IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 3 1, NO. 1, FEBRUARY 1989 21

Plane Wave Coupling to Multiple Conductor Transmission Lines Above a Lossy Earth

GREG E. J . BRIDGES. STUDENT MEMBER, IEEE AND L. SHAFAI, FELLOW, IEEE

Abstract-The current induced on an infinite multiple conductor transmission line located above a lossy homogeneous medium due to a transient plane wave is presented. An exact solution is formulated in the frequency domain using a spacial transform technique. The widely utilized quasi-TEM approximation is derived directly from the exact solution with emphasis on the physical consequences of the assumptions made. Both frequency domain and time domain numerical results are presented for typical transmission structures and documented EMP excitations. Comparison of the quasi-TEM approximation is made with the exact solution in order to study the validity of its application in EMP coupling problems. As well, the modeling of the EMP source as an incident plane wave is examined by comparing the induced current due to a dipole source with its steepest descent contribution.

Index Terms-EMP radiation effects, electromagnetic coupling, trans- mission lines, electromagnetic transient analysis.

I. INTRODUCTION

HE STRESS produced on power transmission systems T due to electromagnetic pulse (EMP) has become an important topic in EMC studies. The formulations and EMP analysis codes developed to solve this problem rely on the accurate determination of the induced currents on overhead lines. Many of the results and techniques are addressed in a recent report by ORNL [I]. The method employed in almost all cases is the quasi-TEM transmission line approach, which assumes that the energy is coupled to only the discrete characteristic propagating modes of the structure, the radiation and surface wave spectra required for an exact solution being ignored. The use of only the discrete modes is logical in view that the exact solution becomes extremely complex and time consuming when finite length transmission lines and complete system networks are considered. The solutions of some special cases based on the exact formulation have, however, been presented in the literature. Extensive studies of the excitation of a single conductor located over a lossy half-space, including the contributions of the discrete as well as continuous spectra of modes, have been performed [2]-[5]. Wait [6] formulated the case of a multiple conductor system over a lossy half space and Kuester [7] studied the discrete modal characteristics of these structures. These studies present a frequency domain solution to the problem using a spacial transform technique. Time domain results can then be obtained through the Fourier

Manuscript received November 30, 1987; revised September 28, 1988. This work was supported by the Natural Sciences and Engineering Research Council of Canada and the University of Manitoba.

The authors are with the Department of Electrical Engineering, University of Manitoba, Winnipeg, Canada R3T 2N2.

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transform. The formulation presented in this paper will follow this approach.

The justification of using the transmission line approach to study EMP coupling problems has been examined by compari- son to various exact solution techniques in [ l] , [8]. These specifically are for a single conductor above a perfectly conducting earth with a temporal step function excitation [9] and through a time domain solution obtained using the scattering theory [lo]. The former comparison assumes a lossless earth and thus the propagation constant of the discrete mode for the structure is equal to the free space value. The exact solution in this case is due to only the radiation spectra contribution into the air half-space. The latter comparison is made only in the early time response region since the scattering theory solution becomes complicated for the late time region. The use of the quasi-TEM theory to study EMP plane wave coupling as well as many other problems has been extensively covered in the literature [11]-[14]. Thus, as this approach is widely utilized a more extensive study of its validity is warranted. Of specific importance is the validation of the quasi-TEM transmission line theory for grazing angles of incidence, and for cases of low earth conductivities. These situations are the ones where maximum induction to transmis- sion systems occurs, and unfortunately are the cases where the quasi-TEM approximation is the poorest.

In the next section an exact solution to the coupling of a plane wave to an infinite multiple conductor transmission line over a lossy earth is developed. The solution is valid for all possible angles of incidence and polarizations of the incident plane wave, arbitrary earth electrical properties (cg, ug), and is exact to the extent that the radii of the conductors are less than the free space wavelength. This completely determines a frequency response into the GHz region and adequately covers the EMP spectrum. In the third section, the quasi-TEM transmission line solution is derived from the exact theory by using appropriate approximations. The induced current on various transmission structures for typical earth properties and EMP excitations is presented in the fourth section. A comparison to the results obtained using the quasi-TEM approach is also made here along with a discussion of its validity. Further, the use of a simpler equivalent single- conductor model, replacing the original multiconductor sys- tem, is examined.

In this paper, the EMP source is modeled as an incident plane wave, whereas in reality, the source will be of finite extent with a complete spectral representation necessary for an exact solution. The plane wave incident model is obtained

OO18-9375/89/02OO-0021$01 .OO 0 1989 IEEE

22 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 1 , FEBRUARY 1989

In this paper it is assumed that the radii of all the conductors are small compared to the free space wavelength (a, 4 he) as well as being small compared to the relevant physical dimensions of the structure (a, 4 pmn, p:,). In this way a thin wire approximation can be made where all conductor currents are assumed to have only an axially directed component as well as being angularly independent. These are the only assumptions made and the solution is otherwise exact.

The determination of the induced currents on the structure due to the plane wave source is facilitated by satisfying the continuity of the tangential fields at the N conductor surfaces. Thus, by matching the axial components of the incident and scattered electric fields a set of N integral equations can be developed as

Y

Sa Zmn(z-z')Z,(z') dz' = (E$,(z)), , = I --

N O m = l , 2, * . a , N

Sm [Z(z-z')l[Z(z')l dz' = [(Es(z))l , - m<z< 03. -a

\' \ \ \ \ \ \ 1 \ \ \ \ \ \ \ L \ \ \ \ \I \ \ \ ,\ , \\ , \ \ , E, ' Fe

Eg 3 P g 3

- X

(1) ;-\ Q - 1

(b)

aLJ ,*V Fig. 1 . EMP excitation of an N conductor transmission line: (a) plane wave

incidence; (b) transmission line geometry.

from the far field approximation of the source, and neglects the surface wave contribution due to the air-earth interface. The error in this assumption has been shown to be worst at grazing angles of incidence [15], and is thus important since this is where maximum induction to the transmission line occurs. For this reason, the validity of the plane wave incidence model for the EMP source is examined in Section V by comparing the full spectral representation of a vertical electric dipole source to its steepest descent contribution.

11. EXACT ANALYTICAL FORMULATION This section formulates the exact frequency domain solution

to the coupling of a plane wave to an above earth transmission line structure. Temporal variation of the source is then accounted for through standard Fourier transform techniques. The problem considered consists of a system of N infinite conductors, all parallel to the z-axis, and located above a homogeneous lossy earth as shown in Fig. 1. The region y > 0 is considered to be free space, characterized by a permittiv- ity E , and a permeability pe. The region y < 0 is designated as the lossy earth, characterized by a permittivity E, , a permeabil- ity p,, and a conductivity U,. A plane wave, which is allowed to have an arbitrary polarization, is incident on the structure as described by the angles 8 and 4. The induced harmonic currents, which are to be determined, are defined by the column vector [I(z)] where an e-;ur time dependence is taken.

This is simply the thin wire electric field integral equation that must be satisfied over the infinite length of the conductor system. The impedance convolution operator Zmn(z - z ' ) is the Green's function representing the scattered axial field at the mth conductor surface (xm, ym, z ) due to the nth conductor current at (x,, y,, 2'). Here (E:m(z)) is the average circumferential value of the axial component of the incident electric field imposed on the surface of the rnth conductor due to the plane wave source. Noting that the physical geometry of the problem is independent with respect to the z dimension, a solution to the integral equation (1) may be obtained by utilizing the spacial Fourier transform pair defined as

"rn

F(k,) = 1- f(z)e-jkzZ dz, -m

1 m 27r -- f(z) = - 1 F(kz)e+jkzz dk,. (2)

Thus, instead of solving the integral equation ( 1 ) in the spacial domain, the unknown currents are solved in the spectral domain using (2) as

The impedance matrix (5 ) consists of two terms, an external impedance term [Ze(kz)] representing the mutual coupling between the conductors, and a self impedance term [Zw(k,)] representing the conductors' surface impedance. In general, the induced current due to any given source can be solved for

BRIDGES AND SHAFAI: PLANE WAVE COUPLING

by transforming the imposed electric field using (2) and then solving through (4).

The next step will be to derive the external impedance matrix elements Zg,(kz). This is accomplished by determin- ing the fields external to the nth conductor, which carries a current I,(kz). Assuming an axial dependence of the form e+jkZz, the fields can be deduced by solving the two- dimensional wave equation in each air and earth half-space [2] . These can be written in terms of potential vectors as

where ne and n g are the two-dimensional Hertz vector potentials in the air and earth regions, respectively. Here k , = -is the propagation constant in the air medium, and kg = do2pgcg + jwpgug is the propagation constant in the ground medium. The associated fields, and thus the elements of [Z'], are determined from

(7) k ; E = vv * n e + k p e , A= - v x n e .

J W P e

A solution to (6) is obtained using the usual transform techniques [2]-[7] and then satisfying the boundary conditions at the air-earth interface. In the case of equal air and ground permeabilities (pg = p,), the external impedance matrix elements are determined as

23

branch cuts have been defined to ensure that the currents and fields decay at infinity. Zo(z), Ko(z) , K l ( z ) are modified Bessel functions of complex argument and n is the refractive index of the air-earth interface. In the derivation of (8), the terms involving K , ( T ~ ~ ~ , ) are due to the primary field of the current source, and the terms involving Ko(7,p*,,,,) are due to its image as if the earth were perfectly conducting. The remaining terms in integral form, (9) and ( IO) , are the corrections due to the imperfectly conducting earth. The modified Bessel function terms Zo(Tearn) account for the average circumferential value of the fields over the conductor surfaces as employed in accordance with the thin wire approximation. Extension of the theory to account for a multiple layered earth model can be facilitated by modifying

The internal impedance matrix elements Z;, ( k Z ) can easily be determined for various conductor types such as solid conductors, Goubau lines, wrapped conductors, etc. [SI, [ 161. For thin solid conductors [ Z w ] is defined as [17]

(9), (10) [61.

where r, = -with k , = t 6 2 p , ~ , + jmp,u, and p,, e, , , U, are the electrical parameters characterizing the nth conductor. Zo(z), Z,(z) are the modified Bessel functions. In this analysis, it has been assumed that the transverse dimen- sions of the conductors are small compared to the free space wavelength, as well as being small compared to all other physical dimensions of the structure, allowing a thin wire approximation to be made. Thus, the quantities in brackets ( ) in ( I ) , (4) that denote the average value of the fields over the transverse dimensions of the conductors can be found by integrating over the surface of the conductors as

U,=-, Re [ U e ] 2 0

U g = w , Re [ U g ] 2 0

where pmn = ~ ( ~ , , - x ~ ) ~ + ( y , - y ~ ) ~ and p:, =

~x,-~~)~+(y,,+y,)~. Here re = J k n and rg = d R are the transverse propagation constants in the air and earth media, respectively. The real parts of the irrationals Re [U,, U,] 2 0 and Re [T,, 7g] 2 0 have been chosen to retain a positive value on the correct Reimann sheet. These

where a, is the radius of the mth conductor and Eim(kZ) is the value of the axial component of the imposed electric field at the center of the mth conductor.

The last remaining task to completely solve the problem is to define the axial components Eim((kZ) of the incident plane wave at the conductor locations. The excitation by a plane wave is a much simpler case than that of a general source since it represents only a single spectral component in the transform (4) and thus, an easy solution is obtainable. The incident plane wave will be represented in terms of its vertical EO and horizontal E+ polarized components as shown in Fig. 1 . In this manner, the imposed electric field at the surfaces of the N conductors can be determined by the direct and reflected fields as

24 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 3 1, NO. 1, FEBRUARY 1989

n2 sin e - J n 2 - cos2 e n2 sin e + dn2- cos2 e

sin e - J n 2 - cos2 e sin e + dn2- cos2 e rv= 7 r H =

( 1 5 )

where rv and rH are the vertical and horizontal Fresnel reflection coefficients at the air-earth interface. The value of the imposed electric field in the spectral domain can be determined by transforming (13) and calculating the average value, defined by (12) for circular conductors, as

= 2?rZo(7 ,~ , )E~~(e , 4 )6 (k , - k, COS 6 COS 4).

(16)

The current induced on the conductors can then be determined by utilizing (4) as

where lo,+ is the axial component of the propagation constant of the incident plane wave and [Z({e,+)] is the impedance matrix previously defined in (8), ( 1 1 ) with its argument specified as k, = {e,+. Here the matrix [I({@,+)] is an N x N diagonal matrix incorporating the average values over the surface of the conductors with elements given by

-jkeJ1 -cos2 0 cos2 4 .

Note that in final equation (17) defining the induced currents on the transmission line structure, the phase reference for all quantities has been chosen as the origin (x = 0, y = 0, z = 0).

For the special case of a single conductor of radius a located at a height y = h in the plane x = 0, the induced current, which is basically the same result as [ 3 ] , is given as

* { 7f [KO (7, a) - 10 ( T e a ) KO (7e2 h ) I -Zo(Tea)[kfJ(Te, 2hY) -k ;G(~e , 2hY)l). (21)

The magnitude of the transverse propagation constant 7, is always less than the free space value I 7, I I k, and approaches zero for grazing angles of incidence. Thus, for thin conductors compared to the free space wavelength, the diagonal matrix [f({e,+)] can usually be approximated by the unit matrix and thus can be neglected in (17). For grazing angles of incidence, re,+ -+ k,, (20) shows that the axial component of the imposed electric field [E:@, 4 ) ] will approach zero. However, examination of (21) also shows that even though the axial field is small, the maximum induced current will occur in the range of small grazing angles since [Z(ro,+)] also approaches zero here. For angles of incidence perpendicular to the conductor axis, {e,+ -, 0, a smaller but nonzero current will be induced.

Calculating the elements of the impedance matrix (8) involves the evaluation of the Sommerfeld type Fourier integrals J(T, , p* ) and G(T, , p*) , which are not identified with known functions and prove difficult to evaluate for all general arguments. Various analytical expressions for the evaluation of the Fourier integrals are documented in the literature [4 ] , [7 ] , [ 131, [ 181, [ 191. Approximate solutions have been developed, in terms of asymptotic expressions for the small argument range, as well as through the method of steepest descent for the large argument range. In this paper, numerical integration, employing a Gaussian quadrature tech- nique, is utilized in the regions where the analytical formula- tions are not adequate.

ID. TRANSMISSION LINE APPROXIMATION

Recent studies determining the induced currents on over- head power cables and stresses produced by an EMP rely mainly on the use of a quasi-TEM transmission line theory. The theory assumes that the currents on the transmission line structure can be completely represented by only exponential traveling waves of a form Zp exp { f j k f z } , where k f is one of the possible characteristic propagating modes of the structure. The representation of the current in this manner neglects the radiation and surface wave contributions, as given by the branch cuts present in the complete solution (4) . In the quasi- TEM theory, the propagation constants are determined from the transmission line circuit parameters [ 121-[ 141, [ 191, the approach being basically the same as that obtained by Carson in 1926 [20]. The per unit length circuit parameters are derived by applying the TEM assumption directly to the wave equation (setting 7, = 0 in (6)). The validity of this approximation has been studied [19], [ 2 1 ] , and the basic observations for reliable application of the quasi-TEM trans-

BRIDGES AND SHAFAI: PLANE WAVE COUPLING 25

mission line approach are that the dimensions of the transmis- sion line structure should be much less than the free space wavelength ( p m , , p,*,, Q A,) and that the refractive index of the air-earth interface should be large (In1 %- 1). Governed by these conditions, the quasi-TEM transmission line theory should be valid up to the megahertz region for typical overhead transmission lines. However, the spectral energy content of a typical EMP is greatest in the MF to HF (300 KHz-30 MHz) range [ l ] and thus, the theory may not be accurate enough for all applications.

In this section the quasi-TEM transmission line approxima- tion is derived directly from the exact formulation presented in the previous section. This approach is different from that usually taken in the literature where the TEM assumption is applied from the start and then the transmission line character- istics are determined from its equivalent circuit parameters. The derivation from the exact formulation allows some insight into the assumptions made. In utilizing the transmission line approach, the current on the structure is generated by an infinite set of delta function voltage sources distributed along the length of the conductors, with the magnitude of the sources proportional to the axial component of the imposed electric field. The current due to each localized source is then assumed to be of exponential form only, the branch cut contributions being neglected. The formulation of the problem in this manner, directly from the exact solution (4), can be developed by utilizing the convolution theorem to represent the source as

I 1 k / , p =N +l,...p ';s" k..

Fig. 2. Location of the P poles and the three branch cuts in the upper half of the complex k, plane.

source as exp ( +jkz 1 z - z, 1 ), where Im [k,] 2 0 is defined when the contour of integration is deformed from the real axis to encompass the poles. The location of the poles is highly dependent on the geometry of the transmission line structure. The contributions due to the branch cuts appearing in the k, plane arise from two sources. The first is due to the requirement on the irrationals Re [U,, U,] L 0 defined so that the fields decay as lyl + 03 in the upper and lower half- spaces. These branch cuts, emanating from the points k, = kk,, -+kg, represent a spectrum of modes radiating into the air half-space and the earth half-space, respectively. Note that if the lower half space is a good conductor, the corresponding branch point f kg = f 4 G i s located far off the real axis and the contribution to the current from this radiation mode is

(22)

Thus, using the expression (4) for the current and replacing the source term using (12), (22)

Here [I(z , z,)] is the current at the observation point z due to a delta function voltage source of strength [Es(z,)] located at z,. The current formulated as in (24), (25) is still an exact solution even though its evaluation in this form would be inappropriate.

The inverse transformation (25) determining the currents on the structure due to the localized delta function source contains a set of branch cuts as well as a set of poles symmetrically located in the k, plane as shown in Fig. 2. The poles, indicated as k$'; p = 1 , 2, . . . , P , arise from the singularities of the impedance matrix [Z(k,)] and can be determined from the solution of the mode equation 1 [Z(k,)] I = 0. Their contribu- tion to the integral transform represents a set of discrete propagating modes for the structure. These currents have fields that decay exponentially in the axial direction from the

negligible. Secondly, an additional branch cut emanating from the point k, = &kzB = + _ k , / J m arises from the singularity in the denominator of the integral G(7,, p*) . This branch cut represents a spectrum of surface waves supported by the interface and can be related to the Zenneck surface wave resulting from a dipole source over a half-space. Whereas the location of the poles is strongly dependent on the geometry of the transmission line structure, the continuous mode spectra associated with the surface wave and radiation branch cuts are not affected strongly by the conductor geometry, but can have a large contribution to the currents under certain conditions.

The path of integration in the complex k, plane will be deformed to separate the contributions due to each of the poles and branch cuts thus giving some insight into the behavior of the structure currents. The integral transform (25) can thus be constructed as a sum of discrete propagating modes as well as a spectrum of continuous modes as

+ [IRADe(z, z ,) + IRADg(z, z ,) + IsuR(z, z,)] (26)

26 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 1, FEBRUARY 1989

where rRADeig are the branch cuts emanating from + kelg and rsuR is the branch cut emanating from +kzB. The separate contributions to the structure currents due to the branch cuts and the discrete modes have been studied by Chang and Olsen [4] . The properties of the discrete modes have also been extensively examined [7] , [18], [22]. It is expected that the discrete modes will be the dominant contributions to the current at distances far from the source since the continuous mode spectra decay algebraically whereas the discrete modes decay as exp ( + j k ; I z - zs I), and Im [k;] can be substan- tially small. In the immediate neighborhood of the source as well as at extremely large distances, the radiation and surface wave spectra are expected to have a significant contribution and should not in general be neglected. In utilizing the transmission line approximation, we are primarily concerned with the contribution to the structure currents due to the discrete modes as defined in (27). The properties of these modes are characterized by the propagation constants k;; p = 1 , 2 , * * * , P and by the magnitude of their associated residue contribution, which determines their relative excitation by a given source. The use of only the discrete modes to represent the structure currents also allows a much simplified transmis- sion line approach to the solution of (25) and the formulation of the plane wave coupling problem. To this extent, consider the contribution to the currents due to the discrete modes only as given by (27). The total induced current on the structure is then derived as

p=1

Note that the number of possible modes for the structure may be greater than the number of conductors (P 1 N). N of the discrete modes arise from the solutions of I [Z(k,)] I = 0 in the region where typically all the terms of the matrix are slowly varying functions of the argument k,. These modes can be considered as the dominant modes for the structure and are usually the major contribution to the current. As shown in Fig. 2, the remaining (P - N) modes occur near the singularity in the impedance matrix due to the G(7,, 6") integral, with their contribution to the current usually being small. The singularity

is identified by the branch point f k Z B in the k, plane. Excitation of these modes at low frequencies is very small for typical sources, and thus usually only the first N discrete propagating modes are important. However, the contribution of all modes becomes important at higher frequencies.

Up to this point, the formulation presented can appropri- ately be denoted as a transmission line solution since only the discrete exponential current modes are used to represent the current on the structure. The values used for the propagation constants k; are solutions of IIZ(kz)]l = 0 as determined using the exact expressions (8)-(11) and thus, the resulting fields are solutions of the wave equation (6). The solution of the mode equation in this form can be considered a generalized eigenvalue problem where an explicit expression cannot be derived since the elements are complicated functions of the unknown eigenvalues and as such are difficult to evaluate. A common approach to simplifying the problem is to assume that the axial variation of the fields is equal to the free space value (k, = k,) when evaluating the mode equation. In this manner, the fields in the upper half-space will be solutions of the two- dimensional Laplace equation. This approach is denoted as the quasi-TEM transmission line approximation and is reasonable if we recognize that the terms J, G, and KO involved in the calculation of the matrix elements in (8) are slowly varying functions of their argument (7, = m) and the axial propagation constant for the structure should be near k,. Applying this approach by assuming 7, + 0 in the arguments of Io, KO, and J in (8) - ( l l ) , an explicit expression for the transmission line parameters will then be given as

(31) IIZser] - [ ( j ~ t , ) ~ ] [ Ysh]- ' l = O

eruke(Ym+Yn) cos (uk,)x,-x, ,I) du. (35)

Here Z,"I: are the series impedance and Y:,, are the shunt admittance terms for the structure and [( j / ~ , ) ~ ] is diagonal. The solution of the standard eigenvalue equation (31) then yields the values of the propagation constants. The logarithmic terms in (32) and (33) represent the field due to the conductor and its image under the conditions of a perfectly conducting earth. The integral term Jc(p;:,) represents the conduction losses in the earth and the contribution of the integral G(7,, p;,,), representing displacement current losses in the earth, has been neglected completely. A good approximation to the terms of the shunt admittance matrix is thus obtained using image theory under static conditions. Many expressions for evaluat-

BRIDGES AND SHAFAI: PLANE WAVE COUPLING 27

ing the integral Jc(p,*,,) are found in the literature [13], [19],

Once the transmission line parameters are determined using the quasi-TEM approximation, the induced currents on the structure can be found through (30). Note that the quasi-TEM approximation yields only N solutions. For the single- conductor case, an expression for the induced current will be found by direct integration of (30) as

1231.

=-ln(;)-. Zo 2h kp

2a ke

Here Zo is the free space impedance and E y ( 0 , 4) was defined in (20). Examination of (36) shows that the maximum induced current occurs near grazing angles as discussed previously. Further, if the earth and wire are assumed to be perfect conductors, the propagation constant for the structure will approach the free space value k f + k, (the transmission line is lossless) and the induced current using (36) can become infinite at grazing angles

I ( z ) - E;‘(0, 4)

k t - k , k,Z,

+ j e+ico,rnz - w. (39) 1 -cos2 0 cos2 d @,$+O

Examining (39), the error in assuming k f + k, becomes greatest for grazing angles of incidence, however, this assumption has been utilized in many time domain methods [ 1 11, [24] and is a fair approximation at nongrazing incident angles [ 8 ] .

In both the exact and quasi-TEM analyses, the transmission line structure has been assumed to be infinite in one dimension and the exciting source has been modeled as an incident plane wave. Thus, the phase variation of the induced current in (17), (19), (36) is characteristic of the axial phase variation of the incident plane wave exp{ + j { o , 4 z } . Note that the expression for the induced current obtained in (36) using the transmission line approach could also have been obtained by directly utilizing the quasi-TEM approximation (7, = 0) in (19)-(2 1).

IV. NUMERICAL RESULTS In this section, the currents induced on various transmission

structures due to the EMP waveforms documented in the

x v

ma3 oo - e = l O . O Y - e=3o.o X - e=~o.o

000 2 0 0 4 00 600 800 100

0 15 1 N u=0.002Sm

0 12

e L5 c U

c :o L I

0 00 0

lL =15 (I# =0.01

4 = 0.00 ---e----= 1.0 - e = 5.0

A e=io.o - e = m o - e=80.0

ii

Fig. 3. Induced currents on a single-conductor system as a function of frequency for various angles of incidence: (a) and (b) are the results plotted for different scales. The exact solution is given by solid curves and the quasi-TEM result is given by dashed curves.

literature are calculated using the exact theory and the quasi- TEM transmission line approach. The first case considered is that of a single conductor of radius a = 2.5 mm located at a height h = 5.0 m above a lossy earth characterized by a relative permittivity crR = 15 and a conductivity aK = 0.01. The magnitude of the induced current for a vertically polarized incident plane wave of unit strength is given in Fig. 3. The solid curves indicate the current calculated using (19) and normalized as

(40)

28 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 3 1, NO. 1, FEBRUARY 1989

3 0 0 0 7 i , N a =o'oo2sm

J"-" q8 =IS U, =0.01

Y lOKHz - lMHr _J_ lOMhz - looMHz

0 0 0 1500 3 0 0 0 4 5 0 0 6 0 0 0 7 5 0 0 9000

6 ( d e 4

Fig. 4. Induced current as a function of incident angle 0 for various frequencies. The exact solution is given by solid curves and the quasi-TEM result is given by dashed curves.

The figure shows the induced current as a function of frequency for various angles of incidence, with the angle 4 fixed at 4 = 0" and only 0 varied. Also shown by the dashed curves in Fig. 3 is the magnitude of the corresponding quasi- TEM quantity

as determined using (36)-(38). The quasi-TEM approximation begins to deviate from the exact solution at about 1 MHz, corresponding to the condition h/X, 4 1 . Further, the quasi- TEM result does not exhibit the proper oscillatory behavior as the free space wavelength becomes comparable to the conduc- tor height. From these figures, the incident angle at which the maximum induced current occurs is dependent on the fre - quency. To examine this relationship, Fig. 4 indicates the induced current as a function of incident angle for various chosen frequencies. The angle at which the maximum induc- tion occurs decreases as frequency increases, and is greatest in the grazing region for most of the strongest part of the EMP frequency spectrum. As also shown by Fig. 3, the quasi-TEM approximation is worst for grazing angles of incidence and in the higher frequency range. For completeness, a plot of the poles k;/k, and branch cuts in the complex kz/k, plane for this single-wire geometry is given in Fig. 5 [18]. The poles are calculated over the frequency rangef = lo2 - 10'O Hz. The branch cut due to the radiation spectra into the upper half-space emanates from the branch point kz/k, = 1 . The branch cut due to the radiation spectra into the lossy earth emanates from the branch point at kJk, = n = derg + jug/wco, which is located far to the right of the region depicted in the figure at all frequencies. It does not make a significant contribution to the current on the structure when the earth behaves as a conductor.

0 4 0 I I

exact solution mode 1

kzB/kr branch point

- z' \ Y 0 Z o 17 - . - - exact quasi-TEM solution solution mode 2// - branch cut E - 0 10

0 00 0 90 100 110 1 ?C 1

R e [ k 2 / h e 1 0

0 03

3

w Y

\,0 15

€ Y U

-

0 0 0 0 97 0 985 1 0 0 1015 ? 03

Re[kz/k , l

Fig. 5. Location of poles and branch cuts in the upper half of the kJk, plane as a function of frequency for a single-conductor system.

The last branch cut, representing the surface wave contribu- tion, emanates from the branch point at kd/k , = n / w , which is located along the dashed line in Fig. 5 as a function of frequency. In the figure there are two possible solutions to the mode equation using the exact formulation. At low frequen- cies, one solution is found near the quasi-TEM result, the other being found near the branch point kzB. The situation reverses as the frequency is increased, and at high frequencies, the solution nearest the branch point crosses the branch cut at kJk, = 1 and passes onto an improper Reimann sheet. The other solution eventually approaches that of the wire as if in free space. Also shown is the propagation constant obtained using the quasi-TEM theory. At low frequencies, the quasi- TEM solution is the same as one of the exact solutions, and for the case considered is valid up to about 1 MHz.

Using the frequency domain data presented for the one wire transmission line, the temporal response of a typical EMP will be determined. Two documented waveforms are presented in the literature [l]. The "BELL" waveform is given as

where Eo = 52.5 x lo3 (V/m), a = 4.0 x lo6 (l/s), and /3 = 4.76 x lo* (l/s). Here fl indicates the polarization of the incident plane wave, from which the respective values of the vertical Eo and horizontal E+ components in (14), (20) can be determined. Only vertical polarized incidence f l = 8, with 4 = 0" will be considered since this is the situation where maximum induction occurs. The time domain response is

BRIDGES AND SHAFAI: PLANE WAVE COUPLING

U =O.O025m += 0.00 -e= 0.4 -e= 1.0

h =5m --+---e= 5.0 -e=io.o __)t_ e = m o Y t,# =I5 U# =0.01 - 8=80.0

a=4.001 IO6 '

8=4.7& loB

0 00 -2000 000 2000 4000 6000 8000

TIME (nsec)

(a) E 00

- A lf C l

0

4 00

b

z & 300 v 3 ti

W

1 E 2 03

n

I3 100 n z

W 0

0 00

-e= 1.0 -e= 5.0 --+-- e=io.o _ i ~ _ e = n o ---+--- B=80.0

003 040 080 120 160 2 00 TIME (nsec) jX1ci3)

(b)

Induced current on a single-conductor system for various angles of incidence: (a) early time response; (b) late time response.

Fig. 6 .

obtained using the standard inverse Fourier transform with

Fig. 6 gives the early and late time responses for various incident angles. Note that the time reference t = 0 is at the origin x = y = z = 0. The maximum induction occurs in the grazing angle range; in this case at about t9 = 10". The induced current in the late time region is largest for grazing angles of incidence also. It is interesting to note the oscillatory behavior of the induced current in the early time region for small angles of incidence. The behavior in this region can be

-4000 000 4000 8000 12000 16 TIME (nsec )

(a)

12 00

n Af h =20m

c,# =I5 U # =0.01 Ln a=4.00r IO6 800 8=4.7& loe

0

+ z

U $= 0.0" - e = 1.0 _ic_ e = 5.0

500 - e=io.o - e=3o.o - B=BO.O v 3 0

W

_1

g 400

n W 0 3 200 n z

0 00 000 040 080 1 2 0 150 2

29

00

0 TIME (nsec ) (x103)

(b)

Induced current as calculated by the exact (solid) and quasi-TEM (dashed) solutions for various angles of incidence: (a) early time response; (b) late time response.

Fig. 7.

explained by noting the magnitude of the term I n sin 0 1 when determining the reflection coefficient r v. For the condition I n sin 0 1 < 1, the reflected field adds to the incident field I'v --*

- 1, causing a large rise in the induced current. When this condition does not hold, 1 n sin 0 1 > 1, as in the later time region when conduction currents in the earth dominate and n becomes large, or when the incident angle 0 is large, then the ground reflected field cancels the incident field rv + + 1, causing a decrease in the induced current.

Fig. 7 gives the time domain response for the same single- wire transmission line as in Fig. 6, except a conductor height of 20 m is chosen here. The maximum induced current is

30 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 3 1, NO. 1, FEBRUARY 1989

15 00

- 12 50 0

X v

A Ln

Ea10 00 0

+ Z $ 7 5 0 Lli 3 0

W

1

v

g 5 0 0

n

0

W 0 3 250

o =0JM2Sm

h =2Om

e =so a = 4 ~ ~ x lo6 p=4.7&. 108

I c,r= ls (Ig

og=0.l - I I i

0004 ' -0 04 0 12 0.28 0.44 0 60 0

TIME (nsec) (x103)

Fig. 8. Comparison of the exact solution (solid) and the quasi-TEM approximation (dashed) for various earth conductivities.

12 00

A

c, 1000 0 X v

Vl 2 e 0 0 0

c z $ 600 cc 3 0 W

1

U

g 4 0 0

n W 0 3 200 n z

0 00 -2

e =50 a =41)(1x 10'

'0 6 0 0 0 14000 2 2 0 0 0 30000 38000 TIME (nsec)

Fig. 9. Comparison of the exact solution (solid) and the quasi-TEM approximation (dashed) for various conductor heights.

larger for this case and again occurs in the grazing angle region; at about 6' = 5 " . Also given by the dashed curves in Fig. 7 are the results calculated using the quasi-TEM theory. The early time quasi-TEM response is not accurate, especially for grazing angles of incidence. Figs. 8-10 show the effects of ground conductivity (og), conductor height (h), and incident field decay rate (a) on the induced current. An incident angle of 4 = O " , 6' = 5" and vertical polarization is chosen for all three cases. The amplitude of the induced current is seen to increase when the height of the conductor is increased and when the ground conductivity is decreased. This occurs since the reflected field does not interfere as strongly with the direct

18 00

- r, I500 0 X v

A m 9 2 00

g 9 0 0

0

c z

(r 3 0

2 6 0 0

v

_1

n W 0 3 3 0 0 0 z

0 00 - 2 IO 60'00 140'00 2 2 0 0 0 30600 3

TiME (nsec) 30

Fig. 10. Comparison of the exact solution (solid) and the quasi-TEM approximation (dashed) for various EMP excitation tail decay rates a.

6 00

5 500 X v

A

Ln

r 4 00 0 U

t Z

Lli 3 0

3 00

W

1 g 2 00

a W 0 3 100 a z

0 00 - 4

m Y

(I, =-

/ I I h =20m

r,r=15 ( I r

! e = x o 10 4 0 0 0 12000 20000 2 8 0 0 0 3E

TIME (nsec)

Fig. 11. Induced current for a temporal step function excitation for various earth conductivities. The quasi-TEM solution for a perfectly conducting earth is given by the dashed curve.

field for these situations. It is important to note that the quasi- TEM approximation underestimates the peak induced current when the ground conductivity (ug) is low or the field decay rate (a) is large. Fig. 11 shows results similar to those of Flammer [8], who calculated the case of a single conductor located above a perfectly conducting earth excited by a temporal step function incident plane wave. The effect of an imperfect earth, as determined using the exact solution (19), is given for comparison.

Fig. 12 gives the induced current on a two-conductor transmission line. The conductors are located at ( X I = 0 m, J+

BRIDGES AND SHAFAI: PLANE WAVE COUPLING 3 1

I

a =0.0025m

-8000 000 8000 160'00 24600 3 2

TIME (nsec) 0

Fig. 12. Induced current on a two-conductor system for a vertically polarized incident plane wave at 4 = 45" and 0 = 30". The dashed curve gives the current I,, on an equivalent single conductor located between the two-conductor structure.

= 20 m) and (x2 = 10 m, y2 = 20 m) and both have a radius of a1 = a2 = 2.5 mm. The earth is characterized by a relative permittivity of erg = 15 and a conductivity of ug = 0.01. The incident plane wave is assumed to be vertically polarized with a temporal variation as given by (42) and is incident at q5 = 45" and 8 = 30". The results of using an equivalent single conductor to model the multiple conductor system are also shown in Fig. 12. The single-conductor equivalent has a radius rcJq = 41 xi - x2 1 a = 0.158 1 m and is located between the two conductors at (x = 5m, y = 20 m). Note that the single- conductor equivalent is capable of modeling only the common mode currents on the transmission structure. For the given situation, the single-wire equivalent is observed to be a good approximation to the two-conductor system, and in general provides a simplified approach to determining the induced currents on multiple-conductor systems. Various other cases as well as the validity of using the quasi-TEM theory to model the differential mode currents have also been examined [25].

V. PLANE WAVE SOURCE MODEL The validity of modeling the EMP source as an incident

plane wave will be examined by considering the excitation of the transmission line by a dipole located in the air half-space ( y > 0). The current induced on a transmission line due to a vertical electric dipole (VED) in the presence of a lossy earth has been formulated by Wait [26] and Olsen and Usta [27], with the latter specifically determining the plane wave contribution to the current using the method of steepest descent. Following this work, the current induced on a transmission line due to a VED will be calculated as shown in Fig. 13. The dipole is located in the upper half-space at (xo, yo, zo), with a moment Id@. The distance ro = d w i s the transverse distance from the dipole to the z-axis and Ro =

A I / ? !4

Fig. 13. Excitation of a transmission line by a vertical electric dipole located in the air half-space.

&; + y i +(z - z,,)~ is the distance from the dipole to some observation point along the z-axis. The induced current is determined using the general expression (4), where the axial component of the imposed electric field [ ( E f ( k , ) ) ] due to the VED is formulated in the spectral domain as [26], [27]

(44)

where rm = d(x0 - x,)~ + ( y o - ynl)L and r; = .\ioro-xm)2 + (yo+ym)2 and (xm, ym) is the position of the mth conductor of the transmission line. Kl(z) and Io(z) are modified Bessel functions and the function G(7,, P ) was previously defined by (10). If the dipole is located electrically far from the transmission line, and far from the air-earth interface, the imposed electric field due to the VED (44) can be determined through the method of steepest descent [27]. Thus, under the conditions I7,r, 1 , I r,r; 1 % 1, yo > xo, the modified Bessel function terms KI ( 2 ) in (44) can be evaluated using their asymptotic expressions [28], and similarly, the remaining integral term can be determined by its steepest descent contribution. Note that the term Zo(z) in (44) and the impedance matrix [Z(kz)] in (8) cannot be evaluated in the same manner since their arguments are not in the asymptotic region. Finally, under the condition 1 keRol % 1, the remaining integral (4) determining the induced current can also be evaluated using the method of steepest descent as

[ I ( z ) ] = [Z(k, cos 8 cos q5)]-l[f(k, cos 0 cos $)]

E:",'(O, $) =Eo sin 0 cos q5 [e -Jke?m 'In 0 - r y e + J k e y m 'In O ]

. e-Jker ,n io\ # \in 4 * (46)

Note that (45) is essentially the same expression as previously

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 1, FEBRUARY 1989 32

1.80

1.50

A W

9 1.20 v

c\ U ) a g 0.90 v A - N =0.60

0 Q 2

- 0.30

0.00 0

P !

1 15.00 30.00 45.00 60.00 75.00 9C

incident angle 0 (deg) Fig. 14. Current induced on a single-conductor transmission line by a VED

located at different heights as determined using the exact (solid) and steepest descent contribution (dashed).

derived for plane wave incidence (17), except the incident field is now modified by the far field factor of a dipole in free space

(47) The effect of the earth is now represented by the Fresnel reflection coefficient rv and the remaining terms in (45) were previously defined by (8), (18).

Fig. 14 gives the induced current on a single-conductor transmission line of radius a = 2.5 mm centered at (x = 0 m, y = 10 m). A frequency of 100 KHz is chosen along with an earth characterized by a relative permittivity erg = 5 and a conductivity a, = 0.01. The exciting VED is located directly above the transmission line (xo = 0 m, zo = 0 m), at three different heights (yo = 3 km, 9 km, 15 km), thesecorrespond- ing to (yo = 1 A,, 3A,, SA,). The magnitude of the induced current I I(z) I as calculated by numerically integrating the exact expression (4), (44) is given by the solid curves, where the dashed curves give the corresponding steepest descent contribution (45). With 4 = O ” , the angle of incidence 8 indicates the position along the conductor axis ( z - zol = yo cot 8; i.e. for yo = 3 km and 8 = 30” then Iz - Z O ~ = 5.196 km. By examining Fig. 14, the steepest descent contribution is accurate when the dipole is electrically far from the interface and not in the grazing angle region. These observations correspond to the condition 17~01 = I -jk, dl - cos28 cos2 4 ro( %. 1 (or 27r sin 8ro/A, %. 1 when 4 = 0) imposed when (44) was evaluated using its asymptotic expressions. For typical EMP bursts, the Compton electron source region ranges from 30 to 50 km above the earth [ 11 and thus, the plane wave model should be valid over the EMP

VI. CONCLUSIONS The exact solution to the induced currents on a multiple

conductor transmission line due to a plane wave excitation was presented. The commonly used quasi-TEM transmission line approximation was derived directly from the exact solution and numerical results were presented to study its validity. As expected, the quasi-TEM approach gives accurate results in the late time response region for typical transmission struc- tures and EMP excitations. The quasi-TEM approach also gives fair results in the early time region for large incident angles, however, in the grazing angle region it does not exhibit the proper behavior. As already discussed in the literature, the quasi-TEM approach is worse for larger conductor heights and smaller earth conductivities. More importantly, the quasi- TEM approach can underestimate the peak induced currents under grazing angle conditions. However, since the applica- tion of the exact theory to finite transmission lines and complete power system networks is extremely complicated, the quasi-TEM method is an acceptable alternative in view that it gives adequate results for most situations. Further, a simpler single-conductor equivalent model was studied as a means of determining the common mode current on the transmission line. The validity of modeling the EMP source as an incident plane wave was also examined with the conclusion that for the EMP sources documented in the literature [l], the model is valid except for the extreme grazing angle region 8 < 1”.

ACKNOWLEDGMENT The authors would like to thank the reviewers for their

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