## 1 Introduction  
Liquid crystals (LCs) are well-known materials in the liquid crystal display (LCD) industry because of their dielectric anisotropic and birefringence nature that permits the tunability of their optical properties. This anisotropic characteristic also can be a promising candidate for use in microwave engineering because of its versatility at high frequency ranges. The tunability of LCs is not only favorable in photonics applications but also for radio frequency band (RF; 3 kHz-300 GHz) applications, especially for reconfigurable millimeter-wave systems (mmWave; 30 GHz-300 GHz). By changing the dielectric characteristic of the LC layer from which a wave is propagating, we can change the propagation constant (specifically, the phase constant \(\beta\)) of that wave. Because of this tunability of LCs, we can implement various microwave devices. The phase shifter is a key component of phased array antennae that are used to implement beam steering in communication systems such as mobile devices [1] and automotive radar [2]. For the implementation of beamforming networks that operate in the millimeter-wave (mmWave) range, phase shifters with high tuning efficiency, low-loss, small size, and low cost have been studied [3, 4]. For example, variable phase shifters that use CMOS [5] or micro-electromechanical systems (MEMS) [6] have been proposed, but they suffer from significant losses and inadequate operational stability in the mmWave range [7]. LCs offer an alternative approach for implementing tunable phase shifters [8, 9]. LCs have dielectric anisotropy controlled by static electric or magnetic fields. In addition, they have relatively low loss characteristics at high frequencies, especially above 10 GHz [10], so they have been usefully used to implement a variable phase shifter at the mmWaverange. Tunable phase shifters using LCs can be implemented using planar transmission lines such as microstrip lines (MS line) and coplanar waveguides (CPW) [11; 12], metallic waveguides, and substrate-integrated waveguides (SIW) [13]. MS lines and CPW phase shifters are commonly used to implement liquid crystal phase shifters in microwave and millimeter-wave applications. However, this configuration suffers from some drawbacks for using high-frequency range applications. The first drawback is the severe dispersion of these planar transmission line topologies, which can cause the phase shift of the transmitted signal to vary with frequency. This can make it difficult to achieve a uniform phase shift over a wide frequency range. The second drawback is the fabricating difficulty at high frequency ranges. Fabricating CPW and MS lines can be challenging, particularly at high frequencies where the dimensions of the components become very small. This can increase the cost and complexity of manufacturing liquid crystal phase shifters using these types of transmission lines. In addition, the limited cell gap range for implementation, caused by impedance matching issues, and unwanted coupling with other components that negatively impact the performance of the phase shifter are key drawbacks of the planar transmission line topology [12]. SIW can be a promising alternative structure for these drawbacks. SIW combines the advantages of planar technologies and conventional metallic waveguides to demonstrate compactness, high-quality factors, wide bandwidth, lower loss, and high-power handling capabilities. The most important advantage of SIW technology is the ability to integrate multiple devices on a single substrate, reducing losses and parasitics [14]. Therefore, LC-based SIW phase shifters are a good candidate for mmWave phase shifters [15; 16]. Research on the existing LC-based SIW phase shifter has been focused on designing to change the effective relative permittivity of the LC \(\varepsilon_{r}\) sealed in the cavity. In this case, the phase shifter length should be extended to increase the differential phase shift.