# Design of True Time Delay Millimeter Wave Beamformers for 5G Multibeam Phased Arrays
## 2 Beamforming Networks Design
### Blass Matrix Architecture
Traditional couplers can exhibit this behavior by using either bondwires and bridges, such as the Lange directional coupler (see Figure 5b) or branch-line directional couplers (see Figure 5a). However, the first approach increases significantly the losses as well as the fabrication complexity especially at mm-Wave frequencies, while the second approach leads into a significantly larger design. To avoid both the bondwires' complexity and minimize the undesired losses, a dual layer directional coupler is used [49], as shown in Figure 6. This coupler is comprised of two rectangular patches (Figure 6) printed on the top and bottom surface of a two-sided substrate with a common ground plane. The two patches are coupled through a rectangular slot etched on thecommon ground plane. The dimensions of the patches and the slot are selected to achieve the desired coupling coefficient. To design the coupler, a quasi-static approach is initially used, utilizing the design formulas given by [PERSON] [50]. Then, the design is optimized by running full-wave simulations. Figure 4: Circular Blass matrix topology. Figure 3: Blass matrix schematic. As mentioned above, one of the most critical components of the Blass topology are the directional couplers. Let us take the case when all couplers have the same coupling coefficient as proposed in [28]. In this case, the higher are the coupling values (see Figure 3), the higher is the circuit efficiency we can achieve. This is expected as more power will be distributed to the radiating elements, instead of getting dissipated at the lines' terminations. Moreover, it can be easily observed that, for every non-spurious path case, the input signal realizes only one coupling value (see in Figure 3 how the signal flows from input port 2 to output port 2) before it reaches any output port. Therefore, an effect of the order of \(C\) [e.g., \(O(C)\)] is introduced on this signal, both in amplitude and phase. On the other hand, for the case of a spurious path, the signal has to travel through at least three \"coupled ports\", introducing an effect of at least \(O(C^{3})\). Thus, the lower the coupling coefficient of the coupler is, the lower the effect of the spurious path is. Therefore, there is a trade-off between the efficiency of the network and the errors introduced on both the amplitude and phase of the excitations. For our case, we choose as a good balance coupling values of \(C=-11.5\) dB [see Figure 7a]. In the next subsection, the design methodology of the proposed \(N\times M\) Blass matrix beamforming network is presented. Figure 5: Traditional implementations of directional cross couplers: (**a**) multi-section branch line cross-coupler; and (**b**) Lange directional coupler. Figure 6: Layout of the dual-layer directional coupler. Blass Matrix Design Methodology  
As mentioned above, a Blass matrix consists of \(M\) traveling wave feed lines (columns) cross-connected to a set of \(N\) transmission lines (rows), each one feeding an antenna element of the array. The first step of the design process is to choose the number of inputs \(M\) and outputs \(N\), respectively (in this work, \(M=3\) and \(N=4\)). Aiming to steer the beams of our antenna array in a specific direction, the desired phase differences \(\Delta\phi^{(m)}_{(n+1,m)}\) between the output ports for each \(m\) beam are computed, utilizing array theory [51]. Recall that, for a linear phased array with interelement distance \(d\), phase difference \(\Delta\phi\), or group delay \(\tau\), to steer the beam toward an angle \(\theta_{0}\) from broadside is (e.g., [51]):  
\[\Delta\phi=\frac{\omega d\sin\theta_{0}}{c}\quad or\quad\tau=\frac{d\sin\theta _{0}}{c} \tag{1}\]  
where \(c\) is the speed of light in vacuum and \(\omega\) is the angular frequency. Although a true-time-delay beamformer is sought, its design is based on the corresponding phase differences. These are first estimated at the center frequency based on standard antenna array synthesis.