# W.C.Chew ECE 350 Lecture Notes A Hertzian dipole is a dipole which

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```					W.C.Chew
ECE 350 Lecture Notes

26. The Fields of a Hertzian Dipole

A Hertzian dipole is a dipole which is muc hsmaller than the wa v elength
under construction so that we can approximate it b ya point current distri-
bution,
J(r) = zIl (r):
^                                   (1)
The dipole may look lik e the follo wing

q                 metallic spheres – charge reservoir
I
+
generator
–
–q

l is the e ective length of the dipole so that the dipole moment p =
ql. The charge q is v arying time harmonically because it is driven b y the
generator. Since dq = I , we hav e
dt

Il = dq l = j!ql = j!p
dt                                                 (2)

for a Hertzian dipole. We already know that the corresponding vector poten-
tial is given b y
A(r) = z 4 Il e;j r :
^ r                                 (3)
The magnetic eld is obtained, using cylindrical coordinates, as

H = 1r A = 1               ^1 @@ Az ; ^ @@ Az                         (4)
p
where @@ = 0 r =         2   + z2 . In the above,   @
@   = @ @r = p 2 +z2 @r = r @r .
@r @           @      @

Hence,
Il
H = ; ^ r 4 ; r12 ; j 1 e;j r :                                (5)
r
1
z

r
θ
y
φ
ρ
φ
ˆ
x

In spherical coordinates, r = sin , and (5) becomes

H = ^ 4 Ilr2 (1 + j r)e;j r sin :                (6)
The electric eld can be derived using Maxwell's equations.
E = j! r H = j! r r sin @@ sin H ; ^ 1 @r rH
1            1 ^ 1
r
@
Ile;j r h                                        i
= j! 4 r3 r2 cos (1 + j r) + ^ sin (1 + j r ; 2 r2 ) :
^                                               (7)

Case I. Near Field, r 1
E = 4 r3 (^2 cos
r                 + ^ sin )       r   1          (8)
H E               when r 1:                   (9)
r could be made very small by making r small or by making ! ! 0. The
above is like the static eld of a dipole.
Case II. Far Field (Radiation Field), r               1
In this case,
E = ^j! 4Ilr e;j r sin                      (10)
and
H = ^j 4Ilr e;j r sin :                 (11)
p
Note that H = !
E
=     = 0 . E and H are orthogonal to each other
and are both orthogonal to the direction of propagation, i.e. as in the case
of a plane wave. A spherical wave resembles a plane wave in the far eld
approximation.
2
The time average power ow is given by
1 <e E H ] = r 1 jH j2 = r 0 Il 2 sin2 :
hSi = 2            ^2 0        ^2 4 r                  (12)
The radiation eld pattern of a Hertzian dipole is the plot of jEj as a
function of at a constant r.
z

θ
|E|
x, y

The radiation power pattern is the plot of hSr i at a constant r.
z

x, y

The total power radiated by a Hertzian dipole is given by
Z 2       Z                                      Z
P=         d           d r sin hSr i = 2
2                                   d 2 4Il 2 sin3 :
0
(13)
0        0                                          0

Since
Z                      Z ;1                                           Z 1
d sin = ;
3                 (d cos ) 1 ; cos ] =      2                               4
dx(1 ; x2) = 3    (14)
0                      1                                           ;1
then
4
P=3 0 4    Il 2 :                                                       (15)
The directive gain of an antenna, D( ), is de ned as
D(       ) = hSr i
P                                         (16)
4       r2

3
where 4 Pr2 is the power density if the power P were uniformly distributed
over a sphere. Substituting (12) and (15) into the above, we have
; Il 2
sin
= 3 sin2 :
0           2
D(     )=        2 4 r
1 4
; Il 2
2               (17)
4 r2 3 0    4

The peak of D( ) is known as the directivity of an antenna. It is 1:5 in this
case. If an antenna is radiating isotropically, its directivity is 1. Therefore,
the lowest possible values for the directivity of an antenna is 1, whereas it
can be over 100 for some antennas like re ector antennas. A directive gain
pattern is a plot of the above function D( ) and it resembles the radiation
power pattern.
If the total power fed into the antenna instead of the total radiated power
is used in the denominator of (16), the ratio is known as the power gain or
just bf gain. The total power fed into the antenna is not equal to the total
radiated power because there could be some loss in the antenna system like
metallic loss.
De ning a radiation resistance Rr by P = 2 I 2Rr , we have
1

Rr = 2P =              l   2

I2         6
0           where 0 = 377 :             (18)
For example, for a Hertzian dipole with l = 0:1 , Rr 8 . For a small dipole
with no charge reservoir at the two ends, the currents have to vanish at the
tip of the dipole.

I(z)

z
–a/2                            a/2

The e ective length of the dipole is half of its actual length due to the
manner the currents are distributed. For example, for a half-wave dipole,
a = 2 , and if we use le = 4 in (18), we have
Rr 50 :                              (19)
However, a half-wave dipole is not much smaller than a wavelength and does
not qualify to be a Hertzian dipole. Furthermore, the current distribution
on the half-wave dipole is not triangular in shape as above. A more precise
calculation shows that Rr = 73 for a half-wave dipole.

4

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