Pyramidal Horn Design - Maple Help

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.

 > $\mathrm{restart}:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{plots}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{ColorTools}\right):$

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:

 > $\mathrm{G__odB}≔22.6:$

Hence

 > $\mathrm{G__o}≔{10}^{\frac{\mathrm{G__odB}}{10}}$
 $\mathrm{G__o}{≔}{181.9700859}$ (2.1)

Frequency (s-1):

 >

Geometrical constraints (cm):

 > $a≔2.286:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}b≔1.016:$

Speed of light (cm/s):

 > $c≔3\cdot {10}^{10}:$

Wavelength (in cm):

 > $\mathrm{λ}≔\frac{c}{f}$
 ${\mathrm{\lambda }}{≔}{2.727272727}$ (2.2)

Governing Equations

These equations are extracted from the reference, and are derived therein.

We require the following for optimum directivity.

 > $\mathrm{cons1}≔\mathrm{G__o}=\frac{2\cdot \mathrm{\pi }}{{\mathrm{λ}}^{2}}\cdot \mathrm{a__1}\cdot \mathrm{b__1}:$
 > $\mathrm{cons2}≔\mathrm{a__1}=\sqrt{3\cdot \mathrm{λ}\cdot \mathrm{ρ__h}}:$
 > $\mathrm{cons3}≔\mathrm{b__1}=\sqrt{2\cdot \mathrm{λ}\cdot \mathrm{ρ__e}}:$

The dimensions pe and ph are equal.

 > $\mathrm{cons4}≔\mathrm{p__e}=\left(\mathrm{b__1}-b\right)\cdot \sqrt{{\left(\frac{\mathrm{ρ__e}}{\mathrm{b__1}}\right)}^{2}-\frac{1}{4}}:$
 > $\mathrm{cons5}≔\mathrm{p__h}=\left(\mathrm{a__1}-a\right)\cdot \sqrt{{\left(\frac{\mathrm{ρ__h}}{\mathrm{a__1}}\right)}^{2}-\frac{1}{4}}:$
 > $\mathrm{cons6}≔\mathrm{p__e}=\mathrm{p__h}:$

Numerically Solve the Governing Equations

 > $\mathrm{res}≔\mathrm{fsolve}\left(\left\{\mathrm{cons1},\mathrm{cons2},\mathrm{cons3},\mathrm{cons4},\mathrm{cons5},\mathrm{cons6}\right\}\right)$
 ${\mathrm{res}}{≔}\left\{\mathrm{a__1}{=}{16.55573668}{,}\mathrm{b__1}{=}{13.01154178}{,}\mathrm{p__e}{=}{27.97911838}{,}\mathrm{p__h}{=}{27.97911838}{,}\mathrm{ρ__e}{=}{31.03837357}{,}\mathrm{ρ__h}{=}{33.50018428}\right\}$ (4.1)

 > $\mathrm{assign}\left(\mathrm{res}\right)$
 > $\mathrm{ρ__1}≔\sqrt{{\mathrm{ρ__e}}^{2}-{\left(\frac{\mathrm{b__1}}{2}\right)}^{2}}:$
 > $\mathrm{t__1}≔\mathrm{θ}\to \sqrt{\frac{2}{\mathrm{λ}\cdot \mathrm{ρ__1}}}\cdot \left(-\frac{\mathrm{b__1}}{2}-\mathrm{ρ__1}\cdot \mathrm{sin}\left(\mathrm{θ}\right)\right):$
 > $\mathrm{t__2}≔\mathrm{θ}\to \sqrt{\frac{2}{\mathrm{λ}\cdot \mathrm{ρ__1}}}\cdot \left(\frac{\mathrm{b__1}}{2}-\mathrm{ρ__1}\cdot \mathrm{sin}\left(\mathrm{θ}\right)\right):$
 > $\mathrm{F}≔\mathrm{θ}\to \mathrm{FresnelC}\left(\mathrm{t__2}\left(\mathrm{θ}\right)\right)-\mathrm{FresnelC}\left(\mathrm{t__1}\left(\mathrm{θ}\right)\right)-I\cdot \left(\mathrm{FresnelS}\left(\mathrm{t__2}\left(\mathrm{θ}\right)\right)-\mathrm{FresnelS}\left(\mathrm{t__1}\left(\mathrm{θ}\right)\right)\right):$

 > $\mathrm{E__θ}≔\mathrm{θ}\to 20\cdot \mathrm{log10}\left(\left|1+\mathrm{cos}(\mathrm{θ})\right|\cdot \frac{\left|F(\mathrm{θ})\right|}{\left|F(0)\right|}\right):$
 > $\mathrm{plot}\left(\mathrm{E__θ}\left(\mathrm{θ}\right),\mathrm{θ}=-\frac{\mathrm{π}}{2}..\frac{\mathrm{π}}{2},\mathrm{thickness}=7,\mathrm{color}=\mathrm{Color}\left("RGB",\left[0,79/255,121/255\right]\right),\mathrm{axes}=\mathrm{frame}\right)$
 >