International Journal of Antennas and Propagation

Volume 2019, Article ID 9582564, 13 pages

https://doi.org/10.1155/2019/9582564

## Design of Multilayer Frequency-Selective Surfaces by Equivalent Circuit Method and Basic Building Blocks

School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Mang He; nc.ude.tib@gnameh

Received 20 March 2019; Accepted 27 June 2019; Published 14 August 2019

Academic Editor: María Elena de Cos Gómez

Copyright © 2019 Yuan Xu and Mang He. 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

An equivalent circuit method (ECM) is proposed for the design of multilayer frequency-selective surfaces (FSSs). In contrast to the existing ECMs that were developed mainly for the analysis of the properties of a given FSS, the presented ECM aims at providing the initial design parameters of an FSS from the desired frequency response. In this method, four types of basic FSS structures are used as the building blocks to construct the multilayer FSSs, and their surface impedances in both the normal- and the oblique-incidence situations are studied in detail in order to achieve more accurate equivalent circuit (EC) representation of the entire FSS. For a general FSS design with expected frequency response, the EC parameters and the geometrical sizes of the required basic building blocks can be synthesized from a few typical *S*-parameter (*S*_{11}/*S*_{12}) samplings of the response curves via a simple least-square curve-fitting process. The effectiveness and accuracy of the method are shown by the designs of a band-pass FSS with steep falling edge and a miniaturized band-pass FSS with out-of-band absorption. The prototype of one design is fabricated, and the measured frequency response agrees well with the numerical results of the ECM and the full-wave simulations.

#### 1. Introduction

As a kind of periodic structures, frequency-selective surfaces (FSSs) are widely used as spatial-frequency filters in many applications, such as hybrid radomes, absorbing materials, and electromagnetic shielding devices [1, 2]. It is important to find efficient and fast methods to expedite the design of FSSs with specified requirements. In the existing literature, the design of FSS mainly relies on the full-wave numerical software, and parametric sweep is an indispensable process. However, although the numerical simulations yield accurate frequency response for a given FSS structure, they cannot provide adequate information on how to start an FSS design and how to initialize the geometrical sizes of the design to fulfill the expected frequency response of a general form. As a powerful analysis tool, the equivalent circuit method (ECM) is often utilized to reveal the operation principles of an FSS and to provide an approximate frequency response with acceptable accuracy to the designers [3–5], but we can rarely find the FSS designs that are implemented by using the ECM as a design tool according to the required frequency response [6].

In general, in order to realize a filter with an arbitrary desired frequency response, one usually needs four types of basic circuits that can provide the high-pass (HP), low-pass (LP), band-pass (BP), and band-stop (BS) filtering functionalities. Among many FSS designs [3–9], the FSSs consisting of the strip grid (SG), the square patch (SP), the square slot (SS), and the square loop (SL) elements present excellent HP, LP, BP, and BS performances, respectively, in terms of insensitivity to the polarization and angle of incidence. Therefore, in principle, these four types of basic structures are good candidates which can be used to construct FSSs with more complicated and required properties.

In this paper, based on the ECM and using the abovementioned four types of basic FSSs as the building blocks, we try to present a fast and simple design method for the multilayer FSS with desired frequency response. To achieve this goal, the EC representations of the basic structures, which are functions of the angle of incidence, polarization, geometrical dimensions, and material properties, should be derived with adequate accuracy first. The surface impedances of the SG and SP were well studied in [10–12], and the formulations that relate the equivalent inductance *L* and capacitance *C* with the geometrical parameters of the SG and SP at normal incidence are given in [7]. However, these works did not consider the effects of the dielectric substrate in the oblique-incidence case. In [13], more accurate derivations for the surface impedances of the SG and SP were proposed for oblique incidence. The surface impedance of other two basic structures, i.e., SS and SL, has been investigated in [14–17]. Based on these studies, one of the objectives of this paper is to derive the EC parameters of the four types of basic structures for both normal and oblique incidence and then to complete the EC representations of the four building blocks. Subsequently, the transmission matrix of the multilayer FSS at arbitrary angle of incidence can be obtained by using the transmission line models for the dielectric substrates [7, 8] and the ECs of the basic building blocks. Through the transmission matrix whose entry is explicit analytical function of the physical sizes of the basic FSSs, the geometrical parameters of the multilayer FSS can be achieved from the samplings of the desired frequency response for different angles and polarizations of incidence by using the least-square curve-fitting method.

In the remainder of the paper, the surface impedances of the four basic structures, as well as the transmission matrix of the multilayer FSS, are derived in the second section. Then, the design procedure is exemplified by two examples, and the effectiveness and accuracy of the proposed method are verified by the full-wave simulations. After that, we present the results of measurement and give the conclusions in the last section.

#### 2. EC Representations of the Multilayer FSS

##### 2.1. Surface Impedance of the Four Types of Basic FSSs

As stated above, in order to get the EC representation of the entire multilayer FSS, we should first derive the formulations of surface impedances for the four basic building blocks at each layer. The structures of the basic FSSs, the SG, the SP, the SS, and the SL, are shown in Figure 1. In general case, these metallic arrays may locate in a homogeneous media or at the interface of two different dielectric substrates. In reference [13], the surface impedances of the SG (Figure 1(a)) and SP (Figure 1(b)) were studied in detail, which are inductive and capacitive, respectively. Based on [13], the characteristics of the latter two basic FSS structures, SS (Figure 1(d)) and SL (Figure 1(c)), are derived here. To derive complete analytical formulations of the surface impedances, we classified the four basic FSSs into two categories: the first one is called the *grid-type* structure, which includes the SG and SS, while the second one is named the *patch-type* structure that includes the SP and SL. As seen in Figure 1(a), the two-dimensional metallic SG resides in the *xoy* plane with the periodicity being *D* in both *x*- and *y*-directions. When the width of the grid is much smaller than *D* ( << *D*), the averaged tangential component of the total electric field can be related with the averaged surface current density along the *ν*-axis (*ν* *=* *x* or *y*) by the surface impedance (*κ* = TE or TM) of the SG as [13]. Therefore, if the *yoz* plane is the plane of incidence, then the relationships become and for the TE and TM polarizations, respectively. For arbitrary angle of incidence, the surface impedance of the SG under the TE- and TM-polarized plane wave isrespectively. is the intrinsic impedance of the effective dielectric substrate in which the SG resides. If the SG locates in a single substrate with electromagnetic (EM) parameters *μ*_{r} and *ε*_{r}, then *μ*_{eff} = *μ*_{r} and *ε*_{eff} = *ε*_{r}; if the SG is at the interface of two substrates with different EM parameters (*ε*_{r1}, *μ*_{r1}) and (*ε*_{r2}, *μ*_{r2}), then the relative effective permittivity and permeability can be averaged as *ε*_{eff} = (*ε*_{r1} + *ε*_{r2})/2 and *μ*_{eff} = (*μ*_{r1} + *μ*_{r2})/2. *α* is defined as the grid parameter in [10] for an electrically dense array of SG (i.e., << *D*, *k*_{eff}*D* << 2*π*):where is the effective wave number and *μ*_{0}, *ε*_{0}, and *k*_{0} are the permeability, permittivity, and wave number in free space, respectively. In equation (2), the SG’s surface impedance for the TM polarization is a function of the angle of incidence *θ*, and the dependence is included in *ξ* as [13].