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1. UEC747: ANTENNA AND
WAVE PROPAGATION
Jan-May 2021
Dr Rajesh Khanna, Professor ECE
and
Dr Amanpreet Kaur, Assistant Professor, ECE
Lecture 13: Linear Wire Antennas

2. • A better approximation of the current distribution of wire antennas, whose lengths are usually λ/50 < l ≤ λ/10, is the
triangular variation of Figure 1.16(a). The sinusoidal variations of Figures 1.16(b)–(c) are more accurate representations of
the current distribution of any length wire antenna.
• The radiation properties of an infinitesimal dipole, which is usually taken to have a length l ≤ λ/50, were discussed in the
previous section. Its current distribution was assumed to be constant.
Linear Wire Antennas

3. 4.3 Small Dipole
• The most convenient geometrical arrangement for the analysis of a dipole is usually to have it positioned symmetrically about
the origin with its length directed along the z-axis, as shown in Figure 4.4(a). This is not necessary, but it is usually the most
convenient. The current is distribution of a small dipole (λ/50 < l ≤ λ/10) is shown in Figure 4.4(b), and it is given by

4. 4.3 Small Dipole
• The maximum phase error in(4-34) by allowing R = r for λ/50 < l ≤ λ/10, will be kl/2 = π/10 rad = 18◦ for l = λ/10.
Smaller values will occur for the other lengths. Performing the integration, (4-34) reduces to
• The potential function given by (4-35) becomes a more accurate approximation as kr →∞. This is also the region of most
practical interest, and it has been designated as the far-field region.
1/2of the infinitesimal dipole case's

5. • Thus we can write the E and H-fields radiated by a small dipole as

6. • Since the directivity of an antenna is controlled by the relative shape of the field or power pattern, the directivity, and
maximum effective area of this antenna are the same as the ones with the constant current distribution given by (4-31) and (4-
32), respectively. Thus the radiation resistance reduces to
• which is also one-fourth ( 1/4 ) of that obtained for the infinitesimal dipole as given by (4-19). Their relative patterns (shapes)
are the same and are shown in Figure 4.3.

8. FINITE LENGTH DIPOLE
Far Field
Near Field

9. Region Separation
• Before we attempt to solve for the fields radiated by a finite dipole of any length, it would be very desirable to discuss
the separation of the space surrounding an antenna into three regions; namely, the reactive near-field, radiating near-
field (Fresnel) and the far-field (Fraunhofer) which were introduced briefly in Section 2.2.4.
• The same approximations used to simplify the formulation of the fields radiated by a finite dipole are also used to
formulate the fields radiated by most practical antennas.
• A very thin dipole of finite length l is symmetrically positioned about
the origin with its length directed along the z-axis, as shown in Figure
4.5(a).

10. 4.4 Region Separation
• For a finite dipole with sinusoidal current distribution, the integral of (4-38) can be reduced
to a closed form that is valid everywhere.
• The length R is defined as the distance from any point on the source to the observation point.
• The integral of (4-38) was used to solve for the fields of infinitesimal and small dipoles in
Sections 4.1 and 4.2.
• in the first case (infinitesimal dipole) R = r and in the second case (small dipole) R was
approximated by r(R ≃ r) because the length of the dipole was restricted to be l ≤ λ/10.
• The major simplification of (4-38) will be in the approximation of R.
• A very thin dipole of finite length l is symmetrically positioned about the origin with its
length directed along the z-axis, as shown in Figure 4.5(a). Because the wire is assumed to be
very thin (x` = y` = 0), we can write (4-38) as

11. Region separation • The wire is assumed to be very thin (x’ = y’ =0)
where

12. 4.4.1 Far-Field (Fraunhofer) Region
Zero
• Using the binomial expansion, we can write (4-40) in a series as
• whose higher order terms become less significant provided r ≫ z`.

13. • It has been shown by many investigators through numerous examples that for most practical
antennas, with overall lengths greater than a wavelength (l > λ), a maximum total phase error of
π/8 rad (22.5◦) is not very detrimental in the analytical formulations.
• Using that as a criterion we can write, using (4-43),
that the maximum phase error should always be
Equation (4-45) simply states that to maintain
the maximum phase error of an antenna equal to
or less than π/8 rad (22.5◦), the observation
distance r must equal or be greater than 2l²/λ.

14. 4.4.1 Far-Field (Fraunhofer) Region
• The usual simplification for the far-field region is to approximate the R in the exponential
(e−jkR) of (4-38) by (4-42) and the R in the denominator of (4-38) by R ≃ r.
These simplifications are designated as the far-field approximations and are usually denoted
in the literature as

15. Thank you for your
attention