Antennas 97

Aperture Antennas

Reflectors, horns.High GainNearly real input impedance

Huygens’ Principle

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

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Antennas 98

Equivalence Principle

Uniqueness Theorem: a solution satisfying Maxwell’s Equations andthe boundary conditions is unique.

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

on , where

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

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

Case 2: PMC. No contribution from .

Infinite Planar Surface

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Antennas 99

To calculate the fields, first find the vector potential due to theequivalent electric and magnetic currents.

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

Since in the far field, the fields can be approximate by spherical TEMwaves,

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Antennas 100

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

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Antennas 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

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Antennas 102

The Uniform Rectangular Aperture

Let the electric field be

Then,

where

Therefore,

At principle planes

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Antennas 103

For large aperture ( ), the main beam is narrow, the factor is negligible. The half-power beam width

.

Also,

Example: a Uniform Rectangular Aperture

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Antennas 104

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Antennas 105

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Antennas 106

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

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

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Antennas 107

Directivity of an Open-Ended Rectangular Waveguide:

Gain and Efficiencies

where : aperture efficiency

: radiation efficiency. (~1 for aperture antennas)

: taper efficiency or utilization factor.

: spillover efficiency. is called : illumination efficiency.: achievement efficiency. : cross-polarization

efficiency. phase-error efficiency.

Beam efficiency

Simple Directivity Formulas in Terms of HP beam width

1. Low directivity, no sidelobe

2. Large electrical size

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Antennas 108

3. High gain

Example 7-5: Pyramidal Horn Antenna (aperture efficiency=0.51)Measured gained at 40 GHz: 24.7 dB.A=5.54 cm, B=4.55 cm.

Example 7-6: Circular Parabolic Reflector AntennaTypical aperture efficiency: 55%.Diameter: 3.66 mFrequency: 11.7 GHzMeasured Gain: 50.4 dB.Measured .

1. Computed by aperture efficiency

2. Computed by half power beam width

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Antennas 109

Rectangular Horn Antenna

High gain, wide band width, low VSWR

H-Plane Sectoral Horn Antenna

Evaluating phase error

thus the aperture electric field distribution

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Antennas 110

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

Directivity

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

factor omitted, and (a measure of the maximum phase

error at the edge)

Figure 7-14: Universal directivity curves.

Optimum directivity occurs at and

From figure 7.13 for optimum case,

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Antennas 111

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Antennas 112

E-Plane Sectoral Horn AntennaThe aperture electric field distribution

See (7-129) for the resulting Directivity

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

factor omitted, and (a measure of the maximum phase

error at the edge)

Figure 7-17: Universal directivity curves.

Optimum directivity occurs at and

From figure 7.13,

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Antennas 114

Pyramidal Horn Antenna

The aperture electric field distribution

At optimum condition:

Optimum gain

For non optimum case,

Design procedure:1. Specify gain , wavelength , waveguide dimension , .

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Antennas 115

2. Using , determine from the following equation

3. Determine from

4. Determine , by ,

5. Determine , by ,

6. Determine , by ,

7. Verify if and , by

,

Example 7-7: Design a X-band (8.2 to 12.4 GHz) standard horn fedby WR90 ( )waveguide.Goal: at 8.75 GHz.1. Solve for A: 2. Solve for the rest parameters:

3. Evaluating the gain by Fig. 7-14 and 7-17:

(exact phase error)

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Antennas 116

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Antennas 117

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Antennas 118

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Antennas 119

Reflector AntennasParabolic 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.Geometrical Optics (GO)

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Antennas 120

Requirements1. The radius curvature of the reflector is large compared to a

wavelength, allowing planar approximation.2. The radius curvature of the incoming wave from the feed is

large, allowing planar approximation.3. The reflector is a perfect conductor, thus the reflect coefficient

.

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 the aperture by

where ,

PO/surface current method

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Antennas 121

PO and GO both yield good patterns in main beam and first fewsidelobes. Deteriorate due to diffraction by the edge of the reflector.PO is better than GO for offset reflectors.

Axis-symmetric Parabolic Reflector Antenna

For a linear polarized feed along x-axis, the pattern can beapproximate by the two principle plan patterns as below.

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

If the pattern is rotationally symmetric, then . We have

Also, the cross-polarization of the aperture field is maximum in the. Leads to cross-polarization.

For a short dipole, , ,

At , only x component exists.

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Antennas 122

Summary:1. F/D increases, cross-polarization decreases. -7809Since the

range of decreases as F/D increases, the term . 2. Fields inverted because of reflection from conductor.3. Cross-polarization cancels each other on principal plane in the

far field.

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Antennas 123

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Antennas 124

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Antennas 125

Approximation formula

Normalized aperture field

Thus, (7-208)

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

Design procedure of axial symmetrical aperture field: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 7-1 for a suitable n.

Example 7-8: A 28-GHz Parabolic Reflector Antenna fed by circularcorrugated horn.

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Antennas 126

Assume , then

Use Table 7.1b for n=2 and interpolate, we have

( measured)

( measured)

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Antennas 127

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.

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Antennas 128

Other types

Design example1. Determine the reflector diameter by half-power beam width.

For the optimum -11 dB edge illumination,

(7-248)

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

(7-249)

Example 7-9: From 7-248

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Antennas 129

Choose F/D=0.5, then .

From (7.249), find q to approximate the pattern by .

Verify EI=-11 dB, by (7-208).

Note: for feed pattern

Consideration of Gain due to taper and spillover:1. The more taper, the less loss due to spillover, but less taperefficiency .

2. The less taper, the more loss due to spillover, but higher taperefficiency .

3. Optimum efficiency for feed: EI= -11 dB.

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A General Approximate Feed Model for Broad Main Beam andPeak at

Assume

That is the feed pattern

For symmetrical feed,

The value of is chosen to match the real feed pattern at (usually

) by

Then,

For symmetrical feed, the illumination efficiency , the spill overefficiency , the feed gain and EI are computed by

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Antennas 134

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Antennas 135

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