loop antena

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