The document discusses antenna theory, focusing on the analysis and design of linear wire antennas, specifically detailing the characteristics and radiation patterns of Hertzian and finite dipoles. It covers topics such as the radiation resistance, directivity, current distribution, and the distinction between various field regions around antennas. Additionally, it explains the importance of the radiation pattern and introduces concepts like isotropic and directional patterns, contributing to a comprehensive understanding of antenna function and design.

1. Linear Wire Antennas
Text Book: Antenna Theory Analysis and Design: C.A. Balanis

2. Block diagram for computing fields radiated by electric and magnetic sources.

3. Magnetic Vector Potential

4. Steps to determine the radiation fields

5. By “Hertzian dipole” we mean an infinitesimal current element
I dl, where dl ≤ λ /10.
Although such a current element does not exist in real life, it
serves as a building block from which the field of a practical
antenna can be calculated by integration
A Hertzian dipole carrying current I = I0 cosw0t

12. To find E

14. Observation of the field equations:
we have terms varying as 1/r3, 1/r2, and 1/r.
The 1/r3 term is called the electrostatic field, since it corresponds to the
field of an electric dipole This term dominates other terms in a region
very close to the Hertzian dipole.
The 1/r2 term is called the inductive field. The term is important only at
near field, that is, at distances close to the current element
The 1/r term is called the far field or radiation field because it is the
only term that remains at the far zone, that is, at a point very far from the
current element.

15. Power Density and Radiation Resistance
the complex Poynting vector
To find the input resistance for a lossless antenna, the Poynting vector is formed in
terms of the E- and H-fields radiated by the antenna.

16. The complex power moving in the radial direction is obtained by integrating the
equation over a closed sphere of radius r.

17. 1. The transverse component W𝜃 of the power density does not contribute to the
integral
2. is purely imaginary, it will not contribute to any real radiated power
The above equation which gives the real and imaginary power that is moving
outwardly, can also be written as

19. • It is clear from the equation that the radial electric energy must be larger than the
radial magnetic energy. For large values of kr (kr ≫ 1 or r ≫ λ), the reactive power
diminishes and vanishes when kr = ∞.
• Since the antenna radiates its real power through the radiation resistance, for the
infinitesimal dipole it is found by equating

20. Example: Find the radiation resistance of an infinitesimal dipole whose overall length
is l = λ∕50.
Since the radiation resistance of an infinitesimal dipole is about 0.3 ohms, it will present a
very large mismatch when connected to practical transmission lines, many of which have
characteristic impedances of 50 or 75 ohms. The reflection efficiency (er) and hence the
overall efficiency (e0) will be very small.

21. Note: The reactance of an infinitesimal dipole is capacitive.
Since the input impedance of an open-circuited transmission line a distance l/2 from its
open end is given by
Zin = −jZccot(𝛽l∕2),
where Zc is its characteristic impedance, it will always be negative (capacitive) for l ≪ λ.

22. The E-field components, Er and E𝜃, are in time-phase but they are in time-phase
quadrature with the H-field component H𝜙; therefore there is no time-average power
flow associated with them.

23. the time-average power density
The condition of kr ≪ 1 can be satisfied at moderate distances away from the antenna
provided that the frequency of operation is very low.

26. The E- and H-field components are perpendicular to each other, transverse
to the radial direction of propagation.
The shape of the pattern is not a function of the radial distance r, and the
fields form a Transverse ElectroMagnetic (TEM) wave whose wave
impedance is equal to the intrinsic impedance of the medium.

27. RADIATION PATTERN
An antenna radiation pattern or antenna pattern is defined as “a mathematical
function or a graphical representation of the radiation properties of the antenna as a
function of space coordinates.”
In most cases, the radiation pattern is determined in the far-field region and is
represented as a function of the directional coordinates. Radiation properties include
power flux density, radiation intensity, field strength, directivity, phase or
polarization.
A trace of the received electric (magnetic) field at a constant radius is called the
amplitude field pattern. On the other hand, a graph of the spatial variation of the
power density along a constant radius is called an amplitude power pattern

29. • Often the field and power patterns are normalized with respect to
their maximum value, yielding normalized field and power patterns
• field pattern (in linear scale) typically represents a plot of the
magnitude of the electric or magnetic field as a function of the
angular space.
• power pattern (in linear scale) typically represents a plot of the
square of the magnitude of the electric or magnetic field as a function
of the angular space.
• power pattern (in dB) represents the magnitude of the electric or
magnetic field, in decibels, as a function of the angular space.

30. • Radiation Pattern Lobe
Various parts of a radiation pattern are referred to as lobes, which may be sub
classified into major or main, minor, side, and back lobes.

31. Radiation lobes and beamwidths of an antenna amplitude pattern in polar form.

32. A major lobe (also called main beam) is defined as “the radiation lobe
containing the direction of maximum radiation.” In Figure the major lobe is
pointing in the 𝜃 = 0 direction.
A minor lobe is any lobe except a major lobe. In Figures, all the lobes with
the exception of the major can be classified as minor lobes. A side lobe is “a
radiation lobe in any direction other than the intended lobe.” (Usually a side
lobe is adjacent to the main lobe and occupies the hemisphere in the
direction of the main beam.)
A back lobe is “a radiation lobe whose axis makes an angle of approximately
180◦ with respect to the beam of an antenna.” Usually it refers to a minor
lobe that occupies the hemisphere in a direction opposite to that of the
major (main) lobe

33. Isotropic, Directional, and Omnidirectional Patterns
An isotropic radiator is defined as “a hypothetical lossless antenna having
equal radiation in all directions.” Although it is ideal and not physically
realizable, it is often taken as a reference for expressing the directive
properties of actual antennas.
A directional antenna is one “having the property of radiating or
receiving electromagnetic waves more effectively in some directions than
in others.
Omnidirectional antenna is defined as one “having an essentially
nondirectional pattern in a given plane (in this case in azimuth) and a
directional pattern in any orthogonal plane (in this case in elevation).” An
omnidirectional pattern is then a special type of a directional pattern.

35. Principal Patterns
• For a linearly polarized antenna, performance is
often described in terms of its principal E- and H-
plane patterns.
• The E-plane is defined as “the plane containing the
electric-field vector and the direction of maximum
radiation,” and
• the H-plane as “the plane containing the magnetic-
field vector and the direction of maximum
radiation.”
• Example: the x-z plane (elevation plane; 𝜙 = 0) is
the principal E-plane
and
• the x-y plane (azimuthal plane; 𝜃 = 𝜋∕2) is the
principal H-plane.
• The omnidirectional pattern of Figure shown has
an infinite number of principal E-planes (elevation
planes; 𝜙 = 𝜙c) and one principal H-plane
(azimuthal plane; 𝜃 = 90◦).

36. The space surrounding an antenna is usually subdivided into three
regions:
(a) reactive near-field
(b) radiating near-field (Fresnel)
(c) far-field (Fraunhofer)

37. Reactive near-field region is defined as “that portion of the near-field region
immediately surrounding the antenna wherein the reactive field predominates.” For
most antennas, the outer boundary of this region is commonly taken to exist at a
distance R < 0.62√D3∕λ from the antenna surface, where λ is the wavelength and D is
the largest dimension of the antenna. “For a very short dipole, or equivalent radiator,
the outer boundary is commonly taken to exist at a distance λ∕2𝜋 from the antenna
surface.
Radiating near-field (Fresnel) region is defined as “that region of the field of an
antenna between the reactive near-field region and the far-field region wherein
radiation fields predominate and wherein the angular field distribution is dependent
upon the distance from the antenna. If the antenna has a maximum dimension that is
not large compared to the wavelength, this region may not exist.
The inner boundary is taken to be the distance R ≥ 0.62√D3∕λ and the outer boundary
the distance R < 2D2∕λ where D is the largest∗ dimension of the antenna. This
criterion is based on a maximum phase error of 𝜋∕8.

38. • Far-field (Fraunhofer) region is defined as “that region of the field of
an antenna where the angular field distribution is essentially
independent of the distance from the antenna. If the antenna has a
maximum† overall dimension D, the far-field region is commonly
taken to exist at distances greater than 2D2∕λ from the antenna, λ
being the wavelength.

40. Directivity
• Directivity of an antenna defined as “the ratio of the radiation intensity
in a given direction from the antenna to the radiation intensity
averaged over all directions. The average radiation intensity is equal to
the total power radiated by the antenna divided by 4𝜋. If the direction
is not specified, the direction of maximum radiation intensity is
implied.”

41. If the direction is not specified, it implies the direction of maximum radiation intensity
(maximum directivity) expressed as

42. Directivity of a infinitesimal dipole
radiation intensity U is given by

43. Directivity of a infinitesimal dipole
radiation intensity U is given by
The maximum value occurs at 𝜃 = 𝜋∕2 and it is equal to
Using equations for Prad and Umax, the directivity reduces to

46. SMALL DIPOLE
The previous section discussed the radiation properties of an infinitesimal dipole, typically with a
length l ≤ λ∕50, assuming a constant current distribution. While constant current distribution is not
feasible except for specific cases like top-hat-loaded elements, it serves as a mathematical
representation for antenna current distributions, often segmented into small lengths.
A better approximation of the current distribution of wire antennas, whose lengths are usually λ∕50 < l
≤ λ∕10, is the triangular variation. The sinusoidal variations are more accurate representations of the
current distribution of any length wire antenna.
The current distribution of a small dipole (λ∕50 < l ≤ λ∕10)

47. The radiation resistance of the antenna is strongly dependent upon the current distribution.

48. E- and H-fields radiated by a small dipole

49. 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 a. The sinusoidal variations of Figures (b)–(c) are more accurate representations of
the current distribution of any length wire antenna.
Current Distribution on Linear Dipole

50. FINITE LENGTH DIPOLE
The techniques that were developed previously for infinitesimal dipole can also be used to analyze the
radiation characteristics of a linear dipole of any length. To reduce the mathematical complexities, it will be
assumed in this chapter that the dipole has a negligible diameter (ideally zero).
Current Distribution: For a very thin dipole (ideally zero diameter)
This distribution assumes that the antenna is center-fed and the current vanishes at the end points (z′ =
±l∕2). Experimentally it has been verified that the current in a center-fed wire antenna has sinusoidal form
with nulls at the end points.

51. Current distribution on a λ∕2 wire antenna for different times.

52. The finite dipole antenna of Figure is subdivided into a
number of infinitesimal dipoles of length Δz′. As the
number of subdivisions is increased, each infinitesimal
dipole approaches a length dz′. For an infinitesimal
dipole of length dz′ positioned along the z-axis at z′, the
electric and magnetic field components in the far field
are given.
Summing the contributions from all the infinitesimal elements, the summation reduces, in the limit,
to an integration. Thus

53. element factor space factor
For this antenna, the element factor is equal to the field of a unit length infinitesimal dipole located at
a reference point (the origin). In general, the element factor depends on the type of current and its
direction of flow while the space factor is a function of the current distribution along the source.
The total field of the antenna is equal to the product of the element and space factors. This is referred to
as pattern multiplication for continuously distributed sources and it can be written as

54. For the current distribution
After some mathematical manipulations

55. Power Density, Radiation Intensity, and Radiation Resistance
The average Poynting vector
and the radiation intensity as

56. Elevation plane amplitude patterns for a thin dipole with sinusoidal current
distribution (l =λ∕50, λ∕4, λ∕2, 3λ∕4, λ).

57. As the length of the dipole increases beyond one wavelength (l > λ), the number of lobes begin to increase.
thin dipole of l = 1.25λ
Current distributions along the length of a linear wire antenna.

59. More specifically, we define the boundary between the near and the far zones by the
value of r given by
r = 2d2 / λ
where d is the largest dimension of the antenna. Thus at far field
Thus at far field

61. The resistance Rrad is a characteristic property of the Hertzian dipole antenna and is
called its radiation resistance.

62. The half-wave dipole derives its name from the fact that its length is half a wavelength L = λ/2

63. The field due to the dipole can be easily obtained if we consider it as consisting of a
chain of Hertzian dipoles.
The magnetic vector potential at P due to a differential length dl (= dz) of the dipole
carrying a phasor current
Is = Io cos βz
We have assumed sinusoidal current distribution:
1. the sinusoidal current assumption is based on the transmission line model
of the dipole. Second, the current must vanish at the ends of the dipole.
2. A triangular current distribution is also possible but would give less accurate results.

64. Note: the significant increase in the radiation resistance of the half-wave dipole over
that of the Hertzian dipole. Thus the half-wave dipole is capable of delivering greater
amounts of power to space than the Hertzian dipole.
The total input impedance Zin of the antenna is the impedance seen at the terminals
of the antenna and is given by
where Rin = Rrad for a lossless antenna.

65. It is found that Xin = 42.5 Ω, so Zin = 73 + j42.5 Ω for a dipole length , ℓ = λ/2. The
inductive reactance drops rapidly to zero as the length of the dipole is slightly reduced.
For , ℓ = 0.485 λ, the dipole is resonant, with Xin = 0. Thus in practice, a λ/2 dipole is
designed such that Xin approaches zero and Zin = 73 Ω.
This value of the radiation resistance of the λ/2 dipole antenna is the reason for the
standard 75 Ω coaxial cable. Also, the value is easy to match to transmission lines.
These factors in addition to the resonance property are the reasons for the dipole
antenna’s popularity and its extensive use

66. The quarter-wave monopole antenna consists of
half of a half-wave dipole antenna located on a
conducting ground plane, as in Figure.
The monopole antenna is perpendicular to the
plane, which is usually assumed to be infinite
and perfectly conducting. It is fed by a coaxial
cable connected to its base.
total input impedance for a l/4 monopole is Zin = 36.5 + j21.25 Ω

67. Example:

71. Example: Hertzian dipole

72. In many practical applications
(e.g., in an AM broadcast station),
it is necessary to design antennas
with more energy radiated in
some particular directions and
less in other directions.

78. Consider an antenna consisting of two Hertzian dipoles placed in free space along the
z-axis but oriented parallel to the x-axis as depicted in Figure.
The total electric field at point P is the vector
sum of the fields due to the individual
elements.
If P is in the far-field zone:

80. Comparing this with eq. of Hertizain dipole shows that the total field of an array is
equal to the field of single element located at the origin multiplied by an array factor
given by

81. This is known as pattern multiplication, and it can be used to sketch, almost by
inspection, the pattern of an array. Therefore, pattern multiplication is a useful tool in
the design of an array.
Also, from above eq. note that |cosθ | is the radiation pattern due to a single element,
whereas the normalized array factor, | cosθ [1/2 {βd cosθ + α | , is the radiation pattern
the array would have if the elements were isotropic. These may be regarded as “unit
pattern” and “group pattern,” respectively.

82. Let us now extend the results on the two-element array to the general case of N element
array:
assumption: array is linear and uniform (each element is fed with current of the same
magnitude but of progressive phase shift α
For the uniform linear array, the array factor is the sum of the contributions by all the
elements. Thus,