62 lEEE TRANSACT'IONS ON ELECTROMAGNETIC COMPATABILITY. VOL. 37. NO. I , FEBRUARY 1995

Transient Plane Wave Coupling to Bare and Insulated Cables Buried in a Lossy Half-space

Greg E. Bridges, Member, IEEE

Abstract-The current induced on an infinite bare or insu- lated cable buried in a lossy earth medium due to a transient plane wave is presented. An exact solution is formulated in the frequency domain using a spatial transform under the thin-wire approximation. The widely used equivalent circuit transmission line model is derived from the exact solution. Results are pre- sented for typical transmission structures under high frequency transient excitation and the exact solution is compared with the transmission line approximation. The transmission line approach provides good results for a wide range of cases. For accurate results in the high frequency situation it is necessary that the correct incident field expressions be used and that a complete representation of the earth's electrical properties (u and 6 ) be retained.

I. INTRODUCTION LECTROMAGNETIC coupling to power and communi- E cation cable systems has become an important topic in

emc studies. This paper formulates the current induced on a buried cable due to a transient plane wave source, considering both the bare and insulated situations. In almost all cases, the solution to this problem has been based on an approximate transmission line approach [ 11, [2]. In this approach the source is assumed to be coupled to only the discrete propagating mode of the structure, ignoring the radiation and surface wave spectra required for an exact solution. In the transmission line model, the per-unit-length equivalent circuit parameters are typically derived assuming the earth behaves as a good conductor, with several formulations available in the literature [I], [3], [4]. Often the effect of the air-earth interface is ignored, this equivalent to assuming the cable is buried at a large depth (homogeneous case). Further, a simplified form for evaluating the transmitted plane wave field into the earth is usually utilized by assuming a large refractive index. The use of the transmission line approximation is logical in view that an exact solution becomes extremely complex and time consuming when finite length cables and complete system net- works are considered. As this approximation is widely utilized a study of its validity is warranted. Of specific importance is the validation of the transmission line theory; over a wide frequency spectrum, for various angles of incidence, for cases of low earth conductivity, and when the cable is buried near the air-earth interface.

Manuscript received February 16, 1993; revised June 20. 1994. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

G. E. Bridges is with the Department of Electrical and Computer Engi- neering University of Manitoba, Winnipeg MB, R3T 2N2 Canada.

IEEE Log Number 9407403.

The problem of plane wave coupling to overhead cable systems [5]-[7] has been examined to a greater extent than the buried counterpart. The solution of the propagation constant of buried cables based on an exact formulation has been developed by several researchers [8]-[ 111. Vertical magnetic dipole excitation of a single cable located in a homogeneous lossy medium was examined by Hill [12] with Wait consider- ing the excitation of a buried cable including the presence of the air-earth interface [8], [9]. These studies presented a frequency domain solution to the problem using a spatial transform technique. Altemative studies of finite length cables using an integral equation approach have also been made [ 131. The formulation presented in this paper for the infinite cable situation will follow the spatial transform approach, with time domain results obtained through the Fourier transform.

In the next section, an exact solution to the coupling of a plane wave to a cable buried in 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 (eg, og), and is exact to the extent that the radius of the cable is less than the propagation wavelength in the earth. In general, the electrical properties of the earth will be frequency dependent. Even though this situation is not considered in this paper, it can be incorporated since a frequency domain formulation methodology is used. The equivalent circuit transmission line solution and several of the commonly used approximate evaluation methods are then derived from the exact theory. The induced current on various cable systems is presented in the final section. It will be shown that the equivalent circuit model for determining the coupling impedance to the transmission line provides good results over a large frequency range. It will also be demonstrated that the approximations commonly used for the incident plane wave transmission coefficient do not give reliable results when the earth does not behave as a good conductor.

11. ANALYTICAL FORMULATION This section formulates the frequency domain solution to the

coupling of a plane wave source to a buried cable structure. Temporal variation of the source is accounted for through standard Fourier transform techniques. As shown in Fig. 1, the problem consists of an infinite bare or insulated single wire cable buried in a lossy half-space at a depth d. The region y > 0 is considered to be free-space, characterized by a permittivity € 0 and permeability p0. The region y < 0 is designated as the lossy earth, characterized by a permittivity

00 I8-Y375/95$04.00 0 1995 IEEE

BRIDGES. TRANSIENT PLANE WAVE COUPLING TO BARE AND INSULATED CABLES BURIED IN A LOSSY HALF-SPACE

Fig. 1. Geometry for plane wave excitation of a buried cable.

eg , conductivity ag, and free-space permeability PO. The cable geometry is specified by an inner conducting core (ew, pw, a , ) of radius a with a lossy insulating jacket ( ~ d , ,ud, ad) of radius b. A plane wave source, which is allowed to have an arbitrary polarization, is incident on the structure as defined by the angles (e, 4). An e P w t time harmonic dependence is used (the equivalent formulation using +jwt is often preferred). In this paper it is assumed that the radius of the cable is small compared to the propagation wavelength in the earth as well as small compared to the burial depth (a, b << A,, d). Under these conditions the thin-wire approximation can be used, where the cable current is assumed to be axially directed and have no azimuthal variation.

The determination of the induced current on the cable due to a general source is facilitated by solving the wave equation in each region (air and earth half-spaces and intemal to the cable) and then satisfying the boundary conditions at their interfaces. Under the thin-wire approximation continuity of only the axial component of the electric field E, at the cable surface is adequate such that

-03

Z(% - z')I(z')dz' = .I, < EfNC(z) >Ip=,:-m < z < 20. (1)

The impedance convolution operator Z(z - z ' ) represents the scattered axial field at the location z due to the current at 2'.

The term < ESNC(z) > is the average circumferential value of the axial component of the electric field due to the source. As the geometry is independent with respect to the z-dimension, a solution to the integral (1) may be obtained by utilizing a spatial Fourier transform as

J--03

. "O?

~

63

Z ( k ) = Z S ( k Z ) - < E:(k*) >I+ / l ( k z ) ( 5 )

The impedance Z is a result of two terms, the axial field < E: > Ip=b just at the exterior surface of the cable taking into account the air-earth interface, and Z" representing the axial field just at the interior surface of the cable assuming a cylindrical geometry. In general, (4) can be used to calculate the induced current due to any given source by transforming the imposed electric field using (2). Specialization to the plane wave situation is developed later in this section. For a zero source field EfNC = 0, the eigenvalue solution of (3) yields the characteristic propagating mode k , = k$ supported by the cable.

The axial field < E: > Ip=b at the outer surface of the cable carrying a current I will be calculated assuming an axial dependence of the form For a cable located at (z = 0,y = -d), the extemal fields ( p > 15) can be deduced by solving

-a where Il and ng are the two dimensional Hertz vector potentials in the air and earth regions, respectively. Here ko = JG is the propagation constant in the air medium, and k , = Jw2potg + iwpoa, is the propagation constant in the earth medium. The associated fields in the earth medium are

The solution to (6) is obtained using the usual transform technique [5] and then satisfying the boundary conditions at the air-earth interface. Following this approach the axial field

64 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 37, NO. 1, FEBRUARY 1995

where TO = ,./= and rg = ,/- are the transverse propagation constants in the air and earth media, respec- tively. The real parts of the irrationals Re[Uo,Ug] 2 0 and R e [ 7 0 , r g ] 2 0 are chosen to retain a positive value. 10 ( z ) , KO ( z ) , K1 ( z ) are modified Bessel functions of complex argument and n = ,./erg + ,io,/wq is the refractive index of the earth. The average value of the fields around the cable <> Ip=b results in the factor 10(7gb) in (8).

The fields at the internal surface of the cable and thus Z" can easily be determined for various cable types such as solid wires, insulated cables, wrapped conductors, etc. [ 141, [15]. For the lossy insulated cable shown in Fig. 1 the surface impedance can be formulated using the procedure in [I61 as

electrical parameters characterizing the cable core and its outer dielectric insulation were previously defined as (tu;, p,. ow) and ( E d , ,%d, Pd) , respectively. (loa) reduces to the bare conduc- tor case (lob) when the insulation thickness approaches zero Zs = ZWla=b. For typical cable diameters, lrgb( . (Tdb( << 1, small argument expressions for the modified Bessel functions in (1 0) can be utilized. As well, the core can usually be as- sumed to behave as a good conductor k, << k, M ,/- such that

( Z d + 2,) + k; (Yd)- ' ; insulated cable z" la=b: bare solid cable

ZS(k , ) M { (1 la)

where Zd + 2" is the per unit length series impedance and Y d is the per unit length shunt admittance for the cable. These are the more commonly used circuit equivalent expressions which are much easier to evaluate and derive for this and more complicated cable structures [l], [17]. For a perfectly conducting core 2"' = 0.

In this paper a plane wave incident source field is con- sidered. 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 is defined in terms of its vertical E0 and horizontal E4 polarized components and the angles (6,4) as shown in Fig. 1. In this manner, the imposed electric field at the surface of the cable can be determined by the transmitted fields into the earth as

(12) EfNC(z) = ET(B, 4 ) e + i k o Z c O S ~ c o S 4

E, T (0, 4) = [EoTv sin Bt cos q5t + E ~ T H sin 4t]e+ik57d

(13a) 2 sin 6

sin B + Jn2 - cos2e ( 13b)

where Tv and T H are the vertical and horizontal transmission coefficients at the interface with n cos Bt = cos 0, n sin Bt = dn2 - cos2B, dt = 4. Note that the phase reference has been chosen as the origin (z = 0, y = 0, z = 0). The axial electric field in the spectral domain can be determined by transforming (12) and evaluating the average circumferential value as

2n sin 0 n2 sin 8 + 4"' Tv = T H =

< EENC(kZ) > 00

ET(^, 4 ) e + i k o z ' c o s e c o s ~ e - i k ; z ' d zI > =< s_, = 27~10(~~b)ET(d , 4)6(k,-ko COS BCOS 4) (14)

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

where <e,+ is the axial component of the incident plane wave wavenumber. The impedance term Z was defined in @)-(IO) with its argument now specified as the single spectral component k , = <e,4. rg is the transverse component of the incident plane wave wavenumber.

111. TRANSMISSION LINE APPROXIMATION

Most studies rely on the use of an equivalent circuit trans- mission line approach to determine the induced currents on buried cable systems [I], [2], [18]. The approach assumes that the currents on the cable can be represented using only exponential traveling waves having the form exp { f i k : z } , k: being the propagation constant of the mode supported by the structure, and thus neglecting the radiation and surface wave contributions in a complete solution (4). The propagation constant (and characteristic impedance) is determined from the transmission line per unit length equivalent circuit parameters, these derived by applying simplifications directly to (6) (e.g., setting k , = 0). The validity of the equivalent circuit approach

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BRIDGES: TRANSIENT PLANE WAVE COUPLING TO BARE AND INSULATED CABLES BURIED IN A LOSSY HALF-SPACE 65

has been studied mainly in the case of overhead transmission lines [7 ] , [ 181, observations being that for reliable application, the transverse dimensions of the transmission line structure should be much less than the free-space wavelength and the refractive index of the earth should be large. For plane wave incidence in the case of overhead systems, the approximation is worst at grazing angles.

In this section the transmission line approximation is derived directly from the exact solution presented in the previous section. In formulating the transmission line approach, the incident field is represented through an infinite set of delta function voltage sources distributed along the length of the cable, the magnitude of the sources at a particular point being proportional to the axial component of the imposed electric field. The formulation is developed from the exact solution (4) by utilizing the convolution theorem as

I(z) = 1“ I(zl z’)EfNC(z’)dz’ (16) -00

where I(z,z’) yields the current at the observation point z due to a delta function voltage source of strength EINC(z’) located at z’. The current formulated as in (16), (17) is still an exact solution. Note that the term lo(.rgb) results from taking the average value of the incident field around the cable.

The inverse transform (17) determining the currents on the structure due to the localized delta function source contains a pole as well as a set of branch cuts symmetrically located in the k , plane as shown in Fig. 2. The pole arises from the singularity in the impedance term Z(k,) and represents a discrete propagating mode having currents and fields which decay exponentially in the axial direction away from the source as exp(+ikclz - z’l), with Im[k:] 2 0. The propagation constant can be determined from the solution of the mode equation Z(kp) = 0. Note that it may be possible for the cable to support more than a single discrete mode as in the case of overhead transmission systems, however, the excitation of these additional modes is expected to be small and are neglected in the buried situation. The contribution due to the branch cuts at f k o , f k g arise from the requirement on the irrationals Re [UO, U g ] 2 0 defined so that the fields decay as IyI + co, these representing a spectrum of modes radiating into the air and earth half-spaces, respectively. An additional branch cut at k , ~ = * k , / d W arises from the singularity in the denominator of the integral G in (9b) and represents a spectrum of surface waves supported at the interface.

By deforming the path of integration from the real axis, the integral transform (17) can be determined as a sum of the discrete propagating mode and spectrums of continuous modes. In the insulated cable case it is expected that the dis- crete mode contribution will dominate the current away from the source [ 121. In the immediate neighborhood of the source the radiation (and surface wave spectra if the cable is located near the interface) may also have a relevant contribution. In utilizing the transmission line approach, however, only the

I I I kf /.

Fig. 2. Pole and branch cut location in the complex k , -plane.

discrete mode contribution to the current is considered. To this extent, the current is given by the residue of (17) as

.EfNC(z’)dz’. (18)

If the limiting value of the impedance term in (18) is viewed as the structure’s generalized characteristic impedance [ 191 and the lo()EfNC() term is viewed as the local excitation voltage, the above form can be solved as

Up to this point, the formulation presented can be denoted as a transmission line solution since only discrete exponential modes are used to represent the current on the structure. The values used for the propagation constants k: are solutions of Z(kf) = 0 as determined using the exact expressions (8)-(10) and thus, the resulting fields are still solutions of the wave (6). The solution of the mode equation in this form, however, still involves the evaluation of complicated integrals (9a, b). To simplify this problem it is useful to cast the impedance Z in terms of the more commonly used transmission line per unit length equivalent circuit parameters as shown in Fig. 3. Specifying the scalar potential at the cable surface as 4 =< -V-ng > lp=b, the circuit parameters can be identified as

66 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY. VOL. 37, NO. I , FEBRUARY 1995

W')

Fig. 3. Equivalent 3 circuit transmission line model.

+K0(7,2d) - G(k:, 0, -41 (24)

where 2""' is the series impedance, Y s h is the shunt ad- mittance, and r = 1.7811. The circuit parameters z", Zd, Y d modeling the cable were previously defined in (1 1). Note that small argument expressions for the appropriate Bessel functions have been utilized as the argument Ir,bl << 1 for typical cable diameters where the transverse propagation con- stant r, = 4- now has a fixed value. The parameters (22)-(24) are generalized circuit quantities in that they are functions of the variable kf. The propagation constant and characteristic impedance of the system can then be expressed as

IC: = 4- -zs"(k:)Y"h(k:), 2, = [ ~ ~ s ~ ~ ~ ~ ~ ] (25)

Note that the evaluation of kf using (25) still yields the same result as exact mode equation. However, the evaluation of 2, using (25) will only equal (20b) under the assumption that 2"" and Ysh are stationary functions at kf [19].

IV. APPROXIMATE EVALUATION METHODS

The transmission line induction formula (19) requires the evaluation of and 2, (usually done through the per-unit- length circuit parameters). The evaluation of the propagation constant of buried cables has been well studied [l], [3], [9], [lo]. Sunde [3] showed that the propagation constant of a filamentary bare wire located at an air-earth interface is equal to the mean-square value of the propagation constants of the two media kf x k$+k;)/2. As the wire depth

constant becomes that of a bare wire in a homogeneous earth medium kf = k,. For coated cables, the propagation constant should be dominated by the insulating jacket as its thickness becomes large kf -+ kd1b-m. In general, kf will depend on the contribution from each of the equivalent circuit components (22) and will approximately lie in the range k d < kcs < k,. With this observation it is reasonable to

is increased to beyond a f-- ew skin depths, the propagation

assume )kfl >> (e,@ = ~ O C O S ~ C O S ~ ~ . Under this assumption and using (25) for k: and Z,, (19) can be simplified as

which is now dependent only on the series impedance term as defined'in (22). The approximation will be worst near grazing angles 6',4 + 0 but is still acceptable as long as Ik:l >> lkol. Note that the excitation field ET becomes very small near grazing, producing a small induced current. This conclusion is drastically different than that arrived at in the overhead cable case, where kf x (e,+? near grazing. In the overhead case the approximation produces a very large induced current which is in error with the exact solution [7].

To determine 2""' the evaluation of Z", Zd, and Zg in (22) is required. The calculation of the series impedance terms 2" and Zd involves simple expressions. The calculation of Z g on the other hand is more difficult, but its accurate evaluation is important in that it is usually the dominating term in the series impedance. Depending on the burial depth two approximate expressions for Zg can be developed. When the cable is buried at a depth near or greater than the skin depth in the earth, the integral J and Bessel function K o ( ~ ~ 2 d ) contributions in (23) become negligible yielding

where 6, = l / d z . i s the skin depth in the earth. Alter- natively, when the cable is buried near the interface relative to the skin depth the integral J and the Bessel function K0(7,2d) can be approximated by their small argument expressions [ 101 as

2

zg(kx) d ~ g 2 [In ($,b) + 2 In (:) - i] . (28)

Both (27) and (28) have a similar behavior, differing only in a constant factor. Zg thus has a form fairly independent of burial depth, as expected, since the magnetic field around the cable is not effected to a great extent by the interface. Ysh on the other hand, decreases to nearly 1/2 its large burial depth value as it approaches the interface. The approximate expressions for Zg still require the evaluation of 7,. Typically the transverse propagation constant is approximated assuming the axial variation of the fields are small compared to their transverse variation r, z -&,. This will also be the situation in the exact case (15) for 171.1 >> 1. This final assumption and the progression of approximations (18)-(28) developed so far lead to a fairly simple form for the coupling impedance as

x iwclo l*l( 2T

where the series impedance of the inner conductor 2"' has been neglected. For a lossless nonmagnetic insulating jacket, the In() terms in Zg and Zd can be combined so that the

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BRIDGES: TRANSIENT PLANE WAVE COUPLING TO BARE AND INSULATED CABLES BURIED IN A LOSSY HALF-SPACE

bare and insulated situations become equal (as done in the last expression of (29)). It is important to note that the standard practice of substituting -ik, x (1 - i ) / h g in (29) 111, [2] can only be used when the earth behaves as a good conductor a g / w t , >> 1. When this is violated the dielectric properties become important.

A further simplification to the induction formula (15) is often made by approximating the incident field EF(0,4) under the assumption of a highly conductive earth 1711 >> 1. For this situation the transmitted axial field (13) will be nearly perpendicular to the interface Ot M 90' so that

2 sin 0 n

--t- 2 TIT - -

et -900 n 1 T H et -900

.2 , / - iw tO/age- ( l - i )d / * , . (30b)

The approximation results in an expression where the excita- tion is independent of incident angle for a vertical polarized field. Since In1 > 1 for all frequencies, the approximation for TH should be adequate for all 0. However, the approximation for T ~ T is accurate only when Insin01 > 1, which may not remain true as 0 -+ 0'. The commonly used approximation (30b) under the condition ag/wtg >> 1 thus leads to a transmitted field which attenuates exponentially with depth and frequency. However, when this condition is violated the magnitude of the field should be independent of frequency.

Finally, using the developed approximations (29), (30), the induced current can be determined using the very simple form

[EO cos 4 + E+ sin 0 sin 41 - -io7 ag/we,Bl d-ln((1 - i)a/6,)

. e - ( l - a ) d l * g e + i S ~ . d z . (31b)

Even though the form (31b) is inaccurate when o g / w t g < 1, it is useful since it is possible to directly obtain a time domain analytical expression for the current for some simple incident field waveforms [2]. The results of the next section examine the validity of the transmission line approximations; (29) with (13), (311, and (31b).

v. NUMERICAL STUDY

In this section the current induced on example cable struc- tures is calculated and the validity of the approximate methods examined. For this study, a cable with a copper inner conductor (crw = 5.7z1O7S/m) of radius a = 1.0 cm will be used. If the insulated situation is considered, a lossless dielectric jacket ( t ,d = 5.0, a d = 0) of outer radius b = 1.25 cm will be added. As discussed in the previous section, the validity of the various approximations will depend on the relationship between; the

~

61

104 3

Fig. 4. Magnitude of 1181 as a function of frequency and conductivity for f , g = 10.

spectral content of the incident field, the electrical properties of the earth, the incident field direction, and to a lesser extent the burial depth of the cable. Figure 4 is a plot of the earth's refractive index as a function of frequency for typical conductivities and a nominal permittivity erg = 10. Several of the approximations were developed under the assumption w < 1 / ~ , , where T~ = €,/ag is defined as the time constant of the earth (not to be confused with the transverse wavenumber T~). However, depending on the spectrum of the incident field, a significant portion of the energy may be in the frequency range w >

To examine the approximations for the coupling impedance, Fig. 5 compares the normalized magnitude Iko/Z(<e,+)l cal- culated using the exact (15), and approximate expression (29) over a 1 H z - l G H z frequency range. A reasonably dry earth a, = 10-3S/7n, E , , = 10 is assumed (1/27vg = 1.8 MHz). This is near the worst case since the approximate methods should be more accurate for higher ground conductivities. Cable burial depths of d = 1 m and d = 100 m are chosen, thus representing both the shallow and deep situations. Perpendicular excitation is assumed with O= 90', q5= 0'. Even though the approximate form (29) is independent of incident angle, burial depth, and whether the cable is bare or insulated, it is a very good estimate of the coupling impedance over the entire frequency range. At very low frequencies, the difference between the exact and approximate cases is due to the series impedance of the inner conductor. This effect will be correctly reflected if 2" is included in the approximation (29). The oscillatory nature of the exact result at high frequencies and shallow burial depths is due to reflections from the interface ( d = A, at 95 MHz). This is not observed at large burial depths due to the rapid decay of the field in the earth. The agreement between the approximate and exact expressions improves when the burial depth becomes greater than the skin depth in the earth (6, = 100 m at 25.3 KHz). The coupling impedance for the insulated situation was also calculated and produced the same result as the bare case. As concluded in the discussion of (29) in the previous section, since the coupling

68 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 31, NO. I , FEBRUARY 1995

Fig. 5. Normalized coupling admittance (k, /Z(C~, , ) (calculated using the exact (15) and approximate expression (29) with H= 90°, o = Oo, e , (I = 10 , u<,= 10-"S/t12.

is dominated by Z"", the bare and insulated cable cases should be equivalent when the insulating jacket is lossless and nonmagnetic, and the outer radius is small compared to the skin depth in the earth.

The transmission of a plane wave into a lossy half-space for both the time harmonic and transient situations has been well studied [20]. To examine the various approximations used for the transmitted field, Fig. 6 compares the magnitude of IET(B,4)/EeI as calculated using the exact (13) and approximate forms (30), (30b). The results are compared over the normalized frequency w x T~ with erg = 10 and gg = 10-3S/7n (1/27r7, = 1.8 MHz). Two depths are examined d = 1 m, 10 m with the incident angle chosen as B= 90"; 4= 0". The approximate form (30b) should not be used when w > l/-rg. As the approximate forms are not a function of 0, Fig. 7 shows the magnitude of the incident field for various angles of incidence. The results of Figs. 6 and 7 indicate that as long as the complete representation of the electrical properties of the earth are retained (both gg and c g ) the approximation (30) should provide adequate results for all frequencies. The approximation is worst for angles of incidence near grazing.

To examine the transient response of the system, an incident field with a single exponential decaying form will be utilized as

where EO and E, are the magnitudes of the vertical and horizontal field components and 7 is the decay time constant. The transient response of the cable is obtained by multiplying the transform of (32) by the impulse response of the cable (15) and then employing the inverse Fourier transform. Note that the time reference t = 0 is located at the origin z = y = z = 0. Many sources produce very low frequency transients, whereas time constants as fast as T = 20 ns have been suggested for an HEMP environment [18]. To include this wide range

10 * 10 ' 1oO 10' Id ztzg (Og'OEg

Fig. 6. exmessions (30). (30b) for d = 1 1 n and lOn~with H= 90'. o = 0'

Transmitted field calculated using the exact (13) and approximate . , , ,

e,.; = 10, ug= lo-:'s/rIl

100 ,__ ,,_,_ & .__.._.__.._.._..-.._.._~

10-15 io" 1 10 ' 1 00 appromate 10' (30) d-an ld

2/'5s (08'wE8

Fig. 7. Transmitted field calculated using the exact (13) and approximate expression (30) for several angles of incidence with d = O i i t , e , (, = 10, oy= 1 0 - 3 s / ~ ~ ~ .

of phenomena, a large variation of the time constant ratio T / T ~ should be examined. The transient response of a cable buried at a depth d = 1 m calculated using the exact (15) and approximate (31) forms is given in Fig. 8. The cable is excited at normal incidence #= 90°, 4 = 0", Eo = IkV/m, E+ = 0 with a fast time constant 7 = 8.854 x lo-' s. The permittivity of the earth is fixed erg = 10 with the conductivity varied as og= lo-', 10-4Ss/m, this corresponding to time constant ratios T / T ~ = 10,1,0.1, respectively. Fig. 9 compares the transient response calculated using the exact (15) and approximate (31) forms for various burial depths d = 1 m,20 m,50 m. A time constant ratio T / T ~ = 1 ( E , ~ = 10,gg= lO-'S/m) with normal incidence B= 90", 4= 0" is chosen. These results indicate that even though the approximate form overestimates the current it may provide a good approximation over a wide bandwidth.

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BRIDGES TRANSIENT PLANE WAVE COUPLING TO BARE AND INSULATED CABLES BURIED IN A LOSSY HALF-SPACE

35

E 30 5 5 25

h

0

4 20 c Q

E 15

-0 10

c

L z 8 z 5 -

0 0 0.2 0.4 0.6 0.8

Time (us)

Fig. 8. Induced current calculated using the exact (15) and approximate expression (31) for several ground conductivities o,with ( 1 = l i i t , H= 90". 0 = O0, f , 9 = 10.

0 0.2 0.4 0.6 0.8 1 Time (us)

Fig. 9. Induced current calculated using the exact (15) and approximate expression (31) for several burial depths dwith H= 90". o = 0". 6 , , = 10, u9= 1 0 - 3 S / ~ i ~ , and T / T ~ = 1.

The approximations (31), (31b) were derived utilizing an approximate form for the transmitted field (30), (30b) so that the coefficient Tv had no 0 dependence. As was indicated in Fig. 7, this approximation was worst near grazing angles. Fig. 10 examines the transient response of a cable at a depth d = 1 m for various angles of incidence 8= 2", 10",90", q5= 0'. A time constant ratio T/T, = 1 with erg = 10, og= lUp3S/m is used. The approximate form (31) yields a single result for all angles. The result calculated using the approximate coupling impedance (29) along with the exact transmitted field (1 3) is also shown in Fig. 10. The result using (13), (29) agrees very well with the exact response and still does not require the evaluation of complex integrals.

Figs. 1 l(a) and 1 l(b) give the transient response of a cable at depths d = 1 m, 10 m calculated using the exact (15) and approximate forms (31) and (31b). Time constant ratios of r/rg = 1.0,O.l are considered with tTg = 10, cg= 10-3S/m

25 .

A

E

!i20 0 A 15 Q C

3 0

D

c

c

t! 10

-0 3 5 c -

0

69

- ~ x a a e - s v - emu e-io - exacl 8-2- - approxmale (31) e q n ~ (299) and (13)

0.2 0.4 0.6 0.8 1 Time (us)

Fig. 10. Induced current calculated using the exact (15) and approximate expressions (31) and (13). (29) for several incident angles Hwith tl = 1 m, f,.<, = 10, cg= lO-'S/ii,, and T / r , = 1.

and an incident angle of 8= 90", q5= 0". Note that (31b) was derived assuming I C , z (1 + i ) / h g , n z d-. These results indicate that for the transmission line approximation to be acceptable in all transient situations, the complete representation of the earth's electrical properties should be considered.

The approximation (3 1) yields a worst case error in the peak current of about 30% for the cases examined in this paper. The transmission line approximation is worst for fast rise time signals and for low earth earth conductivities (T/T, < 1) and for shallow burial depths. The use of (29) in conjunction with (13) provides much better results with very little extra computational burden. In many practical situations accurate knowledge of the earth's properties is not available or varies due to climatic conditions. The error due to these variations may be much greater than the error introduced when using the approximate methods.

VI. CONCLUSION

The transient current induced on a buried bare or insulated cable was formulated using both an exact theory and an approximate equivalent circuit transmission line theory. The transmission line model proved to be adequate over a large frequency bandwidth as long as a complete representation of the earth's electrical properties was retained (both og and tg). Commonly used simplifications for the plane wave transmission coefficient at the air-earth interface become inac- curate near grazing angles of incidence. For accurate coupling results at all angles of incidence and frequencies, the exact transmission coefficient (1 3) should be used in conjunction with the approximate coupling coefficient (29). This does not greatly increase the computational burden as the final frequency domain result is still in a closed form. The induced current for the insulated situation will yield the same result as the bare case when the insulating jacket is lossless and nonmagnetic.

70

25

h

E 2 20

3 0 4 15 c

4- m C a, t 10 a -0 a, 0

-0 C 3 5 -

0

~

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 37, NO. I, FEBRUARY I995

0 100 200 300 400 500 600 Time (ns)

(a)

0 50 150 200 Time (ns)

(b)

Fig. 1 1 , Induced current calculated using the exact (15) and approximate expressions (31) and (31b) for several burial depths dwith 19= 90°, o = Oo, f,g = 10, Ug= 10-3s/Jil; a) r / r g = 1, b) T/Tg = 0.1.

ACKNOWLEDGMENT Suggestions made by the reviewers for improving the clarity

of this paper are appreciated.

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Greg E. Bridges (S’82, M’88) received the B.Sc.E.E., M.Sc., and Ph.D. degrees in electrical engineering from the University of Manitoba, Canada in 1982, 1984, and 1989, respectively.

He joined the Department of Electrical and Computer Engineering at the University of Man- itoba in 1989, where he is currently an Associate Professor. His present research interests are in computational electromagnetics, wave interactions with transmission lines, and the development of microscopy techniques for microwave integrated circuit diagnostics.