Antennas 97

Antennas 97

Aperture Antennas Reflectors, horns. High Gain Nearly real input impedance Huygens’ Principle Each point of a wave front is a secondary source of spherical waves.

97

Antennas 98

Equivalence Principle

Uniqueness Theorem: a solution satisfying Maxwell’s Equations and the boundary conditions is unique. 1. 2.

Original Problem (a): 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 .

98

Antennas 99

Infinite Planar Surface

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

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

99

Antennas 100

Since in the far field, the fields can be approximate by spherical 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 100

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

101

Antennas 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 102

Antennas 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 : illumination efficiency. : achievement efficiency. : crosspolarization efficiency. phase-error efficiency. Beam efficiency

103

Antennas 104

Simple Directivity Formulas in Terms of HP

beam width 1.

Low directivity, no sidelobe

2.

Large electrical size

3.

High gain

Rectangular Horn Antenna

104

Antennas 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

105

Antennas 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 From figure 7.13,

E-Plane Sectoral Horn Antenna The aperture electric field distribution

See (7-129) for the resulting Directivity 106

and

Antennas 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 From figure 7.13,

Pyramidal Horn Antenna

107

and

Antennas 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

,

, 108

by

Antennas 109

Reflector Antennas Parabolic Reflector

Parabolic equation: or

Properties 1. 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. 109

Antennas 110

Geometrical Optics (GO) Requirements 1. 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

,

110

Antennas 111

PO/surface current method

PO and GO both yield good patterns in main beam and first few sidelobes. 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 be approximate by the two principle plan patterns as below.

where

,

are E-plane and H-plane patterns.

If the pattern is rotationally symmetric, then have 111

. We

Antennas 112

Also, the cross-polarization of the aperture field is maximum in the . For a short dipole,

At

,

,

, only x component exists.

F/D increases, cross-polarization decreases. Since the range of term .

decreases as F/D increases, the

112

Antennas 113

Approximation formula Normalized aperture field

Thus, where EI=edge illumination (dB) =20 log C ET=edge taper (dB)=-EI FT=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.

113

Antennas 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.

114

Antennas 115

Other types

Design example 1. Determine the reflector diameter by half-power beam width. For -11 dB edge illumination, 2. 3.

Choose F/D. Usually between 0.3 to 1.0. Determine the required feed pattern using model.

Example 7-9 115