# Rethinking Figure-of-Merits of Liquid Crystals Shielded Coplanar Waveguide Phase Shifters at 60 GHz
## 3 Results  
LCs' intrinsic dielectric properties (i.e., tunability, maximum dissipation factor) and hence the materials' FoM\({}^{\prime}\) were quantified for the 16 types of microwave-enabled LCs, as depicted in Figures 2 and 3, respectively. Among the LCs characterised, the MDA group exhibited high tunability and moderate dissipation factors, with the overall FoM higher than the QYPD group, but less competitive than the GT series. In particular, GT5-28004 exhibited the highest material FoM\({}^{\prime}\), followed by TUD-566, GT5-26001, and GT7-29001. Comparing Figure 4 (below) against Figure 3 (below), however, patterns for the device's FoM\" were reshuffled significantly, with GT7-29001 and MDA-03-2838 ranked at the top. The difference was attributed to the perturbed wave-occupied volume ratio of the tunable-to-non-tunable dielectrics confined in the SCPW mode, with LCs of diverse permittivity values, whereas the PTFE substrate material and SCPW geometry remained unchanged. It is interesting to note that the concept of wave-occupied-volume ratio (WoVR) was first described in our prototype [7] centred on LC-based planar devices with a non-tunable dielectric substrate. WoVR concerns the ratio of the volume taken up by the tunable medium (LC) to the total volume of dielectrics (i.e., tunable LC plus the non-tunable substrate). There are two nonlinear relationships here: first, differential phase shift versus WoVR, and second, insertion loss versus the dielectric constant ratio of tunable to non-tunable media. Future work may explore these complex interactions via machine learning (efforts underway). Figure 3: Various LC materials’ FoM\({}^{\prime}\) at 60 GHz (results derived from Figure 2). Figure 2: Various LCs’ dielectric properties characterised for 60 GHz. It is worth noting that it is difficult for a highly integrated FoM metric to tell the physical phenomena and the effect of scale. To this end, a geometrical factor (GF) can be defined in Equation (4) and is computationally quantified in Figure 5 below. The unit for the GF here is intended to be relaxed, rendering this a relative magnitude for relevant comparisons. \[\text{GF}=\text{FoM''}/\text{FoM''} \tag{4}\]  
Figure 4: Devices’ FoM” with different LCs at 60 GHz (simulated with HFSS). Figure 5: Geometrical factor of SCPW with diverse LCs at 60 GHz (results derived from Figures 3 and 4). With a direct connection to WoVR, GF represents the efficiency of the specific type of LC material fitting into the device geometry. The results in Figure 5 reveal that a large part of the LCs with high material FoM\({}^{\prime}\) essentially exhibited unsatisfyingly low GF (e.g., GT5-28004 and TUD-566), which limits their potential to achieve a higher device FoM\({}^{\prime\prime}\) of interest in practice. Another insight that might have long been overlooked by existing literature is that an LC-based phase shifter's FoM\({}^{\prime\prime}\) is not invariable concerning the line length effect. Although the achievable phase shift increases linearly with the effective line length, the insertion loss's dependency on the effective line length is nonlinear. As a consequence, for device designs with the same cross-sectional geometry, FoM\({}^{\prime}\) can vary with the line length, as evidenced in our 67 GHz experiments [29] with two LC-based phase shifter demos of the same design that differed in line length. A 0-\(\pi\) phase shifter with a length of \(L\) reported a maximum insertion loss of \(-4.37\) dB, i.e., FoM\({}^{\prime\prime}\) (\(L\)) of \(41.2^{\circ}/\)dB. By upgrading the effective length to \(2L\) and achieving a phase-shifting capability of 0-\(2\pi\), the updated maximum insertion loss became \(-8.28\) dB, i.e., FoM\({}^{\prime\prime}\) (\(2L\)) of \(43.5^{\circ}/\)dB, a 5.6% increase as compared to the FoM\({}^{\prime\prime}\) (\(L\)). This illustrates another limitation of a device's FoM\({}^{\prime\prime}\) being a direct performance comparison metric among devices of different phase-tuning ranges.