## 2 Novel Phase Shifter Design
The substrate material used is Rogers RT/duroid 5880 LZ (\(\epsilon_{\rm r}=2.2\), tan \(\delta=0.002\) and thickness = 1.27 mm). To mimic, the magnetic alignment of silver-coated ferrite particles in the MRS structure for P-CRLH phase shifter simulation, grounded vias with circular layouts were inserted at the MRS locations. The activation of vias connects the symmetrical stubs with the ground plane, representing the magnetic alignment of particles in the MRS, and the deactivation of vias disconnects the stubs from the ground plane,which represents the non-alignment of particles in the MRSs. The simulated S parameters for single-unit cells and cascaded-unit cells in the ADS Momentum layout are shown in Figure 4. The impedance matching characteristics at the ports (\(S_{11},~{}S_{22}\), ) are below \(-10\) dB. The insertion loss is 0.087 dB at 5 GHz and goes up to 0.164 dB at 6 GHz. Therefore, it is concluded that the P-CRLH phase shifter supports the propagation of the electromagnetic wave in the frequency band of interest. Figure 1: Proposed unit cell P-CRLH phase shifter with Magneto-static field Responsive Structures (MRSs). All dimensions are in mm (\(\text{L}_{6}=6.8\), \(\text{W}_{8}=1\), \(\text{L}_{t}=6\), \(\text{W}_{t}=0.3\,\text{L}_{\text{HS}}=1\), \(\text{W}_{\text{HS}}=4\), \(\text{s}=0.2,~{}\text{d}=0.9\)). The overall size is \((0.24\lambda_{g}\times 0.22\lambda_{g})\cong\frac{2\lambda_{g}}{4}\) at the design frequency of 5.5 GHz. Figure 2: Isometric view of the proposed unit cell P-CRLH phase shifter with (**a**) magnetic alignment of particles in Magneto-static field Responsive Structures (MRSs); (**b**) Surface current distribution at 5.5 GHz when particles inside the MRS are activated. Figure 4: Simulated S parameter results of the proposed unit cell and cascaded unit cells P-CRLH phase shifter with MRSs embedded on the symmetrical stubs using \(-45^{\circ}\) unit cell phase shift. Figure 3: Illustration of connection/disconnection of a stub with the ground plane using magnetic particle activation/deactivation inside the MRS cavity. **(a,c)** Surface current distribution and S parameters for a disconnected stub with the ground plane, **(b,d)** Surface current distribution and S parameters for a connected stub with the ground plane. The simulated phase responses for the unit cell and cascaded unit cells P-CRLH phase shifter are shown in \[\Phi=\beta L, \tag{2}\]0 At the design frequency of 5.5 GHz, the phase shift from port P1 to P2 for a single-unit cell is \(-46.7^{\circ}\). For cascaded unit cells, the total phase shift simply adds as demonstrated in [33]. Therefore, for cascaded two-unit cells, the calculated phase shift is \(2\times(-46.7^{\circ})=-93.4^{\circ}\), for cascaded three-unit cells, the calculated phase shift is \(3\times(-46.7^{\circ})=-140.1^{\circ}\) and the calculated phase shift for cascaded four-unit cells is \(4\times(-46.7^{\circ})=-186.8^{\circ}\). The simulated phase shifts in Figure 5 for cascaded two-, three- and four-unit cells are \(-92.2^{\circ},-137.8^{\circ}\) and \(-183.9^{\circ}\), respectively. The range of phase error is \([1.2^{\circ}-2.9^{\circ}]\). The effects of the phase error on phased array antenna will be discussed in Section 3. Phase Response of Single- and Cascaded Unit Cells P-CRLH Phase Shifter \(-90^{\circ}\) Phase Shift  
To achieve a \(-90^{\circ}\) phase shift, the unit cell P-CRLH phase shifter in Figure 1 with \(\mathrm{L_{\mu s}}=3.5\) mm (all other dimensions remain the same) is simulated in the ADS 2016 Momentum environment. The same electromagnetic (EM) setup parameters are used as for the \(-45^{\circ}\) case in Section 2.1. The simulated S parameters and phase response results are shown in Figures 6 and 7, respectively. The results in Figure 6 show good impedance matching characteristics and insertion loss indicating the propagation of the electromagnetic wave from port 1 to port 2.