Hertzian Dipole - Auburn Universitymikeb/Antennas/Antennas-p-2.pdf · 1 Antennas • Hertzian...
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1 Antennas • Hertzian Dipole – Current Density – Vector Magnetic Potential – Electric and Magnetic Fields – Antenna Characteristics Hertzian Dipole Let us consider a short line of current placed along the z-axis. do ds j R s I S e β − = z J a ( ) i(t) cos o I t ω α = + j o s I Ie α = The stored charge at the ends resembles an electric dipole, and the short line of oscillating current is then referred to as a Hertzian Dipole. Where the phasor The current density at the origin seen by the observation point is A differential volume of this current element is d dv Sdz = do ds d j R s dv I dz e β − = z J a Step 1: Current Density
Transcript of Hertzian Dipole - Auburn Universitymikeb/Antennas/Antennas-p-2.pdf · 1 Antennas • Hertzian...
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Antennas
• Hertzian Dipole– Current Density– Vector Magnetic Potential– Electric and Magnetic Fields– Antenna Characteristics
Hertzian Dipole
Let us consider a short line of current placed along the z-axis.
dods
j RsIS
e β−= zJ a
( )i(t) cosoI tω α= +j
o sI I e α=
The stored charge at the ends resembles an electric dipole, and the short line of oscillating current is then referred to as a Hertzian Dipole.
Where the phasor
The current density at the origin seen by the observation point is
A differential volume of this current element is
ddv Sdz=
dods d
j Rsdv I dze β−= zJ a
Step 1: Current Density
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Hertzian DipoleStep 2: Vector Magnetic Potential
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doj Ro
osdo
sI dz eR
βµπ
−
−
= ∫ zaA
l
l
The vector magnetic potential equation is
A key assumption for the Hertzian dipole is that it is very short so
doR r≅
4
j ro
ossI e
r
βµπ
−
= zA al rcos sin θθ θ= −za a a
The unit vector az can be converted to its equivalent direction in spherical coordinates using
the transformation equations in Appendix B.
( )rcos sin4
j ro
ossI e
r
β
θ
µθ θ
π
−
= −A a al
This is the retarded vector magnetic potential at the observation point resulting from the Hertzian dipole element oriented in the +az direction at the origin.
Hertzian DipoleStep 3: Electric and Magnetic Fields
The magnetic field is given by
o os s= ∇×B A
1sin
4
j r
ossI e
jr r
β
φβ θπ
−
= +⎛ ⎞⎜ ⎟⎝ ⎠
H al
( )1oo o
ss s
o oµ µ= ∇×
BH A=
( )
2
2 sin 4
1j r
ossI e j
r r
β
φ
βθ
π β β
−
=⎡ ⎤
+⎢ ⎥⎢ ⎥⎣ ⎦
H al
It is useful to group β and r together
2r
λπ
sin 4
j r
ossI e
jr
β
φ
βθ
π
−
=H al
( )2
1 1r rβ β
Far-field condition:
In the far-field, we can neglect the second term.
.os o osη= − ×rE a H
The electric field is given by
sin .4
j r
os osI e
jr
β
θ
βη θ
π
−
=E al
3
( ) *1, , Re
2 o os sr θ φ = ×⎡ ⎤⎣ ⎦P E H
Hertzian DipoleStep 4: Antenna Parameters
( )2 2 2
2r2 2
, sin32
oo Ir
rη β
θ θπ
=⎛ ⎞⎜ ⎟⎝ ⎠
P al
2 2 2
max 2 232oo I
Pr
η βπ
=l
2 2sin sin sin d dp dθ θ θ θ φΩ = Ω =∫ ∫ ∫ ∫
Antenna Pattern Solid Angle:
83p
πΩ =
max4
1.5p
Dπ
= =Ω
Power Density:
Directivity:
Maximum Power Density:
22 2 22 2 2
2 240
32o o
rad P o
IP r I
rη β
ππ λ
= Ω =⎛ ⎞ ⎛ ⎞
⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
l l
2280radR π
λ= ⎛ ⎞
⎜ ⎟⎝ ⎠l
Total Radiated Power and Radiation Resistance :
Hertzian DipoleStep 4: Antenna Parameters
The total power radiated by a Hertzian dipole can be calculated by
2maxrad pP r P= Ω
2rad o radP I R=
The power radiated by the antenna is
Circuit AnalysisField Analysis
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Example
Electric Field:
Power density:
Maximum Power density:
Hertzian Dipole - Example
Normalized Power density
Example
Antenna Pattern Solid Angle:
Radiated Power:
Radiated Resistance:
( ) 3 2sin d d sin cos d d,p nP dθ θ φ θ φ θ φθ φΩ = Ω =∫ ∫ ∫ ∫( )( )3 2sin d cos dp θ θ φ φΩ = ∫ ∫
