What is the best method to compute the antenna input impedance?

Computing accurately the input impedance of a transmitting antenna as a function of frequency is crucial for the matching network design and the bandwidth calculations.

Do you know how the antenna input impedance is being calculated in your simulations?

First of all, let's begin with a standard definition.

Conventional definition of input impedance 

No matter what numerical method we use to determine the antenna current distribution, we will always need a model for the excitation at the feed point. A source model consists of a field distribution in a neighborhood of the antenna feed point and boundary conditions on the input current or voltage. The most used source models and their pros and cons are described in this article: source models. The input impedance can then be obtained by first calculating the input voltage Vi as a line integral of the excitation E-field about the feed point and then dividing it by the input current Ii, this being the conventional definition of an input impedance, Zi = Vi/Ii.

Depending on the antenna feed point boundary conditions, the input impedance may not be well defined as the ratio of input voltage to input current. Power calculations may be necessary to obtain a realistic input impedance that is close to experimental results. So let's briefly review some methods for determining the antenna input impedance that are not based on the conventional definition. We assume that the antenna current distribution has already been computed or that we have a reasonably good approximation.

The Energy method

The Energy method is purely based on first principles and it doesn't depend on the numerical technique used to obtain the antenna current distribution (moment methods, finite elements, etc.).  If we take a closed surface S that contains the antenna inside, we must first calculate the flux of the complex Poynting's vector P through S as well as the magnetic and electric energies stored per unit time in the volume inside the surface (Wm-We term in the above figure). Then, by invoking the principle of conservation of energy, the sum of these powers (flux and volumetric) must be equal to the antenna input power, Wi. Thus, the input impedance Zi will be obtained as the ratio of this complex input power to the square of the absolute value (rms) of the input current, Ii.

Using a sphere as S is a good choice in numerical computations. It is also customary to compute the Poynting's flux in the antenna far field zone to take advantage of the TEM nature of the radiated field. This procedure is known as the Poynting-Vector method and it allows to obtain the radiation resistance, which must coincide with the real part of the input impedance. In fact, the error between both resistances is used to validate a model.

The Induced EMF method

We can view this method as a limiting case of the Energy method, where the surface S surrounding the antenna is reduced to the antenna surface itself. The volume between the antenna structure and S is then reduced to zero, so there is no stored energy to be computed. We save a lot of time and effort in this way because we avoid calculating volume integrals.

The complex Poynting's vector flux through the antenna surface can then be rewritten as the dot product of the induced E-field (the electromotive force, EMF) and the complex conjugate of the current density Js flowing on the antenna structure, the E-field itself being a function of Js. The input impedance is then obtained in the same way as in the Energy method.

The Induced EMF method has been one of the traditional procedures to calculate the input impedance of linear antennas since the involved integrals can be evaluated in closed form for sinusoidal current distributions.

The Variational method

A reaction functional for the input impedance as a function of the current distribution can be obtained from the Reciprocity theorem. This functional has a convenient extremal property: its first variation is zero, which means that the input impedance variational expression is insensitive to deviations from the correct antenna current. So we are able to compute the input impedance accurately just using a rough approximation to the current distribution. In this way, we can use a modest spatial discretization in our numerical method to obtain a practical impedance value, saving simulation time and computer memory.

Notice in the figure above that the input impedance functional resembles the Induced EMF method, but they are not the same.

Then, what is the best method?

As a first approximation to the antenna input impedance we can apply the conventional definition after employing a source model at the feed point, so we have to calculate the ratio of the input voltage and current. When we need to validate this result or obtain further refinement (e.g. to fit experimental data), a good approach would be to apply the Energy, the Induced-EMF or the Variational method.

As an example, the figure below shows the input resistance of a center-fed cylindrical antenna (length L = 0.5 m, radius a = 5 mm, gap g = L/200) as a function of frequency calculated using the conventional definition. On the other hand, the Poynting-Vector method has been applied to obtain the radiation resistance. Previously, the current distribution was obtained by the Method of Moments using a charge-discontinuity source model at the antenna feed point. A rms error < 0.02% between both resistances is obtained, which validates the source model used.

For further reading, see "The Theory of Linear Antennas" by Ronold W. P. King (Harvard Univ. Press, 1956), where the Induced EMF and Poynting-Vector methods are covered on page 258 for linear antennas but can be easily generalized to any other antenna type. For a more general approach, see "Time-Harmonic Electromagnetic Fields" by Roger F. Harrington (IEEE Press, 2001). There is an interesting article by O. Pekonen, K. Nikoskinen and S. Goodwin, "Applying Variational Principles in FDTD Antenna Input Impedance Simulations" (Helsinki Univ. of Tech., Electromagnetics Lab., Report 268, 1998) where the authors show the advantage of using the reaction functional in FDTD simulations.

Tony Golden