example, the interconnections with more than two ports and self-loops with the ports coming from the same network. Here, the ILSRNs are introduced to resolve the problem. Assume a three-port ILSRN has a generalized form ofIn terms of the reciprocity and symmetry, it generatesThen, as to a lossless network, it is satisfied by

The above complex equation set contains four unknowns, and more conditions are needed to resolve it. At the three-port node, Kirchhoff’s voltage law and current law can be applied to givewhere and , , are the normalized voltage and current at the th port, respectively, and are defined byThus, this yieldsFrom (8) and (11), the solutions can be found asSimilarly, the ILSRNs with an arbitrary number of ports can be constructed for particular problems. For example, the scattering matrices of two-port and four-port ILSRNs are given by

To illuminate the decomposition of the complex interconnections, a sketch is shown in Figure 1. We find it contains three nonstandard interconnections (marked by circle lines). By inserting the two-port and three-port ILSRNs (marked by elliptical lines), extra virtual ports are introduced and the complex interconnections are converted into the standard one-by-one case as in the hypothesis without changing the network’s characteristics. Also, we observe it is more advisable to replace the combination of two three-port ILSRNs by a single four-port ILSRN. Consequently, by inserting the ILSRNs, two arbitrary networks interconnected with arbitrary forms can be resolved with the standard algorithm as shown from (1) to (5). From (3), the scattering parameters at the remaining ports can be acquired. They contain the return loss, insertion loss, isolation, and phase distribution at each port, while the incident waves after the interconnection at the vanished ports can be obtained from (4). In this paper, just the incident waves will be utilized to calculate the excitation distribution of antenna elements when the antenna array is interconnected with a feeding network.