Proposal of Technique for Analysis of Complementary Frequency Selective Surfaces

A specific class of frequency selective surface (FSS) is the complementary frequency selective surface (CFSS) that has interesting characteristics such as high angular stability, multiple transmission and/or reflection bands, and the possibility of miniaturization. The analysis and design of this sort of structure are commonly performed using commercial software, which demands a high computational effort, impacting a longer optimization time. The equivalent circuit model combined with a cascading technique emerges as an alternative method to the use of these softwares, in the optimization of the physical dimensions of these structures, as they model the behavior of a CFSS with low computational effort and optimization time, in addition to being able to be implemented in various programming languages. Thus, this work proposes a CFSS analysis technique that combines the equivalent circuit method with the ABCD matrix. To the best of our knowledge, this is the first reported research on approximate techniques for CFSS analysis. The chosen geometry was the circular ring, due to high angular stability and polarization independence. The modeling of the equivalent circuit for patch and aperture geometries is presented. Some structures are simulated, and the results are compared with results obtained with the HFSS software. Finally, two prototypes are built to validate the analyses performed.


10.1029/2022RS007621
2 of 8 responses under angular incidence or under different polarizations were not analyzed.For this, two equivalent circuit models are proposed in order to be used later in the final ECM proposal for the complementary structure.The intention is to develop a method for analysis, with agility and simplicity, in addition to the possibility of implementing it in open-source languages.

Circular Ring and Its ECM
In this work, the focus is on a CFSS structure with patch-type and aperture-type circular rings, each FSS is printed on one side of a dielectric.Figure 1 illustrates the unitary geometry of the patch-type FSS.On each unit cell, there is a printed conductive circular ring.A substrate is used to facilitate the fabrication of the FSS.The substrate thickness and relative permittivity are h and ɛ r , respectively.The physical dimensions are illustrated in Figure 1.
In Varkani et al. (2017) the authors proposed the equivalent circuit model for the circular conducting ring, which is illustrated in Figure 2.
The equations for obtaining the element values, normalized to the free space impedance, are: where a factor π/2 owing to the length of the half circular loop (πd/2) is compared with the straight line for the square loop (d) and g a is the average gap between two adjacent unit cells and can be calculated as: and ɛ eff is the effective permittivity of the substrate calculated as: where x = 10 hr/p, h is the thickness of the substrate, and N is an exponential factor that varies for different cell shapes in terms of the unit cell filling factor.
Conductive circular rings are used for band-stop frequency response.If a bandpass type response is desired, aperture-type circular rings should be used.Figure 3 illustrates the aperture-type circular ring unit cell.The gray-colored part is a conductive material.The outer diameter of the ring is d, w is the aperture thickness, g is the distance between two adjacent rings, and p is the cell periodicity.
In da Silva et al. (2020) the authors proposed an equivalent circuit for aperture-type circular rings, obtaining good results.This same circuit will be considered in this work.The proposed circuit can be seen in Figure 4.The equations for obtaining the element values, normalized to the free space impedance, are:  10.1029/2022RS007621 3 of 8 The subscript int indicates an intermediate variable, to obtain the final parameter.Equation 5 concerns the conductive strip formed by the separation between adjacent rings.At this point, it is worth mentioning the added correction factor (CF), which is related to the average value of g, according to the chosen diameter d, as (da Silva et al., 2020): As can be seen, Equation 11deals with the average value of a function referring to a semicircle and proved to be an interesting approach in the adequacy of the proposed ECM.

ABCD Matrix
One of the concepts of the Transmission Line Theory that can be used combined with the ECM is the ABCD transmission matrix that allows the calculation of the transmission coefficient of CFSSs, as is the case of the frequency selective complementary surfaces (Krushna Kanth & Raghavan, 2018;Munk, 2000;Xu & Mang, 2019).The ABCD parameters are generalized circuit constants used in the transmission line model.More specifically, ABCD parameters are used in two-port transmission lines.
The voltage and current at the input and output terminals of a two-port network are related according to: where V e and I e are the voltage and current at the input terminal, respectively, while V s and I s are the voltage and current at the line output, respectively.Equations 12 and 13 can be rearranged in matrix form, for ease of calculation as: The ABCD matrix is called the transmission matrix or network transfer matrix.This technique can be applied in CFSS in order to calculate the transmission and reflection coefficients.For a multilayer FSS composed of n layers of FSS and (n − 1) layers of dielectric, the transmission matrix can be realized using the transmission line model.For this case, the ABCD matrix for a single substrate layer, as is the case for a CFSS, can be written as: where h is the thickness of the dielectric substrate, v = k d cosθ′ is the vertical component relative to the wavenumber k d , and θ′ is the angle of refraction considering an angle of incidence θ.
The transmission matrix of the multilayer FSS can be obtained by cascading the ABCD matrices of each dielectric layer, as well as, each surface impedance of the resonant metallic layers, as: Thus, the reflection and transmission coefficients of the CFSS can be calculated respectively as:

Result and Discussions
The cascading technique is applied in an unprecedented way in the literature together with the equivalent circuit method, for the analysis of complementary frequency selective surfaces with circular rings.The results obtained with this technique are compared with the results obtained with the commercial software HFSS.
As a first example, we will consider a CFSS whose unitary elements make up a complementary array that has a periodicity p of 30 mm, rings with a diameter d of 28 mm, and a thickness w of 3 mm, for both patch-and aperture-type elements.The chosen substrate was FR-4, with a thickness h of 0.8 mm and electrical relative permittivity of 4.4.Figure 5 illustrates the comparison between the results obtained with the proposed technique and with the HFSS.It can be seen that the results converge.The technique shows a good agreement with respect to the resonant frequency and for the first passband.However, there is a divergence in the reject bandwidth and the second center frequency of the second passband.
Figure 6 illustrates the comparison between the results obtained for the second example.The CFSS has the following constitutive parameters: p = 20 mm, d = 18 mm, w = 2 mm, and h = 1.2 mm and a relative electrical permittivity of 4.4.The comparative results show a good convergence.The resonant frequency of both methods showed close values, while the equivalent circuit calculated 3.7 GHz, the HFSS stipulated 3.85 GHz.For the first passband, the ECM predicted a center frequency of 2.9 GHz while the HFSS obtained  3.2 GHz.As for the second band, the values were 5.5 and 4.85 GHz.The proposed method, again, showed good agreement with the HFSS and we can observe that there was a divergence in the bandwidth and at the second center frequency of the second passband.
A third comparison is illustrated in Figure 7.The CFSS has a periodicity p of 15 mm, a ring diameter d equal to 12 mm, and a thickness w of 1 mm.The dielectric has a thickness h of 1.2 mm and a relative electrical permittivity of 4.4.As for the previous examples, the frequency response obtained with our proposed technique showed good agreement with the frequency response of the HFSS.The results are illustrated in Figure 7.The central frequency of the first transmission band, according to our technique, is 4.4 GHz, while according to the HFSS it is 4.9 GHz.The resonant frequency is 5.4 GHz for the proposed technique and 5.35 GHz for the HFSS.Finally, the second transmission band had a central frequency of 7.3 GHz for our technique and 7 GHz for the HFSS.Once again, the proposed method showed good agreement when compared to the HFSS, but we can observe, again, a divergence in the reject bandwidth and in the second center frequency of the second passband.
For measurements, two ultra-wideband (UWB) 1-10 GHz horn-type Shuwarzbeck model BBHA 9120 B antennas were used for transmission and reception and linear polarization.The antennas have a maximum gain of 18 dBi.The VNA used was Rohde & Schwarz ZND.The distance between the two antennas and the center of the structure was about 1.30 m in both measurements, ensuring that the CFSS will be in the far-field region.Figure 8 illustrates the measurement setup.
In order to validate the analyses, two prototypes were built, and their frequency responses were measured.Figure 9 illustrates the prototypes built.
Figure 10 compares the results obtained through the proposed technique, with the HFSS ones and measured, for prototype 1.The proposed structure is 20 cm × 20 cm containing 6 × 6 elements for both layers, with 2 mm spacing between adjacent elements.The CFSS has a periodicity p of 30 mm, a diameter d of 28 mm, and a thickness w of 3 mm.The dielectric substrate has a thickness h of 0.8 mm and a relative permittivity of 4.4.The experimental characterization of the proposed array, which has an overall size of 20 cm × 20 cm and contains 6 × 6 unit cells, shown in Figure 9, was carried out using the free space method, as shown in Figure 8.
Due to limitations in the measurement setup, the measured values start at 2 GHz.We can observe a good agreement between the results.The first passband has a central frequency of 1.9 GHz, for the results obtained with the proposed technique, while for the results obtained with the HFSS, the central frequency is 2.15 GHz, and the measured result is 2.13 GHz.In the rejection band, the resonant frequency was 2.5 GHz, for the proposed technique and for the HFSS, and for the measured results the value was 2.6 GHz.The second transmission band had a central frequency of 4 GHz, for the proposed technique, while for the HFSS and for the measured results it was 3.2 GHz.
The measured results confirmed the divergence in the reject bandwidth and in the second center frequency of the second passband.
Figure 11 illustrates the simulated results, with our technique and with the HFSS, and measurements, for prototype 2. The proposed structure is 20 cm × 20 cm containing 10 × 10 elements for both layers, with 2 mm spacing between adjacent elements.The CFSS has a periodicity p of 20 mm, a diameter d of 18 mm, and a thickness w of 2 mm.The dielectric substrate has a thickness h of 1.2 mm and a relative permittivity of 4.4.The experimental characterization of the proposed 10 × 10 array, shown in Figure 9, was carried out using the free space method, as shown in Figure 8.We can observe a good agreement between the results.The first passband has a central frequency of 2.9 GHz, according to the equivalent circuit, while for the results obtained with the HFSS, the central frequency is 3.2 GHz and the measured result  10.1029/2022RS007621 6 of 8 is 3.15 GHz.rejection band, the resonant frequency was 3.85 GHz, for our technique, for the HFSS the value was 3.7 GHz, and for the measured results the value was 3.75 GHz.The central frequency of the second passband was 5.5 GHz, for our technique, while for the HFSS it was 4.85 GHz and for the measured results it was 4.78 GHz.For this prototype, the measured results also confirmed the divergence in the reject bandwidth and at the second center frequency of the second passband.

Conclusion
In this work, the Equivalent Circuit Method (ECM) was used to model FSS with patch and aperture circular rings.These circuits allow obtaining the transmission and reflection coefficients, considering the lossless structures.With these coefficients, the ABCD matrix technique for analyzing CFSS with circular rings was proposed, aiming at predicting the frequency response of the CFSS.As a final validation of the proposed equations, results obtained with the ECM were compared with results and simulated with the commercial software HFSS.The results obtained with the ECM showed a good agreement with the finite element method, used by the commercial software HFSS.For validation purposes, two prototypes were built and measured.The  experimental results showed good agreement with the simulated ones.The proposed technique showed divergence in the prediction of the rejection bandwidth and the second center frequency of the second passband.However, the technique can be used to reduce the optimization time of the physical dimensions of a CFSS.We can just use the technique and then perform a more precise adjustment of the dimensions with commercial software, saving computational time for optimization.

Figure 1 .
Figure 1.Conductor circular ring geometry and its dimensions.

Figure 2 .
Figure 2. Equivalent circuit for conductor circular ring.

Figure 3 .
Figure 3. Unit cell of a circular ring aperture.

Figure 4 .
Figure 4. Equivalent circuit for circular ring aperture.

Figure 5 .
Figure 5.Comparison between results obtained with the proposed technique and HFSS: example 01.

Figure 6 .
Figure 6.Comparison between results obtained with the proposed technique and HFSS: example 02.

Figure 7 .
Figure 7.Comparison between results obtained with the proposed technique and HFSS: example 03.

Figure 10 .
Figure 10.Comparison between simulated and measured results for prototype 1.

Figure 11 .
Figure 11.Comparison between simulated and measured results for prototype 2.