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In the last approximation we have retained the first two terms in the series expansion of the square root term and ignored the d in the numerator of the second but last term. The result thus obtained is identical to Eq. (2.28).

The behaviour of Eqs. (2.25), (2.28) and (2.29) is illustrated in Fig. 2.7 [Mat.2.4], where we show the pathlength difference as function of aperture radius for an antenna with focal ratio 0.4 and a defocus of 1 mm.

The approximation of Eq. (2.28) is essentially indistinguishable from the exact result of Eq. (2.29) and hence is to be preferred over the approximation of Eq. (2.25). The difference between these solutions reaches only 1 mm for a defocus of 4 mm. Thus for all purposes Eq. (2.28) represents the path error adequately. If we adjust the maximum phase error at the aperture edge in Eq. (2.25) from tan2 (-2-) to (1 - cos y0), as given in Eq. (2.28), and just maintain the single quadratic dependence on r of Eq. (2.25), we find that this curve is reasonably close to the exact one and will often be acceptable for practical values of d.

The discussion has dealt with the movement of the feed near the primary focus of the paraboloidal reflector. It is valid without any change for axial defocus of the feed in the secondary focus of a Cassegrain reflector configuration. We only have to consider the equivalent primary-fed paraboloid with a focal length of m times that of the real primary, where m is the magnification factor of the Cassegrain system, given by the expressions of Eq. (2.16) or (2.18). Also, it should be mentioned that an axial shift of the secondary reflector ds from the primary focus causes a phase error over the primary reflector as with a primary feed plus a small phase error over the second ary from its displacement with respect to the feed. We will discuss these aspects further in Section 4.3.3.

aperture radius (m)

Fig. 2.7. Top: pathlength difference in mm from Eq. (2.29)-red, Eq. (2.28)-blue and Eq. (2.25)-magenta, as function of aperture radius for an axial defocus of 1 mm. The green curve is Eq. (2.25) but adjusted to the correct maximum path error at the aperture edge (see text). Note that the dashed blue curve effectively suppresses the exact red curve, showing their near identity. This is shown in the bottom plot of the difference between the red and blue curve with the vertical scale in micrometers.

aperture radius (m)

Fig. 2.7. Top: pathlength difference in mm from Eq. (2.29)-red, Eq. (2.28)-blue and Eq. (2.25)-magenta, as function of aperture radius for an axial defocus of 1 mm. The green curve is Eq. (2.25) but adjusted to the correct maximum path error at the aperture edge (see text). Note that the dashed blue curve effectively suppresses the exact red curve, showing their near identity. This is shown in the bottom plot of the difference between the red and blue curve with the vertical scale in micrometers.

This concludes the description of the geometry of the paraboloidal reflector antenna. In the following chapters we shall treat the electromagnetic characteristics of such antennas. This will include the calculation of the influence of defocus on the radiation pattern, for which the equations derived here will be needed. As we shall see, any defocus quickly deteriorates the beam pattern and being able to determine the correct focus is of great practical importance.

■ 2.3. The Mathematica Routines

This section contains the Mathematica routines used in the text of this chapter. They have been numbered Mat.2.x in the text and are identified by the same number in the first line of the expression.

Mat .2.1 - ALMA antenna geometry; dp = 12.0; f = 4.8; m = 20.; ds = 0.75; Y0 = 2 ArcTan [dp /(4 f)] (180 / p) Fo = 2 ArcTan [dp /(4 m f)] (180 / p) e = (m + 1) / (m - 1) m = (e + 1) / (e - 1)

fc = (ds / 2) (Cot[Y0 p / 180] + Cot[F0 p / 180]) l = fc (e - 1)/ 2 e

Mat .2.2 - lateral defocus pathlength error; f = 4.8; c = 0; 5 = 0.001; dl = (5 r / f) / (1 + (r /(2 f)) "2) ; Plot [1000 dl, {r, 0, 6}, PlotRange 0 {0, 1}, Frame 0 True, GridLines 0 Automatic, FrameLabel 0 { "Radius (m)", "Pathlength (mm)"}]

Mat .2.3 - Exact vs approx path error; |

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