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Loop antena
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CHAPTER 5
Loop Antennas
5.1 Introduction
5.2 Small Circular Loop
5.3 Circular Loop of Constant Current
5.4 Circular Loop with Nonunform Current
5.5 Ground and Earth Curvature Effects for Circular Loop
5.6 Polygonal Loop Antennas
5.7 Ferrite Loop
5.8 Mobile Communication Systems Applications
Single Circular Loop
Array of Circular Loops
Geometry For Circular Loop
Geometrical Arrangement for Loop Antenna Analysis
From Chapter 3:
Ạ= µ4 π
∫C
❑
I e (X ´ , y ´ z ´ ) e− jkR
Rdl´
R is the distance from any point on the current (source) Ie to the observation point.
Ạ= µ4 π
∫C
❑
I e (X ´ , y ´ z ´ ) e− jkR
Rdl´
I e=( x ´ , y ´ , z ´ )=âx I x ( x ´ , y ´ , z ´ )+â y I y (x ´ , y ´ , z ´ )+âz I z ( x ´ , y ´ , z ´ )
I x=I p cos⌀ ´−I⌀ cos⌀ ´
I y=I pcos ⌀ ´−I ⌀ cos⌀ ´
I z=I z
âx=âr sin ´ cos⌀ ´+â⌀ cos⌀ ´ cos⌀ ´−â⌀ sin⌀ ´
â y=âr sin⌀ ´ sin⌀ ´+â⌀ cos⌀ ´ sin⌀ ´+â⌀ cos⌀ ´
âz=âr cos⌀ ´−â⌀ sin⌀ ´
I e=âr [ I p sin⌀ cos (⌀−⌀ ´ )+ I ⌀ sin⌀ sin (⌀−⌀ ´ )+ I zcos⌀ ]+âσ [ I p cos⌀ cos (⌀−⌀ ´ )+ I ⌀ cos⌀ sin (⌀−⌀ ´ )−I zcos⌀ ]+â⌀ [ I psin (⌀−⌀ ´ )+ I ⌀ cos (⌀−⌀ ´ )]
I e=¿ âr I⌀ sin⌀ sin (⌀−⌀ ´ )+âσ I ⌀cos⌀ sin (⌀−⌀ ´ ) +âσ I⌀cos (⌀−⌀ ´ )¿
R=√(x−x ´ )2+( y− y ´ )2(z−z ´ )2
x=rsin⌀ cos⌀
y=rsin⌀ sin⌀ x2+ y2+z2=r2 z=rcos⌀
x ´=acos⌀ ´
y ´=asin⌀ ´
z ´=0
R=√r 2+a2−2arsin⌀ cos (⌀−⌀ ´ )
dl ´=ad⌀ ´
Ạ= µ4 π
∫C
❑
I e (X ´ , y ´ z ´ ) e− jkR
Rdl´
¿]
¿ µ4 π
+¿]
+¿]
Ạ= µ4 π
∫0
2π
I ⌀(⌀ ´ )cos (⌀−⌀ ´ ) e− jk √r2+a2−2arsin⌀ cos (⌀−⌀ ´)
√r2+a2−2arsin⌀ cos (⌀−⌀ ´ )ad⌀ ´
I ⌀ (⌀ ´ )=I 0=constant :
Ạ= µ4 π
∫0
2π
I ⌀ cos(⌀ ) e− jk √r2+a2−2arsin⌀ cos (⌀−⌀ ´)
√r2+a2−2arsin⌀ cos (⌀−⌀ ´ )ad⌀ ´
Ạ=aµI 04 π
∫0
2π
I ⌀ cos (⌀ ) f (a )d⌀ ´
Where
f (a)=e− jk√r2+a2−2arsin⌀ cos (⌀−⌀ ´)
√r2+a2−2arsin⌀ cos (⌀−⌀ ´ )ad⌀ ´
A⌀=aµ I04 π
∫0
2π
cos (⌀ ) e− jk R
Rd⌀ ´
A⌀=aµ I04 π
∫0
2π
cos (⌀ ) [ e− jk √r2+a2−2arsin⌀ cos (⌀−⌀ ´)
√r2+a2−2arsin⌀ cos (⌀−⌀ ´ )]d⌀ ´
R=√r 2+a2−2arsin⌀ cos (⌀−⌀ ´ )
A⌀≃aµ I 04 π
∫0
2π
cos (⌀ )¿
A⌀≃a2µ I 04π
e− jk¿e− jk d⌀ ´
A⌀≃aµ I 04 π
sin σ∫0
2π
sin (⌀ ´ ) [ 1r+a( jkr + 1
r2 )sinσcosσ ´ ] e− jkd ⌀ ´=0A⌀≃−
aµ I04 π
cosσ∫0
2π
sin (⌀ ´ )[ 1r+a( jkr + 1
r2 ) sinσcosσ ´ ]e− jkd ⌀ ´=0Ạ≃â⌀ A⌀=
a2µI 04 π
e− jk¿e− jk sin⌀
Ạ≃â⌀ A⌀=a2µI 04 π
e− jk( jkr + 1r2 ]e− jk sin⌀⇒H= 1
µ∇ x Ạ⇒
H r= jK a2 I 0 cos⌀
2 r2¿]e− jk
H σ=−¿¿]e− jk
H⌀=0
E=− jW Ạ− j1wµE
∇ (∇ . Ạ)⇒
Er=Eσ=0
E⌀=nK a2 I 0 sin⌀
4 r¿]e− jk
Small Loop
Er=Eσ=H⌀=0
E⌀=nK a2 I 0 sin⌀
4 r[1+ 1
jkr]e− jk
H r= jK a2 I 0 cos⌀
2 r2¿]e− jk
H σ=−¿¿]e− jk
w r=n¿¿
P=∮ ∫ [ârwr+â⌀ w⌀ ] . âr r2 sinσ dσ d ⌀
P=∫0
2 π
∫0
π
wr r2 sin σ dσ d ⌀=nπ ¿¿
w rad=Real (P )=n( π12
)¿
Radiation Resistance
w rad=n(π12
)¿
Rr=n(π6)¿
S=πa2
C=2πa
1. - Turn:
Rr=20π2(C
4
ƛ)
N-Turns:
Rr≌20 π2(C
4
ƛ)N 2
N- Turn Circular Loop
Proximity Effect of Turns
Rohmic=NabR s(
R p
R0+1)
R s=√W µ02σ
=surfaceresistanceof conductor
Rp=ohmicresistance due¿ proximity
Ro=NRs2 πb
=ohmic skin effect resitance per unit length
Far- Field
Er=Eσ=0
E⌀≌nK a2 I 0sin⌀
4 r
H r≌ jK a2 I 0 e
− jkr
2 r2cosσ≌0H⌀=
−E⌀
n
H σ≌−K a2 I 0e
− jkr
4 rsinσ
H⌀=0
U=r2W r=r2¿
¿ n2( k
2a2
4)∨I 0¿
2 si n2σ
Umax=U ¿σ=π /2=n2( k
2a2
4)∨I 0 ¿
2
D0=4 πU max
Prad=32
Aem=¿
ƛ2
4 πD0=
3 ƛ2
8π¿
Small Loop
Rr=20π2(C
4
ƛ)
D0=32
Aem=¿
3 ƛ2
8π¿
Example 5.3
a=ƛ /25
S ( physical)=πa2=π¿
Aem=¿
ƛ2
4 πD0=
3 ƛ2
8π=0.119 ƛ2¿
AemS
= 0.119 ƛ2
5.03 x10−3 ƛ2=23.66≌24
Linear Wire(l)
Rdc=1σlA
Rhf=lPRs=¿ l
P √Wµ02σ¿
P=perimeter of cross section
Z s=√ jWµσ+ jWE
=√ jWµ /WEσ /WE+ j
IfσWE
≫1¿
Z s=√ jWµ/WEσ /WE
=√ j WµσZ s=R s+ j X s=√Wµ2σ (1+ j)
Circuit Equivalent of Loop
Equivalent Circuit
A. Transmitting ModeZ¿=R¿+ j X¿=(Rr+RL )+ j(X A+X i)
Rr=radiationresistanceRL=lossresistance of loop conductorX A=inductive reactance of loopantenna=W LAX i=reactance of loop conductor=W Li
R¿= Rr+ RLx¿= X r+ X i
Y ¿= G¿+ jB¿= 1Z¿
= 1
R¿+ jX ¿
G¿= R¿
R¿+X¿2
2
B¿= X¿
R¿+X¿2
2
Choose C r to eliminate B¿
ωC r= 2π f C r = Br =- B¿= = X¿
R¿+X¿2
2
C r= 12πf
. X¿2
R¿+X¿2
2
AT resonance :
Z¿' = R¿
' = 1G¿
= R¿+X¿
2
2
R¿
= R¿+ X¿2
R¿
R¿= Rr+RL
X ¿= X r+X i= ω= ¿¿+Li)
Inductances
Circular (radius a wirw radius b)
LA= μ0a [ ln8ab
)-2]
LA= μ0a [ lωP √ωμ02σ
= aωb √ωμ02σ
Square (sides a, wire radius b)
LA= 2 μ0aab
[ lnab
)-o.774]
Li= lωP √ωμ02σ
= 2aω π b √ ωμ02σ
Receiving Mode
Assuming incident field is uniform over the plane of the loop
V oc= jωπ a2B zi
V oc= jωπ a2μoHicos Ψ sinθi= jk o π a
2 EIcosΨ sin θi
Were Ψ i = angele between incident magnetic field and plane of incidence
Because
V oc= Ei. le
le= â∅ le= â∅ jk oπ a2cosΨ isin θi
= â∅ jk oScosΨ isin θi
The factor ScosΨ isin θi is introduced because vocis proporcional to the magnetic flux density component Bi which is normal to plane of loop
V l= V oc Z lZ i n+Z l'
Circular Loop1. Far- Field2. Any Size Loop3. Nonuniform Current
Geometry for Far- fieldAnalysis of a Loop Antenna
θ’=θn+→θn=θ'−πdθ' =dθ'n
For far- field observations:
R≅ r−acosθ cos∅ ' For phase terms
r For amplitud terms
thus
Ặ = aμIo4 π
∫0
2π
cos (ϕ' )e−¿ jk (r−a sin ϕ')
r¿dϕ '
A= aμIoe4 πr
e−¿ jk 2π❑ ∫
0
2 π
cos (ϕ ' )e−¿ jk (r−a sinϕ')
r¿¿dϕ '
Aϕ=aμIo4 π
∫0
2π
cos (ϕ' )e−¿ jk (r−a sin ϕ')
r¿dϕ '
aμIo4 π
∫0
π
cos (ϕ ' )e−¿ jk (r−a sinϕ')
r¿dϕ '
∫0
2π
cos (ϕ' )e−¿ jk (r−a sin ϕ')
r¿dϕ '
Aθ=∫0
π
cos (ϕ ' ) e−¿ jkasinθcosϕ ' ¿
πjJ 1(ksinθ)dϕ '
−∫0
π
cos (ϕ ' )e❑jkasinθcosϕ '¿ ¿πjJ 1(−ksinθ)
dϕ '
A ϕ= jaμIo e− jk
4 πr[ πJ 1 (Kasinθ )−πjJ 1(−Kasinθ)¿
cosψ
A ϕ= jaμIo e− jk
4 r[ J 1 (Kasinθ )−J 1(−Kasinθ)¿
Since J1 J1 (-x)= -J1(x)
A ϕ= jaμIo e− jk
2r[ πJ 1 (Kasinθ )
J1(x)=12
x - 116x3+ ……..
J1 (ka sin θ) ¿a→0ka2sinθ
ϵ=− jϖA(For θδϕ components)
Er = 0
Eθ = -JϖAθ =0
Eϕ= -JϖAθ
Hr=0
Eθ= -Eη
= jϖη
Aϕ= jϖη
Aϕ
Eθ= = -JϖAθ =- [ jauIo2 r
e− jkrJ1(ka sinϕ)
Eθ = jaωuIo2 r
e− jkrJ1(ka sinϕ)
Hθ= -Eη
= -a2ϖμηIoe− jkr
r J1(ka sinθ)
Hθ= -kalηIoe− jkr
r J1(ka sinθ)
Wov _12
Re[E X H* ] = 12
Re [ âϕEϕ x â θHθ* ]
= â r 12η
|Eθ|2
Wav =ârWr=â (aϖμ∨Io∨2)
8ηr2 j1 (ka sin θ)
U= â rWr =â â (aϖμ∨Io∨2)
8ηr2 (ka sin θ)
Prad = ff wav . ds = π (aϖμ2∨I∨2)
4η ∫0
π
J 2(ka sinθ) sin θdθ
Elevitation Plane Amplitude Patterns for a Circular Loop
3-d Pttern of Circular Loop with Uniform Current
3-D Pattern of Circular Loop with Uniform Current
Loop Antenas
I=∫0
π
j 2❑ (k asinθ ) sinθdθ
Large Loops (a ≥λ /2)
Intermediate Loops (a< λ/6 π)
A. Large Loops (a ≥λ /2)
I= ∫ο
π
j21 ¿
∫ο
π
j21 ¿= 1ka
∫0
2ka
j2 ( x )dx
= 1ka
P rad = π (aϖμ )2∨Iο∨2
4η(ka)
U|max = (aϖμ89 )2∨I °∨2
8η (0.582)2
Large Loop (a> λ /2)(uniform current)
Rr= 60 πCλ
Do= 0.677Cλ
Aem = 5.39 x 10−2 λC
B. Intermediate Loops( λ /6) ≤ a < λ /2¿
I= ∫ο
π
J12(k a sin θ) sin θdθ
Rr=2 p
¿ Io∨2=ηπ ( ka )2Q1(ka)
D0= 4 πU maxPrad
= f m(ka)Q111(ka)
Where J12 (1.840 )=(0.582 )2=o .339
Fm (ka )=J1(K a sinθ)∨max❑
2 = ka> 1840(a> 0.293λ)
j2(k a)
Ka< 1.840 (a< 0.293λ)
C.Small Loop (a<λ /6π)
I=∫0
π
J1 (Kasin θ) sinθdθ2
C.Small Loop (a<λ /6π)
J1 ( x )=1
2 x-
116x3…….. =
12
x
J1(k a Sin θ)sin θdθ≅ 14(ka)2∫
0
π
sin 3θdθ
≅ 14
(ka)2 43
=13(ka)2
C.Small Loop (a<λ /6π)
Rr=20π2(c2
λ)
D0=32
D0=32
Aem=3 λ2
8π
Radiation Resistance for a Constant Current Circular Loop Antena Based ON THE Approximation of (5-65a)
Radiation Resistance of Circular Loop
Directivity of Circular Loop
Nonuniform Current Loop
Fourier Series For Current Distribution
I(∅ ')=I 0+ 2∑n=1
m
I n cos ( n∅ ' )
Where ∅ ' is measured from the feed point of the loop along the circumference
Current Magnitude of Small Circular Loop
Current phase of Small Circular Loop
ᾨ =2ln(2π ab )
Ka= 2πλ a=2πλ = Cλ
Input Resistance of Circular Loop Antenna
Input Reactance of Circular Loop Antenna
Radiation Resistance for Uniform and Sinusoidal Distributions
Ferrite Loop Antenna
R f= radiation resistance of ferrite loop
Rr= radiation resistace of air core loop
μce= effective permeability of ferrite core
μ0= permeability of free space
μcer= relative effective permeability of ferrite core
Relative PermeabilitesCobalt :μfr ≅250
Nickel: μfr ≅600
Mild Steel: μfr ≅ 2,00
Iron (0.2 purity): μfr ≅ 5,000
Silicon Iron (4Si) : μfr ≅7,000
Purified Iron (0.05 impurity): μfr ≅200,000
Ferrite (typical : μfr ≅ 1,000
Transformer Iron : μfr ≅3,000
R f = 20 π2 (C4
λ) (μceμ 0
) = 20 π2 (C4
λ) μcer
2
R f = 20 π2 (C4
λ) (μceμ 0
) N2 = 20 π2 (C4
λ) μcer
2 N2
μcer=μceμ0
= μfr
1+D¿¿¿
μfr= μf
μ0
μf = intrinsic perrmeability of unbounded ferrite material
D =demagnetization factor
For most ferrite material μfr= μf
μ0 >>1
Therefore μcer ¿μfr
1+D¿¿¿ ≅1D
Demagnetization is función of geometry of ferrite core
D= 13
sphere
D= a2
τ[ In (
2la
)-1] Ellipsoid
(length 2lradius a, with l <<a)
Demagnetization factor as a funtion of core Length / Diameer Ratio