Application of Green's Function to Application of Green's Function - PowerPoint PPT Presentation

Application of Green's Function to Application of Green's Function to Analysis of Grounding Systems Placed in Analysis of Grounding Systems Placed in Nonhomogeneous Nonhomogeneous Soil Nonhomogeneous Soil Nonhomogeneous Soil Soil SPEAKER: dr Nenad Cvetković SPEAKER: dr Nenad Cvetković University of Niš, Faculty of Electronic Engineering, University of Niš, Faculty of Electronic Engineering, Serbia Serbia http://nenadcvetkovic.elfak.rs http://nenadcvetkovic.elfak.rs

CONTENT CONTENT  Semi-spherical ground inhomogeneity  Semi-cylindrical ground inhomogeneity  Semi-cylindrical ground inhomogeneity  Multilayered ground 2

PART I PART I Semi Semi-spherical Semi Semi-spherical spherical spherical inhomogeneity inhomogeneity 3

Analysis of grounding systems in the presence of semi- spherical ground inhomogeneity-CONTENT  Why semispherical domain?  Brief procedure presentation  Green function for sphere  Green function for semi-sphere  Examples 4

Why semispherical domain? PROBLEMS WITH SEMI-SPHERICAL GROUND INHOMOGENITIES Grounding system in the vicinity of vertical container (silage,  reservoir) having semi-spherical bases with a lower one buried in the ground; ground; Influence of large holes in the ground (pond, small lake) filled with  water on grounding systems; Analysis of pillar ground electrode, when concrete found is  approximated with semi-spherical ground inhomogenity. 5

Brief procedure presentation Single wire grounding electrode outside inhomogenity   =0,    ,      ,    ,   =    ,    ,    = 2 r 0 Point current source Point current source   =0,    ,   DODATI SLIKU DODATI SLIKU    ,    ,    =   ,    ,    = 2 r 0 I 6

Single wire grounding electrode outside inhomogenity R I s' 10 ( ) r 1 i   ,  0 =0, 0 0 x y r  r Outside semi Outside semi- -sphere sphere   ,  s , r s s 0 2 a P         r s           ( r ) d I ( r ) G ( r , r ) ( r ) d I ( r ) G ( r , r ) r 1 r   ,  ' 11 11 1 , 11 11 1 0 l l l l ( ) ( )     I s' I s' z z     G ( r , r ) ( r , r ) /d I 11 11 s'   r     r  Inside semi- Inside semi -sphere sphere d I ( r ) I ( r ) d s s le ak                  ( r ) d I ( r ) G ( r , r ) I ( r ) I ( s ) s s 1 s 1 leak l         G ( r , r ) ( r , r ) /d I s 1 s 1 7

Single Single wire grounding electrode wire grounding electrode outside outside inhomogenity inhomogenity INTEGRAL INTEGRAL EQUATION EQUATION    R I s' 10 ( )       U d I ( r ) G ( r , r ) r 1 i 11   ,  0 =0, 0 0 y x l   ,  s , r s 0 2 a   P    r s  d I ( r ) I ( r ) d s r 1 r leak leak     ,  ,  ' 1 , 1 , 1 1 0 0 ( ) I s' z  s'        I ( r ) I ( s ) s leak   I ( s ) ? 8

PROCEDURES FOR SOLVING INTEGRAL EQUATION  Method of Moments  Variational method  Average potential method  Equivalent electrodes Method 9

GREEN’S FUNCTION GREEN’S FUNCTION Point current source outside/inside semi Point current source outside/inside semi- -conducting sphere conducting sphere SOURCE OUTSIDE SPHERE SOURCE INSIDE SPHERE 10

BRIEF OVERWIEV OF THE PREVIOUS RESEARCHES Stratton 1941.  Hannakam, 1971.  Reiss, 1990.  Lindell, 1992.  Sten/Lindell, 1992.  Veličković, 1994.  Rančić, 2006.  11

THREE CONDITONS THREE CONDITONS Stratton solution (exact but it is not in closed form) Laplace or Poisson’s equation Approximate Veličković solution    I T ( r ) (2 of 3 conditions)         0  Boundary condition of potential continuity        ( r r ) ( r r ) s s s s 11 11 s s 1 1 Boundary condition of normal component of conducting current Approximate Rančić solution (2 of 3 conditions)        s 1 11 1  s      r r r r r r s s 12

Stratton Stratton (“S”) (“S”) solution solution - - source outside the sphere source outside the sphere Laplace and Poisson equation                1 1 I ( r r ) ( ) 2 11   11   T      r sin , r r s         2 2 2 r r  2      r r sin r sin 1           1 1 2 s 1   s 1       r sin 0 , r r s       2 2 r r      r r sin Two boundary conditions Two boundary conditions            ( r r ) ( r r ) 1 s ss    s s   1 s   1 s ss   r r r r r r s s Solution    n 1     I 1 1 T R 1 1 r         S T 1 s 1 s    T R P cos  1 s 1 s 1 s n    4 r r 2 r ' n T 2  r       n 1 1 1 1 s      n  I T 1 T r 1 R T R 1 r r           S T 1 s R 1 s s 1 s 1 s 1 s P cos   ss 1 s  n    2 n 1 4 T r T r r r 2 n T 2 r    n 1 1 s 1 1 s 1 2 s 1 s s 13

Stratton (“S”) solution Stratton (“S”) solution - - source source outside outside the sphere the sphere Poisson and Laplace equation           1 1 2 1 s   1 s       r sin 0 , r r s       2 2 r r      r r sin                1 1 I ( r r ) ( ) 2 ss   ss   T      r sin , r r s 2   2       2   r  r    2 r r sin r sin s Two boundary conditions Two boundary conditions            ( r r ) ( r r )    11 s 1 s s   11 s 1 1  s    r r r r r r s s Solution     2 n 1   I 1 r 1 1 T R 1 r    S   T  s     1 s 1 s s  R P cos     11 1 s   n      n 1 4 r r r r 2 n T 2      r r    1 2 1 s 1 n 1   n  I T R   1 1 1 1 r         S T 1 s 1 s    T R P cos   s 1 1 s 1 s n    4 r r ' 2 r ' n T 2  r       n 1 1 1 1 s 14

Stratton Stratton solution solution overwiev overwiev Obtained solving the Poisson, i.e. Laplace partial differential - equation (separating variables method) Boundary conditions for the electrical scalar potential and - normal component of the conducing curent at the boundary of discontinuity (semi-conducting sphere surface) are of discontinuity (semi-conducting sphere surface) are satisfied - Solution includes infinite series needed to be numerically summed 15

Approximate Approximate Veličković Veličković (“V”) (“V”) solution solution - - source outside the sphere source outside the sphere Assumed form of the potential solution I C C   T  1  2  , r r 11 s   4 r r r 1 2 1 C   3   C , r r s 1 4 s r 1 The boundary condition for potential        ( r r ) ( r r ) s s 11 s 1        I 1 r 1 1     T  1 s s   , r r     11 s       4 r r r r       1 1 1 s 2       2 I 1 1   1 s   T 1   , r r s 1   s          4 r r   1 1 1 s 1 s 16