Design and Implementation of a Pillbox Antenna for an Airborne ...

angle was ψ B = 45 ◦ . A simple MATHCAD calculationgave the length of the feed horn in the offset planeto achieve an equal edge illumination of approximately-10 dB at the edges. The length of the horn was calculatedto be 6.5 cm. It was only later discovered thatthere was an error in the initial calculation of the hornazimuth dimension, it should have been 6 cm instead.• The additional taper due to the space loss in ψ Lis small and can be neglected. The SL at ψ U iscalculated as follows:[ ( )] 80SPL U = −20 log cos 2 2= 4.5 [dB] (4)The space loss will introduce an additional 4.5 dBtaper in the secondary aperture field distribution at theupper edge of the reflector.• The length D az of the aperture required to producean edge illumination of -10 dB at the reflectoredges is calculated by the following equation [16][19]:D az = (1.05A edge + 55.95) λθ az= ((1.05 ◦ × 10) + 55.95) 3.23.8= 56 [cm] (5)• The width of the aperture for a uniform illuminationin the elevation plane is calculated by the followingequation:D el = 57λθ el=57 × 3.225= 7 [cm] (6)• The offset distance was set to h = D/8 so that’h = 7 cm’.2.2.3 Directivity of Pillbox AntennaG D ≈ 1 k · 4πθ az · θ ele=4π0.44 × 0.066 × 1.12= 25.6 [dBi] (7)where θ az and θ ele are the principal azimuth and elevationbeamwidths respectively, ‘k’ is the beam broadeningfactor relative to a uniform distribution, which isalso the loss in directivity. The value of k=0.44 was approximatedfrom the taper imposed on the upper edge ofthe parabolic reflector relative to a uniform distribution.The gain of an antenna which includes the directivityand power gain are well documented in [1].2.2.4 Aperture Field Method:Predicted AzimuthPatternWe used the aperture field method to determine the offsetreflector pattern in the azimuth plane at ψ f = 45 ◦ .We evaluated the equivalent Huygens sources at theaperture and integrated these sources to obtain the reflectorfar-field pattern[2][9] [14]. This section showsthe steps taken to predict the reflector far-field.The equation of the parabolic cylinder in polar coordinatesis given as [6]:ρ (ψ) ==fcos 2 (ψ/2)2f1 + cosψ(8)The primary and secondary power flows are equaland given as [6]:P (y) = I (ψ) dy (9)where I az (ψ) is the feed far-field power pattern ofthe horn in units of watts per radian-meter. P (y) is thesecondary power flow in units of watts per radian-meter.The feed radiation pattern in the H-plane is modelledby the following Fourier Transform expression at thefeed pointing angle ψ f = 45 ◦ [9]:E az (ψ − ψ offset ) =∫ a1/2−a 1/2E 0 cos( πx)e jα dxa