APPLICATION OF INVARIANT APPROXIMATIONS TECHNIQUE

TO ELECTRICAL FIELD ANALYSIS

Mariya HowykowyczDepartment ofElectrical Engineering,National University "Lviv Polytechnic"

Ukraineemail: howykowycz(gieee.org

AbstractThe paper deals with the solution of the boundary value

problem of a steady-state electric field distributed in a linearuniform substance, provided by applying the finite-differencesmethod and the methodology of invariant functions'approximation. The research constitutes the first attempt toapply theforesaid methodology, that guarantees the invarianceofthe solution in respect to the group oflinear transformationsof Cartesian co-ordinate system, i. e. preserves the tensorcharacteristics ofinitial Maxwell's equations, to the simulationof electrical grounds 'field that is inherently three-dimensionaland cannot be substituted by the series of two-dimensionaltasks. In result, the considered domain for electric fieldanalysis comprises the internal subset ofnodes located strictlyinside the domain, the boundary subsets of nodes located ongrounds' surface, the boundary subset ofnodes with Neumannrelation, and the boundary subset of nodes with Dirichletrelation.

The proposed model was tested by means of comparisonof its results with analytical solution obtained by application ofmirror refractions method and average potential method Thetechnique proved to be e]ficient]or arbitrary configuration ofelectrical field distribution domain and can be extended tonon-linear case infuture research.

Keywords. Boundary value problem; invariant approximation;electrical ground'sfield

1. Introduction

The methodology of invariant functions' approximationswas proposed [ 1 ] to simulate the distribution of electro-magnetic fields of arbitrary origin in areas with arbitraryconfiguration. It was used for analysis of plane two-dimensional fields [2]. However, the electric grounds' field isinherently three-dimensional; that is why the task ofapplication of the foresaid methodology to spatial domains offield distribution was formulated [3]. At first it wasimplemented for magnetic circuit analysis [4, 5] and afterwardsextended to electrical grounds' field analysis.

2. Assumptions

In order to reduce the enormous task of finding of electricalsteady-state field distribution, induced by electrical current

1-4244-0038-4 2006 4IEEE CCECE/CCGEI, Ottawa, May 2006

flow from grounds, in a linear uniform substance, thesesimplifying assumptions are made:

1) The conductivity of grounds' material is indefinitelylarge, therefore, the electrical field inside the groundsdoes not exist and intensity of electric field on thesurface of grounds comprises only a normalcomponent;

2) The conductivity of air is equal to zero, therefore,Neumann relation is applicable to air/earth interface;

3) The conductivity of earth is constant and independentfrom field intensity;

4) The domain of electric field analysis is confined by thesurface that is situated such large distance from thegrounds that its electrical potential is equal to zero.

3. Mathematical model of the problem

If the location ofK electrical grounds of arbitrary shapeis given, then according to the made assumptions the domainG of electric field's distribution is chosen in such way that itsboundary F comprises K 2 following portions: the portion,Fk (k = 1, K), that coincides with the surface of the k -thground; the portion, FN, that coincides with air/earth interfaceand is applicable for Neumann relation; the portion, FD , that islocated such distance from the grounds that Dirichlet relationmay be applied. The mathematical model of the consideredproblem may be written down in the form:

A(orr]=O; (r-e G) [1]E = -gradp; (r- E G u F)[r]= E [rI- (r-E G u F)

fgdr]h[r[]dS=Ik-; (reEFk)Sk

[2]

[3][4]

[r-]=const; (r e Fk) [5]n[r]EIFr]=O; (re FN) [6]

4[r]=O, (r EFD) [7]where r = ix + jy + kz is the radius-vector of a pointbelonging to the domain G; qor] is electrical potential as asought scalar function of radius-vector r of a point inside the

.90

domain G and on its boundary F; E[r]= iEx + jEy + kE

and 3[r] = i 3x + j3y + + kd are vector of electrical fieldintensity and vector of electrical current density as soughtvector functions of radius-vector r of a point inside thedomain G and on its boundary F; y is specific conductivity

of earth; Sk (k = 1, K) is the surface of the k -th ground;

n [r] = inx + jny + knz is an external normal unit vector in a

point on the surface of a ground; Ik is the given current of thek -th ground.

4. Nonlinear algebraic analogy of the system

The domain G shall be covered with a grid. Let us allotnumbers from 1 to MG to the inside nodes, numbers fromMG+1 to MG+ MI to the nodes belonging to the partF1 uF...uk uF...K of the boundary (i.e. the portion

KFk (k = 1, K) contains Mk (k = 1, K) nodes; MI =, Mk);

k=l

numbers from MG + MI +1 to MG + MI +MN to the nodesbelonging to the part FN of the boundary, and numbers from.

MG+MI+MN+l to M=MG+MI+MN+MD to the

nodes belonging to the part FD of the boundary. Furthermore,k-l

the denotation MBk = Mi (k = 2,K) and MBk = 0 fori=l

k = 1 will be also used. Such numeration has been called abasic numeration. Every node corresponds to a unique n-thorder mesh consisting ofP nodes where P = (n i 3)! (n!3!) isequal to the number of terms in the Taylor's polynomial of then-th degree; within the mesh this node is called a basic meshnode. For nodes of every mesh we introduce a doublenumeration in which the first part of the node's denotation isequal to the number of an appropriate basic mesh node in thebasic numeration and the second part is equal to the localnodal number within the mesh. Meshes which include nodesbelonging to the boundary are called contiguous (ifcorresponding node does not belong to F7) or boundarymeshes (if corresponding node also belongs to F )Let us assume that grid steps are sufficiently small to

approximate dependencies q = q [x, y, z]; E = E [x, y, z];3 = 3[x, y, z] within a mesh by Taylor's polynomials [4].According to the methodology of invariant function'sapproximation, an algebraic analogy of the consideredboundary problem can be written down as follows:

RAmOm = °; (m = ',MG) [8]

Exm =-RxmOm; Eym =-Rymim;

Ezm =-R~Zm m; (m = I'M) [9]

Sxm Exm,ym =7Eym, 9m =Em; (m=1, M) [10]Mk

,ai3MG+MBk+i = Ik; (k = 1,K)i=l

YMG+MBk+i = (MG+MBk+i+1; (i 1, Mk -1)

(k = 1,K)

nxmExm + nymEym + nzmEzm = 0;

(m = MG +MI +1, MG +MI +MN)

[1 1]

[12]

[13]

(m=0, (m=MG+MI+MN+L,M) [14]where RAm,Rxm,Iyin' are algebraic analogies ofLaplacian operator and algebraic analogies of partialderivatives with respect to independent variables x, y, z,respectively, for the m -th mesh; M = (m,..., mp)t is thenodal column of potentials belonging to the m -th mesh; ai isa coefficient of quadrature formula for numerical integration

over the surface of the k -th ground; gm = '5f2 +2 +z2m(m = MG + MBk + i) is the magnitude of current densitythrough the i -th point located on the surface of the k -thground.As it can be seen, the system comprising 7 M equations

has been obtained. Taking into account the transformationsdeveloped in [2] and partially replacing the basic numerationwith the double numeration, this system can be reduced to thesystem of MG + MI +MN unknown nodal potentials:

p

XRAmn,jYmj =0; (m=l,MG) [15]j=1

MG+MB,k+l P P

E, aM-MG-MBk (( Rxm,j nmj)2 + ( Rym,jYomj )2 +M=MG+MBk+l j=1 j=1

[16](Om

(Om-fm+l = °; (m = MG + MBk + 1, MG + MB,k+1-l)(k = 1,K) [17]

P P PnxmEn Rxmi,jYimj +nym E Rymi,jYimj +fnzm E Rzm,1m =0;

j=1 j=1 j=1

(m=MG +MI +1, MG +MI +MN) [18]where mj is the denotation of j -th local node within the m -thmesh while using the double numeration. The obtained systemis non-linear due to the nonlinearity of the equation [16]. Forits solving a variety of nonlinear methods can be used (e.g. theNewton's method with the prolongation after parameter).

5. Testing of the model

To prove the adequacy of the proposed model amathematical experiment boundary has been conducted. Thetask to calculate the resistance of a rectangular ground was

491

formulated. The ground (its length I = 1.57 m) was buried in a

parallel to the earth surface position as deep as of H = 0.5 m;the earth conductivity was assumed of y= 10-4 Sm/m. Theportion FD u FN of the boundary coincided with the surfaceof a parallelepiped; its one side was appointed to the portionFN and all others were appoimted to the portion FD. Thesteps hx, hy ,.hzof the grid covering the domain G were

assumed to be small enough to use the mesh of the 4th order( P = 35) for constructing the algebraic analogies of differentialoperators, i.e. to approximate the electric potential within a

mesh with relation

=u1 +u2x+u3y+U4Z+...+UX axy yzaZ(ax!ay!az!)+...+uPz 14! [19]

The algebraic analogy of Laplacian operator has in that case

the approximation error of the 3rd order and, using the localco-ordinate system [2], is calculated by means of the formula

Ap =u5 + U8 + U10. [20]

The algebraic analogy of Hamilton's operator has in that case

the approximation error of the 4th order and, using the localco-ordinate system [2], is calculated by the formula

Vq,=iu2 +ju3 +ku4-

Fig. 1. Algebraic analogy of the operator 12h2iA .

.y

[21]For example, the algebraic analogies of Laplacian operator

and Hamilton's operator for the regular mesh of the 4th order,whose co-ordinates are given in Table 1, withhx = hy = hz = h are shown on Figures 1, 2.

Tab. 1. Co-ordinates of the strictlv regilar mesh.1 2 3 4 5 6 7

x 0 hx 0 0 - hx hx hxY 0 0 hy 0 O hy hy 0

z 0 0 0 hz 0 0 hz8 9 10 11 12 13 14 150 0 0 2 hx - hx - hx hx hx- hy hy 0 0 hy 0 - hy hy0 hz - hz 0 0 hz 0 hz16 17 18 19 20 21 22 23hx 0 0 0 0 -2 hx 2 hx 2 hx0 2 hy - hy hy 0 0 hy 0

- hz 0 hz - hz 2 hz 0 0 hz24 25 26 27 28 29 30 31

-hx - hx - hx hx hx hx hx 0

-hy hy 0 2 hy -h hy 0 -2 h0 hz -hz 0 - hz 2 hz 032 33 34 35

x

8i

~ ;

Fig. 2. Algebraic analogy of the operator 12hV.

The analytical solution for the value of the ground'sresistance was obtained by means of the method of electro-static analogy, the method of mirror refractions, and themethod of average potentials. According to the method ofelectrostatic analogy, at first the electrical capacity of an

isolated metal bar of I -length, 2b -width, and 2c -thicknessshall be found. The bar is submerged into a dielectric substancewith permittivity £ in such way the rectangular side2c - 2b - 2c - 2b of the bar is parallel to the surface of thedielectric and it is located at the H -depth from the surface.According to the method of mirror refractions, the equivalentscheme constitutes infinite space filled with the said dielectric,which includes not only the discussed bar but also its mirrorrefraction located as far as 2H from it. Let us choose theorthogonal coordinate system in such way that its axe

Ox coincides with the axe of the bar, the origin 0 of co-

ordinates is located on the bottom side of the bar, the co-

492

t k

=1

16

.tl<16(16flLk~. fi

0 0 0 02 hy - hy hy 0

hz --hz 2 hz -2 hz

ordinate plane Oxy is parallel to the side 2c - I - 2c - I, andthe coordinate plane Oxz is parallel to the side 2b - 1- 2b - 1.According to the method of average potentials, the assumptionof uniform distribution of electric charge over the bar's axe isto be made and the bar's potential is assumed to be equal to itsaverage value; the linear charge density was noted by r .

The expression for electrical potential of arbitrary point ofthe bar, with coordinates xc, xYC ZC, can be written down inthe form:

=J (1/ x/(xc-x)2+y2+Z2)+0

+1/ (xcx)2 + (yc 2H)2 + Z2 )dx= x

x (arcsh[ 'X ]-arcsh| +z +

The average potential of the bar can be calculated by theformula:

1 c 1 c 1 b

yc = (Jdxc 1 =bdYC +JdxCJ|fX=bdYC +JdxCJY Y =Cdzco -c 0 -c 0 b

+JdXc Jf yC=bcfdz(4bl +4cl)= d +(2Jdyc (Yc+bdo b

-Iarcsh 1 + 12+y +b2 + (YC -2H)2 +b2yC +b2

-lI-arcsh 1 + 12+(y 2H)2 +b2)d(y - 2H)2 + b2

b

fdzc (c2+ z2 - I - arcsh 1 + 2+C2+Z)b c +z2

+ fdzc( (c-2H +Z2 -I arcsh 1-b j(c-2H) +zc

b+ l2 +(c-2H)2 +Z2)+ dz ( (c+22H)2 +Z2

-b

-arcsh 1 + I+(c+2H)2 + Z2

+ +(c+ 2 +Z2)/(4bl+ /2 +(c+2H)2 +z2) 1(4b + 4c1).The expression for the bar's capacity is

C =d YIcThe bar's resistance can be calculated by means

R =e 1(C) .The ground's resistance, found by means of the

model, is calculated by formula

[19]

R =koIk, [21]where ok is the sought value of potential on the surface of thek -th ground. The values received from formulas [20], [21] arecompared to estimate the adequacy of the proposed model.

6. Conclusions

Applying the methodology of invariant functions'approximations preserves the tensor nature of Maxwell'sequations describing electrical field during their transformationinto the system of algebraic equations used for numericalsolution. The methodology offers a general algorithm forconstructing non-singular meshes of n -th order used forcalculation of algebraic analogies of differential operators andcan be applied to areas with arbitrary configuration ofboundary surfaces. The methodology was implemented foranalysis of electric fields distributed in linear substances; itsimplementation for non-linear case is a task of a futureresearch.

References

[1] R. Filc, "Discrete analogy of the Hamilton's operator,"Mathematical Methods and Physical-Mechanics Fields (inRussian), Ukraine, vol. 24, pp. 20-25, May, 1986.P. 04JIJlbU, <<AHCKpeTHbIH aHajior orepaTopa FaMHJbTOHa>>MamemamutIecKue memo&bl u tj/u3uKO-mamemamutlecKuenoJiM, YKpauHa, BbIH. 24, c. 20-25, 1986.

[2] R. Filc, M. Kotsyuba, "Solving of nonlinear problems ofmagnetostatics by means of finite differences method andmethodology of invariant approximations of functi-ons,"Electrical Engineering, Ukraine, vol. 54, pp. 127-139,December, 1998.P. )iJIbIm, M. Koiio6a, <<PO3B'A3yBaHHA MeTOjOM CKiH-1ieHH4X pi3HH11b HeJiiHiHHHX 3aja-a'i MaaHeTO CTaTHKH HaHIijCTaBi Teopil iHBapiaHTHOrO Ha6JIH)KeHH5 (yHKUiH,>>Teopemu-IHa efleKmpomexHiKa, YKpai-a, BMH.. 54, c. 127-139, 1998.

[3] M. Kotsyuba, "Application of invariant approximations toground's design, " Digests 3rd Conf Mathematicalmodeling in electrical andpower engineering, Ukraine, p.130-131, 1999.

[4] M. Kotsyuba, V. Horyachko, "Application of invariantapproximation of functions to 3-D magnetic fieldanalysis," Digests 4th International Workshop "Comput-ational Problems ofElectrical Engineering Poland, p.135-136, 2004.

[5] M. Kotsyuba-Howykowycz, "Application of methodo-logyof invariant approximations to magnetic circuit analysis, "Digests 7th International Symposium on Signals, Circuits,and Systems, Romania, p. 585-588, 2005.

[20]proposed

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