3050 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-23, NO. 5, SEPTEMBER 1987

GENERALISED SOLUTION OF TEE FIELD DUE TO A BURIED DIPOLE INSIDE A CONDUCTING MEDIUM BY USING T-R METHOD

C.S.Antonopoulos E.E.Kriezis University of Thessaloniki

Thessaloniki-Greece

Abstract. In this work an attempt is made for the cal- culation of the electromagnetic field inside a condu- cting material by using the T-R method. As an excita- tion a current source is considered which is buried within the conducting medium. The T-S2 method provides flexibility in comparison with the classical method of the magnetic vector potential. For the numerical evalu- ation of integral equations of the method the boundary element method is used.

INTRODUCTION

The need for solving more complicated electromagnetic field problems gave rise to more sofisticated methods. The classical method in which the magnetic vector po- tential 7i is used for solving three-dimensional pro- blems is less atractive because the vector components are coupled and the magnetic and electric interface conditions are given with greater complexity and have to be computed throughout the region of interest. The method T-R which uses a generalized R vector expressed by the electric vector potential T and the scalar ma- gnetic potential R is a flexible method controling the coupled conditions [ 1 1. In previous works [2], [3],[4] the problem of calcula- ting the electromagnetic field due to current sources embedded in a conducting medium was examined for the case of horizontal or vertical dipoles. In this work the current dipole is placed in an arbitrary position and an attemp is made of giving a generalized solution for the determination of the field in this case. The problem is not limited to the power frequencies and it gives a description of the propagation phenomenon in- side the conducting medium. For the numerical treatment of the integral equations of the analytical formulation of the problem the bound- ary elements technique is applied [ 51.

DEVELOPMENT OF EQUATIONS

A current dipole is embedded in a conducting half-space medium at a distance h from the boundary and it is pla- ced in itn arbitrary position (Fig. 1).

Fig. 1. Geometrical configuration

The excitation is a current source, which diffuses a current density

Js = I6(r)e ro jut - ( 1 )

or

Considering the generalised A as a combination of T and R, it is written

A = T-on ( 3 )

Reffering to the basic expression of the magnetic vec- tor potential V d = B , an analogous expression for the electric vector potential is written OxT=J and the fol- lowing equation are formed

where K = jwpo-w LIE

The use of a Green function method needs the definition of the equation

2

V G-K G = -b(x’-x,y’-y,z’-z)

where G = = 4 m the Green function for air,

r=[(x’-x) 2 t(y‘-y) 2 + ( z ’ - z ) ~ ] ~ ’ ~ , (x,y,z) is the field point and (x’,y’,z’) is the source point. Applying Green’s integral theorem, the above Helmholtz equations turn to the following integral equations

2 2

-KT

0018-9464/87/0900-3050$01.00@1987 IEEE

305 1

R(x,y,z)= I (a - - G %)as a~ an s az ( 9 )

Considering eqs. (2) and (8) the components of T on si- de 2 of the dividing surface ss' are written:

T (x,y,z) = (case - -sinesin@ - aG aG X aY

I (T - G -)ds aTX s X az az (IO)

T (x,y,z) = (cos$sin8 - cos8 %) - aG aG Y

TZ(x,y,z) = (singsin$ x - cos@sin8 -) - aG aG ax

(T - - G >)ds aG aT s z az az (12)

Fqs. (9),(10),(11) and (12) form a system of integral equations. A similar system of integral equations is written for side 1 of the dividing surface ss ' , which belongs to, the air. The two sets of integral equations, which are formed in the two half-spases 1 and 2 have the following 16 m- known quantities

Tk,i (~=x,y,z and i=1,2)

aT,,i

9 (i=q ,2)

az (~=x,y,z and i=1,2)

ani az - (i=I ,2)

The above mentioned two sets of integral equations provide eight equations. Three equations are formed by the condition R=-VQ for the half-space 1, in which P O

and one equation is given by using the assumption of the continuity of Q on the surface ss', as an analog of the continuity of 3. For the next two equations the boundary conditions on the surface ss' are applied. These conditions are given by the continuity of the tangential component of the vector E. For the last two equations the gauge condition of eq. (6) and the continuity of the normal components of B on the surface ss' are used.

For the numerical treatment of the analytical equations the boundary element technique is applied [ 51, [6]. Ac- cording to the method the surface ss'. is divided into subsections f d ky ccm&ric circles hxing as mter -be

point 0' and radii starting also from 0'. The following numerical equations are written.

Tk m=2%,i-2 (Tk,i,k aG1,R az - 1 , 1 m

-G aTk,i,R az )As

i # m

where k=x,y,z, i=1,2 and m.R integers and

Lz,2 = (sinOsin@ aG - cosbsin8 -1 aG aY (19)

From eq. (13) and the conditions R,=-Vn, TI =O and the continuity conditions as well as the assumption =Q 1 2 on the surface ss', it is proved that T =T =O on the

surface ss'.

After this result the quantities aTx2/az, aT ./az are calculated. For the calculation of the remaining unknowns from eqs. (13) and (14) a matrix is fromed with (3) ele- ments . Using this matrix formulation the unknowns

x2 Y2

Y2

2

dTZ ani T Z , 2 , e , .Q . - i=l,2 are calculated. l'az

For the evaluation of the numerical results a condu- cting medium with o=IO-~ m-l and E ~ = I is taken. The different quantities are computed by considering the frequencies fl=50Hz and f2=50KHz. In fig. 2 the varia- tion of the component HZ (versus x) is presented. Fig. 3 shows the component Hz versus y. In fig. 4 the varia- tion of the component H (versus x) is presented. The component Hx is smaller than H because the orientation of the dipole favors the component Hy.

Y Y

3052

1 50KHz\

\ 10 DO 60 '70 00 110 130 150 x ( m )

Fig. 2. Hz component versus x for 5OHz and 5OKHz. y=l Om, z=o .

Fig. 3. Hi component versus y for 50Hz and 50KHz. x=z=o.

CONCLUSION

The application of the T-Q method has the capability of solving ?-dimensional problems in a tractable way in comparison with classical analysis. The use of techni- ques for the numerical treatment of integral equations does not provide serious problems. The analyses of this work are extented to problems of propagation through media with limited conductivity. The possibility of strengthening a component depends on the apriori choice of the orientation of the dipole.

- I IO

IO 40 60 80 100 120 140 X(mjb

Fig. 4. H component versus x, for f=SOKHz. y=z=O. Y

RF;FERENcEs

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[2] W. Barnes, K. Davey. "A boundary integral formula-

1

tion of generated sources L&&E'. in conducting nte- dial1 IEEE Southeastcon, Destin,-FL, pp. 88-91, April 5 9 1982.

K. Davey, W. Barnes. "Three dimensional eddy cur- rent problems using the T-R method and Fredholm integral equations". IEEE Tran. Magn. Vol. MAG 19, no 2, pp. 120-125, Mar. 1983.

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C. W. Trowbridge. "Applications of integral equa- tion methods to the numerical solution of magneto- static and eddy-current problems". Chari and Silve- ster (Eds) John Wiley 1980, pp. 191-213.

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[7] M. D. Greenberg. "Application of Green functions in science and engineering". Prentice Hall, Ehgewood Cliffs N. J..