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Antenna Theory and Design
Antenna Theory and Design
Associate Professor
WANG Junjun 王珺珺
School of Electronic and Information
Engineering
Beihang University
[email protected]
13426405497
New Main Building F1025
1
Chapter 4
Wire Antennas
Chapter 4 Wire Antennas
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Infinitesimal Dipole
Small Dipole
Finite length Dipole
Half-wavelength Dipole
Antenna above a perfect ground plane
Folded Dipole
Yagi Uda Antenna
Travelling-Wave Antenna
Wire Antennas
• Wire antennas, linear or
curved, are some of the oldest,
simplest, cheapest, and in
many cases the most versatile
for many applications.
• we begin our analysis of
antennas by considering some
of the oldest, simplest, and
most basic configurations.
Fig.1 Dipole and monopole antennas
1. Infinitesimal Dipole
Fig2 Infinitesimal dipole
• An infinitesimal linear wire (l <<λ, r<< λ) is
positioned symmetrically at the origin of the
coordinate system and oriented along the z axis.
• Although infinitesimal dipoles are not very
practical, they are used to represent capacitor-plate
(also referred to as top-hat-loaded) antennas.
• In addition, they are utilized as building blocks of
more complex geometries. The end plates are used
to provide capacitive loading in order to maintain
the current on the dipole nearly uniform.
Since the end plates are assumed to be small, their radiation is usually negligible. The
wire is very small and thin, the spatial variation of the current is assumed to be constant.
1. Infinitesimal Dipole
• The radiation resistance is:
2𝜋
𝑅𝛾 = 𝜂
3
𝑙
𝜆
2
= 80𝜋 2
𝑙
𝜆
2
• The input reactance is capacitive.
• This can be seen by visualizing the antenna as
an open circuited transmission line. The
distance from the end of the antenna to the
feed point is much less than a quarter
Fig3 Three-dimensional radiation pattern
wavelength and thus the input impedance is
of infinitesimal dipole
capacitive.
• The maximum effective aperture:
• The directivity of this antenna:
𝑈𝑚𝑎 𝑥 3
𝐷0 = 4𝜋
=
𝑃𝑟𝑎 ⅆ 2
𝐴𝑒𝑚
𝜆2
3𝜆2
=
𝐷 =
4𝜋 0
8𝜋
1. Infinitesimal Dipole
• For a wire antenna to be classified as an infinitesimal dipole, its overall length must be
very small (usually l ≤ λ/50).
Example:
• Find the radiation resistance of an infinitesimal dipole whose overall length is 𝑙 = 𝜆 50 .
• Solution:
2
2
𝑅𝑟 = 80𝜋 2
𝑙
𝜆
= 80𝜋 2
1
50
= 0.316 𝑜ℎ𝑚𝑠
• 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.
2. Small Dipole
• For infinitesimal dipole, its current distribution
was assumed to be constant. Although a constant
current distribution is not realizable (other than
top-hat-loaded elements), it is a mathematical
quantity that is used to represent actual current
distributions of antennas that have been
incremented into many small lengths.
• For small dipole antenna whose lengths are
Fig4 current distribution of small dipole
usually λ/50 < l ≤ λ/10, the current must
smoothly go to zero at the ends of the wave. For
′ ′ ′
𝐼𝑒 𝑥 , 𝑦 , 𝑧
a thin wire (diameter<<wavelength), this current
2 ′
distribution is approximately sinusoidal. The
𝑎𝑧 𝐼0 1 − 𝑧 , 0 ≤ 𝑧 ′ ≤ 𝑙 2
𝐿
decreasing current toward the wire ends requires
=
2
that charges peel off and appear on the wire
𝑎𝑧 𝐼0 1 + 𝑧 ′ , − 𝑙 2 ≤ 𝑧 ′ ≤ 0
𝐿
surface.
2. Small Dipole
• As l becomes extremely small, the sinusoidal
type current distribution is well approximated by
the triangular distribution.
• The radiation resistance using the
distribution is:
2𝑃𝑟𝑎ⅆ
𝑙
2
𝑅𝑟 =
=
20𝜋
𝐼0 2
𝑥
Fig5 current distribution of small dipole
triangular
2
which is also is one-fourth of that obtained for
the infinitesimal dipole.
2. Small Dipole
Fig6 capacitor-plate antenna
• To realize a uniform current distribution in practice a
mechanism must be provided for charge storage at the
ends of the short wire. One method of accomplishing
this is to place metal plates at the ends of the wire.
This is called a capacitor-plate antenna, or top-hatloaded dipole antenna.
• If Δ𝑧 ≪≫ 𝜆, the radial current on the plates will
produce fields that almost cancel in the far field, since
the currents are opposite directed and the phase
difference due to separation is small. If, in addition, Δ
𝑧≪Δr, the plates will provide for charge storage such
that the current on the wire is constant. The capacitorplate antenna then closely approximates the uniform
current ideal dipole model.
2 Small Dipole
• The input reactance is capacitive. If end loading is added, this capacitance is reduced as
the loading is increased. This is because the distance from the open circuit in the
transmission line analog is increased.
• 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 infinitesimal dipole.
𝐷0 = 4𝜋
𝐴𝑒𝑚
𝑈𝑚𝑎𝑥 3
=
𝑃𝑟𝑎ⅆ
2
𝜆2
3𝜆2
=
𝐷 =
4𝜋 0
8𝜋
3 Finite-length Dipole
Fig7 Finite-length Dipole
• To reduce the mathematical complexities, it will be
assumed that the dipole has a negligible diameter. For
dipoles, we assume that the current distribution is
sinusoidal(l > λ/10). The current must, of course, be
zero at the ends.
• We are, in effect, using the current distribution which
is found on an open-circuited parallel wire
transmission line. It is assumed that if the end of such
a transmission line is bent out to form a wire antenna,
the current distribution along the bent portion is
essentially unchanged.
• Although this is not strictly true it is a good
approximation for thin antennas, for which the
conductor diameter is on the order of 0:01 or smaller.
3 Finite-length Dipole
• For a very thin dipole (ideally zero diameter), the current distribution can be written,
to a good approximation, as
𝑎𝑧 𝐼0 sin 𝑘
𝐼𝑒 𝑥 ′ = 0, 𝑦 ′ = 0, 𝑧 ′ =
𝑎𝑧 𝐼0 sin 𝑘
𝑙
− 𝑧′
2
𝑙
+ 𝑧′
2
, 0 ≤ 𝑧′ ≤ 𝑙 2
, − 𝑙 2 ≤ 2′ ≤ 0
• 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.
3 Finite-length Dipole
• For different length, the current distribution is shown plotted in Fig 8.
Fig8 the current distribution of Finite-length Dipole
3 Finite-length Dipole
• It is found that the 3-dB beamwidth of
each is equal to
𝑙≪𝜆
3-dB beamwidth=90°
𝑙 = 𝜆/4
3-dB beamwidth=87°
𝑙 = 𝜆/2
3-dB beamwidth=78°
𝑙 = 3𝜆/4
3-dB beamwidth=64°
𝑙=𝜆
3-dB beamwidth=47.8°
Fig9 Elevation plane amplitude patterns for a thin dipole with
sinusoidal current distribution (l = λ/50,λ/4,λ/2,3λ/4, λ).
3 Finite-length Dipole
• As the length of the dipole increases beyond one wavelength (l > λ), the number of
lobes begin to increase.
• The normalized power pattern for a dipole with l = 1.25λ is shown
Fig10 Three- and two-dimensional (elevation pattern) amplitude patterns for a thin dipole of l = 1.25λ
and sinusoidal current distribution.
3 Finite-length Dipole
• By the definition, the radiation resistance is referred to the
maximum current which for some lengths (l = λ/4, 3 λ/4, λ, etc.)
does not occur at the input terminals of the antenna, the antenna
itself is first assumed to be lossless (Rl=0). Then the power at the
input terminals is equated to the power at the current maximum.
𝐼𝑖𝑛
2
• or
𝑅𝑖𝑛
Fig11 Current distribution of a linear
wire antenna when current maximum
does not occur at the input terminals.
2
𝑅𝑖𝑛
𝐼0 2
=
𝑅
2 𝑟
𝐼0
=
𝐼𝑖𝑛
2
𝑅𝑟
𝑅𝑖𝑛 =radiation resistance at input (feed) terminals
𝑅𝑟 =radiation resistance at current maximum
𝐼0 =current maximum
𝐼𝑖𝑛 =current at inpuy treminals
3 Finite-length Dipole
• The input radiation resistance
𝑅𝑖𝑛 =
𝑅𝑟
𝑘𝑙
sin2
2 input (feed) terminals
Rin = radiation resistance at
Rr = radiation resistance at current maximum
Fig12 Current distributions along
the length of a linear wire antenna
Fig13 Radiation resistance, input resistance and directivity
of a thin dipole with sinusoidal current distribution
3 Finite-length Dipole
• Simple Formulas for the Input Resistance of Dipoles
Length L
Input resistance R in (Ω)
0 < 𝐿 < 𝜆/4
20𝜋 2 (L/𝜆)2
𝜆/4 < 𝐿 < 𝜆/2
24.7(𝜋L/𝜆)2.4
𝜆/4 < 𝐿 < 0.637𝜆
11.14(𝜋L/𝜆)4.17
• Input resistance can be related to radiation resistance. There are several ways to
define radiation resistance by using different current reference points.
▫ Usually radiation resistance is defined using the current distribution maximum 𝐼𝑚 , whether
or not it actually occurs on the antenna. We shall use the symbol 𝑅𝑟𝑚 for this definition.
▫ It is also useful to refer the radiation resistance to the terminal point. In this case the
symbol 𝑅𝑟𝑖 is used.
▫ A third radiation resistance, denoted by Rr, is often used, it is the radiation resistance
relative to the maximum current that occurs on the antenna.
4 Half-Wavelength Dipole
• One of the most commonly used antennas is the half-wavelength (l = λ/2) dipole.
Because its radiation resistance is 73 ohms, which is very near the 50-ohm or 75-ohm
characteristic impedances of some transmission lines, its matching to the line is
simplified especially at resonance.
Fig14 sinusoidal current distribution
• This is the well known value for the
radiation resistance of a thin, linear,
center-fed, half-wavelength antenna
with sinusoidal current distribution.
• The half-wave dipole antenna is a
linear current whose amplitude varies
as one-half of a sine wave with a
maximum at the center.
4 Half-Wavelength Dipole
• The advantage of a half-wave dipole is that it can be made to resonate and present a
zero input reactance, thus eliminating the need for turning to achieve a conjugate
impedance match.
• To obtain a resonant condition for a half-wave dipole, the physical length must be
somewhat shorter than a free space half-wave length and as the antenna wire thickness
is increased the length must be reduced more to achieve resonance.
Fig15 The electric and magnetic field of a half-wavelength dipole
4 Half-Wavelength Dipole
• For comparison, the half-power beamwidth of a half-wave dipole pattern is 78, while
that of an ideal dipole pattern is 90 . Thus, there is a small increase in the directivity of
the half-wave dipole over the short dipole.
• The current distribution and radiation pattern of the half-wave dipole is shown.
Fig16 The current distribution and radiation pattern of the half-wave dipole
4 Half-Wavelength Dipole
• The maximum directivity of the half-wavelength dipole
𝐷0 = 4𝜋
𝑈 𝜃=𝜋
𝑈max
= 4𝜋
𝑃𝑟𝑎ⅆ
𝑃𝑟𝑎ⅆ
2
=
4
4
=
≈ 1.643
𝐶𝑖𝑛 2𝜋
2.435
• The radiation resistance, for a free-space medium (η  120π),
𝑅𝛾 =
2𝑃𝑟𝑎ⅆ
𝜂
=
𝐶 2𝜋 = 30 2.435 ≈ 73
𝐼0 2
4𝜋 𝑖𝑛
• The radiation resistance is also the radiation resistance at the input terminals (input
resistance) since the current maximum for a dipole of l = λ/2 occurs at the input
terminals.
• The imaginary part (reactance) associated with the input impedance of a dipole is a
function of its length (for l = λ/2, it is equal to j 42.5). Thus the total input impedance
for l = λ/2 is equal to
𝑍𝑖𝑛 = 73 + 𝑗42.5
4 Half-Wavelength Dipole
• To reduce the imaginary part of the input impedance to zero, the antenna is matched or
reduced in length until the reactance vanishes. The latter is most commonly used in
practice for half-wavelength dipoles. To make the reactance zero, requires that the
antenna be shorted a few percent less than half-wavelength. This shortening results in a
reduction in the value of the radiation resistance to about 65 ohm.
• Depending on the radius of the wire, the length of the dipole for first resonance is about
l = 0.47λ to 0.48λ; the thinner the wire, the closer the length is to 0.48λ. Thus, for
thicker wires, a larger segment of the wire has to be removed from λ/2 to achieve
resonance.
5 Antennas Above A Perfect Ground Plane
• Thus far we have considered the radiation characteristics of antennas radiating into
an unbounded medium.
• Antenna are frequently operated in the presence of other structure. One such
structure that is commonly encountered is a ground plane. A ground plane in its
ideal form is infinite in extent and perfectly conducting, often referred to as a
perfect ground plane. A solid metal sheet or a planar wire grid system that is large
compared to the antenna size is, in most cases, well approximated as a perfect
ground plane.
• Any energy from the radiating element directed toward the ground undergoes a
reflection. The amount of reflected energy and its direction are controlled by the
geometry and constitutive parameters of the ground.
5.1 Image Theory
• To analyze the performance of an antenna near an infinite plane conductor, virtual
sources (images) will be introduced to account for the reflections. As the name implies,
these are not real sources but imaginary ones, which when combined with the real
sources, form an equivalent system.
• For analysis purposes only, the equivalent
system gives the same radiated field on and
above the conductor as the actual system itself.
Below the conductor, the equivalent system
does not give the correct field. However, in this
region the field is zero and there is no need for
the equivalent.
• To begin the discussion, let us assume that a
vertical electric dipole is placed a distance h
above an infinite, flat, perfect electric
Fig17 Vertical electric dipole
conductor.
5.1 Image Theory
• The amount of reflection is generally determined by the respective constitutive
parameters of the media below and above the interface. For a perfect electric conductor
below the interface, the incident wave is completely reflected and the field below the
boundary is zero. According to the boundary conditions, the tangential components of
the electric field must vanish at all points along the interface.
• Thus for an incident electric field with vertical polarization shown by the arrows, the
polarization of the reflected waves must be as indicated in the figure to satisfy the
boundary conditions. To excite the polarization of the reflected waves, the virtual
source must also be vertical and with a polarity in the same direction as that of the
actual source (thus a reflection coefficient of +1).
5.1 Image Theory
• Vertical Electric dipole
𝑘𝐼0 𝑙𝑒 −𝑗𝑘𝑟
𝐸𝜃 ≈ 𝑗𝜂
sin 𝜃 2 cos 𝑘ℎ cos 𝜃 , 𝑧 ≥ 0
4𝜋𝑟
𝐸𝜃 = 0, 𝑧 < 0
• It is evident that total electric filed is equal
to the product of the field of a single source
positioned symmetrically about the origin
and a factor which is a function of the
antenna height (h) and the observation
angle (). This is referred to as pattern
multiplication and the factor is known as
the array factor.
Fig18 Vertical electric dipole above infinite
perfect electric conductor
5.1 Image Theory
• Vertical Electric dipole
2ℎ
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑜𝑏𝑒𝑠 ≈
+1
𝜆
Fig19 Elevation plane amplitude patterns of a vertical infinitesimal electric dipole for different
heights above an infinite perfect electric conductor
5.1 Image Theory
• Horizontal Electric dipole
Fig20 horizontal electric dipole is placed a distance h
above an infinite, flat, perfect electric conductor
• The image of a current element oriented in any
direction with respect to a perfect ground plane
may be found by decomposing it into
perpendicular and parallel components, forming
the images of the components, and constructing
the image from these image components.
• The perfectly-conducting infinite ground plane
is, of course, an idealization. The perfectly
conducting assumption is valid when good
conductors such aluminum or copper are used.
The infinitely large assumption is more severe.
However, generally speaking, if the conducting
plane extends beyond the source by several
times the length of the source and if the source
is not too far away from the conducting plane,
this assumption is also valid.
5.1 Image Theory
• Horizontal Electric dipole
• The total field
𝑘𝐼0 𝑙𝑒 −𝑗𝑘𝑟
ⅆ
𝑟
𝐸𝜓− = 𝐸𝜓 + 𝐸𝜓 = 𝑗𝜂
1 − sin2 𝜃 sin2 𝜙 2𝑗 sin 𝑘ℎ cos 𝜃
4𝜋𝑟
• It again consists of the product of the field of a single isolated element placed
symmetrically at the origin and a factor (within the brackets) known as the array
factor.
Fig21 Horizontal electric dipole above an infinite perfect electric conductor
5.1 Image Theory
• Horizontal Electric dipole
ℎ
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑜𝑏𝑒𝑠 ≈ 2( )
𝜆
Fig22 Elevation plane (φ = 90◦) amplitude patterns of a horizontal infinitesimal electric dipole for different heights
above an infinite perfect electric conductor.
5.2 Principle of Duality
• If antenna structures are dual it is possible to write the fields for one antenna from the
field expressions of the other by interchanging parameters using the principle of duality.
• First we will discuss the general principle of duality as applied to antennas.
• Suppose we have an electric current source with current density J1 and boundary
conditions on materials present 𝜀1 , 𝜇1 , 𝜎1 Maxwell's equations for this system are:
𝛻 × 𝐸1 = −𝑗𝜔𝜇1 𝐻1
𝛻 × 𝐻1 = 𝑗𝜔𝜀1 𝐸1 + 𝐽1
• Now suppose a fictitious magnetic current source with magnetic current density M2 exists
with materials present 𝜀2 , 𝜇2 , 𝜎2 Maxwell equations for this system are :
𝛻 × 𝐻2 = 𝑗𝜔𝜀2 𝐸2
𝛻 × 𝐸2 = −𝑗𝜔𝜇2 𝐻2 − 𝑀2
• The electric and magnetic systems are duals if the procedure in the table on the following
can be performed. Replace the following in system #2 𝑴 𝜺
𝜇
𝑬
𝑯
𝟐
By the following in system #1
𝑱𝟏
𝟐
2
𝜇1
𝜺𝟏
𝟐
𝟐
−𝑯𝟏 𝑬𝟏
5.3 Monopoles
Fig23 The monopoles and their
images in a perfect ground plane
• A monopole is a dipole divided in half at its center
feed point and fed against a ground plane. Highfrequency monopoles are often fed from coaxial
cables behind the ground plane.
• The currents and charges on a monopole are the
same as on the upper half of its dipole counterpart,
but the terminal voltage is only half that of the
dipole.
• The voltage is half because the gap width of the
input terminals is half that of the dipole, and the
same electric field over half the distance gives half
the voltage.
5.3 Monopoles
• Then the input impedance for a monopole is :
𝑍𝑖𝑛,𝑚𝑜𝑛𝑜
1
𝑉𝑖𝑛,𝑚𝑜𝑛𝑜 2 𝑉𝑖𝑛,ⅆ𝑖𝑝𝑜𝑙𝑒 1
=
=
= 𝑍𝑖𝑛,ⅆ𝑖𝑝𝑜𝑙𝑒
𝐼𝑖𝑛,𝑚𝑜𝑛𝑜
𝐼𝑖𝑛,ⅆ𝑖𝑝𝑜𝑙𝑒
2
• This is easily demonstrated for the radiated
resistance. Since the fields only extend over a
hemisphere the power radiated is only half
that of a dipole with the same current.
• Therefore, the radiation resistance
monopole is half that of the dipole.
Fig24 The monopole simulated by HFSS
of
5.3 Monopoles
• The radiation pattern of a monopole above a perfect ground plane is the same as that of
a dipole similarly positioned in free space since the fields above the image plane are
the same. Therefore, a monopole above a perfect ground plane radiates one-half the
total power of a similar dipole in free space because the power is distributed in the
same fashion but only over half as much space.
• The beam solid angle of a monopole above a perfect ground plane is one-half that of a
similar dipole in free space. Then, we can get:
𝐷𝑚𝑜𝑛𝑜 =
4𝜋
𝛺𝐴,𝑚𝑜𝑛𝑜
=
4𝜋
1
𝛺
2 𝐴,ⅆ𝑖𝑝𝑜𝑙𝑒
= 2𝐷ⅆ𝑖𝑝𝑜𝑙𝑒
• The directivity increase does not come from an increase in the radiation intensity but
rather from a decrease in average radiation intensity.
6 Folded Dipole
• An extremely practical wire antenna is the
folded dipole. It consists of two parallel dipoles
connected at the ends forming a narrow wire
loop with dimension d much smaller than L and
much smaller than a wavelength. The feed point
is at the center of one side.
• The folded dipole antenna is a very popular
wire antenna. The reasons for this are its
impedance properties and ease of construction.
In addition, it has wider bandwidth.
Fig25 The folded dipole array
6 Folded Dipole
• The folded dipole is essentially an unbalanced transmission line with unequal currents.
Its operation is analyzed by considering the current to be composed to two modes: the
transmission line mode and the antenna mode.
Fig26 The folded dipole antenna
Fig27 The current modes on a folded dipole antenna.
(a) Transmission line mode. (b) Antenna mode.
6 Folded Dipole
Fig28 Mode excitation and current for a voltage
V applied to the terminals of a folded dipole.
(a) Transmission line mode. (b) Antenna mode.
• The currents in the transmission line mode have
fields that tend to cancel in the far field since d
is small. The input impedance for this mode is
given by the equation for a transmission line
with a short circuit load.
𝐿
𝑍𝑇 = 𝑗𝑍0 tan 𝛽
2
• In the antenna mode the fields from the currents
in each vertical section reinforce in the far field
since they are similarly directed.
• Suppose a voltage V is applied across the input
terminals. The total behavior is determined by
the superposition of the equivalent circuits for
each mode.
6 Folded Dipole
• The transmission line mode current is
𝐼𝑇 =
𝑉 2
𝑉
=
𝑍𝑇
2𝑍𝑇
• For the antenna mode, the antenna current is
𝐼𝐴 =
𝑉 2
𝑉
=
𝑍𝐷
2𝑍𝐷
where to a first-order approximation ZD is the input
impedance for an ordinary dipole of the same wire size.
1
• The total current is 𝐼𝑇 + 2 𝐼𝐴 and the total
voltage is V; so the input impedance of the
folded dipole is
Fig28 Mode excitation and current for a voltage
V applied to the terminals of a folded dipole.
(a) Transmission line mode. (b) Antenna mode.
𝑍𝑖𝑛 =
𝑉
4𝑍𝑇 𝑍𝐷
=
1
𝐼𝑇 + 𝐼𝐴 𝑍𝑇 + 2𝑍𝐷
2
6 Folded Dipole
• As an example consider the popular half-wave
folded, this gives
𝑍𝑖𝑛 = 4𝑍𝐷 𝐿 =
𝜆
2
• The input impedance of a half-wave folded
dipole (at resonance) is four times that of an
ordinary dipole.
• Zin = 280 is very close to the 300 of common
twin-lead transmission line.
Fig28 Mode excitation and current for a voltage
V applied to the terminals of a folded dipole.
(a) Transmission line mode. (b) Antenna mode.
7 Yagi-Uda Antenna
• Array antennas can be used to increase directivity. The arrays we have examined have
had all elements active, requiring a direct connection to each element by a feed
network. The feed networks for arrays are considerably simplified if only a few
elements are fed directly. Such an array is referred to as a parasitic array. The elements
that are not directly driven (called parasites) receive their excitation by near-field
coupling from the driven elements. A parasitic linear array of parallel dipoles is called
Yagi-Uda antenna, Yagi-Uda array, or simply "Yagi".
Fig29 Yagi-Uda antenna
7 Yagi-Uda Antenna
• Yagi-Uda Antennas are very popular because of their simplicity and relatively high
gain. In this section the principles of operation and design data for Yagi will be
presented.
• The basic unit of a Yagi consists of three elements. To understand the principles of
operation for a three-element Yagi we will begin with a driven element (driver) and add
parasites to the array.
Fig30 three element Yagi-Uda antenna
7 Yagi-Uda Antenna
• Consider a driven element that is a resonant half-wave dipole. If a parasitic element is
spaced very close to it, it is excited by the driven element with roughly equal
amplitude, so the field incident on the parasite is
𝐸𝑖𝑛𝑐𝑖ⅆ𝑒𝑛𝑡 = 𝐸ⅆ𝑟𝑖𝑣𝑒
• A current is excited on the parasite and it will radiate electric field Eparasite.
• From the boundary condition for good conductor, 0 = Eparasite + Eincident. Then
𝐸𝑜𝑎𝑟𝑎𝑠𝑖𝑡𝑒 = −𝐸𝑖𝑛𝑐𝑖ⅆ𝑒𝑛𝑡 = −𝐸ⅆ𝑟𝑖𝑣𝑒
• From array theory we know that two closely spaced, equal amplitude, opposite phase
elements will have an endfire pattern.
7 Yagi-Uda Antenna
• Reflector
▫ The numerical methods demonstrates the general trend of a parasite which is longer
than the driver: a single main beam occurs in the endfire direction from the parasite
to the driver along the line of the array. Such a parasite is called a reflector because
it appears to reflect radiation from the driver.
• Director
▫ If the parasite is shorter than the driver, but now placed on the other side of the
driver, the pattern effect is similar to that when using a reflector in the sense that
main beam enhancement is in the same direction. The parasite is then referred to as
a director since it appears to direct radiation in the direction from the driver toward
the director.
7 Yagi-Uda Antenna
• The single endfire beam created by the use of a reflector or a director alone with a
driver suggests that even further enhancement could be achieved with a reflector and a
director on opposite sides of a driver. The maximum directivity obtained from a threeelement Yagi is about 9dB.
• The optimum spacing ( for maximum directivity) are on the order of 0.15 to 0.25
wavelength between the reflector and driver and also between the driver to director.
Typically the reflector is lengthened 5% or more and the director is shortened 5% or
more from the length of resonant length driver.
8 Travelling-Wave Wire Antenna
• The wire antennas we have discussed thus far have been resonant structures. The
wave traveling outward from the feed point to the end of the wire is reflected, setting
up a standing-wave type current distribution. This can be seen by
𝐼𝑚 sin 𝛽
𝐿
−𝑧
2
=
𝐼𝑚 𝑗𝛽𝐿
𝑒
2𝑗
2
𝑒 −𝑗𝛽𝑧 − 𝑒 𝑗𝛽𝑧
• The first term in brackets represents an outward traveling wave and the second term a
reflected wave. The minus sign is the current reflection coefficient at an open circuit.
• If the reflected wave is not present on an antenna it is referred to as a traveling-wave
antenna.
8 Travelling-Wave Wire Antenna
Standing wave antennas:
• Center-fed linear wire antennas were discussed whose amplitude current distribution
was
1. constant for infinitesimal dipoles (l ≤ λ/50)
2. linear (triangular) for short dipoles (λ/50 < l ≤ λ/10)
3. sinusoidal for long dipoles (l > λ/10)
• In all cases the phase distribution was assumed to be constant. The sinusoidal current
distribution of long open-ended linear antennas is a standing wave constructed by two
waves of equal amplitude and 180◦ phase difference at the open end traveling in
opposite directions along its length.
8 Travelling-Wave Wire Antenna
• The voltage distribution has also a standing wave pattern except that it has maxima
(loops) at the end of the line instead of nulls (nodes) as the current.
• In each pattern, the maxima and minima repeat every integral number of half
wavelengths. There is also a λ/4 spacing between a null and a maximum in
each of the wave patterns.
• The current and voltage distributions on open-ended wire antennas are similar
to the standing wave patterns on open-ended transmission lines. Linear
antennas that exhibit current and voltage standing wave patterns formed by
reflections from the open end of the wire are referred to as standing wave or
resonant antennas.
8 Travelling-Wave Wire Antenna
• Antennas can be designed which have traveling wave (uniform) patterns in current and
voltage. This can be achieved by properly terminating the antenna wire so that the
reflections are minimized if not completely eliminated.
• An example of such an antenna is a long wire that runs horizontal to the earth. The
input terminals consist of the ground and one end of the wire. This configuration is
known as Beverage or wave antenna.
Fig31 Long wire above ground
8 Travelling-Wave Wire Antenna
• In general, all antennas whose current and voltage distributions can be represented by
one or more traveling waves, usually in the same direction, are referred to as traveling
wave or nonresonant antennas. A progressive phase pattern is usually associated with
the current and voltage distributions.
• Standing wave antennas, such as the dipole, can be analyzed as traveling wave
antennas with waves propagating in opposite directions (forward and backward) and
represented by traveling wave currents If and Ib.
• Besides the long wire antenna there are many examples of traveling wave antennas
such as dielectric rod (polyrod), helix, and various surface wave antennas. Aperture
antennas, such as reflectors and horns, can also be treated as traveling wave antennas.
In addition, arrays of closely spaced radiators (usually less than λ/2 apart) can also be
analyzed as traveling wave antennas by approximating their current or field distribution
by a continuous traveling wave. Yagi-Uda, log-periodic, and slots and holes in a
waveguide are some examples of discrete-element traveling wave antennas.
8.1 Long Wire
• An example of a slow wave traveling antenna is a long wire. An antenna is usually
classified as a long wire antenna if it is a straight conductor with a length from one to
many wavelengths. A long wire antenna has the distinction of being the first traveling
wave antenna. The traveling-wave long wire antenna is shown in with a matched load
RL to prevent reflections from the wire end.
Fig32 Long wire antenna
8.1 Long Wire
• The height h of the antenna above the ground must be chosen so that the reflected wave
(or wave from the image), which includes the phase due to reflection is in phase with
the direct wave at the angles of desired maximum radiation.
• Simplifying assumptions:
▫ the ground plane effects is ignored and free space;
▫ the details of the feed are assumed to be unimportant, the vertical section of length
d is assumed not to radiate;
▫ the radiative and ohmic losses along the wire are small.
• Then we can write
𝐼𝑡 𝑧 = 𝐼𝑚 𝑒 −𝑗𝛽𝑧
8.1 Long Wire
• To verify some of the derivations and illustrate some of the principles, a number of
computations were made.
Fig33 Three-dimensional free-space amplitude pattern for traveling and standing wave wire antennas of l = 5λ.
8.1 Long Wire
• To verify some of the derivations and illustrate
some of the principles, a number of computations
were made.
• The pattern of Figure (a) is formed by the forward
traveling wave current If = I1e −jkz of Figure(a)
while that of Figure (b) is formed by the forward If
plus backward Ib traveling wave currents of Figure
(a).
• The two currents If and Ib together form a standing
wave; that is,
Is = If + Ib = I1e−jkz − I2e+jkz = −2jI0 sin(kz )
when I2 = I1 = I0.
Fig34 Two-dimensional free-space amplitude
pattern for traveling and standing wave wire
antennas of l = 5λ.
8.1 Long Wire
• As expected, for the traveling wave antenna of
Figure (a) there is maximum radiation in the
forward direction while for the standing wave
antenna of Figure (b) there is maximum radiation
in the forward and backward directions. The lobe
near the axis of the wire in the directions of travel is
the largest.
• The magnitudes of the other lobes from the main
decrease progressively, with an envelope
proportional to cot2(θ/2), toward the other direction.
Fig34 Two-dimensional free-space amplitude
pattern for traveling and standing wave wire
antennas of l = 5λ.
8.1 Long Wire
• The traveling wave antenna is used when it is desired to radiate or receive
predominantly from one direction. As the length of the wire increases, the maximum of
the main lobe shifts closer toward the axis and the number of lobes increase.
Fig35 Two-dimensional free-space amplitude pattern for
traveling wave wire antenna of l = 5λ and 10λ.
8.1 Long Wire
• For traveling wave wire antennas the radiation in the opposite direction from the
maximum is suppressed by reducing, if not completely eliminating, the current
reflected from the end of the wire. This is accomplished by increasing the diameter of
the wire or more successfully by properly terminating it to the ground.
• Ideally a complete elimination of the reflections (perfect match) can only be
accomplished if the antenna is elevated only at small heights (compared to the
wavelength) above the ground, and it is terminated by a resistive load. The value of the
load resistor, to achieve the impedance match, is equal to the characteristic impedance
of the wire near the ground (which is found using image theory).
8.1 Long Wire
• For a wire with diameter d and height h above the ground, an approximate value of the
termination resistance is obtained from
𝑅𝐿 = 138 log10 4
ℎ
𝑑
• A long-wire antenna is linearly polarized, and it is always parallel to the plane formed
by the wire and radial vector from the center of the wire to the observation point. The
direction of the linear polarization is not the same in all parts of the pattern, but it is
perpendicular to the radial vector (and parallel to the plane formed by it and the wire).
• when its height above the ground is small compared to the wavelength and its main
beam is near the ground, is not an effective element for horizontal polarization. Instead
it is usually used to transmit or receive waves that have an appreciable vector
component in the vertical plane. This is what is known as a Beverage antenna which is
used more as a receiving rather than a transmitting element because of its poor
radiation efficiency due to power absorbed in the load resistor.
8.2 V Antenna
• For some applications a single long-wire antenna is not very practical because (1) its
directivity may be low, (2) its side lobes may be high, and (3) its main beam is inclined
at an angle, which is controlled by its length. These and other drawbacks of single
long-wire antennas can be overcome by utilizing an array of wires.
• One very practical array of long wires is the V antenna formed by using two wires each
with one of its ends connected to a feed line.
Fig36 The radiation pattern of V antenna
8.2 V Antenna
• In most applications, the plane formed by the legs of the V is parallel to the ground
leading to a horizontal V array whose principal polarization is parallel to the ground
and the plane of the V. Because of increased side lobes, the directivity of ordinary
linear dipoles begins to diminish for lengths greater than about 1.25λ, However by
adjusting the included angle of a V-dipole, its directivity can be made greater and its
side lobes smaller than those of a corresponding linear dipole. Designs for maximum
directivity usually require smaller included angles for longer V’s.
Fig37 V antenna
8.2 V Antenna
• In theory, the reflections can even be eliminated by properly terminating the open ends
of the V leading to a purely traveling wave antenna. One way of terminating the V
antenna will be to attach a load, usually a resistor equal in value to the open end
characteristic impedance of the V-wire transmission line, as shown in Figure (a). The
terminating resistance can also be divided in half and each half connected to the ground
leading to the termination of Figure (b).
Fig38 Two way of leading to a purely traveling wave antenna
8.2 V Antenna
• If the length of each leg of the V is very long (typically l > 5λ), there will be sufficient
leakage of the field along each leg that when the wave reaches the end of the V it will
be sufficiently reduced that there will not necessarily be a need for a termination. Of
course, termination with a load is not possible without a ground plane.
• For optimum operation, typically the included angle is chosen to be approximately θ0
0.8θm.
𝜃𝑚 = cos−1 1 −
θm is the angle of maximum radiation.
0.371
𝐿 𝜆
8.3 Rhombic Antenna
• Two V antennas can be connected at their open ends to form a diamond or rhombic
antenna as shown in Figure (a), another configuration of a rhombus is that of Figure (b)
which is formed by an inverted V and its image (shown dashed).
Fig39 Rhombic antenna formed by 2V
8.3 Rhombic Antenna
• The antenna is usually terminated at one end in a resistor, usually about 600–800 ohms,
in order to reduce if not eliminate reflections. However, if each leg is long enough
(typically greater than 5λ) sufficient leakage occurs along each leg that the wave that
reaches the far end of the rhombus is sufficiently reduced that it may not be necessary
to terminate the rhombus.
• Rhombic antennas are usually preferred over V’s for nonresonant and unidirectional
pattern applications because they are less difficult to terminate.
Fig40 Rhombic antenna
66
Homework:
• A resonant center-fed dipole is connected to a 50-ohm line. It is desired to maintain the
input VSWR = 2.
(a) What should the largest input resistance of the dipole be to maintain the VSWR = 2?
(b) What should the length (in wavelengths) of the dipole be to meet the specification?
(c) What is the radiation resistance of the dipole?
• Group discussion and write a report in teamwork:
Conclude the similarity and difference between each wire antenna