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CH1:
Radiation
Principles
10/12/2015
Radiation
1
Contents:
1
2
3
4
5
6
7
8
10/12/2015
Potential functions and the electromagnetic fields
Potential functions for sinusoidal oscillations
The alternating current element
POWER RADIATED BY CURRENT ELEMENT
Application to Short Antenna
Assumed current distribution
Radiation from quarter wave monopole or Half wave Dipole
Near Field and far field
Radiation
2
POTENTIAL FUNCTIONS
AND
THE ELECTROMGNETIC FIELDS
Relation of fields with their sources
Fields: Electric Field E, Magnetic Field H
Sources: Charge density ρ , Current density J
Required is:
To generate expressions of E and H from ρ and J
How?
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Contents
POTENTIAL FUNCTIONS
AND
THE ELECTROMGNETIC FIELDS
To generate expressions of E and H from ρ and J:
First setup potentials from sources in terms of ρ and J
Then obtain fields from these potentials
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4
Contents
POTENTIAL FUNCTIONS
AND
THE ELECTROMGNETIC FIELDS
Three ways to derive expressions of E and H for to
generate fields:1
...
Second Method: From Maxwell Equations drive differential equations
that potentials must satisfy and the potentials are guessed to satisfy
the differential equations
3
...
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6
Contents
Heuristic approach:
In steady magnetic fields: vector potential A was obtained from source
Currents density J , then required field H
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7
Contents
Heuristic approach:
Sources of EM fields; current and charges are time function
So potentials are also functions of time:
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8
Contents
Heuristic approach:
Potentials are instantaneous functions of time
Distance between source and observation point
Function of the path distance delay
Retarded potentials
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9
Contents
Heuristic approach:
Potentials are instantaneous functions of time
Distance between source and observation point
Function of the path distance delay
Retarded potentials
Magnetic and Electric fields are derived from A and V
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10
Contents
Heuristic approach:
Potentials are instantaneous functions of time
Distance between source and observation point
Function of the path distance delay
Retarded potentials
Magnetic and Electric fields are derived from A and V
Potentials A and V, are not independent, but they can be
related by Maxwell’s equation
...
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11
Contents
Heuristic approach:
In general only A is evaluated
H is obtained from A
E is obtained by curl of H and taking integration with time
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12
Contents
Maxwell’s equations approach:
Maxwell’s equations are
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13
Contents
Maxwell’s equations approach:
Relation of current density and source charge density
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14
Contents
Maxwell’s equations approach:
Relation of current density and source charge density
To satisfy equation
it is required that H must satisfy
Identity: div of curl of A=0
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15
Contents
Maxwell’s equations approach:
Relation of current density and source charge density
To satisfy equation
Using the equation
10/12/2015
Radiation
it is required that H must satisfy
above equation becomes
16
Contents
Maxwell’s equations approach:
Relation of current density and source charge density
To satisfy equation
Using the equation
Then
10/12/2015
it is required that H must satisfy
above equation becomes
must satisfy that it is gradient of a scalar
Radiation
17
Contents
Maxwell’s equations approach:
Relation of current density and source charge density
To satisfy equation
Using the equation
Then
10/12/2015
it is required that H must satisfy
above equation becomes
must satisfy that it is gradient of a scalar
Radiation
18
Contents
Maxwell’s equations approach:
Both equations
in equation
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and
and it gives
Radiation
are substituted
19
Contents
Maxwell’s equations approach:
Both equations
in equation
10/12/2015
and
and it gives
Radiation
are substituted
20
Contents
Maxwell’s equations approach:
Both equations
in equation
and
and it gives
Using the identity
10/12/2015
are substituted
it gives
Radiation
21
Contents
Maxwell’s equations approach:
Both equations
in equation
and
and it gives
Using the identity
are substituted
it gives
Helmholtz theorem says any vector field due to finite source is specified
if both curl and divergence of the field are specified
...
So if we put
is
equal to zero then divergence of A is also specified
...
Curl of A is specified but not divergence
...
And above equation is now:
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23
Contents
Maxwell’s equations approach:
Both equations
in equation
and
and it gives
Using the identity
are substituted
it gives
Helmholtz theorem says any vector field due to finite source is specified
if both curl and divergence of the field are specified
...
So if we put
is
Equal to zero then divergence is specified
...
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33
Contents
THE ALTERNATING CURRENT
ELEMENT
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Contents
THE ALTERNATING CURRENT
ELEMENT
It is required that:
•Current element should very short length
•Such short that flowing current is considered
constant over length of the element
...
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35
Contents
THE ALTERNATING CURRENT ELEMENT
The alternating current element = I dl cos ωt
and general expression for A is:
In
, dV’ = dS x dl, current density J is
integrated over cross section area of wire, so
J dV= J x dS x dl=dI x dl
And it becomes I dl cos ω(t - r/v)
Current element is oriented in z direction so
magnetic potential A is also designated as Az
...
and
Aφ is zero always
...
Hence Hφ in not zero and Hθ, Hr are always zero
...
Hence Hφ in not zero and Hθ, Hr are always zero
...
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Contents
POWER RADIATED BY
CURRENT ELEMENT
The average value of sin 2 ωt’ and cos 2 ωt’ over a complete cycle is
zero
...
This power goes farward and
backward
...
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49
Contents
POWER RADIATED BY
CURRENT ELEMENT
Removing the sin 2 ωt’ and cos 2 ωt’, terms containing these are
removed and average value of the radial Poynting vector over a cycle
will be due to the last left term only:
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50
Contents
POWER RADIATED BY
CURRENT ELEMENT
The amplitudes of the radiation fields Eθ , Hφ ;of an electric
current element I dl , can be derived from Pr below equation,
which are:
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Radiation
51
Contents
POWER RADIATED BY
CURRENT ELEMENT
The total power radiated by current element can be computed by
integrating he radial Poynting vector over a spherical surface
centered at the element
...
5
ohms and for half wave dipole is 73 ohms
...
To determine any near distant parameter of antenna, it is required to
know near fields
2
...
Near field , infact
it is more complex
...
Electric field is parallel to z direction & antenna, hence for
determination of fields; cylindrical or all cylindrical, spherical and
rectangular coordinate systems are preferred
...
The first term represents spherical wave originating at the top end
of the antenna
2
...
Second term represents also spherical wave originating from
bottom end of the antenna
4
...
The third term, represents a wave originating at the center of the
antenna, at base in case of the monopole
...
The sources of the wave are isotropic
...
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Contents
Near Field and far field/ Electromagnetic field close to antenna
Compares the relative magnitudes of the quadrature terms of parallel and
perpendicular components of E
...
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Contents
Near Field and far field/ Electromagnetic field close to antenna
In-Phase & Quadrature Sinusoidal Components
From the trig identity,
we have
From this we may conclude that every sinusoid can be expressed
as the sum of a sine function (phase zero) and a cosine function
(phase π/2)
...
In general, “phase quadrature” means “90 degrees out of phase,”
i
...
, relative phase shift of ± π/2
...
The proof is obtained
by working the previous derivation backwards
...
Note that they only
differ by a relative degree phase shift