Dipolo Dipolo Infinitesimal Infinitesimal

Sandra Cruz-Pol, Ph. Sandra Cruz-Pol, Ph. D.D.INEL 5305INEL 5305ECE UPRMECE UPRMMayagüez, PRMayagüez, PR

OutlineOutline

Maxwell’s equationsMaxwell’s equations Wave equations for A and for Wave equations for A and for

Power: Poynting VectorPower: Poynting Vector Dipole antennaDipole antenna

Maxwell EquationsMaxwell Equations

Jt

DH

0 B

t

BE

vD

Ampere:

Faraday:

Gauss:

Relaciones de Relaciones de ContinuidadContinuidad

EJt

J

HB

ED

Potencial Magnético Potencial Magnético Vectorial AVectorial A

Si lo sustituimos en la Ley de Si lo sustituimos en la Ley de Faraday:Faraday:

HAB

0

t

AE

0

: vectorial

Identidad

t

AE

At

At

E

is the Electric Scalar Potential

Usando Ley de Ampere:Usando Ley de Ampere:

Jt

EB

Jt

DH

t

A

tJAA

21

AAA 2

2

221

t

A

tJAA

Jt

AA

tA

2

22

Lorentz’ conditionLorentz’ condition

Si escogemosSi escogemos

Queda la Ecuacion de OndaQueda la Ecuacion de Onda

donde J es la densidad de una fuente donde J es la densidad de una fuente de corriente, si hay.de corriente, si hay.

t

A

Jt

AA

tA

2

22

Jt

AA

2

22

de la Ec. de Gauss de la Ec. de Gauss ElElééctricactrica

vD

vE

v

t

A

2

v

t

A

v

tt

2

v

t

2

22

Wave equationWave equation

For sinusoidal For sinusoidal fields (harmonics):fields (harmonics):

wherewhereJAkA

JAjA

Jt

AA

22

22

2

22

)(

22 k

OutlineOutline

Maxwell’s equationsMaxwell’s equations Wave equations for A and for Wave equations for A and for

Power: Poynting VectorPower: Poynting Vector Dipole antennaDipole antenna

Poynting VectorPoynting Vector *

2

1HES

*Re2

1HESave

yx EyExE ˆˆ )(

2

)(1

ztjy

ztjx

eEE

eEE

)(2

)(1

ztj

y

ztjx

eHH

eHHyx HyHxH ˆˆ

*ˆˆˆˆRe2

1yxyxave HyHxEyExS

Average SAverage S yxyxave HyHxEyExS ** ˆˆˆˆRe

2

1

zHEHES xyyxave ˆRe2

1 **

zeHEHES zave ˆcos

2

1 2*12

*21

zeZ

E

Z

ES z

ooave ˆcos

2

1 2

2

2

2

1

zEER

So

ave ˆ2

1 2

2

2

1

Para medios sin pérdidas:

OutlineOutline

Maxwell’s equationsMaxwell’s equations Wave equations for A and for Wave equations for A and for

Power: Poynting VectorPower: Poynting Vector Infinitesimal Dipole antennaInfinitesimal Dipole antenna

Find A from Dipole with Find A from Dipole with current Jcurrent J

Line charge Line charge w/uniform charge w/uniform charge density, density, LL

Jt

AA

2

22

x

z

r

r

0

Jz

)()( yxIoo

Az zyxJAkA zzz ,,22

Assume the simplest solution Az(r):

To find….To find….

a

sin

1aa

AA

r

AA r

F

r

F

rr

Fr

rF r

sin

1sin

sin

11 2

2

zz rAdr

d

rrA

2

22 1

)(

AA 2

Assume the simplest solution Az(r):

Fuera de la Fuente (J=0)Fuera de la Fuente (J=0)

01 2

2

2

zz AkrAdr

d

r

022

2

zz rAkrAdr

d

jkrjkrz ececrA 21

Which has general solution of:

Apply B.C.Apply B.C. If radiated wave travels outwards from the If radiated wave travels outwards from the

source:source:

To find C2, let’s examine what happens near the To find C2, let’s examine what happens near the source. (in that case k tends to 0)source. (in that case k tends to 0)

So the wave equation reduces to So the wave equation reduces to

r

ecrA

jkr

z

2)(jkrz ecrrA 2)(

r

crAz

2)(

)()(2 yxIAA oozz

Now we integrate the Now we integrate the volume around the dipole:volume around the dipole:

And using the And using the Divergence Divergence TheoremTheorem

zIr

Arddr

r

Aoo

zz

22 4sin

ddrdS sin2

dxdydzyxIdvA ooz )()(

zIdSA ooz

Comparing both, we get:Comparing both, we get:

42

zIc oo

24 r

zI

dr

dA ooz

22

r

c

dr

dAz

jkrooz e

r

zIrA

4)(

Now from A we can find Now from A we can find E & HE & H

Using the Victoria Using the Victoria IDENTITY:IDENTITY:

And And

Substitute:Substitute:

zrzzz aarAr

aAH ˆˆ)(1

ˆ1

zzaAAH ˆ11

zzzzzz aAaAaA ˆˆˆ

)( GfGffG

sinˆˆˆ aaa zr

ˆsinˆ1

Har

AH z

ˆsinˆ1

Har

AH z

sin

1

4 2jkro e

rr

jklIH

The magnetic filed intensity from the dipole is:

jkrooz e

r

zIrA

4)(

Now the E field:Now the E field:

t

EH

Jt

DH

rr ArA

rrrA

r

A

r

AA

rrA

1ˆsin

1ˆsin

sin

1ˆ

HEj

ˆˆ

ˆsin

sin

1ˆ

ErEj

rHrr

HrrH

r

The electric field The electric field from infinitesimal dipole:from infinitesimal dipole:

cossin21

4sin

1

sinsin

1

2jkro

r

err

jklI

r

Hr

Ej

cos1/

2 32jkro

r erjr

lIE

sin1

4

1

22 jkro e

rr

jkk

r

ljI

rHrr

Ej

sin

1/

4 32jkro e

rjrr

jlIE

GeneralGeneral @Far field @Far field rr> > 2D2D22//

cos1/

2 32jkro

r erjr

lIE

sin

1/

4 32jkro e

rjrr

jlIE

sin

1

4 2jkro e

rr

jklIH

0rE

sin

4jkro e

r

jlIE

sin

4jkro e

r

jklIH

Note that the ratio of E/H is the intrinsic impedance of the medium.

Power :Hertzian DipolePower :Hertzian Dipole

ˆˆ

2

12

1

**

*

HErHE

HES

r

ddrHErdASAdSP sin2

1ˆ 2*

2

0 0

2322

2

211/

sin32 rr

jk

rjrr

jlIS or

sin1

4sin

1/

42

12

*

32jkrojkro

r err

jklIe

rjrr

jlIS

3

222

21

1sin8 krr

lIS or

dr

krr

lIdP o sin

11sin

82

32

22

2

3

2

13 kr

jlIP o

Resistencia de RadiacionResistencia de Radiacion

La potencia tiene parte real y parte reactivaLa potencia tiene parte real y parte reactiva

La potencia irradiada es:La potencia irradiada es:

Comparando con la P total, se halla la Comparando con la P total, se halla la ImpedanciaImpedancia

COmparando cno la COmparando cno la PPradrad, halla la , halla la RRradrad..

reactivarado PP

kr

jlIP

3

2

13

radoo

rad RIlI

P2

2

2

1

3

3

2

13

2

kr

jlZ

2

280

lRrad