# Design of True Time Delay Millimeter Wave Beamformers for 5G Multibeam Phased Arrays
## 3 Numerical Results
### Blass Matrix Results  
Figure 8 shows our proposed compact double-layer semi-circular Blass matrix that can achieve uniform amplitude distribution and steer its beam between the following three directions: broadside and \(\pm 34^{\circ}\). Notably, the design rules of the proposed Blass architecture are based on the design equations given in the original paper by [PERSON] [28]. The 11.5-dB dual-layer directional coupler used in our design is implemented in a double 0.127-mm-thick duroid 5870 substrate with a dielectric constant of \(\epsilon_{r}=2.33\), as shown in Figure 16. The slot width is 0.4744 mm and the patch width is 0.5277 mm, which are estimated according to the methods in [49; 50] to ensure an 11.5-dB coupling. The analytically estimated dimensions are 0.48 and 0.533 mm for the slot and the patch, respectively. These design parameters are optimized through the ANSYS HFSS simulation software, and the final slot and the patch lengths (see Figure 6) are 1.8 mm. Curved 50 \(\Omega\) microstrip lines are attached to each side of both patches with a radius of 1.3 mm and a 45\({}^{\circ}\) arc angle. As shown in Figure 7a, the 3-dB bandwidth in terms of coupling coefficient \((S_{41})\) ranges from 15 to 45 GHz, while both the return loss \(S_{11}\) and isolation \(S_{31}\) are satisfactory. Finally, the phase of the coupling coefficient with respect to the direct path is designed to be \(\Delta\phi=phase(S_{41})-phase(S_{21})=90^{\circ}\) for all desired bands of operation, as shown in Figure 7b. Both the second and third columns of the Blass matrix are similar to the first but rotated by 31\({}^{\circ}\) and 62\({}^{\circ}\), respectively. The required delay lines of multiples of \(\tau_{h}\) according to Figure 4 are designed and implemented as microstrip lines, and the results are presented in Figure 17. In addition, \(\tau_{0}\) and its multiples are achieved through 1.987 mm straight microstrip line segments, while \(\tau_{h}\) is achieved with curved microstrip lines with a length of 2.01 mm and an arc angle of 31\({}^{\circ}\). Multiples of \(\tau_{h}\) have the same arc angle but double the length. The vertical delay lines are chosen to have zero electrical length as, based on our simulation analysis, the coupler inserts a 18.08\(ps\) delay on its own. Therefore, there is enough physical separation and no extra vertical lines are needed. The desired delays \(\tau_{h}\) for an array with an interelement distance \(d=4.5\) cm at the maximum operating frequency of \(f_{max}=33\) GHz are illustrated in Figure 18. Notably, \(d=\lambda_{min}/2\) at \(f_{max}\) to avoid grating lobes. However, this could be related to higher values due to the restricted angular deflection of the radiated beam from 0\({}^{\circ}\) (broadside) to \(\theta_{max}=34^{\circ}\) degrees as \(d=\lambda_{0,min}/(1+\theta_{max})\). Figure 16: Blass matrix cross section. Figure 17: Group delay response of the delay lines of the Blass matrix. The simulation results of the Blass matrix topology are presented in Figure 19a-f. Specifically, Figure 19a shows that, for the first beam, which aims towards \(+34^{\circ}\), the amplitudes at the corresponding elements exhibit only a \(\pm 0.4\) dB or \(\pm 9.6\%\) deviation from the desired uniform amplitude distribution. As expected, the time delays of the first beam are flat for the total operational bandwidth (see Figure 19b) since spurious paths are avoided by utilizing directional couplers. As shown in the inset of Figure 19b, the time delay differences between adjacent ports are fairly constant, deviating by no more than \(\pm 0.7 ps\) or \(\pm 8.4\%\). Regarding the second beam, which aims towards broadside, the amplitudes at the output ports deviate from the uniform distribution by \(\pm 0.35\) dB or \(\pm 8.4\%\), as shown in Figure 19c. Figure 19d presents the corresponding time delay of the output ports, showing a maximum difference of no more than \(\pm 2 ps\) between them.