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Analysis of Hertzian Dipole
Antenna Antenna EngineeringEngineering
CHO, Yong HeuiCHO, Yong Heui
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EM Wave LabEM Wave Lab2
E and H fields
1. Field analysis
Vector potential
Vector potential approach
AB
V
jkR
VdJR
eA
4
222 )()()( zzyyxxrrR
AH 1
Hj
E 1
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Hertzian dipole
1. Field analysis
Current density approximation
2/2/for
ˆ)()(
zzz
zyxIJ
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Vector potential
1. Field analysis
Current vs. vector potential
zr
zeI
zIdR
ezA
jkr
z
z
z
jkR
ˆ4
4ˆ
0
2/
2/
sin
cos
z
zr
AA
AA
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H field
1. Field analysis
ˆsin1
4
1ˆ11
rjk
r
zeI
ArArr
AH
jkr
r
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E field
1. Field analysis
ˆsin1
4
ˆcos12
1ˆsinsin
1ˆ
1
1
2
2
kr
j
rjk
r
zeI
rkr
j
r
zeI
rHrr
Hrr
j
Hj
E
jkr
jkr
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Electric dipole moment
1. Field analysis
Current vs. electric dipole moment
dt
dp
dt
zdQz
dt
dQzI
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Far-field
1. Field analysis
Dr
ˆsin4
jkr
zeIH
jkr
ˆsin4
jkr
zeIE
jkr
*Re2
1HES
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Radiated power
1. Field analysis
2
22
0 0 2
22
22
*
3
sinsin
42
sin42
1
2
1
Re2
1
zI
ddrr
kzI
dSjkr
zeIdSH
SdHEP
S
jkr
S
Sr
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Radiation resistance
1. Field analysis
rr RIzI
P2
2
2
1
3
2
3
2
zRr
Open transmission line
)2/cot(0in zjZZ
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Simple calculation
1. Field analysis
Far-field calculation
r
zeIA
jkr
z
4
zz AjE sinzEE
ErH ˆ1
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Radiation pattern
1. Field analysis
Power and field pattern dB scale: Half-power beamwidth: Main beam:
sin),( E
sin),( H
2sin),( rS
22Dr
Far-field condition: phase condition
9090(BW)3dB
),(log10 10 rS
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Antenna gain
1. Field analysis
Directivity: 3/2 for Hertzian dipole Gain and efficiency Isotropic radiation dBi
DG
lossloss RR
R
PP
P
r
r
r
r
210
44),(
rP
U
U
UD SrU 2
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(Lorentz) Reciprocity theorem
1. Field analysis
The same propagation characteristics: Tx and Rx
Antenna measurement
1I 2V
1V 2I
21Z
12Z
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
DG
1. Field analysis
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Friis power transmission formula
Microwave radio link
R
tP rP
2. Microwave link
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Transmitted power
Power density:
EIRP (Effective Isotropic Radiated Power)
Radiation pattern [dBi]: dB isotropic
22
/4
mWGR
PS t
t
2. Microwave link
tP
ttGPEIRP
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Received power
Received power:
Effective area
-
er SAP
4
2r
e
GA
WR
GGP
Prt
t
r
2
4
Friis transmission formula
2. Microwave link
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Effective area
Reciprocity theorem:
Effective area:
1221 AGAG
2. Microwave link
er SAP
22
rms1 8
3
4
SR
V
S
PA
r
r
r
rr R
VRIP
44
2rms
2rms zEV rmsrms
2rmsE
S
2
31 G
4
22
2
GA
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Free space loss
Free space loss:
Received power
2
FS
4
R
L
]dB[EIRP FS rr GLP
WL
GG
P
P rt
t
r
FS
2. Microwave link
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0.6
dishdiameter 4
]GHz[14
]W[100
m
f
P
u
t
0.55
2
]GHz[12
]W[10
3dB
θ
f
P
d
t
Uplink and downlink
2. Microwave link
][000,40 kmR
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tt
et
tt
DG
AD
GP
2
4
EIRP
Uplink calculation
2. Microwave link
rr
r
DG
D
21
4
2
FS
4
R
L
]dB[EIRP FS rr GLP
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rr
r
DG
AD
max2
4
Downlink calculation
2. Microwave link
tt
t
tt
DG
D
GP
21
4
EIRP
2
FS
4
R
L
]dB[EIRP FS rr GLP
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Noise power
Thermal noise: white noise, Nyquist formula
Thermal noise
BNN
kTBPn
0
Noise temperature
kB
NT
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 rGLC
G/T: sensitivity of receiver
2. Microwave link