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Dipolo Infinitesimal. Sandra Cruz-Pol, Ph. D. INEL 5305 ECE UPRM Mayagüez, PR. Outline. Maxwell’s equations Wave equations for A and for F Power: Poynting Vector Dipole antenna. Maxwell Equations. Ampere:. Faraday:. Gauss:. Relaciones de Continuidad. Potencial Magnético Vectorial A. - PowerPoint PPT Presentation
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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
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