Lecture Overview uPrinciple
of duality
uRadiation
parameters of a small loop
antenna l
derived using duality principle
uSquare lsolving
loop antenna radiation problem by using the vector potential)
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Duality Theorem When two equations that describe the behavior of two different variables are of the same mathematical form, their solutions will also be identical. The variables in the two equations that occupy identical positions are known as dual quantities and a solution of one can be formed by a systematic interchange of symbols to the other. This concept is known as the duality theorem Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Dual Networks I V
R
L
~
C
I G
V = RI + j ωLI I=
V R + jωL
I = GV + j ωCV
V=
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I G + jωC 3
Dual Antennas
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Suppose that we have an electric current source with current J 1 and boundary condition on materials (ε1 ,µ 1 ,σ1 ) and a magnetic current source M2 with materials (ε 2 ,µ 2 ,σ 2 )
Maxwell’s equations
Maxwell’s equations
for electric system # 1
for magnetic system # 2
ε1,µ1 ,σ1
ε 2 ,µ2 ,σ 2
r r ∇ ×E = − jω µ H 1 1 1
r r r ∇ × H = jωε E + J 1 1 1 1
r r ∇× H = j ωε E 2 2 2
r r r ∇×E = −jωµ H −M 2 2 2 2 5
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M2 ≡ J1
If the sources of two systems are dual
If the boundary conditions are also dual
µ 2 ≡ ε1 and ε 2 ≡ µ1 Then the fields of system # 2 can be found from the solution of system #1 by substitutions
E 2 ≡ −H1 and
H 2 ≡ E1
in the field expressions for system # 1 Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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A current loop can be represented as a magnetic dipole with uniform magnetic current Im and length ∆ z. The sources are dual as required; M2 ≡ J 1 . Let I m ≡ Ie For the ideal electric dipole
For the dual magnetic dipole
r r r E = θE + r E 1 θ,1 r,1
r rr E = −H = −ϕH 2 1 ϕ,1
r r H = ϕH 1 ϕ,1
r r r H = E =θ E + r E 2 1 θ,1 r,1
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H1 = jβ I∆l sin(4πθr)e
− jβr
[1+ ]ϕr 1 jβr
E2 = −H1 = −jβ I
∆zsin(θ)e− jβr 4πr
m
[1+ ]ϕr 1 jβr
−jβr jωµ η r −jβr η 1 θ+ I∆lcos(θ)e 1 rr E1 = I∆lsin(θ)e r + + + 4π 2π 2 jωεr3 r2 jωεr3 r
Im∆zsin( θ)e−jβr 4πr
H2 =E1 = jωε
[1+
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1 jβ r
+
1 ( jβr)2
]
r m θ)e−jβr θ+ jωεI ∆zcos( 2πr
[
1 jβ r
]
r + ( jβ1r)2 r 8
The far-field components are
Eϕ = − j
Hθ = j
βI m ∆z sin( θ )e − jβ r 4 πr
ωεIm ∆z sin( θ) e− jβr 4 πr
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The small square loop antenna u
In solving radiation problem for a small square loop antenna, we will establish a relationship between the current “ I “ in the loop and its equivalent magnetic dipole Im .
u
We assume that the current has constant amplitude and zero phase around the loop
u
Each side of the square loop is a short uniform electric current segment that is modeled as an ideal dipole.
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The small square loop antenna in coordinate systems
Each side of the square loop is a short uniform electric current segment that is modeled as an ideal dipole.
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The total vector potential that is x-directed and y-directed and is given by
A = µ4Iπl
[(
e− jβR4 R4
−j βR
) (
− e R2 2 ax +
where
e−j βR1 R1
− jβR3
) ]
− e R3 ay
R1 = r − 2l sin( θ) cos(ϕ) R 2 = r − 2l sin( θ) sin( ϕ)
R 3 = r + 2l sin( θ) cos(ϕ)
R 4 = r + 2l sin(θ) sin(ϕ)
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By approximation the total vector potential can be written as − jβr r A = jβ S µIe4πr sin θ ϕ
The far-zone radiated fields are given by:
r r E = − j ω A = ηβ 2 S r r 1 r H = r × ϕ E η
ϕ
= −β 2S
Ie − j β r 4π r Ie − j β 4π r
r
r sin( θ ) ϕ sin(
r θ )θ
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
The Radiated electric field
The Radiated electric field
for a magnetic dipole is
for a square loop antenna is
Eϕ = − j
βIm∆zsin(θ)e− jβr 4πr
− jβr
Eϕ = ηβ2S Ie4πr sin(θ)
Comparing the two equations, we conclude that
Im∆z = jωµ IS • u
The field depends only on the magnetic moment SI (current and area) en not on the loop shape. The pattern for a small loop is independent of its shape and it is similar to that of an ideal electric dipole.
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Small Loop antenna u
A closed loop current whose maximum dimension is less than about a tenth of a wavelength is call a small loop antenna
u
Small means; electrically small Using the relationship between the current “I” in the small loop and its equivalent magnetic dipole “Im”
Im∆z = jωµ IS
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Prad =10 ( ISβ2 ) S 2 Rr =31,200 λ
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The radiation resistance of a loop antenna can be increased
Using multiple turns
For a magnetic moment of a N turn loop is NIS , where S is the area of single turn,
R r ≈ 31.200( NS )2 λ2
the radiation resistance is then Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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