IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 33, NO. 1 , FEBRUARY 1991 65

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Tutorial Note on the General Transmission Line Theory for a Thin Wire Above the Ground

James R. Wait, Life Fellow, IEEE

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Abstrucf-Transmission along an overhead wire or cable conductor is considered. It is shown that the rigorous solution of the problem can be put in the context of classical transmission line theory that has been employed by power engineers for many years. However, the effective line parameters are spatially dispersive in the sense that they depend on the actual propagation constant of the mode in question.

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Fig. 3. Measured E- and H-plane aperture distributions at top diameter of conical section (DH = 85.5”, Bo = 15.75”): (a) 3.95 GHz absorber-lined wall; (b) 6.175 GHz absorber-lined wall and smooth metal wall; (c) 11.20 GHz absorber-lined wall.

More recently, because of the significant EM1 reduction achieved by these low WAR levels, CC’s of the above types are now gradually being incorporated into military radar/ECM/communica-

The temptation to apply transmission line formulations to over- head wire lines is often overwhelming. Unfortunately, pitfalls abound when one postulates or assumes that the effective line parameters are independent of the axial wave number or propaga- tion constant [1]-[3]. Rigorous formulations have been made, and cogent and useful numerical results are now available [4]-[9]. However, it is not obvious from such studies how the simple classical transmission line formulation emerges as a special case. We show how this can be done very simply for an assumed time harmonic excitation where the time factor is exp ( j w t ) , where w is the angular frequency.

The model is simplified here to the case where the ground is represented as an homogeneous halfspace (for x 5 0) with permit- tivity e2 and permeability p2. To allow for a dissipative ground, e2 can be replaced by O 2 - j a / w , where a is the ground conductivity. The upper halfspace (for x > 0) is also homogeneous with permit- tivity O1 and permeability pI, which may be replaced by their respective free space values q, and po in the present context. A single infinitely long thin wire is located at heighth (i.e., at x = h and y = 0 for - w < z < + w), as is indicated in Fig. 1:

To simplify the discussion, we assume that a plane electromag- netic wave is incident from the upper halfspace with arbitrary polarization. This planewave will be reflected from the air-earth interface at x = 0. If the wire were not present, it is a simple matter to deduce the form of the axial electric field at the wire location. We may specify this “applied” field as follows;

( 1 ) E:pp = E, exp ( - jgz) tion systems and are also being studied for possible use in high-power microwave (HPM) systems to reduce/eliminate EM1 and the

applications can be made to occupy small volumes and to meet various gain/bandwidth requirements.

where

= O. When the wire is present, we assert that the current on the wire

is the axial wavenumber of the planewave and

potential patricide problem. Special designs for many of these where EO is the value Of the at = h7 y = O7

will have the form REFERENCES

[l] C. M. h o p and Y. B. Cheng, “A note on the mitigation of EM1 caused by wide-angle radiation from parabolic-dish antennas,” IEEE Trans. Electrornagn. Cornpat., vol. 30, no. 4, pp. 583-586, Nov. 1988.

[2] J. N. Hines, T. Li, and R. H. Tumn, “The electrical characteristics of the conical horn-reflector antenna,” Bell Syst. Tech. J., vol. 42, pt. 2, pp. 1187-1211, July 1963 (see also A. W. Love, Electrornag- netic Horn Antennas. New York IEEE Press, 1976, pp. 422-446). H. T. Friis, “Microwave repeater research,” Bell Syst. Tech. J., Apr. 1948. A. B. Crawford, D. C. Hogg, and L. E. Hunt, “A horn reflector for space communication,” Bell Syst. Tech. J . , vol. 40, pp. 1095-1116, July 1961 (see also A. W. Love, Electromagnetic Horn Antennas. New York IEEE Press, 1976, pp. 400-421).

[SI US. Patent 4 410 892, Oct. 18, 1983. [6] C. M. b o p , Y. B. Cheng, and E. L. Ostertag, “On the fields in a

conical horn having an arbitrary wall impedance,” IEEE Trans. AntennasPropagat., vol. AP-34, no. 9, pp. 1092-1098, Sept. 1986.

[7] A. Z. Elsherbeni, J. Stanier, and M. Hamid, “Eigen values of propagating waves in a circular waveguide with an impedance wall,” Proc. Inst. Elec. Eng., vol. 135, pt. H, no. 1, pp. 23-26, Feb. 1988.

[3]

[4]

I ( z) = 1, exp ( - jgz) (2) for - w < z < + 00, where I, is a constant. The principal objec- tive now is to deduce Z, given E, and g. Here, we also need to say something about the characteristics of the wire itself. In the present context, we employ the impedance condition stated as follows:

(3) E? = Zint( g ) I ( Z )

where E,” is the average resultant tangential electric field at the wire surface, and Z’”‘(g) is the specified internal impedance per unit length [ 101.

Now, the resultant E,, for - w < y I + w and for any x > 0, is the superposition of the applied and secondary fields produced by the wire current in the presence of the half space. An expression for E r , which is the secondary field, was derived earlier [11]-[13]. It

Manuscript received January 23, 1990; revised July 16, 1990. The author is with the Electromagnetics Laboratory/ECE Department,

IEEE Log Number 9040737. University of Arizona, Tucson, AZ 85721.

0018-9375/91/0200-05$01.00 0 1991 IEEE

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66 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 33, NO. 1 , FEBRUARY 1991

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Fig. 1. Thin infinitely long wire at height h over the interface between two halfspaces.

is given conveniently in the form

E," = - ( j p w / 4 r ) B ( g ) e x p ( - j g z ) ~ , (4)

where B( g ) is a rather complicated function that we do not need to write out here.

The wire impedance boundary condition, as stated by (3), should, of course, be applied at the actual surface specified by

( x - h)' +y' = U' ( 5 ) where a is the radius of the wire. However, if we choose h > > U corresponding to the thin wire assumption, it is permissible to choose the matching point at x = h and y = a, which is to hold for all values of z . Thus, it follows that

1, = E , / [ Z ' " ' ( g ) + Z e x ' ( g ) ] (6) where Zext( g ) is the external impedance of the wire in the presence of the lower half space (i.e., the ground). In the case where p, = pz = p, it is possible to write [11]-[12]

Z""'(g) = z ( g ) + g'/ y ( g )

Z ( g ) = ( j ~ w / 2 r ) [ A + 2 ( Q -P)]

Y ( g ) = ( j e w 2 1 ) [ A + 2( N - jM)]

(7)

(8)

(9)

where

and

where

A = KO[ j ( k: - g2)1 /2a ] - KO[ j ( k: - g2)'/'(4h2 + a')"'] (10)

Q - j ~ = J, ( U 1 + u,)- 'exp(-2u1h)cosfadf (11 ) 00

and

N - j M = -L k' /"[U, + ( k ~ / k ~ ) u , ] ~ 1 e x p ( - 2 u , h ) c o s f u d ~

k2' 0

(12) where

U, = (g' +f2 - k:)l/'and uz = ( g ' + f' - k:) 1 /'

where

k: = E,FW' and k$ = e2pwz.

In the case where I(k: - g')'/'aI 4 1 and h * U , the modified Bessel functions in (10) can be replaced by their small argument approximations. Then, we have the quasi-static form

A P In [(4h2 + a2)1/2/a] P 1n(2h /a ) . (13) If, in addition the ground is sufficiently well conducting such that I k l / k 2 1 ' 4 1 and I k , h J 4 1, we see that

Q - jPz -J,/2 (14)

where

J, = (2/k;)/"(u - f ) e x p ( - 2 f h ) df (15) 0

where 1 /'

U = (f'- k;) . Here, J, is the famous Carson integral [14]. Under the same conditions, the integral N - j M can be neglected. As a conse- quence, the impedance and admittance parameters Z( g ) and Y( g ) are approximated by their nondispersive forms

Z ( g ) = Z , = ( j p w / 2 ~ ) [ l n ( 2 h / a ) - J,] (16)

Y ( g ) I Y, = ( j e w 2 ~ ) [ l n ( 2 h / a ) ] - ' . (17)

In this quasi-static regime, a simple expression for the internal series impedance Z'"'(g) suffices if, for example, the wire is a metallic conductor [ 101. Such a form is

Z'"' = ( j p , , , ~ / u ~ ) ~ " Z , ( j k , a ) / [2~aZ , ( jk,a)] (18)

and

where 11' jk, = ( j p m w u m )

in terms of the metal conductivity U, and metal permeability p,. Here, Io and Zl are modified Bessel functions. If I k,a( 4 1, we retreive the dc limit

zint = (TU'U,) - l . (18')

To solve generally for the modal propagation constant(s), we return to (6) and examine the zeroes of the denominator. These solutions are the discrete set g = g , , where n is the order of the mode. For example, in the case of a perfectly conducting wire, Zint = 0, and the mode equation is simply Zex ' (g ) = 0, or from (7) g , are the solutions of

j g = [ z ( g ) ~ ( g ) ] ~ " . (19)

jgo = ( zcyc)1 /2 (20)

In the quasi-static or Carson limit, this becomes an explicit formula for the transmission line mode given by

where Z , and Y, are given by (16) and (17) respectively. The Carson integral J, occurring in (16), as defined by (15), is rela- tively simple to deal with. In fact, Carson [14] provided numerical values many years ago. In addition, it can be expressed conveniently in terms of Struve functions as pointed out elsewhere [ 101 - [ 1 11.

To fully comprehend the propagation characteristics of the ele- vated conductor, for an unlimited frequency range, we need to deal directly with the poles of the right-hand side of (6) and thus solve for the roots g,. The complexity of this task is nicely covered by Kuester et al., [6] who also review the related Soviet contributions to the subject.

An early paper by Kikuchi [15] is also worthy of note. He employs an ingenious circuit approach to obtain an extension to the Carson formulation. In the present context, it would appear to be equivalent to solving the mode equation such as (19) by replacing g on the right-hand side by the wavenumber k, of the upper half space. This perturbed transmission line mode is really a first-order correction to the Carson mode. Of course, such a procedure does not predict the existence of fast wave modes, as is demonstrated in more recent studies [4]-[7].

Other attempts to improve the transmission line formulations have also been made. Degauge and Zeddam [16] indicate that the vertical risers at the ends of the elevated conductors or cables should be taken into account. Paul [17] presents a general matrix formulation for multiple overhead conductors allowing for the ground perme-

T -

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 33, NO. 1, FEBRUARY 1991 67

ability contrast, but the restrictions appear to be the same as for the Carson theory.

To deal explicitly with localized sources in the upper half space, we should formulate the problem as a spectrum over all wavenum- bers g. This subject is beyond the scope of this review. However, interested readers may wish to consult [4]-[13] for further insight. In particular, useful numerical data appear in [4]-[9].

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REFERENCES J. G. Anderson and J. R. Doyle, Transmission Line Reference Book, 345 kV and Above. Electric Power Research Institute, 1975. E. Sunde, Earth Conduction Eflects in Transmission Systems. New York: Van Nostrand, 1949. Y. Kami and R. Sato, “Transient response of a transmission line excited by an electromagnetic pulse,” IEEE Trans. Electromagn. Compat., vol. 30, no. 4, pp. 457-462, Nov. 1988. R. G. Olsen and D. C. Chang, “Current induced by a plane wave on an infinite thin wire near the earth,” IEEE Trans. Antennas Propa- gat., vol AP-22, no. 4, pp. 586-589, July 1974. R. G. Olsen and D. C. Chang, “The excitation of an infinite horizontal wire above earth by a vertical electric dipole,” IEEE Trans. Antenna Propagat., vol. AP-25, no. 4, pp. 560-565, July 1977. E. F. Kuester, D. C. Chang, and S . W. Plate, “Electromagnetic wave propagation along horizontal wire systems in or near a layered earth,” Electromagn., vol. 1, no. 3, pp. 243-266, July-Sept. 1981. G. E. J. Bridges pnd L. Shafai, “Plane wave coupling to multiple conductor transmission lines above a lossy earth,” IEEE Trans. Electromagn. Compat., vol. 31, no. 1, pp. 21-38, Feb. 1989. J. M. Fontaine, A. Umbert, B. Djebari, and J. Hamelin, “Ground effects in the response of a single wire transmission line illuminated by an EMP,” Electromagn., vol. 2, no. 1 , pp. 43-54, Jan-Mar. 1982. H. P. Neff and D. A. Reed, “The effect of secondary scattering for a long wire over an imperfect ground from an incident EMP,” IEEE Trans. Antennas Propagat., vol. 37, no. 12, pp. 1554-1558, Dec. 1989. J. R. Wait, Introduction to Antennas and Propagation. London: Peter Peregrinus Ltd./Institute of Electrical Engineers, 1986, ch. 7. J. R. Wait, “Theory of wave propagation along a thin wire parallel to an interface,” Radio Sri., vol. 3, pp. 29-37, Mar. 1972. J. R. Wait, “Excitation of a coaxial cable or wire conductor located over the ground by a dipole radiator,” Archiv Electron Uebertra- gungstechnik, vol. 31, pp. 121-127, 1977; Errata, p. 230. J. R. Wait, “Excitation of an ensemble of parallel cables by an external dipole over the earth,” Archiv Electron. Uebertragung- stechnik, vol. 31, pp. 489-493, 1977. J. R. Carson, “Wave propagation in overhead wires with ground return,” Bell Syst. Tech. J . , vol. 5, pp. 539-554, 1926. H. Kikuchi, “Etude de la transition entre un circuit de retour par la terre et un guide d’onde de surface,” L’Onde Electrique, vol. 383, no. 376, supplement special, 1957/1958. P. Degaugue and A. Zeddam, “Remarks on the transmission-line approach to determining the current induced on above-ground cables, IEEE Trans. Electromagn. Compat., vol. 31, no. 1, pp. 77-80, Feb. 1988. C. R. Paul, “Solution of the transmission line equations for lossy conductors and imperfect earth, Proc. Inst. Elec. Eng., vol. 122, no. 2, pp. 177-182, Feb. 1975.

Calculation of Electric Fields Due to Lines of Charge

Antonio R. Panicali

Abstract-The amplitude of the electric field due to a straight line segment of uniform electric charges is shown to be given by E

Manuscript received February 15, 1990; revised August 15, 1990. A. R. Panicali is with CPqD Telebras S.A., Campinas, Brazil. IEEE Log Number 9040738.

I x

Fig. 1. Basic geometry,

=-- ‘ sen !!, where q is the linear charge density, a is the viewing 4 m , D 2

angle, from the observation point to the Pament extremitks, and D is the distance to the filament; furthermore, the direction of E lies on the bisector of the viewing angle. This result can greatly reduce the compu- tation time in the analysis of EMC problems involving static or quasi- static charge distributions.

I. INTRODUCTION Sabaroff [l], back in 1973, derived a modified form of the

Biot-Savart law of particular use in the calculation of the magnetic fields due to a filament of dc electric current. In this paper, the electric version of that formula is derived, and it has application to the calculation of the electric field due to a static filament of electric charges.

Although solutions to this problem can be found in the literature [2]-[3], it is believed that the simplicity of the expressions obtained with the present formulation allows a better visualization of the general properties of the electric field due to filaments of static charges.

II. ANALYSIS The basic geometry is shown in Fig. 1. A segment of uniform

linear charge density q[ C / m] extends betyeen the points A and B of the line L. The resulting electric field E at an observation point P is desired. As indicated, D denotes the distance between P and L. The angular variable 0, 0, I 8 I 8, is the polar coordinate of an arbitrary point Q(e) on L. Let p ( 0 ) = Q ( e ) P . The auxiliary rectangular coordinates X, Y are used to reference the electric field at P ; as indicated, the Y axis contains the segment D.

Clearly, each differential element dz along L can be expressed as

where

Due to Coulomb’s law, each charge element qdz produces at P a differencial electric field

where $(e) i s the unit vector pointing from Q(8) to P.

0018-9375/91/0200-07$01.00 0 1991 IEEE