# Design of True Time Delay Millimeter Wave Beamformers for 5G Multibeam Phased Arrays
## 2 Beamforming Networks Design
### Blass Matrix Architecture
In turn, phase differences are transformed to the corresponding group delays to implement the desired wideband true-time-delay antenna array. Note that, since we are proposing a Blass matrix implemented in a semi-circular dual-layer topology, which to our knowledge has never been reported before, significant modifications have to be done on the design approach compared to the classical Blass matrix [28]. Namely, the directional couplers as well as the feed-lines' lengths between the couplers have to be appropriately chosen and designed. The Blass matrix design methodology is a serial process, which is performed column-wise. Namely, we start with the design of column 1 adding the appropriate couplers and lengths of lines at each row \(1\cdots M\), respectively. A dual-layer directional coupler in Figure 6 is used herein to accommodate our novel implementation. Since this Blass matrix has \(N=4\) output ports, four directional couplers are needed at each column. Between the couplers, a transmission line with a group delay \(\tau_{o}\) is added. The purpose of this line is to introduce some physical separation between the couplers for fabrication purposes and is usually chosen to be around \(8.3 ps\) at the frequency band we have chosen to operate. In addition, the couplers have to be symmetric and form a \"cross\", as shown in the schematic of Figure 3. To achieve this geometrical configuration, the top and bottom patches of the coupler are tilted at \(45^{\circ}\), while in addition \(45^{\circ}\) arcs are introduced at the ends of the patches, as shown in Figure 6. Note that the radius of the arcs has to be chosen wisely as firstly the characteristic impedance of the corresponding lines have to remain unaffected, and secondly the coupler has to remain compact. The length as well as the width of the slot and the patch are designed following the methodology in [49; 50]. The total phase difference between the coupler's input-port and its through-port is denoted as \(\Delta\phi_{c}\) and the corresponding delay \(\tau_{c}\). In addition, the coupling value is decided by the patch and slot width, and are always \(90^{\circ}\) in terms of electrical length. Thus, the phase  
Figure 3.0 Frequency response of the double layer directional coupler: (**a**) amplitude response; and (**b**) phase difference between the output ports of the double layer directional coupler. introduced due to the coupling value is negligible. Therefore by looking at the Blass matrix schematic (see Figure 8) the total phase introduced at the \(n\)th output-port, when input port \(m=1\) is used as a reference, is:  
Figure 3.1  
where \(\Delta\phi_{0}\) is the phase introduced by line \(\tau_{0}\). From Equation (2), it can be seen that the phase difference between two adjacent output-ports \((n,n+1)\) is:  
Figure 3.2  
Using Equation (3), the electrical length of line \(\tau_{0}\) can be specified. Since \(\tau\) is the desired group delay estimated from the antenna array in order for the beam maximum to point toward the pre-specified direction \(\theta_{0}\), it can similarly be seen that for column 2 the phase differences between adjacent output ports are:  
Figure 3.3  
where \(\Delta\phi_{h}\) is the phase introduced by line \(\tau_{h}\). In a similar manner, the rest of the matrix columns are designed. The last step of the design methodology is to choose the coupling coefficient of the couplers. As mentioned above, all beams except from the first one suffer from spurious excitations. These excitations degrade the performance of the Blass matrix by introducing perturbations in amplitude and phase, as shown in [28] (Equation (6)). It can be seen that the higher the coupling value is, the stronger the effect of these perturbations is, ultimately degrading SLL. On the other hand, if a  
Figure 3.4 Layout of the proposed circular mm-Wave Blass matrix topology. low coupling value is chosen, only a small portion of the power will be coupled to the output ports, and the rest will be consumed by the terminations (see Figure 3).