Antennas
Some Properties and Principles of Antennas
An antenna may be viewed as a transducer used to match the transmission line to the
surrounding medium or vice versa. (Sadiku 588)
Transmission lines are designed to guide electromagnetic energy with a minimum of
radiation. All antennas involve the same basic principle that radiation is produced by
accelerated (or decelerated) charge. (Kraus 5th ed 247) Note: Time-varying current means
that electrons are accelerated and decelerated.
248 In a transmission line, if the spacing between wires is a small fraction of wavelength,
waves are guided along the sine with little loss by radiation. As the separation approaches
the order of a wavelength or more, the wave tends to be radiated so that the opened-out
line acts like an antenna which launches a free-space wave.
Ulaby Fig 9.1
Note: This is consistent with our model of transmission lines in which the energy is
passed on to inductive and capacitive elements. When the energy gets to the antenna, the
inductive and capacitive elements become the permittivity and permeability properties of
space, and Maxwell’s equations require that the varying electric field between the
conductors creates a magnetic field which creates an electric field, etc. in a selfperpetuating fashion.
The currents in the conductors were created by an electric field. In order to create a
current in a receiving conductor (antenna), there must be an electric field induced in that
conductor. That electric field can be created either by the received electric field
component or by the magnetic field component or a combination of both. We will focus
on the electric field component.
We saw before that E and H are at right angles to each other and to the direction of
propagation.
Ulaby Fig. 7.5
So if a sending antenna is oriented so that the electric wave is vertical, a vertical
receiving antenna will be congruent with the vertical electric field. (RFID Fig 3.33)
http://www.rfdesignline.com/howto/202404293;jsessionid=AYVJ1XMI12AROQSNDLP
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Generalization: Parallel antennas will maximize received signal because the induced
current is due only to the component of the electric field parallel to the wire. (Similarly, a
magnetic field varying perpendicular to a conductor induces the greatest emf. Since the
orientation of the magnetic field is perpendicular to the electric field, one can think of the
electric field as having been generated by the magnetic field.)
Polarization of an electromagnetic wave refers to the orientation of the electric field
component of the wave. For a linearly polarized wave, the orientation stays the same as
the wave moves through space. If we choose our axis system such that the electric field is
vertical, we say that the wave is vertically polarized. If our transmitting antenna is
vertically oriented, the electromagnetic wave radiated is vertically polarized since, as we
saw before, the electric field is in the direction of the current in the antenna.
The convention is to refer to polarization with reference to the surface of the earth.
Precise orientation is less problematic than one might think, since waves bounce of the
ground and other objects so do not maintain their original orientation anyway. In space,
horizontal and vertical lose their meaning, so alignment of linearly polarized sending and
receiving antennas is more difficult to achieve. These difficulties are somewhat
circumvented by circular polarization of waves. (RFID 3.34) With circular polarization,
the tip of the electric field vector traces out a circle when viewed in the direction of
propagation.
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Notice that a vertical linear antenna intercepts only the vertical component of the electric
field and, therefore, the intercepted power is one-half of that that passes through the circle
that bounds the circle containing the antenna. With this configuration, however,
orientation of the receiving antenna is less critical; there will be no “dead signal”
orientations.
Actual antennas have different radiation patterns, i.e. they radiate more strongly in some
directions than in other directions. Generally, antennas are reciprocal devices, i.e. an
antenna that radiates strongly in a given direction is more sensitive to reception of signals
from that direction. For example, a dipole antenna one-half wavelength long has its
strongest radiation in a direction perpendicular to its axis and zero radiation in the
direction of its axis. (RFID Fig 3.27)
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Thus, the strength of signal transfer depends on a combination of distance, polarization
and direction of the receiver with respect to the transmitter, as well as several other
factors, such as reflections, wavelength, weather, etc.
See Ulaby M9.1 for cell phone example
Quantitative Characterization of Antennas
The Short Dipole or Hertzian Dipole (Ulaby Fig 9-4)
The dipole length, L << λ so L/λ is very small.
It is shown in Ulaby p. 376 that the vector magnetic potential of the Hertzian dipole is:
where Io is the peak current of Io cos ωt.
The figure below shows the electric field in a plane of the dipole at an instant in time.
Ulaby Fig 9.6
9-1.2 Radiation Power Density Field
We have seen that the Poynting Vector S = E X H is a measure of the rate of energy flow
per unit area through any point on a surface, S. Also, SAV = ½ Re[EXH*] is the time
average power density. (Ulaby, p 378) For a Hertzian, or short, dipole:
Which describes mathematically the toroid-shaped radiation field of a short dipole.
We see that Smax is directly proportional to the square of the current, the square of L/λ
and inversely proportional to R2. S is a function of sin2θ as well, in the third dimension.
The shape of the radiation field for a real dipole is similar and is given by:
which also has a maximum at θ = 90 degrees.
Normalized Radiation Intensity (Ulaby p 379)
The normalized radiation intensity provides us with a convenient expression for the
shape of the radiation pattern.
F(θ,φ) = S(R, θ,φ)/Smax
where Smax is the maximum value of S(R,θ,φ) at a specified range, R.
For the Hertzian dipole: F(θ,φ) = sin2θ
For the real dipole: F(θ,φ) = (cos2[(π/2)cosθ])/sin2θ
Below is a normalized radiation pattern of a microwave antenna. (Ulaby Fig 9-10)
9-2.3 Antenna Directivity (Ulaby p 383)
Antenna directivity is a measure of the degree to which the radiation from an antenna is
in particular directions in comparison to an isotropic antenna, which has the same total
power distributed evenly in all directions. Mathematically, antenna directivity is the ratio
of the maximum power density, Smax, radiated by the antenna to the power density
radiated by an isotropic antenna, which is the same in all directions. An antenna that has a
large amount of power directed in a particular direction has a greater directivity than one
that has its power distributed over a greater range of directions. To find the power density
of the isotropic antenna, we find the total radiated power and divide it by the total solid
angle of a sphere, 4π.
i.e. for the Hertzian dipole, the power density in the direction of maximum power density
is 1.5 times the average.
The directivity of a real dipole is 1.64 or 2.15 dB.
Antenna Gain
We are considering only passive antennas, which don’t have any ability to increase the
output power over the input power. Antenna gain is the relative increase in radiation at
the maximum point expressed as a value in dB above a standard. The real standard is the
dipole so the referenced gain is in dBd. Another standard is the fictitious isotropic
antenna for which the referenced gain is in dBi.
Types and Characteristics of Antennas
All characteristics figures are from:
http://www.kyes.com/antenna/navy/rpatterns/radiapat.htm
Dipole http://www.electronics-tutorials.com/antennas/antenna-basics.htm
Gains are relative to an isotropic antenna.
Helix
Parabolic
http://www.radio-electronics.com/info/antennas/parabolic/parabolic_reflector.php
Note: polarization is same as feed type, frequently circular.
Yagi
http://www.electronics-tutorials.com/antennas/antenna-basics.htm
9-2.5 Radiation Resistance
To the transmission line connected to its terminal, an antenna is merely an impedance.
If the transmission line is matched to the antenna, part of the power supplied by the
generator is radiated into space and the remainder is dissipated as heat in the antenna.
(Ulaby 386) Radiation resistance is the power radiated by the antenna divided by the
square of the rms current. In general, a difficulty in analyzing or designing antennas is
that the current distribution in the antenna is not known, so it is necessary to express the
current magnitude in terms of some reference. (Mott, p 22)
We can find the radiated power by: Prad = ∫Sav = ½ Re ∫[ExH] ∙ dS
i.e. by integrating the time-average Poynting vector over an entire closed spherical
surface surrounding the antenna. To do this, we express E and H in terms of the current
in the antenna, as we have done above, starting with the vector magnetic potential. A.
Since the Hertzian dipole is so short, we assume that the current is equal to Io along its
entire length. For the Hertzian dipole:
We can then calculate the radiation resistance for the Hertzian dipole using
Prad = ½ Io2Rrad
where Io is the peak of the sinusoidal current at the input to the antenna.
Rrad = 0.08 ohms, a very low resistance and therefore low radiated power.
The calculation of radiated power and radiation resistance for a dipole antenna is
somewhat more complicated; but is simpler than for most other antennas. In the case of
linear wire antennas, it has been found that the current behaves approximately as it does
on a transmission line. (Schwarz 353) It has been found from experimental
measurements that the current distribution on a dipole antenna is approximately:
I = Io sin[(2π/λ)(L/2 +-y)]
Where Io is the current at its point of maximum, and where +y is used when y<0 and –y
when y>0. For the half-wave dipole, L=λ/2 and I=Io at the center (y=0) and I = 0 at the
ends. (Kraus p 285)
It is shown (Ulaby p 389) that for a half-wave dipole, Rrad = 73 ohms, approximately
equal to the standard 75 ohm transmission line, designed to match a half-wave dipole.
Antenna Impedance and Matching
Radiation resistance and ohmic resistance is only part of the antenna impedance.
Inductive and capacitive reactance can be present. Energy that is transferred to the near
field relates to the reactive component of the current in the antenna. This is the 1/R2
component of the electric and magnetic fields that we neglected when deriving the
expressions for the far fields.
There are a variety of complex models for antenna impedance.
See http://www.ewh.ieee.org/r6/scv/aps/index_files/Stearns_APS_031406.pdf
We will use simple approximate models – series and parallel RLC circuits.
See http://www.borg.com/~warrend/guru.html
When the dipole is very short (in terms of physical length relative to wavelength), the
dipole can be modeled as a series RLC circuit in which the impedance is dominated by
radiation resistance and capacitive reactance. As the antenna is made longer, Rrad and XL
increase and XC decreases. When the physical length equals λ/2, XL = XC with a resulting
impedance of Rrad. As the antenna is made longer than λ/2, the model is a parallel RLC
circuit. When length equals λ, the tank LC circuit has infinite impedance, leaving the
parallel Rrad as the net impedance. Between a length of λ and 3λ/2, the model is a series
RLC circuit, and so forth.
See the figure for the variation in Rrad, which itself varies with wavelength. Note that Rrad
is about 73 ohms at λ/2.
Lonngren p. 448
So we want to use a 75 ohm transmission to match the impedance of a half-wave dipole
in order to have maximum power transfer from the generator to the antenna. Notice that
we used the impedance of space, ήo, in the calculation of Rrad. If we do not use a length of
λ/2, there will be impedance mismatch and reflections, leading to a standing wave ratio
greater than unity, i.e. less than maximum power transfer to the antenna.
Bandwidth
Note that the system is designed for specific frequency; i.e. at any other frequency it will
not be one-half wavelength. The bandwidth of an antenna is the range of frequencies over
which the antenna gives reasonable performance. One definition of reasonable
performance is that the standing wave ratio is 2:1 or less at the bounds of the range of
frequencies over which the antenna is to be used.
References
Ulaby
Lonngren
Kraus
Sadiku
Mott, Harold Antennas for Radar and Communications: A Polarimetric Approach John
Wiley & Sons, 1992 (In USM library)