Analysis of Hertzian Dipole

A Hertzian dipole is a starting point of antenna theory. Since most of antennas can be understood with a Hertzian dipole, we need to thoroughly study this kind of an infinitesimal antenna that is not real in practical applications.

1. Analysis of Hertzian
Dipole
Antenna EngineeringAntenna Engineering
CHO, Yong HeuiCHO, Yong Heui

2. Antenna EngineeringAntenna Engineering
EM Wave LabEM Wave Lab2
E and H fields
1. Field analysis
Vector potential
Vector potential approach 
AB ´Ñ=
ò
¢=
-
V
jkR
VdJ
R
e
A
p
m
4
222
)()()( zzyyxxrrR ¢-+¢-+¢-=¢-=
AH ´Ñ=
m
1
H
j
E ´Ñ=
we
1

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EM Wave LabEM Wave Lab3
Hertzian dipole
1. Field analysis
Current density approximation
2/2/for
ˆ)()(
zzz
zyxIJ
D<¢<D-
¢¢= dd

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EM Wave LabEM Wave Lab4
Vector potential
1. Field analysis
Current vs. vector potential
z
r
zeI
zId
R
e
zA
jkr
z
z
z
jkR
ˆ
4
4
ˆ
0
2/
2/
p
m
p
m
-
®D
D
D-
-
D
=
¢=ò
q
q
q
sin
cos
z
zr
AA
AA
-=
=

5. Antenna EngineeringAntenna Engineering
EM Wave LabEM Wave Lab5
H field
1. Field analysis
 ( )
fq
p
q
f
mm
q
ˆ
sin
1
4
1
ˆ
11
ú
û
ù
ê
ë
é
+
D
=
ú
û
ù
ê
ë
é
¶
¶
-
¶
¶
=´Ñ=
-
r
jk
r
zeI
ArA
rr
AH
jkr
r

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E field
1. Field analysis

( ) ( )
qq
p
h
q
p
h
qq
qqwe
we
ff
ˆ
sin
1
4
ˆcos1
2
1
ˆ
sin
sin
1
ˆ
1
1
2
2
ú
û
ù
ê
ë
é
-+
D
+
ú
û
ù
ê
ë
é
-
D
=
ú
û
ù
ê
ë
é
¶
¶
-
¶
¶
=
´Ñ=
-
-
kr
j
r
jk
r
zeI
r
kr
j
r
zeI
rH
rr
H
r
r
j
H
j
E
jkr
jkr

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EM Wave LabEM Wave Lab7
Electric dipole moment
1. Field analysis

Current vs. electric dipole moment
dt
dp
dt
zdQ
z
dt
dQ
zI =
D
=D=D

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Far-field
1. Field analysis

 Dr>>
fq
p
ˆ
sin
4
jk
r
zeI
H
jkr-
D
=
qq
p
h
ˆ
sin
4
jk
r
zeI
E
jkr-
D
=
[ ]
*
Re
2
1
HES ´=

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EM Wave LabEM Wave Lab9
Radiated power
1. Field analysis
 [ ]
2
2
2
00
2
2
2
2
2
*
3
sin
sin
42
sin
42
1
2
1
Re
2
1
÷
ø
ö
ç
è
æD
=
D
=
D
==
·´=
òò
òò
ò
-
l
ph
fqq
q
p
h
q
p
hh
pp
zI
ddr
r
k
zI
dSjk
r
zeI
dSH
SdHEP
S
jkr
S
S
r

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Radiation resistance
1. Field analysis

rr
RI
zI
P
2
2
2
1
3
=
D
=
l
ph
2
3
2
÷
ø
ö
ç
è
æD
=
l
phz
R
r

Open transmission line
)2/cot(
0in
zjZZ D-= b

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Simple calculation
1. Field analysis

Far-field calculation
r
zeI
A
jkr
z
p
m
4
-
D
=
zz
AjE w-= q
q
sin
z
EE-=
ErH ´=ˆ
1
h

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Radiation pattern
1. Field analysis

Power and field pattern
dB scale:
Half-power beamwidth:
Main beam:
qfq
q
sin),(µE
qfq
f
sin),(µH
qfq
2
sin),(µ
r
S
l
2
2D
r>
Far-field condition
: phase condition

90=q

90(BW)
3dB
=q
),(log10
10
fq
r
S

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EM Wave LabEM Wave Lab13
Antenna gain
1. Field analysis

Directivity: 3/2 for Hertzian dipole
Gain and efficiency
Isotropic radiation
dBi
DGh=
lossloss RR
R
PP
P
r
r
r
r
+
=
+
=h
210
44
),(
qq
pp
fq »==
r
P
U
U
U
D
SrU
2
=

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(Lorentz) Reciprocity theorem
1. Field analysis

The same propagation characteristics: Tx
and Rx
Antenna measurement
1
I
2
V
1
V
2
I
21
Z
12
Z
1221
ZZ=

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Antenna characteristics
Antenna gain: anisotropic radiation (G > 1)
isotropic radiation (G = 1)
Directivity and efficiency:
Angular beamwidth: 3dB
Radiation pattern [dBi]: dB isotropic
DGh=
1. Field analysis

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Friis power transmission formula
Microwave radio link
R
t
P r
P
2. Microwave link

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Transmitted power
Power density:
EIRP (Effective Isotropic Radiated Power)
Radiation pattern [dBi]: dB isotropic
[ ]
2
2
/
4
mWG
R
P
S
t
t
p
=
2. Microwave link
t
P
tt
GP=EIRP

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EM Wave LabEM Wave Lab18
Received power
Received power:
Effective area
-
er
SAP=
p
l
4
2
r
e
G
A=
[]W
R
GG
P
P
rt
t
r
2
4
÷
ø
ö
ç
è
æ
=
p
l
Friis transmission formula
2. Microwave link

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EM Wave LabEM Wave Lab19
Effective area

Reciprocity theorem:
Effective area:
1221
AGAG=
2. Microwave link
er
SAP=
2
2
rms
1
8
3
4
l
p
===
SR
V
S
P
A
r
r
r
r
r
R
VRI
P
44
2
rms
2
rms
== zEV D=
rmsrms
h
2
rms
E
S=
2
3
1
=G

p
l
4
2
2
2
G
A=

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EM Wave LabEM Wave Lab20
Free space loss
Free space loss:
Received power
2
FS
4
÷
ø
ö
ç
è
æ
=
l
pR
L
]dB[EIRP
FS rr
GLP +-=
[]W
L
GG
P
P
rt
t
r
FS
=
2. Microwave link

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0.6
dishdiameter 4
]GHz[14
]W[100
=
=
=
h
m
f
P
u
t
0.55
2
]GHz[12
]W[10
3dB
=
=
=
=
h

θ
f
P
d
t


Uplink and downlink
2. Microwave link
][000,40 kmR=

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EM Wave LabEM Wave Lab22
tt
et
tt
DG
AD
GP
h
l
p
=
=
=
2
4
EIRP


Uplink calculation
2. Microwave link
rr
r
DG
D
h
qq
p
=
=
21
4
2
FS
4
÷
ø
ö
ç
è
æ
=
l
pR
L
]dB[EIRP
FS rr
GLP +-=

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EM Wave LabEM Wave Lab23
rr
r
DG
AD
h
l
p
=
=
max2
4

Downlink calculation
2. Microwave link
tt
t
tt
DG
D
GP
h
qq
p
=
=
=
21
4
EIRP
2
FS
4
÷
ø
ö
ç
è
æ
=
l
pR
L
]dB[EIRP
FS rr
GLP +-=

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EM Wave LabEM Wave Lab24
Noise power

Thermal noise: white noise, Nyquist formula

Thermal noise
BNN
kTBP
n
0
=
=
Noise temperature

kB
N
T=
2. Microwave link

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Carrier to noise ratio
TkB
C
BN
C
N
C
==
0
C/No: related to carrier to noise ratio
G/T: Figure of merit

Carrier to noise ratio

]dB[EIRP
FS r
GLC +-=
G/T: sensitivity of receiver

2. Microwave link