A new approach to nearly constant input impedance of self-complementary antennas

Abstract

Conformal techniques, based on the covariance of Maxwell's electrodynamics under the full conformal group in four spacetime dimensions, are discussed in relation with the constant input impedance properties of frequency-independent antennas. In particular, we show that by applying suitable conformal transformations to existing self-complementary planar structures like logarithmic spirals, the constant input impedance property is considerably improved, especially in the low-frequency domain, where it usually fails due to the large wavelength involved. This procedure enables to perform a more intelligent truncation with the aim to capture some of the properties left behind in the cutting and to design realistic ultra wideband antennas with a more controlled value of the input impedance in the whole operation range.

Keywords:

• Conformal group
• electrodynamics
• antennas
• frequency-independent antennas
• logarithmic spiral antennas

Acknowledgments

This paper was partially supported by ANPCyT (PICT 2017-4581), CONICET (PIP 2017-1035), UNCUYO (SIIP 2019-06/C560) and CNEA. F.F. and P.A.C.C. are professors at IB and researchers of CONICET; and L.M. is assistant at IB and researcher of CNEA.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We shall adopt throughout the work the Einstein's summation convention.

2 We adopt here the common attitude in physics of considering $\mathrm{\Omega }(x)$ positive in order to prevent causality violations.

3 We will take, then, D = 4 all along this section.

4 The electron charge e is insensible to conformal changes of any sort.

5 Standard references on this topic are, among many other, Chapter one of [15] and Chapter seven of [16].

6 Here, conformal refers to the group briefly exposed in Section 2.1, and not to ‘adapted to a given surface’, as in [19].

7 This procedure is, however, non-unique. In fact, the kind of compact manifold obtained in the text comes from the specific coordinate change (34(34) $v=\tan p,u=\tan q,$(34) ). Other changes are admissible in order to get a boundary.

Additional information

Funding

This paper was partially supported by ANPCyT [grant number PICT 2017-4581], CONICET [grant number PIP 2017-1035], UNCUYO [grant number SIIP 2019-06/C560] and CNEA. F.F. and P.A.C.C. are professors at IB and researchers of CONICET; and L.M. is assistant at IB and researcher of CNEA.