Image Theory: Radiation From Hertzian Dipole Above Ground Plane

This article presents the derivation of the radiated far fields from a Hertzian dipole antenna above the ground plane using image theory.

Hertzian Dipole Antenna

An electric dipole, often referred to as a Hertzian dipole, shown in Figure 1, consists of a short, thin wire of length l carrying a constant current positioned symmetrically at the origin of the coordinate system and oriented along the z-axis.

Ideally, the wire is infinitely short; practically, a wire of the length l <<  λ /50 (λ = wavelength) can be considered a Hertzian dipole.

The far field of a Hertzian dipole has only a θ component (in a spherical coordinate system) and is given by [1].

Eq. 1

The complete derivation of this result is given in [2].

Image Theory

In image theory, a radiating antenna (actual source) is placed at some distance h from a perfect conducting plane. An image of this antenna (virtual source) is placed below the conducting plane at the same distance, h, as shown in Figure 2 [3].

Because of the reflecting ground plane, the total field at an observation point P is the sum of the direct wave and the reflected wave. Obviously, there are no fields below the ground plane.

Instead of obtaining the total field by summing the actual direct and reflected waves, we add the direct wave from the actual source and the direct wave from its image (virtual source) to obtain the same result (above the ground plane). When considering the virtual direct wave, we pretend that the ground plane does not exist and, therefore, the virtual wave has a direct, unobstructed path to the observation point.

Why are we using this approach? Because the calculation of the fields using the actual waves is quite complicated, whereas the calculations using image theory are quite simple, as we shall see.

Radiation from Hertzian Dipole above Ground Plane

Consider the geometry shown in Figure 3.

The source is an infinitesimal dipole of length l, carrying a constant current I₀. The observation point P is in the far field. The direct component of the E field at the observation point is, [2]

 Eq. 2a

The virtual component is

Eq. 2b

The total field at the observation point is

Eq. 3

In the far field, r >> h and the lines from the actual and virtual dipoles to the observation point may be considered parallel, as shown in Figure 4.

This is often referred to as the parallel-ray approximation.

Obviously, we have

Eq. 4a

Eq. 4b

From Figure 5, we have

Eq. 5a

Eq. 5b

We will use the approximations in Equations 4 and 5 when substituting for r₁ and r₂ in the phase component of the expressions in Equation 3. That is,

Eq. 6a

Eq. 6b

When approximating the amplitude components, we may further approximate r₁ and r₂ as

Eq. 7a

Eq. 7b

Thus,

Eq. 8

Using the Euler identity for a cosine function, we arrive at

Eq. 9

References

1. Paul, C. R., Introduction to Electromagnetic Compatibility, 2^nd Edition, John Wiley and Sons, New York, 2006.
2. Bogdan Adamczyk, Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017.
3. Balanis, C. A., Antenna Theory Analysis and Design, 3^rd Edition, Wiley Interscience, Hoboken, New Jersey, 2005.