Design, analysis and fabrication of the CPW resonator loaded by DGS and MEMS capacitors

In this paper, an analytical method of resonant frequency, tuning range, the effective relative dielectric constant, and characteristic impedance is proposed, which is for coplanar waveguide (CPW) resonator loaded by defected ground structure (DGS) and micro-electromechanical systems (MEMS) capacitors. The analytical solution is achieved by an employing equivalent method. The resonant frequency, tuning range, and effective relative dielectric constant are obtained by the analytical solution. For verifying the effectiveness of the proposed method, a CPW tunable bandstop filter (BSF) with DGS and MEMS capacitors is designed, simulated, and fabricated, results show good effectiveness of the proposed analysis method. By changing the height of the MEMS beam with the actuation voltage, the designed BSF can switch the center frequency among three states (i.e. 18.5 GHz, 18.2 GHz, 17.5 GHz, respectively), and fractional bandwidth is changed as well (i.e. 37.8%, 45.6%, 49.5%, respectively).


Introduction
In the planar radio frequency circuits and systems, the coplanar waveguide (CPW), periodically loaded by defected ground structure (DGS) [1,2], have attracted much attention, owing to its attractive features such as compact structure [3], easy * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. insertion of shunt or series active devices, and low substrate dependence [4]. CPW loaded by DGS has been studied in many aspects such as resonant frequency and equivalent circuit and been widely employed in filter design [5][6][7][8][9][10][11]. Microstrip bandstop filters (BSFs) also often use DGS and stepped impedance resonators to obtain the stopband [12]. However, little work, to date, has been done in CPW resonator [13] loaded by both DGS and micro-electromechanical systems (MEMS) capacitors synchronously [14,15]. Most of the articles only describe the performance and lack some corresponding analysis methods. MEMS capacitive devices have demonstrated some superiorities [16,17], such as lower loss, lower parasitics and high linearity, in comparison to other varactor devices [18][19][20], it can be used in tuned filter [21] and reconfigurable antenna design [8,[22][23][24]. Bandstop DGS resonator is usually studied by using equivalent circuit model due to its slow-wave effect and band resistance characteristics [25,26], which is obtained by fitting resonant response curve [10,27]. In literature [11], the bandstop resonant characteristic of CPW with DGS was modeled by a parallel RLC resonator in parallel connection, but this characteristic is contradictorily described by a series connection of a parallel RLC resonator in [6,9]. All those methods fail to reveal the inherent reason of the resonant characteristic [28]. To date, the analysis method of CPW loaded by DGS and MEMS capacitors is rarely reported in available works of literature.
In this paper, the CPW is loaded by DGS and MEMS capacitors, and its bandstop resonant characteristic is analyzed by using an equivalent method, the resonant frequency, tuning range, and effective relative dielectric constant are obtained by an analytical solution. The paper uses the finite element method of the commercial software ANSOFT HFSS for simulation, and the simulation data are exported and plotted by MATLAB. The main contributions of this paper are (a) the MEMS capacitive device is analyzed; (b) the effective relative dielectric constant of the DGS and MEMS capacitors loading CPW are formulated; (c) the resonant frequency and tuning range are calculated, those expressions are associated with the structural parameters of the DGS and the MEMS capacitor. For verifying the effectiveness of the proposed method, we designed, simulated, and fabricated a CPW tunable BSF with DGS and MEMS capacitors, the results illustrate the good effects of the proposed analysis method.

MEMS bridge capacitor
A MEMS bridge capacitor is presented in figures 1(a) and (b), it consists of a MEMS bridge with releasing holes, a dielectric layer, the CPW transmission line, and the silicon substrate. Its equivalent circuit is shown in figure 1(c). The wave port of the characteristic impedance Z 0 denotes the characteristic impedance of the transmission line between the wave port and the edge of the MEMS bridge. R M and L M are the equivalent resistance and equivalent inductance of the capacitor, respectively. Hence, the capacitance C M , is expressed as: where ε 0 is the dielectric constant in free space, h is the initial air gap between the MEMS bridge and the Si 3 N 4 dielectric layer when no actuation voltage is applied, and ε r is the relative dielectric constant of the Si 3 N 4 dielectric. t d is the thickness of the dielectric layer, and A is the area where the upper and downer metallic bridge and the dielectric layer overlap. C f is the fringe capacitive. Because the dielectric layer is very thin, the edge capacitive is ignored in the calculation. The phase delay of the signal current will be generated when the current passes over the bridge, and this phase delay has been taken into account by adding a certain additional line with length (W+∆l)/2. When the operating frequency is far below the selfresonance of the bridge, one may absorb its reactive effective in C M [20]. The length of ∆l and the value of capacitance C M can be calculated by using software simulation.
In this paper, the length of W and the L are 100 and 320 µm, respectively. The circuit parameters C M , L M , and R M can be extracted accurately from the S21 and S11 results. ∆l is the distance from the reference surface to the edge of the MEMS bridge, which is 15 µm. Thus, the simulated result of C M is 65 fF. The simulated S parameters of the MEMS bridge capacitor is shown in figure 2, under several different air gap between MEMS bridge beam and the signal transmission line.

DGS and MEMS bridge capacitors loading CPW tunable resonator
As we know, the phase velocity and the characteristic impedance of the transmission line are determined by the inductance and capacitance per unit length (whether the transmission line is loaded by DGS and MEMS bridge capacitors or not). The variations of the phase velocity, propagation constant, and characteristic impedance should be attributed to the changes of the effective relative dielectric constant because the relative permeability keeps unchanged for the non-magnetic media. However, little work, to date, has been done in the relationship between the effective relative dielectric constant and the transmission line structure (including the capacitance per unit length). The phase velocity, propagation constant, and characteristic impedance are expressed by the inductance and capacitance per unit length, where, the ω is the operating angular frequency, L 0 and C 0 are the inductance and capacitance per unit length of the unloaded CPW transmission line, respectively. The CPW transmission line is designed with a size of 60/100/60 µm, and the characteristic impedance is 50.58 Ohms. Similarly, when the CPW was loaded by DGS, the phase velocity, propagation constant, and characteristic impedance parameters v 1 , β 1 and Z 1 can be expressed by the inductance and capacitance per unit length L 1 and C 1 ; the phase velocity, propagation constant and characteristic impedance parameters v 2 , β 2  According to the comparisons of resonant frequency changes shown in figure 4, the resonant frequency of structure figure 3(c) is far below the structure figure 3(b), this demonstrates the conclusion that the phase delay of MEMS capacitors and DGS loading CPW transmission line is greater than the only DGS loading CPW transmission line.
According to the equivalent circuit analysis of the transmission line loaded by DGS and CPW bridge capacitors, the inductance per unit length remains almost unchanged (because there no magnetic materials), i.e. L 0 = L 1 = L 2 . Thus, the characteristic impedance Z 1 and the effective relative dielectric constant are as follows, where the C DGS is additional capacitance when the CPW transmission line is loaded by DGS. The characteristic impedance Z 1 and effective relative dielectric constant ε r1 can be determinate by simulation or the conformal transformation method [29].
The characteristic impedance Z 2 and the effective relative dielectric constant ε r2 are as follows, where, c is the light velocity in the free space, C MEMS is the additional capacitance when the CPW transmission line is loaded by the MEMS bridge capacitors, respectively. When the characteristic impedance Z 1 and the effective relative dielectric constant ε r1 are determined, one can insert the equation (3) into the equation (4) and obtain the value of the characteristic impedance Z 2 and the effective relative dielectric constant ε r2 . To a first-order approximation (i.e. the load effects are neglected without loss of generality), the DGS and MEMS bridge capacitor loading CPW operates at a resonant state when the phase delay reaches π radian. Therefore, the center frequency is expressed as, Hence, the range of the resonant frequency is expressed as, the length tuning factor χ l and the equivalent relative dielectric constant tuning factor χ ε are as follows, where the ∆l max and ∆l min are the maximum and minimum value of the additional length ∆l, ε r2,max and ε r2,min are maximum and minimum value of the equivalent relative dielectric constant ε r2 , respectively. As shown in figure 4(b), the range of the resonant frequency is 17.0-18.5 GHz, the fractional bandwidth is 8.5%.

Results verification and fabrication
For verifying the effectiveness of the proposed method, a CPW tunable BSF with DGS and MEMS capacitors is designed. The simulation structure of the proposed BSF is based on substrate silicon, with relative permittivity of 11.9, the thickness of 400 µm, and tangent angle loss of 0.01. The BSF consists of two proposed DGS and four MEMS bridge capacitors, xoz is their symmetrical plane. The structure size values of the filter marked in figure 5(a). The structure of the proposed BSF is shown in figure 5. We cascade two unit cells (containing DGS and MEMS bridge capacitors) to construct a second-order BSF. The unit cell can be cascaded to form an Nth order BSF. Figure 6 shows the equivalent circuit diagram of the BSF. A part of the electromagnetic signal flows to the open shortcut of the DGS, so two open resonators are realized. A BSF with two resonant frequency points is formed.
By changing the actuating voltage applied to the MEMS capacitors, the gap between the CPW signal line and the MEMS bridge beam is adjusted. Therefore, the capacitance value is changed accordingly. When the CPW transmission line with DGS is loaded by the different capacitance values, the characteristic impedance and equivalent relative dielectric constant is changed. Thus, the resonant frequency is tuned, and the center frequency and fractional bandwidth of the BSF are tuned as well.
The comparison between simulation and measurement results from figure 7 shows that S-parameters at the second resonance point are smaller than the simulation value. The resonant frequency and bandwidth are in good agreement with the simulation results. The measured stopband rejection is poorer than the simulation. This is mainly due to the incomplete release of the polyimide and the fact that the surface of the MEMS bridge and dielectric layer is not flat. When the manufacturing process has a good release and flatness, stopband rejection will be greatly improved. Because the actuation voltage is difficult to precisely control the height of the bridge beam, the value of the MEMS capacitor is difficult to get accurately.
As shown in table 1 and figure 7, the measured results revealed a −3 dB bandwidth of 14.5 GHz, a −10 dB bandwidth from 15 to 22.5 GHz, a rejection level of −32.71 dB, and a Q factor of 1.245. By changing the gap between CPW signal line and MEMS bridge beam, the designed BSF can switch the center frequency among three states (i.e. 18.5 GHz, 18.2 GHz, 17.5 GHz, respectively), and fractional bandwidth is changed as well (i.e. 37.8%, 45.6%, 49.5%, respectively). Table 2 shows a comparison of filters from recent years. The fabrication technology was shown in figure 8. The proposed BSF can be applied to the satellite communication system.

Conclusions
DGS and MEMS capacitors loading CPW transmission line are analyzed in this paper, and the characteristic impedance and the equivalent relative dielectric constant of unit length are obtained for the implementation of the tuning resonator. The closed-form expressions of equivalent relative dielectric constant, characteristic impedance, and tuning range of the resonator are formulated. The proposed method is verified by designing a center frequency and fractional bandwidth tuning BSF, which shows that this method is a good way to analyze and calculate the parameters of DGS and MEMS capacitors loading CPW.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). When the signal pass over the DGS and MEMS capacitors loading CPW, the phase shift θ total θ total = β e l = ωn(s + w + ∆l) . (A2) The equivalent relative dielectric constant ε r2 of the DGS and MEMS capacitors loading CPW. . (A3) The maximum and minimum resonant frequency of the DGS and MEMS capacitors loading tuning resonator ω r,max , ω r,min                ω r = πv p nl sec = π n(s + w + ∆l) c √ ε r2 ω r,max = π n(s + w + ∆l min ) c √ ε r2,min ω r,min = π n(s + w + ∆l max ) c √ ε r2,max ∆ω r = ω r,max − ω r,min . (A4) ORCID iDs