# 9.1 The Hertzian Dipole

By regarding a linear antenna as consisting of a large number of infinitesimally short conducting elements where each is so short that current may be considered uniform over its length, the field of the entire antenna may be obtained by integrating the fields from all these differential antennas with the proper magnitudes and phases taken into account. We shall first examine the radiation properties of such a differential antenna, known as a Hertzian dipole, and in Section 9-3, we will extend the results to compute the fields radiated by a half-wave dipole, which is commonly used as a standard antenna for many applications.

In this case, the wire is oriented along the z direction in Fig. 9-4 and carries a sinusoidally varying current given by

i(t)=I 0 cos ω t=R e[I 0 e j ω t](A), (9.1)

where I 0 is the current amplitude. From Eq. (9.1), the phasor current I˜=I 0. Even though the current has to go to zero at the two ends of the dipole, we shall treat it as constant across its entire length.

# Figure 9-4 Short dipole placed at the origin of a spherical coordinate system.

Figure 9 4 Full Alternative Text

The customary approach for finding the electric and magnetic fields at a point Q in space (Fig. 9-4) due to radiation by a current source is through the retarded vector potential A. From Eq. (6.84), the phasor retarded vector potential A˜(R) at a distance vector R from a volume υ′ containing a phasor current distribution J˜ is given by

A˜(R)=μ 0 4 π∫υ′J˜e−j k R′R′d υ′,(9.2)

where μ 0 is the magnetic permeability of free space (because the observation point is in air) and k=ω/c=2 π/λ is the wavenumber. For the dipole, the current density is simply J˜=z ˆ(I 0/s), where s is the cross-sectional area of the dipole wire. Also, d υ′=s d z, and the limits of integration are from z=−l/2 to z=l/2. In Fig. 9-4, the distance R′ between the observation point and a given point along the dipole is not the same as the distance to its center, R, but because we are dealing with a very short dipole, we can set R′≈R. Hence,

A˜=μ 0 4 π e−j k R R∫−l/2 l/2 z ˆ I 0 d z=z ˆ μ 0 4 π I 0 l e−j k R R, (9.3)

The direction of A˜ is the same as that of the current (z direction).

Because our objective is to characterize the directional character of the radiated power at a fixed distance R from the antenna, antenna pattern plots are presented in a spherical coordinate system (Fig. 9-5). Its variables, R, θ, and ϕ, are called the range, zenith angle, and azimuth angle, respectively. To that end, we need to write A˜ in terms of its spherical coordinate components, which is realized with the help of Eq. (3.65c) by expressing z ˆ in terms of spherical coordinates:

z ˆ=R ˆ cos θ−θ θ ˆ sin θ.(9.4)

# Figure 9-5 Spherical coordinate system.

Figure 9 5 Full Alternative Text

Upon substituting Eq. (9.4) into Eq. (9.3), we obtain

A˜=(R ˆ cos θ−θ ˆ sin θ)μ 0 I 0 l 4 π e−j k R R=R ˆ A˜R+θ θ ˆ A˜θ+ϕ ϕ ˆ A˜ϕ,(9.5)

with

A˜R=μ 0 I 0 l 4 π cos θ e−j k R R,(9.6a)
A˜θ=−μ 0 I 0 l 4 π sin θ e−j k R R,A˜ϕ=0.(9.6b)

With the spherical components of A˜ known, the next step is straightforward; we simply apply the free-space relationships given by Eqs. (6.85) and (6.86) as

H˜=1 μ 0∇×A˜(9.7a)

and

E˜=1 j ω ε 0∇×H˜(9.7b)

to obtain the expressions

H˜ϕ=I 0 l k 2 4 π e−j k R j k R+1(k R)2 sin θ, (9.8a)
E˜R=2 I 0 l k 2 4 π η 0 e−j k R 1(k R)2−j(k R)3 cos θ, (9.8b)
E˜θ=I 0 l k 2 4 π η 0 e−j k R j k R+1(k R)2−j(k R)3 sin θ, (9.8c)

where η 0=μ 0/ε 0≈120 π (Ω) is the intrinsic impedance of free space. The remaining components (H˜R, H˜θ, and E˜ϕ) are everywhere zero. Figure 9-6 depicts the electric field lines of the wave radiated by the short dipole.

# Figure 9-6 Electric field lines surrounding an oscillating dipole at a given instant.

Figure 9 6 Full Alternative Text

# 9.1.1 Far-Field Approximation

As was stated earlier, in most antenna applications, we are primarily interested in the radiation pattern of the antenna at great distances from the source. For the electric dipole, this corresponds to distances R so that R≫λ or, equivalently, k R=2 π R/λ≫1. This condition allows us to neglect the terms varying as 1/(k R)2 and 1/(k R)3 in Eqs. (9.8a) to (9.8c) in favor of the terms varying as 1/k R, which yields the far-field expressions

and E˜R is negligible. At the observation point Q (Fig. 9-4), the wave now appears similar to a uniform plane wave with its electric and magnetic fields in phase, related by the intrinsic impedance of the medium η 0, and their directions orthogonal to each other and to the direction of propagation (R ˆ). Both fields are proportional to sin θ and independent of ϕ (which is expected from symmetry considerations).

# 9.1.2 Power Density

Given E˜ and H˜, the time-average Poynting vector of the radiated wave, which is also called the power density, can be obtained by applying Eq. (7.100); that is,

For the short dipole, use of Eqs. (9.9a) and (9.9b) yields

S av=R ˆ S(R,θ),(9.11)

with

The directional pattern of any antenna is described in terms of the normalized radiation intensity F(θ,ϕ) and is defined as the ratio of the power density S(R,θ,ϕ) at a specified range R to S max, which is the maximum value of S(R,θ,ϕ) at the same range. Thus,

F(θ,ϕ)=S(R,θ,ϕ)S max.(dimensionless)(9.13)

For the Hertzian dipole, the sin 2 θ dependence in Eq. (9.12) indicates that the radiation is maximum in the broadside direction (θ=π/2), corresponding to the azimuth plane, and is given by

S max=S 0=η 0 k 2 I 0 2 l 2 32 π 2 R 2=15 π I 0 2 R 2 l λ 2(W/m 2),(9.14)

where use was made of the relations k=2 π/λ and η 0≈120 π. We observe that S max is directly proportional to I 0 2 and l 2 (with l measured in wavelengths) and that it decreases with distance as 1/R 2.

From the definition of the normalized radiation intensity given by Eq. (9.13), it follows that

F(θ,ϕ)=F(θ)=sin 2 θ.(9.15)

Plots of F(θ) are shown in Fig. 9-7 in both the elevation plane (the θ plane) and the azimuth plane (ϕ plane).

# Figure 9-7 Radiation patterns of a short dipole.

(a) Elevation pattern

Figure 9 7-a Full Alternative Text

(b) Azimuth pattern

Figure 9 7-b Full Alternative Text

Concept Question 9-1: What does it mean to say that most antennas are reciprocal devices?

Concept Question 9-2: What is the radiated wave like in the far-field region of the antenna?

Concept Question 9-3: In a Hertzian dipole, what is the underlying assumption about the current flowing through the wire?

Concept Question 9-4: Outline the basic steps used to relate the current in a wire to the radiated power density.