ece5318 ch4

Chapter 4Chapter 4

Linear Wire AntennasLinear Wire Antennas

ECE 5318/6352ECE 5318/6352Antenna EngineeringAntenna Engineering

INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE

(only electrical current present)

(constant current)

l ≤ λ/50

θImpinging

oIz zaI ˆ)( ' =

; thin wire ;λ<<l

00 =⇒= FIm

INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)

222 zyxr ++=

Fig. 4.1(a) Geometrical arrangementof an infinitesimal dipole

l ≤ λ/50

mixed coordinates in mixed coordinates in expression expression -- change to change to

sphericalspherical

222 zyxR ++≅

'''' ),,(4

ezyx(x,y,z)jkR

∫≅ IAπ

λ<<for

(x,y,z)

(x’,y’,z’)

source points

l ≤ λ/50

( ) ( ) ( )2 2 2' ' 'R x x y y z z= − + − + −

mixed coordinates in expression mixed coordinates in expression change to sphericalchange to spherical

∫−

4ˆ zd

re(x,y,z)

jkroo IaA z π

jkroo er

(x,y,z) −≅π

ˆ IaA z

(x,y,z)

(x’,y’,z’)

source points

l ≤ λ/50

( ) ( ) ( ) 2'2'2' zzyyxxR −+−+−≅

μθθ sin4

sin jkrooz e

rIAA −=−=

μθ cos4

cos jkroozr e

rIAA −=−=

∫cd ' along source

(x,y,z)

(x’,y’,z’)

source points

mixed coordinates in expression mixed coordinates in expression need to change to sphericalneed to change to spherical

l ≤ λ/50

Using Vector Potential Using Vector Potential A A , , calculate calculate HH & & EE fields fields

[ ] ⎥⎦⎤

⎢⎣⎡

∂∂

−∂∂

=×∇θθφ

rAArrr

[ ]φφ μμAaAH ×∇=×∇=

l ≤ λ/50

Using Vector Potential Using Vector Potential A A , , calculate calculate HH fields fields

AH ×∇=μ1

jkro ejkrr

IkjH −⎥⎦

⎤⎢⎣

11sin4

θπφ

l ≤ λ/50

Using MaxwellUsing Maxwell’’s s EqnsEqns totocalculate calculate EE fields fields

[4-10]

HE ×∇=ωεj1

jkror e

jkrrIE −

⎥⎦

⎤⎢⎣

11cos2 2 θ

jkro erkjkrr

IkjE −⎥⎦

⎤⎢⎣

⎡−+= 22

111sin4

Fig. 4.1(b) Geometrical arrangementof an infinitesimal dipole and its associated electric-field componentson a spherical surface

l ≤ λ/50

Using Using HHφφ, , EErr, , EEθθ,, calculate the complex Poynting vectorcalculate the complex Poynting vector

( )∗∗∗ −=×= φθφθ HEHE rr aaHEW ˆ21)(

⎥⎦⎤

⎢⎣⎡= −⎥

⎤⎢⎣

⎡3)(

roW θλ

[4-12]( )2cos sin 112 3 216 ( )

k Ioj j

r krW η θ θ

θ π+

⎡ ⎤= ⎢ ⎥⎣ ⎦

l ≤ λ/50

Find total outward flux through a closed sphereFind total outward flux through a closed sphere

(only contributions from Wr)

[4-14]∫∫ •=s

⎥⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣⎡= 3

3 krjIo

λπη

θθφπ

φdrWd r sin

0 ∫∫ ===

l ≤ λ/50

Find total outward flux through a closed sphereFind total outward flux through a closed sphere

rad RIIP 22

3=⎥⎦

⎤⎢⎣⎡=

ληπ

316.002.050

=⇒== rRλλ

λπ=rR

2120 πη =

Real P = total radiated power Prad

ExampleExample [Ω]

Radiation resistance

for free space where

[4-19]

[4-16]

l ≤ λ/50

(Impedance would also have a large capacitive term that is not calculated here.)

2 13 kr

Io⎥⎦⎤

⎢⎣⎡−=

λπη

Imaginary part of P = reactive power in the radial direction

(Note: this → 0 as kr → ∞, so it is essentially not present in far field; only important in near field considerations)

[4-17]

l ≤ λ/50

Near Field approximations Near Field approximations [ [ krkr <<<< 1 ]1 ]

(field point very close or very low frequency case)

θπφ sin

4 2reIH

Dominant terms ⇒

[4-20]

η cos2 3rk

eIjEjkr

−≅

ηθ sin

4 3rkeIjE

−≅

Like ‘quasistationary” fields

E near static electric dipole

H near static current element

l ≤ λ/50

Near Field approximations Near Field approximations [ [ krkr <<<< 1 ]1 ]

Biot – Savart Law : infinitesimal current element in directionaz

(same as above when kr →0)

(note E and H are 90° out of phase)

NO RADIAL POWER FLOW --REACTIVE FIELDS

θπφ sin

2rIoaH ≅

][Re21 ∗×= HEWavg

0=avgW

[4-21]

[4-22]

l ≤ λ/50

Intermediate FieldsIntermediate Fields[ [ krkr >> 1]1]

(beginnings of radial power flow; still have radial fields)

1Erθ ∼

1Erφ ∼2

l ≤ λ/50

r = λ/2π (Radian Distance)

(Radius of Radian Sphere)

Energy basically imaginary (stored)

Energybasically

real(radiated)

Fig. 4.2 Radiated field terms magnitude variation versus radial distance

l ≤ λ/50

Far Field Far Field [ [ krkr >>>> 1 ]1 ]

Dominant terms ⇒

[4-26]θπφ sin

4 reIkjH

0r rE E H Hφ θ≅ ≅ ≅ ≅

ηθ sin

4 reIkjE

l ≤ λ/50

Far Field Far Field [ [ krkr >>>> 1 ]1 ]

E( both E and H are TEM to )

Similar to plane wave but propagation in direction

With and variationsr1

[4-27]

l ≤ λ/50

DirectivityDirectivity (use Far Field approx.)

RADIATION INTENSITY

][Re21 ∗×= HEWavg 2

22 sin

⎥⎥⎥

⎢⎢⎢

η 242 sin22

⎥⎥⎦

⎢⎢⎣

⎡== oIkavgWrU

( Note: as before for )2

22 sin8 r

oIavgW θ

⎥⎦

⎤⎢⎣

⎡= )( Real rW

[4-28]

[4-29]

l ≤ λ/50

(in θ = 90° direction) [4-31]

rado P

UD max4π=

max 42 ⎥⎦

⎤⎢⎣

πη oIkU

3 ⎥⎦⎤

⎢⎣⎡=

λπη o

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

λπη

DirectivityDirectivity

l ≤ λ/50

SMALL DIPOLESMALL DIPOLE

Uniform current assumption - only valid for ideal case( approximated by capacitor plate antenna)

value of fields compared to constant current case

λ/50 < l < λ/10

ηθ sin

8 reIkjE

θπφ sin

8 reIkjH

[4-36]

SMALL DIPOLESMALL DIPOLE(CONT)(CONT)

For physical small dipole triangular current distribution

value of case of constant current

same as constant current case

λ/50 < l < λ/10

[4-37]

12 ⎥⎦⎤

⎢⎣⎡=

ληπ o

2220 ⎥⎦

⎤⎢⎣⎡=λ

5.1=oD

FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(length comparable to λ)

(max error where θ = 90° ; 4th term = 0 there)

approx. error

2' 21cos sin

2zR r z

rθ θ

′⎛ ⎞= − + +⎜ ⎟

⎝ ⎠

[4.41]

Fig. 4.5 Finite dipole geometryand far-field approximations

Phase and Magnitude considerationsPhase and Magnitude considerations

In calculating phase assumecan tolerate phase error of π/8 (22°)

Must choose r far enough away so that ….

FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)

Phase and Magnitude considerationsPhase and Magnitude considerations

2max ' =z

2 8k z

rπ′

ORIGIN OF DEFINITION OF FAR FIELD

22>r⇒≤

882 2 πλπ

jkre−For phase term ⇒ use θcos'zrR −=

For magnitude term ⇒ user1 rR =

[4-45]

Finite dipole Current distributionFinite dipole Current distribution

(“thin” wire, center fed, zero current at end points)

λ / 2 < l < λ

[4-56]⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ − '

2sinˆ zkIoza

20 ' ≤≤ z

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ + '

2sinˆ zkIoza 0

2' ≤≤− z

=== ),0,0( ''' zyxeI

(see Fig. 4.8)

Current distribution for linear wire antennaCurrent distribution for linear wire antenna

Fig. 4.8 Current distribution along the length of a linear wire antenna

DIPOLE

Radiated fields at (Radiated fields at (x, y, zx, y, z) ) of finite dipoleof finite dipole

)( zdRezkjEd

jkRe θ

πηθ

( ) 2'22 zzyxR −++=⇒

For infinitesimal dipole at z’ of length Δ z’

Since source is only along the z axis ( )0,0 '' == yx

Radiated fields of finite dipole at (Radiated fields of finite dipole at (x, y, zx, y, z))

In far field regionin phase term

θcos'zrR −=( let )⇒

)( zderezkjEd jkz

jkre θ

θ θπ

[4-58]

Far Field Far Field E & H E & H Radiating fields Radiating fields

∫−=

2/ θθ EdE

'cos2/

)(sin4

zdezIr

ekjE jkze

θ θπ

η∫−

Total Field

[4-58a]

Far Field Far Field E & H E & H Radiating fields Radiating fields

⎥⎥⎥⎥

⎢⎢⎢⎢

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

≅−

θ sin2

coscos2

For sinusoidal current distribution

[4-62]

φEH ≅

Power DensityPower Density

cos cos cos2 2

8 sino

θηπ θ

⎡ ⎤⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

[4-63]

Radiation IntensityRadiation Intensity

coscos2

8⎥⎥⎥⎥

⎢⎢⎢⎢

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

kkIWrU o

avg [4-64]

l ≥ λ/2

33--dB BEAMWIDTHdB BEAMWIDTH

90°87°

78°64°

.25 1.75.5 λ

33--dB BEAMWIDTHdB BEAMWIDTH

λ>If allow new lobes begin to appear

Fig. 4.7(b) 2-D amplitude pattern for a thin dipolel = 1.25 λ and sinusoidal current distribution

Elevation plane amplitude patterns for a thin dipole with sinusoElevation plane amplitude patterns for a thin dipole with sinusoidal current distributionidal current distribution

Fig. 4.6

Radiated power Radiated power

Results of integration give terms involving Ci & Si [4-68]

∫∫ •=s

avgrad dP sW [4.66]

Radiated power Radiated power

sin and cos integrals (tabulated functions like trig. functions, but not as common)

Can find Rr and Do in terms of Ci and Si

Do, Rr, Rin plotted in fig. 4.9

[4-75][4-70]

Radiation resistance, input resistance and directivity of a thinRadiation resistance, input resistance and directivity of a thin dipole with sinusoidal dipole with sinusoidal current distributioncurrent distribution

Fig. 4.9

FINITE LENGTH DIPOLE

Input ResistanceInput Resistance

(note that Rr uses Imax in its derivation)

≥for

oin II ≠

at input terminalsI

VZin =

Ie (z’)

maxIIo =

Input ResistanceInput Resistance

So, even for lossless antenna ( RL = 0 )

[4-77a]

⎥⎦

⎤⎢⎣

⎡=inr RR ≠ ⇒

⎟⎠⎞

⎜⎝⎛

2sin2 k

Ie (z’)

maxIIo =

Input Resistance (cont)Input Resistance (cont)

Not true in practical case, current not exactly sinusoidal at the feed point(due to non-zero radius of wire and finite feed gap at terminals)

Numerous ways to account for more exact current distribution, result in currents that are both in and out of phase, and in Rin + j Xin

(subject of extensive research, numerical and analytical)

Note: when ; andλn= ∞→inR0→inI

Empirical formula for Empirical formula for RinRin

)12max( ⎥⎦⎤

⎢⎣⎡Ω<inR

≤≤4

0 π≤≤ G220GRin ≅

17.414.11 GRin ≅

5.27.24 GRin ≅24λλ

≤≤

λλ 64.02

≤≤

24ππ

≤≤ G

≤≤ Gπ )200max( ⎥⎦⎤

⎢⎣⎡Ω<inR

)76max( ⎥⎦⎤

⎢⎣⎡Ω<inR

let 2kG = for dipole of length

[4-107] → [4-110]

For MONOPOLEFor MONOPOLE

[ ]5.427321 jZin +≅

21Rin (monopole) = Rin (dipole)

[ ]Ω+≅ 2.215.36 jZin

for wavelength monopole14

same current; voltage ⇒ impedance21

[4-106]

HALF WAVE DIPOLEHALF WAVE DIPOLE

⎥⎥⎥⎥

⎢⎢⎢⎢

⎡⎟⎠⎞

⎜⎝⎛

≅−

θ sin

2 reIjE

8⎥⎥⎥⎥

⎢⎢⎢⎢

⎡⎟⎠⎞

⎜⎝⎛

⎥⎥⎥⎥

⎢⎢⎢⎢

⎡⎟⎠⎞

⎜⎝⎛

≅−

πφ sin

2 reIjH

ll = = λλ/2/2

0 20 40 60 80 100 120 140 160 1800

θ (deg)

θ2sin

θ3sin

[4-84]

πη π

dIP orad ∫

⎟⎠⎞

⎜⎝⎛

4[4-88]

[4-86]

[4-85]

ll = = λλ/2/2

HALF WAVE DIPOLE HALF WAVE DIPOLE (CONT)(CONT)

Slightly moredirective thaninf. dipole withDo = 1.5

64.14 max ≅=rad

o PUD π

where 435.2)2( ≅πinC)2(

rad CIP = [4-89]

[4-91]

l l = = λλ/2/2

HALF WAVE DIPOLE HALF WAVE DIPOLE (CONT)(CONT)

since (if lossless)rin RR ≅ inII =max

[ ] [ ]Ω+≅⇒Ω≅ 5.42735.42 jZX inin

[ ]Ω≅== 734

2 )2(2 ππ

radr C

[4-93]

PRACTICAL DIPOLEPRACTICAL DIPOLE

[ ]Ω≅ 300inR

[ ]Ω≅ 300oZ

Folded dipole

Useful for matching to two-wire

lines where

l l slightly < slightly < λλ/2/2

Usually choose ll slightly less than so that is totally real.2λ

ininX Z0 &→

PRACTICAL DIPOLEPRACTICAL DIPOLE(CONT)(CONT)

Resistance and Reactance Variations

2λ(pure real for length slightly less than )

l l slightly < slightly < λλ/2/2

0.5 1.0 λ

IMAGE THEORYIMAGE THEORY

Can calculate the fields in the UHP by removing the conductorand finding the field due to the actual and image sources.

Linear antennas near an infinite ground plane could approximate case of earth.

Direct

Reflectedh2

IMAGE THEORYIMAGE THEORY(CONT)(CONT)

In the Lower Half Plane, E = H = 0→→

μο, εο

σ = ∞

Actual Problem Equivalent Problem

Observation Point

IMAGE THEORY IMAGE THEORY (CONT)(CONT)

Fields due to image source are actually produced by the induced currents in the ground plane

⇓ ⇓

actual

Iactual

VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE

Electric dipoles above an infinite perfect electric conductorElectric dipoles above an infinite perfect electric conductor

Fig. 4.12(a) Vertical electric dipole above anInfinite, flat, perfect electric conductor

Fig. 4.24 Horizontal electric dipole, and its associated image, above an infinite, flat, perfect electric conductor

Electric dipoles above ground planeElectric dipoles above ground plane

Fig. 4.14(a)

Fig. 4.25(a)

Far FieldFar Field

Electric dipoles above an infinite perfect electric conductorElectric dipoles above an infinite perfect electric conductor

Fig. 4.14(b) Fig. 4.25(b)

FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS

approx. in phase terms

θcos1 hrr −≅θcos2 hrr +≅

in magnitude terms321 rrr ≅≅[4-97]

[4-98]

Summing two contributions

total = incident + reflected total = actual + imaginary

rd EEE21 θθθ +≅

ηθ r

eIkjEjkr

ηθ r

eIkjEjkr

ηψ sin

reIkjE

ηψ sin

reIkjE

−≅

rd EEE21 ψψψ +≅

[4-94]

[4-95]

[4-111]

[4-112]

FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS(CONT)(CONT)

VERTICAL DIPOLEVERTICAL DIPOLEHORIZONTAL DIPOLEHORIZONTAL DIPOLE

[ ]θθθ θ

πη coscossin

4jkhjkh

jkro ee

reIkjE +≅ −

[ ]θθψ ψ

πη coscossin

4jkhjkh

jkro ee

reIkjE −≅ −

φθψ 22 sinsin1sin −=

φθψ sinsincos =

[4-99]

[4-116]

Single source at origin array factor

( )sin 2 cos cos4

jkrok I eE j kh

rθη θ θ

≅ ⎡ ⎤⎣ ⎦

Single source at origin array factor

( )[ ]θφθπ

ηψ cossin2sinsin1

422 khj

reIkjE

jkro −≅

for 0=θE 0<z

Amplitude patterns at different heightsAmplitude patterns at different heights

Fig. 4.15Fig. 4.26

Number of lobes

Note minor lobes that are

formed for

HORIZONTAL DIPOLEHORIZONTAL DIPOLEVERTICAL DIPOLEVERTICAL DIPOLE

Number of lobes

Note minor lobes that are

formed for

12+≅

≥h2λ

≅[4-100] [4-117]

Amplitude patterns at different heightsAmplitude patterns at different heights(CONT)(CONT)

Note max radiation is in θ = 90° direction

Fig. 4.16Fig. 4.28

VERTICAL DIPOLEVERTICAL DIPOLE

HORIZONTAL DIPOLEHORIZONTAL DIPOLE[4-102]

[4-118]

RADIATION POWERRADIATION POWER

( )( )

( )( ) ⎥

⎤⎢⎣

⎡+−⎥⎦

⎤⎢⎣⎡= 32

khkhIP o

rad λπη

( )( )

( )( ) ⎥

⎤⎢⎣

⎡+−−⎥⎦

⎤⎢⎣⎡= 32

khkhIP o

rad λπη

[4-104]

[4-123]

DIRECTIVITYDIRECTIVITYVERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE

Fig. 4.29 Radiation resistance and max. directivityof a horizontal infinitesimal electric dipole as afunction of its height above an infinite perfectelectric conductor.

( )( )

( )( ) ⎥

⎥⎥⎥⎥

⎢⎢⎢⎢⎢

⎥⎦

⎤⎢⎣

⎡+−

⎟⎠⎞

⎜⎝⎛ ≥≥

42R(kh)4 λπ hkh

o PUD max4π

( )⎟⎠⎞

⎜⎝⎛ ≤≤

42R(kh)sin4 2 λπ hkhkh

Fig. 4.18 Directivity and radiation resistanceOf a vertical infinitesimal electric dipole as afunction of its height above an infinite perfectelectric conductor.

DIRECTIVITYDIRECTIVITY(CONT)(CONT)

VERTICAL DIPOLEVERTICAL DIPOLEHORIZONTAL DIPOLEHORIZONTAL DIPOLE

Limiting case of kh→ 0

!5!3sin

53 xxxx +−=

!421cos

42 xxx +−=

2345611sin 2

23 ⋅⋅⋅+−=

234211cos 2

22 ⋅⋅+−=

sincos31

+−⇒

==−+≅

⎥⎦

⎤⎢⎣

⎡⋅⋅⋅

+−+⎥⎦

⎤⎢⎣

⎡⋅⋅

+−−≅23456

112342

1131 2

Note: direction of maximum radiationchanges as “h” is varied. Dg (θ=0)

Dg(θ=0)

6.0∞∞

6.57.4582.88

Doh/λkh

6.0∞

slightly

> 6.0.615+n/2(n=1,2,3…)

Doh/λ

∞→

0lim )(

sin5.7 ⎥⎦

⎤⎢⎣

[4-124]

DIRECTIVITYDIRECTIVITY(CONT)(CONT)

Input Impedance of a Input Impedance of a λλ/2 dipole above a /2 dipole above a flat lossy electric conductive surfaceflat lossy electric conductive surface

Fig. 4.20

ininin XRZ +≅ [ ]Ω+≅ 5.4273 jZin

HORIZONTAL DIPOLEHORIZONTAL DIPOLE

Input Impedance of a Input Impedance of a λλ/2 dipole above a /2 dipole above a flat lossy electric conductive surfaceflat lossy electric conductive surface

Fig. 4.30ininin XRZ +≅ [ ]Ω+≅ 5.4273 jZin

GROUND EFFECTSGROUND EFFECTS

Finite conductivity σearth

(“real” earth as ground plane)

Direct

Reflected

σearth

Assume earth flat (ok. for Rearth >> λ)

10 → 1 [S/m]

GROUND EFFECTSGROUND EFFECTS(CONT)(CONT)

(real earth as ground plane)

Fig. 4.31 Elevation plane amplitude patterns of an infinitesimal vertical dipole above a perfect electric conductor σ=∞ and a flat earth σ= 0.01 [S/m]

HORIZONTAL DIPOLEHORIZONTAL DIPOLE

Fig. 4.32 Elevation plane ( φ = 90°)amplitude patterns of an infinitesimal horizontal dipole above a perfect electric conductor σ=∞ and a flat earth σ= 0.01 [S/m]

(real earth as ground plane)

σσ = = ∞∞

σσearthearth

For low and medium frequency applications when height is comparable to skin depth [ δ = 2/ωμσ ]of the ground ⇒ increasing changes in input impedance; less efficient; use of ground wires)

GROUND EFFECTS GROUND EFFECTS (CONT)(CONT)

Usually negligible effect for observation angle ψgreater than 3°.

EARTH CURVATUREEARTH CURVATURE

Fig. 4.34 Geometry for reflections from a spherical surface

EARTH CURVATUREEARTH CURVATURE

Curved surfaces spreads out radiation (divergent) that is reflected more than from flat surface.(can introduce a divergence factor)

Fig. 4.35 Divergence factor for a 4/3 radius earth(ae = 5,280 mi = 8,497.3 km) as a function ofgrazing angle ψ.

reflected field from spherical surface

reflected field from flat surface___________________=

DDivergence factor

l=λl=λ/2l=λ/10l=λ/50

0 (-∞ dB)1.5746 (1.972 dB)0.2181 (-6.613 dB)0.0374 (-14.27 dB)G0abs

0 (-∞ Db)

0.9642 (-0.158 dB)

0.1556 (-8.08 dB)

0.0271(-15.67 dB)

10.18929-0.9189-0.9863Γ

2.4026(3.807 dB)

1.6331(2.13 dB)

1.4009(1.464 dB)

1.3782(1.393 dB)

2.411 (3.822 dB)

1.6409 (2.151 dB)

1.5 (1.761 dB)

0.9965 (-0.015 dB)

0.9952 (-0.021 dB)

0.9339 (-0.296 dB)

0.9188(-0.368 dB)

∞731.97390.3158Rin

199731.97390.3158Rr

0.69810.3490.13960.0279RL

1.39620.6980.27920.0279Rhf

DIPOLE SUMMARYDIPOLE SUMMARY(Resonant ⇒ XA=0; f = 100 MHz; σ = 5.7 x 107 S/m; Zc = 50; b = 3x10-4l)

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