analysis and design of transmission line structures by means of the geometric mean distance_ieee

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ANALYSIS AND DESIGN OF TRANSMISSION-LINE STRUCTURESBY MEANS OF THE GEOMETRIC MEAN DISTANCE

A. J. Sinclair J. A. FerreiraIndustr ial Electronics Research Group, Laboratory for Energy,

Rand Afrikaans Universi ty, P.O. ox 524, Auckland Park, 2006, SOUTH AFRICAABSTRACT

This article describes the Geometric Mean Distance(GMD), a tool for calculating the various direct-currentinductances in any multi-conductor transmission line.Two examples of its use are given. For a two-conductortransmission line consisting of rectangular conductors, theGMD is calculated by means of integration and also bydiscretizing the conductors and treating them as compos-ite conductors. The obtained values for the GMD is thenused to calculate the inductance of each conductor. Atwo-conductor foil transmission line is also investigatedand the GMD is used to determine the optimal foil shapefor a given conductor inductance and minimum conduc-tion losses. Finally, the advantages and limitations of theuse of the GMD are discussed.

INTRODUCTIONThe Geometric Mean Distance (GMD) is a means of r e presenting the total effect with regard to inductance, oftwo conductor cross-sections on each other such that thetw o conductor cross-sections can be replaced by two fil-amentary conductors. The system of two filaments withtheir center points separated by this GMD, will have thesame mutual inductance as the original two conductors.See Figure 1for an illustration of this statement.The formula used to calculate the GMD between the twoconductors with cross-sections Si and S2, is [l]:

where 0 1 2 is the GMD between the two conductors, andIn D12 is called the logarithmic mean distance. D lz r e presents the weighted effect of all the filamentary conduct-ors constituting the two cross-sections.

S,

Fig. 1. Th e Geom e t r i c Me an Dis tance be tween two Gross -Sec tions

Fig. 2. The Geome t r i c Mean Dis tance be tween two Rec tangularC onduc t or s

The inductance of one conductor of a two-conductor go-and-return transmission line (see Figure 2) is given by

PO Dl2 D12L~ = -In - 2 . 1 0 - ~ 1 ~-~ - Dii D11where Dll is the so-called self geometric mean distanceof conductor 1, i.e. the arbitrary points with coordinates( 2 , ~ )nd ( X , Y ) are located in the same cross-section5'1 = 5'2 (coinciding), in equation (1) and Figure 2. Higgins [l]mentions that the concept of the GeometricMean Distance dates back to 1872when it was originatedby J. C. Maxwell, and it has since been used extensivelyin the design of busbars [2] [3] and power systems analysisand design in general [4]. Its strength is the fact that itcan be used to calculate the direct-current inductance ofconductors of any shape and arrangement.

SINGLEPHASE TRANSMISSION LINECONSISTING OF TWO RECTANGULARCONDUCTORS

Consider the two-conductor transmission line in Figure 2. Two methods will now be shown by which the inductanceof conductor 1 can be calculated.Direct IntegrationWhen equation (1) is applied to Figure 2, the followingcomplicated formula for the inductance of conductor 1arises:

In J ( X - z)*+ (Y y)2dXdzdYdy

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-x - / ( X - r)*+ (Y y )2A - - x+(Y - y ) tan-' GI dxcdYdy ( 3 )

and after an inordinately long integration procedure hasbeen performed, the following general formula, first de-rived by Gray , is obtained [5]:

4h2w2 nD12

+ (s - U)' h2 - - -In [(s - ? u ) ~ h 2 ][ (1 112 61 4 htan-' -2 3 S + W+ - ( s + w ) 4 ~ n ( s + w ) - -s41ns+- (S- w ) ~n(s - w) + - h ( s +

h3 h 3 S+ 8 h s 3 tan-' -+ - h 3 (s + w) tan-' -

s 4 h- z h 3 s t a n - 1 - + - h ( s + w ) 3 t a n - 1 __3 h 3 s - ws - 1u

3 h- + -hA (s - U ) an-' - L5w2 2 (4)An empirical formula obtained for the self GMD of arectangular section, is

A s can be seen from ( 3 ) and (4) , calculating inductancesof conductors of arbitrary shapes and arrangements byusing (1) is fairly cumbersome. Fortunately. there is analternative.DiscretizationStevenson [4] gives the following expression for the in-ductance of one composite conductor in a single-phasetransmission line consisting of two composite conductors,each in turn consisting of a number of identical roundconductors (see Figure 3 which illustrates the process ofdiscretization)

Each rectangular conductor in Figure 2has been replacedby a number of identical round conductors arranged insuch a way that they resemble the shape of the originalconductor. Each of the thin round conductors constitutingconductor 1 is denoted by a letter Q to n and each of thoseconstituting conductor 2 is denoted by a letter U' to m' .The GMD between two round conductors is simply thedistance between their center points (denoted by Dab',for

Fig. 2. Discre t i zat zon of Transmission L i ne o j t u o RectangularCo n d u c t o r s

example) and the self geonietric mean distance (denotedby D,,, for example) of a round conductor with radius Tis j u s t r e - + .Equation (6 ) can therefore be described as follows: Theterm v m c ~ ..Dam!) in the numerator ofequation (6 ) represents the GMD of all the round con-ductors constituting conductor 2 , to conductor a, whichis part of conductor 1, and this term therefore representsthe total effect of conductor 2 on the round conductor a.There are n round conductors constituting conductor 1,and there are therefore n of these distance. The geometricmean of these distances (the nth root of the product ofthese distances) represents the GMD between conductors1 and 2.The denominator contains terms liked (DaaDabDac . . .Da,a,which represents the total effectof all the other round conductors in conductor 1. on theround conductor a . This term is a distance at which oneother round conductoir should be placed from conductora , so that it would have the same effect as all of the roundconductors 6 to n. on conductor a . The geometric meanof all these distances is the self GMD of conductor 1.The remarks in t,he previous two paragraphs can be sum-marized by pointing to the similarities between equations(6 ) and (2).Calculation ResultsThe calculations were performed for a single-phase trans-mission line consisting of two rectangular conductors withthe following dimensions:

w = 2 cm, h = 10 cm, s = 10 cmDirect calculation yielded t.he following results:

Dll = 0, %235(w+ h ) = 2 6 , 8 mm

L l = 2 7 6 , 9 nH.m-'A program written in MathCad and implementing the dis-cretization algorithm yielded the results shown in Table1.A finite element analysis package w a s also used to cal-culate the inductances L1 and La, and values of 272,2nH.m-' and 272,3 H.m-l were obtained.

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Table 1Results of Discretization of Rectangular Conductors

No. of points L1, L2 [nH.m-l]6 x 2 271,411 x 3 261.321 x 5 264.6

MINIMIZING CONDUCTION LOSSES OF ATRANSMISSION LINE WITH A FIXEDREQUIRED INDUCTANCEConsider a transmission-line structure consisting of twofoils, the thicknesses of which have already been optim-ized with respect to the operating frequency, i.e. thereis no variation in the current distribution in the thick-ness of the foils. The current distribution therefore onlyvaries laterally (along each foils width). Assume furtherthat the transmission line is required to have a specificinductance, for example to limit faul t currents.First, it is necessary to consider why some foil shapes arecharacterized by higher losses than other shapes. Con-sider the two-foil transmission line in Figure 4 . The foilsare shown in dashed lines. while a few of the filamentsof which the foils are considered to be constituted. areshown as round conductors. A number of flux lines, res-ulting from the current flowing in the foils, are also shown.The filaments toward the centers of the foils are linked byall of the flux shown, whereas those toward the edges ofthe foils are linked by fewer flux lines, so that they arecharacterized by lower inductances.Everywhere in nature, systems seek out states of min-imum energy, and these systems tend to gravitate towardthose states. Since the energy associated with the mag-netic field of the current flowing in the outer filaments,W = + L i 2 , s less than that of the field associated withcurrent flowing in the center filaments, more current willtend to flow in the outer filaments. This variation inlateral current distribution gives rise to an increased res-istance, since the cross-sectional area of each foil is notfully utilized.There are again at least two ways to determine the o ptimal shape of the foils, viz. discretization and finite ele-ment analysis.DiscretizationInitially each foil conductor is represented by a horizontalrow of evenly-spaced thin round conductors, which will

Fig. 4. Transmission Line consisting of tw o Flat Foils

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0 00 0 0 0 0 0 01

Fig. 5. Iterative Process to find Optimal Shape of Discretazed Foils

be called filaments, for the sake of brevity. Each fila-ments inductance is then calculated. The inductance ofconductor a in Figure 4 , for example, is given by

I

( 7 )VDaalDab, . . .Dam,VDaaDab . . . D a nL~ = 2.10-~1nand the total inductance of each foil conductor can becalculated using equation (6) .Due to the symmetrical nature of the transmission line,the calculations need only be performed for one of thefoils. As the final step of each iteration, the mean andstandard deviation of the inductances of the filaments rep-resenting the foil must be calculated.The second iteration involves moving the two outer fila-ments of each foil away from the other foil by an angle 8,but keeping the distance between the outer filament andthe adjacent filament second from the outside constant,so a s not to alter the width of the foils.Once again the inductance of each filament is calculated,as well as the values of the mean and standard deviation ofthe filamentary inductances in each foil. The standard de-viation should decrease as more iterations are performed.As soon as the standard deviation of the filamentary in-ductances in a foil starts to increase again, the previousconfiguration is such that the filamentary inductances ineach foil conductor is most nearly equal. This means thatthe current distribution will be the closest to uniform forthat configuration and the foil should be shaped like thecurve passing through all of the filaments.The process described above, involves only the moving ofthe outermost filaments in each foil, but the same processcan be carried out after the filaments second from theoutside in each foil have been moved by a certain angle,and so on toward the center filaments.This method is of a recursive nature and can easily beimplemented in a computer program. All of the iterationsdescribed above, can be incorporated into the program, sothat the user only needs to specify the foil width and theshortest separation distances between the two foils. Theprogram would then calculate and display the optimalpositions of the filaments.Finite Element AnalysisAnother method to determine the optimal shape of thetwo-foil transmission line, is to use a two-dimensionalfiniteelement analysis package that solves for the mag-netic vector potential after current sources and the con-ductor geometries have been specified.

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_..._.._._._.... .-._-.._........ ..._..... .--.. ..-.-__-Am?/ ~ results can be obtained more rapidly than with othermethodsi t is cheaper to use than methods such as finite ele-ment analysisit is as flexible as more expensive methods in termsof conductor shapes and arrangements that can bedealt with.The main limitation of the use of the GILZD is that i t is re-str icted to direct-current conditions where the current isdistributed uniformly across the conductor cross-section.or cases where the current distribution is known exactly(for example, at very high frequencies where current onlyflows on the conductor surfaces, or when the conductorsare so thin that the skin effect, does not influence the cur-rent distribution noticeably).

Fig. 6. I t er a t i ve Process t o f ind Optimal Foil Shape by S i m u l a t i o nCONCLUSIONS

Lines of constant magnetic vector potential are, in fact,flux lines, and these lines should not cross the foil cross-sections if all filaments are to link the same amount offlux, and hence have equal inductances.Initially the two flat foils separated by a certain distance,would be simulated and the calculated direct-current fluxlines plotted.The user would then have to alter the geometry in sucha way that the foils do not cross any direct-current fluxlines, and the process would be repeated.Once the foil shapes need not be changed any more toprevent flux lines from crossing the foil surfaces, the op-timal shapes have been determined.The first two steps in the process are shown in Figure 6.Application to BusbarsThe foil shapes arrived at by means of the above method,are usually not easily manufacturable, but there are somepractical busbar shapes that closely resemble the calcu-lated shapes for various inter-conductor distances. Figure7 [6] shows some theoretical foil shapes and the closestcorresponding practical busbar shapes.

The Geometric Mean Distance as a tool for analysis anddesign has long been discarded in favor of Finite-ElementMethods (FEM), but in certain cases it can still be usedto great advantage to yield approximate results morequickly and cheaply than finite-element methods.

REFERENCES[I ] T. J . Higgins. Theory and Application of Com-

plex Logarithms and Geometrical Mean Distances,Trans. AIEE, vol. 66, pp. 12- 16, 1947.PI Copper Development Association,Copper for Busbars. Potters Bar, Hertfordshire:Copper Development Association, ch. 9 , pp. 33-36, 1984.131 P. J. H. Rata and A. G. Thomas,Aluminium Busbar. London: Northern AluminiumCO. (Ltd.), ch. 5, pp. 37- 40, 1960.

[4]W. D. Stephenson, J r . . Elementsof Power System Analysis New York: McGraw-Hill

[ 5 ] A. H. M. Arnold, The Inductance of Linear Con-ductors of Rectangular Section, JIEE, vol. TO, pp.Book CO..ch. 3 , pp. 52- 55 , 1982.

,579- 586. 1932.[6] A. H. M. Arnold, The Transmission of Alternating-

Current Power with small Eddy-Current Losses,DVANTAGES AND LIMITATIONS OF THEGMD JIEE. vol. 80. pp. 395- 400, 1937.The advantages of using the G I ID for inductance calcu-lations are that in some cases:

F i g . 7. Practical Busbar Shapes mo s t c lo se ly r es embl ing T heor e t -ical Foil Shapes

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analysis and design of transmission line structures by means of the geometric mean distance_ieee

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