How to model the antenna feed point correctly

Do you know how the feeding point is being modeled in your antenna simulation?

Excitation modeling is a critical step in the simulation of any kind of antenna and it is still not completely resolved. The input impedance of a transmitting antenna is directly impacted by the source model, but errors can also propagate to the radiation pattern, directivity and gain. On the other hand, we need a simple source model that is independent of the antenna structure, so that we don't have to repeat the calculations every time we change the source. There are several feeding systems for the excitation of real world antennas, so the question is, Is it possible to model a complete feeding system using just a single parameter?

Fortunately, in most cases, the neighborhood of the feeding point is small compared to the wavelength, therefore the antenna will "feel" the characteristic impedance Zc given by the ratio between the E and H excitation fields in that neighborhood. This is the case when two-wire or coaxial transmission lines are used to feed the antenna. So, the source model that we choose should consider the characteristic impedance of the feeding system as an input parameter, this being the single parameter we are looking for.

Next, let's briefly review the most popular source models, widely used in simulations by the Method of Moments (MoM).

Delta-gap or applied-field model

The feedpoint is reduced to a null width gap where the applied E-field is assumed to be infinite. It seems like an overly extreme simplification, however a finite input voltage V is obtained as an integral of the field represented by a Dirac delta function. The problem arises when the gap width tends to zero. The characteristic impedance of such a system is zero, and the derivative of the current distribution is infinite at the feeding point. As a consequence, the input impedance does not converge as finer discretization is used in the MoM. Then it was realized that the gap should be finite (and therefore its characteristic impedance). This model is the simplest, but it does not give correct results.

Magnetic frill or coaxial-line model

The feedpoint is surrounded by a magnetic field ring, which is equivalent (applying the image method) to the TEM field distribution at the open end of a coaxial transmission line crossing a ground plane. The excitation field is distributed throughout the space, but is extremely concentrated at the feedpoint. The model parameters are the inner and outer radii of the coaxial transmission line, so a non-zero and finite characteristic impedance is obtained. A problem arises when sampling the E-field, since a high degree of discretization is needed because it is very concentrated. To use this model, we have to calculate the inner and outer radii that give us a characteristic impedance equal to that of our feeding system (not all antennas are fed by coaxial lines). This model is not so simple, but it gives correct results.

Bicone or charge-discontinuity model

Here we gain simplicity and correct results. The "bicone" name comes from the fact that the characteristic impedance Zc of a biconic transmission line is used as an input parameter, which depends on a finite gap width and wire radius. The voltage V applied between the antenna terminals and the characteristic impedance Zc impose a boundary condition on the derivative of the current distribution (the electric charge) at the feedpoint. To the right of the feedpoint (at x = 0 in the image above) the derivative has the opposite sign than to the left (they can also differ in absolute value if there is asymmetry), hence the name of "charge-discontinuity" source model.

The biconic line source model could be replaced with any other equivalent transmission line as long as it has the same characteristic impedance. It is customary and convenient to replace the bicone with a wire segment in the MoM. However, only an approximate formula for the characteristic impedance of thin wires (a well-known log formula) is implemented in many simulation programs, which loses validity for thicker wires. To solve this, we can apply a semi-analytical method to obtain this impedance "exactly". The discrepancy between both methods can be seen in the following plot. For Zc = 50 Ohm, it is clear that the approximate formula cannot be used.

What source model are you using in your antenna simulations?

For further reading, you can have a look at the book by Ronold W. P. King, "The Theory of Linear Antennas" (Harvard Univ. Press, 1956), where "The Gap Problem" is discussed in page 844. This problem dates back to the 1940s, see e.g. "The Influence of the Width of The Gap Upon The Theory of Antennas" by L. Infeld (Quarterly of Applied Mathematics, July 1947). In the book by Roger F. Harrington, "Field Computation by Moment Methods" (IEEE Press, 1992) the gap is not treated in detail but difficulties in modeling are mentioned on page 71. Among modern books, we can see "The Method of Moments in Electromagnetics" by Walton C. Gibson (Chapman & Hall/CRC, 2008), chapter 4.

Tony Golden