Characterization of Hyperbolic Characterization of Hyperbolic SubreflectorsSubreflectors in Cassegrain Reflector in Cassegrain Reflector Antennas Using NonAntennas Using Non--Linear Least Linear Least

Squares Curve FittingSquares Curve Fitting

Frank N. Bauregger, W6QIFrank N. Bauregger, W6QI

OutlineOutline

IntroductionIntroductionCassegrain Antenna GeometryCassegrain Antenna GeometryThe ParabolaThe ParabolaThe HyperbolaThe HyperbolaCassegrain Antenna SetupCassegrain Antenna Setup

Least Squares Curve FittingLeast Squares Curve FittingTheoryTheoryImplementationImplementation

ResultsResultsTheoryTheoryExperimentalExperimental

ConclusionsConclusions

The Cassegrain Reflector AntennaThe Cassegrain Reflector Antenna

Main Reflector (36”)

Feed Horn(Rectangular)

Subreflector

Advantages (Compared to Prime Focus Advantages (Compared to Prime Focus Reflector Antennas)Reflector Antennas)

Feed spillover is directed toward cold sky in Feed spillover is directed toward cold sky in EME/satellite applicationsEME/satellite applicationsBulky high frequency electronics conveniently located Bulky high frequency electronics conveniently located behind main reflectorbehind main reflectorCan achieve much greater focal length than prime focus Can achieve much greater focal length than prime focus antennasantennas

Reduced crossReduced cross--polarizationpolarizationUseful for Useful for monopulsemonopulse feeds, which require a large feed feeds, which require a large feed apertureaperture

Main beam may be broadened or scanned by moving Main beam may be broadened or scanned by moving only the small subreflectoronly the small subreflector

Disadvantages (Compared to Prime Focus Disadvantages (Compared to Prime Focus Reflector Antennas)Reflector Antennas)

Added complexity of second reflecting surfaceAdded complexity of second reflecting surfaceProper mounting and alignmentProper mounting and alignment

Potential to over/underPotential to over/under--illuminate two surfaces illuminate two surfaces instead of just oneinstead of just oneMore difficult to tuneMore difficult to tune

Two degrees of freedomTwo degrees of freedom

The ParabolaThe Parabola

• The distances from the focal point to all points on the aperture plane are equal• Plane wave is focused to a point• One focal point at (a,0)

fxy 42 =

The HyperbolaThe Hyperbola

12

2

2

2

=−by

ax

• The set of all points whose difference in distance between two foci is constant• Two foci of hyperbola at (±c,0)

x

y

222 bac +=

Putting it all TogetherPutting it all Together

Fm

FVFR

• Fm – Main reflector focal point• FR – Subreflector “real focal point”• FV – Subreflector “virtual focal point”

The ProblemThe Problem

You make a great deal on a surplus precision You make a great deal on a surplus precision Cassegrain reflector antenna!Cassegrain reflector antenna!You measure the main reflector diameter and You measure the main reflector diameter and focal length, but you know nothing about the focal length, but you know nothing about the hyperbolic subreflectorhyperbolic subreflectorSolution:Solution: Take measurements of the surface of Take measurements of the surface of the subreflector, and fit a hyperbola to the the subreflector, and fit a hyperbola to the measured datameasured data

NonNon--Linear Least SquaresLinear Least Squares

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛+== 11),(

2

byayfx s

ss α

⎥⎦

⎤⎢⎣

⎡=

ba

αunknowns

measurements ( )ii ss yx ˆ,ˆ

[ ]2ˆ)ˆ,(∑ −i

ss iixyf

minα

α

• We must find the set of unknowns which minimizes the square of the error between the function and the set of measurements

NonNon--Linear Least SquaresLinear Least SquaresInitialGuess

atUnknowns

ComputeResidual

ComputeAdjustment

toUnknowns

ResidualBelow

Threshold?Finished

ApplyAdjustment

toUnknowns

yesno

*

Theoretical Results from the Curve FittingTheoretical Results from the Curve Fitting

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

"856.1"089.1

ba

α

ResidualsResiduals

"0009.0=σ"0002.0−=μ

Final Cassegrain Antenna SystemFinal Cassegrain Antenna System

Experimental Results from the Curve FittingExperimental Results from the Curve Fitting

• Antenna checked out OK when tuned to a distant 47 GHz beacon

• Single, sharp main lobe• HPBW was a fraction of a degree• -20 dB sidelobes

•• Antenna was used to set two 47 GHz world records at 290 Antenna was used to set two 47 GHz world records at 290 and 313 km, oneand 313 km, one--way at 345 kmway at 345 km

ConclusionsConclusions

Demonstrated a nonDemonstrated a non--linear least squares curve linear least squares curve fitting technique for fully characterizing fitting technique for fully characterizing hyperbolic hyperbolic subreflectorssubreflectorsFitted hyperbola matched measured data with Fitted hyperbola matched measured data with ss < 0.001” for a 3< 0.001” for a 3--foot Cassegrain antennafoot Cassegrain antennaThis technique is also applicable to Gregorian This technique is also applicable to Gregorian reflector antennasreflector antennas

AcknowledgementsAcknowledgements

Thanks to Gary AD6FP for use of his threeThanks to Gary AD6FP for use of his three--foot foot Cassegrain antenna during the 2004 ARRL 10 Cassegrain antenna during the 2004 ARRL 10 GHz and Up Cumulative ContestGHz and Up Cumulative ContestThanks to Lars AA6IW and Gary AD6FP for Thanks to Lars AA6IW and Gary AD6FP for inspiring me to write this paperinspiring me to write this paper

The EndThe End

Hartebeesthoek Radio Astronomy Observatory