# Design of True Time Delay Millimeter Wave Beamformers for 5G Multibeam Phased Arrays
## 3 Numerical Results
### Tree Topology Results  
Figure 13 shows the proposed three-layer tree beamforming network. The implementation of each layer is in stripline technology and the substrate is RT duroid 5870 with \(\epsilon_{r}=2.33\) and \(tan\delta=0.0012\). The thickness of each board is 0.127 mm and the signal line of each tree network is sandwiched between two single plated boards, as shown in Figure 27.  
Figure 27: Cross section of the multilayer tree beamforming network. Figure 26: Proposed Rotman lens with absorbers in the side walls reflection coefficients. All the input ports are located at the bottom layer of the network (see detail in Figure 13). Microstrip to stripline transitions are used for each layer separately. The first tree beamformer is located at the feeding layer and is designed to excite beam 1, as shown in Figure 13. The second tree beamformer at the middle of our multi-layer topology steers the beam to the left and is excited through a via that connects layer 1 with layer 2, as shown in detail in Figure 13. Finally, the third tree beamformer, at the top of our topology, is responsible for the broadside beam and is excited by utilizing two cascaded vertical transitions, as shown in detail in Figure 13.  
As mentioned above, the power dividers are of high importance in the proposed topology. Two types of power dividers are incorporated in the network; an equal split Wilkinson power divider [57] and an unequal split power divider as the one reported in [56]. They are both implemented using stripline technology. Figure 28a shows the layout of the equal power divider. As shown in Figure 28b, the reflection coefficient is below 20 dB and the power is equally split through the desired bandwidth. Regarding the unequal power divider (shown in Figure 14), based on the design specifications of our tree topology, a 5-dB power division ratio has to be achieved. To achieve this power division ratio, the divider is designed to have a lower arc of an electrical length of \(L_{1}^{\epsilon}=151.2^{\circ}\) and an upper arc of \(L_{2}^{\epsilon}=106.21^{\circ}\), as shown in Figure 29a. In addition, the lines connecting the 50 \(\Omega\) resistor to the branches of the divider are designed with electrical lengths of \(L_{3}^{\epsilon}=14.85^{\circ}\) and \(L_{4}^{\epsilon}=59.95^{\circ}\), respectively. Figure 29a shows the amplitude response of the unequal power division power divider. It can be seen that return loss is kept low for a large bandwidth, whereas the power division ratio limits the bandwidth of this design. As shown in Figure 29b, the phase difference between the two output ports of the divider is approximately equal at the center frequency. In the proposed design (Figure 13), the physical lengths of the transmission lines connecting the output ports with the unequal power division power dividers on layer 1 are chosen to be 2.36 and 4.05 mm for the left and the right branches, respectively, and they consist of two right angle curved bends. The lengths of the lines connecting the Wilkinson power divider of the first stage to the power dividers of the second stage are 6.24 mm for the left and 2.86 mm for the right branch. In both branches, the arcs attached to the unequal division power divider have a radius of 0.372 mm and an angle of 130\({}^{\circ}\). The tree network of layer 2 is a the same as that of layer 1 but mirrored over the vertical axis. Finally, for the tree network in layer 3, the transmission lines connecting the output ports with the unequal power division power dividers again consist of two right angle curved bends, whose combined physical length is 2.36 mm, while the interconnecting lines between the two stages have a total length of 3.02 mm. Figure 28: Equal power Wilkinson power divider: (**a**) layout of the equal power split Wilkinson power divider; and (**b**) amplitude response of the equal power split Wilkinson power divider. Figure 30a-f shows the simulated results of the proposed multilayer tree topology.