Unitcell design

The topology of the proposed single layer FSS is illustrated in Fig. 1. The low-profile dual band FSS consisting of unitcell made of MDSL is printed on the flexible/conformable PI material with an overall thickness of 0.1 mm, dielectric constant of 3.5, and loss tangent value of 0.06. The proposed MDSL is evolved from the double square loop (DSL) design by convoluting the inner Square inwards bending c-shaped stub structure which is designed on all sides of the square loop.

Fig. 1. Unitcell element of the proposed design.

MDSL equivalent circuit model

The equivalent circuit model (ECM) is used to analyze the proposed design of the MDSL FSS structure. This structure consists of DSL with C-stub which is designed to provide dual band frequency response.

Consider, the normalized inductive impedance expression of strip grating which were given by Marcuvitz as [11–15]

(1)$$X_{TE} = F( {\,p, \;w, \;\lambda } ) = \displaystyle{{\,p\;\cos \theta } \over \lambda }\;\left[{\ln {\rm cosec}\;\left({\displaystyle{{w\Pi } \over {2p}}} \right) + G( p, \;w, \;\lambda , \;\Theta ) } \right], \;$$

where p represents the periodicity of the proposed structure, w is the width of the inductive strip (MDSL), λ is the wavelength, X _TE, the inductive impedance of TE mode, and G(p, w, λ Θ,) is the correction factor.

From the Babinet's duality condition, the susceptance for TM-incidence can be found

(2)$$B_{TM} = F( {\,p, \;w, \;\lambda } ) = 4\displaystyle{{\,p\;\cos \Phi } \over \lambda }\left[{{\rm ln\ cosec}\;\left({\displaystyle{{{\rm g\Pi }} \over {2{\rm p}}}} \right) + G( {\,p, \;g, \;\lambda , \;\Phi } ) } \right]$$

Equations (1) and (2) are valid only if $w\ll p, \;d\ll p$, and p ≪ λ. Similarly, the inductance of the TM-incidence and the capacitance of the TE-incidence can be written as follows:

(3)$$X_{TM} = F( {\,p, \;w, \;\lambda } ) = \displaystyle{{\,p\;\sec \theta } \over \lambda }\;\left[{{\rm ln\ cosec}\;\left({\displaystyle{{w\Pi } \over {2p}}} \right) + G( p, \;w, \;\lambda , \;\Theta ) } \right], \;$$

(4)$$B_{TE} = F( {p, \;w, \;\lambda } ) = 4\displaystyle{{\,p\;\sec \Phi } \over \lambda }\left[{\ln {\rm cosec}\;\left({\displaystyle{{g\Pi } \over {2p}}} \right) + G( p, \;g, \;\lambda , \;\Phi ) } \right]$$

Here the G factor is neglected due to minor deviation occurrence. The given FSS structure is to change the response when the angle of incidence of the wave is changed and to avoid the grating lobes, equation (5) periodicity is related to the wavelength equation that is used in [Reference Jha, Singh and Jyoti16, Reference Reed17].

(5)$$p( 1 + \sin \,\Theta ) < \lambda $$

In general, the above equation for TE mode is also applicable for TM mode because of the square loop structure, the polarization-independent character is occurred and achieved the same response for TE and TM mode and for angle of incidence (Θ,Φ) can be used, respectively.

From the above basic equations, the proposed structure inductance and susceptance values are represented as,

(6)$$X_{L1} = \displaystyle{\,p \over {d1}}\;F ( p, \;2w1, \;\lambda 1) , \;$$

(7)$$X_{L2} = \displaystyle{\,p \over {d2}}\;F( p, \;2w2, \;\lambda 2) , \;$$

(8)$$B_{C1} = 4\displaystyle{\,p \over {d1}}\;F ( p, \;g1, \;\;\lambda 1) , \;$$

(9)$$B_{C2} = 4\displaystyle{\,p \over {d2}}F( p, \;g2, \;\;\lambda 2).$$

In Fig. 2, the L ₁ and C ₁ are used for outer square loop structure and the frequency response is calculated using fr ₁ value, and the inner square loop with C-stub FSS structure having series of combination of L ₂, L ₃, and C ₂. The L ₃ for C-stub is added as the inductance value and is used to achieve the desired frequency response. Multiple simulations have been done to find the L and C values using ADS tool. Since the thickness of the substrate is very small the effect of the transmission line is neglected.

Fig. 2. Equivalent circuit of the modified double square loop structure.

Where Z _FSS, the impedance of the proposed structure and the corresponding resonant frequency of fr ₁ and fr ₂ are calculated using the numerator of equation (11) equals to zero.

(10)$$Z_{{\rm FSS}} = \left[{\displaystyle{{1-\omega 2L1C1} \over {\,j\omega C1}}} \right] + \left[{\,j\omega L2 + j\omega L3 + \displaystyle{1 \over {\,j\omega C2}}} \right], \;$$

(11)$$Z_{{\rm FSS}} = \displaystyle{{1-\omega 2L1C1} \over {\,j\omega C1}} + \displaystyle{{1-\omega 2C2( {L2 + L3} ) } \over {\,j\omega C2}}, \;$$

(12)$$fr_1 = \displaystyle{1 \over {2\Pi \sqrt {L1C1} }}, \;$$

(13)$$fr_2 = \displaystyle{1 \over {2\Pi }}\;\sqrt {\displaystyle{1 \over {( {L2 + L3} ) C2}}}.$$

Finally, the admittance of the MDSL design is calculated for more compact form in equation (14) by using the equivalent circuit seen in Fig. 2,

(14)$$Y = j\left[{\displaystyle{{B1} \over {1-X1B1}} + \displaystyle{{B2} \over {1-B2( {X2 + X3} ) }}} \right].$$

The optimization parameter of the proposed structure is set as follows: 2.5 ≤ p ≤7.67 mm, 1.3 ≤ d1 ≤6.0 mm, 3.57 ≤ d2 ≤ 4.36 mm, 2.6 ≤ d3 ≤ 4.16 mm, 0.1 ≤ w1 ≤ 2.0 mm, 0.1 ≤ w2 ≤ 2.0 mm, 0.1 ≤ w3 ≤ 2.0 mm, 0.2 ≤ g1 ≤ 2.26 mm, 0.2 ≤ g2 ≤ 2.0 mm, 0.02 ≤ g3 ≤ 0.6 mm. The obtained parameters are p = 5.4 mm, d1 = 5.0 mm, d2 = 4.2 mm, d3 = 3.8 mm, w1 = 0.4 mm, w2 = 0.4 mm, w3 = 0.4 mm, g1 = g2 = g3 = 0.4 mm. The transmission characteristics of both computed and optimized results are in good agreement and are shown in Fig. 3. The resonant frequencies of the simulated bandstop response achieved at 9.4 GHz and 16.7 GHz using CST tool and the measured result achieved 9.42 and 16.72 GHz using ADS tool. Table 1 represents the LC values of the proposed structure.

Fig. 3. Comparison of ECM result and simulation result.

Table 1. LC Values of the proposed design.

Simulation results

Using a commercially available CST microwave studio software, the proposed single layer PI material-based FSS is designed and simulated, and the simulation was done in frequency domain solver. A single layer PI substrate is used with a thickness of 0.1 mm and dielectric constant of 3.3. The transmission coefficient (bandstop) and reflection coefficient (band pass) of the proposed MDSL FSS simulation result are shown in Fig. 4. The transmission coefficient (S ₁₂) result provides dual bandstop response at X-band 9.4 GHz and Ku-band 16.7 GHz at 20 dB attenuation and it provides 924 MHz and 1.34 GHz bandwidth, respectively. The corresponding reflection coefficient (S ₁₁) provides band pass in transmission coefficient frequency response. Figure 5, inference the same frequency response for both TE and TM modes because of the identical unitcell design and it enables the polarization-independent characteristic. Thus, the proposed conformal FSS design provides dual band frequency response that is used for effective shielding applications.

Fig. 4. S-Parameters of the proposed FSS unitcell.

Fig. 5. Transmission characteristics of both TE and TM mode.

Effect of changing angle of incidence

The performance of the proposed FSS under oblique incidence for TE and TM modes upto 60° angle of incidence is presented in Figs 6 and 7. C-Stub or shaped analyze are mainly used for angular stability [Reference Parker and El Sheikh18]. It is observed that the bandstop response is changed depending on the propagation constant of TE and TM mode. In Fig. 6, the transmission coefficient of varying an angle of incidence upto 60° for TE Mode is shown, there is a minute frequency change that has occurred at 60° and the other angles are the same. In Fig. 7, the transmission coefficient of varying an incidence angle for TM mode is shown. It is observed that the frequency response is shifted little high at 30° and the other angles are the same. So, from the effect of changing the incidence angles the frequency response is slightly changed but the changes are within the limits so the overall FSS is stable for both TE and TM modes and the comparison of bandwidth are detailed in Table 2.

Fig. 6. Effect of changing different angle of incidence on TE mode response.

Fig. 7. Effect of changing different angle of incidence on TM mode response.

Table 2. Bandwidth response for TE and TM modes.

Effect of changing materials

The performance of the proposed FSS is tested for different materials using three different materials which are chosen, FR4, Silicon, and PI material. In Fig. 8, the silicon substrate has a high dielectric constant (11.9) compared to others and the second highest is FR4 (4.3) and the final one is PI material (3.3). From Fig. 6, it is observed that the bandstop frequency is shifted towards the lower frequencies in both silicon and Fr4 material because of the dielectric constant. If the dielectric constant increases, the frequency will decrease. This is because of the electrical length “l” and dielectric constant “ɛeff” equation $( l\;\propto 1/\sqrt {\varepsilon eff} )$. So, from this, the PI material achieves exact dual band frequency at X- and Ku-band response with high transmission coefficient.

Fig. 8. Effect of changing substrate material.

Effect of changing substrate thickness

The parametric study has been done for the proposed design using the substrate thickness. By varying the thickness of the substrate, the frequency response will change accordingly. From Fig. 9, it is observed that the proposed design with 0.1 mm thickness has achieved high bandwidth and good transmission coefficient. While increasing the thickness of the substrate the frequency shifts to lower response. For conformal application, the lower substrate thickness is used.

Fig. 9. Effect of changing substrate thickness.

Bending analysis

For various applications like parabolic reflector and aerospace, the conformal behavior is analyzed. To test the flexibility/conformability of the proposed design the bending analysis has been done with different scenarios using CST software and is shown in Fig. 10. When the curvature (bent) increases, the surface area of the unitcell also increases and the EM field exposes more. So, depending on the curvature the operating frequency of the bent analysis is varied and it is evident that the proposed FSS structure provides negligible shift at its operating frequencies for X-bent and Y-bent. So, the proposed design compared the results of X and Y-bent analysis with no bent results, hence it provides the same response. So, it is able to use for conformal applications.

Fig. 10. Bending analysis for testing conformability.

Measurement setup

For the verification of the simulation results, the proposed design is fabricated and tested and the overall dimension of the fabricated prototype is 35cm × 35cm and is shown in Fig. 11. The fabricated prototype is tested using an experimental setup as shown in Fig. 12. The proposed FSS is fabricated on the flexible PI material with a thickness of 0.1 mm with 3.5 dielectric constant. The experimental setup consists of two pairs of horn antennas, one to measure the frequency between 1 and 12 GHz (JR-12) and the other to measure the frequency between 12 and 18 GHz (KU5086). Generally, the horn antenna used to transmit and receive the signal and the antennas connected through the Agilent technologies (VNA) N9917A which is used in-between the anechoic chamber stand is used to place the fabricated prototype and to avoid distortion. Initially, the measurement is carried without a fabricated prototype and then the measurement is carried with FSS prototype. Comparison of measured and simulation results are shown in Fig. 13.

Fig. 11. (a) Fabricated flexible FSS prototype, (b) enlarged view.

Fig. 12. Measurement setup for the proposed design.

Fig. 13. Comparison of simulation and measured result.

From Figs 14 and 15, the measured results of a different angle of incidence for both TE and TM mode are shown. Little deviation occurs in frequency response for both TE and TM mode but it is an acceptable value. Due to the edge reflections and scattering of EM waves, a lack of smoothness is encountered during the measurements.

Fig. 14. Measurement results for different angle of incidence in TE mode response.

Fig. 15. Measurement results for different angle of incidence in TM mode response.