I. Introduction
In the field of modern electromagnetic (EM), frequency selective surfaces (FSSs) with excellent filtering characteristics have recently been widely applied in the design of absorbers [1–4], antennas [5, 6], lenses [7–9], and sensors [10, 11]. Notably, the EM performance of FSSs is dependent on the different structural patterns.
To address the increasing demand for FSS devices, numerous patterns that tackle various requirements have been put forward and efficient approaches to analyzing and designing FSSs have been proposed. Among these, the equivalent circuit model (ECM) is an intuitive and simple approach for analyzing FSSs [12–16]. In the ECM, complex structures are considered equivalent to integrated circuits composed of resistors, inductors, and capacitors. For instance, Marcuvitz [17] implemented the ECM by considering infinite strips irradiated by EM waves as inductance and capacitance, deducing formulas for both. Over time, ECM has been widely promoted, and its mathematical formulas have been extended to be applicable to finite strips. However, despite its achievements, discrepancies between the true and numerical values calculated by the formulas persist. To handle these issues, an accurate and efficient equivalent circuit model (AE-ECM) was proposed, which attempted to correct the errors related to the specific scopes of strip-gap FSSs by introducing effective widths, leading to improvements in the accuracy of the experimental results [18]. Furthermore, the correction factor of capacitive reactance was introduced in the ECM to better represent the frequency response of square slot FSSs [19]. Overall, it is evident that the literature largely focused on studying the limited scopes of FSSs, including their limited size, narrow bandwidth, and limited normal incidence.
Conventional EM calculation methods, such as the finite-difference time-domain method and the method of moment, are some of the most straightforward approaches for analyzing FSSs. However, solving the Maxwell equations consumes a huge amount of time and computer memory. Recently, artificial intelligence technology that uses neural networks has emerged as a significant tool for studying FSSs. For instance, artificial neural networks were utilized to establish a reverse analysis model that predicted structural parameters from an input of S-parameters, while avoiding the complex calculation of the Maxwell equations [20]. A lightweight physical model based on neural networks, which introduced two deep parametric factors obtained from multi-layer perceptron networks to compensate for the absence of nonlinearity, has also been proposed [21]. However, the model generation process was entirely data-driven, largely dependent on the capacity of the training data. This suggests that as FSS structures become increasingly complex, the amount of sampling data required will increase dramatically, which in turn will prolong the time taken to generate training samples and train models manifold. Moreover, many researchers employ simulation software for optimization and analysis. Computer Simulation Technology (CST) Microwave Studio and High Frequency Structure Simulator (HFSS) are two popular simulation software that are widely utilized by researchers [6, 22]. However, using these software to analyze complex structures poses several obstacles for modeling. Additionally, the demand for higher calculation precision makes even subtle scale division more difficult to achieve.
In this work, a generalized and efficient equivalent circuit model (GE-ECM) that is applicable to multi-size, broadband, and oblique incidence FSSs is proposed. The application limitations of ECM are discussed in view of electron motion theory by expounding the distribution of surface current. In particular, two factors are introduced in the GE-ECM to compensate for the irregularity of coupling electrons—the gap factor and the width factor. The former recalibrates the impedance expression, while the latter redefines the admittance deduction. In addition, normalized sizes of structure periods are introduced to further generalize the GE-ECM. To verify GE-ECM performance, four simulation experiments were conducted to examine the accuracy in broadband, multi-size flexibility, the feasibility of application in different kinds of FSSs, and stability with regard to wide incident angles. Furthermore, the fabrication and measurement of a practical FSS were carried out, with the comprehensive experimental results demonstrating the excellent performance of the proposed GE-ECM.
II Theoretical Background and Model Design
1. Classic ECM Theory
The strip-gap (SG) is a basic FSS structure whose diverse combinations can be employed to achieve desired EM performances. Thus, an EM analysis model for SG can offer basic but vital theoretical guidance for designing arbitrary FSSs.
As mentioned before, the ECM [17] regards structures as integrated circuits. In Fig. 1, the intrinsic impedance and admittance in free space are denoted as Z0 and Y0, respectively, while e represents electrons and k is the wave vector of incidence. According to the theory of electron motion, different circuit elements generate different EM wave phenomena. Therefore, in Fig. 1(a), the direction of the electric field is parallel to the extended direction of the metal strips, which means that the strips are characterized by low-frequency reflection and high-frequency transmission. As a result, they can be considered equivalent to the inductance L, while the corresponding impedance can be expressed as jXL. However, when the magnetic field is parallel to the extended direction of the metal strips, as shown in Fig. 1(c), the strips exhibit low-frequency transmission and high-frequency reflection. Consequently, the equivalent capacitance C and the corresponding admittance jBC are aquired.
An SG FSS originates from the periodic cut-off of infinite strips, which means that both strips and gaps are in the direction of the electric field. Due to the formation of L and C in the direction of electric fields, the continuous strips and gaps can be regarded as inductance and capacitance, respectively. Hence, the ECM of SG FSS can be regarded as a series of L and C, as shown in Fig. 1(e) and 1(f). Therefore, similar to infinite strips, the classic formulas for XL and BC of SG FSS are as follows:
where p,d, and g are the size parameters of the SG FSS, while λ and θ denote the wavelength and incident angle of the EM wave, respectively [18, 19]. Notably, in Eqs. (3)–(6), noted below, d can be replaced by different inputs.
Moreover, transmittance, which accounts for the accuracy of the resonant frequency and the stability of the frequency response, was utilized to assess the performance of the GE-ECM. It can be expressed as follows:
where ZL = jXL and ZC = jBC refer to the impedance of inductance and capacitance, respectively, while ZFSS is the equivalent impedance.
Although weighting coefficients were considered the cut-off sizes of d and g in Eqs. (1) and (2), the ECM still imported significant deviations. Assuming the size parameters of the SG FSS to be d = 5 mm, g = 3 mm, p = 15 mm, resonant frequency discrepancies between the numerical calculation conducted using Eqs. (1)–(7) and the values obtained from the EM software CST were found to be 2.72 GHz, which could cause unpredictable problems when designing FSSs for practical applications.
2. Design of the GE-ECM
The ECM aims to interpret the EM performance of infinite strips. Therefore, to reconstruct an optimized ECM, the physical differences between the XL and BC of SG FSS and infinite strips were considered.
To analyze and optimize the XL model, a comparison of the surface current distribution of the SG FSS and infinite strips was conducted, as demonstrated in Fig. 2(a) and 2(b). It is evident that the surface current is continuous and homogeneous for infinite strips, whereas it is discontinuous and inhomogeneous for SG FSS. This difference can be explained by electron motion theory. When irradiated by an EM wave, the electrons in infinite strips continuously move along the direction of the electric field. In contrast, due to the limitation in length in SG FSS, the electrons pause and aggregate at both terminal sides, leading to an inhomogeneous distribution of current. Therefore, in the case of the XL model of SG FSS, it would be inappropriate to refer directly to the formulas of infinite strips. Instead, a gap factor (GF) δg needs to be introduced to compensate for the differences between the infinite and finite strips. Effectively, the optimized XL model can be expressed as follows:
To analyze the BC model, current distributions were simulated, as illustrated in Fig. 2(c) and 2(d). The distribution of current of SG FSS was observed to be similar to that of vertical strips. In particular, the electrons aggregated at the terminal sides along the electric field. The positive and negative charges at the adjacent sides formed capacitance between the adjacent strips. However, for the infinite strips, the distribution was homogeneous. Therefore, to compensate for this difference, a width factor (WF) δd was introduced. The optimized BC model can be expressed as follows
Furthermore, to expand the application of the proposed GE-ECM, the normalization of sizes of SG FSS relative to structure periods, such as d1 for d/p and g1 for g/p, was conducted. The WF and GF were obtained for different periods, as long as d1 and g1 could be calculated from different d and g values. Optimal values of the GF and WF considering multiple sizes of SG FSS were obtained by conducting CST simulations and optimal calculations, the results of which are listed in Tables 1 and 2, respectively.
In the optimal procedure, the ranges of GF and WF were restricted using Eqs. (10) and (11). By carrying out fitting and optimization experiments drawing on a large number of mathematical operation rules, the mathematical relations between GF, WF, and the normalized sizes of structure periods were established using Eq. (11), thus generating an effective mathematical model for GE-ECM.
Where
where m1 and n1 (i = 1,2,3) are the weighting coefficients. Based on data fitting, the values of m and n were obtained as follows: m = [m1,m2,m3] = [−0.7244,0.2914,−0.4684] and n = [n1,n2,n3] = [0.8891,0.0011,0.1861].
III. Results and Discussions
To verify the efficiency and accuracy of the proposed GE-ECM, simulation experiments were conducted, after which the results were compared to those of the classic ECM [17] and AE-ECM [18]. In particular, the root mean squared error (RMSE), relative error (RE), and absolute error (AE) were considered. Furthermore, a typical square loop FSS was fabricated and measured to validate the practicability of the GE-ECM. Notably, the dielectric used in the simulations was air, while the metal used in all structures was copper, with a thickness of 0.0175 mm and conductivity of 5.5 × 107 S/m.
1. Accuracy in Broadband
The frequency responses of the SG FSS with fixed sizes of d = 4 mm, g = 3 mm, and p = 15 mm when using different ECMs are illustrated in Fig. 3. Compared to the values attained using the professional simulation software CST, the resonant frequencies of the classic ECM, AE-ECM, and proposed GE-ECM deviate by 3.12 GHz, 0.28 GHz, and 0.01 GHz, respectively. This confirms that the GE-ECM demonstrates better accuracy than the others. Moreover, the REs attained by the classic ECM, AE-ECM, and proposed GE-ECM compared to CST were 25.91%, 2.32%, and 0.08%, respectively, thus further verifying the accurate performance of the GE-ECM, since the deviation between the GE-ECM and CST was the smallest. Although the difference was larger at 18–20 GHz than at lower frequencies, the RMSE verified the overall stability. In this context, it is also worth noting that the RMSE of the GE-ECM was the smallest among the three models—only 0.0043— considerably smaller than the 0.0472 and 0.3469 attained by the AE-ECM and ECM, respectively. A comparison of the bandwidth of transmittance is demonstrated in Fig. 3(b). Compared to existing methods, the proposed model shows only the slightest bandwidth errors with respect to the CST, thus verifying the accuracy of the GE-ECM in broadband.
2. Multi-Size Flexibility
To verify the multi-size flexibility of the proposed GE-ECM, the REs obtained for the resonant frequency when considering different size ratios are illustrated in Fig. 4(a), with the different colors representing different error values. Compared to the classic ECM and AE-ECM, the colors for the GE-ECM are more uniform, with fewer mutations, demonstrating a large blue area. In contrast, the colors for the ECM and AE-ECM show large regional inconsistencies and obvious green areas. Notably, the blue color indicates small errors, while green refers to large errors. In particular, the average REs of the GE-ECM, AE-ECM, and ECM were 1.19%, 2.36%, and 10.96%, respectively, consistent with the colors of the three models in Fig. 4(a). Furthermore, in the case of specific g and d values, with 0 < d1 < 0.18 and 0 < g1 < 0.15, Fig. 4(a) shows that the errors are higher than the other values. The small width of the square loop and loop gap can cause strong electromagnetic coupling between adjacent square loops, which is difficult to describe and predict using parametric physical models. Therefore, although the maximum RE attained by the GE-ECM for a specific size was 11.56%, it still emerged as the most accurate model.
To further verify the universality of the GE-ECM, SG FSSs with 10 mm, 12 mm, 14 mm, 15 mm, and 18 mm periods were considered to examine the stability and flexibility of the structure at different periods. As shown in Fig. 4(b), the frequency responses of the GE-ECM are closer to those of the CST than the AE-ECM for various periods. This implies that the proposed model is suitable not only for analyzing SG FSSs with fixed sizes but also for analyzing multi-size and multi-period SG FSSs.
3. Feasibility in Different Kinds of FSSs
FSS patterns usually feature different sizes, rotation angles, and SG materials. Thus, ensuring the feasibility of GE-ECM for complex FSSs is important. Therefore, a square loop FSS, which is composed of four strips, was considered in this study. Its equivalent circuit formulas had to be rewritten based on the optimized strip formulas, as follows [19, 21]:
The L and C values of a square loop FSS are related to many parameters, including the period of unit, width of loop, gap length of square loop, wavelength, incident angle, and impedance in free space. Therefore, the L and C values can be changed by adjusting any parameter, including the resonant frequency, frequency response curve, and so on. Tables 3 and 4 list the WF and GF values of the square loop FSS with p = 10 mm. Furthermore, the REs of the resonant frequency for the square loop FSS with p = 10 mm are illustrated in Fig. 5. It is observed that the REs are generally quite small and stable. In particular, the average relative errors of the GE-ECM and ECM were 3.56% and 5.32%, respectively, which was substantially improved by the proposed model. Moreover, when considering small values of g, the colors are abundant, meaning that the errors are relatively large at these stages. This phenomenon can be attributed to the unpredictable resonance of small-sized structures. However, although errors exist, the feasibility of using the proposed GE-ECM for analyzing square loop FSS is considered acceptable, illustrating that the GE-ECM can be extended for use in more complex structures.
4. Stability in the Presence of Wide Incident Angles
The transmittances of different oblique incidences in the TE and TM polarization modes by the GE-ECM, AE-ECM, and CST are presented in Fig. 6. It is observed that as the oblique incidence increases from 0° to 36°, the resonant frequency errors increase gradually. However, at high frequencies, the transmittances decrease, which can be attributed to the fact that the direction of the magnetic field is no longer parallel to the FSS plane at oblique incidence. Despite some differences, the frequency response trends of the GE-ECM is closer to those of the CST than the AE-ECM, exhibiting only slight deviations. Overall, the feasibility of oblique incidence in both the TE and TM polarization modes further proves the high efficiency and universality of the GE-ECM.
5. Practicability for Real FSS
To further verify the practicality of the proposed model, a typical square loop FSS was fabricated and measured in a professional microwave darkroom. The upper and lower layers of the FSS were made of copper, and the intermediate medium was FR4. The period, loop width, and gap width of the square loop were set to 10 mm, 2 mm, and 1 mm, respectively. Copper metal was used to cover the bottom layer. The experimental results obtained using the CST, GE-ECM, and the measurement results are shown in Fig. 7, showing resonant frequencies of 4.95 GHz, 5.1 GHz, and 4.97 GHz, respectively, along with slight deviations. At a reflection coefficient of 0.95, the bandwidths of the three curves are 1.7 GHz, 1.8 GHz, and 1.1 GHz, respectively. The bandwidth difference between CST and the proposed model is only 0.1 GHz, while that between CST and the measured results is 0.6 GHz, caused by slight unavoidable errors during fabrication and measurement. It is evident that, although there are slight differences in the overall frequency response curves, a high consistency is still maintained. In addition, although there are some inevitable fluctuations in insertion loss near the resonant frequencies, the overall average value of insertion loss is only 0.04, which meets the requirements for practical applications. Therefore, the feasibility of using the proposed model in practical applications is verified.
IV. Conclusion
In this work, a GE-ECM for analyzing both SG FSS and more complex FSSs is proposed. Furthermore, theoretical analysis, model design, and experiments are conducted for the GE-ECM’s related aspects, ultimately arriving at the following conclusions:
• It was found that nonlinear electron motion in adjacent structures causes difficulties in EM characterization. The proposed GE-ECM compensates for nonlinear coupling by enhancing physical models through the construction of a more adaptive and accurate mapping relationship between metasurfaces and EM behavior.
• Two enhancement factors—GF and WF—are introduced in the proposed GE-ECM to compensate for irregular coupling electrons. The GF is added to the formula for impedance calculation, while the WF is incorporated to develop a more reasonable formula for admittance expression. Furthermore, the normalization operation served to generalize the application scope of the GE-ECM.
• Experiments were conducted on structures with various parameters using the GE-ECM and other contrasting models. The experimental results showed that the GE-ECM maintains close similarity with the CST in terms of multi-size flexibility and broadband from 0 GHz to 20 GHz. The experiments also showed that the GE-ECM is feasible for oblique incidence (0°–36°). The RE and RMSE results further verified the advantages and accuracy of the GE-ECM. Overall, the experimental results verified the stable performance of GE-ECM for multi-size, broadband, and large oblique incidence FSSs.
• This paper not only constructs a generalized and efficient physical model for EM representation but also provides a novel strategy for FSS design, thus offering guidance for the efficient design of more complex metasurfaces.
Fig. 1
Metal strips and ECM: (a) formation of inductive strips, (b) ECM of inductive strips, (c) formation of capacitive strips, (d) ECM of capacitive strips, (e) schematic of SG FSS, and (f) ECM of SG FSS.
Fig. 2
Surface current of metal strips: (a) current of vertical SG FSS, (b) current of vertical infinite strips, (c) current of horizontal SG FSS, and (d) current of horizontal infinite strips.
Fig. 3
Comparison of the transmittance of fixed-scale SG FSS: (a) comparison of resonant frequencies and frequency curves and (b) comparison of bandwidth.
Fig. 4
Comparison of relative errors of the resonant frequency for different sizes of SG FSS: (a) multi-size SG FSS and (b) multi-period SG FSS.
Fig. 6
Frequency responses of different oblique incidences: (a) TE polarization mode and (b) TM polarization mode.
Table 1
Optimal values of δg
Table 2
Optimal values of δg
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