## Parabolic Reflecting Antennas

Actual antennas are not perfect. Some unwanted signals will enter the head unit. Waves are diffracted and scattered at the rim of the dish. Unwanted signals may thus converge on the focal point. Surface irregularities cause reflection errors. The head unit and support structure block signals. Galactic noise will enter along with the desired signal (background radiation from the "big-bang" is most pernicious at 10-15 GHz5. The sun and the earth are also sources of electrical noise at all frequencies). Signals may be absorbed rather than reflected by the antenna dish. Some of these real life problems are illustrated in the diagram at right.

In spite of these complications, we achieve accurate approximations and considerable understanding through our idealized geometric models. So we shall study the shapes of the antenna in detail here.

We all know how to derive the formula for a parabola. But for quick reference, here it is:

y = x^2/4p (#)

With x = the radius of the dish, (i.e. aperture/2), we may rearrange (#) above as follows,

f = D^2/16d (!)

Where D is the aperture and d is the depth of the dish. In this way, we may locate the focal point, f, given a particular dish. These parameters are easy to measure and the calculation is easy to perform by hand.

A dish antenna may be shallow or deep depending on the slice of the paraboloid envisaged during manufacture. The figure to the right illustrates three possibilities. Practically speaking, it is difficult to illuminate the dish uniformly with the feed inside the aperture plane (A). This is because waves arriving from opposite directions tend to cancel through superposition. This is also why our eye peers in one direction only. On the other hand, placing the focal point well outside the aperture plane increases the chance of receiving unwanted signals and noise. The feed point is not well shielded, and this configuration increases the chance of transmission loss. Signals from the feed horn may miss the edge of the dish. This effect is known as "over-illumination". The ratio of the focal distance to the dish diameter, denoted f /D is a standard component parameter used by systems installers. For a feed point at the aperture plane, parabola geometry dictates that radio to be 0.25. Observe that if f = d in (!), we have

f = D^2/16f

f^2 = D^2/16

f=D/4.

Therefore, f /D = .25, as we said.

Deep dishes with low f /D ratios tend to have higher efficiency and are more shielded from noise. In practice, the f /D ratios are greater than .25 but less than 1 for manufacturing reasons. It is much easier to fabricate, finish, and transport a shallow dish. There are also practical problems with illuminating angles greater than 180 degrees. The problem of over-illumination is countered by the design of the head unit, which must compensate by restricting the beamwidth there.

The gain of a dish antenna, (the ratio of output to input power and denoted Ga) is dependent on its size and the length of the waves being transmitted/received. An expression for the gain which takes into account the efficiency (rho) of the system is:

Ga = 10 ln(((pi*d)^2*rho)/lam^2 dB

Where d = the reflector aperture (m)

rho = the normalized antenna efficiency (typically .60-.80)

lam = wavelength of radiation (m)

and dB indicates decibels, the usual units for power.

The expression above comes from solutions to Maxwell's equations. We can see why dish antennas are only common for work with very short waves. Note that for a given efficiency, gain for a parabolic reflector increases as the aperture increases and as the radiation wavelength decreases. We generally find dish antennas used in the micro-wave range of frequencies, where 1-40 GHz corresponds to wavelengths about 30 cm - 7.5 mm.

Typical parabolic type antennas, called "prime focus" antennas, position the focal point at the center of the dish. Most large antennas use this method because of the mechanical stability inherent with this geometry. For medium to high power systems operating at or above 10 GHz, this configuration is rare. For these small antennas, the offset focus configuration is most often used

The figures above show that the offset focus dish is really a section cut out of a much larger parabola (this does not imply that they are manufactured this way!).

This geometry has several advantages. The feed head is out of the way of incoming signals and is inclined away from the earth toward a cool sky. Further, snow and other debris can slide off the dish more easily. We see a specimen of this type in the photo above.

As we noted, system efficiency is affected by the conformity of the actual

reflecting surface to the ideal model: the parabola. However, large local deviations from this shape are not as critical as small deviations overall. Above we see reflectors which have large voids in them. They continue to function efficiently because the size of the voids are small compared to the length of the waves which they reflect. The cylindrically parabolic reflector focuses in one spatial direction only. This makes aiming simpler, but reduces gain.

In the photo above we have another type of antenna. It is a radio lens, analogous to an optical lens. Although this paper is not describing refracting components, it is interesting to note that these lenses have voids in their structure as well, just as the reflector mounted below it does. Here we see an actual lens of this type. The radiation it focuses has a wavelength which is comparable to the spaces between the plates in the lens.

The fact that large local deviations are insignificant to performance is good news for any amateur who wants to build a parabolic antenna. Precision on a small scale is not a limiting concern. As amateur astronomers discover when grinding their mirrors, the "Rayleigh limit" for telescopes is that little gain increase is realized by making the mirror accuracy greater than 1/8 wavelength. An eighth wave is 3.4 inches at 432 MHz, 1.1 inches at 1296 MHz and 0.64 inch at 2300 MHz, (UHF and microwave radio frequencies, respectively).

Below, we have a photo of a common design used by amateurs. The so-

called "stressed" parabolic antenna. It is lightweight, portable, and easy to build. Each spoke is basically a cantilevered beam with end loading. The equations of beam bending predict a near perfect parabolic curve for very small deflections (actually any engineering student will remind you that an end loaded cantilever deforms as a cubic function of distance from its support - but cubics are enough like quadradics for small displacements). However, here the deflections are not that small and the loading is not perpendicular. The uncorrected surface is enough for 432 and 1296 MHz bands, but bending the supports a little to fit a pattern is a good suggestion.

By placing the transmission line inside the central pipe that supports the feed horn, the area of the shadows or blockages on the reflector surface is much smaller than in other feeding and supporting systems, thus increasing
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