Use of the Method of Moments to Find the Charge Densities and Capacitances of a Shielded T.wisted Pair Transmission Line

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1 Use of the Method of Moments to Find the Charge Densities and Capacitances of a Shielded T.wisted Pair Transmission Line Craig James Rome Laboratory / ERST 525 Brooks Rd. Rome, NY John Norgard University of Colorado at Colorado Springs College of Engineering and Applied Science Department of Electrical & Computer Engineering Colorado Springs, CO Abstract The surface charge densities and associated capacitances of a shielded twisted pair transmission line are found for quasi static conditions, under balanced and longitudinal excitations. The unknown charge density is expanded in a Fourier series. The series is used as basis functions for a method of moments solution to find the unknown series coefficients. This paper describes the geometric structure and the formation of the integral equation that describes the potential distribution as a function of the unknown charge densities. The simplifying approximations made am pointed out, as well as the limitations to the solution caused by the approximations. Figure Cable Cross-Section. Geometric Considerations Figure is a perpendicular cross section of a shielded cable with a pair of straight wires. The dielectric filling the shield and surrounding the inner conductors is considered lossless, with a dielectric constant E,. The inner conductors of radius rw are equally spaced about the central axis a distance d from the axis. The shield is of radius rs about the central axis. When twisted, each wire forms a helix about the central axis. The twist is assumed constant along the length of the cable, producing a helical symmetry to the structure. The cross-sectional surface of the twisted wire in the plane as shown in Figure will actually be kidney shaped. The distortion from a circular cross-section becomes greater as the twist becomes tighter. However, for moderate twist lengths, the distortion from a circular cross-section is not a great source of error. (The twist length is the length of transmission line required for the imier conductors to rotate one complete turn about the central axis.) The solution developed in this paper assumes a circular cross-sectional surface of radius rw, independent of twist length.. Historical Background n general, a transmission line is characterized by the per-unit-length values of the resistance of the conductors, the conductance of the insulation between conductors, the series inductance, and the shunt capacitance associated with the transmission line. t is the intention of this work to model the capacitance per meter of a shielded twisted pair transmission line. A previous take-up model based on surface area, and the separation of the surfaces has been considered. However, the model predicts a steep increase in capacitance as the twist length of the wires decreases. The general feeling has been that predictions made with the take-up model are not accurate. The most recent work in this area has been done by J.D.Norgard and R.M.Sega of the University of Colorado at Colorado Springs, and M.G.Harrison of USAF %illips Lab, Kirtland AFB. The results were presented at the EMC Zurich symposium in March, 993. [] The predicted capacitance did not increase as radically as predicted by the take-up model, however, the model still predicted a steep increase with twist. The case of the straight shielded pair has been dealt with in several papers. [2],[3], [4],[5],[6] The charge distribution on the conductors are assumed to be described by Fourier series. O /96/$ EEE 277

2 Expressions are then written for the voltage throughout the cable as functions of these charge distributions, with the series coefficients as unknowns. The expressions are evaluated for points on the surface of the conductors where the potentials are known. The coefficients are then found by solving the resulting system of equations. For the case of the straight shielded pair, the geometry is simple enough that the problem is two dimensional, and this technique is straight forward. Norgard, Sega, and Harrison expanded the case of the straight wires. A perturbation was introduced to the equations, due to the twist of the wires. For an infinite twist length, the perturbation was zero. For shorter twist lengths the modification was small, therefore, the analysis was considered a good approximation. However, as shown in Reference, as the twist length becomes short, the charge on the surface of the conductors is predicted to vary greatly around the wires and the shield. A more exact solution is required for the shorter twist lengths. V. DeveloDment of the Solution Consider the perpendicuhu cross section of the cable at an arbitrary location along the length of the cable. Let this cross section define the z=o plane, with the z axis of the circular cylindrical coordinate system defining the central axis of the cable. The cross section is shown in Figure 3 below with the same dimensions as shown in Figure.. Cable Canacitances Figure 2 is the cross section with the associated capacitances shown schematically. Figure 3 Coordinate System for the Method of Moments Solution Figure 2 Capacitances Associated with the Cable Throughout this paper, the shield is always at ground potential. A capacitance matrix can be defined as: Cl, = ;q = 0 2 c,, = %v* =o where pl,z are the charges per unit length on wires and 2 due to the excitation voltage V,,,. By symmetry C,, = CZ2, and by reciprocity C,, = C,,. The direct capacitance between the inner conductors is then C, = -C,> The capacitance to ground (recall the shield is grounded) is C, = C,, + C,,. The mutual capacitance of the cable with respect to the inner conductors is then C, in parallel with the capacitance to ground, (the series combination of the two values of C,) or C, = C, + (%)C,. Therefore, the problem of determining the mutual capacitance reduces to finding the charge densities on the conductors as functions of the applied voltages. A. Wire # The point P is an observation point in the cross sectional plane with coordinates (p,<p,o). The point P is a source point in the cross sectional plane on the surface of wire #l_ The coordinates of the point P are (p,@,o). Recall the radius of the inner conductors is r, and the distance from the center of the cable to the center of the inner conductor is d. The angle 0 is defined as the angle between the line connecting the centers of the inner conductors, and the line from the center of wire #l to the point P. With the circular cross section approximation, the coordinates of P can be written as a function of 0. These expressions are given in equation. sin@ ) = ($llo~) P cos(@q = (+o( p - r$$os(q )) p = &P + r, - 2&,cos(Q ) () The charge density on wire #l is expressed as a Fourier series in 0. Because of symmetry, the coefficients for the sine terms will all be zero. The charge density is then written as, m o(q ) = C a, cos(no/). The unknown function has been?=0 reduced to a series of unknown constants. 278

3 With movement in the z direction, the inner conductors rotate about the z axis. With the definition of 0 as given above, the expression for o(0 ) is an expression for the charge density in any z plane. Note, however, that 0 is defined only in the z=o plane. The point P is then the point in the z=o plane on a line charge running along the inner conductor with a constant value of p. Therefore, the coordinates of an arbitrary source point ), where T! = (@,+2nz /e), and Q is the twist length. The distance s shown in Figure 3 is found using the law of cosines. s= p2+@ )2-2pp cos(q, - 0 ) (2) Using the Pythagorean theorem, the distance from any arbitrary source point on wire #l to an observation point in the z=o plane is Therefore, the potential due to the charge distribution on wire # can be approximated by XY r, + (p > - 2@ cos(@ - cs ) Y@ l p2 + (p > - 2pp cos(@ - W) Using the expressions in equation, the potential charge distribution on wire #l can be written as (7) due to the S= P2 + (P > + (0-2pp cos[(@ - a? ) - T (3) Taking advantage of the fact that the shield is grounded, the potential at point P due to the charge distribution on wire #l can be written as given in equation 4. v,= -& - o(q )r, dq u!z J d* &-al sp 2r - - s e =cl z - O(QjYw dq c?!?i The distance sp is from the source point to the observation point, and s,, is from the source point to a point on the shield. This development, to this point, parallels the work done previously, found in the references. The difference being in the expressions for S, and S,. As z becomes large, the contributions of the cosine terms in sp and So drop off. Therefore, the integrals over z can be evaluated by expanding the cosine term about z =O. The distance then takes the form in equation 5. The coefficients are: s PO (4) sp = a& 3 + qj (5) ao = p + (p, - 2pp coqd 4 = 2PP 2%Ll((D y- ) a-2 = + pp+%os(@ - Q ) The distance spa is found by replacing p with p,,, and Cp with Q,. Each integral over z is divergent in the infinite limits. The difference of these divergent integrals can be found when the coefficients %(~,a,) and %((~b,$) are forced to be equal. This can be done by setting P* = rs a, = cos-l[(~)co~ip - cp )] + a s (6) x M (rs2 d2 r; 2pd cos(@)) 2rJd co@) )) PO (p2 d2 r; 2pd cos(fd)) 2r,(d COT@ ) pcos(cp ti)) B. Wire #2 A similar expression can be written for the potential due to the cosine series charge density on wire #2. l$= rwcos(no) LO p(t)2(d cos(@) r,,,cos@ 0 )) r, 2pd cos(@)) 2rJd co@ ) pcos(@ 0 )) )dq 2rJd cos(@ ) pcd,@ 8 )) d d2 r, 2pd cos(@)) The angle 0 in the expression for V, is defined as the angle between the line connecting the centers of the conductors, and the line connecting the center of wire #2 and the source point on wire #2, in the z=o plane. This is the same definition as used for V,. C. Boundary Conditions The potential at all points within tire cable is the superposition of these two expressions. The unknown series coefficients are found by evaluating the expression for the potential at points on the inner conductors where the voltages are known. For example in or&r to determine the first ten coefficients in each series, ten points must be considered on each conductor. The approximation for s, and G is good as long as Q is large. However, if Q = 2n(pp ) there is a singularity in the integrand of equation 7 at (a - a, ) = T. Therefore, the accuracy of this solution is good for only a limited range of twist lengths. Given this limitation, the circular cross section surface approximation is not a great source of error. 279

4 V. Results of The Model The solution was used to evaluate the charge densities on n order to evaluate the Fourier coefficients, the previous the inner conductors and the capacitances of a c&nmonly used integration is performed at each of a set of observation points 9 gauge twisted pair cable. The dimensions of the cable are on the surface of one of the conductors where the value of the given below in Table. The value of the dielectric constant potential is known. Half the points in the set of observation used is E, = 2.. With these dimensions, the model will fail points are on the surface of the wire over which the integration because of the singularity for twist lengths tighter than 0.48 in. is performed. The ln function in the integrand of equation 7 becomes undefined when the observation point and the source Table Cable Dimensions point coincide. Commonly, this problem is avoided by assuming that the source is a small distance inside the surface and the observation is taken at the surface. [7] n this paper, the integration is evaluated numerically, dividing the integration pjq$$jg range into N equal divisions. The problem of the source point and observation point coinciding is avoided by letting N be a prime number, and taking the observation points at even divisions of 0 between 0 and x. A. Balanced Excitation Evaluating the integral of equation 7 at M different Figure 4 is a plot of the charge distribution on wire # observation points on each conductor results in the matrix with a balanced excitation (ie. with voltage on wire # (V,) of equation, common to the Method of Moments solutions, given + volt, and the voltage on wire #2 (VJ of - volt), for twist in equation 9 below. lengths of infinite, 2.0 in,.5 in,.0 in, and 0.5 in. AC=B (9) Using balanced excitation, the charge on wire #2 is the negative of the charge on wire #l. The independent axis is 0 The matrix A is a 2M X 2M matrix of the evaluated integral. The matrix C is a column vector of the unknown cosine series coefficients. The matrix B is a column vector of the values of as defined in Figure 3. As the twist length decreases, the charge density increases on the surface of the conductor near 0 = 80. the potential at the observation points used to evaluate the integrals. B. Longitudinal Excitation A FORTRAN routine was written to carry out the integration on a desk top personal computer. The routine included the matrix reduction required to evaluate the unknown Figure 5 is a plot of the charge density on wire #l with longitudinal excitation, with V, = V, = + volt. Again, as the twist length decreases, the charge density increases on the coefficients of the column vector C. t was found that only five surface of the conductor near 0 = 80. With the appropriate coefficients were required to accurately represent the charge scaling and superposition of the balanced and longitudinal density on the wires. Note that only the first coefficient, or the excitation, the charge densities for any excitation voltages can dc term, is needed to find the capacitances. be found., 0.7, Charge Density VS Twist Length on Wire # o* Twist Length = infinite! ,.*-. = i Twist Length 2.0 in \ _** s Twist Length=.5in 4. : N = :.( E a-* *\ *. 4 $ * a _.!! Angle Theta, in Degrees Figure 4 Charge Density on Wire #l, Balanced Excitation 280

5 Charge Density VS Twist Length on Wire # Twist Length=.5in ----Twist Length=.0 in / # TwistLength=OSin ; Angle Theta, in Degrees Figure 5 Charge Density on Wire #l, Longitudinal Excitation C. Canacitances Table 2 lists the capacitances of the cable at these same twist lengths. For the straight wire case, the exact solution is known. The model agrees very well with the exact solution. Figure 6 is aplot of the predicted mutual capacitance as a function of twist length. The predicted solution shows an increase in capacitance with shortening twist length, but the increase is not as great as that predicted with previous take-up models. The data is not reliable for twist lengths less than.o because of the singularity that occurs, rendering the model meaningless, at a twist length of 0.48 (with the other dimensions of the structure as stated previously). At the twist length of l-o, the model is very good Twist Length in nches Figure 6 Mutual Capacitance VS Twist Length V. Conclusions The solution found in this paper is usable over a limited, but wide range of twist lengths. This limit is well defined. Previous work done in this area has predicted an increase in the mutual capacitance of the transmission line with shortening twist lengths. However, the predicted increase has been larger than that found in reality. The solution developed in this paper also predicts an increase in the mutual capacitance with shortening twist lengths. However, the increase is not as great as previous models have shown. A future paper will present another solution which takes into effect the kidney shaped cross sections of the twisting inner conductors, and which has no geometric limitations. Both of these solutions will be compared to measured data. Table 2 Cable Capacitances 28

6 References [l] J.D.Norgard, R.M.Sega, and M.G.Harrison, The Capacitance Matrix and Surface Charge Distribution of a Shielded Twisted Pair Cable Using a Quasi-Static Perturbatiqnal Method, n Proceedings of the 993 Electromagnetic Compatibility Symposium, Zurich, Switzerland, March, 993. [2] J.W.Craggs, The Determination of Capacity for Two- Dimensional Systems of Cylindrical Conductors, Quarterly Journal of Math. (Oxford), Series, Vol. 7, 946, p 3. [3] J.W.Craggs and C.T.Tranter, The Capacity of Conductors and Dielectrics with Circular Boundaries, Quarterly Journal of Math. (Oxford), Series, Vol 7, 946, p 39. [4] C.M.Miller, Capacitance of a Shielded Balanced-Pair Transmission Line, Bell System Tech. Journal, Vol. 5, No. 3, March 972, p 759. [5] J.D.Norgard, The Capacitance and Surface Charge Distributions of a Shielded Balanced Pair, EEE Transactions on Microwave Theory and Techniques, Vol. 24, No. 2, February 976, pp [6] J.D.Norgard, The Capacitance and Surface Charge Distributions of a Shielded Unbalanced Pair, EEE Transactions on Microwave Theory and Techniques, Vol. 25, No. 2, February 977, pp [7] Balanis, Constantine A. Advanced Engineering Electromagnetics, John Wiley and Sons, nc., 989.

ELECTRO MAGNETIC FIELDS

ELECTRO MAGNETIC FIELDS SET - 1 1. a) State and explain Gauss law in differential form and also list the limitations of Guess law. b) A square sheet defined by -2 x 2m, -2 y 2m lies in the = -2m plane. The charge density on the