## 3 Inductive Posts in the LC-Based SIW Phase Shifter  
The cylindrical IP inserted into the SIW (Figure 2) is equivalent to a T-network high-pass filter [23]. The high-pass filter produces a phase advance when used in a phase shifter and is widely used in SIW phase shifters [24,25]. The phase advance \(\phi\) can be calculated as [19]:  
\[\phi=-tan^{-1}\left(\frac{2X_{b}}{Z_{0}}+\frac{X_{a}}{Z_{0}}\right)-tan^{-1} \left(\frac{X_{a}}{Z_{0}}\right) \tag{3}\]  
where \(X_{a}\) is the reactance, \(X_{b}\) is the susceptance, and \(Z_{0}\) is the line's characteristic impedance. To gain a deeper understanding of parameters related to phase modulation, we need the following detailed equations [19]:  
\[\frac{X_{a}}{Z_{0}}-\frac{X_{b}}{Z_{0}}=\frac{a}{2\lambda_{g}}csc^{2}\frac{ \pi p}{a}\Bigg{[}S_{0}-\left(\frac{\pi 2r}{2\lambda_{g}}\right)^{2}-\left( \frac{\pi 2r}{2a}\right)^{2}\left(S_{0}cot\frac{\pi p}{a}-S_{1}\right)^{2}\Bigg{]}, \tag{4}\]  
\[\frac{X_{b}}{Z_{0}}=\frac{a}{\lambda_{g}}\left(\frac{\pi 2r}{a}\right)^{2} sin^{2}\frac{\pi p}{a}, \tag{5}\]  
\[S_{0}=\ln\left(\frac{4a}{\pi d}sin\frac{\pi p}{a}\right)-2 sin^{2}\frac{\pi p}{ a}+2\sum_{n=2}^{\infty}sin^{2}\frac{n\pi p}{a}\Bigg{[}\frac{1}{\sqrt{n^{2}- \left(\frac{2n}{\lambda_{g}}\right)^{2}}}-\frac{1}{n}\Bigg{]}, \tag{6}\]  
Figure 1: Relative permittivity of LC for the RF field: (**a**) director, (**b**) perpendicular state of LC, (**c**) parallel state of LC.  
\[S_{1}=\frac{1}{2}cot\frac{\pi p}{a}-sin\frac{2\pi p}{a}+\sum_{n=2}^{\infty}sin\frac {2n\pi p}{a}\left[\frac{n}{\sqrt{n^{2}-\left(\frac{2a}{\lambda_{g}}\right)^{2}}}- 1\right], \tag{7}\]  
where \(a\) is the width of the waveguide, \(\lambda_{g}\) is the guide wavelength, \(r\) is the radius, and \(p\) is the offset of the inductive posts in the waveguide (Figure 2). \(X_{a}\) and \(X_{b}\) are functions of radius \(r\) and offset \(p\) of the IP (Figure 2). \(\lambda_{g}\) is the relative dielectric constant of the substrate \(\varepsilon_{r,sub}\)[14]. From the equations, the phase advance caused by the insertion of inductive posts depends on their \(r\) and \(p\) and the substrate's permittivity \(\varepsilon_{r,sub}\). \(\phi\) increases as \(r\) and \(p\) increase, but decreases as \(\varepsilon_{r,sub}\) increases [23]. We inserted an IP into the LC-based SIW phase shifter that makes a phase advance depending on \(\varepsilon_{r,sub}\). The LC-based SIW phase shifter with IPs produces the following phase shift as  
\[\phi_{phase\ shiftter}=\text{Re}\left(\phi_{l,phase\ shiftter}\right)=\beta l _{physical}\approx\frac{2\pi}{c_{0}}fl_{physical}\sqrt{\varepsilon_{r,eff}}-\phi, \tag{8}\]  
where \(\phi_{phase\ shiftter}\) is the resultant phase shift of phase shifter, \(\phi_{l,phase\ shiftter}\) is the electrical length of the phase shift, and \(\beta\) is the propagation constants. Therefore, the differential phase shift of the LC-based SIW phase shifter with inductive posts inserted can be written as  
\[\begin{array}{c}\Delta\phi_{phase\ shiftter}=\text{Re}\left(\phi_{l,\|} \right)-\text{Re}\left(\phi_{l,\bot}\right)\\ =(\frac{2\pi}{c_{0}}fl_{physical}\sqrt{\varepsilon_{r,eff,}}-\phi_{\|})-( \frac{2\pi}{c_{0}}fl_{physical}\sqrt{\varepsilon_{r,eff,}}-\phi_{\bot}).\end{array} \tag{9}\]  
\(\phi_{\|}\) is the phase advance produced by the IP when the LCs are perpendicular to the RF field, and \(\phi_{\bot}\) is produced by the IP when the LCs are parallel to the RF field. The inserted IP produces a greater \(\phi\) when the LCs are perpendicular to the RF field than when they are parallel. This phenomenon results in a difference in \(\phi\) and, thereby, an increased \(\Delta\phi\). In this work, we propose the effectiveness of inserting IPs for phase advance at LC-based SIW phase shifter.