IEEE Transactions on Power Apparatus and Systems, Vol. PAS-97, no. 3, May/June 1978

ADVANCES IN FINITE ELEMENT TECHNIQUES FOR CALCULATING CABLE RESISTANCES AND INDUCTANCES

Robert LucasPower Program

Carnegie Institute of TechnologyCarnegie-Mellon University

Schenley Park, Pittsburgh, PA 15213

ABSTRACT

In theory, finite element procedures can be applied to arbitraryconductor configurations in order to determine their frequency de-pendent resistances and inductances to an arbitrarily high degree ofaccuracy. In practice, limitations are imposed by the shapes of thefinite elements used and by the accuracy of the formulae invoked toestimate their inductances. This paper reviews the concepts underlyingextant finite element procedures, proposes a more efficient set ofelement shapes than are used in extant procedures and develops theinductance formulae necessary to implement these shapes. This resultsin a considerably more capable algorithm.

INTRODUCTION

Transmission lines (the term is used in its most general sense) inwhich the conductors are relatively close together are becoming in-creasingly important constituents of electric energy delivery systems.Pipe-Type-Cables and Gas-Insulated-Buses are cases in point. The per-

formance of such lines, particularly their effective resistances andinductances, are markedly influenced by skin and proximity effectsand by any assymmetry in the conductor arrangements. Techniquesfor assessing these influences can be divided into two major categories:

* Analytical techniques in which attempts are made to express

the solutions of the appropriate field equations in terms offunctions such as Bessel function[ 1] -[8].

* Numerical techniques in which the solutions of the filed equa-tions are obtained by finite element or finite differencemethods[9]-[ 13 ].

Analytical techniques suffer from a fundamental limitation - they arenot generally applicable, particularly to complicated conductorarrangements.

In contrast, numerical techniques are, at least in theory, quitegenerally applicable. However, they are subject to computationallimitations that arise from two sources:

* The presence of nonlinearities particularly magnetic saturationin the pipes enclosing certain cables

* the extremely large numbers of finite elements that can berequired to produce accurate results.

The work of Dugan and Talukdar [14] indicates that nonlineareffects in cables have relatively little influence on their performanceand that linear models give good results over large ranges of behavior.

F 77 694-3. A paper reca:mrnded and approved bythe IE: Insulated Condudtors Camnittee of the IUE

Pcwer Engineering Society for presentation at the IEEf'PES Sumrer Meeting, Mexico City, Mex., July 17-22,1977. Manuscript submitted January 31, 1977; madeavailable for printing April 29, 1977.

Sarosh Talukdar

This paper is devoted to the remaining problem - the large numberof elements required by extant finite element procedures for theproduction of accurate results [11] [121. Our approach will be parttutorial part developmental. First we will review extant finite elementprocedures for calculating conductor effective resistances and induc-tances and point out that the need for large numbers of elementsstems from the use of simple element cross sections. Next we willpropose better but more elaborate cross sections. The use of thesecross sections is contingent on the availability of accurate and com-

putationally efficient formulae for estimating their inductances. Suchformulae cannot be found in the literature so we will devote the lastpart of the paper to developing and verifying them.

FINITE ELEMENT METHODS - AN OVERVIEW

Conductors subject to skin and proximity effects have nonuniformcurrent distributions. This makes the direct evaluation of their effec-tive resistances and inductances difficult. Finite element approachesdeal with this situation by dividing each conductor into a largenumber of subconductors or filaments. It is then assumed that thecurrent distribution over each filament is constant. By making thenumber of filaments suitably large the error in this approximation canbe made arbitrarily small.

Philosophy and Governing Assumptions

Consider a long N-conductor transmission line. Let each conductorbe divided into a number of parallel filaments and let the total numberof these filaments be n. If it is assumed that:

* the transmission line's cross section is uniform, that is, theelectromagnetic effects of irregularities such as spacers incables and towers in open wire lines are negligible,

* current flows only longitudinally in the filaments,

* the current density is constant over each filament's cross

section,

* the conductors and intervening spaces have the same, constantpermeability,

* the conductivity of each filament is constant (thought notnecessarily the same as for any other filament),

and if a "Return Path" is designated for the filament currents, theneach filaments can be represented by constant, lumped resistors andinductors as shown in Figure 1. The resistance of the k-th filament isrk. The self inductance of the k-th "loop" is 1Lkk while 1k is themutual inductance between the k-th and j-th "loops." Each "toop" is

formed by a filament and the "Return Path."

The value of the resistance, rk, is determined directly from theassociated filament's resistivity, length and cross sectional area. Theinductance determinations, however, are more involved and requirethat a "Return Path" be designated - useful values of inductances can

only be defined for closed current paths. One of the filaments can beselected to serve as this "Return Path." However, it is usually moreconvenient to create a fictitious conductor which exists only for theinductance calculation process. To prevent it from interfering else-where its current must be constrained to be always zero, that is: 2ik =

0, at all times.Naturally, the location and shape of the "Return Path" affect the

values of the loop inductances. However, in circuit calculations theseeffects cancel when the zero-current-constraint is imposed on the"Return Path." Theoretically then, the "Return Path" could be placed

0018-9510/78/0500-0875$00. 75 ©) 1978 IEEE

875

1 St

* a Conductor

I

I

I

Ret urn

Path

n N.i =k E:k =kk-l k = 1

Figure A Lumped, Constant Parameter Model of an N-ConductorTransmission Line.

anywhere and have any shape. From a numerical point of view,however, some locations and shapes are more- auspicious than others

[151. One of the best choices is a cylindrical "Return Path" thatencloses all the filaments.Once the filaments and "Return Path" have been selected, the next

problem is to calculate the "loop" inductances. If the filaments have

simple cross sections (circles or squares, for instance) formulae for

doing this can easily be found in the literature [28] [29]. Accurateformulae for non-simple cross sections, however, are not as easy to

come by. Later we will show that there are profound advantages to

using certain non-simple cross sections and will develop appropriateinductance formulae for them. Meanwhile, we will complete the more

general description of finite difference methods that was embarked on

in this section.

Finite Element Models

The network of Figure1 is described by the matrix equation:

E = RI + L dt (1)

where I is a current vector whose elements, il,i2....in are the fila-

ment currents andE is a voltage vector whose elements, el,e2,. en,are the filament voltages (note: since filaments 1 through ml belong

to conductor 1 and are in parallel, e =e2=. . =em1 =v1, the voltage

applied across the first conductor; and so on). R is a matrix of the

filament resistances and, if the "Return Path" is fictitious, R is a

diagonal. L=[Qkj], is the "loop inductance matrix." each "loop" being

formed by a filament and the "Return Path."

Model (1) is quite general, can be applied to the computation of

both transients and steady state values and intcludes the influences of

both skin and proximity effects (subject to limitations imposed by

discretization errors). Theoretically the discretization errors can be

made arbitrarily small by sufficiently reducing the filament crosssections.

The R and L. matrices in (1) are constant. However, their large sizetends to make model (1) inconvenient to use. Size reductions can beachieved under steady state conditions by submerging the separateidentities of the filaments and preserving only the identitites of theconductors. Efficient matrix manipulation procedures for doing thisare described in the Appendix. The result is a set of equations of theform:

V ZZRI =c[Rr (w) + j LeRW) I I

where V is a complex vector whose elements, V1,V2,.--VN are the

conductor-steady-state-voltages, I is a complex vector whose elements,

1112...,2nare cond uctor-steady-state-currents and Z = (RR + j LR) isan (N X N) impedance matrix representing the conductors at the

prevailing frequency,co. The entries of RR(CO) and LR(w) representthe conductor equivalent resistances and inductances. They are fre-

quency dependent in physical terms, that is the result of skin and

proximity effects; mathematically, it is the result of reducing the large

(n x n) matrices in (1) to the much smaller (N X N) matrices in (2)while preserving model fidelity.

PERFORMANCE CHARACTERISTICS OFFINITE ELEMENT PROCEDURES

The performance of finite element and other discretization pro-

cesses is in general, assessed in terms of accuracy, computing time andthe amount of storage required. The performance of the methodsconsidered here is critically affected by two factors:

The size, shape and disposition of the filaments, that is, by thescheme used to divide each conductor into filaments;

* The accuracy of the formulae invoked to determine the

"loop" inductances.

Lesser influences are exerted by the matrix procedures used to reduceequation (1) to the form of (2) and by the shape and location of the

"Return Path." The latter set of influence is too subtle to be ade-

quately covered here a discussion of it may be found in [ 151 Thematrix reduction procedures mainly influence storage requirementsand computing time - their adverse effects may be minimized by

using the efficient procedures such as those described in the Appen-dix.

Returning to the two major influences we note that the distributionof the currents among the filaments is determined almost exclusivelyby the inductance matrix, L. Therefore, the accuracy of the overallresults is critically determined by the accuracy of the formulae used to

calculate these "loop" inductances.

In contrast, the effects of filament cross sections have on perfor-mance that are not quite as obvious, they are best illustrated by an

example.

Consider a coaxial cable. Its steady state current density profiles at

various frequencies are shown in Figure 2. As frequency increasescurrent tends to concentrate at the outer surface of the inner con-

ductor and the inner surface of the outer conductor. As a result theeffective resistances of these conductors increases monotonically withfrequency. Since the cable's fields are cylindrically symmetric theycan be calculated by analytic (closed form) techniques [26]. Typicalcurrent density profiles and an effective resistance curve are displayedin Figures 2 and 3.

Suppose now that a finite element procedure is invoked to calculatethe effective resistances. The cylindrical symmetry of the coaxial cablecan be taken advantage of by selecting thin-walled-concentric-cylindrical shapes for the filaments. Thegenerated valuesof the inner conductor's effective resistance are shown in Figure 3.They are asymptotic to a value we will call the Cut-Off-Resistance(COR). This is a characteristic of all finite element procedures. TheCOR in this case, is a little greater than the d.c. resistance of the outer

filament of the inner conductor. The COR and this resistance wouldbe equal were it not for certain secondary effects involving the manner

in which filament inductances are defined and calculated. A heuristic

explanation follows. As the frequency increases, more and morecurrent tends to flow in the outer filament of the inner conductor. At

876

ii

-4 ~~VI

i1 r2211

_ ~~~~~~i2 r2 It,22

(2)

877

H OOzwicici:

CO

Figure 2 Plots, of the Absolute Values of Typical Current DensityProfiles for a Coaxial Cable at Various Frequencies.

TIRUE RESISTANCE

COR (CUT-OFF-RESISTANCE)

/ / ~~OUTER FILAMENT

< ESISTANCE FROM FINITE ELEMENT PROCEDURE

.,,. ,. ..50 100 200

VFREOUENCY -

Figure 3 The Effective Resistance of the Inner Conductor of a CoaxialCable - A Comparison of the Actual Values to ThoseGenerated by a Finite Element Procedure.

high frequencies virtually all the current is flowing in it, that is, itconstitutes the only path for current flow in the inner conductor andits d.c. resistance is seen as the conductor's effective resistance. Thefinite element procedure cannot generate substantially higher resist-ances because it assumes that the current density in each filament isuniform.

In general, the COR of any finite element procedure is closelyrelated to the d.c. resistance of the filaments covering the conductorareas in which current tends to concentrate at high frequencies. Thus,to increase the COR's and thereby, to increase the usable frequency

band of a finite element procedure, one must increase the d.c. resist-ances of these filaments. In other words, one must decrease theircross-sectional areas. The question is: How best should this be done?To provide an answer we need additional information.

In all discrete variable procedures (including finite elementmethods), the amount of discretization error introduced depends onthe interval size and the rate of change of the dependent variables withrespect to the independent ones - the higher this rate the samller theinterval required. In our case the dependent variable is current densityand the independent variable is position in the conductor's crosssectional plane. In the coaxial cable the current density and its rate ofchange vary exponentially with distances along radii and are constanton circles that are concentric with the conductors. Consequently,optimum (in terms of the number of filaments needed to produce agiven level of accuracy) discretization schemes consist of concentriccylindrical filaments whose thicknesses decrease exponentially to theedge of the inner conductor and increase from the inner edge of theouter conductor to its outer edge (Uniform thicknesses consistentwith the largest rate of change of current density could be used butwould be watesful in areas of lower rates).

In asymmetric conductor arrangements one cannot predict, a priori,exactly where the highest rates-of-change-of-current-density willoccur. However, as a rule of thumb, extrema appear at conductorsurfaces and in the vicinities of straight lines joining the conductors'centers of gravity. Current density tends to change far more rapidlyalong normals to conductor surfaces than in directions parallel tothese surfaces. In other words, contours of constant current densitytend to be roughly parallel to conductor surfaces.

Optimum filament arrangement schemes would closely follow theshapes of the constant current density contours if they were known.However, they are not, and in any case, the resulting filament shapeswould be very irregular. Therefore, the next best alternative is to fitthe filament cross sections to estimates of the contours while pre-serving some regularity of shape. One such arrangement for a trans-mission line consisting of two solid conductors placed side by side isshown in Figure 4. (This arrangement anticipates current densitieschanging most rapidly at the conductor surfaces along the line joiningtheir centers.

Figure 4 An Optimal Arrangement of Filaments for a Two ConductorTransmission Line.

The edges of most conductor cross sections are formed fromstraight lines and circular arcs. Therfore, the types of filaments neededto assemble optimal arrangements are of the forms shown in Figure 5.For brevity, we will refer to these filaments henceforward as Ele-mentals. The computation of elemental inductances is considerablemore difficult than for simpler filaments of exclusively square orcircular cross sections (for which standardized formulae can be foundin the literature [271-[29]. In fact, most of the rest of the paper isdevoted to developing adequate formulae for the Elementals whichraises the question: What compensating advantages do they offer? Theanswer is: for the same level of accuracy far fewer Elementals arerequired than simpler filaments. In multiconductor lines (for instance,

v loz

ujnaw8a:

a:6

z

>: 4<)

L2

Ile

878

in three-phase-pipe-type-cables with neutral wires), this can be a deter-mining factor - memory limits often prohibit the use of simplefilaments. A related effect is the poor marginal efficiency of simplefilament schemes. Discouragingly large increments in their numbersare required to produce relatively samll increases in overall accuracy.This was noted by Comellini, et al. [13] though they mistook thisslow convergencee for proximity to the true solution.

FILAMENT INDUCI ANCESThis section is devoted to a review of some concepts underlying the

computation of inductances and to the development and verificationof inductance formulae for Elemental loops (loops formed by a pair oflong straight filaments with cross sections of the type shown in Figure5). The reviewed material though relatively simple, is included becauseit is germane and not readily available in the modern literature. Infact, some of it is in danger of being lost to antiquity.

l I

0Figure 5 Elementals - Filaments with Cross Sections of the Type

Shown Here.

GMD (Geometric'Mean Distance) [281 - [30]

Integrating over the length of AB to get its total contribution to thefield strength at point P gives

L S.dAr.sin e L3.dA .k

H dfo=ds,0 0

ds

6 dA= 1 1 Q_- Lk \Ig2 + k2 I( -L)2 + k2

If we assume that the fields produced by 1B are cylindrically symmetricabout filament AB, then the flux linkages, ;/ of filament CD are:

LHHd.dk

f

where ,.t is the permeability of the medium containing the filaments.Evaluating the integrals gives:

= 2X10 L. . dA1 1|(d + +1)( )2+I ]

when , = MO, the permeability of free space, ,B is in amps, L and d arein meters and ; is in webers. If L >> d (as is the case withtransmission lines), we may neglect d/L to give:

= 2X10 7 L3. dA1 En (d 1] (3)

Now consider a pair of parallel monolithic filaments with arbitrarycross sectional areas, A1 and A2 as shown in Figure 7. We wish todetermine the flux linkages, 4A 1A2 of the second filament as a resultof a current, i, flowing in the first filament. Thereby, we will deter-mine the mutual inductance between the filaments.

Consider the pair of parallel, conducting filaments AB and CDshown in Figure 6. Suppose that the filaments have infinitesimal crosssectional areas, dA1 and dA2. Let the current density in AB be , Themagnetic field intensity, dhp, at point P due to the current in theinfinitesimal length, ds, is given by:

$ .dA1.ds.sin 0dH =

r

dA I dA2B

;d -

L 1:} - k

:4 ; 93-ds-r

A

Figure 6

DFigure 7 A Pair of Monolithic Filaments With Arbitrary Cross

Sections, A1 and A2.

_ _ _.p

Assume that the current is uniformly distributed over A andconsider two infinitesimal filaments, dA and dA2 so that formula (3)can be directly applied to them. Note that the assumption that bothconducting and intervening media have the same constant perme-ability causes the system to be linear for magnetic phenomena. There-fore, we can invoke superposition to determine the flux linkages,4A, dA2 of the infinitesimal filament, dA2, produced by the entirecur6ent in the monolithic filament, A1:

C

A Pair of Long Conducting Filaments. *Aj,dA2= 2xlO L.(A ) !n (d- _] dA1

1

(4)

co

. =V f411

d

Some observations are now in order. Obviously the flux linkages ofdA2 vary with its location within A2. Threfore, a varying current inA1 will induce different voltages in different parts of A2. We cannotaccommodate this level of detail -our ultimate goal is to model eachmonolithic filament by lumped, constant circuit elements. In ortherwords, we must approximate each multivalued quantity that variesover the cross section of A2 bu a single representative value. The moststraightforward way to this is by averaging the multivalued quantitiesover the area A2. In the case of flux linkages this averaging processyields:

=A 2X10 L.(A I I Fin(-)-idA dA1A'A2 ~ A1,A2 A A L\d/ 1 2 (5)

= 2X10 L.i djin(2L)-fJ - (G) (6)

where G, the GMD (Geometric Mean Distance) is given by:

G = AA lndn . dA1. dA(2 1

(This expression is general in the sense that if A and A2 are identicaland coincident, it will yeild their self GMD's). lIhus, the flux linkagesper unit length are:

1 2 = 2X10 7 .i [in (2L -L L G1

THE LOOP INDUCTANCESThe GMD concept can be applied to determining loop inductances.

Consider four long, parallel filaments denoted by: a,b,c and d, respec-tively. A loop is formed by connecting the ends of a and b andanother by connecting the ends of c and d. The mutual inductance,Q 12, between these loops is given by:

t12 'P12/i1 (9)

where 42 is the flux linkages of loop #2 resulting from a current il inloop# 1. invoking superposition again, it is seen that

12 ~(ac Wad) bc- bd)

879

Processes that make an adequate compromise between efficiencyand accuracy for the Elementals (filaments with cross sections shownin Figure 5) are given below. Several different situations must beconsidered: 1. Elementals that are relatively far apart as shown inFigure 8. In this case each long narrow elemental is divided into foursectors. Pairs of points are symmetrically placed about the linesbetween sectors at an angular distance of f.A/6, where A is the anglespanned by the elemental and f = 0.564, is a factor that has beenempirically chosen to minimize errors [ 151. (In the special case whenan Elemental is a circle, the six points degenerate to a single locationat the centre of the circle.) The mutual GMD is taken to be thegeometric mean of the 36 distances between the set of six points inone elemental and the set in the other. This process works well for allbut two situations: calculating the self GMD of Elementals and cal-culating the mutual GMD between long narrow Elementals that arevery close together as shown in Figure 9. 2. Self GMD's of Ele-mentals. The self GMD of a rectangle of lenth 1 and width w is verynearly 0.2335 (1 + w), regardless of the ratio of 1/2w[27], [28]. Thisexpression with the constant changed to .2315 is a good approxima-tion for long narrow curved Elementals [ 151 . Expressions for circularElementals are given in Table I. 3. The Mutual GMD between narrowElementals that are very close together as shown in Figure 9. It can beshown [15] that the mutual GMD between the pair of straight linesdisplayed in Figure 10 is given by:

Figure 8 The The Discretization Scheme for Calculating the MutualGMD Between Non-circular Elementals That Are RelativelyFar Apart.

(10)

where 4ac and 'ad are the flux linkages of filaments c and d resultingfrom a current in filament a, and so on. (The negative signs areintroduced because current in filaments b and d flow in oppositedirections to those in a and c. End effects have been neglected.Combining (8), (9) and (10) gives:

91 =2X10 7 ln12 G~ac Gbdj (11)

where Gad is the GMD between the cross sections of filaments a andd, and so on. Expression (11) is quite general and can be applied toloops with a filament in common (that is, loops with a common returnpath) and even loops that are coincident (the mutual inductancebetween spatially coincident loops is the self inductances of theseloops).

THE NUMERICAL COMPUTATION OF GMD's

To determine filament GMD's one must evaluate the integrals inequation (7). For non-simple filament cross sections, A1 and A2 thiscan only be done by numerical integration processes. Naturally theseprocesses must be both efficient and a curate - a system containing n

filaments requires a total of about n /2 GMD evaluations. Errors inthese evaluations are reflected directly in the loop inductances whichin turn directly affect the eventually computed values of effectiveconductor resistance and inductance.

Figure 9 A Pair of Non-circular Elementals That Are Close Together.

n d 1 .

1~J

Figure 10 A Pair of Parallel Straiglit Lines.

(8)

880

l n 2 2d2dln(GMD) = L(2 1;dn (x2 + d2)+

(x2 _ x2)( x2 )2nX(x22 d2d1 2 2ST2nZ n (x1 +d) +

2dx2 -1 -2)- tan '.d)

ndxi) tan ( Xd +

2+ x 2 Q (12)

wherexl1 = Q-n_andx2 =Q+n2 2 . This expression can be adapted to the

calculation of the GMD between the Elementals in Figure 9 byreplacing the straight line lengths, Q and n, with the lengths of the arcsalong the "centers" of the Elementals and multiplying Elementals, by.955 to compensate for curvature and non-zero thickness [ 15]. 4. Allother situations involve circular or cylindrical Elementals for shichGMD formulae are well known. However, for the readerer's conven-ience, they have been assembled in Table I.

Verification

The previously suggested processes for calculating GMD's wereproduced by a fairly long investigatory and developmental project.They have been verified in considerable detail. Specimen verificationswere presented below.

r= 0.03048mr2 °. 0419 I m

181

Figure 11 An Example Used in the Verification of the GMD Calcula-tion Processes.

Consider the example shown in Figure 11. It depicts a typicalconductor divided into 36 Elementals. The self GMD of the shadedElemental and the mutual GMD's between it and every other Ele-mental were calculated by the suggested procedures and compared tothe true GMD's (these were obtained by another numerical procedurethat generates virtually exact results but requires too much computingtime to be useful for any but verificational purposes [ 151 ). Thepercentage errors in the results from the suggested procedures aregiven in Table II. For instance, the entry in the 3rd row and 4thcolumn is the percentage error in the calculated mutual GMD betweenthe shaded Elemental and the Elemental in the 2nd layer that is at1800.

As the numbers indicate, the suggested procedures generate resultsof admirable accuracy. Sensitivity studies [ 151 indicate that this levelof accuracy is sufficient for our purposes.

TABLE 1I Errors in GMD Calculations For the Elementals inFigure I1

AngleLayer

01

2

3

4

00 600 1200

-0.88% .13% .005% .003%

0.4% 0.07% 0.007% 0.005%

0.088% 0.017% 0.006% 0.006%

-0.079% 0.005% 0.007% 0.005%

-.231% 0.007% 0.008% 0.007%

CONCLUDING REMARKS

This paper has:

1. Reviewed the concepts underlying the use of finite elements forcomputing the frequency dependent, effective resistances andinductances of multiconductor cables and cable like structures.

2. Suggested a non-simple set of filaments called Elementals, for cablecalculations.

3. Developed and verified the inductance formulae needed to imple-ment these Elementals.

A dramatic illustration of the relative advantages afforded by theuse of Elementals is provided by simple example, namely: a hollow,cylindrical, conductor with an inner radius of 0.03048 m and an outerradius of 0.0381 m. The number of simple filaments (constant radiuscircles, in this case) and the number of Elementals required to produceresults of comparable accuracy over two frequency ranges are con-trasted in Table III. The numbers indicate that for arrangementsinvolving several such conductors the use of simple filaments wouldconstitute an impractical approach while the use of Elementals wouldbe eminently practical. In fact, algorithms embodying the Elementalsand the efficient matrix reduction procedures described in the Appen-dix have been programmed and found to be convenient for analyzing anumber of configurations that would severly strain conventional pro-cedures, if they were applicable at all. (These configurations were

TABLE III THE Numbers of Simple Circular Filaments and Ele-mentals Required to Produce Results of ComparableAccuracy For a Hollow Thin Walled CylindricalConductor

FrequencyRange

0 - 100 Hz

0 - 10 kHz

Number ofElementals Required

36

36

Number of SimpleFilaments Required

450

1007

encountered in studies undertaken to generate sensitivity and perform-ance data for typical Pipe-Type-Cables. These data will appear in aforthcoming publication).

Finally we reiterate that the procedures presented here are appli-cable only when the conducting and insulating media have the same,constant permeability. However, available information [14] suggeststhat this is not a severe restriction. Apparently, the non-linear effectsintroduced by saturation in components such as steel pipes are smalland can be accommodated by linear (constant permeability) models.

ACKNOWLEDGEMENT

The work reported here was supported by a grant from MAPRC(Middle Atlantic Power Research Committee).

881

where UT is the transpose of U since all the entrivs of the vector ENare equal to the single voltage VN across the Nt conductor. Com-bining (14) (15) and (16) gives:

--J -1 [-T7 -IN NN [N ZNa Ia]

-- -1- (17)orV= NN Na I + N

N --T a -- -1TNN NN

In a similar fashion [ 151 and beginning with the upper partition of(13) we can replace ZaNIN with expressions involving 'a and 'N. Theresult is:

Ea

NJwhr N a

where z =Zaa aa

zaa

, Na ZNN I:N- - -1--T-- -1-z z uuz z+aNNN N Na

(TN- U )uzNN -T

(18)

_ - -1 -

ZaN NN Na

- -T aN NN TaN

uNa Uj UNN

1ZNN -- -1-=TUz=NUNN NN

APPENDIX

We consider here the problem of reducing equation (1) to the form

of (2). For steady state conditions equation (1) can be rewritten in the

form:

IEa

LEN

Ia

IN i(13)

where Ea and 1a are vectors containing the voltages across and currentsthrough, all the filaments in the first N-1 conductors EN and 'N are

voltage and current vectors for the filaments in the last conductor Z =

(R + jwL) is the overall impedance matrix for the filaments and ispartitioned into Zaa, ZaN, etc.

It follows that

- -1 --

IN =NN [EN -ZNaIal (14)

But IN the total steady state current of the N-th conductor is givenby:

(15)IN N

whereU-.l 1. 1].

- -TEN = N (16)

In (18) the filaments in the N-th conductor have been parallel andtheir separate identities lost. The same procedure can be repeated toparallel the filaments in each of the other conductors and obtain thereduced impedance matrix of equation (2). The reduction could alsobe acheived by a relatively straightforward inversions and summationsbut such direct processes can involve twice as much computation[151.

REFERENCES

[ 11 A. Russel, "The Effective Resistance and Inductance of a Con-centric Main, and Methods of Computing the Ber and Bei andAllied Functions," Philosophical Mag. Vol. 17, 1909, pp.524-553.

[ 2] H.B. Dwight, "Skin Effect in Tubular and Flat Conductors,"AIEE Trans., Vol. 37, 1918, pp. 1379-1403.

[ 3] A.H.M. Arnold, "The Alternating-Current Resistance of TubularConductors," Journal of IEEE, London, Vol. 78, 1936.

[ 41 C. Manneback, "An Integral Equation for Skin Effect in ParallelConductors," J. Math. Phys., Vol. 1, 1922, p. 123.

[ 5] W.A. Lewis, P.D. Tuttle, "The Resistance and Reactance ofAluminum Conductors, Steel Reinforced," AIEE Trans., Vol.77, PAS 1958, pp. 1189-1215.

[ 6] C.I. Mocanu, "The Equivalent Schemes of Cylindrical Con-ductors at Transient Skin Effect," paper #7 1TP667 presented atthe IEEE PES Summer Meeting, Portland, July, 1971.

1 71 G.W. Brown, "Surge Propagation in Pipe-Type Cable," Confer-ence paper presented at the IEEE PES Winter Meeting,Jan./Feb. 1974.

[ 8] G.W. Brown, R.G. Rocamora, "Surge Propagation in ThreePhase Pipe-Type-Cables, Part I: Unsaturated Pipe," Paper no.F75 424-2, presented at the PES Summer Meeting, 1975.

[ 9] P. Graneau, "Alternating and Transient Conduction Currents inStraight Conductors of Any Cross-Section," InternationalJournal of Electronics, Vol. 19, 1965, pp. 41-59.

[10] P. Silvester, "Nodal Network Theory of Skin Effect in FlatConductors," Proc. of IEEE, Vol. 54, #9, Sept. 1966, pp.1147-1151.

882

[1 1] P. Silvester, "The Accurate CAlculation of Skin Effect in Con-ductors of Complicated Shape," IEEE Trans. on PAS, Vol. 87,#3, March, 1968, pp. 735-742.

[12] E. Comellini, A. Invernizzi, G. Manzoni, "A Computer Programfor Determining Electrical Resistance and Reactance of AnyTransmission Line," IEEE Trans. on PAS, Jan./Feb. 1973, pp.308-314.

[13] P. Silvester, M.V.K. Chari, "Finite Element Solution of Satur-able Magnetic Field Problems," IEEE Trans. on PAS, Vol. 89,Sept./Oct. 1970, pp. 1642-1651.

[14] R.C. Dugan, S.N. Talukdar, "The Effects of Saturation on thePerformance of Pipe-Type Cables," paper No. A76 340-0 pre-sented at the PES Summer Meeting, Portland, Oregon, 1976.

[15] R.N. Lucas, "On Modelling Pipe-Type-Cables and Gas-Insulated-Buses," MS Project Report, Carnegie-Mellon Univer-sity, 1977.

[16] A.B. Niemoller, "Skin Effect in Round and Tubular Con-ductor," Conference paper presented at the IEEE PES WinterMeeting, New York, NY, January 30-February 4, 1972.

[171 A.N. Talukdar and J.C. Wang, "On Simulating Transient SkinEffects," presented at the Pittsburgh Modeling And SimulationConference April 23-24, 1973.

[18] S.N. Talukdar and J.C. Wang, "Including Frequency DependentParameters in Transmission Line Simulation Studies," Confer-ence Paper, present at the IEEE PES Winter Meeting, New YorkNY, January 28-February 2,1973.

[191 H.B. Dwight, "Skin Effect and Proximity Effect in TubularConductors," Presented at the 10th Midwinter Convention ofthe A.I.E.E., New York, NY, February 15-17, 1922, pp.189-198.

[20] E.J. Tuohy, T.H. Lee, and H.P. Fullerton, "Transient Resistanceof Conductors," IEEE Transactions on Power Apparatus andSystems, Vol. PAS-87, No.2, February 1968, pp. 455-462.

[21] A.H.M. Arnold, "Proximity Effect in Solid and Hollow RoundConductors," Journal of the Institute of Electrical Engineering,pp. 349-359.

[221 R.L. Jackson, "Measurement of Skin and Proximity Effects inCircular Conductors," Proc. IEE, Vol. 117, No. 7, July 1970.

[23] W.Scott Meyer and H.W. Dommel, "Numerical Modelling ofFrequency-Dependent Transmission-Line Parameters in anElectromagnetic Transients Program," IEE Trans. on PAS,Sept./Oct. 1974, pp. 1401-1409.

124] E.R. Thomas and R.H. Kershaw, "Impedance of Pipe-Cable,"IEEE Transaction on Power Apparatus and Systems, Vol.PAS-84, No. 10, October 1965, pp. 953-965.

[25] Hans-Jurg Sigg and M.J.O. Strutt, "Skin Effect and ProximityEffect in Polyphase Systems of Rectangular Conductors CA1-culated on an RC Network," IEEE Transactions of PowerApparatus and Systems, Vol. PAS-89, No. 3, March 1970, pp.470-477.

[26] B.D. Popovic, Introductory Engineering Electromagnetics,Addison-Wesley, 1971.

[27] W.D. Stevenson, Jr., Elements of Power System Analysis,Second Edition, McGraw Hill, 1962.

[28] F.W. Grover, Inductance Calculations - Working Formulas andTables, D. Van Hostrand, 1946.

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[30] E.W. Kimbark, Electrical Transmission of Power and Signals,John Wiley and Sons, 1949.

Discussion

J.R. Brauer (A.O. Smith Corp., Milwaukee, WI): This interestingpaper gives an excellent summary of certain types of numerical techni-ques for calculation of cable resistance and inductance. Despite thetitle, however, the paper does not discuss the finite element method, ex-cept for listing reference (13).

Reference (13) shows that the finite element method calculatesmagnetic fields and inductances by minimizing an energy functional.No network (as in Figure 1) need be established, and no formulas forinductance (as in Equations 1 through 11) need be derived.

When resistance and inductance are functions of frequency, thefinite element method including the effects of induced currents can beused (31). For example, Figure 12 shows the finite element mesh for aportion of a cable that contains seven conductors. The cable resistanceand inductance versus frequency can be calculated without the need fornetworks or inductance formulas.

I encourage using a finite element program such as reference (31) toanalyze the cables in this paper, for comparison with the authors'method.

(31) J.R. Brauer, "Finite Element Analysis of Electromagnetic Inductionin Transformers", paper A 77-122-5 presented at IEEE Winter PowerMeeting, February 1977.

Manuscript received July 13, 1977.

Figure 12. Concept of finite element model of a seven-conductor cable in freespace.

Carlos Katz and David Silver (General Cable Corporation, Union, NJ):We certainly hope that the approach presented by the authors will beup-graded to the point where it can be used for the calculation of pipe-type cable resistances and inductances in practical systems.

The assumptions made in the paper appear to be extremelysimplified and at least one of the assumptions requires clarification. Ina pipe-type cable, normally made with Milliken conductors, the currentflows in a helical path around the center of the conductor. Therefore,when the authors indicate in their second assumption that the "currentflows only longitudinally in the filaments" they should clarify if thesefilaments have been considered to be parallel or helical to the center ofthe conductor.

We would like to ask the authors if in reality, the proposed solu-tion is or will be of use in pipe-type cables when consideration is givento the following characteristic design details of the great majority ofpipe-type cables:

a) These cables are generally made with segmental conductors.b) Two, opposite segments are insulated from the other two by paper

tapes.c) The strands in each segment continuously change rotational posi-

tion within the segment, therefore, forcing the current to flow fromstrand to strand.

d) Each segment changes continuously rotational position around thecenter of the conductor.

e) The distribution of current in each segment is affected by skin andproximity effects.

f) Not all segmental conductors are made the same way. Some arepreformed, others are not preformed, thereby affecting the degree ofstrand contact and consequently the interstrand electric resistance.

g) Coatings such as tin, lead alloy, natural oxidation may influence toa smaller or greater degree the interstrand contact resistance.

Manuscript received August 11, 1977.

Sarosh Talukdar and Robert Lucas: The discussers have raised some in-teresting issues. We will offer our thoughts on them beginning with

Manuscript received October 28, 1977.

883several rather general observations.

Finite element procedures may be used with advantage to tacklecomplex phenomena in large regions of multi-dimensional spaces. Theprinciple is to divide a region into a number of pieces or finite elementsthat are small enough so that the phenomena in them can be describedby relatively simple models; yet not so small that the total number ofpieces becomes inconveniently large. To implement a finite element pro-cedure one must select a set of state variables in terms of which todescribe the phenomena and a scheme by which to divide the region intopieces. Both choices have critical effects on the performance of theoverall procedure.

We have selected voltages, currents and magnetic fluxes as ourstate variables and finite elements shaped as shown in Figure 5. Theresult is a procedure that requires a relatively small number of elementsthough the calculations associated with each can be faily involved. Anadvantage in dealing with a number of frequencies is that much of thecalculations need be done only once and can then be reused for eachseparate frequency to be investigated.

Mr. Brauer uses another procedure that requires far more elements(c.f. his Figure 12) but less calculations per element. Which procedure isbetter? It would, as Mr. Brauer suggests, be most interesting to in-vestigate this question.

Turning now to another issue we note that, to be useful, modelsmust be easier to deal with than the physical systems they represent.They cannot duplicate all the features of a physical device or system.The trick is to build them so that they include the important features.Since a considerable amount of developmental effort can be involved inadding features to a model such efforts should not be undertaken unlessthe need for them has been demonstrated.

Messrs. Katz and Silver have pointed out that our models do notexplicitly include certain physical features of certain cable structures.How important are the differences? The algorithmic work done in thepaper makes it possible to now address this question. We join Messrs.Katz and Silver in calling for investigative work in the area. We suspectthat these "differences" will affect cable behavior to some extent. It isprobable that the effects can be included in models by the addition ofemperically determined constants and factors.