Lecture 25
Radiation by a Hertzian Dipole
25.1 Radiation by a Hertzian Dipole
Radiation by arbitrary sources is an important problem for antennas and wireless communi-
cations. We will start with studying the Hertzian dipole which is the simplest of a radiation
source we can think of.
25.1.1 History
The original historic Hertzian dipole experiment is shown in Figure 25.1. It was done in 1887
by Heinrich Hertz [18]. The schematics for the original experiment is also shown in Figure
25.2.
A metallic sphere has a capacitance in closed form with respect to infinity or a ground
plane. Hertz could use those knowledge to estimate the capacitance of the sphere, and also,
he could estimate the inductance of the leads that are attached to the dipole, and hence, the
resonance frequency of his antenna. The large sphere is needed to have a large capacitance,
so that current can be driven through the wires. As we shall see, the radiation strength of
the dipole is proportional to p = ql the dipole moment. To get a large dipole moment, the
current flowing in the lead should be large.
251
252 Electromagnetic Field Theory
Figure 25.1: Hertz’s original experiment on a small dipole (courtesy of Wikipedia [18]).
Figure 25.2: More on Hertz’s original experiment on a small dipole (courtesy of Wikipedia [18]
25.1.2 Approximation by a Point Source
A Hertzian dipole is a dipole which is much smaller than the wavelength under consideration
so that we can approximate it by a point current distribution, mathematically given by [31,38]
J(r) = ˆzIlδ(r) (25.1.1)
The dipole may look like the following schematically. As long as we are not too close to the
dipole so that it does not look like a point source anymore, the above is a good model for a
Radiation by a Hertzian Dipole 253
Hertzian dipole.
Figure 25.3: Schematics of a small Hertzian dipole.
In (25.1.1), l is the effective length of the dipole so that the dipole moment p = ql. The
charge q is varying in time harmonically because it is driven by the generator. Since
dq
dt = I,
we have
Il = dq
dt l = jωql = jωp (25.1.2)
for a Hertzian dipole. We have learnt previously that the vector potential is related to the
current as follows:
A(r) = μ
˚
dr′J(r′) e−jβ|r−r′|
4π|r − r′| (25.1.3)
Therefore, the corresponding vector potential is given by
A(r) = ˆz μIl
4πr e−jβr (25.1.4)
The magnetic field is obtained, using cylindrical coordinates, as
H = 1
μ ∇ × A = 1
μ
(
ˆρ 1
ρ
∂
∂φ Az − ˆφ ∂
∂ρ Az
)
(25.1.5)
where ∂
∂φ = 0, r = √ρ2 + z2. In the above,
∂
∂ρ = ∂r
∂ρ
∂
∂r = ρ
√ρ2 + z2
∂
∂r = ρ
r
∂
∂r .
254 Electromagnetic Field Theory
Hence,
H = − ˆφ ρ
r
Il
4π
(
− 1
r2 − jβ 1
r
)
e−jβr (25.1.6)
Figure 25.4: Spherical coordinates are used to calculate the fields of a Hertzian dipole.
In spherical coordinates, ρ
r = sin θ, and (25.1.6) becomes [31]
H = ˆφ Il
4πr2 (1 + jβr)e−jβr sin θ (25.1.7)
The electric field can be derived using Maxwell’s equations.
E = 1
jω ∇ × H = 1
jω
(
ˆr 1
r sin θ
∂
∂θ sin θHφ − ˆθ 1
r
∂
∂r rHφ
)
(25.1.8)
= Ile−jβr
jω4πr3
[
ˆr2 cos θ(1 + jβr) + ˆθ sin θ(1 + jβr − β2r2)
]
(25.1.9)
25.1.3 Case I. Near Field, βr 1
E ∼ = p
4πr3 (ˆr2 cos θ + ˆθ sin θ), βr 1 (25.1.10)
H E, when βr 1 (25.1.11)
where p = ql is the dipole moment, and βr could be made very small by making r
λ small or
by making ω → 0. The above is like the static field of a dipole. The reason being that in
Radiation by a Hertzian Dipole 255
the near field, the field varies rapidly, and space derivatives are much larger than the time
derivative.1
For instance,
∂
∂x ∂
c∂t
Alternatively, we can say that the above is equivalent to
∂
∂x ω
c
or that
∇2 − 1
c2
∂2
∂t2 ≈ ∇2
In other words, static theory prevails over dynamic theory.
25.1.4 Case II. Far Field (Radiation Field), βr 1
In this case,
E ∼ = ˆθjωμ Il
4πr e−jβr sin θ (25.1.12)
and
H ∼ = ˆφjβ Il
4πr e−jβr sin θ (25.1.13)
Note that Eθ
Hφ = ωμ
β = √ μ
= η0. Here, E and H are orthogonal to each other and are both
orthogonal to the direction of propagation, as in the case of a plane wave. A spherical wave
resembles a plane wave in the far field approximation.
25.1.5 Radiation, Power, and Directive Gain Patterns
The time average power flow is given by
〈S〉 = 1
2 <e[E × H∗] = ˆr 1
2 η0 |Hφ|2 = ˆr η0
2
( βIl
4πr
)2
sin2 θ (25.1.14)
The radiation field pattern of a Hertzian dipole is the plot of |E| as a function of θ at a
constant r. Hence, it is proportional to sin θ, and it can be proved that it is a circle.
1This is in agreement with our observation that electromagnetic fields are great contortionists: They will
deform themselves to match the boundary first before satisfying Maxwell’s equations. Since the source point
is very small, the fields will deform themselves so as to satisfy the boundary conditions near to the source
region. If this region is small compared to wavelength, the fields will vary rapidly over a small lengthscale
compared to wavelength.
256 Electromagnetic Field Theory
Figure 25.5: Radiation field pattern of a Hertzian dipole.
The radiation power pattern is the plot of 〈Sr 〉 at a constant r.
Figure 25.6: Radiation power pattern of a Hertzian dipole which is also the same as the
directive gain pattern.
Radiation by a Hertzian Dipole 257
The total power radiated by a Hertzian dipole is given by
P =
ˆ 2π
0
dφ
ˆ π
0
dθr2 sin θ〈Sr 〉 = 2π
ˆ π
0
dθ η0
2
( βIl
4π
)2
sin3 θ (25.1.15)
Since
ˆ π
0
dθ sin3 θ = −
ˆ −1
1
(d cos θ)[1 − cos2 θ] =
ˆ 1
−1
dx(1 − x2) = 4
3 (25.1.16)
then
P = 4
3 πη0
( βIl
4π
)2
(25.1.17)
The directive gain of an antenna, G(θ, φ), is defined as [31]
G(θ, φ) = 〈Sr 〉
P
4πr2
(25.1.18)
where
P
4πr2
is the power density if the power P were uniformly distributed over a sphere of radius r.
Substituting (25.1.14) and (25.1.17) into the above, we have
G(θ, φ) =
η0
2
( βIl
4πr
)2
sin2 θ
1
4πr2
4
3 η0π
( βIl
4π
)2 = 3
2 sin2 θ (25.1.19)
The peak of G(θ, φ) is known as the directivity of an antenna. It is 1.5 in the case of a
Hertzian dipole. 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 reflector antennas (see Figure 25.7). A directive gain pattern is a plot
of the above function G(θ, φ) and it resembles the radiation power pattern.
258 Electromagnetic Field Theory
Figure 25.7: The gain of a reflector antenna can be increased by deflecting the power radiated
in the desired direction by the use of a reflector (courtesy of racom.eu).
If the total power fed into the antenna instead of the total radiated power is used in the
denominator of (25.1.18), the ratio is known as the power gain or just 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.
25.1.6 Radiation Resistance
Defining a radiation resistance Rr by P = 1
2 I2Rr , we have [31]
Rr = 2P
I2 = η0
(βl)2
6π ≈ 20(βl)2, where η0 = 377 ≈ 120π Ω (25.1.20)
For example, for a Hertzian dipole with l = 0.1λ, Rr ≈ 8Ω.
The above assumes that the current is uniformly distributed over the length of the Hertzian
dipole. This is true if there are two charge reservoirs at its two ends. For a small dipole with
no charge reservoir at the two ends, the currents have to vanish at the tip of the dipole as
shown in Figure 25.8.
Radiation by a Hertzian Dipole 259
Figure 25.8: The current pattern on a short dipole can be approximated by a triangle since
the current has to be zero at the end points of the short dipole.
The effective 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 = λ
2 , and if we use leff = λ
4 in (25.1.20),
we have
Rr ≈ 50Ω (25.1.21)
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 = 73Ω for a half-wave
dipole [48].
The true current distribution on a half-wave dipole resembles that shown in Figure 25.9.
The current is zero at the end points, but the current has a more sinusoidal-like distribution
like that in a transmission line. In fact, one can think of a half-wave dipole as a flared,
open transmission line. In the beginning, this flared open transmission line came in the
form of biconical antennas which are shown in Figure 25.10 [124]. If we recall that the
characteristic impedance of a transmission line is √L/C, then as the spacing of the two
metal pieces becomes bigger, the equivalent characteristic impedance gets bigger. Therefore,
the impedance can gradually transform from a small impedance like 50 Ω to that of free
space, which is 377 Ω. This impedance matching helps mitigate reflection from the ends of
the flared transmission line, and enhances radiation.
Because of the matching nature of bicone antennas, they tend to have a broader band-
width, and are important in UWB (ultra-wide band) antennas [125].
260 Electromagnetic Field Theory
Figure 25.9: A current distribution on a half-wave dipole (courtesy of electronics-notes.co).
Figure 25.10: A bicone antenna can be thought of as a transmission line with gradually
changing characteristic impedance. This enhances impedance matching and the radiation of
the antenna (courtesy of antennasproduct.com).
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