Validation of NEC2 and Zo by Feedpoint Reactance of Monopoles

Modeler's Notes

Gerald J. Burke
Lawrence Livermore National Laboratory
Published in: ACES Newsletter, Vol. 10 No. 2, July 1995

For this issue of Modeler's Notes we have a contribution from Ralph Holland, VK1BRH, on NEC results for the feedpoint reactance of monopoles. This will be followed by an update of NEC subjects from the last issue and a do-it-yourself change to NEC to reduce the solution time for large matrices.

Holland's results show good agreement in feedpoint reactance between NEC and the simple formula attributed to Howe. I do not have access to references [1] and [2] that Holland cites, but it would be interesting to trace the history of the bicone approximations for dipole or monopole capacitance.

R. E. Collin, in Antennas and Radiowave Propagation (McGraw-Hill, 1985), gives the form of Holland's equation (8) derived from the biconical transmission line analysis of Schelkunoff, Advanced Antenna Theory. Collin says that it is "not very accurate but does provide a useful estimate."

In the earlier book Antennas, Theory and Practice by Schelkunoff and Friis (Wiley, 1951, p.305), the form of Holland's equation (9) (Howe's form) is obtained for the antenna capacitance by including the effect of the wire ends on the charge distribution.

Holland's results show that the end effect is worth including at zero cost in complexity.

Of course Schelkunoff's and Howe's equations do not take account of the source-gap width, and NEC does so in a "fuzzy" sense, so the agreement would no doubt be different for thicker wires.

Anyone who has comments on material covered here, suggestions or observations on using NEC or other modeling codes, or can submit an article on EM modelling topics, is encouraged to submit them to:

Gerald J. Burke

Lawrence Livermore National Laboratory

PO Box 5504, L-156

Livermore, CA 94550

Any contributions will be very welcome, and thanks to Ralph Holland for his results on input reactance.

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Editorial Note

Gerald J. (Jerry) Burke was the principal author of the Numerical Electromagnetics Code (NEC) at Lawrence Livermore National Laboratory. His Modeler's Notes in the ACES Newsletter provided the community's primary forum for NEC guidance and validation. Burke's commentary above constitutes the peer review of the accompanying paper by Ralph Bruce Holland VK1BRH, and represents his independent corroboration that Holland's results validating NEC2 and the Howe characteristic impedance formula are technically sound. Jerry Burke is remembered with great warmth and respect. See: In Memoriam — Gerald J. Burke, LLNL

Validation of NEC2 and Zo by Feedpoint Reactance of Monopoles

Ralph Bruce Holland, VK1BRH
8 Hardy Place
Kambah, ACT 2902
Australia

Published in: ACES Newsletter, Vol. 10 No. 2, July 1995

Abstract

It is a well known fact, in practical circles, that the reactance of a short linear radiator is related to its static or low-frequency capacitance. This relationship holds true for lengths up to 1/20 of a wavelength [1]. This capacitance can be used to determine the average characteristic impedance of that radiator and to validate the performance of NEC [3].

Introduction

Short radiators can be considered as a uniform transmission line with little or no losses per unit length [1] — i.e. the propagation constant is nearly pure imaginary and the characteristic impedance is nearly pure real. For such cases, the transmission line formula [1][2][4][5] can be simplified with the relationship between the input impedance, characteristic impedance and length given by:

$X_{i}=-Z_{0}\cot(\beta h)$

where $h$ is height and $\beta =2\pi /\lambda$.

When $\beta h$ is small ($h\leq \lambda /20$):

$X_{i}\approx -{\frac {Z_{0}}{\beta h}}$

The characteristic impedance is also related to the inductance and capacitance per unit length:

$Z_{0}=R_{0}(1-\alpha /\beta ,j)$

where $\alpha$ is the attenuation constant.

For small $\alpha$:

$R_{0}={\sqrt {\frac {L}{C}}}$

where $L$ is in H/m and $C$ is in F/m.

For free space:

$LC=\mu _{0}\epsilon _{0}={\frac {1}{c^{2}}}$

Combining the equations:

$C={\frac {1}{cR_{0}}}={\frac {3.3356\times 10^{-9}}{R_{0}}}$

Capacitive reactance is:

$X_{c}=-{\frac {1}{\omega C_{0}}}$

where $C_{0}$ is capacitance and $\omega =2\pi f$.

Several formulae have been provided by various authors to determine the characteristic impedance $Z_{0}$ of monopole radiators. There is some contention about which formula should be used because the characteristic impedance of a linear element is not uniform along its length.

Characteristic-Impedance Formulae

Schelkunoff [1] states, from biconical dipole theory, that for a monopole:

$R_{0}={\frac {\eta }{2\pi }}\left[\ln \left({\frac {2h}{a}}\right)-1\right]$

where $\eta ={\sqrt {\mu _{0}/\epsilon _{0}}}=377\ \Omega$.

Howe [1] states:

$R_{0}={\frac {\eta }{2\pi }}\left[\ln \left({\frac {h}{a}}\right)-1\right]$

Verification

NEC-81 [3] was used to simulate the feedpoint reactance $X_{i}$ for various vertical dipoles in free space. The corresponding reactances were divided by two to relate them to monopoles over ideal gro.nd.

(The impedance of a monopole of length $h$ over perfect ground is half that of a dipole of length $2h$.)

The negative reactances were transformed into equivalent capacitances using the reactance equation above.

• Figure 1 shows input capacitance $C_{0}$ versus frequency for several antenna heights.
• Figure 2 shows capacitance per meter $C$ versus $h/a$, comparing NEC2 results with Schelkunoff and Howe formulae.
• NEC2 agrees reasonably well with Schelkunoff's theory.
• Howe's formula agrees even better with NEC2 for small $h/a$.

Conclusion

The good correspondence between feedpoint reactance and derived characteristic impedance $Z_{0}$ indicates that:

1. NEC2 is reasonably accurate in determining the reactive component of short radiators.
2. Howe's formula corresponds more closely to NEC2 than Schelkunoff's over a wide range.
3. Characteristic-impedance calculations for short antennas and loading applications should preferably use Howe's formula.

References

1. H. Paul Williams, Antenna Theory and Design, Volume II, 2nd Ed., 1966.
2. H. Paul Williams, Antenna Theory and Design, Volume I, 2nd Ed., 1966.
3. G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC-81), UCID-18834, Lawrence Livermore National Laboratory, 1981.
4. John D. Kraus, Antennas, 2nd Ed., 1988.
5. Simon Ramo, John R. Whinnery, Theodore Van Duzer, Fields and Waves in Communication Electronics, 3rd Ed., 1994.