Antenna first unitt

CHAPTER 2 Fundamental Parameters of Antennas
2.1 INTRODUCTION...........................................................................................................................3
2.2 RADIATION PATTERN................................................................................................................3
2.2.1 Radiation Pattern Lobes ........................................................................................................ 6
2.2.2 Isotropic, Directional, and Omnidirectional Patterns ............................................................ 9
2.2.3 Principal Patterns ................................................................................................................ 10
2.2.4 Field Regions ....................................................................................................................... 12
2.2.5 Radian（弧度）and Steradian（立体弧度） .................................................................... 16
2.3 RADIATION POWER DENSITY................................................................................................18
2.4 RADIATION INTENSITY...........................................................................................................22
2.5 BEAMWIDTH ..............................................................................................................................25
2.6 DIRECTIVITY..............................................................................................................................28
2.6.1 Directional Patterns ............................................................................................................ 38
2.7 ANTENNA EFFICIENCY............................................................................................................43
2.8 GAIN.............................................................................................................................................45
2.9 BEAM EFFICIENCY ...................................................................................................................53
2.10 BANDWIDTH ............................................................................................................................55

2.11 POLARIZATION........................................................................................................................57
2.11.1 Linear, Circular, and Elliptical Polarizations ....................................................................... 61
2.11.2 Polarization Loss Factor and Efficiency .............................................................................. 68
2.12 INPUT IMPEDANCE.................................................................................................................75
2.14 ANTENNA RADIATION EFFICIENCY...................................................................................79
2.15 ANTENNA VECTOR EFFECTIVE LENGTH AND EQUIVALENT AREAS........................82
2.15.1 Vector Effective Length ..................................................................................................... 83
2.15.2 Antenna Equivalent Areas
.................................................................................................. 87
2.16 MAXIMUM DIRECTIVITY AND MAXIMUM EFFECTIVE AREA .....................................94
2.17 FRIIS TRANSMISSION EQUATION AND RADAR RANGE EQUATION ..........................97
2.17.1 Friis Transmission Equation ............................................................................................... 97
2.17.2 Radar Range Equation ..................................................................................................... 101
2.17.3 Antenna Radar Cross Section .......................................................................................... 108
Problems............................................................................................................................................110

2
a
2
a
r
t
c
p
d
d


2.1 INT
To de
are necess
2.2 RAD
An an
as “a mat
representa
the ante
coordinate
pattern is
Radia
density, r
directivity
 Amplitu
 Amplitu
TRODUC
escribe the
sary.
DIATIO
ntenna rad
thematica
ation of th
enna as
es.” In m
determine
ation prop
radiation
, phase or
ude field p
ude power
CTION
e perform
ON PAT
diation pa
l function
he radiatio
a funct
most cases
ed in the f
perties inc
intensity,
r polarizat
pattern.
r pattern.
N
mance of a
TTERN
ttern is de
n or a gra
on propert
tion of
s, the rad
far field re
lude powe
field stre
ion.”

an antenna
efined
aphical
ties of
space
diation
egion.
er flux
ength,
a, definitio
ons of var
rious parameters

y
l
a
Often
yielding n
ogarithmi
accentuat
n the field
ormalized
ic scale (
e in more
d and pow
d field and
dB). This
details th
wer patte
d power p
scale is
ose parts

erns are n
patterns. T
desirable
of the pat
normalized
The patte
e because
ttern of ve
d to the m
rn is usua
e a logarit
ery low va
maximum
ally plotte
thmic sca
lues.
value,
ed on a
ale can

 Field pattern typically represents a plot of the magnitude of the electric or
magnetic field as a function of the angular space.
 Power pattern typically represents a plot of the square of the magnitude of the
electric or magnetic field as a function of the angular space.
 Power pattern (in dB) represents the magnitude of the electric or magnetic field,
in decibels, as a function of the angular space.

2
m
w
2.2.1 Ra
Vario
minor, sid
Figur
with a nu
adiation
ous parts
de, and ba
re 2.3(a) d
mber of r
n Pattern
of a radia
ack lobes.
demonstr
radiation l
n Lobes
ation patte

rates a sy
lobes.

ern are re
mmetrica
eferred to
al three d
o as lobes
imension

：
major or
al polar p
r main,
pattern

l
p

t
t
e

Some
obes. Fig
pattern ch
 A majo
the direct
the θ = 0
exist more
 A mino
e are of g
ure 2.3(b
haracteris
r lobe (m
tion of ma
direction
e than on
or lobe (旁
greater rad
) illustrate
stics are in
main beam
aximum r
n. In some
ne major l
旁瓣) is any
diation in
es a linea
ndicated.
m，主瓣)
radiation.”
e antenna
obe.
y lobe exc

ntensity th
r two‐dim
is defined
” In Figur
as, such a
cept a ma
han other
mensional
d as “the
e 2.3 the
s split‐be
ajor lobe.
rs, but all
pattern w
radiation
major lob
eam anten

are classi
where the

lobe cont
be is poin
nnas, ther
ified as
e same
taining
nting in
re may

 A side lobe(副瓣) is “a radiation lobe in any direction other than the
intended lobe.” Usually a side lobe is adjacent to the main lobe
 A back lobe is “a radiation lobe whose axis makes an angle of approximately
180o
with respect to the beam of an antenna.”
Minor lobes usually represent radiation in undesired directions and should
be minimized. Side lobes are the largest minor lobes.

2.2.2 Isotropic, Directional, and Omnidirectional Patterns
 An isotropic radiator is defined as “a hypothetical lossless antenna having
equal radiation in all directions.” Although it is ideal and not physically
realizable, it is often taken as a reference for expressing the directive properties
of actual antennas.
 A directional antenna is one having the property of radiating or receiving
electromagnetic waves more effectively in some directions than in others.
 An omnidirectional antenna is defined as one “having an essentially
nondirectional pattern in a given plane and a directional pattern in any
orthogonal plane. An omnidirectional pattern is then a special type of a
directional pattern.

2
o
2.2.3 Pr
For a
of its prin
 The E‐
contai
and t
radiat
 The H
plane
vector
maxim

rincipal
a linearly
cipal E‐ an
‐plane is d
ining the
the direc
ion.”
H‐plane
containin
r and
mum radia
Pattern
polarized
nd H‐plan
defined as
electric f
ction of
is define
ng the ma
the dir
ation.”
ns
d antenna
ne pattern
s “the pla
field vect
maximu
ed as “t
agnetic‐fie
rection

a, perform
ns.
ne
tor
um
he
eld
of
mance is o
often des
scribed in terms

(
p
s
p
(
An il
(elevation
plane; θ=
selected.
principal
(azimutha
llustration
n plane; 
= /2) is t
The omn
E‐planes
al plane; θ
n is show
 = 0) is
the princi
nidirection
s (elevati
θ= 90o
).
wn in Fig
the princ
ipal H‐pla
nal patter
on plane

gure 2.5.
cipal E‐pla
ane. Othe
rn of Figu
es;  =
For this
ane and t
er coordin
ure 2.6 ha
c) and
example
the x‐y p
nate orien
as an infi
one pr

, the x‐z
lane (azim
ntations c
inite num
incipal H
plane
muthal
can be
mber of
H‐plane

2
r
(
1
b
0
i
a
b
2.2.4 Fie
The s
reactive n
(Fraunhof
1. React
For
boundary
0.62 /
s the large
a. The re
b. For
equiv
bound
/2
eld Regi
space sur
near‐field,
fer 夫琅和
tive near‐
most an
of thi
/ , is th
est dimen
eactive fie
a very
valent ra
dary is c
.
ions
rounding
, radiatin
和费) regio
field regio
ntennas,
s region
he wavele
sion of th
eld predom
short d
diator, t
commonly
an anten
g near‐fie
ns
on
the oute
n is
ength and
e antenna
minates
dipole, o
the oute
y taken t

na is usu
eld (Fresn
er
D
a.
or
er
to
ally subdi
nel 菲涅
ivided int
耳) regio
o three r
on and f
egions:
ar‐field

2
n
a
b
c
2. Radiati
Defin
near‐field
a. Radiati
b. The an
is de
distanc
c. If the a
overall
very sm
wavele
may no
The r
0.62
ing near‐f
ned as “t
d region an
on fields
ngular fie
ependent
ce from th
ntenna h
dimens
mall com
ength, thi
ot exist.
region is l
2 /
field (Fres
hat regio
nd the far
predomin
ld distrib
upon
he antenn
as a maxi
ion whic
mpared to
is field re
imited by
2
snel) regio
n of the
r‐field reg
nate
ution
the
na.
mum
ch is
o the
egion
y
/ .

on
field of a
gion
an antenn
na betwe
een the re
eactive

3
a
b
m
n
p
a
f
p
u
m
m
3. Far‐fiel
a. The an
the ant
b. The far
the ant
As t
moved t
near‐field
pattern b
and form
far‐field
pattern
usually c
minor lob
more, ma
d (Fraunh
gular field
tenna.
r‐field reg
tenna.
he observ
to the r
d region
begins to
m lobes.
region
is well
onsisting
bes and
ajor lobes.
hofer) reg
d distribu
gion is ta
vation is
radiating
n, the
smooth
In the
, the
formed,
of few
one, or
.
gion
ution is es
ken to ex

ssentially
xist at dis
independ
stances g
dent of th
reater th
e distanc
an 2 /
e from
from

p
o
a
d
t
B
r
u
f

Figure
parabolic
of R 2
It is
almost
difference
the first
Because
realizable
used crit
far‐field o
2.9 show
reflector
2D /, 4D
observed
identical,
es in the p
null and
infinite
e in pract
erion for
observatio
ws three
r calculat
D /, and
d that th
, excep
pattern st
at a leve
distanc
ice, the m
r minimu
ons is 2
patterns
ed at dis
d infinity.
he patter
pt for
tructure a
el below
ces are
most com
um distan
/.

s of a
stances

rns are
some
around
25 dB.
e not
mmonly
nce of

2
a
o
s
c
s
T
2.2.5 Ra
The m
One
angle with
of radius
surface ar
Since
4
closed sph
The
surface of
Therefore
adian（弧
measure o
steradian
h its verte
that is
rea .
e the area
, there a
here.
infinites
f a sphere
e, the solid
弧度）a
of a solid
n is defin
ex at the c
s subtend
a of a sphe
are 4 sr
simal are
e is given b

d angle d
nd Stera
angle is a
ned as t
center of
ded by a s
ere of rad
r 4 /
ea
by

d can be
/

adian（
a steradia
he solid
a sphere
spherical
dius is
in a
on the
  
e written
 
（立体弧度
n.
(m2
)
as
  (sr)
度）

(2‐1)
(2‐2)

Example 2.1
For a sphere of radius r, find the solid angle (in square radians or steradians)
of a spherical cap on the surface sphere over the north‐pole region defined by
spherical angles of 0    30o
, 0    360o
. Do this
a. exactly.
b. using A  1 2, where 1 and 2 are two perpendicular angular
separations of the spherical cap passing through the north pole.
Compare the two.
Solution:
a. Using (2‐2), we can write that
Ω Ω
/
0.83566
b. Ω ΔΘ ∙ ΔΘ | ∙ 1.09662
The approximate beam solid angle is about 31.23% in error.

2.3 RADIATION POWER DENSITY
Instantaneous Poynting vector is a power density and is used to describe
the power associated with an electromagnetic wave
(2‐3)
：instantaneous Poynting vector (W/m2
)
： instantaneous electric field intensity (V/m)
： instantaneous magnetic field intensity (A/m)
The total power, crossing a closed surface, can be obtained by integrating the
normal component of the Poynting vector over the entire surface
∯ ∙ ∯ ∙ (2‐4)
P: instantaneous total power (W);
n: unit vector normal to the surface
da: infinitesimal area of the closed surface m

For time varying fields, average power density is needed, which is obtained
by integrating the instantaneous Poynting vector over one period and dividing
by the period. For the form 
, , ; , ,  ; , , ; , ,  2‐5, 6
Using the definitions of (2‐5) and (2‐6) and the identity
Re , ,  , ,   /2
(2‐3) can be written as
(2‐3) 
∗  (2‐7)
Finally, the time average Poynting vector (average power density) is
, , , , ; ∗
/2 (W/m2
) (2‐8)
Note:
 The real part of ∗
/2 represents the average (real) power density
 The imaginary part represents the reactive (stored) power density

The 1/2 factor appears in (2‐7) and (2‐8) because the and fields
represent peak values, and it should be omitted for RMS values
Based upon the definition of (2‐8), the average power radiated power can
be written as
∙
∯ ∙ ∯ Re ∗
∙ (2‐9)

Example 2.1
The radial component of the radiated power density of an antenna is
(W/m2
)
is the peak value of the power density,  is the spherical coordinate,
and is the radial unit vector. Determine the total radiated power.
SOLUTION
For a closed surface, a sphere of radius is chosen. To find the
total‐radiated power, the radial component of the power density is
integrated over its surface.
∙ ∙

2.4 RADIATION INTENSITY
Radiation intensity in a given direction is defined as "the power radiated
from an antenna per unit solid angle."
The radiation intensity is a far‐field parameter, and it can be obtained by
multiplying the radiation density by the square of the distance.
(2‐12)
Where
=radiation intensity (W/unit solid angle);
=radiation density (W/m2
)
The radiation intensity is also related to the far‐zone electric field of an
antenna by
,
2
, ,
2
| , , | , ,
| , | | , | (2‐12a)
Where

, , : far‐zone electric field intensity of the antenna
,

E , E : far‐zone electric field components of the antenna
η : intrinsic impedance of the medium
Thus the power pattern is also a measure of the radiation intensity.
The total power is obtained by integrating the radiation intensity, as given
by (2‐12), over the entire solid angle of 4. Thus
Ω (2‐13)
Comparison: ∯ ∙

Example 2.2
For Example 2.I, find the total radiated power using (2‐13).
SOLUTION
Using (2‐12)  and by (2‐13)

For an isotropic source, will be independent of the angles  and ,
as was the case for . Thus (2‐13) can be written as
∯ Ω ∯ Ω 4 (2‐14)
or the radiation intensity of an isotropic source as
/4 (2‐15)

2
i
1
2
b
H
t
l
d
a
c
2.5 BEA
The b
identical p
1. Half‐Po
2. First‐Nu
 Of
beamwidt
HPBW.
 Th
trade‐off
obe le
decreases
and vice v
 Th
capabilitie
AMWID
eamwidth
points on
ower Beam
ull Beamw
ften,
th usua
he bea
between
evel. Th
s, the sid
versa.
he beamw
es to dist
DTH
h of a pat
opposite
mwidth (H
width (FN
the
ally refe
mwidth
n it and t
he bea
e lobe in
width of t
inguish tw
tern is de
side of th
HPBW).
BW).
term
ers to
is a
he side
mwidth
creases
the anten
wo adjace

efined: the
he pattern
nna is also
ent radiat
e angular
n maximu
o used to
ing sourc
separatio
um.
o describe
es or targ
on betwee
e the reso
gets.
en two
olution

The most common resolution criterion is FNBW/2, which is usually used to
approximate HPBW.
 That is, two sources separated by angular distances equal or greater than
FNBW/2 ≈ HPBW of an antenna can be resolved.
If the separation is smaller, then the antenna will tend to smooth the
angular separation distance.
Example 2.4
The normalized radiation intensity of an antenna is represented by
U θ cos θ cos 3θ , 0 θ 90 , 0 ϕ 360
Find the
a. half‐power beamwidth HPBW (in radians and degrees)
b.first‐null beamwidth FNBW (in radians and degrees)
Solution:

a. Since the represents the power pattern, to find the half‐power
beamwidth. Let
U θ | 3 | 0.5 ⟹ θ cos 3θ 0.707
⟹ θ 0.25 rad 14.3250

Since is symmetrical about the maximum at  0, then the HPBW is
HPBW 2θ 0.5 rad 28.65
b.To find the first‐null beamwidth (FNBW), let the equal to zero
U θ | cos θ cos 3θ | 0
This leads to two solutions for θ
θ
2
90 , θ
6
30
The one with the smallest value leads to the FNBW. Again, because of the
symmetry of the pattern, the FNBW is
FNBW 2θ
π
3
radians 60

2.6 DIRECTIVITY
 The directivity of an antenna defined as the ratio of the radiation intensity in
a given direction from the antenna to the radiation intensity averaged over
all directions.
 The average radiation intensity is equal to the total power radiated by the
antenna divided by 4.
If the directionis not specified, the direction of maximum radiation intensity
is implied. Directivity can be written as
D ,
|

D = directivity (dimensionless); D0 = maximum directivity (dimensionless)
U = radiation intensity (W/unit solid angle);
= total radiated power (W)
= maximum radiation intensity (W/unit solid angle);
= radiation intensity of isotropic source (W/unit solid angle);

For antennas with orthogonal polarization components, define the partial
directivity of an antenna for a given polarization in a given direction as:
Partial directivity
Part of the radiation intensity with a given polarization in a given direction
the total radiation intensity averaged over all directions

So, in a given direction “the total directivity is the sum of the partial directivities
for any two orthogonal polarizations.” For a spherical coordinate system, the
total maximum directivity for the orthogonal  and  components of an
antenna can be written as
while the partial directivities and are expressed as
,
where
= radiationin tensity in a given direction contained in  field component
= radiationin tensity ina givendirectioncon tained in  field component

= radiated power in all directions contained in  field component
= radiated power in all directions contained in  field component
Example 2.5
Find the maximum directivity of the antenna whose radiation intensity is
that of Example 2.2. Write an expression for the directivity as a function of the
directional angles  and .
Solution:
The radiation intensity is given by

The maximum radiation is directed along:  /2
Thus
In Example 2.2 it was found that
P π A
We find that the maximum directivity is equal to

4 4
1.27
Since the radiation intensity is only a function of , the directivity as a
function of the directional angles is represented by
1.27
Example 2.6
The radial component of the radiated power density of an infinitesimal
linear dipole of length l <<  is given by
(W/m2
)
where is the peak value of the power density,  is the usual spherical
coordinate, and is the radial unit vector. Determine the maximum
directivity of the antenna and express the directivity as a function of the
directional angles  and .
Solution:

The radiation intensity is given by
2
The maximum radiation is directed along /2. Thus

The total radiated power is given by
Ω 8 /3
Using (2‐16a), the maximum directivity is equal to
4 4
8 /3
1.5
greater than 1.27 in Example 2.5. Thus the directivity is represented by
1.5
Figure 2.12 shows the relative radiation intensities of Example 2.5
(U A sin) and Example 2.6 (U A sin )

a
b

a. Both p
b. Examp
elevat
patterns a
ple 2.6 h
tion plane
are omnid
has more
e.
directiona
e directio

l
onal char
racteristic
cs (is na

rrower) in the

A
a
T
i
Another e
approxim
The value
in Figure 2
example:
ated by
es represe
2.13(a, b)
Examin
ented by (
.
ne the dir

(2‐18) and

ectivity o
 1.67
d those o
of a half‐w
7 
f an isotr
wavelengt

opic sour
th dipole,

rce (
which
(2.18)
1) are

 It is apparent that when sin 1/1.67 /
57.44 θ 122.56 , the
dipole radiator has greater directivity than that of an isotropic source.
 Outside the range, the isotropic radiator has higher directivity. The maximum
directivity of the dipole is 1.67 2.23 /2.
 The directivity of an isotropic source is unity
 For all other sources, the maximum directivity will be greater than unity.
 The directivity can be smaller than unity: in fact it can be equal to zero. For
Examples 2.3 and 2.4, the directivity is equal to zero in the  0 direction.
A more general expression for the directivity can be developed to include
sources with radiation patterns that may be functions of both spherical
coordinate angles  and .
Let the radiation intensity of an antenna be of the form
, , | , | | , | 2‐19

where is a constant, and and are the antenna's far‐zone electric
field components. The maximum value of (2‐19) is given by
, | , (2‐19a)
The total radiated power is found using
∯ , , 2‐20
We now write the general expression for the directivity and maximum
directivity using (2‐16) and (2‐16a), respectively, as
D θ, ϕ
,
,
, D
, |
,
2‐21, 22
Equation (2‐22) can also be written as
D
,
, |
2‐23
where Ω is the beam solid angle, and it is given by

1
, |
,
, 2‐24
,
,
, |
(2‐25)
Dividing by , | merely normalizes the radiation intensity
, , and it makes its maximum value unity.
The beam solid angle Ω is defined as the solid angle through which all
the power of the antenna would flow if its radiation intensity is constant (and
equal to the maximum value of U) for all angles within Ω .

2
d
m
m
a
o
t
p
b
p
a
2.6.1 Di
Instea
derive sim
For a
major lo
minor lob
approxim
of the h
two perpe
For a
pattern,
beamwidt
perpendic
as illustrat
rectiona
ad of usin
mpler expr
antennas
obe and
bes, the be
ately equ
alf‐power
endicular
a rotatio
the
ths in
cular plan
ted in Fig
al Patter
ng (2‐23)
ressions a
with one
very n
eam solid
ual to the
r beamw
planes.
onally sy
hal
n any
nes are th
ure 2.14(
rns
to compu
approxima
e narrow
egligible
angle is
product
widths in
mmetric
lf‐power
two
he same,
b).

ute the d
ately.
(a)Nonsym
Figure 2.1
an
irectivity,
mmetrical pa
14 Beam so
nd symmetr
it is ofte

ttern (b) S
lid angles fo
ical radiatio
n conven
Symmetrica
or nonsymm
on patterns.
ient to
l pattern
metrical

With this approximation, (2‐23) can be approximated by
D
,
, |
2‐23

(2‐26)
The beam solid angle Ω has been approximated by
Ω Θ Θ (2‐26a)
Where
= half‐power beamwidth in one plane, (rad)
= half‐power beamwidth in a plane at a right angle to the other, (rad)
If the beamwidths are known in degrees, (2‐26) can be written as
/ ,
(2‐27)
where
1d= half‐power beamwidth in one plane (degrees)
2d=half‐power beamwidth in a plane at a right angle to the other (degrees)

The validity of (2‐26) and (2‐27) is based on a pattern that has only one
major lobe and any minor lobes.
 For a pattern with two identical major lobes, the value of the maximum
directivity using (2‐26) or (2‐27) will be twice its actual value.
 For patterns with significant minor lobes, the values of maximum
directivity obtained using (2‐26) or (2‐27), which neglect any minor
lobes, will usually be too high.

E
a
w
r
(
F
a
b
S
b
ϕ
Example
The
antennas
where
radiation
(0
Figure. Fin
a. beam s
b. maximu
Solution:
beamwidt
ϕ coordin
2.7
e radiation
can be ad
is the
intensity
/2, 0
nd the
olid angle
um direct
The half
th in the
nate,
n intensity
dequately
maximum
exists on
2 )
e; exact an
tivity; exac
f‐power p
directi
y of the m
y represen

m radiati
ly in the
), and it
nd approx
ct using (2
point of t
ion is 12
Θ

major lobe
nted by
ion inten
upper he
is show
ximate.
2‐23) and
the patte
0 . Since
, Θ
e of many
nsity. The
misphere
wn in the
d approxim
ern occur
the patte
,
y
e
e
e
mate using
rs at
ern is inde
g (2‐26).
60 . Th
ependent
us the
t of the

a. Beam solid angle Ω
/
π steradians (2‐24)
Approximate: Using (2‐26a)
4.386 steradians

(2‐26)
b. Directivity :
Exact: 4 10 log 4 6.02dB
Approximate: 2.865 4.57

2
a
i
a
S
1
2
a
2.7 ANT
Ass
antenna
into acco
and with
Such loss
1. Reflec
betwe
antenn
2. lo
The
as
TENNA
ociated w
efficien
ount losse
hin the st
ses may b
tions be
een the t
na
osses (co
overall e
A EFFIC
with an a
cy is
es at the
tructure
be due to
ecause o
ransmiss
onduction
efficiency
CIENCY
antenna
s used
e input te
of the a
o
f the m
sion line
n and die
y can be

Y
are a nu
to take
erminals
ntenna.
ismatch
and the
electric)
written
umber of

f efficien

ncies. The

e total
(2‐44)

where
total efficiency (dimensionless)
reflection(mismatch) efficiency = 1 || (dimensionless)
= conduction efficiency (dimensionless)
= dielectric efficiency (dimensionless)
= voltage reflection coefficient at the input terminals of the antenna
 Z Z / Z Z
Where
Z = antenna input impedance,
Z = characteristic impedance of the transmission line
Usually and are very difficult to compute, but they can be
determined experimentally.

2.8 GAIN
The gain of the antenna is closely related to the directivity, it is a
measure that takes into account the efficiency of the antenna as well as
its directional capabilities.
Directivity is a measure that describes only the directional
properties of the antenna, and it is controlled only by the pattern.
Absolute gain of an antenna is defined as "the ratio of the intensity,
in a given direction, to the radiation intensity that would be obtained if
the power accepted by the antenna were radiated isotropically. The
isotropically radiated power is equal to the power accepted by the
antenna divided by 4."
gain 4π

4π
,
(dimensionless)
(2‐44)

Relative gain is defined as "the ratio of the power gain in a given
direction to the power gain of a reference antenna in its referenced
direction." The power input must be the same for both antennas.
The reference antenna is usually a dipole, horn, or any other
antenna whose gain can be calculated or it is known. In case the
reference antenna is a lossless isotropic source. Then
4
,

(dimensionless) (2‐44a)
When the direction is not stated, the power gain is usually taken in
the direction of maximum radiation.
Referring to Figure 2.17(a), we can write that the total radiated
power (P ) is related to the total input power (P ) by
(2‐45)

Both are very important losses and need to be included in the link
calculations of a communication system to determine the received or
radiated power.
Using (2‐45) reduces (2‐44a) to
,
,
(2‐46)
which is related to the directivity of (2‐21) by
, , (2‐47)
In a similar manner, the maximum value of the gain is related to the
maximum directivity by
According to the IEEE Standards, "gain does not include
losses arising from impedance mismatches (reflection losses)
and polarization mismatches (losses)."

G G θ, ϕ | e D θ, ϕ | e D (2‐47a)
The partial gain of an antenna for a given polarization in a given
direction is "that part of the radiation intensity corresponding to a given
polarization divided by the total radiation intensity that would be
obtained if the power accepted by the antenna were radiated
isotropically." The total gain in a given direction is the sum of the partial
gains for any two orthogonal polarizations.
For a spherical coordinate system, the total maximum gain for
the orthogonal  and  components of an antenna can be written as
  (2‐48)
while the partial gains  and  are expressed as
 4 / , 4 /

(2‐48a)
where

U = radiation intensity in a given direction contained in  field component
U = radiation intensity in a given direction contained in  field component
Pin = total input (accepted) power
For many practical antennas an approximate formula for the gain,
corresponding to (2‐27) or (2‐27a) for the directivity, is
,
(2‐49)
Usually the gain is given in terms of decibels instead of the
dimensionless quantity of (2‐47a). The conversion formula is given by
G dB 10 log e D dimensionless
(2‐50)

Example 2.8
A lossless resonant half‐wavelength dipole antenna, with input impedance
of 73 ohms, is to be connected to a transmission line whose characteristic
impedance is 50 ohms. Assuming that the pattern of the antenna is given
approximately by , find the overall maximum gain of this antenna.
SOLUTION
Let us first compute the maximum directivity of the antenna. For this
|
P U θ, ϕ sinθdθdϕ 2πB θsinθdθ 3π B /4
4
Prad
16
3π
1.697
Since the antenna was lossless, then the radiation efficiency
= 1
Thus, the total maximum gain, as defined in this edition and by IEEE, is equal to

G D 1 1.697 1.697
G dB 10 log 1.697 2.297
which is identical to the directivity because the antenna is lossless.
There is the loss due to reflection or mismatch losses between the antenna
(load) and the transmission line. This loss is accounted for by the reflection
efficiency of (2‐51) or (2‐52), and it is equal to
1 || 1
Z Z
Z Z
0.965
10 log 0.965 0.155 (dB)
Thus, the overall efficiency is
0.965
0.155 dB
Thus, the overall losses are equal to 0.155 dB.
The gain in dB can also be obtained by converting the directivity and
radiation efficiency in dB and then adding them. Thus,

dB 10 log 1.0 0
D dB 10 log 1.697 2.297
G dB dB D dB 2.297
which is the same as obtained previously.

2
t
d

p
2.9 BEA
Beam
transmitt
directed
BE
 is the
percenta
Equat
B
AM EFF
m efficie
ting and
along th

e half‐an
age of the
tion (2‐5
E
FICIEN
ency is
receivin
e z‐axis (

gle of th
e total po
3) can be
,
,
NCY
frequen
g antenn
( = 0), th

he cone
ower is to
e written

ntly use
nas. For a
he beam

within w
o be foun
n as

ed to j
an anten
efficienc

(2‐53
which the
nd.
(2‐54
udge th
nna with
cy (BE) is

(d
)
e
)
he quali
its majo
s defined
dimensio
ity of
or lobe
d by
onless)

If  is chosen as the angle where the first null or minimum occurs
(see Figure 2.4), then the beam efficiency will indicate the amount of
power in the major lobe compared to the total power.
A very high beam efficiency is necessary for antennas used in
radiometry, astronomy, radar, and other applications where received
signals through the minor lobes must be minimized.

2.10 BANDWIDTH
The bandwidth of an antenna is defined as "the range of frequencies
within which the performance of the antenna, with respect to some
characteristic, conforms to a specified standard."
The bandwidth can be considered to be the range of frequencies, on
either side of a center frequency, where the antenna characteristics
(such as input impedance, pattern, beamwidth, polarization, side lobe
level, gain, beam direction, radiation efficiency) are within an acceptable
value of those at the center frequency.
 The bandwidth is usually expressed as the ratio of the
upper‐to‐lower frequencies of acceptable operation. For example,
a 10 : 1 bandwidth indicates that the upper frequency is 10 times
greater than the lower.

 For narrowband antennas, the bandwidth is expressed as a
percentage of the frequency difference (upper minus lower) over
the center frequency of the bandwidth. For example, a 5%
bandwidth indicates that the frequency difference of acceptable
operation is 5% of the center frequency of the bandwidth.

2.11 POLARIZATION
Polarization of an antenna in a given direction is defined as "the
polarization of the wave transmitted (radiated) by the antenna.
Polarization of a wave is defined as "that property of an
electromagnetic wave describing the time varying direction and relative
magnitude of the electric‐field vector "
 When the direction is not stated, the polarization is taken to be the
polarization in the direction of maximum gain."
 Polarization of the radiated energy varies with the direction from the
center of the antenna, so that different parts of the pattern may have
different polarizations.
Polarization is the curve traced by the end point of the arrow
representing the instantaneous electric field. A typical trace as a
function of time is shown in Figures 2.18(a) and (b).

P
1
Figure 2.18
Polarizat
1. Elliptic
The f
Linea

8 Rotation of a
ion may
cal
igure tha
r and cir
a plane electr
be classi
at the ele
rcular po
omagnetic wa
ified as
ectric fiel
larization

ave and its po
ld traces
ns are sp
olarization elli
is an elli
ecial cas

ipse at z = 0 a
pse.
es of elli
s a function o
ptical
of time.

2. Linear
If the vector that describes the electric field at a point in space as a
function of time is always directed along a line, the field is said to be
linearly polarized.
 Vertical polarization
 horizontal polarization
3. Circular
The figure of the electric field is traced in a
Clockwise (CW): the electric field vector is right‐hand polarization
Counterclockwise (CCW): the electric field vector is left‐hand
polarization.

4. Co‐polarization and cross polarization
At each point on the radiation sphere the polarization is usually
resolved into a pair of orthogonal polarizations, the co‐polarization and
cross polarization. Co‐polarization represents the polarization the
antenna is intended to radiate (receive) while cross‐polarization
represents the polarization orthogonal to the co‐polarization.
2.11.1 Linear, Circular, and Elliptical Polarizations
The instantaneous field of a plane wave, traveling in the negative z
direction, can be written as
, , , (2‐55)
According to (2‐5), the instantaneous components are related to their
complex counterparts by
, ωt+kz) ωt+kz+ )

ωt+kz+ ) (2‐56)
, ωt+kz) ωt+kz+ )

ωt+kz+ ) (2‐57)
where and are, respectively, the maximum magnitudes of the
x and y components.
A.Linear Polarization
For the wave to have linear polarization, the time‐phase difference
between the two components must be
   n, 0, 1, 2, 3, … (2‐58)
Which means that the phases of z, t and z, t are the same or
reverse.
B. Circular Polarization
Circular polarization can be achieved only when

 the magnitudes of the two components are the same
 the time‐phase difference between them is odd multiples of /2.
That is,
| | E E
(2‐59)
   2n π, n 0, 1, ,2, … for CW (2‐60)
   2n π, n 0, 1, ,2, … for CCW (2‐61)
If the direction of wave propagation is reversed (i.e., +z direction), the
phases in (2‐60) and (2‐61) for CW and CCW rotation must be
interchanged.
C. Elliptical Polarization
Elliptical polarization can be attained only when
 the time‐phase difference between the two components is odd

multiples of /2 and their magnitudes are not the same
 or, when the time‐phase difference between the two components is
not equal to multiples of /2 (irrespective of their magnitudes). That
is,
| |  E  E
When
   2n π, n 0, 1, ,2, … for CW 2‐62a
   2n π, n 0, 1, ,2, … for CCW 2‐62b
Or
    π 0, 0, 1, ,2, … 2‐63
    π 0, 0, 1, ,2, … 2‐64

t
t
2
m
w
For e
traced at
time is a
2.18(b). T
minor ax
AR
where
OA
OB
The t

elliptical
t a given
a tilted e
The ratio
xis is the

E
B E
ilt of the
τ
polariz
n position
ellipse, as
o of the
axial rati
E
E
ellipse,
tan
ation, t
n as a fu
s shown
major ax
o (AR),
, 1AR
E E
E
relative t
n

he curv
unction o
in Figur
xis to th
 2‐65
E 2E
E 2E
to the y a
cos
ve
of
re
e
5
E cos
E E cos
axis, is re
2ϕ
2Δϕ /
s 2Δϕ /
epresente

/

/ /

ed τ giv

2‐66
2‐67
ven by
2‐68

SUMMARY
1. Linear Polarization
A time‐harmonic wave is linearly polarized at a given point in space if the
electric field (or magnetic field) vector at that point is always oriented along the
same straight line at every instant of time. This is accomplished if the field
vector (electric or magnetic) possesses:
a. Only one component, or
b.Two orthogonal linear components that are in time phase or 180o
(or
multiples of 180 o
) out of phase.
2. Circular Polarization
A time‐harmonic wave is circularly polarized at a given point if the electric
(or magnetic) field vector at that point traces a circle as a function of time. The
necessary and sufficient conditions to accomplish this are:
a. The field must have two orthogonal linear components, and
b. The two components must have the same magnitude, and

c. The two components must have a time‐phase difference of odd multiples
of 90 o
.
3. Elliptical Polarization
A wave is elliptically polarized if it is not linearly or circularly polarized.
Although linear and circular polarizations are special cases of elliptical, usually
in practice elliptical polarization refers to other than linear or circular. The
necessary and sufficient conditions to accomplish this are if the field vector
(electric or magnetic) possesses all of the following:
a. The field must have two orthogonal linear components, and
b.The two components can be of the same or different magnitude.
c. (1) If the two components are not of the same magnitude, the time‐phase
difference between the two components must not be 0 or multiples of
180 (because it will then be linear). (2) If the two components are of the
same magnitude, the time‐phase difference between the two components
must not be odd multiples of 90 o
(because it will then be circular).

2.11.2 Polarization Loss Factor and Efficiency
The polarization of the receiving antenna will not be the same as the
of the incident wave. This is stated as "polarization mismatch." The
amount of power extracted by the antenna from the incident signal will
not be maximum because of the polarization loss. Assuming the electric
field of the incident wave is
E ρ E 2‐69
where ρ is the unit vector of the wave. Assuming the polarization of
the electric field of the receiving antenna is
E ρ E 2‐70
where ρ is its unit vector (polarization vector). The polarization loss
factor (PLF) is defined
PLF |ρ ∙ ρ | cos|ψ | dimensionless 2‐71

w
i
i
m
p
t
e
w
t
where ψ
incoming
If the
its PLF
maximum
Anoth
polarizat
that of
efficiency
Pola
"The
wave of
the same
ψ is th
g wave an
e antenna
1 and
m power
her figur
ion char
an ante
y.
rization e
e ratio of
arbitrary
e antenn
he angle
nd of the
a is pola
d the ant
from the
re‐of‐me
racteristic
enna is
efficiency
f the pow
y polariz
a from a
e betwee
e vectors
rization
tenna w
e incomi
rit descr
cs of a w
the po
y is defin
wer rece
ation to
plane w

en the
shown i
matched
ill extrac
ng wave
ribing th
wave an
olarizatio
ned as:
ived by a
the pow
wave of th
two uni
n Figure
d,
ct
.
e
d
n Figu
vectors
an anten
wer that
he same
it polari
2.19.
ure 2.19 Po
s of inciden
antenn
nna from
would b
power fl
zation o
olarization
nt wave (ρ
na (ρ )
m a given
be receiv
ux densit
of the

unit
ρ ) and
plane
ved by
ty and

direction of propagation, whose polarization has been adjusted for a
maximum received power."
| ∙ |
| | ∙| |
2‐71a
Where
= vector effective length of the antenna
= incident electric field
The vector effective length is a vector that describes the
polarization characteristics of the antenna. Both the PLF and lead to
the same answers.

Example 2.9
The electric field of a linearly polarized electromagnetic wave given by
,
is incident upon a linearly polarized antenna whose electric field polarization
can be expressed as
, ,
Find the polarization loss factor (PLF)
SOLUTION
For the incident wave and the antenna
,
√

The PLF is:
PLF | ∙ | | ∙
√
| 10 log 0.5 3dB
1. Even the incoming wave and the antenna are linearly polarized, there is a
3‐dB loss in extracted power because the polarization of the incoming wave is

2
t
not alig
2. If the
the a
the i
In Fig
types of a
gned with
e polariza
antenna,
ncoming
gures 2.2
antennas;
(a
h the pola
tion of th
then the
wave and
0(a, b) we
wires and
a) PLF for tra
rization o
he incomin
re will be
d the PLF w
e illustrate
d apertur
ansmitting a

of the ante
ng wave is
e no powe
will be ze
e the pola
es.
and receivin
enna.
s orthogo
er extract
ro or ‐ d
arization l
ng aperture
onal to the
ted by th
dB.
oss factor
antennas
e polariza
e antenn
rs (PLF) of
ation of
a from
f two

c
a
a
p
l
Fi
The
calculatio
a very crit
Link
are very
power is a
oss facto
(b
igure 2.25 P
polarizat
ns design
tical facto
calculatio
stringent
a limiting
rs to ensu
) PLF for tra
Polarization l
tion loss
n of a com
or.
ons of co
because
considera
ure a succ
nsmitting a
loss factors
must al
mmunicati
ommunica
of limita
ation. The
cessful ope

nd receiving
(PLF) for ap
ways be
on system
ation syst
ations in
e design m
eration of
g linear wire
erture and
taken in
m because
ems for o
spacecraf
must prop
f the syste
e antennas
linear wire a
nto accou
e in some
outer spa
ft weight
perly take
em.

antennas.
unt in th
cases it m
ace explo
. In such
into acco
he link
may be
rations
cases,
ount all

a
f
(
A
p
An a
as shown
field comp
If th
(perpendi
Also, if th
polarizatio
antenna th
in Figure
ponents.
Figure
he two di
icular to t
he two dip
on along z
hat is ellip
e 2.26. Th

2.26 Geome
poles are
the plane
poles wer
zenith wo
ptically po
he two cr
etry of ellipt
e identica
of the tw
re fed wit
ould be cir

olarized is
rossed dip
tically polar
al, the fie
wo dipole
th a 90◦ d
rcular and
s compose
poles pro
ized cross‐d
eld intens
s) would
degree tim
d elliptica
ed of two
vide the

dipole anten
sity of ea
be of the
me‐phase
l.
crossed d
two orth
na.
ch along
e same int
e differenc
dipoles,
ogonal
zenith
tensity.
ce, the

2
t
o
t
i
t
r
t
d
a
w
2.12 INP
Inpu
terminals
of the app
Inpu
terminals
interestin
terminals
ratio of th
terminals,
defines
antenna a
where

PUT IM
ut impeda
or the ra
propriate
ut impeda
of the a
g. In F
are desi
he voltag
, with n
the im
as
ZA = ante
RA = ante
MPEDA
ance is de
tio of the
compone
ance at
ntenna is
igure 2.2
gnated a
ge to curr
no load
pedance
enna impe
enna resist
ANCE
efined as
e voltage t
ents of the
the inpu
s primaril
27(a) th
s a‐b. Th
rent at th
attached
of th
edance at
tance at t

"the impe
to current
e electric
ut
y
e
e
e
d,
e
terminal
terminals
edance p
t at a pair
to magne

s a‐b (o
a‐b (o
resented
r of termi
etic fields

hms)
ohms)
at an ant
nals or th
at a poin

tenna’s
he ratio
t."
(2‐72)

w
w
B
a
a
w

Assum
with inter
where
R = r
X = r
Being use
and gener
To f
amount d
within the
XA = ante
me the an
rnal imped
resistance
reactance
ed in the
rator can
find the
dissipated
e loop wh
enna react
ntenna is
dance
e of gener
e of gener
transmit
be repres
amount
in as
hich is give
tance at t
attached

rator impe
rator impe
tting mod
sent by Fig
of powe
s heat (
en by

erminals
d to a gen

edance
(
edance (
de, the an
gure 2.27
r deliver
/2), w
a‐b (o
nerator
2‐74
ohms)
ohms)
ntenna
7(b).
ed to
we first fin
hms)
for rad
nd the cur

diation an
rrent deve
A
nd the
eloped
2‐75

and its magnitude by | |
| |
| | ／
2‐75a
where is the peak generator voltage.
 The power delivered to the antenna for radiation is given by
| | |
| |
W 2‐76
 and that dissipated as heat in antenna by
| | |
| |
W 2‐77
 The power dissipated as heat on the internal resistance Rg of the generator is
| |
W 2‐78
The maximum power delivered to the antenna occurs when conjugate
matching is achieved
; 2‐79, 20‐80
For this case

| |

| |
2‐81
| |
2‐82
| |

| |

| |
2‐83
From (2‐81)‐(2‐83), it is clear that
2‐84
The power supplied by the generator during conjugate matching is
W 2‐85

2.14 ANTENNA RADIATION EFFICIENCY
The conduction‐dielectric efficiency e is defined as the ratio of the
power delivered to the radiation resistance , to the power delivered to
. The radiation efficiency can be written as
dimensionless 2‐90
For a metal rod of length and area , the dc resistance is given by
ohms 2‐90a
If the skin depth 2/ of the metal is very small compared to
the smallest diagonal of the cross section of the rod, the current is confined to a
thin layer near the conductor surface. Therefore the high‐frequency resistance
can be written, based on a uniform current distribution, as
ohms 2‐90b
Where

2 is the perimeter of the cross section of the rod ( : the radius of wire)
is the conductor surface resistance
 is the angular frequency
 is the free‐space’s permeability
 is the metal’s conductivity.
Example 2.13
A resonant half‐wavelength dipole is made out of copper ( 5.710 s/m)
wire. Determine the conduction‐dielectric (radiation) efficiency of the dipole
antenna at 100 MHz, the radius of the wire b is 310 , and the
radiation resistance of the /2 dipole is 73 ohms.
SOLUTION
At 10 Hz
3 ; 3/2 ;
2 2 3 10 6 10

For a /2 dipole with a sinusoidal current distribution RL where
is given by (2‐90b). Therefore,
1
2
0.25
6 10
10 4 10
5.7 10
0.349
e
.
0.9952 99.52; e dB 10log 0.9952 0.02

2
c
a
d
w
2.15 ANT
ARE
An a
capture (
as shown
For e
defined t
wave is in
TENNA
EAS
antenna
(collect)
n in Figur
each ant
to descri
ncident u
VECTOR
in the r
electrom
res 2.29(
enna, its
ibe the r
upon the
R EFFEC
receiving
magnetic
a) and (b
s equival
receiving
e antenna

CTIVE L
g mode,
waves a
b).
ent lengt
g charact
a.

LENGTH
whateve
nd to ext
th and e
eristics o
H AND E
er its for
tract pow
quivalen
of an ant
EQUIVA
rm, is us
wer from
nt areas c
tenna, w
ALENT
sed to
m them,
can be
when a

2.15.1 Vector Effective Length
The effective length of an antenna is a quantity to determine the voltage
induced on the open‐circuit terminals of the antenna when a wave
impinges upon it. It should be noted that it is also referred to as the
effective height.
The vector effective length for an antenna is a complex vector
represented by
, , , (2‐91)
It is a far‐field quantity and it is related to the far‐zone field
radiated by the antenna, with current in its terminals
(2‐92)
The effective length is particularly useful in relating the open‐circuit voltage
of receiving antennas. This relation can be expressed as

∙ (2‐93)
where
= open‐circuit voltage at antenna terminals
= incident electric field
= vector effective length
In (2‐93) can be thought of as the voltage induced in a linear antenna
of length when and are linearly polarized.
From the relation of (2‐93) the effective length of a linearly polarized
antenna receiving a plane wave in a given direction is defined as

In addition, the antenna vector effective length is used to determine
the polarization efficiency of the antenna.
The ratio of the magnitude of the open‐circuit voltage developed at the
terminals of the antenna to the magnitude of the electric‐field strength in the
direction of the antenna polarization.

Example 2.14
The far‐zone field radiated by a small dipole of length

and
with a triangular current distribution is given by
8
Determine the vector effective length of the antenna.
Solution: According to (2‐92), the vector effective length is
(2‐92)
⇒
2

 This indicates the effective length is a function of the direction angle .
 The maximum open circuit voltage at the dipole terminals occurs when the
incident direction of the wave is normal to the dipole ( 90 ).
 In addition, the effective length of the dipole to produce the same output

open‐circuit voltage is only half (50%) of its physical length if it were replaced
by a thin conductor having a uniform current distribution
 The maximum effective length of an element with an ideal uniform current
distribution is equal to its physical length.

2.15.2 Antenna Equivalent Areas
 Antenna Equivalent Areas
 The scattering area
 The loss area
 the capture area
Antenna Equivalent Areas are used to describe the power
capturing characteristics of antennas when wave imping on them, which
in a given direction is defined as “the ratio of the available power at the
terminals of a receiving antenna to the power flux density of a plane
wave incident on the antenna from that direction, the wave being
polarization‐matched to the antenna. If the direction is not specified, the
direction of maximum radiation intensity is implied.”It is written as
| | /
(2‐94)
Where = effective area (effective aperture) (m2
);
= power delivered to the load (W);
= power density of incident wave (W/m2
)

c
m
W
p
Using
can write
| |
Unde

maximum
When (2‐9
power of
g the equ
(2‐94) as
|
er condit
an
m effective
96) is mu
(2‐89).
uivalent o
| | /
ions of
nd
e aperture
|
ltiplied by
of Figure
/

maximum
, the
e given by
|
y the incid

2.28, we
(2‐94)
(2‐95)
m power
e effectiv
y
|
dent powe
e
)
)
r transfer
ve area o
|
er density
r (conjug
of (2‐95)

y, it leads
gate mat
reduces

to the ca

tching),
to the
(2‐96)
ptured

∗
∗
| |
(2‐97)
All captured power by an antenna is not delivered to the load. In
fact, under conjugate matching only half of the captured power is
delivered to the load; the other half is scattered and dissipated as heat.
To account for the scattered and dissipated power, in addition to the
effective area, the scattering, loss and capture equivalent areas are
defined.
The scattering area is defined as the equivalent area when
multiplied by the incident power density is equal to the scattered or
reradiated power. Under conjugate matching this is written as
A
| |
(2‐97)
multiplied by the incident power density gives the scattering power.

The loss area is defined as the equivalent area, which when
multiplied by the incident power density leads to the power dissipated
as heat through . Under conjugate matching this is written as
| |
(2‐98)
multiplied by the incident power density gives the dissipated power.
The capture area is defined as the equivalent area, which when
multiplied by the incident power density leads to the total power
captured by the antenna. Under conjugate matching this is written as
| |
(2‐99)
multiplied by the incident power density, it leads to the captured power.
The total capture area is equal to the sum of the other three, or

Capture Area Effective Area Scattering Area Loss Area

The aperture efficiency  of an antenna, which is defined as the
ratio of the maximum effective area A of the antenna to its physical
area A , or


(2‐100)
 For aperture type antennas, such as waveguides, horns, and
reflectors, the maximum effective area cannot exceed the physical area
but it can equal it (A A or 0  1). Therefore the maximum
value of the aperture efficiency cannot exceed unity (100%).
 For a lossless antenna (R 0) the maximum value of the
scattering area is also equal to the physical area. Therefore even though
the aperture efficiency is greater than 50%, under conjugate matching
only half of the captured power is delivered to the load and the other
half is scattered.

E
s
r
l
S
F
o
S
c
Example 2
A unifo
shown in
radiation
inearly po
Solution:
For R
of (2‐96) r
|
Since the
constant a
2.15
orm plane
Figure 2
resistanc
olarized a
0, the m
reduces to
|
e dipole i
and of un
e wave is
.29(a). Fi
e of the
along the
maximum
o
|

| |
8
1
is very sh
iform pha
incident u
nd the m
dipole is
axis of the
effective
|
(
1

hort, the
ase. The in

upon a ve
maximum
R 80
e dipole.
e area

2‐96)
induced
nduced vo
ery short l
effective
0 / ,
d current
oltage is
lossless d
area ass
and the
can be
ipole (l <<
suming th
incident
assumed
< ), as
hat the
field is

to be

where
VT = induced voltage on the dipole
E = electric field of incident wave, l = length of dipole
For a uniform plane wave, the incident power density can be written as
/2
where  is the intrinsic impedance of the medium ( 120 ohms for a free‐space
medium). Thus
| |
8 /2 80 /
0.119

2
e
e
r
B
o
T
l
2.16 MAX
To d
effective
effective
, . If
radiated
Because o
of the ant
The pow
load wou
XIMUM
derive th
area, le
areas a
f antenn
power d
of the di
tenna, its
wer collec
uld be
DIRECT
he relat
et Anten
and direc
a 1 wer
density at

rective p
actual de

cted (rec
TIVITY A
tionship
nna 1 be
ctivities
e isotrop
t a distan
(2‐101)
properties
ensity is
(2‐102)
ceived) b

AND MA
betwee
e a trans
of each
pic, and
nce w
)
s
)
by the a
AXIMUM
en direc
smitter
are des
the t
would be
ntenna a
M EFFECT
ctivity a
and 2 a
ignated
total radi
and tran
TIVE AR
nd max
receive
as ,
ated pow
sferred t
REA
ximum
r. The
and
wer. its
to the

⟹ 4 (2‐103)
If antenna 2 is used as a transmitter, 1 as a receiver, and the
intervening medium is linear, passive, and isotropic, we can write that
4 (2‐104)
Equating (2‐103a) and (2‐104) reduces to
(2‐105)
Increasing the directivity of an antenna increases its effective area in
direct proportion. Thus, (2‐105) can be written as
(2‐106)
where A and A (D and D ) are the maximum effective areas
(directivities) of antennas 1 and 2, respectively. If antenna 1 is isotropic,

then D 1 and its maximum effective area can be expressed as
(2‐107)
For example, let the antenna be a very short  dipole
whose effective area = 0.119 and maximum directivity = 1.5. The
maximum effective area of the isotropic source is then equal to
.
.
(2‐108)
Using (2‐108), we can write (2‐107) as
(2‐109)
In general then, the maximum effective aperture (A ) of any
antenna is related to its maximum directivity (D ) by
(2‐110)

2
E
r
E
2
p
t
2.17 FR
EQUAT
The
require t
Equation
2.17.1 Fr
The
power tr
the large
RIIS TRA
TION
analysis
the use o
n.
riis Trans
Friis Tra
ransmitte
est dimen
ANSMIS
s and d
of the Fr
smission
nsmissio
ed betwe
nsion of e
SSION E
esign of
iis Trans
Equatio
on Equat
een two
either an

EQUAT
f radar
mission
n
tion relat
antenna
ntenna.
ION AN
and com
Equation
tes the p
as separa
ND RAD
mmunica
n and the
power re
ated by R
DAR RA
ations sy
e Radar
eceived t
R

,

ANGE
ystems
Range
to the
, D is

Assuming the transmitting antenna is isotropic. If the input power at the
terminals of the transmitting antenna is P , then its isotropic power
density W at distance R from the antenna is
W e 2‐113
e : the radiation efficiency of the transmitting antenna.
For a nonisotropic transmitting antenna, the power density of
(2‐113) in the direction , is
W
,
e
,
(2‐114)
Since the effective area of the receiving antenna is related to its
efficiency and directivity by
; , (2‐115)
the power collected by the receiving antenna is

P e D θ , ϕ W e e
, ,
|ρ ∙ ρ |
(2‐118)
The ratio of the received to the input power as
, ,
(2‐117)
The power received based on (2‐117) assumes that the transmitting
and receiving antennas are matched to their respective lines or loads
and the polarization of the receiving antenna is polarization‐matched to
the impinging wave.
If these two factors are also included, then the ratio of the received
to the input power of (2‐117) is represented by
, , 1 |Γ | 1 |Γ | |ρ ∙ ρ |
(2‐118)

For reflection and polarization‐matched antennas aligned for maximum
directional radiation and reception, (2‐118) reduces to
2‐119
Equations (2‐117), (2‐118), or (2‐119) are known as the Friis
Transmission Equation, and it relates the power (delivered to the
receiver load) to the input power of the transmitting antenna . The
term


is called the free‐space loss factor, and it takes into
account the losses due to the spherical spreading of the energy by the
antenna.

2
s
d
s
t
2.17.2 Ra
Assu
shown in
The
defined
scattered
to that sc
adar Ran
uming th
n Figure 2
radar c
as the a
d isotrop
cattered
nge Equa
at the tr
2.32.
ross sec
area inte
pically, pr
by the a
ation
ransmitte
tion (RC
ercepting
roduces a
ctual tar

ed powe
CS) or ec
g that am
at the re
get. In eq
er is incid
cho area
mount o
ceiver a
quation f
dent upo

a ( ) o
of power
density
form
on a targ
of a targ
r which,
which is
get, as
get, is
when
equal

→       (2‐120)
→
4
→
4
| |
| |

→ 4
| |
(2‐120a)
 = radar cross section or echo area (m2
)
R = observation distance from target (m)
W = incident power density (W/m2
)
W = scattered power density (W/m2
)
E (E ) = incident (scattered) electric field    (V/m)
H H ) = incident (scattered) magnetic field   (A/m)
Using the definition of radar cross section, we can consider that
 the transmitted power incident upon the target is initially
captured；

 then it is reradiated isotropically, insofar as the receiver is
concerned.
The amount of captured power is obtained by multiplying the
incident power density of (2‐114) by the radar cross section, or
W
,
e
,
   (2‐114)
, ,
   (2‐121)
The power captured by the target is reradiated isotropically, and the
scattered power density can be written as
,
   (2‐122)
The amount of power delivered to the receiver load is given by
, ,
   (2‐123)

Equation(2‐123) can be written as the ratio of the received power to the
input power, or
, ,
(2‐124)
Expression (2‐124) is used to relate the received power to the input
power, It does not include reflection losses and polarization losses. If
these two losses are also included, then (2‐124) must be expressed as
, ,
1 |Γ | 1 |Γ | |ρ ∙ ρ |
(2‐125)
For polarization‐matched antennas aligned for maximum directional
radiation and reception, (2‐125) reduces to
2‐126
Equation(2‐124 ), or (2‐125) or (2‐126) is known as the Radar Range

Equation. It relates the power P (delivered to the receiver load) to the
input power P transmitted by an antenna, after it has been scattered
by a target with a radar cross section (echo area) of .
Example 2.16
Two lossless X‐band 8.2– 12.4 GHz horn antennas are separated by a
distance of 100 . The reflection coefficients at the terminals of the
transmitting and receiving antennas are 0.1 and 0.2 , respectively. The
maximum directivities of the transmitting and receiving antennas (over
isotropic) are 16 dB and 20 dB, respectively. Assuming that the input power in
the lossless transmission line connected to the transmitting antenna is 2W, and
the antennas are aligned for maximum radiation between them and are
polarization‐matched, find the power delivered to the load of the receiver.
Solution:
For this problem
e e 1 because antennas are lossless.

|ρ ∙ ρ | 1 because antennas are polarization‐matched
D D , D D because antennas are aligned for maximum
radiation between them
D 16 dB➱39.81 (dimensionless)
D 20 dB➱100 (dimensionless)
Using (2‐118), we can write
2 1 1 39.81 100 1 0.1 1 0.2 1 =4.777mW

2.17.3 Antenna Radar Cross Section
The radar cross section is a far‐field parameter, which is used to
characterize the scattering properties of a radar target.
 monostatic or backscattering RCS
 bistatic RCS
The RCS of a target is a function of
1. polarization of the incident wave,
2. the angle of incidence,
3. the angle of observation,
4. the geometry of the target,
5. the electrical properties of the target,
6. the frequency of operation.
The units of RCS of three‐dimensional targets are m2
,or dBsm, or RCS/2
in
dB.
The RCS of a target can be controlled using primarily two basic methods:

shaping and the use of materials.
 Shaping is used to attempt to direct the scattered energy toward
directions other than the desired. However, for many targets shaping has
to be compromised in order to meet other requirements, such as
aerodynamic specifications for flying targets.
 Materials are used to trap the incident energy within the target and to
dissipate part of the energy as heat or to direct it toward directions other
than the desired.
Usually both methods, shaping and materials, are used together in order to
optimize the performance of a radar target.
One of the“golden rules”to observe in order to achieve low RCS is to
“round corners, avoid flat and concave surfaces, and use material treatment in
flare spots.”

Problems
2.4. Find the half-power beamwidth (HPBW) and first-null beamwidth (FNBW),
in radians and degrees, for the following normalized radiation intensities:
a U θ cos θ b U θ cos θ
c U θ cos 2θ d U θ cos 2θ
e U θ cos 3θ f U θ cos 3θ
0 θ 90 , 0 φ 360
2.7. The power radiated by a lossless antenna is 10 watts. The directional
characteristics of the antenna are represented by the radiation intensity of
a U B cos θ
b U B cos θ
(watts/unit solid angle) and (0 ≤ θ ≤ π/2, 0 ≤ φ ≤ 2π)
For each, find the
(a) maximum power density (in watts/square meter) at a distance of 1,000
m(assume far-field distance). Specify the angle where this occurs.

(b) exact and approximate beam solid angle 2A.
(c) directivity, exact and approximate, of the antenna (dimensionless and in dB).
(d) gain, exact and approximate, of the antenna (dimensionless and in dB).
2.8. You are an antenna engineer and you are asked to design a high
directivity/gain antenna for a space-borne communication system operating at 10
GHz. The specifications of the antenna are such that its pattern consists basically
of one major lobe and, for simplicity, no minor lobes (if there are any minor
lobes they are of such very low intensity and you can assume they are negligible
/zero). Also it is desired that the patternis symmetrical in the azimuthal plane. In
order to meet the desired objectives, the mainlobe of the patternshould have a
half-power beamwidth of 10 degrees. Inorder to expedite the design, it is
assumed that the major lobe of the normalized radiation intensity of the antenna
is approximated by
U θ, φ cos θ
and it exists only in the upper hemisphere (0 ≤ θ ≤ π/2, 0 ≤ φ ≤ 2π). Determine
the:
(a) Value of n (not necessarily an integer) to meet the specifications of the major
lobe. Keep 5 significant figures in your calculations.

(b) Exact maximum directivity of the antenna (dimensionless and in dB).
(c) Approximate maximum directivity of the antenna based on Kraus’ formula
(dimensionless and in dB).
(d) Approximate maximum directivity of the antenna based on Tai & Pereira’s
formula (dimensionless and in dB).
2.9. In target-search ground-mapping radars it is desirable to have echo power
received from a target, of constant cross section, to be independent of its range.
For one such application, the desirable radiation intensity of the antenna is given
by
   
1 0 20
U , 0.342csc 20 60 0 360
0 60 180
o o
o o o o
o o

    

 
 
 
    
 
 
 
 
Find the directivity (in dB) using the exact formula.
2.15. The radiation intensity of an antenna is given by

U θ, φ cos θ sin φ
for 0≤θ≤π/2 an d 0≤φ≤2π(i.e., inthe upper half-space). It is zero in the lower
half-space. Find the
(a) exact directivity (dimensionless and in dB)
(b) elevationplan e half-power beamwidth (in degrees)
2.17. The maximum gain of a horn antenna is +20 dB, while the gain of its first
sidelobe is −15 dB. What is the difference in gain between the maximum and
first sidelobe:
(a) in dB
(b) as a ratio of the field intensities.
2.34. A 300 MHz uniform plane wave, traveling along the x-axis in the negative
x direction, whose electric field is given by
 
0
ˆ ˆ
3 jkx
w y z
E E ja a e
 
r
where Eo is a real constant, impinges upon a dipole antenna that is placed at the

o
d
w
(
i
(
Y
(
2
originan
direction i
where Ea
(a) Polariz
if any). Yo
(b) Polariz
You must
(c) Polariz
2.35. The
d whose
is givenby
is a real c
zation of
ou must ju
zation of
justify (s
zation los
e electric
electric
y
E
constant. D
the incide
ustify (sta
the antenn
state why？
ss factor (d
field of a
field rad

a a
E E

r
Determin
ent wave
ate why？
na (includ
？).
dimension
a uniform

iated tow
ˆ ˆ
2
y z
a a

ne the follo
(includin
).
ding axial
nless and
plane wa
ward the
 jkx
z e
owing:
ng axial ra
l ratio and
in dB).
ave travel
x-axis in
atio and se
d sense of
ling along
n the pos
ense of ro
f rotation,
g the neg
itive x
otation,
if any).
gative z

directionis given by
  0
ˆ ˆ
i jkz
w x y
E a ja E e
 
and is incident upon a receiving antenna placed at the origin and whose radiated
electric field, toward the incident wave, is given by
  1
ˆ ˆ
2
jkr
a x y
e
E a a E
r

 
Determine the following:
(a) Polarizationof the incident wave, and why？
(b) Sense of rotation of the incident wave.
(c) Polarization of the antenna, and why？
(d) Sense of rotation of the antenna polarization.
(e) Losses (dimensionless and in dB) due to polarization mismatch between the
incident wave and the antenna.

Antenna first unitt

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