On the impedance of a vertical half-wave antenna above an earth of finite conductivity

Abstract

The complex impedance of a vertical half-wave antenna located any distance above an earth of given conductivity and dielectric constant is calculated by the “induced electromotive force” method. Based on the assumption k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ≪ |k| (where k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> and k are the wave numbers for the atmosphere and earthy respectively), the results are applicable down to about 10 meters for any earth except a very dry soil. The calculation is based on the Sommerfeld-von Hoerschlemann expression for the field of a dipole above a half space of arbitrary electrical character. After splitting the total impedance into three parts, Z = Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> + Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> + Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> , the component Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> is shown to be the self-impedance of the antenna, Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> the mutual impedance between the antenna and its perfect image, and Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> an impedance component due to the finite conductivity of the earth. Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> is found to be proportional to two factors, one of which depends on the conductivity and dielectric coefficient of the earth and the wavelength and the other of which depends only on the ratio h/λ, where h = antenna height and λ = wavelength. Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> is put in a form suitable for the computation of any given case and curves are shown for four typical examples. For λ > 10 meters and all except very dry soil, the effect of the finite conductivity is quite small and the assumption, often made, of a perfectly reflecting earth thus is justified for a large number of cases. The impedance is, except for very short waves or exceedingly dry soil, substantially that obtained for a perfectly conducting earth. A principle of similitude is stated, in which two antennas over the same kind of earth and having equal values of h/λ have identical impedances.