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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.
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Analysis of Hertzian Dipole
Antenna Antenna EngineeringEngineering
CHO, Yong HeuiCHO, Yong Heui
Antenna Antenna EngineeringEngineering
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab3
Hertzian dipole
1. Field analysis
Current density approximation
2/2/for
ˆ)()(
zzz
zyxIJ
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab4
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab5
H field
1. Field analysis
ˆsin1
4
1ˆ11
rjk
r
zeI
ArArr
AH
jkr
r
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab6
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab7
Electric dipole moment
1. Field analysis
Current vs. electric dipole moment
dt
dp
dt
zdQz
dt
dQzI
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab8
Far-field
1. Field analysis
Dr
ˆsin4
jkr
zeIH
jkr
ˆsin4
jkr
zeIE
jkr
*Re2
1HES
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab9
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab10
Radiation resistance
1. Field analysis
rr RIzI
P2
2
2
1
3
2
3
2
zRr
Open transmission line
)2/cot(0in zjZZ
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab11
Simple calculation
1. Field analysis
Far-field calculation
r
zeIA
jkr
z
4
zz AjE sinzEE
ErH ˆ1
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab12
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab13
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab14
(Lorentz) Reciprocity theorem
1. Field analysis
The same propagation characteristics: Tx and Rx
Antenna measurement
1I 2V
1V 2I
21Z
12Z
1221 ZZ
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab15
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab16
Friis power transmission formula
Microwave radio link
R
tP rP
2. Microwave link
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab17
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab18
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab19
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab20
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab21
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab22
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab23
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
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab24
Noise power
Thermal noise: white noise, Nyquist formula
Thermal noise
BNN
kTBPn
0
Noise temperature
kB
NT
2. Microwave link
Antenna Antenna EngineeringEngineering
EM Wave LabEM Wave Lab25
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
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