lecture 28

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


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


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


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

=

=+


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


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


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


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:


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:


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


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


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 =


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:


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


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)


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


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

10


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


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


Working in spherical coordinates

y

11


Near-Fields of a Hertzian Dipole - I

x

z


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

12


Far-Fields of a Hertzian Dipole

x

z


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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lecture 28

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