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

A modeling of the resonant behavior of a continuously reconfigurable 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 ( ) 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 ( ) 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.

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.
After introducing the FSS and varactors basics in this Section, four arms star geometry and FSS Fig. 1. FSS geometry and parameters that affect the FSS frequency response, [5].

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  Without the gap, Fig. 3(a), the FSS resonant frequency can be approximately determined by (1), with good results, principally for ℎ ≪ 0 , [5], [19], [21].
With the gap and the bias lines, Fig. 3 (d), roughly speaking, the resonant frequency, − , can be estimated by (2). a) Typical frequency response -y polarization b) Equivalent circuit 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.
Considering 0 known, and 吠 can be effortlessly determined. Inserting the varactor, a series variable capacitance is introduced, Fig. 5, in which the equivalent capacitance is given by (5).
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. Keysight E3633A power supply, and a measurement window as shown in Fig. 6. The wave incidence is considered normal to the reconfigurable FSS. 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. 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.       In order to achieve the FSS equivalent circuit, Fig. 5, it is necessary to determine the adequate values of 0 , 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, f 0 , are extracted for the lowest varactor capacitance, C v (highest reverse voltage).
2-An impedance value Z 0 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. 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 Z 0 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, If 吠 = 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

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 ≤ 吠 ≤ 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 ≤ 吠 ≤ 9.63 pF ). Once again, a good agreement was observed between