Modeling the Resonant Behavior of Continuously Reconfigurable FSS Based on Four Arms Star Geometry

Abstract

A modeling of the resonant behavior of a continuously reconfigurable frequency selective surface, FSS, is described in this paper. The FSS is based on four arms star geometry and its reconfigurability is achieved by the use of varactors. FSS, four arms star geometry and varactor principles are explained. Considering available a set of measured or numerical data, with frequency responses for different varactor capacitance values, the FSS equivalent circuit is stablished. Then, the varactor capacitance effect is included and the resonant behavior can be easily determined. After stablished, the FSS equivalent circuit can be applied even for a different varactor. In order to validate the proposed modeling, the equivalent circuit of a reconfigurable FSS using the varactor SMV1231 (0.466 pF ≤ C_v ≤ 2.35 pF is obtained. The equivalent circuit results are compared to numerical (ANSYS) and measured results, verifying a good agreement. Following, for the same geometry, the equivalent circuit is applied to a reconfigurable FSS using the SMV1234 varactor (1.32 pF ≤ C_v ≤ 9.63 pF) and once more a good agreement between the results is observed, indicating the applicability of the proposed modeling, which is especially attractive for optimization process.

Index Terms
Four arms star; modeling; reconfigurable FSS; varactor

I. INTRODUCTION

Frequency selective surfaces, FSS, consist of conductive patches or aperture elements, etched on a dielectric substrate, arranged in a planar periodic structure, providing filtering properties, which frequency response depends on the polarization of the incident wave, the geometry and periodicity of the elements within the FSS structure, and the substrate thickness and permittivity, Fig. 1. As FSS can pass or block electromagnetic waves in free space for different frequencies, they are also known as spatial filters [¹]–[⁵]. FSS have been investigated for more than five decades, with applications from microwave to terahertz frequencies. Initially, FSS were employed in radomes, absorbers and dual-band antennas sub-reflectors systems [¹]–[⁴]. Nowadays, FSS have been reported in diverse applications, such as flat lens [⁶], [⁷], radar-cross-section-reduction [⁸], [⁹], polarization converters [¹⁰], [¹¹], smart antennas [¹²], [¹³], and wireless security [¹⁴], [¹⁵].

Fig. 1
FSS geometry and parameters that affect the FSS frequency response, [⁵].

When reconfigurable FSS are desired, two main approaches are considered: mechanical and electronic tuning. In the mechanical tuning authors exploit mechanical modifications, such as stretching, folding or rotate the basic element to achieve the frequency tuning [³],[¹⁶]–[¹⁸]. In the electronic tuning, discrete components, such as varactors, PIN diodes or MEMS switches, are incorporated in the FSS basic geometry [³], [⁵], [¹⁹]–[²¹]. Reconfigurable FSS have been especially attractive for implementation of smart antennas, an increasing demand of the wireless communication systems [³], [¹²], [¹³], [²²], [²³]. The electronic reconfigurability can be discrete or continuously. In discrete manner, the FSS assumes a limited number of frequency responses, and PIN diodes or MEMS switches are commonly used. Usually, a FSS with continuously variable frequency response is obtained by the use of varactors as tuning components.

Varactors are PN junction diodes in which the depletion region that is formed at the junction acts as a nearly-ideal insulator, which separates the highly-doped anode from the cathode layer, thus forming a parallel plate capacitor, which capacitance can be controlled by the reverse bias voltage [²⁴]. Despite a more accurate model of the varactor includes inductances and resistances, Fig. 2, in many applications a simplified model, considering only the variable capacitance, can be adopted. Reconfigurable FSS frequency response, including varactor effects and bias lines, can be numerically simulated using commercial software [¹³], [²⁵], [²⁶]. However, the required computational processing time imposes limits, principally for optimization techniques. In this paper we introduced a modeling procedure from which the FSS equivalent circuit is obtained. Then, the varactor capacitance effect is included and the continuously reconfigurable FSS can be easily characterized. To the best of the authors' knowledge, this is the first time that the equivalent circuit of the FSS based on four arms star geometry is described. Furthermore, the proposed modeling procedure may be applied to other FSS geometries.

Fig. 2
Varactor models.

After introducing the FSS and varactors basics in this Section, four arms star geometry and FSS equivalent circuit are described in Section II. The proposed modeling, including numerical and measured results, is detailed in Section III. The procedures, results and conclusions are summarized in Section IV.

II. FOUR ARMS STAR GEOMETRY AND FSS EQUIVALENT CIRCUIT

Four arms star geometry was introduced in [²⁷], with very interesting characteristics, such as miniaturization and switching. In Fig. 3, the four arms star geometry is depicted and the procedure to obtain it is detailed in [⁵], [¹⁹], [²¹]. Without the gap, Fig. 3 (a), it is polarization independent. With the gap insertion, Fig. 3 (b) and Fig. 3 (c), the geometry becomes polarization dependent and only the frequency response for the y polarization is affected by the presence of the varactor. Including the bias lines, Fig. 3 (d), basic cell is complete. If the insertion of the varactor is not considered, the FSS has a band-stop frequency response, as it is shown in Fig. 4.

Fig. 3
Four arms star geometry.

Fig. 4
Typical four arms star FSS frequency response, y polarization.

Without the gap, Fig. 3 (a), the FSS resonant frequency can be approximately determined by (1), with good results, principally for h ≪ λ₀, [⁵], [¹⁹], [²¹].

(1) $$f_{res}\left(GHz\right)=\frac{0.3}{2L_{efe}},$$

in which L_efe = Lx + Ly.

With the gap and the bias lines, Fig. 3 (d), roughly speaking, the resonant frequency, f_res-gb, can be estimated by (2).

(2) $$1.5f_{res}\le f_{res-gb}\left(GHz\right)\le 2.0f_{res},$$

Despite the resonant frequencies obtained by (1) and (2) are not exact, these equations are specially interesting as a first step for a numerical optimization.

Taking into account the FSS frequency response and its equivalent circuit, Fig. 4, the following equations can be obtained.

(3) $$f_{res-gb}=f_0=\frac{1}{2\pi \sqrt{LC}}$$

(4) $$C=\frac{\left|1-{\left(\frac{f_c}{f_0}\right)}^2\right|}{2\pi f_c{Z}_0}$$

Considering Z₀ known, L and C can be effortlessly determined. Inserting the varactor, a series variable capacitance is introduced, Fig. 5, in which the equivalent capacitance is given by (5).

(5) $$C_{eq}=\frac{C\times C_v}{C+C_v}$$

Fig. 5
FSS equivalent circuit considering the varactor insertion.

Note that if C_v ≫ C, C_eq → C. Otherwise, if C_v ≪ C, C_eq → C_v. Therefore, in order to obtain a variation in the FSS frequency response, the varactor must present a capacitance small enough when compared to the FSS intrinsic capacitance, as it will be verified numerically and experimentally in the next Section.

III. MODELING THE FSS RESONANT BEHAVIOR

As previously mentioned, in the proposed modeling it is presumed available a set of measured or numerical data, with frequency responses for different varactor capacitance values. Thus, initially it is described how numerical and measured results were acquired.

Numerical results were obtained using the commercial software ANSYS. The measured results were obtained at the GTEMA/IFPB microwave measurements laboratory using an Agilent VNA E5071C two ports vector network analyzer, two A.H. Systems double ridge horn antennas SAS-571, a Keysight E3633A power supply, and a measurement window as shown in Fig. 6. The wave incidence is considered normal to the reconfigurable FSS.

Fig. 6
Measurement setup.

The reconfigurable FSS based on four arms star geometry was designed and fabricated using a low cost fiber-glass substrate FR-4 (ε_r = 4.4, loss tangent 0.02, thickness 1.2 mm, 35 μm copper thickness) with the following basic cell dimensions: W_x = W_y = 30mm, L_x = L_y = 20mm, S_x = S_y = d_x = d_y = 3mm, m_x = m_y = g = 1 mm, , as depicted in Fig. 3. The whole reconfigurable FSS has 6 × 6 basic cells, arranged in 6 lines, with 180 mm × 180 mm. Despite the basic cell lines could be individually controlled, in this work they were connected. The resistor R, 150 Ω, was introduced as a protection to avoid any damage due to a wrong connection. The voltage applied in varactors is directly controlled by the power supply. A photograph of the fabricated FSS is presented in Fig. 7. The used varactor is SMV1231-079LF [²⁸], which Capacitance (pF) × Reverse voltage (V) curve is presented in Fig. 8.

Fig. 7
Continuously reconfigurable FSS.

Fig. 8
Varactor SMV1231-079LF, Capacitance (pF) × Reverse voltage (V) curve [²⁸].

Numerical results for the FSS frequency response, without the varactor insertion, are presented in Fig. 9–Fig. 11. In Fig. 9, only the four arms star geometry is considered, with x and y polarization showing the same frequency response. The resonant frequency obtained by (1) is 3.75 GHz, a difference of 7.4% when compared to the numerical result, 4.05 GHz, a good approximation for a numerical optimization first step. Fig. 10 presents the FSS frequency response for the four arms star geometry with the gap. As expected, the x polarization frequency response remains practically unchanged. The frequency response, after adding the bias lines, is shown in Fig. 11, with a resonant frequency of 6.33 GHz for y polarization. In this case, resonant frequency for x polarization is out of the frequency range of interest.

Fig. 9
FSS frequency response – Four arms star geometry.

Fig. 10
FSS frequency response – Four arms star geometry with gap.

Fig. 11
FSS frequency response – Four arms star geometry with bias lines.

In Fig. 12, for y polarization, numerical and measured results for the frequency response are presented for different reverse voltages, 1 V (1.58 pF), 5 V (0.68 pF), and 10 V (0.50 pF), considering the varactor SMV1231. Each reverse voltage is associated to a capacitance in accordance with Fig. 8. This capacitance value was used in the lumped RLC boundary (ANSYS) to numerically determine the FSS frequency response. For the reverse voltages, 1 V, 5 V and 10 V, the measured/numerical resonant frequencies are, respectively, 3.89 GHz/4.01 GHz, 4.31 GHz/4.48 GHz, 4.79 GHz/4.70 GHz, which correspond to a maximum difference of 3.8%, a good agreement, despite the resonance intensity differences.

Fig. 12
|S21| (dB) × Frequency (GHz), for different reverse voltages, varactor SMV1231, numerical results (continuous lines), measured results (dashed lines).

In Fig. 13 the reconfigurable FSS resonant frequency is presented as a function of the reverse voltage, for the varactor SMV1231. This result is superimposed with the capacitance curve. A first feature to be highlighted is the good agreement between numerical and measured results, both curves presenting a similar behavior. Furthermore, a continuous variation of the resonant frequency from 3.750 GHz to 4.795 GHz is achieved (measured), a bandwidth of 1.045 GHz, which is very interesting. When the capacitance achieves the minimum value saturation, around 10 V (0.50 pF), it can be observed a similar behavior of the resonant frequency, limiting the reconfigurable FSS resonance to 4.795 GHz, measured/4.700 GHz, numerical. Fig. 14 presents the same result, but now with the resonant frequency as a function of the capacitance.

Fig. 13
Resonant frequency (GHz) × Reverse voltage (V), varactor SMV1231.

Fig. 14
Resonant frequency (GHz) × Capacitance (pF), varactor SMV1231.

In order to achieve the FSS equivalent circuit, Fig. 5, it is necessary to determine the adequate values of Z₀, C and L. Considering available a set of measured or numerical data, with frequency responses for different varactor capacitance values, the following steps are adopted:

1. The cutoff frequency, f_c, and the resonant frequency, f₀, are extracted for the lowest varactor capacitance, C_v (highest reverse voltage).

2. An impedance value Z₀ is assumed and from (3) and (4), C_eq and L are calculated. Note that this capacitance includes the varactor capacitance and the FSS intrinsic capacitance.

3. From (5) the FSS capacitance, C, is determined.

4. With the FSS equivalent circuit, from (3), the resonant frequencies are calculated for the other varactor capacitance values.

5. If the obtained resonant frequency curve fits the available measured/numerical curve, the procedure is completed.

6. If a good fitting is not achieved, a new Z₀ is assumed and the procedure is repeated, step 2.

It must be highlighted that after the FSS capacitance and inductance be obtained, a new varactor can be inserted and the reconfigurable FSS frequency response can be effortless calculated. To exemplify the proposed procedure, be considered the measured frequency response for VR=10.0V, C_v = 0.497 pF, in which f_c = 3.86 GHz and f₀ = 4.795 GHz. Resonant frequency curves are shown in Fig. 15 for different Z₀ values. Choosing Z₀=62.5 Ω, we obtain C_eq = 0.231 pF, L = 4.77 nH, C = 0.430 pF. Fig. 16 presents the frequency responses for different reverse voltages, as shown Fig. 12, but now including the frequency responses obtained with the FSS equivalent circuit. When compared to numerical and measured results, equivalent circuit results present a good agreement.

Fig. 15
Resonant frequency (GHz) × Capacitance (pF), varactor SMV1231, for different Z₀ values.

Fig. 16
|S21| (dB) × Frequency (GHz), for different reverse voltages, varactor SMV1231, numerical results (continuous lines), measured results (dashed lines), equivalent circuit (dashed dotted lines).

Still considering the equivalent circuit obtained, and the resonant frequency curve, Fig. 15, another interesting result is the interpolation equation for the resonant frequency as a function of C_v:

(6) $$f_{res}\left(C_v\right)\left(GHz\right)=0.156C_v{}^4-1.153C_v{}^3+3.232C_v{}^2-4.289C_v{}^{}+$$

If C_v = 0 pF, we get the resonant frequency of the FSS without the varactor insertion, in this case 6.240 GHz, a very good result when compared to the numerical one, 6.33 GHz, as shown in Fig. 11.

In order to evaluate the equivalent circuit for another varactor, be considered the results present in [²¹], in which the same FSS geometry is adopted, but using a varactor SMV1234 [²⁸], that presents the resonant frequency curves as shown in Fig. 17. As the FSS equivalent circuit has been determined, the frequency response can be effortless determined, Fig. 18. Fig. 19 presents the frequency responses for different reverse voltages, including the frequency responses obtained with the FSS equivalent circuit. When compared to numerical and measured results, equivalent circuit results present a good agreement.

Fig. 17
Resonant frequency (GHz) × Reverse voltage (V), varactor SMV1234, [²¹].

Fig. 18
Resonant frequency (GHz) × Capacitance (pF), varactor SMV1234.

Fig. 19
|S21| (dB) × Frequency (GHz), for different reverse voltages, varactor SMV1234, numerical results (continuous lines), measured results (dashed lines), equivalent circuit (dashed dotted lines).

IV. CONCLUSIONS

A modeling of the resonant behavior of a continuously reconfigurable FSS based on the four arms star geometry was presented in this paper. Initially the FSS and the four arms geometry principles were described. The FSS reconfigurability is achieved electronically, by the use of varactors, for which the adopted model was also described. The proposed modeling take account the availability of initial set of FSS frequency response for different varactor capacitances, which can be obtained numerically or experimentally. However, after the FSS modeling has been carried out, the obtained equivalent circuit can be applied to different capacitance values, and new frequency responses can be determined in fast manner. The modelling procedure was described and successfully applied to a reconfigurable FSS using the SMV1231 varactor (0.466 pF ≤ C_v ≤ 2.35 pF). The equivalent circuits results were compared to numerical (ANSYS) and measured results, verifying a good agreement. After, for the same geometry, the equivalent circuit was applied to a reconfigurable FSS using the SMV1234 varactor (1.32 pF ≤ C_v ≤ 9.63 pF). Once again, a good agreement was observed between numerical, measured and equivalent circuit results, indicating the applicability of the proposed modeling.

The proposed modeling is especially attractive for optimization techniques, since after the FSS equivalent circuit has been achieved, the varactor capacitance can be varied and different frequency responses can be readily obtained.

ACKNOWLEDGMENT

The authors would like to thank to Leanne Vincent, from Skyworks, by the provided varactors. This work was supported in part by the CNPq (Project 407028/2016-1) and CAPES, Brazilian Federal Agencies, and by the IFPB, Federal Institute of Paraíba (PPGEE, PRPIPG 01/2019).

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Publication Dates

• Publication in this collection
  16 Sept 2020
• Date of issue
  Sept 2020

History

• Received
  20 June 2020
• Reviewed
  29 June 2020
• Accepted
  13 Aug 2020