A wire electrode inside parallelepipedicaly-shaped ground inhomogeneity: comparison of two solutions.
The pillars realized in practice (parts of the overhead power network or the lightning protection system) necessarily include a corresponding grounding system. Such system is usually formed of a basic star-, rectangularly- or circularly-shaped electrode ,  connected to an iron armature of the concrete pillar foundation (usually of the shape of parallelepiped ), which can be treated as a second part of the grounding system. Official publications as  usually neglect influence of the concrete foundation. One simple procedure for approximate modelling of the described influence was proposed in  and provides reduction of the analysed grounding system, which includes concrete foundation (ground inhomogeneity), into a problem of a grounding system in the homogeneous ground, which can be solved using well known procedures (in this case Method of Moments (MoM)). The approach is based on approximation of parallelepipedical concrete domain with circle cross-section cylinder and analysis of the current density field. Level of the described influence depends on concrete's and ground's electrical parameters, which can have different values depending on the ground's structure, humidity, etc. The described procedure has been applied in -.
In this paper, the resistance of the single wire electrode inside parallelepipedically-shaped domain surrounded by homogenous ground is determined and compared with available results obtained with hybrid boundary elements method (HBEM). This problem was often analysed , . The HBEM was recently proposed in , based on the idea to substitute boundary segments with total charges placed in free space at the centres of boundary segments. This approach includes using of corresponding Green's function for electric scalar potential and point matching method (PMM)  for matching values of potential and boundary condition for normal components of the electric field. It has been applied for electrostatic as well as magnetostatic problems . The results which are used for comparison in this paper are directly adopted from  where the method has been applied for analysis of quasi-stationary grounding systems with complex conductivity of multilayered media.
II. The Procedure for Approximating Single Wire Conductor Inside Parallelepiped
The idea to reduce problem of a vertical conductor (which models armature's conductors system) inside the concrete foundation on the problem of wire electrode having equivalent length and cross-section radius placed in homogeneous ground, resulted with the procedure proposed and in details explained in . The basics of this procedure (with additionally included description of the complex function theory application) are briefly presented in the text that follows.
Firstly, using the complex function theory, the system of vertical conductors having a circular cross-section of radius [r.sub.0] and placed on the circle of radius a, can be replaced by a single wire vertical conductor of circular cross-section of radius [a.sub.C], Fig. 1 . The complex function
[w.bar] = [C.sub.1]1n[z.bar] + [C.sub.2], [z.bar] = r[e.sup.j[theta]], (1)
is applied on analysis of the cage system formed from N conductors placed in linear homogeneous media of electrical conductivity a and having leakage current of density per unit length [I.sub.leak]. Now, the complex potential is
[w.bar] = -([I.sub.leak] / 2[pi][sigma])[N.summation over (n = 1)]1n([z.bar] - [[z.bar].sub.n]) + C. (2)
In previous expression are [[z.bar].sub.n] = a[e.sup.j(n - 1)[alpha]] and [alpha] = 2[pi]/N, while C is constant depending on the referent level of zero potential. Since is
[[z.bar].sup.N] - [a.sup.N] = [N.[product] over (n = 1)][z.bar] - [[z.bar].sub.n], (3)
[w.bar] = -([I.sub.leak]/2[pi][sigma])1n([[z.bar].sup.N] - [a.sup.N]) + C. (4)
The electrical scalar potental is
[phi] = Re([w.bar]) = -[[I.sub.leak]/(4[pi][sigma]N)]1n([r.sup.2N] + [a.sup.2N] -2[(ra).sup.N] cos N[theta]) + C. (5)
Applying previous expression for determining the potential of the single conductor at the point A (Fig. 1) defined with x = a + [r.sub.0], y = 0 = 0, i.e. r = a + [r.sub.0], [theta] = 0, obtains
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
In (6), with [I.sub.tot] = [NI.sub.leak] the density per unit length of total leakage current from the cage system is labeled. Using condition [r.sub.0] << 2a[pi]/N one can write
[(a + [r.sub.0]).sup.N] [approximately equal to] [a.sup.N] + [Na.sup.N - 1][r.sub.0] (7)
and consequently, the potential of the single cage conductors' system is approximately
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
If instead of the cage the single cylindrical electrode of cross-section having radii [a.sub.C] , and leakage current of density per unit length [I.sub.tot] is observed, the following expression can be formed
U = -[[I.sub.tot]/ (2[pi][sigma])] 1n([a.sub.C]) + C. (9)
Comparing expressions (8) and (9) for the equivalent radius of the conductors' system from Fig. 1 obtains 
[a.sub.C] = [a.sup.N][square root of ([Nr.sub.0]/a.]
In the second step, Fig. 2, adopting that the vertical conductor has a semi-spherically shaped end, applying the equation of continuity and determining the electrical scalar potential in the surrounding ground, the resistance of the vertical electrode is obtained as ((5) in )
[[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where [[sigma].sub.1] and [[sigma].sub.C] are electrical conductivities of the ground and the concrete, respectively, and the rest of parameters in (11) can be understood from Fig. 2. Applying the same procedure on a single electrode of equivalent length [l.sub.e] = [K.sub.e][l.sub.c] and of cross-section radius [a.sub.e] = [K.sub.e][a.sub.C] , placed in a homogeneous ground of electrical conductivity [[sigma].sub.1], the electrode's resistance is obtained ((6) in )
[R.sub.ge] = (1/[K.sub.e])(1/2[pi][[sigma].sub.1][l.sub.C])1n([l.sub.C]/[a.sub.C]), (12)
and consequently, the constant K e can be calculated using the expression ((7) in )
[K.sup.-1.sub.e] = [[sigma].sub.1]/[[sigma].sub.C] + (1 - [[sigma].sub.1]/[[sigma].sub.C])[1n([l.sub.C]/b)] / 1n(1 + [1.sub.c]/[a.sub.C]). (13)
The general expression for calculating the electric scalar potential in the vicinity (point defined with field vector [??]) of the single wire electrode of the length [l.sub.e] from Fig. 1 is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)
In previous expression, with I(s ') is labeled unknown longitudinal current distribution along conductor axis (s' coincide with z axis), K([??], s') = (1/[r.sub.1]) + (1/[r.sub.2])is the kernel, while [r.sub.1] and [r.sub.2] are distances from the conductor element, i.e. its image in the flat mirror from the point where the potential is determined, respectively. The antenna model is used and the longitudinal current is assumed in polynomial form having unknown coefficients [I.sub.m], m = 0,1, ..., M 
I(s') = [M.summation over (m = 0)][I.sub.k][(s'/[l.sub.e]).sup.m] (15)
Taking in consideration a fact that quasi-stationary regime is assumed, and after applying partial integration, the expression for calculating the electric scalar potential gets the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
Obviously, there are in total M + 1 unknown coefficients. One equation can be obtained from the condition (Eq. (9c) in ):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], 
where [I.sub.leak](s') is leakage current density per unit length.
The rest M equations are formed matching potential value (16) in the points at the conductors' surface 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)
Since is I(s' = 0) = [I.sub.0] = [I.sub.g], the electrode resistance is
[R.sub.g] = U/[I.sub.g] = U/[I.sub.0]. (19)
III. The Application of Hybrid Boundary Elements Method
The application of the hybrid boundary elements method (HBEM) is presented in Fig. 3. Unknown total charges distribution at the boundary surface total charges are modelled with point charges [Q.sub.tn] n = 1, ... N, placed at the points [r.sub.tn] and [z.sub.tn], n = 1, ... N. (In this case with r = [square root of [chi square] + [y.sup.2] is labeled radial coordinate of the cylindrical coordinate system).
The longitudinal current distribution is assumed in polynomial form (15). Potential of the system is expressed as 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
[k.sub.1n.sup.2] = [4rr.sub.t n] / [(r + [r.sub.t n).sup.2] + [(z + [z.sub.t n]).sup.2], (21)
[k.sub.2n.sup.2] = [4rr.sub.t n] / [(r + [r.sub.t n).sup.2] + [(z + [z.sub.t n]).sup.2] (22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
is a complete elliptic integral of the first kind.
The system of charges and currents in Fig. 3 and potential expression (20) is formed using image theory in the flat mirror and symmetrical features of the structure from Fig. 3. Total M equations (the number of unknown coefficients in leakage current distribution (17)) are obtained matching potential value at the conductors' surface, and N equations (total number of point charges) is formed based on the relation between normal electric field component and surface charge density . Since feeding current can be obtained as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
The electrode resistance can be calculated as
[R.sub.g] = 1/[G.sub.g] = U/[I.sub.g] (25)
IV. Numerical Results
Based on the described models and procedures, the single resistance of wire electrode having length [l.sub.C] = 1m and cross-section radius [a.sub.C] = 1m placed in parallelepiped concrete cylinder of electrical conductivity [[sigma].sub.c] and surrounded by ground of electrical conductivity [[sigma].sub.1] = 0.001S/m is determined. The other parameters values are [b.sub.s] = [b.sub.1] = 0.3m. The results obtained with procedure of approximating wire conductor inside parallelepiped (AWCP) are compared with those ones available from , which are calculated applying HBEM. Parameter M in (15) is M = 6. Normalized values of the conductance obtained using both described procedures are presented in Table I. It can be noticed that matching between results obtained using described procedures increases with decreasing of electrical conductivity of parallelepiped. The graphs obtained based on the values from Table I are shown in Fig. 4.
The resistance of the single wire electrode inside parallelepipedically-shaped domain surrounded with homogeneous ground is determined in the paper. The results obtained applying one recently proposed procedure for approximating wire electrode inside parallelepiped with one equivalent single wire electrode placed in homogeneous ground are presented. In this way obtained values for the resistance of the observed system are compared with recently published results, realized applying hybrid boundary elements method. Agreement of the results improves with decreasing electrical conductivity (i.e. increasing electrical resistivity) of the parallelepipedically-shaped domain. The procedure of approximating wire conductor inside parallelepiped is more "engineering" oriented and simpler for application. Since this kind of a problem is very realistic model of the armature conductors inside concrete pillar foundation, one can conclude that for large values of the concretes' electrical resistivity, the procedure based on application of HBEM gives sufficiently correct results for the resistance of the observed structure.
Manuscript received June 08, 2012; accepted February 16, 2013.
This paper is supported in part by project grant HI44004 (2011-2014) financed by the Ministry of Education and Science of Republic of Serbia.
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N. N. Cvetkovic (1)
(1) University of Nis, Faculty of Electronic Engineering Aleksandra Medvedeva 14, 18000 Nis, Serbia firstname.lastname@example.org
TABLE I. NORMALIZED CONDUCTANCE OF THE WIRE ELECTRODE INSIDE PARALLELEPIPED. [sigma]c(S/m) [G.sub.g]/(4[pi][[sigma].sub.1]lc) AWCP HBEM 5 * [10.sup.-2] 0.2420 0.3798 [10.sup.-2] 0.2229 0.2864 5 * [10.sup.-3] 0.2028 0.2412 [10.sup.-3] 0.1180 0.1172 5 * [10.sup.-4] 0.0774 0.0726 [10-.sup.4] 0.0207 0.0181
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|Publication:||Elektronika ir Elektrotechnika|
|Date:||May 1, 2013|
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