Pyramidal Horn Design
Introduction
This application calculates the optimum design parameters for an X-band pyramidal horn.
Reference: Based on example 13.6, page 782 of Antenna Theory, Analysis and Design, Constantine A. Balanis, 3rd Edition.
restart:withplots:withColorTools:
Parameters
The design equations require that the gain be unitless and that the wavelength, l, be in cm. So our first equations which address design criteria (1) and (2), are
Gain in dB at design frequency:
G__odB≔22.6:
Hence
G__o≔10G__odB10
G__o≔181.9700859
Frequency (s-1):
f≔11.0⋅109.:
Geometrical constraints (cm):
a≔2.286:b≔1.016:
Speed of light (cm/s):
c≔3⋅1010:
Wavelength (in cm):
λ≔cf
λ≔2.727272727
Governing Equations
These equations are extracted from the reference, and are derived therein.
We require the following for optimum directivity.
cons1≔G__o=2⋅πλ2⋅a__1⋅b__1:
cons2≔a__1=3⋅λ⋅ρ__h:
cons3≔b__1=2⋅λ⋅ρ__e:
The dimensions pe and ph are equal.
cons4≔p__e=b__1−b⋅ρ__eb__12−14:
cons5≔p__h=a__1−a⋅ρ__ha__12−14:
cons6≔p__e=p__h:
Numerically Solve the Governing Equations
res≔fsolvecons1,cons2,cons3,cons4,cons5,cons6
res≔a__1=16.55573668,b__1=13.01154178,p__e=27.97911838,p__h=27.97911838,ρ__e=31.03837357,ρ__h=33.50018428
Plot the E-Plane Radiation Pattern
assignres
ρ__1≔ρ__e2−b__122:
t__1≔θ→2λ⋅ρ__1⋅−b__12−ρ__1⋅sinθ:
t__2≔θ→2λ⋅ρ__1⋅b__12−ρ__1⋅sinθ:
F≔θ→FresnelCt__2θ−FresnelCt__1θ−I⋅FresnelSt__2θ−FresnelSt__1θ:
Radiation Pattern
E__θ≔θ→20⋅log101+cos(θ)⋅F(θ)F(0):
plotE__θθ,θ=−π2..π2,thickness=7,color=ColorRGB,0,79/255,121/255,axes=frame
polarplotE__θθ+70,θ=0..2 π, thickness=0,color=ColorRGB,0,79/255,121/255,filled=true,transparency=0,title=E-Plane Radiation Pattern,size=800,800
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