## 2 Novel Phase Shifter Design
\[\Phi=\beta L, \tag{2}\]  
where \(L\) is the physical length of the CRLH transmission line unit cell given in Figure 1. Two Magneto-static field Responsive Structures (MRSs) [32] are placed in the two symmetrical stubs to produce the effects of \(L_{L}\). The work in [24] has outlined the detailed procedure for the development of MRS, and it is briefly mentioned here. As shown in Figure 2, a cylindrical cavity of diameter d is drilled inside the dielectric substrate material. This cavity is then partially filled with micron-sized silver-coated ferrite particles [23]. The top of the cavity is the printed portion of the CRLH stub transmission line and the bottom of the cavity is the ground plane. When a magnetic field (using a tiny magnet beneath the ground plane at the MRS location) is applied, these particles are magnetically aligned in the \(z\)-direction, as shown in Figure 2a. This magnetic alignment of silver-coated ferrite particles inside the MRS provides a path for the flow of current from the top of the stub to the ground plane producing the effects of \(L_{L}\). Since the CRLH-TL is a composite structure, changing the path of current flow along the stubs with MRSs will also affect the other structural parameters (\(L_{R}\), \(C_{R}\)). This in turn will determine the phase shift through Equations (1) and (2). The activation of particles inside the MRS along the length of a stub transmission line is equivalent to a shunted stub in a standard CRLH transmission line with a conducting via. The connection/disconnection of the stub transmission line with the ground plane through particles in the MRS is further investigated using 5 parameters simulations. The surface current distribution at the design frequency of 5.5 GHz when particles in the MRS cavities are aligned in the presence of a magnetic field is shown in Figure 2b, which shows that the structure is smoothly propagating an RF signal from port 1 to port 2. Similar surface current distributions are observed for other frequencies in the band of interest and are not shown here. The zoom-in views of the surface current along the length of the stub and MRS in the absence and presence of particle alignment inside the MRS cavity are shown in Figure 3. The corresponding plots for \(S_{11}\) and \(S_{12}\) are also shown in Figure 3. There is a mismatch at the ports (\(S_{11}\) is greater than \(-5\) dB) with high isolation between the ports (\(S_{12}\) is greater than \(5\) dB) in the absence of a magnetic field. When particles in the MRS structure are activated in the presence of a magnetic field, the \(S_{11}\) is greater than \(-10\) dB and \(S_{12}\) is \(0.13\) dB. This shows excellent matching characteristics with low insertion loss when particles in the MRS structure are activated. This behavior is similar to that observed in [32] for a simple microstrip transmission line. We denote the novel CRLH transmission line with embedded magnetic particles as a 'P-CRLH' phase shifter. As compared to the recently published work in [29], the proposed P-CRLH phase shifter has the following advantages: (1) A 60% size reduction. The size comparison of the proposed P-CRLH phase shifter with [29] is discussed in Section 4. (2) In the proposed work, the entire phased array design is on a single PCB as compared to [29], where the antenna array is excited with phase shifters separately resulting in an overall larger size. (3) The size requirement of printed microstrip transmission lines on both ends of the phase shifter in the proposed work is 73.68% less than the phase shifter in [29]. Next, the phase response analysis of single-unit cells and cascaded unit cells of the P-CRLH phase shifter are discussed. Phase Response of Single- and Cascaded Unit Cells P-CRLH Phase Shifter \(-45^{\circ}\) Phase Shift  
The unit cell P-CRLH phase shifter with dimensions in Figure 1 is simulated using the ADS 2016 Momentum simulator with the following EM setup: 50-ohm single-mode ports P1 and P2 with TML calibration, mesh frequency is 6 GHz with a mesh density of 20 cells/wavelength.