Communications Propagation Along a Two-Wire Line Located at the Air-Earth Interface

A simple quasi-static expression has been derived for the propagation constant of a two-wire transmission line located at the airearth interface. A numerical solution of the mode equation shows that the quasi-static approximation is valid when the wire separation is much less than a free-space wavelength. The quasi-static approximation can he used to determine the complex dielectric constant of the earth from measurements of either the propagation constant or the characteristic impedance of the transmission line. I . INTRODUCTION The theory of propagation along a single wire [ 1 1 , [2] or multiple wires [3], [4] located above the earth has been studied thoroughly, and the appropriate mode equations have been solved numerically to determine the complex propagation constants. In this communication we analyze the special case of a two-wire transmission line located at the air-earth interface and derive a simple quasi-static expression for the complex propagation constant of the differential mode. The limitations of the quasi-static result are determined by solving the rigorous mode equation numerically. The primary application of this work is to the in situ measurement of the complex dielectric constant of various soil types. Most soil measurements are made by packing samples into coaxial lines or waveguides [ 5 ] , [6], and the sample preparation must be done very carefully. Other in situ methods involving various antenna configurations have been suggested [7], but the two-wire transmission line configuration is attractive because of its large bandwidth and simplicity. 11. MODE EQUATION The geometry for a symmetrical two-wire line centered in the air-earth interface is shown in Fig. 1 . Each circular wire has radius a and is assumed to be perfectly conducting. The air has free-space permittivity E ” , and the earth has complex permittivity E,? . Both the air and the earth have free-space permeability w O . For the differential mode, the two wires carry oppositely phased currents / ( z ) of the form: I ( z ) = flu exp ( -jyz) ( 1 ) where z is the coordinate along the wires in the direction of propagation, /, is a constant, and y is the unknown complex propagation constant. The exp ( j u t ) time dependence is suppressed. In general, the complex propagation constant can be written, y = P j a ( 2 ) where p is the phase constant and a is the attenuation rate. [4] to the geometry in Fig. 1 , we can write the mode equation as If we specialize the differential mode equation of Olsen and Wu


I. INTRODUCTION
The theory of propagation along a single wire [ 11, [2] or multiple wires [3], [4] located above the earth has been studied thoroughly, and the appropriate mode equations have been solved numerically to determine the complex propagation constants.In this communication we analyze the special case of a two-wire transmission line located at the air-earth interface and derive a simple quasi-static expression for the complex propagation constant of the differential mode.The limitations of the quasi-static result are determined by solving the rigorous mode equation numerically.
The primary application of this work is to the in situ measurement of the complex dielectric constant of various soil types.Most soil measurements are made by packing samples into coaxial lines or waveguides [ 5 ] , [6], and the sample preparation must be done very carefully.Other in situ methods involving various antenna configurations have been suggested [7], but the two-wire transmission line configuration is attractive because of its large bandwidth and simplicity.

MODE EQUATION
The geometry for a symmetrical two-wire line centered in the air-earth interface is shown in Fig. 1 .Each circular wire has radius a and is assumed to be perfectly conducting.The air has free-space permittivity E", and the earth has complex permittivity E,?.Both the air and the earth have free-space permeability wO.
For the differential mode, the two wires carry oppositely phased currents / ( z ) of the form: where z is the coordinate along the wires in the direction of propagation, / , is a constant, and y is the unknown complex propagation constant.The exp ( j u t ) time dependence is suppressed.In general, the complex propagation constant can be written, where p is the phase constant and a is the attenuation rate.where .
The square roots in uo and u R are taken so that the real parts are positive.The derivation of (3) requires the thin-wire assumption that both koa and a / d are much less than 1.In general, the X integration must be evaluated numerically, and a numerical search for y in the complex plane is required.

QUASI-STATIC SOLUTION
To obtain a quasi-static solution to (3) we approximate the X integration for the case where kod is small.When Ad is small, the two cosine terms in (3) are both near 1 and tend to cancel.Consequently, when k,d is small the main contribution to the X integral comes from large values of A. For large A, we can write: ( 4 ) For the quasi-static case (small kod ) we retain only the A-' term in ( 4 ) and write the approximate quasi-static mode equation: F,(Y) = 0 ( 5 ) where .im A-' (cos hacos A d ) dh. (6) The X integral in (6) is a known form [8] whose value is In ( d / a ) .Thus (6) can be written, Thus the quasi-static solution for y from ( 7) is The quasi-static result in (8) is the same which Coleman [9] and Wait 111 obtained for a single wire located at the interface, and is the same as the form assumed by Wen [IO] for a pair of strips U.S. Government work not protected by U.S. copyright (9) In the general case where E* is complex, the measurement of y involves measuring both the phase constant P and the attenuation rate a as indicated in (2).
The best way to determine the range of validity of the quasistatic expressions (8) and ( 9) is to solve the general mode equation (3) without the quasi-static approximation.A computer program was written to solve (3) numerically by Newton's method using (8) For a given frequency, (10) gives an upper limit on the wire separation d.For example, the upper limit on d is 48 cm at 10 MHz, 4.8 cm at 100 MHz, and 4.8 mm at 1 GHz.If the upper limit on d is exceeded, then (8) and (9) are no longer valid and a numerical solution of (3) for y is required.The difficulties of oper-ating in the range, kod > 0.1, are that the transmission line might support an additional propagation mode [3] and that no simple relationship exists for E~ in terms of the measured y.

IV. CHARACTERISTIC IMPEDANCE
The quasi-static approximation for the characteristic impedance of the two-wire line can be obtained by replacing the half-space geometry in Fig. 1 by a homogeneous medium with an effective dielectric constant [lo].From (8), it can be seen that is The characteristic impedance Z, of the two-wire line is The above expression assumes that d / a >> 1, and this is consistent with the thin-wire approximation which was used in the derivation of the propagation constant.
If the characteristic impedance is measured, then E* can be determined from In the general case where E* is complex, the measurement of Z,. involves measuring both the amplitude and phase.
All of the previous analysis can also be applied to a pair of perfectly conducting strips of zero thickness located in the air-earth interface if the strip width w equals 4a.Thus a should be replaced by w/4 in the previous expressions to treat a transmission line made up of two identical strips.For example, the characteristic impedance becomes 112 This expression is identical to that given by Frazita [ 1 11 for d / w >> 1 .(14) z, = (PLO/%) In (4d/w)/a.

V. CONCLUSIONS
A simple quasi-static expression has been derived for the propagation constant of a two-wire line located at the air-earth interface.This expression is valid for sufficiently small kod as given by (10).The quasi-static analysis provides a simple expression for the complex dielectric constant of the earth if the propagation constant is measured.An alternative is to measure the characteristic impedance of the line and to determine the complex dielectric constant from (13).The analysis was performed for circular wires, but the results also hold for equivalent flat strips.
There are a number of possible extensions to this work.The feasibility of measuring the propagation constant or the characteristic impedance could be studied experimentally.The analysis could be extended to allow the magnetic permeability of the earth to differ from that of free space.In this case, it might be possible to determine both the permittivity and permeability from simultaneous measurements of the propagation constant and characteristic impedance.The effect of finite conductivity of the wires could be included by applying an axial impedance condition [4].The practical question of whether wires lying on the earth's surface are approximately equivalent to wires symmetrically located in the interface (as in Fig. 1) could be analyzed, but this type of analysis would require an examination of the validity of the thin-wire assumption [ 121.
as a starting value for y.Numerical values for p and a are shown in Figs. 2 and 3 as a function of kod for d / u = 20 and for two representative values of E , .Calculations for other values of d / a indicate that the results are only weakly dependent on d / a .It can be seen from Figs. 2 and 3 that the quasi-static approximation is valid for: