Download - Lecture 28
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1ECE 303 Fall 2005 Farhan Rana Cornell University
Lecture 28
Antennas and Radiation and the Hertzian Dipole
In this lecture you will learn:
Generation of radiation by oscillating charges and currents
Hertzian dipole antenna
ECE 303 Fall 2005 Farhan Rana Cornell University
Maxwells Equations and Radiation
Maxwells equation predict outgoing radiation from sinusoidallytime-varying currents (and charges - recall that current and charge densities are related through the continuity equation:
( ) 0,. = trHo rr ( ) ( )t
trHtrE o= ,,
rrrr
( ) ( )trtrEo ,,. rrr = ( ) ( ) ( )t
trEtrJtrH o+= ,,,
rrrrrr
Time-varying currents as the source or the driving term for the wave equation:
( ) ( ) ( ) ( )
( ) ( ) ( )t
trJt
trEc
trE
ttrE
cttrJ
ttrHtrE
o
oo
=
+
==
,,1,
,1,,,
2
2
2
2
2
2
rrrrrr
rrrrrrrr
( ) ( ) 0,,. =+
ttrtrJ
rrr 911
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2ECE 303 Fall 2005 Farhan Rana Cornell University
Electro- and Magneto-quasistatics and Potentials
( ) ( )trtrEo ,,. rrr =( ) ( )trJtrH ,, rrrr =
( ) 0,. = trHo rr( ) 0, = trE rr
Electroquasistatics Magnetoquasistatics
( ) 0, = trE rrSince:
One could write:
( ) ( )trtrE ,, rrr =
Since:
One could write:
( ) ( )trAtrH ,, o rrrr =
( ) 0,. = trHo rr
Scalar potential Vector potential
ECE 303 Fall 2005 Farhan Rana Cornell University
Electrodynamics and Potentials - I( ) 0,. = trHo rr ( ) ( )
ttrHtrE o
= ,, rrrr
( ) ( )trtrEo ,,. rrr = ( ) ( ) ( )t
trEtrJtrH o+= ,,,
rrrrrr
One still has:( ) 0,. = trHo rr
Therefore, one can still introduce a vector potential:
( ) ( )trAtrH ,, o rrrr =Faradays law then becomes:
( ) ( )
( ) ( )
( ) ( ) 0,,
,,
,,
=
+
=
=
ttrAtrE
ttrAtrE
ttrHtrE o
rrrr
rrrr
rrrr
Vector Potential
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3ECE 303 Fall 2005 Farhan Rana Cornell University
Electrodynamics and Potentials - II
Using the vector and scalar potentials, the expressions for E-field and H-field become:
( ) ( )trAtrH ,, o rrrr =( ) ( ) ( )tr
ttrAtrE ,,, r
rrrr =
Scalar Potential
Since: ( ) ( ) 0,, =
+t
trAtrErrrr
One may introduce a scalar potential as follows:
( ) ( ) ( )( ) ( ) ( )tr
ttrAtrE
trt
trAtrE
,,,
,,,
rrr
rr
rrr
rr
=
=+
ECE 303 Fall 2005 Farhan Rana Cornell University
Choosing a Gauge in Electromagnetism
The vector potential A is not unique only the curl of the vector potential is a well defined quantity (i.e. the B-field):
( ) ( )trAtrH ,, o rrrr =Demonstration: suppose we change the vector potential - such that the new vector potential is the old vector potential plus the gradient of some arbitrary function
( ) ( ) ( )trtrAtrAnew ,,, rrrrr +=
Non-uniqueness of the vector potential
A vector field can be uniquely specified (up to a constant) by specifying the value of its curl and its divergence
To make the vector potential A unique, one needs to fix the value of its divergence a process that usually goes by the names gauge fixing or fixing the gauge or choosing a gauge
Making the vector potential unique
( ) ??,. = trA rr
Then: ( ) ( ) ( )( ) ( )trAtrA
trtrAtrA
new
new
,,,,,
rrrrrrrrr
=+=
0
The new vector potential is just as good as it will give the same B-field
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4ECE 303 Fall 2005 Farhan Rana Cornell University
Gauges in Electromagnetism
Coulomb Gauge:
This gauge is commonly used in electro- and magnetoquasistatics
This gauge is not commonly used in electrodynamics
( ) 0,. = trA rr
Lorentz Gauge:
( ) ( )t
trc
trA = ,1,. 2
rrr Relates divergence of the vector potential to the time derivative of the scalar potential
This gauge is commonly used in electrodynamics
ECE 303 Fall 2005 Farhan Rana Cornell University
Vector Potential Wave Equation
Using the vector and scalar potentials, the expressions for E-field and H-field were:
( ) ( )trAtrH ,, o rrrr = ( ) ( ) ( )trttrAtrE ,,,
rrr
rr =
Amperes Law becomes:
( ) ( ) ( )
( ) ( ) ( ) ( )
=
+=
ttr
cttrA
ctrJtrA
ttrEtrJtrH
o
o
,1,1,,
,,,
22
2
2
rrrrrrr
rrrrrr
( ) ( )[ ] ( )trAtrAtrA ,,., 2 rrrrrr =Remembering that:
( ) ( )t
trc
trA = ,1,. 2
rrr and
We finally get:
( ) ( ) ( )trJt
trAc
trA o ,,1, 2
2
22 rr
rrrr =
This is the vector potential wave equation with current as the driving term
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5ECE 303 Fall 2005 Farhan Rana Cornell University
Scalar Potential Wave Equation
Using the vector and scalar potentials, the expressions for E-field and H-field were:
( ) ( )trAtrH ,, o rrrr = ( ) ( ) ( )trttrAtrE ,,,
rrr
rr =
Gauss Law becomes:
( ) ( )( ) ( ) ( )
o
otr
ttrAtr
trtrE
,,.,
,,.2
rr
rrr
=
=
( ) ( )t
trc
trA = ,1,. 2
rrr
This is the scalar potential wave equation with charge as the driving term
( ) ( ) ( )o
trt
trc
tr ,,1, 2
2
22
rrr =
Remembering that:
We finally get:
ECE 303 Fall 2005 Farhan Rana Cornell University
Scalar and Vector Potential Wave Equations
( ) ( ) ( )trJt
trAc
trA o ,,1, 2
2
22 rr
rrrr =
( ) ( ) ( )o
trt
trc
tr ,,1, 2
2
22
rrr =
Given any arbitrary time-dependent charge and current density distributions one can solve these two wave equations to get the potentials:
911
Scalar potential wave equation
Vector potential wave equation
( ) ( )trAtrH ,, o rrrr =( ) ( ) ( )tr
ttrAtrE ,,,
rrr
rr =
And then find the E- and H-fields using:
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6ECE 303 Fall 2005 Farhan Rana Cornell University
Superposition Integral Solution of the Scalar Wave Equation
We know that Poisson equation in electrostatics:
( ) ( )o
rr
rr =2
Has the solution:
( ) ( ) ''4
' dvrr
rro
= rrrr
rr
'rr
'rrrr
( )'rrThe wave equation (which looks somewhat similar to the Poisson equation):
Has the solution:
( ) ( ) ''4',' , dv
rrcrrtrtr
o
= rrrrrr
( ) ( ) ( )o
trt
trc
tr ,,1, 2
2
22
rrr =
Retarded Potential: The potential at the observation point at any time corresponds to the charge at the source point at an earlier time Electromagnetic disturbancestravel at the speed c
ECE 303 Fall 2005 Farhan Rana Cornell University
Superposition Integral Solution of the Vector Wave Equation
We know that the vector Poisson equation in magnetostatics:
Has the solution:
rr
'rr
'rr rr
( )'rJ r
The wave equation (which looks somewhat similar to the vector Poisson equation):
Has the solution:
( ) ( )rJrA o rrrr =2
( ) ( ) ''4
' dvrr
rJrA o = rrrrrr
( ) ( ) ''4
',' , dvrr
crrtrJtrA o = rr
rrrrrr
( ) ( ) ( )trJt
trAc
trA o ,,1, 2
2
22 rr
rrrr =
Retarded Potential: The potential at the observation point at any time corresponds to the current at the source point at an earlier time Electromagnetic disturbancestravel at the speed c
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7ECE 303 Fall 2005 Farhan Rana Cornell University
Time-Harmonic Fields and Complex Wave Equations
( ) ( ) ( )trJt
trAc
trA o ,,1, 2
2
22 rr
rrrr =
Assuming time harmonic currents, charges, and fields, the wave equations:
become:
( ) ( ) ( )rJrAkrA o rrrrrr =+ 22
( ) ( ) ( )o
trt
trc
tr ,,1, 2
2
22
rrr =
( ) ( ) ( )o
rrkr
rrr =+ 22
( ) ( )[ ]tjerEtrE rrrr Re, = ( ) ( )[ ]tjerHtrH rrrr Re, =
( ) ( )[ ]tjertr rr Re, = ( ) ( )[ ]tjerJtrJ rrrr Re, =( ) ( )[ ]tjerAtrA rrrr Re, = ( ) ( )[ ]tjertr rr Re, =
ck =
ECE 303 Fall 2005 Farhan Rana Cornell University
Superposition Integral Solutions of the Complex Wave Equations
The solutions to the complex wave equations are found as follows:
( ) ( )[ ]tjertr rr Re, = ( ) ( )[ ]tjerJtrJ rrrr Re, =( ) ( )[ ]tjerAtrA rrrr Re, = ( ) ( )[ ]tjertr rr Re, =
( ) ( ) ''4',' , dv
rrcrrtrtr
o
= rrrrrr
For time harmonic signals:
( ) ( ) ''4
' ' dverr
rr rrkjo
= rrrr
rr
( ) ( ) ( )[ ] ( )[ ]tjcrrjcrrtj eerercrrtr '' 'Re'Re',' rrrr rrrrr ==
( ) ( ) ''4
' ' dverr
rJrA rrkjo = rrrr
rrrr( ) ( ) '
'4',' , dv
rrcrrtrJtrA o
= rrrrrrrr
( ) ( ) ( )[ ] ( )[ ]tjcrrjcrrtj eerJerJcrrtrJ '' 'Re'Re',' rrrr rrrrrrrr ==
And:
Where we have used:
Where we have used:
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8ECE 303 Fall 2005 Farhan Rana Cornell University
Wave Equations and Methods of Solution
Suppose we need to find the radiation emitted by some collection of sinusoidally time varying charges and currents
(1) We can solve these two equations:
(2) And then find the E-field and the H-field through:
( ) ( )rArH rrrr =o ( ) ( ) ( )trrAjrE ,rrrrr =
(1) We can solve just one equation:
(2) And then find the H-field and the E-field through:
( ) ( ) ( )rEjrJrH o rrrrrr +=
911
( ) ( ) ( )o
rrkr
rrr =+ 22( ) ( ) ( )rJrAkrA o rrrrrr =+ 22
( ) ( ) ( )rJrAkrA o rrrrrr =+ 22
( ) ( )rArH rrrr =o
Method 1
Method 2
ECE 303 Fall 2005 Farhan Rana Cornell University
Hertzian Dipole Antenna - I A Hertzian dipole is one of the simplest radiating elements for which analytical solutions for the fields can be obtained
A Hertzian dipole consists of two equal and opposite charge reservoirs located at a distance d from each other, as shown below:
x
z
d
The two charge reservoirs are electrically connected and a sinusoidal current I(t) flows between them
Consequently, the charge in the reservoirs also changes sinusoidally:
t
I(t)q(t) -q(t)
q(t)
-q(t)
( ) ( )dt
tdqtI =
I(t)
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9ECE 303 Fall 2005 Farhan Rana Cornell University
Hertzian Dipole Antenna - II
x
z
dq(t)
-q(t)I(t)
Suppose we could write a current density for the Hertzian dipole
Then:
( )rJ rr
( ) ( ) dIzdzdydxrJdvrJ == rrrr
The integral represents the total weight or strength of the dipole If the size of the dipole is much smaller than the wavelength then one may write:
( ) ( ) ( ) ( )( )rdIz
zyxdIzrJr
rr
3
==
( ) 33 m1rrAmpImd
( ) 2mAmprJ rr
The above expression will give the same strength for the dipole, i.e.
Check units:
( ) dIzdvrJ = rr
ECE 303 Fall 2005 Farhan Rana Cornell University
Hertzian Dipole Antenna - III A Hertzian dipole is represented by an arrow whose direction indicates the positive direction of the current and also the orientation of the dipole in space:
x
z
dq(t)
-q(t) x
z
Example: If then:( ) ( ) += tItI o cos ( ) ( )redIzrJ jo rrr 3 =
I(t)
jo edI
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10
ECE 303 Fall 2005 Farhan Rana Cornell University
Radiation Emitted by a Hertzian Dipole - I
Need to solve:
( ) ( ) ( )rJrAkrA o rrrrrr =+ 22Use the superposition integral:
( ) ( )
( ) rkjorkjorrkjo
erIdze
rIdzrA
dverr
rJrA
==
=
4
4
''4
' '
r
rr
rrr
rrrrrr
Finding the H-field:
( ) ( ) ( )[ ] rkjo erIdrrA =
4
sincos rr
( ) ( )( ) ( )
sin11
4
o
+==
rkj
erIdkjrH
rArH
rkjrr
rrrr
x
z
( ) ( )rdIzrJ rrr 3 =
Working in spherical coordinates
y
ECE 303 Fall 2005 Farhan Rana Cornell University
Radiation Emitted by a Hertzian Dipole - II
H-field was:
( ) ( ) sin11
4
+=
rkje
rIdkjrH rkj
rr
Finding the E-field:
Use Amperes Law: ( ) ( ) ( )rEjrJrH o rrrrrr += Away from the dipole the current density is zero, therefore:
( ) ( )rHj
rEo
rrrr = 1
( ) ( )
( )
++
+
+=
sin111
cos2114
2
2
rkjrkj
rkjrkjre
rIdkjrE rkjo
rr
x
z
( ) ( )rdIzrJ rrr 3 =
Working in spherical coordinates
y
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11
ECE 303 Fall 2005 Farhan Rana Cornell University
Near-Fields of a Hertzian Dipole - I
x
z
( ) ( )rdIzrJ rrr 3 =
y
( ) ( ) sin11
4
+=
rkje
rIdkjrH rkj
rr
( ) ( )
( )
++
+
+=
sin111
cos2114
2
2
rkjrkj
rkjrkjre
rIdkjrE rkjo
rr
Near-field is the field close to the dipole where: kr
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12
ECE 303 Fall 2005 Farhan Rana Cornell University
Far-Fields of a Hertzian Dipole
x
z
( ) ( )rdIzrJ rrr 3 =
y
( ) ( ) sin11
4
+=
rkje
rIdkjrH rkj
rr
( ) ( )
( )
++
+
+=
sin111
cos2114
2
2
rkjrkj
rkjrkjre
rIdkjrE rkjo
rr
Far-field is the field far away from the dipole where: kr >> 1(or more accurately where: d /AntiAliasGrayImages false /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /Unknown
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