Modeling a Wire Antenna with Insulation

Ralf Schlatterbeck

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[Update 2024-08-20: Correction of formula for L and equivalent radius]

[Update 2024-09-19: Correction of citation of article by Stearns, Supplement to Antenna Book [5]]

For my antenna modeling code pymininec (which is a re-implementation of the original Mininec code in Basic) I was researching how to model insulated wires. I had asked Roy Lewallen, W7EL, the author of EZNEC what he is using in EZNEC and received a pointer to a paper by J. H. Richmond [1] which seems the first paper to propose an algorithm for modeling insulated wires in an antenna and actually implementing it. The algorithm was implemented by Jerry McCormack in his masters thesis [2] and later incorporated into a contractor report [3] (the contractor report is almost identical with the thesis except for updated Fortran code but has a different author). The original Fortran code is not only listed in the reports but is still made available by Ray L. Cross on his "Antenna Scatterers Analysis Program" (ASAP) page. The subroutine "DSHELL" implements the insulated wire algorithm. With some modifications I was able to run this code today. Many thanks go to Roy, W7EL for sending me this information.

To test my implementation in pymininec against real measured data, I dug up a paper by David Lamensdorf [4] where the effect of insulation were actually measured. Since at the time the insulation could only be modelled as an infinite cylinder, the experiment used insulation that goes beyond the end of the wire until no more changes in the measured parameters were observed.

Experiments with the formulation by Richmond [1] indicated that the resulting impedance matrix is very ill-conditioned: For short wire segments the formulation yields very large absolute values that are added to the main diagonal of the impedance matrix and subtracted from the second diagonals (the matrix is symmetric and the second diagonal is the same for the lower and upper triangle). Since these large values are subtracted during the matrix inversion this leads to Catastrophic Cancellation and results that are dominated by cancellation effects.

So I investigated further and found a paper [5] and a presentation [6] by Steve Stearns, K6OIK which gives distributed inductance and an equivalent-radius formula for modeling insulation. His presentation also has a great history of various approaches to modeling insulation with programs that originally do not support it (like the original NEC-2 and the original Mininec):

\begin{align*} a_e &= a \left(\frac{b}{a}\right)^{\left(1- \frac{1}{\varepsilon_r}\right)} \\ L &= \frac{\mu_0}{2\pi}\left(1-\frac{1}{\varepsilon_r} \right)\log\left(\frac{b}{a}\right) \\ \sigma_e &= \sigma\left(\frac{a}{b}\right)^{2\left(1-\frac{1}{\varepsilon_r}\right)} \\ \end{align*}

where \(a\) is the original radius of the wire, \(b\) is the radius of the wire including insulation, \(\sigma\) is the conductivity of the wire, \(\varepsilon_r\) is the relative dieelectric constant of the insulation, \(a_e\) is the equivalent radius and \(\sigma_e\) is the equivalent conductivity that is used for removing the effect of the changed radius on the computation of the skin effect loading of the wire. The inductance \(L\) is the inductance per length of the insulated wire (or wire segment).

I then searched for a paper that would confirm the formulas by Steve Stearns and found an even older paper by Tai Tsun Wu [7] which has exactly the formula for the equivalent radius \(a_e\) and for the distributed inductance \(L\).

The formula for the equivalent conductivity is just used to cancel the effect of the equivalent radius on the computation of the skin effect load. In a previous blog post [8] I had given the formula for the skin effect load. In that formula, the conductivity \(\sigma\) appears as its reciprocal \(\rho\), the resistivity. The square root of \(\sigma\) is always used as a product with the radius. So we can undo the equivalent radius correction by multiplying \(\sigma\) with the reciprocal of the equivalent radius factor. We get

\begin{align*} \sqrt{\sigma}a &= \sqrt{\sigma_e}a\left(\frac{b}{a}\right)^{\left (1-\frac{1}{\varepsilon_r}\right)} \\ \frac{\sigma_e}{\sigma} &= \frac{1}{\left(\frac{b}{a}\right)^{2\left (1-\frac{1}{\varepsilon_r}\right)}} = \left(\frac{a}{b}\right)^{2\left(1-\frac{1}{\varepsilon_r}\right)} \\ \end{align*}

In pymininec I don't need to correct the equivalent radius for the skin effect computation, I can simply use the original radius there.

In the following three graphics I'm comparing various implementations of insulated wires with the measurements by Lamensdorf [4]. The wire has a diameter (not radius!) of 0.25in, the frequency is 600MHz. I'm using only the measurements with \(\varepsilon_r = 3.2\), the higher relative permittivities are unrealistic in practice and will deviate from the formulas due to the unrealistic experimental setup. I'm plotting the real- (conductance) and imaginary (susceptance) parts of the admittance (reciprocal of the impedance). The three graphics are:

  • Without any insulation

  • Diameter of wire with insulation of 0.375in (insulation 0.125in)

  • Diameter of wire with insulation of 0.5in (insulation 0.25in)

The traces in the plots are:

  • ASAP with two, four, eight and 16 segments for each half of a dipole, the traces are marked 'AS02', 'AS04', 'AS08' and 'AS16', respectively, I'm not showing all four traces in each plot

  • 4NEC2 which, according to Stearns [6], implements a formula derived by Cebik [9], these traces are marked '4N2'

  • NEC-2 using Wu's [7] formula for equivalent radius and distributed inductance as well as Stearns' [6] formula for equivalent \(\sigma\) marked as 'NEC'

  • My pymininec implementation using Wu's [7] formula for equivalent radius and distributed inductance (but not the \(\sigma\) correction because I can use the original radius for the skin effect directly) marked as 'mini'

  • The measurements from Lamensdorf [4] are the blue and orange bubbles (for the real and imaginary parts of the impedance, respectively)

  • In the graphics with insulation I'm also showing the results of EZNEC, the traces are greyed out and can be turned on in the legend on the right side

The Formula by Cebik [9] and by Wu [7] are very similar, the traces '4N2' and 'NEC' are almost identical.

I'm greying out the EZNEC traces and some of the ASAP traces because they exhibit numerical problems due to the ill-conditioned impedance matrix resulting from Richmond's formulation. It is interesting to see that EZNEC also exhibits numerical problems for smaller segments. You can see the traces by clicking into the legend on the right side.

The segmentation in all examples is 21 segments for the wire. This means small segments for the lower end of the lengths computed. It triggers a small segment warning in EZNEC for the first three points (up to and including \(\frac{4.2}{32} = 0.06558125\) wavelength) but not for the clearly numerically invalid 6th value in the first graphics with insulation and the points exhibiting numeric instability in the second graphics with insulation.

Note that the measurements by Lamensdorf [4] were performed on monopole-antennas. I'm simulating dipoles. The impedance of monopole is half that of a dipole, so for admittance I'm multiplying by 2. All admittances are in milli-Siemens, called milli-mhos (mho being ohm spelled backwards) with the symbol ℧ at the time of the Lamensdorf [4] paper.