# A Gap Waveguide-Based Compact Rectangular Waveguide to a Packaged Microstrip Inline Transition
## 4 Transition Design  
For the design of the Chebyshev transformer, which matches the standard rectangular waveguide to the first section of the GGW, the procedure presented in [27] has been followed. The impedances of the different transformer sections were calculated by the standard formulation [31]. The order of the transition, 5, has been calculated to reach 20 dB return loss in the full W-band. The impedance of the groove gap waveguide sections was adjusted by changing their height according to the waveguide electric impedance formula [32].  
\[Z_{gg}=2\frac{\eta}{\sqrt{\epsilon_{e}}}\frac{b}{a} \tag{1}\]  
where \(\eta\) is the free-space impedance and \(\epsilon_{e}\) is the effective dielectric constant of the partially filled gap waveguide section. The waveguide width, \(a\), corresponds to the width of the equivalent rectangular waveguide that provides the same propagation constant as the gap waveguide. Finally, \(b\) is the height of each waveguide section, and its value is used to control the impedance of each section. Since \(\epsilon_{e}\) depends also on \(b\), an iterative process is followed until the height that provides the desired impedance is obtained.  
For the first transition it must be taken into account that the GGW is partially filled with the dielectric substrate. Therefore, the effective dielectric constant, \(\epsilon_{e}\), changes with different GGW heights.  
Figure 3: Comparison of the propagation constant of width (\(w\)) groove gap waveguides and a WR10 standard rectangular waveguide. The gray areas correspond to the limits of the bandgap of the bed of nails.  
As a consequence, if \(\epsilon_{e}\) changes, the cutoff frequency (\(f_{c}\)) of the second order mode of this waveguide, i.e., the \(TE_{2,0}\), also does: the smaller the GGW height, the higher the \(\epsilon_{e}\), and consequently, the lower the cut-off frequency. The main implication of this behavior is that for small waveguide heights the effective dielectric constant will be close to \(\epsilon_{r}\) and the second order mode of the partially filled GGW can be in propagation in the W-band. Therefore, an analysis of the cut-off frequency of the second order mode of the GGW, \(f_{c}\), versus the waveguide height (\(b\)) was carried out. The results are shown in Figure 4 for a constant width GGW, with \(w=2.65\) mm [33]. It can be observed that for waveguide heights lower than 0.6 mm, \(\epsilon_{e}\) is so high that the second mode propagates within the W-band; therefore, those GGW sections whose height, \(b_{n}\), allows propagation of this second order mode are substituted by ridge gap waveguide sections in order to achieve the required impedance without the presence of higher-order modes.  
This is not a problem for the second transition, in which the dielectric substrate does not fill any of the steps of the Chebyshev transformer. In this case we have one metal step that connects the microstrip and the groove gap waveguide transformer calculated by the standard formulation. The metal step is calculated as a dielectric-filled ridge gap waveguide in order to have the same impedance as the microstrip line. That way, the function of the ridge metal step is just to ensure the connection between the microstrip and the Chebyshev transformer based on groove gap waveguide.  
Finally, the second part of the transition is a microstrip taper that matches the 50 \(\Omega\) microstrip line to the impedance of the dielectric-filled ridge gap waveguide (DFRGW) [34] for the first design. For the second case there is no need for a microstrip taper and the 50 \(\Omega\) microstrip line is directly connected to a ridge waveguide with the same width (and impedance) as the microstrip line.