Dr. Mohamed Ouda Electrical Engineering Department

Islamic University of Gaza

2013

Antenna Theory

EELE 5445

Lecture3: Radiation from Infinitesimal Source

Radiation from an infinitesimal dipole

Definition: The infinitesimal dipole is a dipole whose length Δl is

much smaller than the wavelength λ of the excited wave, i.e.

The infinitesimal dipole is equivalent to a current element IΔl, where

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Magnetic vector potential

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Power density and overall radiated

power of the infinitesimal dipole

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Duality in Maxwell’s equations

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DUAL QUANTITIES IN

ELECTROMAGNETICS

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Radiation from an infinitesimal

magnetic dipole

Using the duality theore for obtaining the vector potential

and the field vectors of a magnetic dipole (magnetic current

element) ImΔl

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Equivalence between a magnetic

dipole and an electric current loop First, we prove the equivalence of the fields excited by the

following sources:

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If the boundary conditions (BCs) for E1 and E2in are the same

and the excitations of both fields fulfill

then both fields are identical i.e.,

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Consider a loop [L] of electric current I.

The integral on the left side is the electric current I. M is assumed

non-zero and constant only at the section Δl , which is normal to

the loop’s plane and passes through the loop’s centre. Then,

The magnetic current Im corresponding to the loop [L] is obtained

by multiplying the magnetic current density M by the area of the

loop A[L] , which yields

Thus, a small loop of electric current I and of area A[L] creates EM field

equivalent to that of a small magnetic dipole (magnetic current

element) Im Δl.

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Field vectors of an infinitesimal loop

antenna By substitution

The far-field terms show the same behaviour as in the case of an

infinitesimal dipole antenna.

The radiated power can be found to be

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Radiation zones – introduction

Reactive near-field region

This is the region immediately surrounding the antenna, where the

reactive field dominates and the angular field distribution is different

at different distances from the antenna.

where D is the largest dimension of the antenna

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For the infinitesimal dipole

This approximated field is purely reactive ,H and E are in phase

quadrature

Since we see that:

(1) Hφ has the distribution of the magnetostatic field of a current

filament IΔl (remember Bio-Savart’s law);

(2) Eθ and Er have the distribution of the electrostatic field of a dipole

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The field is almost purely reactive in the near zone is obvious

Its imaginary part is

The radial near-field power flow density Pr has negative

imaginary value and decreases as 1/r5

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The near-field Pθ power density flow component is also

imaginary and has the same order of dependence on r but it is

positive:

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Radiating near-field (Fresnel)

region

This is an intermediate region between the reactive near-field

region and the far-field region, where the radiation field is

more significant but the angular field distribution is still

dependent on the distance from the antenna.

In this region,

For most antennas, it is assumed that the Fresnel region is

enclosed between two spherical surfaces

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The fields of an infinitesimal dipole in the Fresnel region are

obtained by neglecting the higher-order (1/r)n-terms:

The radial component Er is still not negligible, but the

transverse components(Eθ and Hφ ) are dominant

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Far-field (Fraunhofer) region Only terms 1/ r are considered when . The angular field

distribution does not depend on the distance from the source any

more. The field is a transverse EM wave.

For most antennas, the far-field region is defined as

The far-field of the infinitesimal dipole is obtained as

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Important features of the far field

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Far-field approximation

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