ELECTROMAGNETIC WAVE PROPAGATION EC 442
Transcript of ELECTROMAGNETIC WAVE PROPAGATION EC 442
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ELECTROMAGNETIC WAVE PROPAGATION
EC 442 Prof. Darwish Abdel Aziz
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CHAPTER 6
LINEAR WIRE ANTENNAS INFINITESIMAL DIPOLE
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INTRODUCTION
Wire antennas, linear or curved, are some of the
oldest, simplest, cheapest, and in many cases the
most versatile for many applications.
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INFINITESIMAL DIPOLE 1 - INTRODUCTION
An infinitesimal linear wire is positioned symmetrically at the origin of the coordinate system as shown in Figure (6-1).
Figure 6-1 Infinitesimal dipole
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INFINITESIMAL DIPOLE
• 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. May, 2015 5 Prof. Darwish
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INFINITESIMAL DIPOLE 2 – CURRENT DISTRIBUTION
• The wire, in addition to being very small , is very
thin .
• The spatial variation of the current is assumed to be
constant and it current element is given by
Where, .
• The remaining two equations are unchanged from their static (non time-varying) form:
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INFINITESIMAL DIPOLE 3 – RADIATION EQUATIONS
• Since
• So,
, and Where, and and
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INFINITESIMAL DIPOLE 3 – RADIATION EQUATIONS
• So , and,
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INFINITESIMAL DIPOLE 4 – AUXILIARY VECTOR POTENTIAL FUNCTION
• So the electric vector potential components are:
• While the magnetic vector potential components are:
, and
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INFINITESIMAL DIPOLE 5 – THE RADIATED FIELD COMPONENTS
• The Magnetic Field Components can be found as follows:
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INFINITESIMAL DIPOLE 5 – THE RADIATED FIELD COMPONENTS
and • The Electric Field Components can be found as follows:
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INFINITESIMAL DIPOLE 5 – THE RADIATED FIELD COMPONENTS
So ,
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INFINITESIMAL DIPOLE 5 – THE RADIATED FIELD COMPONENTS
and and
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INFINITESIMAL DIPOLE 6 – THE RADIAL AND TRANSVERSE POWER DENSITY
• For the infinitesimal dipole, the complex Poynting vector can be written using (6-6a) - (6-6b) and (6-8a) - (6-8c) as
Whose radial and transverse components are given,
respectively, by May, 2015 14 Prof. Darwish
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INFINITESIMAL DIPOLE 7 – THE RADIAL POWER
• The complex power moving in the radial direction is obtained by integrating (6-9) – (6-10b) over a closed sphere of radius . Thus it can be written as
which reduces to May, 2015 15 Prof. Darwish
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INFINITESIMAL DIPOLE 8 – THE REACTIVE POWER
• The transverse component of the power density does not
contribute to the integral. Thus (6-12) does not represent
the total complex power radiated by the antenna.
• Since , as given by (6-11b), is purely imaginary, it will not
contribute to any real radiated power.
• However, it does contribute to the imaginary (reactive)
power which along with the second term of (6-12) can be
used to determine the total reactive power of the antenna.
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INFINITESIMAL DIPOLE 9 – THE TOTAL OUTWARDLY RADIAL POWER
• The reactive power density, which is most dominant for
small values of , has both radial and transverse
components.
• Equation (6-11b), which gives the real and imaginary
power that is moving outwardly, can also be written as
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INFINITESIMAL DIPOLE 9 – THE TOTAL OUTWARDLY RADIAL POWER
Where
From (6-12)
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INFINITESIMAL DIPOLE 9 – THE TOTAL OUTWARDLY RADIAL POWER
• It is clear from (6-15) that the radial electric energy must
be larger than the radial magnetic energy.
• For large values of , the reactive
power diminishes and vanishes when .
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INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE
for infinitesimal dipole, as represented
by (6- 6a) - (6- 6c) and (6- 8a) - (6- 8b), are valid everywhere
(except on the source itself). An inspection of these
equations reveals the following:
• At a distance ,
which is referred to as the radian distance,
the magnitude of the first and second terms within the
brackets of (6-6c) and (6-8a) is the same.
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INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE
Also at the radian distance the magnitude of all three terms
within the bracket of (6 – 8b) is identical; the only term that
contributes to the total field is the second, because the first
and third terms cancel each other.
• At distances less than the radian distance ,
the magnitude of the second term within the brackets of
(6 - 6c) and (6 – 8a) is greater than the first term and begins
to dominate as .
o
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INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE
For (6-8b) and , the magnitude of the third term
within the brackets is greater than the magnitude of the
first and second terms while the magnitude of the second
term is greater than that of the first one; each of these terms
begins to dominate as .
The near-field region, is defined as the region
, and the energy in that
region is basically imaginary (stored).
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INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE
• At distances greater than the radian distance ,
The first term within the brackets of (6-6c) and (6-8a) is
greater than the magnitude of the second term and begins to
dominate as .
For (6-8b) and , the first term within the brackets is
greater than the magnitude of the second and third terms
while the magnitude of the second term is greater than that of
the third; each of these terms begins to dominate as .
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INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE
The intermediate - field region is defined as the region
The far- field region is defined as the region
, and the energy in that region
is basically real (radiated).
• The radian sphere is defined as the sphere with radius
equal to the radian distance .
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INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE
• The radian sphere defines the region within which the
reactive power density is greater than the radiated power
density.
For an antenna, the radian sphere represents the volume
occupied mainly by the stored energy of the antenna’s
electric and magnetic fields.
Outside the radian sphere the radiated power density is
greater than the reactive power density and begins to
dominate as .
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INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE
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INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE
• The radian sphere can be used as a reference, and it defines
the transition between stored energy pulsating primarily in
the direction [represented by (6-10b)] and energy
radiating in the radial direction [represented by (6-10a); the
second term represents stored energy pulsating inwardly
and outwardly in the radial direction].
• Similar behavior, where the power density near the antenna
is primarily reactive and far away is primarily real, is
exhibited by all antennas, although not exactly at .
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INFINITESIMAL DIPOLE 11 – NEAR FIELD REGION
• An inspection of (6-6a)- (6-6b) and (6-8a)- (6-8c) reveals that for or they can be reduced in much simpler form and can be approximated by
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INFINITESIMAL DIPOLE 11 – NEAR FIELD REGION
• The components, are in time- phase but
they are time- phase quadrature with the
component ; therefore there is no time-average power
flow associated with them. This is demonstrated by forming
the time- average power density as
which by using (6-16a)- (6-16d) reduces to
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INFINITESIMAL DIPOLE 12 – INTERMEDIATE FIELD REGION
• As the values of begin to increase and become greater
than unity, the terms that were dominant for become
smaller and eventually vanish. For moderate values of
the components lose their in-phase condition and
approach time-phase quadrature. Since their magnitude is
not the same, in general, they form a rotating vector whose
extremity traces an ellipse.
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INFINITESIMAL DIPOLE 12 – INTERMEDIATE FIELD REGION
• At these intermediate values of , components
approach time-phase, which is an indication of the
formation time-average power flow in the outward (radial)
direction (radiation phenomenon).
• As the values of become moderate , the field
expression can be approximated again but in a different
form. In contrast to the region where , the first term
within the brackets in (6-6b) and (6-8a) becomes more
dominant and the second term can be neglected.
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INFINITESIMAL DIPOLE 12 – INTERMEDIATE FIELD REGION
• The same is true for (6-8b) where the second and third
terms become less dominant than the first.
• Thus we can write for
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INFINITESIMAL DIPOLE 13 – FAR - FIELD REGION
• Since (6-19a) - (6-19d) are valid only for values of ,
then will be smaller than because is inversely
proportional to where is inversely proportional to .
• In a region where , (6-19a) - (6-19d) can be simplified
and approximated by
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INFINITESIMAL DIPOLE 13 – FAR - FIELD REGION
• The ratio of to is equal to
where
The components are perpendicular to each
other, transverse to the radial direction of propagation, and
the variations are separable from of variations.
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INFINITESIMAL DIPOLE 13 – FAR - FIELD REGION
• The shape of the pattern is not a function of the radial
distance , and the fields form a Transverse ElectroMagnetic
(TEM) wave whose wave impedance is equal to the intrinsic
impedance of the medium.
• As it will become even more evident, this relationship is
applicable in the far-field region of all antennas of finite
dimensions.
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INFINITESIMAL DIPOLE 14 – FAR FIELD RADIATED COMPONENTS
• The far field components of (6-20a) - (6-20c) can also
be derived using the procedure outlined and
relationships developed in chapter-5 of auxiliary
vector potential functions.
• The far field radiated components using the radiation
equations can be
written as:
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INFINITESIMAL DIPOLE 14 – FAR FIELD RADIATED COMPONENTS
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INFINITESIMAL DIPOLE 15 – POWERE DENSITY AND RADIATION RESISTANCE
• The input impedance of an antenna, which consists of real and imaginary parts as discussed in Chapter- 4 (Fundamental Parameters of Antenna).
• For a lossless antenna, the real of the input impedance was designated as radiation resistance, through which the radiated power is transferred from the guided wave to the free space wave.
• To find the input resistance for a lossless antenna, it is required to find the time average poynting vector as
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INFINITESIMAL DIPOLE 15 – POWERE DENSITY AND RADIATION RESISTANCE
• The total radiated power in the radial direction is
obtained by integrating (6-23e) over a closed sphere of
radius . Thus it can be written as:
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INFINITESIMAL DIPOLE 15 – POWERE DENSITY AND RADIATION RESISTANCE
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INFINITESIMAL DIPOLE 15 – POWERE DENSITY AND RADIATION RESISTANCE
• Since the antenna radiates its real power through the
radiation resistance, for the infinitesimal dipole it can be written that
• For free space medium, , where is the intrinsic impedance, so
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INFINITESIMAL DIPOLE 16 – DIRECTIVITY
• As was shown before, the average power density of the infinitesimal dipole is given by (6-23e) as
• As was discussed in Chapter- 4 (Fundamental Parameters
of Antenna), the radiation intensity can be obtained from
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INFINITESIMAL DIPOLE 16 – DIRECTIVITY
• The maximum value of the radiation intensity occurs at and it is equal to
• The real power radiated by the infinitesimal dipole is
given by (6-24e) as
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INFINITESIMAL DIPOLE 16 – DIRECTIVITY
• As was discussed in Chapter- 4 (Fundamental Parameters of Antenna), the directivity is given as
• As was discussed in Chapter- 4 (Fundamental Parameters of Antenna), for lossless antenna, the relation between the directivity and the maximum effective aperture area is given as
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