r2 sin dOd4d
=
sins 0d0
cpn=d21w
= I Idlj2
=16d1
t[icos O(sin' 0+2)]1"
Ifd1l 2
='IQ
71k 2
r7
i
J,=0
s2
1
676
Radiation
As far as the dipole is concerned, this radiated power is lost
in the same way as if it were dissipated in a resistance R,
(30)
= i l2R
where this equivalent resistance is called the radiation resistance:
(k d 2 27 dl(31)
R =
) ,
k=--In free space '70-/ Lo/EO0
1207r, the radiation resistance is
2
Ro = 8012()
(free space)
(32)
These results are only true for point dipoles, where dl is
much less than a wavelength (dl/A << I). This verifies the validity of the quasi-static approximation for geometries much
smaller than a radiated wavelength, as the radiated power is
then negligible.
If the current on a dipole is not constant but rather varies
with z over the length, the only term that varies with z for the
vector potential in (5) is I(z):
S+d1/2 li(z)
A,(r)= Re
Sd/2
2
1
e- jkrQp,
dz
~-QP
e-jkrQ'
Re
4
Q
rrQP
+d1/2
-dU2
(z)dz
(33)
where, because the dipole is of infinitesimal length, the distance rQp from any point on the dipole to any field point far
from the dipole is essentially r, independent of z. Then, all
further results for the electric and magnetic fields are the
same as in Section 9-2-3 if we replace the actual dipole length
dl by its effective length,
1
+dl/2
10
di/2
dleff -
I(z) dz
(34)
where 0ois the terminal current feeding the center of the
dipole.
Generally the current is zero at the open circuited ends, as
for the linear distribution shown in Figure 9-4,
I(z) = Io(1-2z/dl),
Io(l+ 2z/dl),
- z- dl/2
-dl/2-z-0
(35)
so that the effective length is half the actual length:
dle=ff
2
1 r+d/
-
J-I/
2
10 d/2
I(z) dz =
dl
2
(36)
677
Radiation from PointDipoles
7i(z)
'Jo
dl1f = d12
-d1/2
d/12
d!.z
7(z) dz
x
(b)
(a)
Figure 9-4 (a) If a point electric dipole has a nonuniform current distribution, the
solutions are of the same form if we replace the actual dipole length dl by an effective
length dl,,. (b) For a triangular current distribution the effective length is half the true
length.
Because the fields are reduced by half, the radiation resistance is then reduced by 1:
JR( :
(dleu]'•
: 201P\'r
(
(37)
In free space the relative permeability /A, and relative
permittivity e, are unity.
Note also that with a spatially dependent current distribution, a line charge distribution is found over the whole
length of the dipole and not just on the ends:
1 di
I=--
jw dz
(38)
For the linear current distribution described by (35), we see
that:
2I/o
j od
9-2-6
0 5 z s dl/2
I-dl/2 <7=
22
12"n'c
(42)
To approximately compute wo, we use the approximate
radius of the electron found in Section 3-8-2 by equating the
energy stored in Einstein's relativistic formula relating mass
to energy:
2
3Q2
3Q 2
105
mc
2e
Ro 20
.IMC
x 10L1.69
m
(43)
Then from (40)
o=
3
/5/3 207EImc
,
3Q 2
-
~2.3 x 10'" radian/sec
is much greater than light frequencies (w
becomes approximately
lim ý 12 2 Eow
o>>
127A mcwo
(44)
1015) so that (42)
(45)
This result was originally derived by Rayleigh to explain the
blueness of the sky. Since the scattered power is proportional
to w 4 , shorter wavelength light dominates. However, near
sunset the light is scattered parallel to the earth rather than
towards it. The blue light received by an observer at the earth
is diminished so that the longer wavelengths dominate and
the sky appears reddish.
9-2-7
Radiation from a Point Magnetic Dipole
A closed sinusoidally varying current loop of very small size
flowing in the z = 0 plane also generates radiating waves.
Because the loop is closed, the current has no divergence so
680
Radiation
that there is no charge and the scalar potential is zero. The
vector potential phasor amplitude is then
0)
A(r) =
e--jr,-
dl
(46)
We assume the dipole to be much smaller than a wavelength,
k(rQp-r)<< 1, so that the exponential factor in (46) can be
linearized to
-
e ikQp = e - jk r e -
lim
j (r
Pg
P- r
T) ,e-i
er[l
- jk( rQp - r)]
k(rqp-r)
21i
i, d=
(-sin i + cos 4i,) di = 0
(49)
the integral is again zero as the average value of the unit
vector i# around the loop is zero.
The remaining integral is the same as for quasi-statics
except that it is multiplied by the factor (1+ jkr) e-i. Using
the results of Section 5-5-1, the quasi-static vector potential is
also multiplied by this quantity:
M= sin 0(1 +jkr) e-k'i,,
4 7tr-
t =
dS
(50)
681
Point Dipole Arrays
The electric and magnetic fields are then
f=lvxA=•
jksei,
2cos
1
+i
Sjkr
x
X
1si
(jkr)
=
WE
(jkr)
(51)
1
(jkr)I
71e-krsin 0
(jkr)
4r
1
+
(jkr)'
S
u
+
(jkr)2Y
The magnetic dipole field solutions are the dual to those of
the electric dipole where the electric and magnetic fields
reverse roles if we replace the electric dipole moment with the
magnetic dipole moment:
p
9-3
q dl
I dl
m
(52)
POINT DIPOLE ARRAYS
The power density for a point electric dipole varies with the
broad angular distribution sin 2 0. Often it is desired that the
power pattern be highly directive with certain angles carrying
most of the power with negligible power density at other
angles. It is also necessary that the directions for maximum
power flow be controllable with no mechanical motion of the
antenna. These requirements can be met by using more
dipoles in a periodic array.
9-3-1
A Simple Two Element Array
To illustrate the basic principles of antenna arrays we
consider the two element electric dipole array shown in
Figure 9-6. We assume each element carries uniform currents
II and i2 and has lengths dll and dl2, respectively. The elements are a distance 2a apart. The fields at any point P are
given by the superposition of fields due to each dipole alone.
Since we are only interested in the far field radiation pattern
where 01 02 0, we use the solutions of Eq. (16) in Section
9-2-3 to write:
+
E2 sin 0 e -kZ
EI sin Oe- '
jkr,
jkr2
where
P,
dl k•
41r
21 dl2 k"y
4.7
(
682
Radiation
Z
11/2
r
r + asinOcoso
asinOcos0
a
= sin 0 cos 0
Figure 9-6 The field at any point P due to two-point dipoles is just the sum of the
fields due to each dipole alone taking into account the difference in distances to each
dipole.
Remember, we can superpose the fields but we cannot
superpose the power flows.
From the law of cosines the distances r, and r 2 are related
as
r 2 = [r2+ a 2 - 2ar cos (7Tr- 6)]j /2 = [r2+ a 2 + 2ar cos
rl = [r2 + a2-
2
] 1/ 2
ar cos]12
(2)
where 6 is the angle between the unit radial vector i, and the x
axis:
= ir, ix = sin 0 cos
cos
4
Since we are interested in the far field pattern, we linearize (2)
to
rf
r i/a 2 2
2a
r + 2-+--sin
s 0 cos
r,
r--
r + a sin 0 cos •
lim
a2
2ar sin 0 cos
)
r - a sin 0 cos 4
In this far field limit, the correction terms have little effect in
the denominators of (1) but can have significant effect in the
exponential phase factors if a is comparable to a wavelength
so that ka is near or greater than unity. In this spirit we
include the first-order correction terms of (3) in the phase
I
I_·_
683
Point Dipole Arrays
factors of (1), but not anywhere else, so that (1) is rewritten as
/E = -/H,
jk- sin Oe=
4 rr
jkr(l
dil ejk si' .
• +
d12
e - k
' -
in
"'.
)
(4)
array factot
eltentlt factol
The first factor is called the element factor because it is the
radiation field per unit current element (Idl) due to a single
dipole at the origin. The second factor is called the array
factor because it only depends on the geometry and excitations (magnitude and phase) of each dipole element in the
array.
To examine (4) in greater detail, we assume the two dipoles
are identical in length and that the currents have the same
magnitude but can differ in phase X:
dl = dl2 -dl
i, = ie' xe)Xfi
i,
ý2
=
e")
(
so that (4) can be written as
0=
, =
e-i
sin
ejx/2 cos (ka sin 0 cos
jkr
2
(6)
Now the far fields also depend on 0. In particular, we focus
attention on the 0 = 7r/ 2 plane. Then the power flow,
lim
S> =
1I
2
1Eol 22
(kr) co s 2 ka cos
-
(7)
depends strongly on the dipole spacing 2a and current phase
difference X.
(a) Broadside Array
Consider the case where the currents are in phase (X= 0)
but the dipole spacing is a half wavelength (2a = A/2). Then,
as illustrated by the radiation pattern in Figure 9-7a, the field
strengths cancel along the x axis while they add along the y
axis. This is because along the y axis r, = r2, so the fields due to
each dipole add, while along the x axis the distances differ by
a half wavelength so that the dipole fields cancel. Wherever
the array factor phase (ka cos 0 -X/ 2 ) is an integer multiple of
nT,
the power density is maximum, while wherever it is an odd
integer multiple of 7'/2, the power density is zero. Because
this radiation pattern is maximum in the direction perpendicular to the.array, it is called a broadside pattern.
684
Radiation
acos2(Tcose),
2
X= 0
cos2(
0
Ecos
2
-
8
'), x = 1!
4
acos2(!cos 02
Broadside
(a)
=
4
), X
2
(c)
a
(
k--x
3
acos2'(2cos0- 1r), X= 32
2
8
4
acos2IIcoS
2
Endfire
(d)
2a =X/2
-I
2
), x= r
(e)
Figure 9-7 The power radiation pattern due to two-point dipoles depends strongly
on the dipole spacing and current phases. With a half wavelength dipole spacing
(2a = A/2), the radiation pattern is drawn for various values of current phase difference
in the 0 = ir/2 plane. The broadside array in (a) with the currents in phase (X = 0) has
the power lobe in the direction perpendicular to the array while the end-fire array in
(e) has out-of-phase currents (X = 7r) with the power lobe in the direction along the
array.
-
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