# Review of Substrate Integrated Waveguide Circuits for Beam-Forming Networks Working in X-Band
## 3 Components for the Beam-Forming Network
### Magic Tee  
The Magic Tee is, in practice, a directional coupler with 3 dB coupling, where the signals at the output ports are in opposition of phase. The device can also be used to make the sum and the difference of two input signals, so it is very important for many applications, for example, radars and mixers. The scattering matrix of the ideal Magic Tee follows [43]:  
\[S=\frac{-j}{\sqrt{2}} \begin{bmatrix}0&1&1&0\\ 1&0&0&1\\ 1&0&0&-1\\ 0&1&-1&0\end{bmatrix} , \tag{7}\]  
where port 1 is the sum port and port 4 is the difference one. Figure 8: Experimental response of the six-port coupler shown in Figure 7. (**a**) Reflection coefficients and isolation among input ports; (**b**) Transmission coefficients. Figures adapted from [36]. Figure 7: Photograph of the prototype of the six-port coupler. Figure adapted from [36]. Figure 9: Experimental phase difference between the output signal on port 4 and 6 when port 1 is fed. The standard waveguide Magic Tee is a 3D component, being a combination of an H-plane junction and an E-plane junction [43]. Therefore, it cannot be easily built using a single layer substrate. In the literature there were three different approaches to solve this problem. The first one made use of two different substrates [44; 45; 46], the second approach consisted of connecting the sum and difference circuits by means of a slotline, and the third approach made use of a phase shifter to adjust the phase of one output [47; 48]. In the first case, the device had to be built by attaching together two substrates, using screws for example. This made the device more complicated and expensive. However, the frequency response was usually better in terms of bandwidth and isolation, with respect to single layer implementations. In [46], a double-layer magic tee was designed starting from the H-plane T junction (Figure 10a). This junction matched well and contained a small slot that was used to connect the difference port fabricated in another board (Figure 10b). The isolation between sum and difference ports was enforced by the different field symmetries when the ports were excited. The two boards were then attached with screws (Figure 11). In the figure, port 1 is the sum port and port 4 is the difference one. The final size of the device, with the transitions to SMA connectors, was 60 \(\times\) 60 mm\({}^{2}\), whereas the core of the coupler without the connectors was 32 \(\times\) 25 mm\({}^{2}\). Figure 12 shows the experimental reflection and transmission coefficients. With this simple design, a return loss of at least 12 dB can be obtained over the band 10.5 GHz-12.7 GHz. The transmission from the difference port to the output ports (i.e., S\({}_{24}\) and S\({}_{34}\)) was affected by additional losses due to the additional length of the waveguide. Figure 11: Assembled double-layer Magic tee. Figure 10: Layers of the Magic Tee: (**a**) First layer with the H-plane T-junction; (**b**) Second layer with the E-plane junction. Figure 13 shows the isolation among the ports. As expected, the isolation between the sum and difference (S41) port was higher. A solution based on the second approach is shown in Figure 14. In this case, the device was built into a single layer, and the H-plane and E-plane junctions were connected by using a slotline between them (Figure 14b). The device was designed in two steps, using the same method of the double-layer magic tee. First, the H-plane junction was optimized with an aperture to make space for the slotline. This aperture was narrow enough in order to avoid any power leakage when the sum port was excited. Then, the  
Figure 14: Single-layer Magic Tee prototype. (**a**) top view; (**b**) bottom view. Figure 12: Experimental response of the Magic Tee prototype shown in Figure 11. (**a**) Reflection coefficients; (**b**) Transmission coefficients. Figure 7.0 Experimental isolation of the Magic Tee prototype shown in Figure 7.1  
E-plane junction was connected using the slotline.