Antennas 97

97

Aperture Antennas

Reflectors, horns.High GainNearly real input impedance

Huygens’ Principle

Each point of a wave front is a secondary source ofspherical waves.

Antennas 98

98

Equivalence Principle

Uniqueness Theorem: a solution satisfying Maxwell’sEquations and the boundary conditions is unique.

1. Original Problem (a): 2. Equivalent Problem (b): outside ,

inside , on , where

3. Equivalent Problem (c): outside , zerofields inside , on , where

To further simplify,Case 1: PEC. No contribution from .Case 2: PMC. No contribution from .

Antennas 99

99

Infinite Planar Surface

To calculate the fields, first find the vector potential dueto the equivalent electric and magnetic currents.

In the far field, from Eqs. (1-105),

Antennas 100

100

Since in the far field, the fields can be approximate byspherical TEM waves,

Thus the total electric field can be found by

Let be the aperture fields, then

Let

Use the coordinate system in Fig. 7-4, then

and

Antennas 101

101

or in spherical coordinate system

Using Eq. (7-8), we have

If the aperture fields are TEM waves, then

This implies

Full Vector Form

Antennas 102

102

Techniques for Evaluating Gain

Directivity

From (7-27), (7-24), (7-61)

Thus, for broadside case,

Total power

Then,

In general, for uniform distribution

If

then

Antennas 103

103

where are the directivity of a line source due to respectively. the main beam direction

relative to broadside.

Gain and Efficiencies

where : aperture efficiency

: radiation efficiency. (~1 for aperture antennas)

: taper efficiency or utilization factor.

: spillover efficiency. is called : illuminationefficiency.

: achievement efficiency. : cross-polarization efficiency. phase-error efficiency.

Beam efficiency

Antennas 104

104

Simple Directivity Formulas in Terms of HPbeam width

1. Low directivity, no sidelobe

2. Large electrical size

3. High gain

Rectangular Horn Antenna

Antennas 105

105

High gain, wide band width, low VSWR

H-Plane Sectoral Horn Antenna

Evaluating phase error

thus the aperture electric field distribution

where is defined in (7-108), (7-109)

Directivity

Antennas 106

106

Figure 7-13: universal E-plane and H-plane pattern with factor omitted, and

Figure 7-14: Universal directivity curves.

Optimum directivity occurs at and

From figure 7.13,

E-Plane Sectoral Horn AntennaThe aperture electric field distribution

See (7-129) for the resulting Directivity

Antennas 107

107

Figure 7-16: universal E-plane and H-plane pattern with factor omitted, and

Figure 7-17: Universal directivity curves.

Optimum directivity occurs at and

From figure 7.13,

Pyramidal Horn Antenna

Antennas 108

108

The aperture electric field distribution

Optimum gain

Design procedure:1. Specify gain , wavelength , waveguide

dimension , .2. Using , determine from the following

equation

3. Determine from

4. Determine , by ,

5. Determine , by ,

6. Determine , by ,

7. Verify if and , by

,

Antennas 109

109

Reflector Antennas

Parabolic Reflector

Parabolic equation:

or

Properties1. Focal point at . All rays leaving , will be

parallel after reflection from the parabolic surface.2. All path lengths from the focal point to any

aperture plane are equal.3. To determine the radiation pattern, find the field

distribution at the aperture plane using GO.

Antennas 110

110

Geometrical Optics (GO)

Requirements1. The radius curvature of the reflector is large

compared to a wavelength, allowing planarapproximation.

2. The radius curvature of the incoming wave fromthe feed is large, allowing planar approximation.

3. The reflector is a perfect conductor, thus the reflectcoefficient .

Parabolic reflector:Wideband.Lower limit determine by the size of the reflector.

Should be several wavelengths for GO to hold.Higher limit determine by the surface roughness of

the reflector. Should much smaller than a wavelength.Also limited by the bandwidth of the feed.

Determining the power density distribution at theaperture by

where ,

Antennas 111

111

PO/surface current method

PO and GO both yield good patterns in main beam andfirst few sidelobes. Deteriorate due to diffraction bythe edge of the reflector. PO is better than GO for offsetreflectors.

Axis-symmetric Parabolic Reflector Antenna

For a linear polarized feed along x-axis, the pattern canbe approximate by the two principle plan patterns asbelow.

where , are E-plane and H-plane patterns.

If the pattern is rotationally symmetric, then . Wehave

Antennas 112

112

Also, the cross-polarization of the aperture field ismaximum in the .

For a short dipole, , ,

At , only x component exists.

F/D increases, cross-polarization decreases.

Since the range of decreases as F/D increases, theterm .

Antennas 113

113

Approximation formula

Normalized aperture field

Thus,

whereEI=edge illumination (dB) =20 log CET=edge taper (dB)=-EIFT=feed taper (at aperture edge) (dB)=Spherical spreading loss at the aperture edge

Example 7-8,1. Estimate EI by the radiation pattern of the feed at

the edge angle of the reflector.2. Calculate due to the distance from the feed to

the edge.3. Estimate ET at the aperture by adding the EI and

. 4. Look up Table 7-1b for n=2.

Antennas 114

114

Offset Parabolic Reflectors

Reduce blocking loss.Increase cross-polarization.

Dual Reflector Antenna

Spill over energy directed to the sky.Compact.Simplify feeding structure.Allow more design freedom. Dual shaping.

Antennas 115

115

Other types

Design example1. Determine the reflector diameter by half-power

beam width. For -11 dB edge illumination,

2. Choose F/D. Usually between 0.3 to 1.0.3. Determine the required feed pattern using

model.

Example 7-9