1. Chapter 4 Linear Wire Antennas ECE 5318/6352 Antenna Engineering
Dr. Stuart Long

2. INFINITESIMAL DIPOLE ; thin wire ; Io z  l  /50 (constant current)
Impinging
Wave z
[4-1]
(constant current)
(only electrical current present)

3. INFINITESIMAL DIPOLE (CONT)
Fig. 4.1(a) Geometrical arrangement
of an infinitesimal dipole

4. mixed coordinates in expression - change to spherical
INFINITESIMAL DIPOLE (CONT)
l  /50
(x,y,z)
(x’,y’,z’)
source
points l
mixed coordinates in expression - change to spherical
for
[4-2]

5. mixed coordinates in expression change to spherical
INFINITESIMAL DIPOLE (CONT)
l  /50
mixed coordinates in expression change to spherical
(x,y,z)
(x’,y’,z’)
source
points l
[4-4]

6. mixed coordinates in expression need to change to spherical
INFINITESIMAL DIPOLE (CONT)
l  /50
mixed coordinates in expression need to change to spherical
(x,y,z)
(x’,y’,z’)
source
points l
along source
[4-6]

7. INFINITESIMAL DIPOLE (CONT)
Using Vector Potential A ,
calculate H & E fields
[4-7]

8. INFINITESIMAL DIPOLE (CONT)
Using Vector Potential A ,
calculate H fields
[4-8] 

9. INFINITESIMAL DIPOLE (CONT)
Using Maxwell’s Eqns to
calculate E fields 
Fig. 4.1(b) Geometrical arrangement
of an infinitesimal dipole and its
associated electric-field components
on a spherical surface
[4-10]

10. Using H, Er, E, calculate the complex Poynting vector
INFINITESIMAL DIPOLE (CONT)
l  /50
Using H, Er, E, calculate the complex Poynting vector
[4-12]

11. INFINITESIMAL DIPOLE (CONT)
Find total outward flux through a closed sphere
(only contributions from Wr)
[4-14]

12. INFINITESIMAL DIPOLE (CONT)
Find total outward flux through a closed sphere
Real P = total radiated power Prad
[4-16]
[4-19]
(Impedance would also have a large capacitive term that is not calculated here.)

13. INFINITESIMAL DIPOLE (CONT)
Imaginary part of P = reactive power in the radial direction
[4-17]
(Note: this  0 as kr  , so it is
essentially not present in far field;
only important in near and intermediate field considerations)

14. Near Field approximations [ kr  1 ]
INFINITESIMAL DIPOLE (CONT)
l  /50
Near Field approximations [ kr  1 ]
(field point very close or very low frequency case) 
Dominant terms
Like ‘quasistationary” fields
E near static electric dipole
H near static current element
[4-20]

15. Near Field approximations [ kr  1 ]
INFINITESIMAL DIPOLE (CONT)
l  /50
Near Field approximations [ kr  1 ]
Biot – Savart Law :
infinitesimal current
element in direction az 
(same as above when kr 0)
[4-21]
(note E and H are 90° out of phase)
[4-22]
NO RADIAL POWER FLOW --
REACTIVE FIELDS

16. Intermediate Fields [ kr > 1]
INFINITESIMAL DIPOLE (CONT)
l  /50
Intermediate Fields [ kr > 1]
(induction zone; still have radial fields)
Er a 1/r
Eq a 1/r
Hf a 1/r

17. INFINITESIMAL DIPOLE (CONT)
Intermediate
Region
Induction Zone
r = l/2p (Radian Distance)
(Radius of Radian Sphere)
Energy
basically
Real
(radiating)
Far field
Energy
basically
imaginary
(stored)
Near field
Fig Radiated field terms magnitude
variation versus radial distance

18. Far Field [ kr >> 1 ]
INFINITESIMAL DIPOLE (CONT)
l  /50
Far Field [ kr >> 1 ]
Dominant terms 
[4-26]

19. Far Field [ kr >> 1 ]
INFINITESIMAL DIPOLE (CONT)
Far Field [ kr >> 1 ]
( both E and H are TEM to )
[4-27]

20. INFINITESIMAL DIPOLE (CONT)
Directivity
(use Far Field approx.)
[4-28]
[4-29]
RADIATION INTENSITY

21. INFINITESIMAL DIPOLE (CONT)
Directivity
[4-31]

22. SMALL DIPOLE /50 < l < /10
Uniform current assumption - only valid for ideal case
( approximated by capacitor plate antenna)
½ value of fields compared
to constant current case
[4-36]

23. SMALL DIPOLE (CONT) For physical small dipole
triangular current distribution
value of case of
constant current 1 _ 4
[4-37]
same as constant
current case

24. (max error where  = 90° ; 4th term = 0 there)
FINITE LENGTH DIPOLE
(length comparable to )
approx.
error
[4.41]
(max error where  = 90° ; 4th term = 0 there)
Fig. 4.5 Finite dipole geometry
and far-field approximations

25. Phase and Magnitude considerations
FINITE LENGTH DIPOLE (CONT)
Phase and Magnitude considerations
In calculating phase assume
can tolerate phase error of /8 (22°)
Must choose r far enough away so that ….

26. Phase and Magnitude considerations
FINITE LENGTH DIPOLE (CONT)
Phase and Magnitude considerations
[4-45]
ORIGIN OF
DEFINITION
OF FAR FIELD

27. Finite dipole Current distribution
FINITE LENGTH DIPOLE (CONT)
 / 2 < l < 
Finite dipole Current distribution
(“thin” wire, center fed, zero current at end points)
[4-56]
(see Fig. 4.8)

28. Current distribution for linear wire antenna
DIPOLE
Fig Current distribution along
the length of a linear wire antenna

29. Radiated fields at (x, y, z) of finite dipole
FINITE LENGTH DIPOLE (CONT)
Radiated fields at (x, y, z) of finite dipole
For infinitesimal dipole at z’ of length  z’
Since source is only
along the z axis
( ) 

30. Radiated fields of finite dipole at (x, y, z)
FINITE LENGTH DIPOLE (CONT)
Radiated fields of finite dipole at (x, y, z)
In far field region
in phase term 
( let )
[4-58]

31. Far Field E & H Radiating fields
FINITE LENGTH DIPOLE (CONT)
Far Field E & H Radiating fields
Total Field
[4-58a]

32. Far Field E & H Radiating fields
FINITE LENGTH DIPOLE (CONT)
Far Field E & H Radiating fields
For sinusoidal current distribution
[4-62]

33. FINITE LENGTH DIPOLE (CONT)
Power Density
[4-63]

34. FINITE LENGTH DIPOLE (CONT)
Radiation Intensity
[4-64]

35. FINITE LENGTH DIPOLE (CONT)
3-dB BEAMWIDTH
90°
87°
78°
64°
48°
3-dB BEAMWIDTH

36. FINITE LENGTH DIPOLE (CONT)
3-dB BEAMWIDTH
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

37. Elevation plane amplitude patterns for a thin dipole with sinusoidal current distribution
Fig. 4.6

38. FINITE LENGTH DIPOLE (CONT)
Radiated power
[4.66]
Results of integration give terms involving Ci & Si
[4-68]

39. FINITE LENGTH DIPOLE (CONT)
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
[4-70]
[4-75]
Do, Rr, Rin plotted in fig. 4.9

40. FINITE LENGTH DIPOLE
Radiation resistance, input resistance and directivity of a thin dipole with sinusoidal current distribution
Fig. 4.9

41. FINITE LENGTH DIPOLE (CONT)
Input Resistance
(note that Rr uses Imax in its derivation) z’
Ie (z’)
at input
terminals
for

42. FINITE LENGTH DIPOLE (CONT)
Input Resistance z’
Ie (z’)
So, even for lossless antenna ( RL = 0 ) 
[4-77a]

43. FINITE LENGTH DIPOLE (CONT)
Input Resistance (cont)
Note: when ; and
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)

44. Empirical formula for Rin
FINITE LENGTH DIPOLE (CONT)
Empirical formula for Rin
let
for dipole of length 
[4-107] [4-110]

45. For MONOPOLE for wavelength monopole Rin (monopole) = Rin (dipole)
[4-106]
same current; voltage  impedance

46. l = /2
HALF WAVE DIPOLE
[4-84]
[4-85]
[4-86]
[4-88]

47. HALF WAVE DIPOLE (CONT)
[4-89]
[4-91]
Slightly more
directive than
inf. dipole with
Do = 1.5

48. HALF WAVE DIPOLE (CONT)
[4-93]

49. l slightly < /2
PRACTICAL DIPOLE
Folded dipole

50. PRACTICAL DIPOLE (CONT)
l slightly < /2
PRACTICAL DIPOLE (CONT)
Resistance and Reactance Variations
(pure real for length slightly less than )
0.5
1.0
G , B G B

51. IMAGE THEORY Linear antennas near an infinite ground plane
Direct
Reflected h2
Linear antennas near an infinite ground plane
could approximate case of earth.
Can calculate the fields in the UHP by removing the conductor
and finding the field due to the actual and image sources.

52. IMAGE THEORY (CONT) In the Lower Half Plane, E = H = 0 Image h mo, eo
s = 
Actual Problem
Equivalent Problem
Observation
Point
In the Lower Half Plane, E = H = 0 

53. IMAGE THEORY (CONT) Fields due to image source are actually produced
by the induced currents in the ground plane      
actual
image I
image
actual I
actual
image I 

54. Electric dipoles above an infinite perfect electric conductor
HORIZONTAL DIPOLE
VERTICAL DIPOLE
Fig. 4.12(a) Vertical electric dipole above an
Infinite, flat, perfect electric conductor
Fig Horizontal electric dipole, and its associated
image, above an infinite, flat, perfect electric conductor

55. Electric dipoles above ground plane
VERTICAL DIPOLE
HORIZONTAL DIPOLE
Fig. 4.25(a)
Fig. 4.14(a)

56. Electric dipoles above an infinite perfect electric conductor
Far Field
Fig. 4.14(b)
Fig. 4.25(b)

57. FAR FIELD RADIATING FIELDS
VERTICAL DIPOLE
HORIZONTAL DIPOLE r1 h r r2 
h cos  x y z h r1 r r2 x y z 
approx. in
phase terms
[4-97]
in magnitude terms
[4-98]

58. FAR FIELD RADIATING FIELDS (CONT)
VERTICAL DIPOLE
HORIZONTAL DIPOLE
Summing two contributions
total = incident + reflected
total = actual + imaginary
[4-111]
[4-94]
[4-95]
[4-112]

59. FAR FIELD RADIATING FIELDS (CONT)
VERTICAL DIPOLE
HORIZONTAL DIPOLE

60. FAR FIELD RADIATING FIELDS (CONT)
VERTICAL DIPOLE
HORIZONTAL DIPOLE
Single source at origin
array factor
Single source at origin
array factor
[4-116]
[4-99]
for

61. Amplitude patterns at different heights
HORIZONTAL DIPOLE
VERTICAL DIPOLE
Fig. 4.15
Fig. 4.26
Note minor lobes that are
formed for
Note minor lobes that are
formed for
Number of lobes
Number of lobes
[4-100]
[4-117]

62. Amplitude patterns at different heights
(CONT)
HORIZONTAL DIPOLE
VERTICAL DIPOLE
Fig. 4.16
Fig. 4.28
Note max radiation is in  = 90° direction

63. RADIATION POWER VERTICAL DIPOLE HORIZONTAL DIPOLE [4-102] [4-118]
R(kh)

64. DIRECTIVITY VERTICAL DIPOLE HORIZONTAL DIPOLE [4-104]
Fig Radiation resistance and max. directivity
of a horizontal infinitesimal electric dipole as a
function of its height above an infinite perfect
electric conductor.
Fig Directivity and radiation resistance
Of a vertical infinitesimal electric dipole as a
function of its height above an infinite perfect
electric conductor.
[4-123]

65. DIRECTIVITY (CONT) VERTICAL DIPOLE HORIZONTAL DIPOLE
Limiting case of kh 0
Note: direction of maximum radiation
changes as “h” is varied. Dg (=0)
Note: Dg
(=0)
h/

66. DIRECTIVITY (CONT) VERTICAL DIPOLE HORIZONTAL DIPOLE h/ Do 7.5
[4-124]
h/ Do
7.5
.615+n/2
(n=1,2,3…)
slightly
 6.0 
6.0 kh
h/ Do 3
2.88
.458
6.57 
6.0

67. Input Impedance of a /2 dipole above a
flat lossy electric conductive surface
VERTICAL DIPOLE
Fig. 4.20

68. Input Impedance of a /2 dipole above a
flat lossy electric conductive surface
HORIZONTAL DIPOLE
Fig. 4.30

69. GROUND EFFECTS (“real” earth as ground plane) earth
Finite conductivity earth
10  1 [S/m] h1 h2
Direct
Reflected
earth
Assume earth flat (ok. for Rearth  )

70. GROUND EFFECTS (CONT) (real earth as ground plane) VERTICAL DIPOLE
HORIZONTAL DIPOLE
Fig Elevation plane amplitude patterns
of an infinitesimal vertical dipole above a perfect
electric conductor = and a flat earth = 0.01 [S/m]
Fig Elevation plane ( = 90°)amplitude patterns
of an infinitesimal horizontal dipole above a perfect
electric conductor = and a flat earth = 0.01 [S/m]

71. GROUND EFFECTS (CONT) (real earth as ground plane)
For low and medium frequency applications when
height is comparable to skin depth [  = 2/ ]
of the ground  increasing changes in input
impedance; less efficient; use of ground wires)

72. GROUND EFFECTS (CONT) Usually negligible effect
EARTH CURVATURE
Usually negligible effect
for observation angle 
greater than 3°.
Fig Geometry for reflections from a spherical surface

73. GROUND EFFECTS (CONT)
EARTH CURVATURE
Curved surfaces spreads out radiation (divergent) that is
reflected more than from flat surface.
(can introduce a divergence factor)
Divergence factor =
reflected field from spherical surface
reflected field from flat surface
___________________ =
Fig Divergence factor for a 4/3 radius earth
(ae = 5,280 mi = 8,497.3 km) as a function of
grazing angle .

74. DIPOLE SUMMARY l=/50 l=/10 l=/2 l= Rhf 0.0279 0.2792 0.698 1.3962
(Resonant  XA=0; f = 100 MHz; s = 5.7 x 107 S/m; Zc = 50; b = 3x10-4l)
l=/50
l=/10
l=/2
l=
Rhf
0.0279
0.2792
0.698
1.3962 RL
0.1396
0.349
0.6981 Rr
0.3158
1.9739 73
199
Rin 
ecd
0.9188
( dB)
0.9339
( dB)
0.9952
( dB)
0.9965
( dB) D0
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) G 1 er
0.0271
( dB)
0.1556
(-8.08 dB)
0.9642
( dB)
(- Db)
G0abs
( dB)
( dB)
(1.972 dB)
0 (- dB)