chapter 4 linear wire antennas 1 ece 5318/6352 antenna engineering dr. stuart long

Chapter 4Chapter 4

Linear Wire AntennasLinear Wire Antennas

1

ECE 5318/6352Antenna Engineering

Dr. Stuart Long

INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE

oIz zaI ˆ)( '

2

(only electrical current present)

(constant current)

l /50

I

l / 2

l / 2

Io

Impinging Wave

z

; thin wire ;l

00 FI

m

[4-1]

INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)

3

222 zyxr

Fig. 4.1(a) Geometrical arrangementof an infinitesimal dipole

l /50

mixed coordinates in mixed coordinates in expression - change to sphericalexpression - change to spherical

222 zyxR

'''' ),,(4

dR

ezyx(x,y,z)

jkR

c

eo

IA

2 2 2' ' 'R x x y y z z

4

for

(x,y,z)

(x’,y’,z’)

source points

l

[4-2]


l /50

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

5

[4-4]

2/

2/

'

4ˆ

zd

r

e(x,y,z)

jkroo I

aA z

jkroo er

(x,y,z)

4

ˆ

IaA z

(x,y,z)

(x’,y’,z’)

source points

l


l /50

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

2'2'2' zzyyxxR

sin4

sin jkrooz e

r

IAA

6

cos cos4

jkro or z

IA A e

r

cd' along source

0A

(x,y,z)

(x’,y’,z’)

source points

l

[4-6]


l /50

7

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

rAAr

rr)(

1A

AaAH

1

ˆ1

[4-7]


l /50

AH

1

jkro ejkrr

IkjH

11sin

4

0rH

0H

8

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

[4-8]


l /50

9

Using Maxwell’s Eqns toUsing Maxwell’s Eqns to calculate calculate EE fields fields

[4-10]

HE

j

1

jkror e

jkrr

IE

11cos

2 2

0E

jkro erkjkrr

IkjE

22

111sin

4

Fig. 4.1(b) Geometrical arrangementof an infinitesimal dipole and its

associated electric-field components on a spherical surface


l /50

Using Using HH, , EErr, , EE,, calculate the complex Poynting vectorcalculate the complex Poynting vector

1 1ˆ( )

2 2

2 2sin 11

2 38 ( )

2cos sin 1

12 3 216 ( )

E H E Hr r

IoW j

r r kr

k IoW j j

r kr

W E H a a

10

[4-12]


l /50

s

dP sW

3

2

)(

11

3 krj

Io

drWd r sin

0

22

0

11

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

(only contributions from Wr)

[4-14]


l /50

12

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

0.02 0.31650

Example: rR

22

22

2

1

3 2

Radiation resistance 80

for free-space where 120

orad o r

r

IP I R

R

Real P total radiated power Prad

[4-19]

[4-16]


l /50

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

13

32

1

3 kr

Io

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 and intermediate field considerations)

[4-17]


l /50

Near Field approximations Near Field approximations [ kr [ kr 1 ] 1 ]

(field point very close or very low frequency case)

sin

4 2r

eIH

jkro

14

Dominant terms

[4-20]

cos

2 3rk

eIjE

jkro

r

sin

4 3rk

eIjE

jkro

Like ‘quasistationary” fields E near static electric dipole

H near static current element


l /50

Near Field approximations Near Field approximations [ kr [ kr 1 ] 1 ]

sin

4ˆ

2r

IoaH

15

Biot – Savart Law : infinitesimal current element in direction az

(same as above when kr 0)

(note E and H are 90° out of phase)

NO RADIAL POWER FLOW -- REACTIVE FIELDS

][Re2

1 HEW

avg

0avgW

[4-21]

[4-22]


l /50

Intermediate FieldsIntermediate Fields[ kr [ kr >> 1] 1]

(induction zone; still have radial fields)

• E 1/r • H 1/r

16


l /50

• Er 1/r

17

r = /2 (Radian Distance)

(Radius of Radian Sphere)

Energy basically imaginary

(stored)Near field

Energybasically

Real(radiating)

Far field

Fig. 4.2 Radiated field terms magnitude variation versus radial distance


l /50

IntermediateRegion

Induction Zone

Far Field Far Field [ kr [ kr >>>> 1 ] 1 ]

18

Dominant terms

[4-26] sin

4 r

eIkjH

jkro

0r rE E H H

sin

4 r

eIkjE

jkro


l /50

Far Field Far Field [ kr [ kr >>>> 1 ] 1 ]

H

E

ˆSimilar to plane wave, but propagation in direction

1With and sin variations

r

r

a

19

( both E and H are TEM to )[4-27]


ra

l /50

DirectivityDirectivity (use Far Field approx.)

21 sin2ˆRe[ ]22 2 4

22 2sin

2 4

k Io

avg r r

k IoU r W

avg

W E H a

20

RADIATION INTENSITY

2 2sinas before for RNote ea: l ( )

28 r

IoW W

avg r

[4-28]

[4-29]


l /50

DirectivityDirectivity

21

[4-31]

2

2

in =90 direct

38max4

io

4 1.52

3

2

max 2n

4

2

3

o

o

o

IU

D Do P Irad

k IoU

IoP

rad


l /50

SMALL DIPOLESMALL DIPOLE

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

sin

8 r

eIkjE

jkro

sin

8 r

eIkjH

jkro

22

½ value of fields compared to constant current case

/50 < l < /10

/50 < l < /10

[4-36]

SMALL DIPOLESMALL DIPOLE(CONT)(CONT)

23

For physical small dipole triangular current distribution

value of case of constant current

14

same as constant current case

/50 < l < /10

[4-37]

2

12

orad

IP

2220

rR

5.1oD

FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE

2' 21cos sin

2

zR r z

r

24

(length comparable to )

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

approx. error

[4.41]

Fig. 4.5 Finite dipole geometryand far-field approximations

Phase and Magnitude considerationsPhase and Magnitude considerations

25

In calculating phase assume can tolerate phase error of /8 (22°)

Must choose r far enough away so that ….

FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)

Phase and Magnitude considerationsPhase and Magnitude considerations

for magnitude term use

for phase term us

1

- coe sjkr

R rr

e R r z

2 22 2 2' max 2 8 2 8 8

k zz r

r r

26

ORIGIN OF DEFINITION

OF FAR FIELD

[4-45]


Finite dipole Current distributionFinite dipole Current distribution

' 'ˆ sin 02 2' ' '( 0, 0, )

' 'ˆ sin 02 2

I k z zz o

x y ze

I k z zz o

a

I

a

27

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

/ 2 < l <

[4-56]

(see Fig. 4.8)


Current distribution for linear wire antennaCurrent distribution for linear wire antenna

28

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

''

sin4

)(zd

R

ezkjEd

jkRe

I

2'22 zzyxR

29

For infinitesimal dipole at z’ of length 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))

'cos'

'

sin4

)(zde

r

ezkjEd jkz

jkre

I

30

In far field region in phase term

cos'zrR ( let )

[4-58]


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

2/

2/

EdE

'cos2/

2/

' '

)(sin4

zdezIr

ekjE jkz

e

jkr

31

Total Field

[4-58a]


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

sin2

coscos2

cos

2

kk

r

eIjE

jkro

E

H

32

For sinusoidal current distribution

[4-62]


Power DensityPower Density

2

2

2 2

cos cos cos2 2

8 sino

r avg

k kI

Wr

33

[4-63]


2

2

22

sin2

coscos2

cos

8

kkI

WrU oavg

34

Radiation IntensityRadiation Intensity

[4-64]


l ≥ /2

3-dB BEAMWIDTH3-dB BEAMWIDTH

35

3-dB

BE

AM

WID

TH

90° 87°

78°

64°48°

.25 0.5 0.75 1


3-dB BEAMWIDTH3-dB BEAMWIDTH

36

If allow new lobes begin to appear

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


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

37

Fig. 4.6

Radiated power Radiated power

s

avgrad dP s

W

38

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


Radiated power Radiated power

39

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 thin dipole with sinusoidal current Radiation resistance, input resistance and directivity of a thin dipole with sinusoidal current distributiondistribution

40

Fig. 4.9

FINITE LENGTH DIPOLE

Input ResistanceInput Resistance

2

oin II

41

(note that Rr uses Imax in its derivation)

for

at input terminalsI

VZ in

z’

Ie (z’)

maxIIo


Input ResistanceInput Resistance

42

So, even for lossless antenna ( RL = 0 )

[4-77a]

rin

oin R

I

IR

2

inr RR

2sin 2 k

Rr

z’

Ie (z’)

maxIIo


43

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 ; andn inR0inI


Empirical formula for Empirical formula for RinRin

)200max(

inR )12max(

inR

44

40

40

G220GRin

17.414.11 GRin

5.27.24 GRin 24

64.0

2

24

G

22

G

)76max(

inR

let2

kG for dipole of length

G

[4-107] [4-110]


For MONOPOLEFor MONOPOLE

5.42732

1jZ in

kG

2.215.36 jZ in

45

2

1Rin (monopole) Rin (dipole)

for wavelength monopole1

4

same current; voltage impedance2

1

2

1

[4-106]

HALF WAVE DIPOLEHALF WAVE DIPOLE

sin

cos2

cos

2 r

eIjE

jkro

2

22

2

sin

cos2

cos

8

r

IW o

avg

sin

cos2

cos

2 r

eIjH

jkro

dI

P orad

0

22

sin

cos2

cos

4

46

ll = = /2/2

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(deg)

No

rma

lize

d P

ow

er

2sin

3sin

[4-84]

[4-88]

[4-86]

[4-85]

64.14 max rad

o P

UD

47

ll = = /2/2

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

Slightly moredirective thaninf. dipole with

Do = 1.5

2

(2 ) where (2 ) 2.4358

IoP C C

rad in in

[4-89]

[4-91]

max

2 (2 ) 732 4

since if lossless

42.5 73 42.5

r

in r in

in in

PradR C

inIo

R R I I

X Z j

48

l l = = /2/2

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

[4-93]

PRACTICAL DIPOLEPRACTICAL DIPOLE

300

Useful for matching to wire lines where

300

in

o

R

Z

49

Folded dipole

l l slightly < slightly < /2/2

2

Usually choose slightly less than 2

so that 0 & Z is totally realXin in

PRACTICAL DIPOLEPRACTICAL DIPOLE(CONT)(CONT)

50

Resistance and Reactance Variations

2

(pure real for length slightly less than )

l l slightly < slightly < /2/2

0.5 1.0

G , B

G

B

IMAGE THEORYIMAGE THEORY

51

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

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

h1

Direct

Reflected

h2

IMAGE THEORYIMAGE THEORY(CONT)(CONT)

52

In the Lower Half Plane, E = H = 0

h

h

h

Image

Actual Problem Equivalent Problem

Observation Point

Observation Point

IMAGE THEORY IMAGE THEORY (CONT)(CONT)

53

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

actual

image

I

I

image

actual

I

Iactual

image

I

I

54

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

55


Electric dipoles above ground planeElectric dipoles above ground plane

Fig. 4.14(a) Fig. 4.25(a)

56

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)

57

FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS

r1

h

h

r

r2

h co

s

x

y

z

h

h

r1

r

r2

x

y

z


approx. in phase terms

cos1 hrr cos2 hrr

in magnitude terms321 rrr [4-97]

[4-98]

rd EEE21

rd EEE21

58

Summing two contributions

total = incident + reflected total = actual + imaginary


11

sin4

1

r

eIkjE

jkrod

22

sin4

2

r

eIkjE

jkror

sin

4 1

1

r

eIkjE

jkrod

sin

4 2

2

r

eIkjE

jkror

[4-94]

[4-95]

[4-111]

[4-112]

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

coscossin

4jkhjkh

jkro ee

r

eIkjE

59

VERTICAL DIPOLEVERTICAL DIPOLEHORIZONTAL DIPOLEHORIZONTAL DIPOLE

coscossin

4jkhjkh

jkro ee

r

eIkjE

22 sinsin1sin

sinsincos


sin 2 cos cos4

jkrok I e

E j khr

60

[4-99]

[4-116]


Single source at origin array factor

Single source at originarray factor

cossin2sinsin1

422 khj

r

eIkjE

jkro

for 0E 0z


12

h

4

h

2

h

61

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

h2

[4-100] [4-117]

62

Amplitude patterns at different heightsAmplitude patterns at different heights

(CONT)(CONT)


Note max radiation is in = 90° direction

Fig. 4.16Fig. 4.28

32

2

2

2sin

2

2cos

3

1

kh

kh

kh

khIP o

rad

32

2

2

2sin

2

2cos

2

2sin

3

1

kh

kh

kh

kh

kh

khIP o

rad

63

VERTICAL DIPOLEVERTICAL DIPOLE

HORIZONTAL DIPOLEHORIZONTAL DIPOLE

[4-102]

[4-118]

R(kh)

RADIATION POWERRADIATION POWER

32

max

2

2sin

2

2cos31

24

kh

kh

kh

khP

UD

rado

42R(kh)

4 hkh

rad

o P

UD max4

42R(kh)

sin4 2 hkh

kh

64

[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.

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

!5!3

sin53 xx

xx

!42

1cos42 xx

x

23456

11sin 2

23

x

xx

x

2342

11cos 2

22

x

xx

x

65

DIRECTIVITYDIRECTIVITY (CONT)(CONT)

VERTICAL DIPOLEVERTICAL DIPOLEHORIZONTAL DIPOLEHORIZONTAL DIPOLE

Limiting case of kh

Note:

32

sincos

3

1

x

x

x

x

3

2

6

4

6

1

2

1

3

1

23456

11

2342

11

3

1 2

2

2

2

x

x

x

x

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

Dg

(=0)

h/

kh h/ Do

0 0 3

2.88 .458 6.57

6.0

h/ Do

0 7.5

.615+n/2(n=1,2,3…)

slightly

6.0

6.0

66


6

312

lim

oDkh

3

322

0lim

oDkh

2

0lim

)(

sin5.7

kh

khoD

kh

[4-124]

DIRECTIVITYDIRECTIVITY (CONT)(CONT)

in in inZ R jX

5.4273 jZ in

67

VERTICAL DIPOLEVERTICAL DIPOLE

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

Fig. 4.20

ininin XRZ

5.4273 jZ in

68

HORIZONTAL DIPOLEHORIZONTAL DIPOLE

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

Fig. 4.30

GROUND EFFECTSGROUND EFFECTS

69

Finite conductivity earth

(“real” earth as ground plane)

h1

h2

Direct

Reflected

earth

Assume earth flat (ok. for Rearth )

10 1 [S/m]

GROUND EFFECTSGROUND EFFECTS (CONT)(CONT)

70

(real earth as ground plane)

Fig. 4.31 Elevation plane amplitude patterns of an infinitesimal vertical dipole above a perfect

electric conductor = and a flat earth = 0.01 [S/m]


Fig. 4.32 Elevation plane ( = 90°)amplitude patterns of an infinitesimal horizontal dipole above a perfect

electric conductor = and a flat earth = 0.01 [S/m]

GROUND EFFECTS GROUND EFFECTS (CONT)(CONT)

71

(real earth as ground plane)

For low and medium frequency applications when height is comparable to skin depth [ ]

of the ground increasing changes in input impedance; less efficient; use of ground wires)


72

Usually negligible effect for observation angle greater than 3°.

EARTH CURVATUREEARTH CURVATURE

Fig. 4.34 Geometry for reflections from a spherical surface

73

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 .

reflected field from spherical surface

reflected field from flat surface

DDivergence factor

rf

rs

E

E


74

l=/50 l=/10 l=/2 l=

Rhf 0.0279 0.2792 0.698 1.3962

RL 0.0279 0.1396 0.349 0.6981

Rr 0.3158 1.9739 73 199

Rin 0.3158 1.9739 73

ecd 0.9188 (-0.368 dB)

0.9339 (-0.296 dB)

0.9952 (-0.021 dB)

0.9965 (-0.015 dB)

D0 1.5 (1.761 dB)

1.5 (1.761 dB)

1.6409 (2.151 dB)

2.411 (3.822 dB)

G0 1.3782 (1.393 dB)

1.4009 (1.464 dB)

1.6331(2.13 dB)

2.4026 (3.807 dB)

-0.9863 -0.9189 0.18929 1

er 0.0271 (-15.67 dB)

0.1556 (-8.08 dB)

0.9642 (-0.158 dB)

0 (- Db)

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

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

chapter 4 linear wire antennas 1 ece 5318/6352 antenna engineering dr. stuart long

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