# A 3-D Metal-Printed Dual-Polarized Ridged Waveguide Slot Array Antenna for X-Band Applications
## 3 Antenna Design and Analysis Procedure
### Parameter Extraction of Radiation Slots  
The first parametric sweep is performed with the width of the radiation slot to find the initial optimal value for all slots in the antenna array. It is observed that the width of the slot has an effect on the bandwidth of the slot [20], and the frequency sweeps are performed in the operating frequency region for three different slot widths of 1.5, 2.0, and 2.5 mm as shown in Figure 5. The variation rate of self-admittance tends to increase as the slot width is larger but decreases the radiated power of the slot. When the slot is wider, the excitation distance between the elements will be larger, and the radiative effect on the slot will be smaller. This restricts the maximum power, which can be radiated by a slot in the array to a lower value. Therefore, the optimal parameter selected for the width of the slot is \(slot\_w\) = 2 mm.  
For the longitudinal slots in the VP array, the iterative approach to find a valid set of initial slot parameters is as follows: an appropriate offset is chosen to perform a slot-length parameter sweep to obtain values for the normalized conductance and resonance length at the design frequency. The curves about \(slot\_l\) versus \(slot\_x\) and the obtained results for normalized resonant conductance \(g\) are summarized in Figure 6.  
Figure 5: Effects of the slot width versus normalized admittance at the resonant frequency of 9.8 GHz.  
The normalized conductivity is uniform and proportional to the increase of the groove deviation and the resonance length as the mutual coupling effect between adjacent waveguides is reduced by the radiation waveguide structures of the HP array interwoven between the VP array, thereby increasing the isolation.  
For the inclined slots in the HP array, through each value of inclination angle selected to sweep the resonant slot depth parameters, the resonance conductivity values are recorded. The extracted parameter curves about \(slot\_d\) versus \(slot\_\theta\) and the obtained results for \(g\) are summarized in Figure 7. It is found that the normalized admittance curves diverge with the increase of the tilted angle. Different from the VP array, the normalized conductivity of the HP array is inversely proportional to the increase in inclined angle but proportional to the increase in resonant depth. In order to excite each slot while being in phase and spaced by a half-guided wavelength, adjacent V-shaped slots are arranged symmetrically and in opposite directions.  
Both the HP and VP linear arrays are designed to follow the uniform distribution array with a side lobe level (SLL) of 13.26 dB. The radiated power and resonant conductance of a slot are proportional to the square of the voltage excitation of the slot. The equivalent value of the resonant conductance is calculated as in [17]:  
\[g_{n}=g_{in}\frac{(V_{n}^{s})^{2}}{\sum\limits_{i=1}^{n}(V_{n}^{s})^{2}} \tag{1}\]  
where, \(g_{in}=2\) is the matching condition for the center feed of the linear slot array. The number of slots \(N=8\), and \(V_{n}^{s}=1\) is the voltage excitation for each slot. Therefore, the normalized conductance of each slot is calculated as \(g_{n}=2/N\). Once the slot conductance is known, the slot parameters can be determined based on the curves in Figures 6 and 7. Thereby, the design parameters for the longitudinal slot are \(slot\_1\) = 19.15 mm and \(slot\_x\) = 1.25 mm and, for the V-shaped slot, are \(slot\_d\) = 4.0 mm and \(slot\_\theta\) = 23.75\({}^{\circ}\).  
Figure 6: Parameter extraction curves of the resonant slot length and slot offset versus the resonant slot conductance for the VP array.