An equivalent circuit method (ECM) is proposed for the design of multilayer frequencyselective 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 obliqueincidence 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
As a kind of periodic structures, frequencyselective surfaces (FSSs) are widely used as spatialfrequency filters in many applications, such as hybrid radomes, absorbing materials, and electromagnetic shielding devices [
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 highpass (HP), lowpass (LP), bandpass (BP), and bandstop (BS) filtering functionalities. Among many FSS designs [
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 [
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 fullwave simulations. After that, we present the results of measurement and give the conclusions in the last section.
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
The four FSS structures as the building blocks of multilayer FSSs. (a) Square grids (SGs). (b) Square patches (SPs). (c) Square loops (SLs). (d) Square slots (SSs). Metallic parts are colored in grey.
Since the SP is a complementary structure of the SG, its surface impedance
Thus,
In equations (
The SS shown in Figure
Similarly, the SL in Figure
In equations (
By now, the equivalent surface impedances of the four basic building blocks have been derived completely for arbitrary angle of incidence, which closely relate the EC representations of the basic FSS structures to their physical dimensions.
For a multilayer FSS composed of
Structure of multilayer FSS.
In equation (
The reflection and transmission coefficients of the multilayer FSS are calculated as
In this section, a singlelayer FSS composed of SL elements is used to validate the accuracy of the proposed ECM. The configuration of the FSS is shown in Figures
Structure of the singlelayer bandstop FSS. (a) SL element of the FSS. (b) Equivalent circuit of the FSS.
Comparison of the reflection coefficients
It has been shown above that the EC can reproduce the characteristics of the multilayer FSSs constructed by the four basic building blocks with reasonable accuracy. Naturally, we try to use the proposed ECM and the basic FSS structures to design the multilayer FSSs with desired frequency response. To determine the dimensions of the desired FSS structure, the design procedure contains the following:
Step 1: obtain the equivalent circuit model of the FSS
Step 2: derive the impedance of each FSS layer
Step 3: build the transmission matrix based on the transmission line principle
Step 4: synthesize the EC parameters and the geometrical sizes of the structure from a few typical
Step 5: based on the ECM data, sweep the FSS parameters using fullwave simulation software to obtain the optimized parameters and finish the design
In the following, two examples are presented to illustrate the design procedure.
The first example is to design a broadband bandpass FSS with a fast falling edge, and the desired transmission coefficient of the FSS is shown in Figure
The desired frequency response of the bandpass FSS with steep falling edge. The samplings marked in red color are chosen to show the main properties of the FSS in the passband and stopband.
In general, the FSS composed of the SS elements is a bandpass filter, but it cannot provide fast rising or falling edges. On the other hand, the SG structure has a wideband highpass response; thus, if it works together with the SL structure that acts as a bandstop filter, then the overall frequency response will consist of a passband and a stopband. To satisfy the steep falling edge requirement, two SL FSS layers with the same element’s sizes are used to make
The configuration and equivalent circuit of the doublelayer bandpass FSS with steep fallingedge stopband. (a) Geometry of one element of the FSS. (b) Equivalent circuit.
Then, from equations (
Samples from the desired frequency response of
Sampling frequencies (GHz)  3.0  4.5  5.0  5.5  6.0  6.5  7.0  8.0  12.0 

−10.0  −1.0  −1.0  −1.0  −1.0  −1.0  −1.0  −20.0  −20.0 
In this example, the two layers of dielectric substrates are of the same thickness of 1.27 mm and same relative permittivity of 2.2. Two different angles of incidence (
Geometrical parameters calculated by the curvefitting method.






0°  8.3  1.0  2.0  10.0 
45° TE  8.4  0.9  1.3  10.2 
45° TM  8.6  1.6  2.0  9.8 
Averaged values  8.4  1.2  1.8  10.0 
Optimal values by HFSS parameter sweep  8.2  1.0  1.0  10.0 
Figures
The simulated results using the ECM and HFSS. (a)
The second example is to design an FSS with a narrow passband near 4.0 GHz in which the transmission efficiency is more than 90% and the reflection coefficient is less than 10% (i.e.,
The desired frequency response of
From the four building blocks, the FSS with SL elements is chosen to form the narrow passband around 4 GHz, and the FSS with SL elements loaded with lumped resistors [
The configuration and equivalent circuit of the miniaturized bandpass FSS with outofband absorption. (a) Geometry of one element of the FSS. (b) Equivalent circuit.
Similar to the former design example, if the dielectric substrate is given, the geometrical and EC parameters
The relative permittivity, loss tangent, and thickness of the dielectric substrate used in this design are 2.2, 0.001, and 2.54 mm, respectively. The width of the SL element
Samples from the desired frequency response of
Frequency (GHz)  2.0  4.0  7.0  10.0 


−0.5  −15.0  −1.0  −20.0 

−15.0  −0.5  −15.0 
Equivalent circuit and geometrical parameters calculated by the curvefitting method.







0°  95.01  0.48  6.3  7.9  8.9 
45° TE  75.00  0.50  6.7  8.3  9.7 
45° TM  73.82  0.54  6.3  7.6  8.5 
Averaged values  81.28  0.51  6.4  7.9  9.0 
Values after HFSS optimization  70.00  0.50  6.0  7.9  9.0 
In Figure
Simulated
Simulated
In order to verify the effectiveness of the proposed ECM, the prototype of the first design example, i.e., a multilayer bandpass FSS with a fast falling edge, is fabricated. Figures
Configuration of the bandpass FSS with steep falling edge. (a) Exploded view of the FSS. (b) Photograph of the fabricated prototype.
The measurement setup is shown in Figure
Schematic of the measurement setup.
Figure
Comparison of the measured and the simulated results. (a)
A fast design method for the multilayer FSS based on four types of basic building blocks and the equivalent circuit representation is proposed in this paper. The method utilizes the four basic FSS structures to construct multilayer FSSs with desired frequency response of general forms, and the geometrical sizes of the FSS can be synthesized in a fast manner using the ECM. In order to derive equivalent circuit representations of the basic FSSs with adequate accuracy, a complete set of formulas for computation of the surface impedances of the four basic structures is given. The synthesized physical dimensions by the proposed ECM are very close to the finetuned values via numerical parameter sweeping process, which indicates that the design method can yield good initials to the fullwave simulators for further optimization. The effectiveness and accuracy of the presented method are verified by numerical and experimental examples.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The research and publication of this article were funded by the National Natural Science Foundation of China under grant 61471040.