Problems on Antenna Theory (1)

The electric field produced by an arbitrary time harmonic current distribution may be expressed as

where is the free space dyadic Green’s function given by the expression

= under the integral sign.

In the above expression stands for the free space scalar Green’s function given by=

Determine the expression for the far-zone field due to a very small (electrically) antenna possessing a uniform current distribution (i.e, a Hertz dipole) oriented along the z – direction by using the expression provided above and making the necessary far-field approximations. You should clearly state the conditions for the existence of the far-zone fields under which the related approximations can be made.

Hint:

(a) One may use the symmetry property of and use instead of in expressing the integral for in a form which will simplify the integrand.

(b) When the far-zone conditions prevail, one may omit “some” of the differentiations coming up in the evaluation of the field entities.

Given:

=

,

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Consider an isotropic and inhomogeneous medium. Prove that, in “geometrical optics”, in the zero wavelength limit, the E-field is perpendicular to the to the ray, which determines the direction of propagation of the wave front.

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Using the tools of Geometrical Optics, determine the equation for the surfaces of a microwave lens antenna having an index of refraction larger than unity. The lens should be capable of convertinf a spherical wavefront into a planar one, and vice versa. The focal point of the lens, measured along the axis of symmetry, lies at a distance F from the apex. Use the polar coordinates in expressing the equation of the surface of the lens.

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Derive the expression for the transformation of polarization of the E-field of an incoming wave in undergoing reflection from a perfectly conducting reflector in using the aperture field integration technique. Comment on the underlying assumptions.

( given : aXbXc = b(a.c)-c(a.b) , where a,b,c are vectors )

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A plane wave is incident on the the concave side of a paraboloidal reflector antenna.

The equation representing the reflector surface is given by

x² + y² = 4 f z

where f stands for the focal length.

The E-field of the incident wave given by

E₀ e^+jk₀^z ; (with e^jwt type of time dependence)

a) Determine the expression for the surface current density J_s induced on the reflector surface using the “Physical Optics” (PO) approach.

b) Briefly discuss the underlying assumptions of the PO approach.

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A parabolic reflector is illuminated by means of a feed antenna. The E-field of the wave incident on the reflector is given by

where C is a constant proportional to the square root of the power radiated by the feed and the subscript ‘f’ indicates the coordinate variables with respect to the feed.

The expression for the unit normal to the reflector is given by

Using the Geometrical Optics approach, determine the expression for the E-field distribution over the focal plane of the reflector; specifying the aperture E-field amplitude, phase and polarization ( expressed in terms of the unit vectors of the feed system). .)

Given : ; sin 2A = 2 sinAcosA

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