CHAPTER4 Linear Wire Antennas

 CHAPTER4 Linear Wire Antennas 4.1 INTRODUCTION ............................................................................................................................................................................................................................... 2 4.2 INFINITESIMAL DIPOLE ................................................................................................................................................................................................................... 2 4.2.1 Radiated Fields ..................................................................................................................................................................................................................... 3 4.2.2 Power Density and Radiation Resistance ............................................................................................................................................................................ 7 4.2.3 Near‐Field (
) Region .............................................................................................................................................................................................. 13 4.2.5 Intermediate‐Field (kr > 1) Region ..................................................................................................................................................................................... 15 4.2.6 Far‐Field (kr >> 1) Region ................................................................................................................................................................................................... 17 4.2.7 Directivity ........................................................................................................................................................................................................................... 19 4.3 SMALL DIPOLE ....................................................................................................................................................................................................................... 21 4.4 REGION SEPARATION .............................................................................................................................................................................................................. 25 4.4.1 Far‐Field (Fraunhofer) Region ............................................................................................................................................................................................ 27 4.4.2 Radiating Near‐Field (Fresnel) Region ............................................................................................................................................................................... 30 4.4.3 Reactive Near‐Field Region ................................................................................................................................................................................................ 32 4.5 FINITE LENGTH DIPOLE ................................................................................................................................................................................................................. 33 4.5.1 Current Distribution ........................................................................................................................................................................................................... 33 4.5.2 Radiated Fields: Element Factor, Space Factor, and Pattern Multiplication ..................................................................................................................... 35 4.5.3 Power Density, Radiation Intensity, and Radiation Resistance ......................................................................................................................................... 37 4.5.4 Directivity ........................................................................................................................................................................................................................... 41 4.5.5 Input Resistance ................................................................................................................................................................................................................. 42 4.6 HALF‐WAVELENGTH DIPOLE ......................................................................................................................................................................................................... 45 4.7 LINEAR ELEMENTS NEAR OR ON INFINITE PERFECT CONDUCTORS ............................................................................................................................................. 49 4.7.1 Image Theory ..................................................................................................................................................................................................................... 50 4.7.2 Vertical Electric Dipole ....................................................................................................................................................................................................... 53 1. Radiation pattern ................................................................................................................................................................................................................ 54 2. Radiation power and directivity .......................................................................................................................................................................................... 57 3. monopole ............................................................................................................................................................................................................................ 61 4.7.4 Antennas for Mobile Communication Systems ................................................................................................................................................................. 63 4.7.5 Horizontal Electric Dipole .................................................................................................................................................................................................. 67 PROBLEMS .......................................................................................................................................................................................................................................... 74 4.1
1 INTROD
DUCTION
Wire antennas,
a
, linear or curved, are somee of the o
oldest, sim
mplest, cheapest, an
nd the mo
ost versatiile for many applicaations. 4.2
2 INFINITESIMAL D
DIPOLE  Infinitesimal dipolles are no
ot practicaal, they arre used to
o represen
nt capacittor‐plate antennass. on, they aare utilized as build
ding moree complexx geometrries.  In additio
The end pllates are
e used to provide ccapacitivee loading to maintaain the current on the dip
pole neaarly uniform. The plates are very small, their radiation is usually negligible. The wire, in ). The spatial variation of addition to being very small (l <<), is very thin ( the current is assumed to be constant ′
; = constant (4‐1) 4.2.1RadiatedFields
To find the fields radiated by the current element, it will be required to determine first and and then find the and . 1. Calculation of Since the source only carries an electric current , therefore and the potential function are zero. To find we write ,
, ′
′ (4‐2) x, y, z : the observation point ; x’, y’, z’ : the source coordinates : the distance from any point on the source to the observation point path C : is along the length of the source Fo
or the problem of FFigure 4.1
,
,
4‐3
0 (infiniteesimal dip
pole) ′ so
o we can w
write (4‐2) as , , /
/
(4‐4) 2. Calculation of and To calculate and spherical components. , it is simpler to transform (4‐4) from rectangular to For this problem, 0
(4‐5) 0, so (4‐5) using (4‐4) reduces to (4‐6) 0
⟹
(4‐7) Substituting (4‐6) into (4‐7) reduces it to 0
1
(4‐8)
Th
he electricc field E caan now bee found. TThat is, ∙
(4‐9) (4‐10) 1
1
0
The and ‐field components aree valid evverywherre, exceptt on the so
ource itself, and th
hey are sketched s
in Figure 4.1(b) 4
on the surfaace of a sp
phere of raadius . 4.2.2PowerDensityandRadiationResistance
The input impedance of an antenna consists of real and imaginary parts. For a lossless antenna, the real part of the input impedance was radiation resistance. To find the input resistance for a lossless antenna, the following procedure is taken. For the infinitesimal dipole, the complex Poynting vector can be written using (4‐8a)–(4‐8b) and (4‐10a)–(4‐10c) as 1
2
∗
∗
1
2
∗
∗
(4‐11) ⟹
1
|
|
(4‐12) 1
Since  is imaginary, it will not contribute to real radiated power. The reactive power density, which is most dominant for small values of , has both radial and transverse components. It merely changes between outward and inward directions to form a standing wave at a rate of twice per cycle. It also moves in the transverse direction. The complex power moving in the radial direction is obtained by integrating (4‐11)–(4‐12b) over a closed sphere of radius r. Thus it can be written as ∯
⟹
∙
∙
1
4‐13 (4‐14) Equation (4‐13), which gives the real and imaginary power that is moving outwardly, can also be written as ∗
∙
1
P
j2ω W
W (4‐15) Where: P power inradialdirection ; Prad time‐averagepowerradiated
W
time‐averagemagneticenergydensity inradialdirection W
time‐averageelectricenergydensity inradialdirection W
2 W
time‐averageimaginary reactive power From (4‐14) P
; 2ω W
It is clear from (4‐17) that When kr
andvanishes.
W
(4‐16, 17) ∞, the reactive power diminishes
1. radiation resistance of the infinitesimal dipole Since the antenna radiates its real power through the radiation resistance, for the infinitesimal dipole it is found by equating (4‐16) to | |
⇒
80
(4‐18, 19) For a wire antenna to be classified as an infinitesimal dipole, its overall length must be very small (usually 
). Example 4.1 Find the radiation resistance of an infinitesimal dipole whose overall length is /50. Solution: Using (4‐19) 1
80
80
0.316
50
Since the radiation resistance of an infinitesimal dipole is about 0.3 ohms, it will present a very large mismatch when connected to practical transmission lines, many of which have characteristic impedances of 50 or 75 ohms. The reflection efficiency ( ) and hence the overall efficiency ( ) will be very small. 2. The reactance of an infinitesimal dipole is capacitive. This can be illustrated by considering the dipole as a flared open‐circuited transmission line. Since the input impedance of an open‐circuited transmission line a distance from its open end is given by where ≪ . 2
is its characteristic impedance, it will always be negative (capacitive) for 4.2.3Near‐Field(
)Region
An inspection of (4‐8) and (4‐10) reveals that for can be approximated by  (4‐8a,10c)  (4‐20c)
(4‐8b)   The E‐field components, /2 they (4‐20d) (4‐10a)  (4‐20a) (4‐10b)  (4‐20b) and are in time‐phase;  They are in time‐phase quadrature with the H‐field component  ;  Therefore there is no time‐average power flow associated with them. This is demonstrated by forming the time‐average power density as W
Re E
H∗
Re
∗
∗
⟹W
Re
|
|
0 (4‐22) Equations (4‐20a) and (4‐20b) are similar to those of a static electric dipole and (4‐20d) to that of a static current element. Thus we usually refer to (4‐20a)–(4‐20d) as the quasi‐stationary fields. 4.2.5Intermediate‐Field(kr>1)Region
As the values of begin to increase and become greater than unity, the terms that were dominant for ≪ 1become smaller and eventually vanish. 1
(4‐8b)  (4‐23d) 1
(4‐10a)  (4‐23a) 1
(4‐10b)
For moderate values of (4‐23b) :  The E‐field components lose their in‐phase condition and approach time‐phase quadrature.  Their magnitude is not the same, they form a rotating vector whose extremity traces an ellipse. This is analogous to the polarization problem except that the vector rotates in a plane parallel to the direction of propagation and is usually referred to as the cross field.  At these intermediate values of , the and components approach time‐phase, which is an indication of the formation of time‐average power flow in the outward direction. (4‐8a, 10c) 
(4‐23c)
(4‐8b) 
(4‐23d)
(4‐10a) 
(4‐23a)
(4‐10b)
(4‐23b)
The total electric field is given by (4‐24) 4.2.6Far‐Field(kr>>1)Region
In a region where ≫ 1 , (4‐23a) – (4‐23d) can be simplified and approximated by (4‐8a, 10c)  The ratio of to (4‐26b) (4‐8b)  (4‐26c) (4‐10a)  (4‐26b) (4‐10b)
(4‐26a) is equal to Z
(4‐27) The E‐ and H‐ field components are perpendicular to each other, transverse to the radial direction of propagation. The fields form a Transverse ElectroMagnetic (TEM) wave，its wave impedance is the intrinsic impedance of the medium. Example 4.2 For an infinitesimal dipole determine and interpret the vector effective length. At what incidence angle does the open‐circuit maximum voltage occurs at the output terminals of the dipole if the electric‐field intensity of the incident wave is 10mV/m? The length of the dipole is 10cm. Solution: Using (4‐26a) and the effective length as defined by (2‐92), we can write that 4
The maximum value occurs at 
maximum voltage is equal to |
∙
|
10
2
26a
92
⟹ 90 and it is equal to . The open‐circuit 10
∙
|
10 volts 4.2.7Directivity
The real power P radiated by the dipole was found in Section 4.2.2, as given by (4‐16). The same expression can be obtained by first forming the average power density, using (4‐26a)–(4‐26c). That is, Re
∗
|
|
(4‐28) Integrating (4‐28) over a closed sphere of radius r reduces it to (4‐16). Associated with the average power density of (4‐28) is a radiation intensity U which is given by |
| ⟹ (4‐29, 30) Using (4‐16) and (4‐30), the directivity reduces to 4
(4‐31) and the maximum effective aperture to (4‐32) 4.3
3 SMALL DIPOLEE  The creation of th
he current distribution on aa thin wiree was disscussed in
n Section 1.4, and it was illu
ustrated w
with somee examplees in Figure 1.16.  The radiaation prop
perties off an infinittesimal diipole weree discusse
ed in the previous section. IIts curren
nt distribution was assumed to be con
nstant. A consttant curreent distrib
bution is n
not realizable. A beetter approximatio
on of the cu
urrent disttribution o
of wire an
ntennas, ((/50
/10) is the trriangular vvariation, wh
hich is sho
own in Figgure 4.4(b
b) ,
,
1
1
, 0
,
0
(4
4‐33)
Th
he vector potential can be w
written using (4‐33) as /
1
/
1
(4‐34) Becausse the len
ngth of th
he dipole is very sm
mall
/10 , for diffferent ’ alo
ong the w
wire are not much different from . TThus can be ap
c
pproximatted by throughoutt the integgration paath. The maximum phase error in (4
4‐34) by allowingg will be /2
/
18 for f
/10. Thiss amount of phase error hass very litttle effect /10
on
n the overrall radiation characteristics. Then, (4
4‐34) redu
uces to (4‐‐35) (4‐35) which is one‐half of that for the infinitesimal dipole. Ref: /
/
, , (4‐4) The potential function (4‐35) becomes a more accurate approximation as kr → ∞. Since the potential function for the triangular distribution is one‐half of the corresponding one for the constant (uniform) current distribution, the corresponding fields of the former are one‐half of the latter. Thus we can write the E‐ and H‐fields radiated by a small dipole as (4‐26b)  (4‐36b) (4‐26a)  (4‐36a) (4‐26c)  (4‐36c) Since the directivity of an antenna is controlled by the relative shape of the field or power pattern, the directivity, and maximum effective area of this antenna are the same as the ones with the constant current distribution given by (4‐31) and (4‐32), respectively. Using the procedure established for the infinitesimal dipole, the radiation resistance for the small dipole is 80
(4‐18) | |
20
(4‐37) The small dipole its radiated power is of (4‐18). Thus the radiation resistance of the antenna is strongly dependent upon the current distribution. 4.4 REGION SEPARATION Before solving the fields radiated by a finite dipole of any length, it is desirable to discuss the separation of the space surrounding an antenna into three regions  The reactive near‐field  The radiating near‐field  The far‐field To solve for the fields efficiently, approximations can be made to simplify the formulation. The difficulties in obtaining closed form solutions that are valid everywhere for any practical antenna stem from the inability to perform the integration of ,
, ′
′ (4‐2, 38) where (4‐38a) In the calculations for infinitesimal dipole and small dipole. The major simplification of (4‐38) will be in the approximation of R. The Figgure show
ws a very tthin dipolle of finitee length ll symmetrically possitioned. Beecause thee wire is vvery thin ((x’ y’ 0), wee can writte (4‐38) aas (4‐39) wh
hich can b
be written
n as 2
2 ′
(4‐40
0)
Ussing the binomial eexpansion, we can w
write (4‐4
40) in a seeries as ⋯ (4‐41) wh
hose higher order tterms beccome lesss significan
nt provideed r >> z’.. 4.4.1Far‐Field(Fraunhofer)Region
The most convenient simplification of (4‐41) is to approximate it by ≃
′
(4‐42) To maintain the maximum phase error of an antenna equal to or less than /8 rad (22.5 ), the observation distance r must equal or be greater than 2 /. 2 / (4‐45) The usual simplification for the far‐field region is ≃
forphaseterms
≃ foramplitudeterms1/
,
Ref: , ′
′ (4‐38) For any other antenna whose maximum dimension is (4‐46) is valid provided r
(4‐46) , the approximation of 2D /λ (4‐47) For an aperture antenna the maximum dimension is taken to be its diagonal. It wou
uld seem that thee approxiimation o
of R in (4‐46) fo
or the am
mplitude is more sevvere than that fo
or the phaase. Exxample 4
4.3 ons are For an
n antenn
na with an overall lengtth 5, the observati
o
made at 60. Find thee errors in phase and amp
plitude ussing (4‐4
46). So
olution: For 
90 , , z’
2
2.5, and
d r
6
60, (4‐40
0) reduce
es to 2

With 60
2.5 2 ′
(4‐40) 60.052 ≃
forphaseterms
≃ foramplitudeterms1/
(4‐46) 60 r
Therefore the phase difference is ∆
∆
2
0.327
18.74
22.5 The difference of the inverse values of R is 1
1.44
1 1
1
60 60.052
which should always be a very small value in amplitude. 1
10
4.4.2RadiatingNear‐Field(Fresnel)Region
If the observation point is chosen to be smaller than 2 / , the maximum phase error by the approximation of (4‐46) is greater than /8 rad (22.5o). ≃
forphaseterms
≃ foramplitudeterms1/
(4‐46) If it is necessary to choose observation distances smaller than 2 / , another term (the third) in the series solution of (4‐41) must be retained to maintain a maximum phase error of /8 rad (22.5o). ⋯ (4‐41) Doing this, the infinite series of (4‐41) can be approximated by (4‐48) A value of greater than that of (4‐52a) will lead to an error less than /8 rad (22.5o). √
0.385
or √
0.62
/ (4‐52, 4‐52a) The region where the first three terms of (4‐41) are significant, and the omission of the fourth introduces a maximum phase error of /8 rad (22.5o), is defined by 2
/
0.62
/ (4‐53) This region is designated as radiating near field because  The radiating power density is greater than the reactive power density  The field pattern is a function of the radial distance r. This region is also called the Fresnel region because the field expressions in this region reduce to Fresnel integrals. 4.4.3ReactiveNear‐FieldRegion
If the distance of observation is smaller than the inner boundary of the Fresnel region, this region is usually designated as reactive near‐field with inner and outer boundaries defined by 0.62
/ > 0 (4‐54) In summary, the space surrounding an antenna is divided into three regions whose boundaries are determined by Reactive near‐field 0.62
/ > 0 (4‐55a) Radiating near‐field (fresnel) 2
/
0.62
/ (4‐55b) Far‐field (fraunhofer) 2
/
0.62
/ (4‐55c) 4.5 FINITE LENGTH DIPOLE The techniques developed previously can be used to analyze the radiation characteristics of a linear dipole of any length. To reduce the mathematical complexities, it will be assumed that the dipole has a negligible diameter. 4.5.1CurrentDistribution
For a very thin dipole (ideally zero diameter), the current distribution can be written, to a good approximation, as 0,
0,
,0
,
(4‐56) 0
This distribution assumes that the antenna is  center‐fed  the current vanishes at the end points.  Experiments have verified that the current in a center‐fed wire antenna has sinusoidal form with nulls at the end points. For /2 an
nd /2
 tthe current distrib
bution of f (4‐56) iss shown plo
otted in FFigures 1.1
16(b) and
d (c), respeectively. TThe geom
metry of th
he antenn
na is that shown in Figure 4.5.
4.5.2 Radiated Fields: Element Factor, Space Factor, and Pattern
Multiplication
Since closed form solutions, which are valid everywhere, cannot be obtained for many antennas, the observations will be restricted to the far‐field region. The finite dipole antenna is subdivided into a number of infinitesimal dipoles of length ’. For an infinitesimal dipole of length dz’ positioned along the z‐axis at z’, the electric and magnetic field components in the far field are given as , ,
(4‐26a)  (4‐26b)  ′ (4‐57a) 
(4‐26b)  (4‐57b) , ,
′ (4‐57c) where R is given by (4‐39) or (4‐40). Using the far‐field approximations given by (4‐46), (4‐57a) can be written as ,
,
′ (4‐58) Summing the contributions from all the infinitesimal elements to integration. Thus /
/
/
/
,
,
′ (4‐58a)  The factor outside the brackets is designated as the element factor  And that within the brackets as the space factor. For this antenna, the element factor is equal to the field of a unit length infinitesimal dipole located at a reference point. The total field of the antenna is equal to the product of the element and space factors. For the current distribution of (4‐56), (4‐58a) can be written as 4
/
/
⇒ ′ 2
′
′ (4‐60) (4‐62a) The total component can be written as (4‐62b) 4.5.3PowerDensity,RadiationIntensity,andRadiationResistance
For the dipole, the average Poynting vector can be written as ∗
∗
∗
|
| |
|
(4‐63) and the radiation intensity as | |
(4‐64) The normalized elevation power patterns, for /4, /2, 3/4, and  are shown in Figure 4.6. The current distribution of each is given by (4‐56). The power  is also included for patterns for an infinitesimal dipole ≪  
comparison. It is found that the 3‐dB beamwidth of each is equal to ≪ :3dBbeamwidth 90 0
0
-10
/4:3dBbeamwidth
87 -20
/2:3dBbeamwidth
78 -30
3/4:3dBbeamwidth
64 :3dBbeamwidth
47.8 As the length of the antenna increases, the beam becomes narrower. Because of that, the directivity should also increase with length. 30
330
300
60
-40 270
90
-30
-20
240
120
-10
0
210
150
180
1/50
3/4
1/4
1
1/2
As the dipole’s length increases beyond one wavelength  , the number of lobes begin to increase. The normalized power pattern for a dipole with 1.25 is shown in Figure 4.7.  Figure 4.7(a) is the
e three‐diimensionaal pattern
n  Figure 4.7(b) is thee two‐dim
mensional pattern
The cu
urrent diistribution
n for th
he dipolees with /4, /2, , 3/2,
/ and2, as given by (4
4‐56), is shown in Figure 4.8.
0
0
30
330
-1
10
-2
20
60
300
-3
30
90
-4
40 270
-3
30
-2
20
240
120
-1
10
0
150
210
Figure 4.8 Current disttributions ength of a liinear wire Figgure 4.7 Thrree‐ and two
o‐dimension
nal amplitud
de patterns ffor a thin along the le
antenna. dipole of l = 1.25
 and sinuso
oidal currentt distribution. 180
To find
d the total powerr radiated
d, the average Po
oynting ve
ector of (4‐63) is inttegrated o
over a sph
here of raadius r. Th
hus ∙
∯
∮
∙
| |
∮
(4‐66) After some extensive mathematical manipulations, it can be reduced to | |
4
1
2
2
2
/2
2
2
(4‐68) where C 0.5772 (Euler’s constant) and Ci x and Si x are the cosine and sine integrals given by ;
4
The radiation resistance can be obtained using (4‐18) and (4‐68) 1
2
2
2
| |
2
2
/2
2
2
68a, b (4‐70) 4.5.4Directivity
The directivity was defined mathematically by (2‐22), or ,
4
|
(4‐71) ,
where F ,  is related to the radiation intensity U by (2‐19), or ,
(4‐72) From (4‐64), the dipole antenna of length has F θ, ϕ
, B
F θ
η
| |
(4‐73,73b) Because the pattern is not a function of , (4‐71) reduces to |
,
(4‐74) The corresponding values of the maximum effective aperture are related to the directivity by (4‐76) 4.5.5Inpu
utResista
ance
The inp
put imped
dance was defined
d as“the ratio of tthe voltagge to currrent at a paair of term
minals orr the ratio of the appropriiate comp
ponents of o the eleectric to maagnetic fie
elds at a p
point.”
The reaal part of the inputt impedan
nce was deefined as the inputt resistancce which for a losslesss antenna reducess to the raadiation resistance. he radiatiion resisttance of a dipolee of lenggth l with Th
sin
nusoidal ccurrent distribution
n is expresssed by (4
4‐70). 2
| |
1
2
2
/2
2
2
2
2
(4‐70
0)
By the definition
n, the rad
diation resistance iis referred
d to the m
maximum
m current wh
hich for so
ome lengths (l = //4, 3/4, , etc.) do
oes not occur at th
he input terminals of the anten
nna. To refe
er the radiation ressistance to
o the inpu
ut terminals of med to bee lossless (RL = the antenna, the anttenna is ffirst assum
he power at the in
nput term
minals is equated e
to
o the 0). Then th
po
ower at th
he curren
nt maximu
um. Referrring to Figure 4.10
0, we can write |
|
| |
⟹
(4‐77)
here wh
R rad
diation reesistance aat input (ffeed) term
minals R = raadiation resistancee at curren
nt maximu
umEq. (4‐‐70) Figure 4.10 Current distribution, m
maximum does not occcur at the input term
minals. I = cu
urrent maximum I = cu
urrent at input term
minals For a d
dipole of length l, the curreent at thee input terminals (I ) is reelated to the current maximum
m (I ) refferring to Figure 4.10, by (4‐78) he input raadiation rresistancee of (4‐77aa) can be written aas Thus th
(4‐79) Radiation resistancce, input rresistancee and directivity of a thin dip
pole with Figgure 4.9 R
sinu
usoidal cu
urrent distribution.. 4.6 HALF‐WAVELENGTH DIPOLE One of the most commonly used antennas is the half‐wavelength (l = /2) dipole. Because  Its radiation resistance is 73 ohms very near the 50/75‐ohm characteristic impedances of some transmission lines,  Its matching to the line is simplified especially at resonance. The electric and magnetic field components of a half‐wavelength dipole can be obtained from (4‐62a) and (4‐62b) by letting l = /2. , (4‐84, 85) The time‐average power density and radiation intensity can be written, respectively, as | |
| |
| |
| |
(4‐86) (4‐86) Figure 4
4.6 and 4.11 show the two‐ and the tthree‐ dim
mensionall pattern. 0
0
30
330
-10
0
-20
0
60
300
-30
0
-40
-40
0 270
90
-30
0
-20
0
240
120
-10
0
0
150
210
180
Th
he total po
ower radiated can be obtain
ned as a special casse of (4‐67
7) | |
| |
| |
2
(4‐88) (4‐89) Byy (4‐69) 2
0.577
72
ln 2

2
2
0.5
5772
1.838
0.02 2
2.435 (4‐90) Using (4‐87), (4‐89) and (4‐90), the maximum directivity of the half‐wavelength dipole reduces to 4
4
|
/
1.643 (4‐91) .
The corresponding maximum effective area is equal to 1.643
0.13 (4‐92) and the radiation resistance, for a free‐space medium (
| |
2
30
2.435
120), is 73 (4‐93) The radiation resistance of (4‐93) is also the radiation resistance at the input terminals (input resistance) since the current maximum for a dipole of /2 occurs at the input terminals. As it will be shown later, the imaginary part associated with the input impedance of a dipole is a function of its length (for /2, it is equal to j42.5). Thus the total input impedance for /2 is equal to 73
42.5 (4‐93a) To reduce the imaginary part of the input impedance to zero, the antenna is matched or reduced in length until the reactance vanishes. The latter is most commonly used in practice for half‐wavelength dipoles.  Depending on the radius of the wire, the length of the dipole for first resonance is about 0.47to0.48; the thinner the wire, the closer the length is to 0.48.  For thicker wires, a larger segment of the wire has to be removed from /2 to achieve resonance. 4.7 LINEAR ELEMENTS NEAR OR ON INFINITE PERFECT CONDUCTORS The presence of obstacles, especially when it is near the radiating element, can significantly alter the overall radiation properties. The most common obstacle is the ground. Any energy from the radiating element directed toward the ground undergoes a reflection. The amount of reflected energy and its direction are controlled by the ground. The ground is a lossy medium ( 0) whose effective conductivity increases with frequency. Therefore it should be expected to act as a good conductor above a certain frequency, depending primarily upon its composition and moisture content. To simplify the analysis,  First assuming the ground is a perfect electric conductor, flat, and infinite.  The same procedure can also be used to investigate the characteristics of any radiating element near any other infinite, flat, perfect electric conductor. The effects that finite dimensions have on the radiation properties of a radiating element can be accounted for by the use of the Geometrical Theory of Diffraction and/or the Moment Method. 4.7.1Imag
geTheorry
To analyze the performaance of an
n antennaa near an
n infinite plane conductor, s, which virrtual sourrces (images) will be introd
duced to account for the reflection
r
wh
hen comb
bined with the real sources, form an
n equivaleent system
m. The eq
quivalent system give
es the same radiated field on and above a
the conducctor as th
he actual system itself. Below the condu
uctor, thee field is zero. (a) Vertical electric dip
pole (b
b) Field com
mponents at point of refflection Figure 4
4.12 Vertical electric dipole abo
ove an infin
nite, flat, p
perfect elecctric condu
uctor The amount of reflection is generally determined by the respective constitutive parameters of the media below and above the interface. For a perfect electric conductor below the interface, the incident wave is completely reflected and the field below the boundary is zero.  Vertical polarization The tangential components of the electric field must vanish on the interface. Thus for an incident electric field with vertical polarization, the polarization of the reflected waves must be as indicated in the figure. To excite the polarization of the reflected waves, the virtual source must also be vertical and with a polarity in the same direction as that of the actual source (thus a reflection coefficient of 1).  Horizontal polarization Another orientation of the source will be to have the radiating element in a horizontal position, the virtual source (image) is also placed a distance h below the interface but with a 180 polarity difference relative to the actual source (thus a reflection coefficient of 1). In addiition to electric e
so
ources, artificial equivalentt“magne
etic”sourrces and maagnetic co
onductorss have been introduced.  Figure 4.13(a) 4
displays th
he sourcees and their imagges for an a electric plane conducto
or. The direction d
of the arrow id
dentifies the polarity. Sincce many problemss can be ssolved usiing dualityy.  Figure 4..13(b) illu
ustrates th
he sourcees and their imagees when the t obstacle is an infinite, fflat, perfeect “magnetic” conducto
or. (a) EElectric con
nductor (b) Magnetic conductorr Figure 4
4.13 Electrric and maggnetic sources and th
heir images near elecctric (PEC) and magnetic (
m
PMC) cond
ductors. 4.7.2VertiicalElecttricDipo
ole
Assumiing a vertical electrric dipole is placed a distancce above an infinite, flat, peerfect elecctric cond
ductor as sshown in Figure 4.1
12(a).  For an ob
bservation point P1, there iss a dirrect wavee.  On the interface, the incid
dent wavee is comp
pletely refflected an
nd the field below o. The tan
ngential ccomponen
nts of thee electric field musst vanish the boundaary is zero
n the interrface. on
1. Radiationpattern (1) Direct component The far‐zone direct component of the electric field of the infinitesimal dipole of length , constant current , and observation point P is given according to (4‐26a) by (4‐94) (2) The reflected component The reflected component can be accounted for by the introduction of the virtual source (image), as shown in Figure 4.14(a), and it can be written as (4‐95, 4‐95a) (3) The total field The total field above the interface (z≥0) is equal to the sum of the direct and reflected components as given by (4‐94) and (4‐95a). In general, we can write that 2
/
, 2
/
(4‐96a, b) For far‐field ob
bservation
ns r ≫ h , (4‐96aa) and (4‐96b) reduce ussing the bin
nomial exxpansion tto , (4‐97a,b) (
2 cos
z
0 0
0
(4‐98) (4‐99) The shaape and aamplitudee of the field is nott only con
ntrolled b
by the field of the sin
ngle elem
ment but also a by th
he positio
oning of the t elemeent relativve to the ground. Th
he normalized pow
wer patteerns for 0, /8, /4, 3 /8, /2, and
a
have been plo
otted in FFigure 4.15
5.. 0
0
0
5
15
15
30
-10
0
30
45
45
-20
0
60
60
-30
0
-40
0
-50
0
5
75
h=0
h=1/8
8
h=1/4
4
75
h=3/8
h=1/2
h=1
9
90
-40
0
90
10
05
105
-30
0
120
-20
0
135
-10
0
0
120
135
150
150
180 165
65
180 16
For h λ/4 more minor lobes, in
n addition
n to the m
major one
es, are forrmed. As h attains values v
grreater than λ, an
n even greater nu
umber of minor lobes is inttroduced.. 0
0
15
30
-10
These are shown in Figure 4.16 for h 2λ and 5λ . In general, the total number of lobes is equal to the integer that is closest to numberoflobes
45
-20
60
-30
75
-40
h=2
h=5
-50
90
-40
105
-30
2
1 120
-20
135
-10
150
0
180 165
2. Radiationpoweranddirectivity
The total radiated power over the upper hemisphere of radius r using 1
2
∙
/
|
|
which simplifies, with the aid of (4‐99), to /
|
|
(4‐101) (4‐102)  As kh → ∞ the radiated power, as given by (4‐102), is equal to that of an isolated element.  As kh → 0, it can be shown that the power is twice that of an isolated element. The radiation intensity can be written as |
|
(4‐103) (4‐103) The directivity can be written as 4
(4‐104) The maximum value occurs when kh 2.881 h 0.4585 , and it is equal to 6.566 which is greater than four times that of an isolated element (1.5). The pattern for h 0.4585 is shown plotted in Figure 4.17 while the directivity, as given by (4‐104), is displayed in Figure 4.18 for 0 h 5. Figure 4.17 Elevation plane amplitu
ude pattern
n of a verticaal infinitesim
mal electric d
dipole at a h
height of 0.4585 ab
bove an infin
nite perfect electric con
nductor. Ussing (4‐10
02), the radiation reesistance can be written as
| |
2
Th
he radiation resistaance is plotted p
in Figure 4.18 4
for0
0 h
an
nd the element is raadiating in
nto free‐sspace (η
120). (4‐105) (4‐19) 5 5 when = /50 Fiigure 4.18 D
Directivity an
nd radiation
n resistance of a vertical infinitesimal electric d
dipole as a fu
unction of its height above an infinite perfectt electric conductor 3. monopo
ole
In pracctice, a wide w
use has been
n made of o a quarrter‐wavelength monopole m
(
λ/4) mounted m
above a
a ground g
plane, and
d fed by a a coaxial line, as shown s
in Figgure 4.19
9(a). For analysis purposess, aλ/4 image is introducced and it forms theλ/2 eq
quivalent of Figurre 4.19(b). It should be emphasize
e
ed that the λ/2 eq
quivalent of Figure 4.19(b) ggives the correct field valuees for the
e actual syystem of 0, 0
Figgure 4.19(a) only above the interfacee (z
θ
/2). Figgure 4.19 Quarter‐‐wavelenggth monopole on aan infinite perfect e
electric co
onductor Thus, the far‐zone electric and magnetic fields for the λ/4 monopole above the ground plane are given, respectively, by (4‐84) and (4‐85). , (4‐84, 4‐85) The input impedance of a λ/4 monopole above a ground plane is equal to one‐half that of an isolated λ/2 dipole. Thus, referred to the current maximum, the input impedance Z is given by Z
monopole
Z
dipole
73
j42.5
36.5
j21.25 (4‐106) 4.7.4AntennasforMobileCommunicationSystems
 The dipole and monopole are two of the most widely used antennas for wireless mobile communication systems. 
An array of dipole elements is extensively used as an antenna at the base station of a land mobile system while the monopole, because of its broadband characteristics and simple construction, is perhaps to most common antenna element for portable equipment, such as cellular telephones, cordless telephones, automobiles, trains, etc. 
An alternative to the monopole for the handheld unit is the loop. Other elements include the inverted F, planar inverted F antenna (PIFA), microstrip (patch), spiral, and others. The variations of the input impedance, real and imaginary parts, of a vertical monopole antenna mounted on an experimental unit are shown in Figure 4.21. Figure 4.21 Input impedance, real and iimaginary parts, of a veertical mono
opole mountted on an expeerimental ceellular teleph
hone devicee. It is apparent a
that the first reso
onance, around 1,0
000 MHz, is slowlyy varying values of im
mpedancee versus frequencyy, and off desirable magnitude, for practical im
mplementaation. Above the firstt resonance, the im
mpedancee is inducttive. The ssecond reesonance rapid changes in th
he valuess of the impedance. Thesee values and variation of im
mpedance are usually undesirable for practical implementation. 4.7
7.5Horizo
ontalElecctricDipo
ole
When
n the lineear elemeent is placed ho
orizontallyy relativee to the infinitte electric grou
und plaane, as sh
hown in Fiigure 4.24
4. Figgure 4.24 Ho
orizontal eleectric dipole,, and its associated imaage, above aan infinite, fflat, perfectt electric con
nductor The analysis a
p
procedure
e of this iss identicaal to the one of th
he verticaal dipole. Inttroducingg an imagge and assuming a
far field observattions, as shown in Figure 4.2
25(a, b), (a) Horizontal electricc dipole abo
ove ground p
plane (b) Far‐‐field observvations orizontal eleectric dipolee above an infinite perfeect electric conductor
Figgure 4.25 Ho
Since the reflection co
oefficient is equal to R
refflect components ccan be wrritten as 1, The direct and the ⟹
(4‐111) (4‐112) To fiind the angle ψ , which is measu
ured from
m the y‐‐axis tow
ward the ob
bservation
n point, w
we first forrm ∙
∙
⟹ (4‐113) 1
1
(4‐114) Since for far‐field observations for phase variations (4‐115a) for amplitude variations (4‐115b) the total field, which is valid only above the ground plane (z≥h; 0≤θ≤/2, 0≤ ≤2), can be written as E
1
sin
sin
2 sin
cos
(4‐116) Equation (4‐116) again consists of the product of the field of a single isolated element placed symmetrically at the origin and a factor (within the brackets) known as the array factor. The two‐dime
t
ensional elevation
e
plaane patte
erns (norrmalized to 0 dB) for  = 900 ((y‐z planee) when h = 0, /8, //4, 3/8, /2, and
d  are plotted p
in Figgure 4.26. Since this antenn
na system is not symm
metric witth respectt to the z axxis, the azzimuthal plane (x‐‐y plane) paattern is not isotrop
pic. Fiigure 4.26 Elevation plaane ( = 900) amplitude )
patterns off a horizontaal infinitesim
mal electric d
dipole for diffeerent heightts above an infinite perfect electricc conductor.. As the height increases beyond o
one waveelength (h
h ), a larger nu
umber of lob
bes is agaain formed, as in Fiigure 4‐16
6 for the vertical d
dipole. The
e total nu
umber of lob
bes is equ
ual to the integer th
hat most closely is equal to
numberoflobes
(4‐117) The radiated power can be written as (4‐118) The radiation resistance as | |

For small values of →

(4‐120) For kh→∞, (4‐119) reduces to that of an isolated element. 80
0
(
(4‐119) The radiation resistance, as given by (4‐119), is plotted in Figure 4.29 for h 5 when λ/50 and the antenna is radiating into free‐space 120 ). Figgure 4.29 Raadiation ressistance and
d maximum directivity o
of a horizonttal infinitesimal electricc dipole as a functtion of its heeight above an infinite p
perfect electric conducttor. The radiation in
ntensity is given by
1
Th
he maximum value
e of (4‐121) depends on thee value off (4‐121) (wh
hether kh≤/2, h ≤/4 or kh
h > /2,h >>/4). It ccan be sho
own that the maxim
mum of (4
4‐121) is:
,
2
,
,
2
4
The directivity can be written as 0
4
4
4
4
0 4
,
122
sin
1 4
122
,
4
123
4
,
4
4
123
where （4‐123c） For small values of kh (kh → 0), (4‐123a) reduces to 4
2 2
3 3
8
15
7.5
sin
For h = 0 the element is shorted and it does not radiate. PROBLEMS 4.1. A horizontal infinitesimal electric dipole of constant current I0 is placed symmetrically about the origin and directed along the x‐axis. Derive the (a) far‐zone fields radiated by the dipole (b) directivity of the antenna 4.2. Repeat Problem 4.1 for a horizontal infinitesimal electric dipole directed along the y‐axis. 4.22. A thin linear dipole of length l is placed symmetrically about the z‐axis. Find the far‐zone spherical electric and magnetic components radiated by the dipole whose current distribution can be approximated by (a) (b) (c) 1
1
,0
,
, , 0
4.23. A center‐fed electric dipole of length l is attached to a balanced lossless transmission line whose characteristic impedance is 50 ohms. Assuming the dipole is resonant at the given length, find the input VSWR when (a) /4 (b) /2 (c) 3 /4 (d) 4.26. A resonant center‐fed dipole is connected to a 50‐ohm line. It is desired to maintain the input VSWR = 2. (a) What should the largest input resistance of the dipole be to maintain the VSWR = 2? (b) What should the length (in wavelengths) of the dipole be to meet the specification? (c) What is the radiation resistance of the dipole? 4.29. A base‐station cellular communication system utilizes arrays of λ/2 dipoles as transmitting and receiving antennas. Assuming that each element is lossless and that the input power to each of the λ/2 dipoles is 1 watt, determine at 1,900 MHz and a distance of 5 km the maximum (a) radiation intensity Specify also the units. (b) radiation density (in watts/m2) 4.30. A /2 dipole situated with its center at the origin radiates a time‐averaged power of 600W at a frequency of 300MHz. A second /2 dipole is placed with its center at a point P r, θ, φ , where r 200m, θ 90 , φ
40 . It is oriented so that its axis is parallel to that of the transmitting antenna. What is the available power at the terminals of the second (receiving) dipole?