TE4109 Lecture06-07-Radiation From Infinitesimal (Elementary) Source

8/10/2019 TE4109 Lecture06-07-Radiation From Infinitesimal (Elementary) Source

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TE4109 Antennas 1

Radiation from Infinitesimal

(Elementary) Source

Radiation from Infinitesimal (Electric) Dipole

Duality in Maxwells Equations

Radiation from Infinitesimal Magnetic Dipole(Electric Current Loop)

Radiation Zones

TE4109 Antennas 2

Radiation from Infinitesimal Dipole (1)

Infinitesimal dipole very short current element

A very short piece of infinitesimally thin wire with constant currentdistribution

, /50l l


2/19

2

TE4109 Antennas 3


Magnetic vector potential (VP) of a current element

( ) ( )4

PQj R

Q

PQL

eA P I Q dl

R

=

Recall that P is the observation point and Q is the source pointwhere integration takes place

( ) ( ) ( ) ( )4 4

PQ PQ

Q

j R j R

Q z

PQ PQL l

e eA P I Q dl A P Ia dl

R R

= =

The dipole is infinitesimally small, so the following approximation ishold

PQR r

TE4109 Antennas 4


Finally, Magnetic VP is given by

/ 2

/ 2( )

4

j rl

zl

I eA P a dl

r

=

4

j r

z eA a I lr

=

The above result is used in applications of the superpositionprinciple

4

j re

dA I dlr

=

The field radiated by any complex antenna in a linear medium canbe represented as a superposition of the fields due to currentelements.


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3

TE4109 Antennas 5


In antenna theory, the preferred coordinate system is the sphericalsystem.

cos( ) ( ) cos( )4

sin( ) ( ) sin( )4

0

j r

r z

j r

z

eA A I l

r

eA A I l

r

A

= =

= =

=

( , , ) ( , , ) ( , , ) ( , , )r rA r a A r a A r a A r = + +

Angular dependence is separable fromradial dependence.

http://www.ece.mcmaster.ca/faculty/nikolova/antenna_dload/current_lectures/L03_RadIS.pdf

TE4109 Antennas 6


Field vectors of a current element (see note for derivation)

They are vectors at a point in free space, specified by

1H A

=

( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , )r r

r r

H r a H r a H r a H rE r a E r a E r a E r

= + += + +

1( )sin( ) 1

4

0

j r

r

eH j I l

j r r

H H

= +

= =

( , , )r

The magnetic field


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4

TE4109 Antennas 7


The electric field

1( )

J jE A j A A

j j

= =

2

2

1 12 ( )cos( )

4

1 1( )sin( ) 1

( ) 4

0

j r

r

j r

eE I l

r j r r

eE j I l

j r r r

E

= +

= +

=

TE4109 Antennas 8


EM fields generated by the current element is rather complicatedunlike the VP.

The use of VP instead of the field vectors is usually advantageous inantenna studies

Features of the field vectors of a current element

Longitudinal components decrease with distance as 1/r2 or faster

Transverse component have a 1/r term dominant at large distances

Transverse electric and magnetic field components are orthogonal toeach other

In the far zone,

andE H

| | | | | | | |E H E H = =


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TE4109 Antennas 9

Radiated Power from Infinitesimal Dipole (1)

Power density (Poynting vector) of a current element

( ) ( )

( )

* *

* *

1 1 ( )

2 2

1

2

r r

r r r r

W E H a E a E a H

a E H a E H a W a W

= = +

= = +

2 2

2 2 3

2

2 3 2

( ) sin ( )1

8 ( )

( ) sin( )cos( ) 1116 ( )

0

r

I l jW

r r

I lW jr r

W

=

= +

=

TE4109 Antennas 10


Total power of a current element is calculated over surface of asphere

S

P W ds=

Surface vector is described by its normal vector and its area

2

sin( )s

r

ds a ds

a r d d

=

=

2 ( ) ( sin( ) )r r rS

P a W a W a r d d = +

Thus,



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TE4109 Antennas 11


Only the radial component of the Poynting vector contributes to theoverall power.

2

31 , W

3 ( )

I l jP

r

=

Radiated power is equal to the real part of the complex power.

2

, W3

rad

I lP

=

Radiation resistance of a current element (ideal dipole)

221 22

r r PP R I RI

= =

22

,3

id

r

lR

=

120 =

2

80 ,idr

lR

=

TE4109 Antennas 12

Duality in Maxwells Equations (1)Substituting the quantities from one set of EM equations with therespective quantities from the dual set produces a valid equation(the dual of the given one).

2 2

0

4

Electric Source ( 0, 0)

1

( )

j r

V

E j H

H j E J

DB

J j

A A J

eA J dv

r

H A

jE j A

M

A

J

=

= +

= =

=

+ =

=

=

=

=

2 2

0

4

Magnetic Source ( 0,

(

1

0)

)

m

m

j r

V

H j E

E j H M

BD

M j

F F M

eF M dv

r

E

J

j

M

F

jH F F

=

=

= =

=

+ =

=

=

=

=


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TE4109 Antennas 13

Duality in Maxwells Equations (2)

Dual Quantities

Given 1/

Dual 1/

E H J M A F

H E M J F A

TE4109 Antennas 14

Radiation from Infinitesimal Magnetic Dipole (1)

Infinitesimal magnetic dipole very short magnetic currentelement (exists in the text book only)

Magnetic current is assumed to be constant along the element

It is the duality of the infinitesimal electric dipole

Magnetic current is in the unit of V (volts)

Magnetic current is related to the magnetic current density by

mS

I M ds=

Electric vector potential

( ) ( )4

PQ

Q

j R

Q

PQv

eF P M Q dv

R

=

( ) ( ) , C/m4

PQ

Q

j R

m QL

PQ

eF P I Q dl

R

=


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TE4109 Antennas 15


In antenna theory, the preferred coordinate system is the sphericalsystem.

cos( ) ( ) cos( )4

sin( ) ( ) sin( )4

0

j r

r z m

j r

z m

eF F I l

r

eF F I l

r

F

= =

= =

=

( , , ) ( , , ) ( , , ) ( , , )r rF r a F r a F r a F r = + +

Angular dependence is separable fromradial dependence.

TE4109 Antennas 16


Field vectors of a magnetic current element

They are vectors at a point in free space, specified by

1E F

=

( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , )

r r

r r

H r a H r a H r a H rE r a E r a E r a E r

= + += + +

1( )sin( ) 1

4

0

j r

m

r

eE j I l

j r r

E E

= +

= =

( , , )r

The electric field


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9

TE4109 Antennas 17


The magnetic field

1( )

M jH F j F F

j j

= =

2

2

2( )cos( ) 1 1

4

( )sin( ) 1 11

( ) 4

0

j r

mr

j r

m

I l eH

r j r r

j I l eH

j r r r

H

= +

= +

=

TE4109 Antennas 18

Magnetic Dipole and Electric-Current Loop (1)

Equivalence of the field excited by a magnetic dipole and anelectric-current loop

Since the magnetic current Im does not exist, we need to find its

physically-realizable equivalence in terms of electric current I

Considering two sets of Maxwells equations

1 1

1 1

21 1

Electric Current

E j H

H j E J

E E j J

=

= +

=

2 2

2 2

22 2

Magnetic Current

H j E

E j H M

E E M

=

=

=

1 2 1 2If , then andj J M E E H H = = =


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10

TE4109 Antennas 19


This result relates the magnetic dipole to the electric-current loop

j J M =

Considering the surface S, enclosed by the closed path C

[ ] [ ]

( )C CS S

j J ds M ds =

Applying Stokes theorem

Cj I M dc =


TE4109 Antennas 20


Magnetic current density is assumed nonzero and constant only at

the section l.

Cj I M dc =

j I M l =

[ ] m LI MA= [ ]L mj IA I l =

Small loop of electric current I and of area A[L] creates EM fieldequivalent to that of a small magnetic dipole (magnetic current

element) (Il)



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11

TE4109 Antennas 21

Fields of Infinitesimal Current Loop (1)

Fields of a small electric-current loop

[ ]L mj IA I l =

2

2

2

2

1( ) sin( ) 1 ,

4

1 12 ( )cos( ) ,

4

1 1( )sin( ) 1 ,

( ) 4

0.

j r

j r

r

j r

r

eE IA

j r r

eH j IA

r j r r

eH IA

j r r r

E E H

= +

= +

= +

= = =

In the far-field region the radial components have no far-field terms

E H

| | | | | | | |E H E H = =

TE4109 Antennas 22

Radiated Power from Infinitesimal Current Loop (1)

Power density (Poynting vector) of a small electric-current loop

( ) ( )

( )

* * *

* *

1 1 ( )

2 2

1

2

r r

r r r r

W E H a E a H a H

a E H a E H a W a W

= = +

= + = +

24 2

2 3

sin ( ) 1( ) 1

2 (4 ) ( )rW IA

r j r

=

Total power of a current loop is calculated over surface of a sphere

Only radial component of the Poynting vector contributes to theradiated power


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TE4109 Antennas 23

Radiated Power from Infinitesimal Current Loop (2)

Total power of a current element is calculated over surface of asphere

2 ( ) ( sin( ) )r r rS

P a W a W a r d d = +

4 2

3

1( ) 1 , W

12 ( )P IA

j r

=

Radiated power is equal to the real part of the complex power.

4 2( ) , W

12radP IA

=

Radiation resistance of a small electric-current loop

2

2

1 2

2 r r

PP R I R

I= = 4 2 ,

6md

radR A

=

TE4109 Antennas 24

Radiation Zones (1)

Space surrounding an antenna is divided into three regions

Reactive near-field region

Region immediately surrounding the antenna

Reactive field predominates

Radiating near-field (Fresnel) region

Radiation field is more significant but the angular field distribution isstill dependent on the distance from the antenna

Far-field (Fraunhofer) region

Angular field distribution does not depend on the distance from theantenna Spherical wave front

The field is a transverse EM wave

3

0.62 D

r

Largest dimension

of the antenna

D=

3 220.62

D Dr


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TE4109 Antennas 25

Radiation Zones (2)Amplitude pattern changes in

shape as the distance is variedfrom the reactive near field to the

far field due to variations of fieldmagnitude and phase

In the reactive near-field region,pattern is nearly uniform

In the far-field region, pattern iswell formed

Constantine A. Balanis, Antenna Theory, Analysis and Design, 3rd Ed., 2005

Constantine A. Balanis, Antenna Theory, Analysis and Design, 3rd Ed., 2005

TE4109 Antennas 26

Radiation Zones (3)

Study the general field behavior of the infinitesimal electric dipole.

2

2

1 12 ( )cos( )

4

1 1( )sin( ) 1

( ) 40

r r

j r

r

j r

E a E a E a E

eE I l

r j r r

eE j I l

j r r r

E

= + +

= +

= +

=

1( )sin( ) 1

4

0

r r

j r

r

H a H a H a H

eH j I l

j r r

H H

= + +

= +

= =

2 2

2 2 3

2

2 3 2

Complex Poynting Vector

( ) sin ( )1

8 ( )

( ) sin( )cos( ) 11

16 ( )

0

r r

r

W a W a W a W

I l jW

r r

I lW j

r r

W

= + +

=

= +

=

{ } { }2

3

Overall Power

Re Im

1 , W3 ( )

P P j P

I l jP

r

= +

=


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TE4109 Antennas 27

Reactive Near-Field Region

It is the region within the sphere having the radius of3

0.62 1D

r r

2

2

1( )sin( ) 1

4

1 12 ( )cos( )

4

1 1( )sin( ) 1

( ) 4

0

j r

j r

r

j r

r

eH j I l

j r r

eE I l

r j r r

eE j I l

j r r r

H H E

= +

= +

= +

= = =

2

3

3

( )sin( )

4

( )cos( )

4

( )sin( )

4

0

j r

j r

r

j r

r

I l eH

r

I l eE j

r

I l eE j

r

H H E

= = =

Electric field and magnetic field are in phase quadrature Field isreactive (ejr can be neglected)

{ } { }2 2

3

1Im Re

3 ( ) 3 rad

I l I lP P P

r

= = =

TE4109 Antennas 28

Radiating Near-Field (Fresnel) Region

Intermediate region between the reactive near-field region and thefar-field region

Radiation field is more significant but the angular field distributionstill depends is the distance from the antenna

3 220.62 1

D Dr r

Fields can be founded by ignoring the higher-order (1/r)n

-terms

2

( )sin( )

4

( )cos( )

2

( )sin( )

4

0

j r

j r

r

j r

r

I l eH j

r

I l eE

r

I l eE j

r

H H E

= = =

2

2

1( )sin( ) 1

4

1 12 ( )cos( )

4

1 1( )sin( ) 1

( ) 4

0

j r

j r

r

j r

r

eH j I l

j r r

eE I l

r j r r

eE j I l

j r r r

H H E

= +

= +

= +

= = =

Er is still not negligible, but the transverse components E and Hare dominant


15/19

15

TE4109 Antennas 29

Far-Field (Fraunhofer) Region (1)

Only terms 1/r are considered22

1D

r r

The field is transverse EM wave

( )sin( )

4

0( )

sin( )4

0

j r

r

j r

r

I l eH j

r

E

I l eE j

r

H H E

= = =

2

2

1( )sin( ) 1

4

1 12 ( )cos( )

4

1 1( )sin( ) 1

( ) 4

0

j r

j r

r

j r

r

eH j I l

j r r

eE I l

r j r r

eE j I l

j r r r

H H E

= +

= +

= +

= = =

TE4109 Antennas 30

Far-Field (Fraunhofer) Region (2)

Complex Poynting vector is in the radial direction

2 2

2 2 3

2

2 3 2


( ) sin ( )

18 ( )

( ) sin( )cos( ) 11

16 ( )

0

r r

r

W a W a W a W

I l jW

r r

I lW j

r r

W

= + +

=

= +

=

2 2

2 2


( ) sin ( )8

0

0

r r

r

W a W a W a W

I lW

r

W

W

= + +

=

=

0

EZ

H

= =2

21 1

2 2r r

EW a a H

= =


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16

TE4109 Antennas 31

Radiation Separation (1)

Consider the VP integral for a linear current source

( )4

j R

L

eA I l dl

R

=

2 2 2( ) ( ) ( )R x x y y z z = + +

Size of an infinitesimalantenna is very small.

Distance between theintegration point and theobservation point isconsidered constant.

2 2 2R r x y z = + +http://www.ece.mcmaster.ca/faculty/nikolova/antenna_dload/current_lectures/L03_RadIS.pdf

TE4109 Antennas 32


As the maximum dimension of the antenna becomes comparable to

the wavelength, the error, especially in the phase term R,

increases.

1 1Amplitude Factor:

Phase Factor: Error in / 8 22.5

j R R r

e

R

r

< =

For an infinitesimally thin wire

2 2 2

2 2 2 2 2 2

0 ( )

( 2 ) 2 cos( )

x y R x y z z

R x y z z zz r z rz

= = = + +

= + + + = +


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17

TE4109 Antennas 33


Using the binomial expansion and ignoring high-order terms

1 2 2 3 3( 1) ( 1)( 2)( ) ...2! 3!

n n n n nn n n n na b a na b a b a b

+ = + + + +

2 2 3 2

2

1 1cos( ) sin ( ) cos( )sin ( )

2 2R r z z z

r r + +

Only the first twoterms are taken into

account


TE4109 Antennas 34

Far-Field Approximation (1)

At the distance very far from the source

Only the first twoterms are taken intoaccount

z r

2 2 3 2

2

1 1cos( ) sin ( ) cos( )sin ( )

2 2

cos( )

R r z z zr r

R r z

+ +



18/19

18

TE4109 Antennas 35

Far-Field Approximation (2)

The minimum distance, at which the phase error is acceptable

32 2 2

2

1cos( ) cos( )sin ( )

2

The mo

1sin ( )

st significant error ter

2

m

zr

R r z zr

+ +

22 22 max

max

( )1 ( ) ( / 2)( ) sin ( ) ( )

2 2 2

zz De r e r

r r r

= = =

2( / 2) 2 8

D

r

2

2Dr

In addition, andr D r

TE4109 Antennas 36

Radiating Near-Field Approximation (1)

This region is adjacent to the Fraunhofer region, so its upperboundary is given by

22Dr

2 2 3 2

2

2 2

1 1cos( ) sin ( ) cos( )sin ( )

2 2

1cos( ) sin ( )

2

R r z z zr r

R r z zr

+ +

+

When the observation point is in this region, the approximation of Ris given by


19/19

TE4109 Antennas 37

Radiating Near-Field Approximation (2)

The minimum distance, at which the phase error is acceptable

2

2 3 221cos( ) sin ( )2

T

1cos( )sin ( )

he most significant error term

2R r zz z

r r + +

32

2

1 ( )( ) cos( )sin ( ) Maximum at arctan( 2) 54.7

2

ze r

r

= =

3

2

( / 2) 1 2

2 3 83

D

r

3 320.62

3 3

D Dr

=

2

1

3

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TE4109 Lecture06-07-Radiation From Infinitesimal (Elementary) Source

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