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Displacement of the feed or source along the focal axis of the parabola has no effect on pointing error because this effect is symmetrical about the RF axis. However, the resulting defocusing reduces the gain and raises the side lobes. If the feed is displaced δc normal to the focal axis, the RF signal reflected from the design parabola will be deflected by θc as shown in Fig. 3. However, the beam deviation angle, θc, depends on and is reduced considerably by the magnification factor M:

where e is the eccentricity of the hyperboloid.

The displaced (virtual) focal point, due to feed translation, is –δc/M, and the corresponding beam deviation or pointing error equation for feed displacement is:

*Sign Convention: Clockwise beam rotation (upward focal point motion) is positive.

The effect of feed translation on pointing error is shown in Fig. 4. The beam deviation angle, θc, is only slightly affected by displacement of the feed phase center. θc is approximately one-fifth the feed displacement angle or δc/F.

Effects of hyperbola displacement

Like feed displacement, hyperbola-vertex displacement along the focal axis of the design parabola does not affect pointing error or beam deviation, because whatever happens is symmetrical about the RF axis.

Displacement of the hyperbola vertex normal to the focal axis is reflected as displacement of the source image from the focus of the design parabola (Fig. 5). The beam deviation angle, θht, resulting from this displacement, depends on and is reduced slightly by M, the magnification factor.

The beam-deviation or pointing-error equation is determined by computing the location of the virtual focal point, X, as follows:

With the feed displaced an amount equal to the displacement (-δht) of the hyperbola, the displaced focal and feed points become coincident. The pointing error becomes simply:

A correction must be applied to this equation to relate the displaced feed point to the design focal axis. As before, this is accomplished by “translating” the displaced focal point an amount equal to +δht back to the original design focal axis. This effect is identical to beam deviation due to feed translation. Therefore:



the resultant beam deviation or pointing error equation for hyperbola translation becomes:

From the virtual focal point, (X), the same equation can be derived as follows:

X = -δht + Y.


X = -δht + δht/M

= -δht (M – 1)/M


The effect of hyperbola translation on pointing error is shown in Fig. 6. Note that the beam deviation angle, θht, is approximately equal to δht/F which is the full hyperbola translational pointing error.