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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:

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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 Soil

SPEAKER: dr Nenad Cvetković SPEAKER: dr Nenad Cvetković University of Niš, University of Niš, Faculty of Electronic Engineering, Faculty of Electronic Engineering, Serbia Serbia http://nenadcvetkovic.elfak.rs http://nenadcvetkovic.elfak.rs

Nonhomogeneous Nonhomogeneous Soil Soil

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CONTENT CONTENT

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

2

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SLIDE 3

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

3

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SLIDE 4

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

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SLIDE 5

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

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Single wire grounding electrode outside inhomogenity

 

=0,    

,  

,      

,

=

,   

  

,

=

Brief procedure presentation DODATI SLIKU DODATI SLIKU

2r0 I  

=0,    

,  

,      

,

=

,   

  

,

=

2r0

Point current source Point current source

6

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SLIDE 7

Single wire grounding electrode outside inhomogenity

rs

z x y

R I s'

10 ( )

r r1 r ' P 2a r1i I s' ( )

 

0=0,

,   

s, s

,   

1, 1

, 

s

r r 

Outside semi Outside semi-

  • sphere

sphere

   

l

r r G r I r ) , ( ) ( d ) (

11 11

   

   

l

r r G r I r ) , ( ) ( d ) (

11 11

   

   

l

r r G r I r ) , ( ) ( d ) (

1 s 1 s

   

I r r r r G /d ) , ( ) , (

1 s 1 s

       

s

r r 

Inside semi Inside semi-

  • sphere

sphere

z

s' I s' ( )

s r I r I     d ) ( ) ( d

ak le

 

s s I r I        ) ( ) (

leak

l

I r r r r G /d ) , ( ) , (

11 11

       

l

7

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SLIDE 8

Single Single wire grounding electrode wire grounding electrode outside

  • utside inhomogenity

inhomogenity INTEGRAL INTEGRAL EQUATION EQUATION

rs

x y

R I s'

10 ( )

r r1 r ' P 2a r1i

 

0=0,

,   

s, s

,   

1, 1

, 

s r I r I     d ) ( ) ( d

leak

 

    

l

r r G r I U ) , ( ) ( d

11

  

z

s' I s' ( )

 

1, 1

, 

leak

? ) ( ) ( ) (

leak

         s I s s I r I 

8

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SLIDE 9

 Method of Moments  Variational method

PROCEDURES FOR SOLVING INTEGRAL EQUATION

 Average potential method  Equivalent electrodes Method

9

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SLIDE 10

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

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SLIDE 11

Stratton 1941.

Hannakam, 1971.

Reiss, 1990.

BRIEF OVERWIEV OF THE PREVIOUS RESEARCHES

Lindell, 1992.

Sten/Lindell, 1992.

Veličković, 1994.

Rančić, 2006.

11

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SLIDE 12

           ) (r IT 

) ( ) (

s 1 s s 11  

     r r r r

THREE CONDITONS THREE CONDITONS Laplace or Poisson’s equation Stratton solution (exact but it is not in closed form) Boundary condition of potential continuity Approximate Veličković solution (2 of 3 conditions)

s 1 s s 11

 

          

s 1 s s s 11 1

r r r r r r

Boundary condition of normal component of conducting current Approximate Rančić solution (2 of 3 conditions)

12

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Stratton Stratton (“S”) (“S”) solution solution -

  • source outside the sphere

source outside the sphere

s 2 1 11 2 11 2 2

, sin ) ( ) ( 2 sin sin 1 1 r r r r r I r r r r r

T

                                     

s 1 s 2 1 s 2 2

, sin sin 1 1 r r r r r r r                             

Laplace and Poisson equation Two boundary conditions

) ( ) (

s s

ss s 1

 

     r r r r

 

          

s ss s s s 1 1

r r r r r r

Two boundary conditions Solution

 

                      

  

cos 2 1 ' 1 2 1 1 4

1 1 s 1 s 1 s 1 s 1 1 s 1 1 T S s 1 n n n

P r r T n r R T r R r T I

   

               

  

cos 2 1 2 1 1 4

1 2 s 1 s 1 s 1 s 1 s s 1 2 s 1 s s 1 s 1 1 1 s s 1 1 T S ss n n n n

P r r r T n R T r R r r r T T R r T T I

13

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Stratton (“S”) solution Stratton (“S”) solution -

  • source

source outside

  • utside the sphere

the sphere

Poisson and Laplace equation Two boundary conditions

s s 1 2 s 1 2 2

, sin sin 1 1 r r r r r r r                             

s 2 s ss 2 ss 2 2

, sin ) ( ) ( 2 sin sin 1 1 r r r r r I r r r r r

T

                                     

) ( ) (

s 1 s s 11  

     r r r r

 

          

s 1 s s s 11 1

r r r r r r

   

                          

   

cos 2 1 2 1 1 1 4

1 1 2 s 1 s 1 s 1 s 1 2 s 1 1 1 T S 11 n n n n s

P r r r T n R T r r r r R r I

 

                      

 

cos 2 1 ' 1 2 ' 1 1 4

1 s 1 s 1 s 1 s 1 1 s 1 1 T S 1 s n n n

P r r T n r R T r R r T I

Two boundary conditions Solution

14

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SLIDE 15

Stratton Stratton solution solution overwiev

  • verwiev
  • 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

  • f discontinuity (semi-conducting sphere surface) are
  • f discontinuity (semi-conducting sphere surface) are

satisfied

  • Solution includes infinite series needed to be numerically

summed

15

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Approximate Approximate Veličković Veličković (“V”) (“V”) solution solution -

  • source outside the sphere

source outside the sphere

The boundary condition for potential

Assumed form of the potential solution

s 2 2 1 1 1 T 11

, 4 r r r C r C r I       

s 4 1 3 1 s

, r r C r C     ) ( ) (

s 1 s s 11  

     r r r r

s 2 s s 1 s 1 1 1 T 11

, 1 1 1 4 r r r r r r r I                              

s 1 s 1 1 s 1 1 1 T 1 s

, 1 1 2 4

s

r r r r I                         

16

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Approximate Approximate Veličković Veličković (“V”) (“V”) solution solution -

  • source outside the sphere

source outside the sphere

The boundary conditions for conducting current

Assumed form of the potential solution

 

s 1 s 2 s s 1 s 1 1 1 T 11

, cos 1 1 1 4 r r P r r A r r r r r I

n n n n

                                     

 

 

s 1 s s 1 s 1 1 s 1 1 1 T 1 s

, cos 1 1 2 4 r r P r r A r r I

n n n n

                                

 

The boundary conditions for conducting current

 

          

s 1 s s s 11 1

r r r r r r

Legendre polynom theory

   

s 2 2 s 1 s 1 1 2 s s 1 s 1 1 1 T V 11

, 2 cos ln 1 1 1 1 4 r r r r r r r r r r r r I                                                  

 

 

s 1 2 s 1 1 1 s 1 s 1 1 s 1 1 1 T V 1 s

, 2 cos ln 1 1 1 2 4

s

r r r r r r r r r I                                             

Solution

17

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SLIDE 18

Approximate Approximate Veličković Veličković (“V”) (“V”) solution solution -

  • source inside the sphere

source inside the sphere

The boundary conditions for potential

Assumed form of the potential solution

s 2 1 1 s 1

, r r r C r C    

s 4 2 3 1 s T ss

, 4 r r C r C r I       

) ( ) (

s ss s s 1  

     r r r r ) ( ) (

s ss s s 1

     r r r r

s 1 s 1 s 1 s 1 s 1 s s T s 1

, 1 1 2 4 r r r r I                          

s s 1 s 1 s 1 s 2 s 1 s 1 s 1 s T ss

, 1 1 1 4 r r r r r r r I                              

18

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Approximate Approximate Veličković Veličković (“V”) (“V”) solution solution -

  • source inside the sphere

source inside the sphere

The boundary conditions for conducting current

Assumed form of the potential solution

 

s 1 s 1 s 1 s 1 s 1 s 1 s s T s 1

, cos 1 1 2 4 r r P r r A r r I

n n n n

                                 

 

 

s 1 s s 1 s 1 s 1 s 2 s 1 s 1 s 1 s T ss

, cos 1 1 1 4 r r P r r A r r r r r I

n n n n

                                       

 

The boundary conditions for conducting current Legendre polynom theory Solution

 

          

s ss s s s 1 1

r r r r r r

s 1 s s 1 s 1 s 1 1 s 1 1 T V s 1

, 2 cos ln 2 1 4 r r r r r r r T R r R r T I                       

s 2 s s 1 s 1 s s 1 2 s 1 s s 1 s 1 1 1 s s 1 1 T V ss

, 2 cos ln 2 1 4 r r r r r r r T R r R r r r T T R r T T I                            

19

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Veličković solution overview Veličković solution overview

  • Closed form solution formed in two steps
  • STEP 1: Solution corresponds to the images in the spherical

mirror and approximately satisfies boundary conditions on the boundary surface

  • STEP 2: Solution is broadened by infinite sums

approximately corresponding to those one occurs in Stratton STEP 2: Solution is broadened by infinite sums approximately corresponding to those one occurs in Stratton solution (Approximately satisfies Laplace’s equation)

  • Now, solution satisfies boundary condition for the normal

component of the conducting current and electrical scalar potential

  • Closed form obtained using Legendre polynomial theory for

summing procedure

20

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Approximate Approximate Rančić Rančić (“R”) (“R”) solution solution -

  • source outside the sphere

source outside the sphere

) ( ) (

 

     r r r r

The boundary conditions for potential

Re-arranged Stratton solution (satysfies Laplace and Poisson equation)  

s 1 1 s 2 1 1 1 T 11

, cos 1 4 r r P r r B r r B r r r C r I

n n n n s s

                       

  

 

s s 1 1 1 1 T 1 s

, cos 1 4 r r P r r A A r D I

n n n n

                       

 

) ( ) (

s 1 s s 11  

     r r r r

s 2 s s 1 s 1 2 1 1 1 T R 11

, 2 cos ln 2 1 1 1 4 r r r r r r r r r T R r r r r R r I

s s

                                    

s 1 s 1 s 1 1 1 1 1 T R 1 s

, 2 cos ln 2 1 4 r r r r r r r T R r R r T I

s s

                         

Legendre polynom theory

21

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SLIDE 22

Approximate Approximate Rančić Rančić (“R”) (“R”) solution solution -

  • source

source inside inside the the sphere sphere

 

    

The boundary conditions for potential

Re-arranged Stratton solution (satysfies Laplace and Poisson equation)

 

s 1 s 2 s 1 1 1 s s 1 1 T ss

, cos 1 1 4 r r P r r A A r r r D r T T I

n n n n

                         

 

 

s 1 1 s s 1 1 1 T s 1

, cos 1 4 r r P r r B r r B r C I

n n n n

                     

  

) ( ) (

s s

ss s 1

 

     r r r r

 

s 1 s 1 s 1 s 1 1 s 1 1 T R s 1

, 2 cos ln 2 1 1 4 , r r r r r r r T R r R r T I r r                            

s 2 s s 1 s 1 s 1 2 s 1 1 s 1 1 1 s s 1 1 T R ss

, 2 cos ln 2 4 , r r r r r r r T R r R r r r T T R r T T I r r

s s s

                              

Legendre polynom theory

22

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SLIDE 23

Rančić solution overview

  • Started form general Stratton solution
  • Terms corresponding to the images in the spherical mirror

having unknown weight coefficients are singled out

  • Remaining part is infinite sums which terms are product of

unknown constant, factored function of radial sphere coordinate and Legendre polynomial of the first kind coordinate and Legendre polynomial of the first kind

  • The constants are determined in a way to completely satisfy

boundary condition for electric scalar potential and approximately satisfy boundary condition for the normal component of total current density.

  • Infinite sums are expressed in closed form

23

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SLIDE 24

COMPARISON OF THE MODELS COMPARISON OF THE MODELS

The point source inside inhomogeneity The point source inside inhomogeneity

Normalized potential Normalized potential distribution distribution

The point source outside inhomogeneity The point source outside inhomogeneity

24

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SLIDE 25

CONCLUSIONS CONCLUSIONS

 Two approximate solutions for Green functions [Rančić,

Veličković] of the point source inside/otside semi-conducting sphere are presented in the paper.

 The approximate solutions are compared with the exact one

[Stratton]

 Based on the numerical experiments, one can conclude that by

using Rančić soulution proposed approximate solution, smaller error related to exact solution [Stratton] in potential evaluation is done, related to Veličković solution

25

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SLIDE 26

Green‘s function for point source in the presence of Green‘s function for point source in the presence of semi semi-

  • shperical ground inhomogeneity

shperical ground inhomogeneity

s

r r 

Potential Potential outside

  • utside inhomogenity

inhomogenity Point source OUTSIDE semi Point source OUTSIDE semi-

  • conducting semi

conducting semi-

  • spherical

spherical inhomogenity inhomogenity

                                                                           r r D r r R T r r r r R r r r r D r r R T r r r r R r r I r r

i i i

2 ' cos ln ' 1 2 ' 1 ' 1 1 2 cos ln ' 1 2 ' 1 ' 1 1 4 d ) , (

2 s 1 s 1 2 s 1 s 1 2 s 1 s 1 2 s 1 s 1 1 11

 

IMAGE IMAGE SOURCE SOURCE

26

slide-27
SLIDE 27

Potential Potential inside inside inhomogenity inhomogenity

s

r r 

Point source OUTSIDE semi-conducting semi-spherical inhomogenity

s

r r 

                                                        ' 2 ' cos ' ln ' 1 2 ' 1 1 ' 2 cos ' ln ' 1 2 ' 1 1 4 d ) , (

1 s 1 s 1 s 1 1 s 1 1 s 1 s 1 s 1 1 s 1 1 1 s

r r r r r R T r R r T r r r r r R T r R r T I r r

i i

 

SOURCE SOURCE IMAGE IMAGE

27

slide-28
SLIDE 28

s

r r 

Potential Potential outside

  • utside inhomogenity

inhomogenity Point source INSIDE semi-conducting semi-spherical inhomogenity

SOURCE SOURCE IMAGE IMAGE

                                                        r r r r r R T r R r T r r r r r R T r R r T I r r

i i s

2 ' cos ' ln 1 2 1 1 2 cos ' ln 1 2 1 1 4 d ) , (

1 s s 1 s 1 s 1 1 s 1 1 s s 1 s 1 s 1 1 s 1 1 1

 

28

slide-29
SLIDE 29

Point source INSIDE Point source INSIDE semi semi-

  • conducting

conducting semi semi-

  • spherical

spherical inhomogenity inhomogenity Potential Potential inside inside inhomogenity inhomogenity

s

r r 

SOURCE SOURCE IMAGE IMAGE

                                                                          D r r D r R T r r r n r R r n D r r D r R T r r r n r R r n I r r

i i s i s

2 ' cos ln 1 2 1 ' 1 1 2 cos ln 1 2 1 ' 4 d ) , (

2 s s 1 s 1 s 2 2 s 1 s 1 1 2 s 1 2 s s 1 s 1 s 2 2 s 1 s 1 1 2 s 1 1 ss

 

29

slide-30
SLIDE 30

Single Single wire ground electrode wire ground electrode outside

  • utside inhomogenity

inhomogenity

14 p =10 R []

EXAMPLES EXAMPLES

2 4 6 8 10 2 4 6 8 10 12 14 Potential distribution ps=10 ps=1 ps=0.1 Re{}/Ig x/rs rs=1m, x0=1.5m h=0.7m, l=2m

1=45

0, r0=25mm

1=0.01S/m, r1=r2=10

N=10 10 20 30 40 50 60 70 80 90 34.5 35.0 35.5 36.0 36.5 37.0 37.5 Resistance of the electrode ps=10 ps=1 ps=0.1 Rg []

1[

0]

rs=1m, x0=1.5m, h=0.7m l=2m, r0=25 mm

1=0.01S/m, r1=r2=10, N=10

Potential distribution Potential distribution Resistance Resistance

30

slide-31
SLIDE 31

Single wire ground electrode outside Single wire ground electrode outside inhomogenity inhomogenity

2.0 Re{I [mA]} 1.0 Real part of lonigitudinal current using segment currents Re{Ilong}/Ig

Leakage current Leakage current Longitudinal current Longitudinal current

0.0 0.2 0.4 0.6 0.8 1.0 0.8 1.0 1.2 1.4 1.6 1.8 Real part of leakage current Re{Ileak [mA]} s/l rs=1m, h=0.7m, l=2m, r0=25mm

1=45

0, 1=0.01 S/m, ps=10

r1=r2=10, x0=1.5m

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Real part of lonigitudinal current s/l using segment currents

  • pol. apr. of 3. degree

rs=1m, h=0.7m, l=2m, r0=25mm

1=45

0, 1=0.01 S/m, ps=10

r1=r2=10, x0=1.5m

31

slide-32
SLIDE 32

3

Resistance of the WGE

p =10

Rg []

Single wire ground electrode inside Single wire ground electrode inside inhomogenity inhomogenity

Resistance Resistance

10 20 30 40 50 60 70 80 90 10

1

10

2

10

3

Resistance of the WGE rs=1m, l=0.5 m, r0=25 mm

1=0.01S/m, r1=r2=10, N=10 ps=10 ps=1 ps=0.1 2[

0]

32

slide-33
SLIDE 33

z x h

l1

I =I +

g 1 2

I

y

 

,    

,  

=0,    

, rs   

s s

, ,

 M r12 l2 I1 I2  r11

r'

2

r'

1

r 2a2 2a 10 20 30 40 50 60 70 80 90 33 34 35 36 37 38 Re{Z11} []

rs=1m, l1=2m, l2=0.9m x0=1.5m, h=0.7m, 1=0.01 S/m,

r1=r2=10, 2=0, r0=25mm

1

ps=100, ps=10 ps=1, ps=0.1, ps=0.01

Two wire ground electrode Two wire ground electrode

Self resistance of external electrode Self resistance of external electrode

2a1

33

10 20 30 40 50 60 70 80 90 6.0 6.2 6.4 6.6 6.8 7.0 7.2

1

rs=1m, l1=2m, l2=0.9m x0=1.5m, h=0.7m, 1=0.01 S/m

r1=r2=10,2=0, r0=25mm

ps=100 ps=10 ps=1 ps=0.1 ps=0.01

Re{Z12} []

10

  • 2

10

  • 1

10 10

1

10

2

10 15 20 25 30 35 40

rs=1m, l1=2m, l2=0.9m x0=1.5m, h=0.7m, 1=0.01 S/m

r1=r2=10, 2=0, r0=25mm

ps

1=90 1=60 1=45 1=30 1=0

Re{Zg} []

Self resistance of external electrode Self resistance of external electrode Mutual resistance Mutual resistance Total resistance Total resistance

slide-34
SLIDE 34

Ring electrode and vertical wire electrode Ring electrode and vertical wire electrode

Ig2

  

s s

, , rK z 2a2 2a4 h y l4

4 3 1

l1

2

x0 2a1 l2

x

2a3 l3

Ig1

rs   

  

, ,  ,  

  

,

16

z

0.5 1.0 1.5 2.0 2.5 3.0 9 10 11 12 13 14 15 16

rs=1m, rK=2m, l2=0.9m, 1=0.01 S/m

r1=r2=10, 2=0

r0=25 mm, rt=8.34mm

ps=100 ps=10 ps=1 ps=0.1 ps=0.01

h[m]

Re{Z11} [] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 h [m] Re{Z12} [] rs=1m, rK=2m, l2=0.9m, 1=0.01 S/m

r1=r2=10, 2=0

r0=25 mm, rt=8.34mm

ps=100 ps=10 ps=1 ps=0.1 ps=0.01 34

Mutual resistance Mutual resistance Ring electrode resistance Ring electrode resistance

slide-35
SLIDE 35

z

y z x

2ak

rs rs sk '

' I s

k k

( )

 

,    

,  

=0,    

,   

s s

, ,

4 6 8 10 12 Rgnor 3 5 p1s=10

System of vertical electrodes inside semi System of vertical electrodes inside semi-

  • sphere

sphere

z

5 10 15 20 2 4 N 1 0.1 0.3 0.5 3

35

Resistance Resistance

slide-36
SLIDE 36

Ig1

z

x y  

,    

,  

=0,    

, rs   

s s

, ,

h l

2

2a2 b 2 Ig2 x0 s'

2

5 6 7 8 9 10 Mutual resistance in [] ps=100 ps=1 rs=1m, l=0.9m, a=1m, b=0.5m, h=0.5m,

r1=r2=10, b=45

0, r0=25mm

Wire electrode and rectangular plate ground electrode Wire electrode and rectangular plate ground electrode

b1 S'

z

b2

36

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2 3 4 Mutual resistance in [ ps=1 ps=0.01 x1-rs

Mutual resistance Mutual resistance

slide-37
SLIDE 37

References…

1.Nenad N. Cvetković, Predrag. D. Rančić, "A Simple Model for a Numerical Determination of Electrical Characteristics of a Pillar Foundation Grounding System", Engineering Analysis with Boundary Elements, Elsevier, Volume 33, pp. 555-560, 2009, ISSN: 0955-7997; DOI: 10.1016/j.enganabound.2008.08.005 2.Nenad N. Cvetković, Predrag D. Rančić, "Influence of foundation on pillar grounding system’s characteristics", COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Group Publishing Limited, Volume 28, Number 2, pp. 471-492, 2009, ISSN: 0332-1649; DOI: 10.1108/03321640910929335 3.Nenad N. Cvetković, Predrag D. Rančić, "The point ground electrode in vicinity of the semi-spherical inhomogenity", Serbian Journal of Electrical Engineering, Volume 2, No. 2, November 2005, pp. 163-172, ISSN 1451-4869, COBISS.SR-ID 111412236. DOI: 10.2298/SJEE0502163C 4.Nenad N. Cvetković, Predrag. D. Rančić, "The Influence of Semi-Spherical Inhomogenity on the Linear Grounding System Characteristics", FACTA UNIVERSITATIS, Series Electronics and Energetics, University

  • f Nis, Serbia, Volume 20, Number 2, pp. 147-161, 2007, DOI:10.2298/FUEE0702147C
  • f Nis, Serbia, Volume 20, Number 2, pp. 147-161, 2007, DOI:10.2298/FUEE0702147C

5.Predrag D. Rančić, Miodrag S. Stojanović, Milica P. Rančić, Nenad N. Cvetković, “A Point Source in the Presence of Spherical Material Inhomogenity: Analysis of Two Approximate Closed Form Solutions for Electrical Scalar Potential”, FACTA UNIVERSITATIS, Series Electronics and Energetics, University of Nis, Serbia, Volume 22, Number 3, pp. 265-284, 2009, DOI:10.2298/FUEE0903265R 6.Nenad N. Cvetković, Predrag D. Rančić, "Conductive Semi-sphere and Two Ring Ground Electrodes as Pillar Foundation Grounding System", Elektrotechnica&Elektronica E+E, The Union of Electronics, Electrical Engineering and Telecommunications, Vol 45, No 1-2, pp.8-11, 2010. ISSN:0861-4717 7.Nenad N. Cvetković, Saša S. Ilić, Dragan D. Vučković, Dejan B. Jovanović, Dejan D. Krstić, "Application of Hybrid Boundary Element Method-Example orf Semispherical Ground inhomogeneity", Serbian Journal of Electrical Engineering, Volume 11, No. 4, December 2014, pp. 617-628.. Printed Version: ISSN 1451 – 4869, Online Version: ISSN 2217 – 7183 UDC: 621.316.99:537.8]:517.544 DOI: 10.2298/SJEE1404617C

37

slide-38
SLIDE 38

…References References… …

  • 8. Nenad N. Cvetković, Predrag D. Rančić, "The point ground electrode in vicinity of the semi-spherical

inhomogenity", Seventh International Conference on Applied Electromagnetics, PES 2005, Niš, Serbia and Montenegro, May 23-25, 2005, CD Proceedings of papers, ISBN: 86-85195-12-8 (abstrakt u Proceedings of extended abstracts, pp. 139-140, ISBN: 86-85195-06-3)

  • 9. Nenad N. Cvetković, Predrag D. Rančić, "Single wire grounding electrode in the presence of semi-spherical

inhomogenity", International PhD Seminar "COMPUTATIONAL ELECTROMAGNETICS AND TECHNICAL APPLICATIONS", Banja Luka, Bosnia and Herzegovina, August 28-September 01, 2006, Proceedings of Full Papers, pp 57-63, ISBN 99938793-5-5, COBISS.BH-ID 92696 10.Nenad N. Cvetković, Predrag D. Rančić, "Influence of the semi-spherical semi-conducting ground inhomogenity on the grounding characteristics", VII International Symposium on Electromagnetic Compatibility-EMC BARCELONA '06, 05-09 Sep. 2006, Proc. of papers, pp 918-923, CD ISBN 84-689- 9438-3 (abstrakt u Book of abstracts, pp. 29) 9438-3 (abstrakt u Book of abstracts, pp. 29) 11.Nenad N. Cvetković, Predrag D. Rančić, "Conductive semi-sphere and linear ground electrode as pillar foundation grounding system", Eight International Conference on Applied Electromagnetics, PES 2007, September 03-05, 2007, Proceedings of papers (CD), Paper O3-6. 3 – 5 September 2007, Nis, Serbia, , CD- proceedings (Session O3-6), 2007, ISBN 978-86-85195-47-0. (abstrakt u Proceedings of extended abstracts, ISBN 978-86-85195-83-9, pp. 43-44) 12.Nenad N. Cvetković, Predrag D. Rančić, "A Simple Model for Numerical Determining of Electrical Characteristics of a Pillar Foundation Grounding System", 16-th International Conference on the Computation of Electromagnetic Fields-COMPUMAG, June 24-28, 2007, Aachen, Germany, Proceedings,

  • pp. 1193-1194.

13.Nenad N. Cvetković, Predrag D. Rančić, "Conductive semi-sphere and two ring ground electrodes as pillar foundation grounding system", 9th International Conference on Applied Electromagnetics, PES 2009, August 31 – September 02, 2009, Niš, Serbia, CD-proceedings (Session O3-2), 2009, ISBN 978-86-85195-84- 6 (abstrakt u Proceedings of extended abstracts, ISBN 978-86-85195-83-9, pp. 43-44.)

38

slide-39
SLIDE 39

…Referenc References es

14.Nenad N. Cvetković, Predrag D. Rančić, "Star-shaped Grounding System in the Vicinity of a Semi- Spherical Inhomogeneity", The 14th International IGTE Symposium, Graz, Austria, 20-22. September 2010, CD Proceedings, pp. 124-129, ISBN: 978-3-85125-133-3. (abstrakt u Abstracts, pp 28.) 15.Nenad N. Cvetković, "Comparison of two different models for a vertical electrode inside a pillar foundation", 10th International Conference on Applied Electromagnetics, PES 2011, September 25– 29, 2011, Niš, Serbia, CD-proceedings (Session O5-2), 2011, ISBN 978-86-6125-042-2 (abstrakt u Proceedings of extended abstracts, ISBN 978-86-6125-035-4, pp. 93-94.) 16.Nenad N. Cvetković, "Modelling of Hemispherical Ground Inhomogeneities", 10th International Conference on Telecomunications in Modern Satellite, Cable and Broadcasting Services-TELSIKS 2011, October 05-08, 2011, Niš, Serbia, Proceeding of papers, Volume 2, pp. 436-439, 2011. 2011, October 05-08, 2011, Niš, Serbia, Proceeding of papers, Volume 2, pp. 436-439, 2011. DOI:10.1109/TELSKS.2011.6143238 17.Dragan D. Vučković, Nenad N. Cvetković, Slavoljub R. Aleksić, Saša S. Ilić, Dragan Tasić, Dejan Krstić, “Application of Hybrid Boundary Element Method on Modelling of Hemispherical Ground inhomogeneity”, International Symposium on Theoretical Electical Engineering, ISTET 2013, Pilsen, Czech Republic, 24-26 June 2013, CD Proceedings, Paper I-7, ISBN: 978-80-261-0246-5, (abstrakt u Proceedings, pp. I-7, I-8 ) 18.Nenad N. Cvetković, Predrag D. Rančić, "Uticaj polusferičnog temelja stuba na električne karakteristike konturnog kružnog lineičnog uzemljivača", Zbornik radova L Konferencije ETRAN- a, Beograd, 6-9. juna 2006. godine, Sveska II, str. 251-254, 2006, ISBN 86-80509-59

39

slide-40
SLIDE 40

PART II PART II Semi Semi-cylindrical cylindrical Semi Semi-cylindrical cylindrical inhomogeneity inhomogeneity

40

slide-41
SLIDE 41

Analysis of grounding systems in the presence of semi- cylindrical ground inhomogeneity-CONTENT

 Why semi-cylindrical domain?  Green function for cylinder  Green function for semi-cylinder  Examples

41

slide-42
SLIDE 42

Why semi-cylindrical domain?...

  • Different kinds of facilities, which can be placed in the

vicinity of the road, necessarily include grounding systems realized according to the corresponding standards. This is also valid for the grounding systems of the lighting poles that are often placed along the highways.

  • Therefore, there is an interest to determine the influence
  • f the road, treated as a ground inhomogeneity.
  • f the road, treated as a ground inhomogeneity.
  • Geometrical

parameters

  • f

the analyzed grounding systems, as well as the used conductivity values of the surrounding ground, and the road (the inhomogeneity), have been taken from the official publications.

42

slide-43
SLIDE 43

…Why semi-cylindrical domain?

  • The used semi-analytical procedure provides possibility for

precise analysis of the influence of a long ground inhomogeneity with parallelepiped shape

  • n

the grounding system characteristics, without a need for applying approximate methods, such as FEM, BEM.

  • It is based on idea of using an estimation method, for

approximation of the road domain by a domain having a approximation of the road domain by a domain having a semi-circular cross section.

  • Further analysis necessary include application of the

quasistationary image theory and Green‘s function for the point source placed inside/outside the cylinder of semi- circular cross section.

43

slide-44
SLIDE 44

Semi Semi-

  • cylindrical domain

cylindrical domain Green’s function for point source Green’s function for point source inside cylindrical domain inside cylindrical domain

Potential inside inhomogenity, r<aeq

eq 21

, a r    

eq 11

; ), ( ) ( ) (

1

a r z z r r

r I

           

Poisson and Laplace equation

 

  

                        

1 2 1 11

)] ( cos[ ) ( ) ( ) ( )] ( cos[ 2 ˆ 1 4

m m m m m

d z z r I r I A m I R R I  Potential outside inhomogenity, , r>aeq

44

... , 2 , 1 , 2 , 1      m

m

 

  

                     

1 2 21

)] ( cos[ ) ( ) ( ) ( )] ( cos[ 2

m m m m m

d z z r K r I B m I

     

B m A m

B A ,

) ( ) ( ) ( ) (

1 2

a K a K a K a K

m m m m A

          

) /( 1 a

B

  

) ( ) ( ) ( ) (

1 2

a K a I a K a I

m m m m

           

slide-45
SLIDE 45

Semi Semi-

  • cylindrical domain

cylindrical domain Green’s function for point source inside inhomogeneity Green’s function for point source inside inhomogeneity

  • J. Ma and F. P. Dawalibi, "Analysis of grounding

systems in soils with cylindrical soil volumes", IEEE Transactions on Power Delivery, vol. 15, no 3, pp. 913-918, 2000; DOI: 10.1109/61.871352

x I 0=0

2

0=0

2

Earth surface

1

I

R 

z

R

M( , , r z

0)

 M( , , x y z )

M’ ( , - ,

x y z

0)

M’ ( , - , r z

0)

aeq

2 , 1 , ,    j i I G

ij ij

Potential outside inhomogenity, r>aeq Potential inside inhomogenity, , r<aeq

... , 2 , 1 , 2 , 1 , )] ( cos[ ) ( ) ( cos cos 1 1 4

1 2 1 11

                                  

 

  

m d z z r I r I A m m I R R R R I

m m m m m m

   

... , 2 , 1 , 2 , 1 , )] ( cos[ ) ( ) ( cos cos

1 2 21

                    

 

  

m d z z r K r I B m m I

m m m m m m

m B m m A m

m m

B A       ,

) ( ) ( ) ( ) (

eq eq eq eq 1 2

a K a K a K a K

m m m m Am

          

) /( 1

eq

a

m

B

  

) ( ) ( ) ( ) (

eq eq eq eq 1 2

a K a I a K a I

m m m m m

           

y

M( , , x y z

0)

45

slide-46
SLIDE 46

Semi Semi-

  • cylindrical domain

cylindrical domain Green’s function for point source Green’s function for point source outside cylindrical domain

  • utside cylindrical domain

Potential inside inhomogenity, r;aeq

2

z

aeq

1

z=0

y

M( , ,

r z

0)

I r

M( , ,

x y z

0)

x

R 

                

 

   2 2 11

)] ( cos[ ) ( ) ( ) ( cos cos d z z r I r I A m m I

m m m m m

eq 12

, a r    

eq 22

), ( ) ( ) (

2

a r z z r r

r I

            

Poisson and Laplace equation

Potential outside inhomogenity, , r:aeq

46 y

  

 2 m

... , 2 , 1 , 2 , 1      m

m

) ( ) ( ) ( ) ( ) ( ) /( ) (

eq eq eq eq 2 1 eq

a K a I a K a I r I a r K A

m m m m m m m

               

                                

 

   2 2 2 12

)] ( cos[ ) ( ) ( ) ( cos cos 1 1 4 d z z r K r I B m m I R R R R I

m m m m m

   

) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) (

eq eq eq eq 2 1 2 1 eq eq

a K a I a K a I r I r K a I a I B

m m m m m m m m m

                            

slide-47
SLIDE 47

Green’s Green’s function for point source function for point source outside semi

  • utside semi-
  • cylindrical domain

cylindrical domain

Potential outside inhomogenity, r>a

Dragan D. Vuckovic, Nenad N. Cvetkovic, Miodrag S. Stojanovic, Ilona Iatcheva, “Approximate model for ground inhomogeneity with rectangular cross-section: application to analysis of grounding systems”, Electrical Engineering Vol. 99, No. 1. 2017, DOI: 10.1007/s00202- 016-0483-1

2 , 1 , ,    j i I G

ij ij

Potential outside inhomogenity, r>aeq Inside inhomogenity, , r<aeq

m D m m C m

m m

D C       ,

... , 2 , 1 , 2 , 1 , )] ( cos[ ) ( ) ( ) ( cos cos

2 2 12

                     

 

  

m d z z r I r I C m m I

m m m m m m

... , 2 , 1 , 2 , 1 , )] ( cos[ ) ( ) ( ) ( cos cos 1 1 4

2 2 2 22

                                     

 

  

m d z z r K r I D m m I R R R R I

m m m m m m

   

) /( ) ( ) (

eq

a r I r K

m m Cm

    

   

 

) ( ) ( ) ( ) ( / ) (

eq eq eq eq 2 1 2

a K a I a K a I r I

m m m m m m

           

                 

2 1 eq eq

1 ) ( ) ( ) ( ) ( r K a I r I a I

m m m m Dm

47

slide-48
SLIDE 48

Application - grounding system of the pillar inside the road

n

ln

2an In

n

(s' ) 1 N

s'n

0=0

z

b Earth surface

x

GS

Ig

n

ln

2an In

n

(s' ) 1 N

s'

n

GS

0=0

z

x

I

g

Earth surface

aint bint bext aext

2b

a

The star-shaped grounding system

  • f the pillar inside the road.

Approximation of the rectangular cross-section using an equivalent semi-circular one.

1

2

b y a

x 2

x y

1

aeq

h

 

  8

2 2 2

eq

b a b a b a a     

Estimation Method

48

slide-49
SLIDE 49

Application- grounding system of the pillar

  • utside the road

h 0=0

2

1

z

x b

lk

2a Ik

k

(s' ) 1 NC y

I

g

Earth surface d

d 0=0

z

x

I

g

Earth surface

The star-shaped grounding system

  • f the pillar outside the road.

Approximation of the rectangular cross-section using an equivalent semi-circular one.

k

2ak

s'

k

Ik

k

(s' ) y a

2

x k

lk

2ak

s'

k

Ik

k

(s' ) 1 NC y

g

1

aeq

h 49

slide-50
SLIDE 50

Application-two pillars inside and outside the road

z

x Earth surface h

n

ln

2an In

n

(s' ) 1 N1

s'

n

GS1 GS2

2

k

lk

2ak

s'

k

Ik

k

(s' ) 1

N2

y

1

aeq

h

50

slide-51
SLIDE 51

MoM application

U  

  

  N k M j mn

k I

I

1 1 g

g g

I U R 

z

x Earth surface

n

ln

2an In

n

(s' ) 1 N1

s'

n

GS1 GS2

... , 2 , 1 , 2 , 1 , )] ( cos[ ) ( ) ( cos cos 1 1 4 1

1 2 1

                                

 

  

m d z z r I r I A m m I R R R R

m m m m m m

   

Integration of function e.g.

m B m m A m

m m

B A       ,

) ( ) ( ) ( ) (

eq eq eq eq 1 2

a K a K a K a K

m m m m Am

          

) /( 1

eq

a

m

B

  

) ( ) ( ) ( ) (

eq eq eq eq 1 2

a K a I a K a I

m m m m m

           

2

x k

lk

2ak

s'

k

Ik

k

(s' ) 1

N2

y

1

aeq

h

51

slide-52
SLIDE 52

Numerical results The pillar outside the road

h 0=0

2 1 z

x b k

lk

2ak

s'

k

Ik

k

(s' ) 1 NC y a I

g

Earth surface d

a

l

2

d

l

1

z y The road x

Normalized resistance

1E-3 0.01 0.1 0.11 0.12 0.13 NC=2, l1=l2=5 m, a1=a2=5 mm, h=0.8 m, a=10 m, b=4.75 m, d=8.88 m

2/1

Rgdnor / 2 6 7 8 9 10 11 12 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 NC=2, l1=l2=5 m, a1=a2=5 mm h=0.8 m, a=10 m, b=4.75 m, d=8.88 m x (m) 2/1=0.1 2/1=0.01 2/1=0.001 2/1=0.0001

/U

11 10 9 8 7 6 0.1 0.2 0.3 0.4 0.5

  • 15
  • 10
  • 5

5 10 15

NC=2, l1=l2=5 m a1=a2=5 mm h=0.8 m, a=10 m b=4.75 m, d =8.88 m

2/1=0.1

/U

z x

Potential distribution at the ground surface

52

slide-53
SLIDE 53

Numerical results The pillar inside the road

2aeq

l

2

l

1

z y The road x 2 1

n

ln

2an I

n n

(s' ) 1 N

s'n

0=0

1

2

z

b y a Earth surface

x

GS

Ig

Normalized resistance Potential distribution at the ground surface

2aeq

10-3 10-2 10-1 100 101 1 2 N=2, l1=l2=10 m, a1=a2=5 mm, h=0.8 m, aeq=11.61m

2/1

Rgdnor / 1

  • 9
  • 6
  • 3

3 6 9 0.2 0.4 0.6 0.8

  • 15
  • 10
  • 5

5 10 15

z x

53

slide-54
SLIDE 54

Numerical results The pillars inside and outside the road

2

z

x k

lk

2ak

s

Ik

k

(s' ) 1

N2

y Earth surface

1

aeq

h

n

ln

2an In

n

(s' ) 1 N1

s'

n

GS1 GS2

Normalized resistance Potential distribution at the ground surface

k

s'

k

54

slide-55
SLIDE 55

REFERENCES

  • D. D. Vučković, N. N. Cvetković, D. D. Krstić, M. S. Stojanović, "Modeling of the Road Influence on

the Grounding System in its Vicinity", The 15th International IGTE Symposium, CD Proceedings, pp. 294-299, Graz, Austria, 16-19 September 2012.  Nenad N. Cvetković, Dragan D. Vučković, Miodrag Stojanović, Dejan Krstić, Dragan Tasić, “Application of a model of the road influence on the lighting pillars’ grounding system”, 11th International Conference on Applied Electromagnetics, PES 2013, September 01–04, 2013, Niš, Serbia, CD-proceedings (Session P1-13), 2013, ISBN 978-86-6125-090-3 (abstract in Proceedings of extended abstracts, ISBN 978-86-6125-088-0, pp. 53-54.)  Nenad N. Cvetković, Dragan D. Vučković, Miodrag Stojanović, Dejan Krstić, Dragan Tasić “The Grounding System of the Pillar on the Road“,11th International Conference on Telecomunications in Grounding System of the Pillar on the Road“,11th International Conference on Telecomunications in Modern Satellite, Cable and Broadcasting Services-TELSIKS 2013, October 16-19, 2013, Niš, Serbia, CD Proceeding of papers, pp. 45-48, 2013. DOI: 10.1109/TELSKS.2013.6704891  Nenad N. Cvetković, Dragan D. Vučković, Miodrag S. Stojanović, Aleksa S. Ristić, “Three-Wire Star- Shaped Grounding Electrode in the Vicinity

  • f

the Semi-Cylindrically Shaped Ground Inhomogeneity”,XLIX International Scientific Conference on Information, Communication and Energy Systems and Technologies – ICEST 2014, June 25 – 27, 2014, Niš, Serbia, Proc. of papers, Volume 2, pp. 360-363, ISBN: 978-86-6125-109-2  Dragan D. Vuckovic, Nenad N. Cvetkovic, Miodrag S. Stojanovic, Ilona Iatcheva, “Approximate model for ground inhomogeneity with rectangular cross-section: application to analysis of grounding systems”, Electrical Engineering Vol. 99, No. 1. 2017, DOI: 10.1007/s00202-016-0483-1

55

slide-56
SLIDE 56

PART III PART III Multilayered soil Multilayered soil Multilayered soil Multilayered soil

56

slide-57
SLIDE 57

Analysis of grounding systems in multilayered soil- CONTENT

 Description of the problem  Green function  Examples

57

slide-58
SLIDE 58

Multilayered soil

  • Great number of publications dealing with procedures for

analyzing grounding systems in nonhomogeneous ground approximated by homogeneous layers has been published

  • One procedure for modeling ground of arbitrary specific

conductivity distribution with multilayered ground will be presented and applied.

  • The procedure includes using of electric potential Green
  • The procedure includes using of electric potential Green

functions (response to unit excitation) for point source in multilayered media obtained by solving Poisson’s i.e. Laplace’s equation for electric scalar potential and Method of Moments.

  • Application is illustrated on the problem of vertical ground

electrode in three-layer ground.

  • The results are compared with those ones obtained using

COMSOL program package

58

slide-59
SLIDE 59

General Green function General Green function

59

Point current source in multilayered ground

slide-60
SLIDE 60

General Green function General Green function

Poisson’s and Laplace’s equation

   

              

 

     1

, , d ) ( , , , d ) (

m kz m kz m kz n kz n

H z z m n k k kr J e B e A m n N n k k kr J e B e A

                    m n H z r r I m n

m

), ( ) ( 2 ,

General solution

60

 

                                        

 

    1

, d ) ( 4 4 , , d ) (

m kz kH m m kz kH m m m m m

z z H k k kr J e e k I B e e k I A H z z m n k k kr J e B e A

       

lim 1 , , 1 , ,

1

                     

       z n n n n n n

N n z z z z z z N n z z z z

N n B A

n n

,... 2 , 1 , , , 

Equations system

Unknown coefficients

slide-61
SLIDE 61

61

Ground electrode in three-layered ground

slide-62
SLIDE 62

Point source in layer 2

   

                                                        

    

         

z h k k kr J e B H z h k k kr J e B e A h z H k k kr J e e k I B e e k I A h z k k kr J e B e A z k k kr J e A

kz kz kz kz kH kz kH kz kz kz 1 2 2 2 2 2 2 2 1 1 1

, d ) ( , d ) ( , d ) ( 4 4 , d ) ( , d ) (

62

Green function

     

z h k k kr J e B

kz 2 3

, d ) (

 

     

 

       

 

     

 

      

 

              2

1 2 2 1 2 2 1 2 2 1 2 2 2 2 3 2 1 2 2 1 2 2 1 2 2 2 2 2 2 2 1 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 1 2 2 1 2 2 2 2 1 1 2 1 2 2 1 2 2 1 2 2 2 2

1 1 2 4 1 1 4 , 1 4 1 ) 1 ( 4 1 ) 1 ( 2

1 1 1 2 1 2 1 1 2 1 1 1 2 1 2 3 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 2 2 1

t t e e t e e t e e e t e e e k I B t t e e t e e t e e t e e k I B t t e e t e e e t e t e k I A t t e e t e e t e t e e k I B A t t e e t e e t e t e e k I A

h k h k h k kh h k kh kH h k h k H h k h k h k h k kh h k kH H h k H k h k h k h k kh kH H h k h k h k h k h k kh kH H h k h k h k h k h k kh kH H h k h k

                                    

    

3 , 2 , 1 , ,

2

   i I G

i i

slide-63
SLIDE 63

Point source in layer 3

     

                        

   

       2 3 3 2 1 2 2 1 1 1

, d ) ( , d ) ( , d ) ( , d ) ( H z h k k kr J e B e A h z h k k kr J e B e A h z k k kr J e B e A z k k kr J e A

kz kz kz kz kz kz kz

63

Green function

3 , 2 , 1 , ,

2

   i I G

i i

 

                 

 

  3 3 2 3 3

, d ) ( 4 z H k k kr J e e k I B

kz kH

 

   

 

                       

3 2 3 2 2 2 1 2 1 1 2 1 2 2 2 1 2 2 1 2 2 1 2 2 ) 2 ( 3 3 3 3 2 1 2 2 2 1 2 ) 2 2 ( 2 1 2 3 2 2 1 2 2 2 1 2 ) 2 ( 2 1 2 3 2 2 1 2 2 2 1 2 ) 2 ( 2 1 2 3 1 1 2 1 2 2 2 1 2 ) 2 ( 2 1 2 3

, , 1 1 4 , 4 1 ) 1 ( 1 4 , 1 ) 1 ( 4 1 ) 1 ( 1 4 1 ) 1 ( 1 2

1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1

                                                    

      

t t t t e e e t e e t t e t e e e k I B e k I A t t e e e t e e t t e k I B t t e e e t e e t t e k I A t t e e e t e e t t e k I B A t t e e e t e e t t e k I A

h k H k h k h k h k kh h k h k H h k kH h k H k h k h k H h h k h k h k H k h k h k H h k h k h k H k h k h k H h k h k h k H k h k h k H h k h k

slide-64
SLIDE 64

MoM application

       

   

 

       

l H h h H l H h h H

dz G h H I dz G H h I U dz G h H I dz G H h I U

2 2 2 1 2 2 2 1

33 2 3 3 32 1 2 2 23 2 3 3 22 1 2 2

3 2 g

I I I  

g

I U R /

g 

64

slide-65
SLIDE 65

Numerical results

TABLE I Electrode resistance versus the value of specific conductivity of the layers for parameters values h1=0.2 m, h2=2m, H1=1 m, l=3 m and a=0.035 m

65

slide-66
SLIDE 66

Numerical results

Potential distribution at the ground surface

66

slide-67
SLIDE 67

Numerical results

Current density lines and potential in the electrode vicinity

67

slide-68
SLIDE 68

REFERENCE

Dejan Jovanović, Nenad N. Cvetković, Miodrag Stojanović, Dragan Tasić, Ilona Iacheva, “Modeling nonhomogeneous ground as multi-layered media”, The 17th International IGTE Symposium, Graz, Austria 18-21 September 2016, CD Proceedings, pp 21-25. ISBN: 978-3-85125-560-7

68

slide-69
SLIDE 69

Application of Green's Function to Analysis of Grounding Systems Placed in Nonhomogeneous Soil

SPEAKER: dr Nenad Cvetković University of Niš, Faculty of Electronic Engineering, Serbia http://nenadcvetkovic.elfak.rs

Nonhomogeneous Soil