IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 31. NO

again forms a reasonable lower bound. An upper bound “cube” curve might be expected around 1&12 dB, in analogy to Fig. 3. In fact, this is what is measured at the MEMPS simulator.

The far-field source data are shown in Fig. 7. Again the low- frequency SE ( < I O MHz) tends to approach the static (Fig. 7(a)) and quasi-static (Fig. 7(b)) enclosure limits (represented here by the MEMPS data), and the higher frequency SE (> I O MHz) approaches the plane wave level. at least to 100 MHz after which the mesh approximations begin to lose validity. Above 100 MHz, the 50-cm mesh provides very little to no shielding. In fact, negative SE values are observed in the electric field case, indicating that resonances may result in fields which are locally higher than those without the cage present.

The 2-m cage of 50-cm mesh can also be simulated using the method of moments (MOM) using 192 wires each 50 cm long (the 5-cm mesh cube would require 19,200 wires which is beyond our present capability). Each wire is subdivided into 3 segments for the computation. A 2.25 mm wire radius and plane wave incidence are assumed. The fields are calculated at the center of the cube and normalized to the incident field. These data are also shown in Fig. 7 (designated with the program name CONCEPT). These data agree well with MEMPS far-field measured data. Note that in the electric field case (Fig. 7(a)) a resonance is predicted around 35 MHz as opposed to the 106 MHz expected for a closed metallic cavity of the same dimensions. Measured data (not shown here, see [5]) confirm the numerical results suggesting that resonance effects in metallic mesh structures may be expected at significantly lower frequencies than for closed metallic cavities structures.

V. CONCLUSION This paper has examined various analytic approximations for the

shielding effectiveness of cage structures made of wire mesh and compared calculated results to a variety of measured data. The following statements may be made.

1) For a given mesh structure, both calculated and measured SE data cover a wide range of values depending on the wave impedance, frequency, antenna type and separation, and structure geometry. The interpretation of SE data without clearly specifying these parameters is difficult.

2) For the mesh cube considered here, near-field source SE tends to fall between an upper limit defined by the cage geometry and a lower limit defined by the antenna separation.

3) Near-field source SE is reduced as the source and observation point are brought together.

4) For the mesh cube considered here, far-field source SE tends to fall between an upper bound defined by the SE of an infinite sheet of the wire mesh and a lower bound defined by the cage geometry.

ACKNOWLEDGMENT

The author would like to thank R. Coray of the Swiss Telecom PTT and the firm EMC Fribourg for their support of this work.

REFERENCES

[I] J. R. Wait, ‘Theories of scattering from wire grid and mesh structures,” in Electromagnetic Scattering, P. L. E. Uslenghi, Ed. New York: Academic Press, 1978, pp. 253-287.

[2] K. F. Casey, “Electromagnetic shielding behaviour of wire-mesh screens,” IEEE Trans. Electromagn. Compat., vol. EMC-30, no. 3, pp. 298-306, Aug. 1988.

I , FEBRUARY 1995 131

131 M. I. Astrakhan, V. P. Akimov, and G. A. Fersman, “Electromagnetic grid shields,” in Proc. Int. WrotJaw Symp. on EMC, Wroclaw, Poland, pp. 707-710, Sept. 1990.

[4] M. Ma, Ed., Special Issue on Electromagnetic Shielding, IEEE Trans. Electromagn. Compat., vol. EMC-30, no. 3, Aug. 1988.

[5] M. Nyffeler, B. Braendli, B. Reusser, E. Doerr, D. Gin, and E. Tomer, “On the interaction of electromagnetic fields with wire cage structures,” IEEE Trans. Electrmagn. Compat., vol. 34, no. 4, pp. 471477, Nov. 1992.

Capacitance Calculations for Cable Harnesses using the Method of Moments

J. S. Savage, Student Member, IEEE. and W. T. Smith, Member, IEEE

Absfruct-A method of moments procedure to obtain the capacitance matrix for a cable harness is presented. The geometry consists of multiple, parallel, insulated wires, possibly above a ground plane. The method uses Fourier harmonic expansion functions for the total charge and Galerkin testing functions. The capacitance matrix is obtained by the relationship of free charge to potential. Computed and experimental results are presented and compared to exact analytical solutions when possible.

I. INTRODUCTION

Cable hamesses are used to connect a variety of electrical equip- ment. The main reason for grouping wires into hamesses is to keep the wires neat, orderly, and hidden from view. However, crosstalk is maximized when wires are placed in such close proximity.

The first step in modeling the crosstalk in any multiconductor transmission line (MTL) is to obtain the per-unit-length (PUL) inductance, capacitance, and resistance parameters. The transverse electromagnetic (TEN) field structure is assumed. Given the correct PUL parameters, computation of the crosstalk is straightforward. In cable harnesses, however, obtaining accurate PUL parameters is dif- ficult due to the inhomogeneities introduced by dielectric insulation, the close proximity of the wires, and the typical presence of a ground plane.

The determination of the per-unit-length parameters has been the focus of much research in recent years. Under the TEM assumption, the problem involves the solution of a two-dimensional electrostatic problem. Unfortunately, closed form solutions for the per-unit-length parameters can be determined for a very limited number of MTL geometries. In the analysis of cable harnesses, approximations or numerical methods must be used.

The wide separation approximation is a common method used to obtain the per-unit-length parameters for cable hamesses. This method assumes a constant charge distribution around the periphery of the wires and yields accurate results as long as the wires are sufficiently separated [ I ] . In addition, wire insulations are ignored when using the wide separation approximation. More accurate results can be obtained through the use of numerical methods. For example, the finite difference and finite element methods are well suited for

Manuscript received July 7, 1994; revised June 15, 1994. J. S. Savage is with the Georgia Institute of Technology. W. T. Smith

is Assistant Professor with the Department of Engineering, University of Kentucky, Lexington, KY 40506 USA.

IEEE Log Number 9407396.

0018-9375/95$04.00 0 1995 IEEE

132 lEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 37, NO. I . FEBRUARY 1995

Fig. I . the distance from the ith wire center to the jth wire surface.

A sample 7) + 1 wire harness cross-section. The test distance, t/( is

closed-region electrostatic problems [2], [3]. The method of moments, however, is well suited for open region geometries such as cable hamesses [4]-[7].

Method of moments solutions have previously been used to deter- mine the per-unit-length capacitances and transverse charge distribu- tions for multiconductor transmission lines [8], [9]. In those works, the method of moments solutions involved the use Fourier harmonic basis functions and point matching. The method presented in this paper incorporates Fourier harmonic basis functions with Galerkin testing and is based on the work of Clements, Paul and Adams [8]. The Galerkin method has two advantages over the point-matching method. First, convergence rates are accelerated for all geometries when Galerkin testing is used. Second, point-matching can yield a singular moment matrix, unless special considerations are given to match-point location. The Galerkin testing procedure requires no special considerations. Computed and experimental results are presented and compared to exact analytical solutions when possible.

11. MOMENT METHOD FORMULATION

A. Uninsulated Wires This method assumes a two-dimensional geometry (invariant in

one direction) of circular conductors. A sample I I + 1 wire hamess cross-section is given in Fig. 1. The unknown charge distribution on each conductor surface is expanded with a set of Fourier harmonic basis functions of the form

N .

where Q , L is an unknown constant coefficient and f,k (0,) is a basis function with known form. The subscripts li and i denote the Xth basis function on the ith wire. The basis functions span the entire domain of each wire and are functions of the wire periphery angle. They have the form

l i = O

I; = 1. . . . , -Yz f , L ( Q l ) = C O S [ l i Q , ] . l i= l:..,S, (2) { sin[l;o,], l.

where S, is the number of Fourier harmonics on the ith wire. Including the constant term, the ith wire has 2 S , + 1 total basis functions. The Green's function for this problem is the voltage at any point due to a line charge. From Gauss's Law, this expression is

(3)

where d , is the distance from the line charge to the test point. The constant of integration is ignored. By convolving this Green's function with each basis function, the potential at any point due to each basis function is obtained [8]. The analytical expressions for these convolutions are summarized in Table I.

The term r, is the radius of the ith wire and I;, ( f , ~ ) indicates the voltage at a point due to the kth basis function on the ith wire. Using a Galerkin testing procedure, the voltage functions are matched with

TABLE I A SUMMARY OF POTENTIALS OUTSIDE FOURIER

HARMONIC BAsls FUNCTIONS FROM [8]

testing functions identical in form to the basis functions. This results in a matrix equation of the form

Z a = V (4)

The vector Q contains the unknown basis function coefficients. The excitation vector V has the form

v = [V,'. . . V T . . . v y ( 5 )

where '"'' denotes vector transpose. Each sub-vector in (5) is

v, = [27r1; O . . . O ] T (6)

The form of (6) is a result of Galerkin testing. Each potential term in the excitation vector is multiplied by the testing function and integrated around the wire periphery. Since

277

cros(lio, )do, = 1 \ in(l io,)do, = o lT (7)

only the constant terms, n,o survive the integration. The moment matrix, 2, has the form

where the submatrix, Z , , , relates the ith wire and the j t h wire. The terms in these sub-matrices are

.2T

[ z , , ] a h = f ~ a 1 ; b f l O ~ (9) 0

where [Z,,lC,b is the entry on the crth row and bth column of the submatrix, Z , , , f J U is the crth testing function on the j th wire, and I ; l is the voltage at a point due to the bth basis function on the ith wire. The integral in (9) generally does not have a closed form solution. Therefore, numerical integration was used when analytical expressions could not be obtained.

The generalized capacitance matrix is defined using

Q = C@ (10)

where is a vector of the total free charge on each wire and !P is a vector of the potential on each wire. The total free charge on the ith wire is simply the integral of the free charge density

Substituting the charge density from ( I ) gives

Q, = 2Tir,n,o

~ ~~

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 31, NO. 1 , FEBRUARY 1995 133

TABLE I1 A SUMMARY OF POTENTIALS INSIDE FOURIER

HARMONIC BASIS FUNCTIONS FROM [8]

Again, only the first coefficient from ( I ) remains. The total potential on the j th wire is already contained within V. The solution of (4) is

rZoo io, . . .

where each submatrix, Z,, , has the same dimensions as the original submatrix, Z , , .

With some manipulation, (13) can put in the form of (10). Since the total charge on any wire involves only the first basis function coefficient (constant) and the potential on any wire involves only the first term in each voltage sub-vector, each term in the generalized capacitance matrix involves only the first term of each submatrix Z,, . The generalized capacitance matrix is

where [Z,,]OO is the first term (upper-left) of the submatrix Z,, and r , is the radius of the ith wire. If desired, the transmission line capacitance matrix can be obtained from (14) by choosing one conductor as a reference conductor [l]. It can be shown that the first order Galerkin moment method solution for the generalized capacitance matrix is identical to the wide separation approximation (see appendix).

B . lnsulated Wires The method of the previous section may be extended to include

dielectric insulations around each wire. The insulations are also assumed to be circular and concentric with respect to the conductor axes. The same basis functions from Table I are used to expand the total (free plus bound) charge density on conductor surfaces. Additional basis functions are used to expand the total (bound) charge density on the dielectric surfaces. The additional basis functions have the form

k=O

where "*" denotes the charge on the dielectric surfaces. These additional unknowns require additional equations to obtain a unique solution. These equations are obtained using the normal boundary condition. The normal components of the electric flux density must be continuous across material interfaces. If this boundary condition is applied, a new matrix equation is obtained,

TABLE 111 A SUMMARY OF THE ELECTRIC FIELD INSIDE AND OUTSIDE

FOURIER HARMONIC BASIS FUNCTIONS FROM [8]

Wire i /

Fig. 2. An illustration of the variables used in TABLE 111.

where P1l and PI,, relate all charge expansions to the conductor voltages and P,I and P 2 2 enforce the continuity of normal electric flux density on dielectric-free space interfaces. The entries in this matrix equation are now considered.

In (16), P11 is identical to 2 from (4). P12, however, is a slightly modified version of 2. In P1;7 calculations, the basis functions lie on dielectric surfaces, while testing functions lie on conductor surfaces. Since dielectrics cannot overlap, only the self matrices in PI? are special cases. For the self terms, expressions for the potential inside a given Fourier harmonic charge distribution are required. These expressions were determined in [8] and are summarized in Table 11.

In P r l and P22. the basis functions lie on conductor surfaces, while the testing functions lie on dielectric surfaces. The entries in Pzl and P22 relate the difference between the normal components of the electric flux density. For these entries, expressions for the electric flux density produced by any basis function are needed. The electric flux density D is found by

(17)

where E,k( f,k ) is the electric field on the j th surface produced by the kth basis function on the ith surface. The expressjons fo' E,A (f,k ) given in [8] are summarized in Table 111, where rl, and of are unit vectors parallel and perpendicular to the vector rl , Lo! as shown in Fig. 2.

Finally, since the normal components of D must be equal on dielectric-free space interfaces, the difference between the normal components of D with and without the dielectric must be zero. Therefore, the excitation vector, V, is augmented with zeros. The entries in P21 and P 2 2 are:

D = :E = : ( -Tl ;~( f ,k) ) = :E,A(~,A)

[[PzI],,],,~ = f 7 fJ,,[fi . : , E , ~ ( f , b ) ] t l o , 0

- 12= fJ,4[il . r o E , b ( f , b ) ] r l o ,

I34 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 31, NO. I . FEBRUARY 1995

[ [ p Z z ] t J ] t 1 6 = rr f;t#[fi ' fjE,6(f:6)]do,

Z7r

- 1 ff;,[fi ' ' O E t b ( f : b ) ] d o , (18)

where f i is the outward unit normal vector on the surface of the j t h dielectric-free space interface. By linearity, the entries in P21 and Pe2 are

21.

[ [ p 2 1 ] , J ] u 6 = ( f J - f o ) / .f;~t[~ ' E t 6 ( f l 6 ) l d O J

[ [ P 2 2 ] r J ] a 6 = ( : J - ' a ) / f;<#[fi .Ez6(.f:b)]do~ P A

(19) 0

This completes the evaluation of the terms in (16). Finally, the generalized capacitance matrix is formed by determining the total free charge on each conductor. The free charge on the conductor surfaces is simply the total charge minus the bound charge on the inner dielectric surface. Since the charge neutral dielectrics have equal but opposite bound charge on the inner and outer surfaces, the free charge on the conductor surface is the total charge plus the bound charge on the outer dielectric surface. Therefore, if Z is defined in a fashion similar to (13),

2 , , J i s in the upper-left quadrant of 2-' and Z,,+l+t., is in the lower-left quadrant of 2-' . The sum in (20) is the result of the sum of the total conductor charge and the outer dielectric bound charge. Once again, the PUL capacitance matrix, C, can be determined by choosing any wire as the reference conductor,

C. Wires Over a Ground Plane

The problem of 1 1 wires over a ground plane is now considered. Using image theory, an equivalent problem is formed where the ground plane is removed and image wires are added. The distance of each wire above the ground is equal to the distance of its image below the ground plane. Since the equivalent problem is just 21, wires, it is solved using the same procedure as in the case of 1 1 + 1 wires. This procedure gives twice as many unknowns as would n wires. The result is a (2t1 x 2 n ) matrix equation of the form Q = CV, or

. . .

(21)

The potential of any wire is opposite to the potential of its image, I ; = -I- ,+, , , for i = I , . . . . i t . Therefore, (21) may be written [':'I (22)

1 - 8 8

D(3 - d = 2 . l r

Fig. 3. ratio is d / r = 2.1.

Two uninsulated wires in free space. The separation-to-wire radius

Two Wire Convergence Study Uninsulated Wires, (d/r=2.1)

90

h 80 E 2 70

60

v P)

5 50

6 u 40

30 0 2 4 6 8 10 12 14 I6 18 20

Number of Basis Functions/Wire

- Galerkin -8- PointMatch

Fig. 4. tween Galerkin matching, point matching, and the exact capacitance.

Capacitance of two uninsulated wires. Comparisons are made be-

which has the form, Q = CV, where C is the desired transmission line capacitance matrix in reference to the ground plane. The transmission line capacitance matrix terms are

ct,J = ct.J - c!,J+#t (23)

111. RESULTS

A . Two Uninsulated Wires

The geometry for two uninsulated, parallel wires is shown in Fig. 3. The separation-to-wire radius ( d / r ) ratio is 2.1. The wide separa- tion approximation is not valid for wires in such close proximity. However, an exact analytical solution for the capacitance exists for this problem. Comparisons were made between the exact analytical solution, the previous moment method using point matching [8], and the modified moment method from this paper using Galerkin testing. The results are presented in Fig. 4. The number of Fourier harmonic functions used for each wire is denoted by NF.

Fig. 4 shows that both methods converge to the exact solution. The Galerkin testing method, however, converges substantially faster than the point matching method. Convergence requires roughly half as many basis functions for the Galerkin method in this example.

With the Galerkin testing method, test points are fixed. With the point matching method, the choice of test point locations is not unique. It was shown in [8] that varying test point locations greatly affects the convergence rate. In fact, with the symmetric geometry in Fig. 4, certain test point locations exist which lead to nonunique solutions [8]. This problem is completely avoided with Galerkin testing.

B . One Insulated Wire over Ground

The geometry for this example is given in Fig. 5. This case involves a single, insulated wire above a ground plane. For this configuration,

~~ ~

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 31, NO. I , FEBRUARY 1995 I35

-10

-20

-30

-40

-50

-60

-70

-80

Fig. 5. h / r = 1.0/0.9.

One insulated wire over ground. The height-to-wire radius ratio is

-

-

-

- - - - - I

260

240

220 3 200

.e 180

4

4 120

0)

160

5 140 6

V 100

80

60

One Wire Convergence Study Insulated Wire over Ground, ( h / r = l . l l l )

I ~~

2 4 8 8 10 12 14 16 Number of Basis Functions/Wire

Fig. 6. Capacitance of one insulated wire over ground.

no analytical expression for the capacitance exists and the wide separation approximation is again invalid. The wire radius is 0.9 times the wire height. The insulation has a relative permittivity of 4.0, and is just touching the ground plane. The results are given in Fig. 6.

The convergence of the Galerkin testing method for the insulated wire is somewhat slower than in the previous examples. This is due to the unusually thin dielectric insulation which allows the wire to be in such close proximity to the ground plane. In more realistic applications, the dielectric insulation's thickness is usually on the same order as the wire radius. This forces conductors to be farther apart, which improves convergence rates.

C . A Sample Cable Bundle

A cable bundle is shown in Fig. 7(a) with terminations shown in Fig. 7(b). The length of this transmission line is 2.2098 meters. The wires each have a radius of 18 mils (0.457 mm). The insulations each have a radius of 41 mils (1.041 mm) and relative permittivity of 2.0. The insulations are touching each other and are resting on the ground plane.

The PUL parameters were computed using the Galerkin moment method (with and without the insulations) and the wide separation approximation. These parameters were used to compute the crosstalk for the given configuration. The crosstalk was computed using the exact multiconductor transmission line equations with modal decomposition [ 11. Although a simple inductive-capacitive model could be used to compute the crosstalk at low frequencies, the full MTL equations are required for high frequency computations. The transmission line was constructed and the crosstalk ratio was measured. Fig. 8 shows the results of the crosstalk predictions and measurements.

The crosstalk prediction from the wide separation approximation is very close to that of the Galerkin moment method without the wire

Far End Near End - - - - - - - - - - ~

I -%a- Generator I ] I ~

@Vin=1V vg I I Receptor 1 I lki-2) 1 - I I

I 1 lkQ{ I Ground Plane I

lkL$ vr I _

I

_ _ _ _ _ _ _ _ _ _ J

(b)

Fig. 7. A sample cable harness for crosstalk computations. (a) Cross-section of two insulated (E,. = 2.0) wires above a ground plane. Units are in mils. (b) Circuit diagram showing transmission line terminations.

Two Wire Crosstalk High Impedance Loads

O r

Galerkin epsr= 1 .O Galerkin [epsr=2.0 Wide Separation

1

10s 104 10s 106 107 108 -9oL-0 ' """" ' """" ' """" ' """" ' """"

Frequency (Hz)

Crosstalk predictions and measurements. The crosstalk is predicted Fig. 8. using the wide separation approximation and the Galerkin moment method.

insulations. The slight deviation is due to the difference in the PUL parameters. Both of these crosstalk predictions are approximately 5 dB lower than the measured crosstalk.

The Galerkin moment method crosstalk predictions including the insulations are very close to the measured crosstalk. Even at higher frequencies, predictions closely match the measured crosstalk. This example demonstrates that ignoring wire insulations can lead to less accurate crosstalk predictions. If the insulation permittivity is higher than 2.0 or the conductors are closer together, it is expected that the crosstalk predictions performed by ignoring the insulations would be even less accurate.

IV. CONCLUSION This paper has presented an analysis of crosstalk in cable harnesses.

The focus has been on capacitance matrix calculations for multiple insulated wires over a ground plane. Such hamesses are common in automobiles, appliances, and many other electrical devices.

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

Fig. 9. Galerkin Method.

The geometry for computing the entries in Z,, using the first order

An improved numerical method was presented which allows accu- rate computation of the per-unit-length parameters for cable harnesses with or without a ground plane. The method places no requirements on conductor spacing. This method uses Fourier harmonic expan- sion functions and Galerkin testing. The improved method is free of moment method matrix singularity problems and always gives monotonic convergence.

A special relationship between the wide separation approximation and the Galerkin moment method was given. If only one constant basis function is used on each conductor (a first order Galerkin method), the Galerkin method is equivalent to the traditional wide separation approximation.

Capacitance matrix computation results were used to demonstrate the Galerkin moment method's validity. The Galerkin method con- verges considerably faster than the previous point matching method.

Crosstalk predictions from the Galerkin method and the wide separation approximation method were presented. It was shown that ignoring dielectric insulations can lead to inaccurate crosstalk predictions. The amount of error depends on insulation permittivities and conductor spacing.

APPENDIX RELATIONSHIP BETWEEN THE GALERKIN SOLUTION

AND THE WIDE SEPARATION APPROXIMATION

This appendix proves a relationship between the wide separation approximation and the Galerkin moment method solution for 11 + 1 uninsulated wires. The first order Galerkin moment method solution uses only a constant term in the charge expansion on each conductor. This is the fewest number of basis functions which can be used in the t i + 1 wire problem. Using only constant terms, the integrals in the Z matrix of (4) all have closed form solutions. Fig. 9 shows the geometry used to determine the entries in 2.

From Table I, the potential, I;, at any test point r p on the j t h wire due to the constant charge expansion on the ith wire is

where n , is the charge coefficient associated with the constant basis function on the ith wire. Applying the Galerkin test procedure to (24) gives

where

r,, = J(d,, - r J C O ~ ioJ 12 + ( r J sin io, )* (26)

Since 1; is constant, (25) may be reduced to

From [lo], (865.73): La 1 ll~(d;', - 2r1,,rJ C O ~ 0, + r ; )d io , = 47rlii(d,,) (28)

Therefore, (27) reduces to

(29) n , r

1; = - L l 1 l ( d t , )

Since the total charge on the ith wire, 4,. is 27ir,n,, (29) may be written

Similarly, the potential on the ith wire due to the charge on the ith wire may be determined using the Galerkin testing procedure:

- n , r , In( r , ) 27i1; =27r - (33)

(34)

(30) and (34) may be combined to form a matrix equation:

Iii(d10) In(r1) ... l i i ( d ~ , , )

ln(d,,o) lll(dl,t) ...

(35)

From the definition of the PUL generalized capacitance matrix in (lo), it is apparent from (35) that

r l l l ( r o ) h ( d l o ) ... I 1 1 ( d , , ~ ) l

(36) is the inverted generalized capacitance matrix obtained from the first order Galerkin moment method applied to I I + 1 uninsulated wires.

A rigorous derivation of the wide separation approximation for 1 1 + 1 wires was performed in [ 111. In this derivation, the following expression for the inverted generalized capacitance matrix was given:

. . (37) I ln(r0) lll(dl0) ... lll(d?to)

Ill(d,,O) lll(dl,, 1 . . . lii(r,, 1

(37) is the inverted generalized capacitance matrix obtained from the wide separation approximation applied to I I + 1 uninsulated wires. Comparing (36) to (37), it is apparent that for t I + 1 wires the wide separation approximation is identical to the first order Galerkin moment method solution. Although it is not proven explicitly in this paper, if can be inferred that the wide separation approximation is also identical to the first order Galerkin moment method solution in the problem of t j wires above a ground plane.

ACKNOWLEDGMENT

The authors would like to thank Dr. C. R. Paul for his insightful discussions on this topic.

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATABILITY, VOL. 37, NO. I , FEBRUARY 1995 137

REFERENCES

[ 11 C. R. Paul, Analysis of Multiconductor Transmission Lines. New York: John Wiley Interscience, 1994.

[2] M. N. 0. Sadiku, Numerical Techniques in Elecrromagnetics. Boca Raton, FL: CRC Press, 1992.

[3] R. L. Khan and G. I. Costache, “Finite element method applied to modeling crosstalk problems on printed circuit boards,” IEEE Trans. Elecrromagn. Compat., vol. EMC-31, pp. 5-15, Feb. 1989.

141 C. Wei, R. F. Harrington, J. R. Mautz, and T. K. Sarkar, “Multicon- ductor transmission lines in multilayered dielectric media,” IEEE Trans. Microwave Theory Tech., vol. MTT-32, pp. 4 3 9 4 9 , Apr. 1984.

[5] T. Lu and R. L. Olesen, “Analysis of transmission line structures using a new image-mode Green’s function,” IEEE Trans. Microwave Theory Tech., vol. MTI-38, pp. 782-785, Jun. 1990.

[6] H. A. N. Hejase, A. T. Adams, R. F. Hanington, and T. K. Sarkar, “Shielding effectiveness of ‘pigtail’ connections,” IEEE Trans. Electro- magn. Compar., vol. EMC-31, pp. 63-68, Feb. 1989.

(71 T. R. Arabi, A. T. Murphy, and T. K. Sarkar, “Electric field integral equation formulation for a dynamic analysis of nonuniform microstrip multi-conductor transmission lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-40, pp. 1857-1869, Oct. 1992.

[8] J. C. Clements, C. R. Paul and A. T. Adams, “Computation of the capacitance matrix for systems of dielectric-coated cylindrical conduc- tors,” IEEE Trans. Electromagn. Compat., vol. EMC-17, pp. 238-248, Nov. 1975.

[9] D. V. Gin, F. M. Tesche, and S. K. Chang, “The transverse distribution of surface charge densities on multiconductor transmission lines,” IEEE Trans. Electromagn. Compat., vol. EMC-21, pp. 22C227, Aug. 1979.

IO] H. E. Dwight, Tables of Integrals and Other Mathematical Data. New York: Macmillan, 1961.

I I ] C. R. Paul, “Application of multiconductor transmission line theory to the prediction of cable coupling-Vol. I-Multiconductor transmission line theory,” Tech. Rep., Rome Air Development Center, Griffiss AFB, RADC-TR-76101, Apr. 1976, pp. 82-97.