The theory of Antenna Radiation
Antennas are everywhere. Whether it be in the form of large towers sitting atop rooftops on buildings, satellite dishes, or even in the form of very tiny printed circuits on your smartphone; antennas play a crucial role in transmitting information across the globe and into outer space without the need for wires or cables spanning ridiculous distances. So how does an antenna work, exactly? To answer this question, we will look at how electromagnetic waves can be produced in a simple model known as the Hertzian dipole antenna.
*Note: this article assumes you are familiar with some vector calculus and with Maxwell’s equations. If you would like a more foundational look at these, you may check out my other articles on topics such as “Deriving Gauss’s law from Coulomb’s law” and “Deriving Gauss’s law of magnetism”
The equations that describe antenna radiation in general
Antenna radiation can be studied in the context of electromagnetic (EM) waves using Maxwell’s equations in frequency-space, namely
These equations represent the electric field E and magnetizing field H (not to be confused with the magnetic field B, although in special cases these two are related by the equation B =𝜇H), along with a current density J and a charge density 𝜌. The constants 𝜇 and 𝜀 represent the permeability and permittivity of the propagation medium, respectively. The parameter 𝜔 represents an arbitrary frequency of oscillation (in units of rad/s, related to the frequency f in Hertz via 𝜔 = 2𝜋f), which can take on multiple values since waves can be constructed by adding multiple waves of different frequencies and amplitudes, so that the electric and magnetic fields are functions of space and frequency. This version of Maxwell’s equations is valid when we assume the EM fields to be time-harmonic (e.g. described by a series of sinusoids in time), and are obtained by taking the Fourier transform of the time-dependent equations (for instance, the Fourier transform of a time-varying quantity x(t) is Fourier[x(t)] = i𝜔X(𝜔), where i is the imaginary unit defined as i² = -1).
To get a solution to these equations, we focus on Eq(1a) first, by taking the curl of both sides
and substituting the left hand side by
by using what is known as the “double-curl identity of vector calculus”. Next, we replace the right-hand side of Eq(2) by Eq(1b) to obtain
which can be re-arranged into
At this point, we can make the substitution
such that we end up with an equation for the electric field E explicitly in terms of the current density J only
Using the method of Green’s functions, we can solve this equation in terms of the following operator corresponding to Helmholtz-type equations
thus integrating both sides of Eq(4) with respect to r’ yields the solution to the electric field
Note that this equation works for any type of current density J, by evaluating the integral
and from this we calculate the magnetizing field as
However, a more enlightening result is derived if we take the curl of both sides of Eq(6), which yields
the 2nd term becomes zero because of the vector identity
Then, we can write the magnetizing field as
and if we now define the vector potential A as
we can write both electric and magnetizing fields as functions of this vector potential
Approximate analysis of a dipole antenna
Eq(9a) and (9b) cannot be solved explicitly in most cases due to the complexity of the potential A in Eq(8). We can, however, consider a very simple case in which we possess a straight filament of length L oriented along the z-axis with current I, for which the current density may be written as
over the range
where 𝛥A is the cross-section of the rod, as shown in Figure 1(a).
This can be easily generated by connecting two L-shaped conducting rods to a voltage source, creating a filament of length L as in Figure 1(b) . The idea is to predict the radiation pattern for the case where L<<𝜆, where 𝜆 is the wavelength of the source. Furthermore, we assume that the width of the antenna itself is very small comparable to its length, such that we can write
Also, the length of the antenna is assumed to be very small compared to the distance in the far-field measurement, such that the displacement vector r - r’ does not vary much along z and can be treated as a constant r inside the integral. Introducing this expression into the integral in Eq(8) yields the vector potential in
However, since we expect the radiation pattern to be better represented in spherical coordinates, we now convert this expression into spherical coordinates via the matrix equation
Given that
we directly
Next, we take the curl of the field
from which we calculate the magnetizing field
Next we calculate the electric field outside the region of charge
and taking the curl of H we obtain
which yields the expression for the electric field
Near-field and far-field approximations
Eq(10) and Eq(11) are very complicated to understand due to the multiple terms they have. For this reason, it is often useful to look at two regions of interest. Firstly, we look at the near-field case where r<<𝜆, where the term r^(-n) with the highest n dominates over all others, which allows us to write
Similarly, we can write equations for the far-field case in which r>>𝜆 such that all terms r^(-n) for n > 0 are neglected, yielding
