The Point Ground Electrode in Vicinity of the Semi-Spherical Inhomogenity

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1 SEBIAN JOUNAL OF ELECICAL ENGINEEING Vo., No., Novembe 005, 63-7 he Point Goun Eectoe in Vicinity of the Semi-Spheic Inhomogenity Nen N. Cvetković, Peg. nčić Abstct: Chcteistion of the point goun eectoe pce in the suounings o insie of the semi-spheic eth inhomogenity n fe by ow fequency (LF) cuent using isote ething conucto, is pesente in this ppe. he goun impence (esistnce n ectnce) n potenti istibution on the goun sufce e etemine. Imge theoy fo two-ye semiconucting mei, s we s fo the one point eectoe pce neby o insie of the spheic boy is use uing the nysis. Keywos: Gouning system, Impence, Potenti, Geen s function. Intouction he spheic n semi-spheic inhomogenity in the suounings of the gouning cn be pesente in nysis n soving of iffeent pobems. Fo exmpe, concete fountion of the pi cn be ppoximtey tete s inhomogenity of this kin. Aso, the obtine esuts cn be use fo chcteistion (etemining impence, step votge, touch votge n potenti istibution) of the ine gouning eectoe pce t the bity iection in the suounings o insie of nye inhomogenity, teting eth s semiconucting meium. Combintion of the qusisttiony imge theoy fo twoye semi-conucting mei n fo semi-conucting spheic inhomogenity, ws ppie. he esut of the peviousy escibe nysis is the Geen s function of the point eectoe pce neby o insie of the semi-spheic eth inhomogenity [, ]. heoetic Bckgoun he point goun eectoe hving ius 0 fe by LF cuent I n pce insie n outsie of the semi-spheic semi-conucting inhomogenity of ius is obseve (Fig. -b). he eectoe is pce t the epth h. he eth is so tete s semi-conucting meium. he nom istnce between the Fcuty of Eectonic Eng. of Nis, A. Meveev 4, 8000 Nis, S&M, E-mi: Fcuty of Eectonic Eng. of Nis, A. Meveev 4, 8000 Nis, S&M, E-mi: 63

2 N.N. Cvetković, P.. nčić eectoe n the cente of the semi-spheic inhomogenity is L. he cente of the spheic coointe system coincies with the cente of semi-spheic inhomogenity. he position of the point eectoe is efine by the vues of the spheic coointes = ( h L ) n α = ctn( L h). he conuctivities, pemitivitties n pemebiities of the system shown in Fig. -b e bee with i, ε,, i = ε0 εi μi =μ0 i = 0,,, whee inexes 0, n coespon to the fee spce, semi-conucting goun n semi-spheic inhomogenity, espectivey. he compex conuctivities e: 0 = jωε0 fo fee spce ( 0 = 0 ), = jωε fo semi-conucting goun n = jωε fo semi-spheic inhomogenity, whee ω is the ngu fequency. ) b) Fig. Point goun eectoe outsie () n insie (b) of the semi-spheic inhomogenity.. etemintion of the potenti n gouning impence Poceue of potenti etemintion of the system shown in Fig. -b is bse on combine using of the qusisttiony imge theoy fo two-ye semiconucting mei [] n imge epesenttion of the point eectoe pce ne the semi-conucting spheic boy []. In gene cse, the equivent system cn be fome combining ppiction of imge theoem in pne semiconuctive mio n semi-conuctive semi-spheic inhomogenity. In obtine cse, the conition, >> 0 is stisfie. Afte ppying the imge theoy, the equivent system is fome of point eectoe, n its imge pce t the istnce h bove the bouny sufce, t nom istnce L fom the 64

3 he Point Goun Eectoe in Vicinity of the Semi-Spheic Inhomogenity 65 cente of the spheic coointe system. he equivent imge cuent is I 0 (when gouning eectoe is insie of the inhomogenity) o I 0 (when souce is outsie of the inhomogenity), whee ) ( ) ( = n ) ( ) ( = e the efection coefficients. By tht, since it is 0, >>, it cn be ssume tht, 0 0. Aso, in this cse, fte ppying the imge theoy, the semi-spheic inhomogenity cn be tete s spheic semi-conucting boy. he potenti of the fie point in the goun is obtine s esut of the potenti supeposition of the souce n its imges. his potenti pesents the Geen s function fo the point eectoe pce neby o insie of the semispheic eth inhomogenity (Fig. -b), when 0, >>. It is ssume tht souce n point in which potenti is etemine e pce in the sme te = c ψ -pne of thespheic coointe system. Now, potenti cn be expesse s θ α θ α π θ α θ α π ϕ = I I, n n 4, n n 4 () when eectoe is pce outsie, n θ α θ α π θ α θ α π ϕ = I I, n n 4, n n 4 (b) when eectoe is pce insie of the semi-spheic inhomogenity. By tht e:

4 N.N. Cvetković, P.. nčić n = [ ( α θ, )] = [ ( α θ, )] = [ α θ ] = [ ( α θ. )] he efection n tnsmission coefficients n in expessions (-b) e s foows: = ( ) ( ). = he istnce of the imges fom the cente of the semi-conucting sphee, obtine using the imge theoy fo the spheic semi-conucting boy, is bee with =. In the fme of the pesente nysis, it is ssume tht LF omin is consiee ( f = 50H, ω = 00π/s ). Afte etemining the potenti on the eectoe sufce ϕ = U, it is possibe to obtin the goun eectoe impence tht hs cpcitive chcte, Z = U I = j X. g g g 3 Numeic esuts he esistnce n ectnce of the point gouning goun eectoe hving ius 0 n fe by LF cuent I shown in Fig. (when point gouning is pce outsie of inhomogenity), vesus the istnce L, fo iffeent tio s pmete, e shown in Fig.. he vues of the pmetes e h = 0.5 m, = m, 0 =.5 cm, ε = 0, ε = 3 n = 0.0 S/m. he potenti istibution on the goun sufce fo point gouning shown in Fig., fo iffeent tio s pmete, is shown in Fig. 3. he pmetes vues e L = m, h = m, = m, 0 =.5 cm, ε = 0, ε = 3 n = 0.0S/m. he esistnce n ectnce of the point gouning fom Fig. b (when point gouning is pce insie of the inhomogenity), vesus the istnce L, fo iffeent tio s pmete, e shown in Fig. 4. he vues of the pmetes e h = 0.5 m, =m, 0 =.5 cm, ε = 0, ε = 3 n = 0.0S/m. 66

5 he Point Goun Eectoe in Vicinity of the Semi-Spheic Inhomogenity Fig. esistnce n ectnce of the point goun eectoe (Fig. ). Fig. 3 Nomie potenti istibution on the goun sufce (Fig. ). 67

6 N.N. Cvetković, P.. nčić he potenti istibution on the goun sufce fo point gouning shown in Fig. b, fo iffeent tio s pmete, is shown in Fig. 5. he pmetes vues e L = 0.5 m, h = 0.3 m, =m, 0 =.5 cm, ε = 0, ε = 3 n = 0.0S/m. he esistnce n ectnce of the point gouning pce t the sufce ( h = 0 ) vesus the istnce L, fo iffeent tio s pmete, e pesente in Fig. 6 (when point gouning is pce on the inhomogenity sufce), n Fig. 7-b (when gouning is on the eth sufce). he vues of the pmetes e =m, 0 =.5 cm, ε = 0, ε = 3 n = 0.0 S/m. When the point gouning is pce in the cente of the inhomogenity, it is possibe to obtin its compex impence in the e fom s 0 Z = j X =. () g g g π0 Fig. 4 esistnce n ectnce of the point goun eectoe (Fig. b). 68

7 he Point Goun Eectoe in Vicinity of the Semi-Spheic Inhomogenity Aso in this cse of h = 0, when the point gouning is pce t ge istnce fom inhomogenity ( L >>> ), with incesing L, the compex impence ppoches the imit vue Z g = g j X g =. (3) π0 he compison of esistnce obtine using expessions (b) n (), when point eectoe is in the cente of the inhomogenity, is pesente in be. he vues of the pmetes e =m, 0 =.5 cm, ε = 0, ε = 3 n = 0.0S/m. Goo geement between compe esuts is notbe. he compe esuts fo esistnce obtine using expessions () n (3), (the point eectoe is t ge istnce fom inhomogenity) e given in be. he vues of the pmetes e = m, 0 =.5 cm, ε = 0, ε = 3 n = 0.0S/m. Vey goo geement cn be notice yet fo L =. 5. ) b) Fig. 5 Nomie potenti istibution on the inhomogenity sufce (), n on the eth sufce (b) fo the gouning eectoe shown in Fig. b. 69

8 N.N. Cvetković, P.. nčić Fig. 6 esistnce n ectnce of the point goun eectoe when eectoe is pce on the inhomogenity sufce (Fig. b). Fig. 7 esistnce of the point goun eectoe when eectoe is pce on the eth sufce (Fig. ). 70

9 he Point Goun Eectoe in Vicinity of the Semi-Spheic Inhomogenity pp. vue Fig. 7b ectnce of the point goun eectoe when eectoe is pce on the eth sufce (Fig. ). be esistnce of the point eectoe g (Ω) pce in the inhomogenity cente n vue obtine using exp. () exp. () be esistnce of the point eectoe (Ω) pce t L =. 5 fom the inhomogenity cente, n vue obtine using exp. (3). he potenti istibution on the goun sufce in the vicinity of the point goun eectoe pce insie n outsie of spheic inhomogenity n fe by LF cuent using the isote ething conucto, s we s point eectoe 7 g pp. vue exp. (3) Concusion

10 N.N. Cvetković, P.. nčić compex impence, e etemine in this ppe. he content of this ppe pesents intouctoy nysis fo soving moe compex pobems such s chcteistion of the ine goun eectoe pce in bity iection in the vicinity of the semi-conucting inhomogenity o insie of the inhomogenity teting goun s semi-conucting meium. 5 Acknowegement Authos eicte the ppe to the te pofesso gutin M. Veičkovic (94-004). Fist utho thnks the AA fountion n the echnic Univesity of Imenu, Gemny fo the significnt suppot of iffeent kin in the fmewok of the Joint Poject. 6 efeences [] P.. ncic: A new concept fo ine gouning systems nysis (invite ppe), Fouth Intention Symposium of Appie Eectosttics, PES 96, Nis, Yugosvi, 4. My, 996, Poceeings of Ppes, pp []. M. Veickovic: Geen's function of spheic boy, Euo Eectomgnetics, EUOEM'94, 30. My 4. Juni, 994, Boeux, Fnce, Confeence Poceeings, Hp [3] P.. ncic, L. V. Stefnovic, j.. jojevic: A New Moe of the Vetic Goun o in wo-lye Eth, IEEE ns. on Mgnetics, Vo. 8, No., pp , Mch [4] P.. ncic, L. V. Stefnovic, j.. jojevic: An Impove Line Gouning System Anysis in wo-lye Eth, IEEE ns. on Mgnetics, Vo. 3, No. 5, pp , Septembe [5] P.. ncic, Z. P. Stjic, j.. jojevic, B. S. osic: Anysis of Line Goun Eectoes Pce in Vetic hee - Lye Eth, IEEE ns. on Mgnetics, Vo. 3, No. 3, pp , My

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