## 2 Liquid Crystal Bandpass Filter Design  
Since dual-mode technology for bandpass filters was introduced by [PERSON] in 1972 [7], these filters have been widely used in microstrip technology because of their advantages, specifically in size reduction. This is a critical issue in LC-based devices, since a size reduction involves a decrease of the amount of used LC and, thus, a lower manufacturing cost. Dual-mode operation consists of a resonant circuit where two electromagnetic degenerated modes are excited due to the presence of some kind of perturbation in the microstrip structure. Therefore, a two-pole filter can be obtained with a single resonator, which implies a significant size reduction. For implementing dual-mode filters in microstrip technology, different topologies have been used, such as square-loops [8] or T-shaped stubs [9]. The topology shown in Figure 1a is a dual-mode filter structure developed by [PERSON] _et al_. [10] achieving a good performance [11]. It consists of a microstrip metallic square patch with a central square notch and two perpendicular feed lines for exciting the two degenerated modes. In this work, a reshaping of the patch of this structure has been proposed. The modification consists of making a square cut of size k etched on the patch, as it is shown in Figure 1b. This new microstrip patch is expected to improve the passband return loss (S\({}_{11}\) parameter) and to narrow the filter bandwidth. The value of the filter central frequency (\(f_{\mathrm{c}}\)) for the structure presented in Figure 1 is a function of the structure dimensions and the effective permittivity (\(\varepsilon_{\mathrm{eff}}\)). It is especially sensible to variations in the \(b\) dimension, while filter bandwidth depends on the \(a\), \(c\) and \(e\) dimensions [10]. Thus, for fixed dimensions of the structure, \(f_{\mathrm{c}}\) can be varied as \(\varepsilon_{\mathrm{eff}}\) is modified. In addition, permittivity depends on the substrate materials of the device, so a suitable choice of the substrate materials has been the strategy to achieve the device tunability. In this work, LC is used as the dielectric substrate of the microstrip line in the device. As LC permittivity changes between two extreme values, \(\varepsilon_{\mathrm{r\perp}}\) and \(\varepsilon_{\mathrm{r\parallel}}\), by applying voltage, the filter central frequency is also voltage-controlled. Therefore, the use of LC as dielectric substrate is the reason which allows the filter to be tunable. LC is a fluid material, so it needs to be confined in a cavity inside the device which will be afterwards sealed. For this reason, the filter is implemented by using an inverted-microstrip structure, as shown in Figure 2. This structure has been already reported in other LC-based devices [12]. The LC cavity is designed of 0.25-mm thick and delimited by the spacers; the dielectric material Taconic TLX-08 (\(\varepsilon_{\mathrm{r}}=2.55\)) is employed for them. This material is also used as the substrate that supports the microstrip line with 0.8-mm thick. Before the fabrication of the device and the filling with the LC, several electromagnetic simulations by using the software Ansoft HFSS have been run for optimizing the filter dimensions. Initially, the LC cavity has been considered to be empty in the simulations, that is, \(\varepsilon_{\mathrm{r}}=1\). As it is recalled in Section 3, LC permittivity extreme values are unknown at microwave frequencies. The dimensions of the patch have been chosen by comparing the spectral response of the filter with a conventional dual-mode patch (Figure 1a) and with the new patch with a square cut (Figure 1b). The results of the patch dimensions for the topology presented in Figure 1a is solved for a filter central frequency nearby 7.5 GHz considering empty the LC cavity. These values are summarized in Table 1. The microstrip line width (\(w\)) has been designed of 0.65 mm in order to obtain input impedance (\(Z_{\mathrm{in}}\)) close to 50 \(\Omega\).