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Equivalent circuit model for array of circular loop FSS structures at oblique angles of incidence

2017; Institution of Engineering and Technology; Volume: 12; Issue: 5 Linguagem: Inglês

10.1049/iet-map.2017.1004

1751-8733

Ali Ramezani Varkani, Zaker Hossein Firouzeh, Abolghasem Zeidaabadi Nezhad,

Antenna Design and Optimization

IET Microwaves, Antennas & PropagationVolume 12, Issue 5 p. 749-755 Research ArticleFree Access Equivalent circuit model for array of circular loop FSS structures at oblique angles of incidence Ali Ramezani Varkani, Corresponding Author Ali Ramezani Varkani a.ramezani@ec.iut.ac.ir Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, 84156–83111 IranSearch for more papers by this authorZaker Hossein Firouzeh, Zaker Hossein Firouzeh orcid.org/0000-0002-0064-0150 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, 84156–83111 IranSearch for more papers by this authorAbolghasem Zeidaabadi Nezhad, Abolghasem Zeidaabadi Nezhad Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, 84156–83111 IranSearch for more papers by this author Ali Ramezani Varkani, Corresponding Author Ali Ramezani Varkani a.ramezani@ec.iut.ac.ir Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, 84156–83111 IranSearch for more papers by this authorZaker Hossein Firouzeh, Zaker Hossein Firouzeh orcid.org/0000-0002-0064-0150 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, 84156–83111 IranSearch for more papers by this authorAbolghasem Zeidaabadi Nezhad, Abolghasem Zeidaabadi Nezhad Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, 84156–83111 IranSearch for more papers by this author First published: 16 February 2018 https://doi.org/10.1049/iet-map.2017.1004Citations: 31AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study aims to introduce a simple equivalent circuit model of circular loop shaped frequency selective surface (FSS), predicting the plane-wave transmission characteristics for oblique angles of incidence. The equivalent circuit model is based on a series of basic equations in order to calculate the inductance and capacitance of strip gratings. Through provided relations, the values of circuit elements can be calculated as well. The proposed relations are applied to many circular loop FSSs with different dimensions, and the results are compared with the full-wave simulations obtained from computer simulation technology microwave studio. Moreover, some comparison graphs between the simulated and analytical resonant frequency for different values of design parameters have been carried out to show the circuit model follows the trend of the simulated responses. Finally, both the FSS structures have been fabricated and measured in an anechoic chamber. A good agreement between the experimental results and the simulated responses demonstrates the validity of the circuit model. 1 Introduction Frequency selective surfaces (FSSs) have recently attracted the researchers’ attention and pave the way for significant research interests due to their variety in terms of electromagnetic applications, such as electromagnetic absorbers, filters, and antennas [[1]]. FSSs constitute periodic structures with either patch or aperture elements implemented within a two-dimensional periodic grid, which contains frequency filtering properties [[2], [3]]. As to the FSS element shapes, the most common types involve straight dipole, cross dipole, Jerusalem cross, circular loop, and square loop [[2]]. As the literature previously demonstrated, circular loop elements for FSS are considered to be less sensitive to incident angles than many other elements shaped differently, such as crossed dipoles, tri-poles, and so on [[4]]. In order to analyse the electromagnetic behaviour of FSSs, the most common and available method is to take the rigorous full-wave technique into account, which is based on several mathematical approaches (finite element method, finite difference time domain, method of moment) [[5], [6]]. Although the above-mentioned techniques are accurate in analysis they require time-consuming simulations not allowing the designer to gain a good insight regarding the physics of the structures. On the other hand, the equivalent circuit model (ECM) is fairly simple and fast [[7], [8]]. However, this is only limited to a very few simple FSS structures [[9]–[11]]. It will be very difficult to derive an ECM for complicated structures [[12]–[14]]. By applying this method, the array of FSS can be modelled to the capacitances and inductances. The frequency characteristics of these structures can be determined based on LC resonant circuits [[15]]. Models have been found to previously be available for the Jerusalem cross [[16]], single and double-square loops [[17], [18]]. Hence, this study aims to present ECM of single and double circular loop (DCL) FSSs for the first time. Moreover, the study looks into the effect of the FSS shape (in terms of the length, width, permittivity, thickness, and unit cell period) on the frequency characteristics of FSS structures. 2 Circular loop FSS and its ECM Here, the focus is on a FSS structure including a two-dimensional periodic array of circular loops. Fig. 1 illustrates the geometry of the FSS which is under investigation. In each unit cell, there is a printed circular conducting loop. A thin layer of substrate is implemented for facilitating the fabrication of the FSS in order to print the conducting loops. The thickness and relative permittivity of the substrate are h and ɛ, respectively, and they should be as small as possible. Table 1 shows the parameters of circular loop FSS. The ECM of circular loop FSS is depicted in Fig. 1c, including a LC circuit where L and C are represented by conducting elements and inter-element spacing, respectively. Hence, the resonant frequency is given by [[8]] (1) Fig. 1Open in figure viewerPowerPoint Geometry of the circular loop FSS and its ECM (a) Perspective view, (b) Top view, (c) ECM Table 1. Parameters of the circular loop FSS Parameter Description p unit cell size g minimum gap between the circle in two adjacent unit cell d outer diameter of the circle w copper layer width t copper layer thickness h thickness of the substrate The difference between the upper and lower frequency responses at −10 dB represents the fractional bandwidth [[19]] (2) The circuit model at oblique angles of incidence requires expressions of inductance and capacitance for both transverse electric (TE) and transverse magnetic (TM) incidence [[20]]. Using the geometry of Fig. 2, the basic equations for calculating the inductance and capacitance of strip gratings with periodicity, p, width, w, and spaced a distance, g, apart at oblique angles of incidence are found in [[21]], and the general forms are given by (3) (4) (5) (6) where and are the angles of incidence, λ is the wavelength, and G is the correction term stated in [[20]]. By varying and , it allows incidence of the plane waves at arbitrary angles. Langley and Parker [[17]] have extracted the lumped-element ECM for the square-loop array. Accordingly, a lumped-element ECM can be obtained for the circular loop array. Fig. 2Open in figure viewerPowerPoint Plane wave incidence upon an inductive strip grating [[21]]. For a capacitive strip grating, replace the incident electric field (E) with a magnetic field (H) The inductance value is determined by the width and length of the metallic strip. Therefore, the reactance X1 of the inductance L1 for TE and TM incidence is as follows: (7) (8) where a factor π/4 due to the environment of the circular loop (πd) compared with square loop environment (4d) and the reactance is reduced by a factor d/p owing to non-continuous feature of the conductor. The capacitance between two adjacent circular loops can be modelled to a pair of parallel plate capacitance. The capacitance value is determined by the average gap width between the two adjacent loops, the half loop length, and the effective dielectric constant of the substrate. Therefore, the susceptance B1 of the capacitance C1 for TE and TM incidence is given by (9) (10) 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), is the effective permittivity of the substrate, and is the average gap between two adjacent unit cells and can be calculated as (11) Regarding the confirmation of the validity of the proposed ECM for the circular loop FSS, the simulated reflectivity and transmission performance of a circular loop FSS are compared by the circuit model with the full-wave simulator computer simulation technology (CST), as demonstrated in Fig. 3. The values of lumped inductance and capacitance (L, C) in the circuit model are then computed from the FSS dimensions (p, w, d, and g) and the thickness and relative permittivity of the substrate based on (7)–(10). The CST results as shown in Fig. 3 are based on the values including: p = 20 mm, w = 1 mm, d = 19 mm, and g = 1 mm. The corresponding values of the lumped inductance and capacitance in the ECM are L = 6.56 nH and C = 0.176 pF. Fig. 3Open in figure viewerPowerPoint Reflectivity and transmission response of the FSS calculated from the ECM and CST (a) Reflectivity response, (b) Transmission response It can be observed that the results of the circuit model and CST simulations are in good agreement. As Fig. 3 depicts, the designed circular loop FSS is a band stop filter. The structure reflects signals at resonant frequency, i.e. 4.7 GHz, but it transmits the signal at lower or higher frequencies than this frequency. The difference between the lower and upper frequencies at S-parameter of −10 dB divided by the resonant frequency defines the relative bandwidth of transmission response. In this regard, the upper and lower frequency at −10 dB transmission response are 5.6 and 3.9 GHz, thus the bandwidth is almost 36%. Table 2 summarises the comparison between the resonant frequencies predicted by the proposed model as well as those simulated using CST for 12 arrays. The thickness and relative permittivity of the substrate for all cases are 0.508 mm and 2.2, respectively. It can be seen that there was a close agreement at the resonant frequencies. The resonant frequencies are adequately predicted (to within 3%) in all 12 cases by the equivalent circuit. Table 2. Simulated and calculated results for different arrays of circular loop FSSs Dimensions Resonant frequency (normal-wave incident) Array p w d g CST simulation Equivalent circuit No. Unit mm mm mm mm GHz GHz 1 20 1 19 1 4.71 4.69 2 20 1 17 3 6.11 5.94 3 20 3 19 1 6.61 6.82 4 12 0.75 11.25 0.75 8.25 8.22 5 12 0.75 10.5 1.5 9.78 9.52 6 12 1.5 11.25 0.75 10.17 10.4 7 6 0.5 5.75 0.25 15.84 16.26 8 6 0.5 5 1 22.17 21.7 9 6 1 5.75 0.25 21.45 21.77 10 3 0.25 2.75 0.25 34.77 34.74 11 3 0.25 2.5 0.5 41.67 42.41 12 3 0.5 2.75 0.25 44.88 45.97 3 DCL FSS and its ECM Fig. 4a indicates the geometry of the DCL FSS. There are two printed circular conducting loops in each unit cell. The parameters of DCL FSS are shown in Fig. 4a. The lumped-element ECM for the DCL array can be obtained and compared to the results by the full-wave simulator. Fig. 4b demonstrates the ECM derived for the array of DCLs FSS consisting of two series circuits in parallel. The admittance of the DCL-FSS is calculated using the equivalent circuit as follows: (12) Fig. 4Open in figure viewerPowerPoint Unit cell of the double circular loop FSS and its ECM (a) Geometry of FSS, (b) ECM Langley and Parker [[18]] have extracted the lumped-element ECM for the double square-loop array. Accordingly, a lumped-element ECM can be obtained for the double circular loop array. The four circuit elements in Fig. 4b are given by (13) (14) (15) (16) where the multiplication factor is obtained based on geometrical considerations and , the average gap between two adjacent unit cells, can be calculated as (17) The normalised admittance, Y, of an array at oblique incidence can currently be calculated, and the transmission coefficient τ can be determined as follows [[22]]: (18) As to the verification of the validity of the ECM for the proposed DCL-FSS, the comparison of the calculated transmission performance of the DCL-FSS can be done based on the parameter values shown in the first row of Table 3. The results of the circuit model are compared to the full-wave simulator, CST microwave studio, as depicted in Fig. 5. The simulation results of the ECM, as indicated in Fig. 4, are obtained through using the values: L1 = 3.46 nH, L2 = 1.41 nH, C1 = 0.019 pF, and C2 = 0.011 pF. Fig. 5Open in figure viewerPowerPoint Transmission response of the DCL FSS calculated from the ECM and CST Table 3. Simulated and calculated results for different array of DCL FSSs No. Dimension, mm Resonant frequency, GHz p d1 g1 w1 d2 g2 w2 f1 CST f1 ECM f2 CST f2 ECM 1 6 5 1 0.5 3 0.5 0.5 19.52 19.68 39.60 39.72 2 6 5.5 0.5 0.5 3 0.75 0.5 16.76 16.14 39.76 38.52 3 6 5.5 0.5 0.25 3 1 0.5 15.08 15.17 39.28 39.56 4 10 9 1 0.5 6 1 0.5 9.61 9.74 18.52 17.71 5 10 9.5 0.5 0.5 6 1.25 0.5 8.51 8.56 18.35 17.21 6 10 9 1 0.25 6 1.25 0.5 9.07 9.25 18.38 18.12 7 10 9 1 1 6 0.5 1 10.72 10.93 22.00 20.91 8 15 14 1 1 8 2 1 6.46 6.33 15.17 14.33 9 15 13 2 1 8 1.5 1 7.43 7.53 15.22 14.92 10 15 14 1 0.5 8 2.5 1 5.89 6 15.02 14.84 The results are found to be in good agreement over the whole frequency range. A comparison between the resonant frequencies predicted by this model and those simulated using CST for ten arrays are summarised in Table 3. It can be seen that there was a close agreement at the first resonant frequency. The model for the second resonant frequency is less precise, but nevertheless predicts f2 to within about 5%. 4 Effect of design parameters This section illustrates the comparison graphs between the simulated and analytical resonant frequency for different values of design parameters to show whether the circuit model follows the trend of the simulated responses or not. There are various parameters including the substrate thickness, h, dielectric constant, , outer diameter, d, width of the circle, w, and gap between two elements, g, through which significant characteristics of the circular loop FSS can be determined. All the simulations are performed using the CST microwave studio software and the results of the proposed ECM are compared with the CST ones to show the validity of the proposed approach. 4.1 Dielectric effects The equivalent capacitance of a FSS printed on a dielectric substrate is modified in proportion to the dielectric permittivity and the thickness of the supporting dielectric. This equivalent capacitance in the presence of a thick dielectric substrate on one side of the FSS is increased by a factor equal to [[23]]. For thin dielectric substrates, the capacitance variation is simulated as a function of the thickness and relative permittivity of the substrate. Through an iterative procedure, the optimum capacitance values as a function of the dielectric thickness was calculated and normalised to the values provided by free standing FSS [[24]] (19) where h is the thickness of the substrate, p is the size of the unit cell, and N is an exponential factor that varies from 1.3 to 1.8 for different cell shapes in terms of the unit cell filling factor. It can be shown that the increase of the filling factor decreases the exponential factor. Concerning to the circular loop array, the right value is chosen N = 1.8. The comparison graphs between the simulated and analytical effective permittivity, for different values of thickness and relative permittivity of the substrate are shown in Fig. 6. The analytical effective permittivity are obtained by (19). It can be observed that the results of the circuit model and CST simulations are in good agreement. Fig. 6Open in figure viewerPowerPoint Effective permittivity as a function of thickness and relative permittivity of the substrate with the geometrical parameters p = 10 mm, w = 0.5 mm, d = 9 mm (a) ɛr = 4.3, h = variable, (b) h = 0.127 mm, ɛr = variable 4.2 Change of the element outer diameter A comparison between the resonant frequency of a circular loop FSS in terms of the circle outer diameter obtained by the ECM and by the CST software is illustrated in Fig. 7. Change in the outer diameter is in line with the change in the gap between copper and the unit cell. Outer diameter increase is in alignment with the increase of both the inductance and capacitance. Capacitance and inductance increase, leading to the lower resonant frequency. According to Fig. 7, it can be inferred that increasing the element outer diameter from 7.5 to 9.8 mm leads to the reduction of the resonant frequency from 14.1 to 7.7 GHz. The resonant frequency of d = 7.5 mm is almost twice that of d = 9.8 mm. The results of the circuit model compared with the results obtained by the CST are in good agreement. Fig. 7Open in figure viewerPowerPoint Comparison between the resonant frequency of a circular loop FSS as a function of circle outer diameter with the geometrical parameters: p = 10mm, w = 0.5mm; substrate: ɛr = 4.3, h = 0.127 mm 4.3 Change of the copper width Fig. 8 indicates the changes in resonant frequency because of the copper width w change. All the parameters are observed to be fixed except the element width. In (7), it was seen that inductance is proportional to unit cell size and is inversely proportional to copper width. Fig. 8Open in figure viewerPowerPoint Comparison between the resonant frequency of a circular loop FSS as a function of circle width with the geometrical parameters p = 10mm, d = 9mm; substrate: ɛr = 4.3, h = 0.127 mm Hence, as the copper width increases, the inductance decreases but the capacitance does not vary. Thus, the resonant frequency will be higher, meanwhile the bandwidth becomes wider. As shown in Fig. 8, the results of the ECM compared with those of CST software are in good agreement, up to 0.05p < w < 0.2p. 4.4 Change of the gap between unit cells A comparison between the resonant frequency of a circular loop FSS as a function of the gap between adjacent elements obtained by the ECM and by the CST software is shown in Fig. 9. For the same loops, although the element spacing is increased, the periodicity of the array is increased, but the capacitance is decreased. Therefore, the gap increment results in higher resonant frequency. As shown in Fig. 9, the results of the ECM and those obtained by the CST software agree very well, up to . Fig. 9Open in figure viewerPowerPoint Comparison between the resonant frequency of a circular loop FSS as a function of inter-elements gap with the geometrical parameters d = 9mm, w = 0.5mm; substrate: ɛr = 4.3, h = 0.127 mm 4.5 Oblique incidence The analytical expressions for the circular loop array's inductance and capacitance are valid for both normal and oblique incidence. As it has been previously presented in the literature [[4], [5]], the circular loop elements for FSS are less sensitive to incident angles than many other elements shaped differently, such as crossed dipoles, tri-poles, and so on. Consequently, their applications are preferred which demands a large variation of incident angles. A comparison between the transmission coefficient of the circular loop FSS obtained by CST and the circuit-model approaches at oblique incidence is shown in Fig. 10. The curves revealed the approximation of the model for oblique incidence was good up to the propagation of grating lobes. It can be found that for TE mode, the increase in the incident angle results in resonant frequency decrease and the bandwidth increase and for TM mode as incident angle increases, the resonant frequency increases and the bandwidth decreases. Fig. 10Open in figure viewerPowerPoint Comparison between the transmission response of the circular loop FSS at different incidence angles for TE and TM modes obtained with the geometrical parameters p = 10 mm, d = 9 mm, w = 0.5 mm substrate: ɛr = 4.3, h = 0.127mm (a) Incident angle is 30°, (b) Incident angle is 60° 5 Experimental verification To verify the validity of the circuit model responses with the measured results, two 20 × 20 unit cells FSS prototypes have been manufactured using the standard photo-lithographic techniques: The first one is a single circular loop FSS (see Fig. 11a), and the second one is a DCL FSS (see Fig. 11b). The photographs of the both prototypes are shown in Fig. 11. The substrates used for both of them are FR4 with a thickness of 0.127 mm and a dielectric constant of 4.3 with the overall dimension of 200 mm × 200 mm. The dimensions of the circular loop and DCL FSS are: p = 10 mm, d = 9 mm, w = 0.5 mm, and p = 10 mm, d1 = 9.5 mm, w1 = w2 = 0.5 mm, d2 = 7.5 mm, respectively. The transmission coefficient of the FSS prototypes was measured in the anechoic chamber. Fig. 11Open in figure viewerPowerPoint Photographs of the FSS structures with the substrate of ɛr = 4.3, h = 0.127 mm (a) Circular loop FSS, (b) DCL FSS The measurement was performed by two steps: The transmission coefficient measurement without the FSS structure. The transmission coefficient measurement with the FSS structure. Then, the transmission coefficient of the FSS structure was normalised to the transmission coefficient measurement without the FSS structure. The measured results and the circuit model responses of the FSS prototypes for normal-wave incident are compared in Fig. 12, where a good agreement between the results is observed. The minor discrepancies between the results can mostly be attributed to fabrication tolerances. Fig. 12Open in figure viewerPowerPoint Measured and ECM transmission coefficients of the both FSS prototypes (a) Circular loop FSS, (b) DCL FSS 6 Conclusion This study aimed to propose a simple ECM of circular loop FSS structures predicting the plane-wave transmission characteristics for oblique incidence. The model can be used for the circular loop FSS structures in order to accurately predict the resonant frequency. The effect of parameter variation based on the ECM was also taken into account. Moreover, the angular stability under variation of incidence angles for TE- and TM- polarised waves was investigated. It was also found that the proposed ECM model can predict the characteristics of circular loop FSS even at oblique angle of incidence. Finally, both the FSS structures have been fabricated and measured in an anechoic chamber. The results obtained by simulation and measurement are in good agreement with the validity of the proposed circuit model. 7 References [1]Ghosh, S., Bhattacharyya, S., Srivastava, K.V.: ‘Design, characterisation and fabrication of a broadband polarisation-insensitive multi-layer circuit analogue absorber’, IET Microw. Antennas Propag., 2016, 10, (8), pp. 850– 855 [2]Munk, B.A.: ‘ Frequency selective surfaces – theory and design’ ( Wiley, New York, NY, USA, 2000) [3]Luukkonen, O., Costa, F., Simovski, C.R., et al.: ‘A thin electromagnetic absorber for wide incidence angles and both polarizations’, IEEE Trans. Antennas Propag., 2009, 57, (10), pp. 3119– 3125 [4]Parker, E.A., Hamdy, S.M.A.: ‘Rings as elements for frequency selective surfaces’, Electron. 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