Electromagnetic modeling of a lightning rod

Transcription

1 Boundry lements nd Other Mesh Reducton Methods XXIX 79 lectromgnetc modelng of lghtnng rod D. Poljk, M. Brkc, D. Kosor 3, C. A. Brebb 4 & V. Murko 5 Deprtment of lectroncs, Unversty of Splt, Crot Techncl Deprtment, Ar Trffc Center Pul, Crot 3 Techncl Deprtment, Ar Trffc Center Splt, Crot 4 Wessex Insttute of Technology, UK 5 Iskr Zscte, Sloven Abstrct Ths pper dels wth frequency domn nlyss of lghtnng rod usng the ntenn theory model. The lghtnng rod struck by lghtnng s represented by strght thn wre ntenn excted by n equvlent current source. The current nduced long lghtnng rod due to drect lghtnng strke s determned by solvng the homogeneous ntegro-dfferentl equton of the Pocklngton type. Once obtnng the current dstrbuton long the rod provdes the clculton of the chrge nduced long the rod nd relted rrdted electrc feld. The correspondng Pocklngton equton nd feld ntegrl reltonshps re hndled v the Glerkn-Bubnov scheme of the Indrect Boundry lement Method (GB- IBM. Introducton ghtnng flsh surges re common sources of electromgnetc nterferences (MI nduced on electrcl nd electronc systems nd equpment, thus producng mny undesred effects, or even mlfuncton of systems nd equpment [ 3]. The purpose of lghtnng protecton system (PS n terms of lghtnng rod s to cpture drect lghtnng strke. The mn prmeter of PS s ts effcency whch mesure s relted to the probblty of drect strke to PS nsted of strke to the object wthn the protected volume defned by the protecton one. The protecton one s consdered to be substntlly mmune to lghtnng strke due to the r termnl,.e. the lghtnng rod. The protecton one round the rod cn be ssessed by pplyng the full wve nlyss []. In prtculr, the protecton one cn be determned f the electrc feld n the vcnty of the rod s known [4, 5]. WIT Trnsctons on Modellng nd Smulton, Vol 44, 007 WIT Press ISSN X (on-lne do:0.495/b0707

2 80 Boundry lements nd Other Mesh Reducton Methods XXIX Ths pper frst dels wth the clculton of electrc feld round the rod usng the ntenn theory nd boundry element nlyss. The ntenn model of the lghtnng rod used n ths work s bsed on the homogeneous Pocklngton ntegrodfferentl equton [, 3] by whch the current dstrbuton long the rod s governed. The Pocklngton equton s solved v the Glerkn-Bubnov vrnt of the ndrect Boundry lement Method (GB-IBM [6]. Once the nduced current long the rod s obtned one my clculte the relted nduced chrge long the rod feturng the contnuty equton nd the rrdted electrc feld from the correspondng ntegrl formuls usng BM formlsm. quvlent ntenn model of the lghtnng rod The geometry of the problem s shown n Fg.. The lghtnng rod of length nd rdus, sts vertclly on perfectly conductng (PC ground. Accordng to the mge theory the equvlent representton n terms of strght wre ntenn s presented n Fg.. The wre s ssumed to be perfectly conductng nd ts dmensons stsfy the thn wre pproxmton (TWA condtons [, 3]. The clculton of the nduced current long the lghtnng rod due to the lghtnng return stroke s crred out by usng the end-fed thn wre model [ 3], Fg... Integrl equton for current dstrbuton long the rod The homogeneous Pocklngton ntegro-dfferentl equton for the current dstrbuton long the lghtnng rod cn be derved by expressng the electrc feld n terms of the mgnetc vector potentl nd by stsfyng certn contnuty condtons for the tngentl feld components t the PC cylnder surfce [5]. Fgure : ghtnng rod. WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

3 Boundry lements nd Other Mesh Reducton Methods XXIX 8 I g d' I ( σ GROU N D PAN IMAG x y I g Fgure : quvlent ntenn model of lghtnng rod. The electrc feld vector expressed n terms of mgnetc vector potentl [6], s gven by: jωµε 0 ( A jωa ( where k s the phse constnt of free spce: k ω µ 0ε ( 0 whle ε nd 0 µ 0 denotes the permtvty nd permeblty of the free spce. Due to rottonl symmetry of the problem the rdted electrc feld does not depend on muth vrble Φ nd the electrc feld components re gven by: ρ jωµε 0 A ρ (3 jωµε 0 A jωa (4 The vector potentl -component s gven by the prtculr ntegrl [6]: WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

4 8 Boundry lements nd Other Mesh Reducton Methods XXIX µ A g( x,, ' I( ' d' (5 4 π where I(' s the unknown current dstrbuton long the rod, g(x,,' s the free spce Green functon of the form: g jkr e, (6 R ( x, ' nd R s the dstnce from the source pont on the rod to the rbtrry observton pont n free spce. The totl tngentl electrc feld on the PC wre surfce (ρ vnshes,.e. the nterfce condton s gven by: where exc s the exctton functon nd exc sct (, + (, 0 (7 sct s the relted scttered feld. Combnng the reltons (4 to (7 results n the Pocklngton ntegro-dfferentl equton for the unknown current: πωε 0 ( exc + k g, ' I( ' d' j4 (8 where g s the ntegrl equton kernel: g jkr e, (9 R ( x, ' R s the dstnce from the source pont n the wre xs to the rbtrry observton pont t the wre surfce: R ( - ' + (0 As the exctton functon s not vlble n the form of electrc feld,.e. the equvlent ntenn s nether drven by voltge source, nor excted by plne wve, the left-hnd sde of the equton (8 vnshes: πωε 0 ( + k g, ' I( ' d' 0 j4 ( WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

5 Boundry lements nd Other Mesh Reducton Methods XXIX 83 As t s well-known from the ntenn theory the ntegro-dfferentl equton kernel becomes qussngulr [6] due to the presence of the second-order dfferentl opertor. Ths problem cn be overcome by pplyng the so-clled wek formulton of the problem nd Glerkn Bubnov ndrect Boundry lement Method (GB-IBM. Thus, utlng the property of the kernel g (, ' g (, ' ' the lterntve form of the ntegro-dfferentl equton s obtned: ( j4πω ε 0 - ( g (, ' I ' ' d' + k - I ( ' g (, ' d' 0 (3 Solvng the Pocklngton equton the ntenn current s obtned. The mthemtcl detls on the BM soluton of equton (3 re presented n Appendx A.. The boundry (end condtons The equvlent ntenn s excted by n del current genertor wth one termnl connected to the ntenn nd the other one grounded n the remote pont n the spce. Ths current source cn be ncluded nto the ntegrl equton scheme through the forced boundry condton ppled t the top of rod. In ccordnce to the mge theory the mged current source s lso njected on the top of the mge rod, s shown n Fg.. Therefore, the ppled symmetrc boundry condton cn be wrtten, s follows: I( I( I (4 g where I g denotes the equvlent current source. The proposed model cn be further upgrded by tkng nto ccount the nfluence of the lossy erth [6, 7], nd by ncludng the lghtnng chnnel ttched to the structure []..3 The nduced chrge long the rod A deeper nsght nto the lghtnng phenomen nd behvour of PS s provded by the ssessment of the chrge dstrbuton nduced long the cylnder rod struck by lghtnng. The lner chrge densty long the rod cn be redly computed from the contnuty equton [7], s follows: where I( s the current dstrbuton long the rod. di( q ( jω d (5 WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

6 84 Boundry lements nd Other Mesh Reducton Methods XXIX.4 The electrc feld ntegrl formuls The electrc feld rrdted by the equvlent ntenn representng the lghtnng rod cn be determned knowng the current dstrbuton nduced long the rod. The rdl (norml feld component cn be determned nsertng the expresson for the mgnetc vector potentl (5 nto equton (3: ρ (, ', ρ g I( ' d' (6 j4πω ε ρ Performng the ntegrton by prts equton (6 becomes: 0 - ρ ( ' g(, ', ρ I d' j4πω (7 ε 0 ' ρ - The -component of the electrc feld s defned by equtons (4 nd (5,.e.: πωε 0 ( ρ + k g0, ', I( ' d' j4 (8 nd fter ntegrton by prts t follows: ( g(, ', ρ ' I ' + d k ( ( I ' g, ', ρ d' (9 j4πω ε 0 ' - - The ntegrls n expressons (7 to (9 contn qus-sngulr kernel due to the presence of dfferentl opertor [6]. Ths qus-sngulrty cn be effcently treted by the boundry element/fnte dfferences pproch [6, 7]. The mthemtcl detls on computton of feld components re gven n Appendx B. 3 Numercl results Frst computtonl exmple s relted to the sngle lghtnng rod of length 0m nd rdus 0.09m. Fgures 3 nd 4 show the nduced current nd chrge long the rod excted by the unt current source t frequency fkh. Fgure 5 shows the tngentl nd the norml component of the rrdted feld ρ0m wy from the rod. Fgure 6 nd 7 show the nduced current nd chrge, respectvely, long the rod wth the sme dmensons excted by the unt current source t the frequency fmh, whle Fg 8 shows the relted tngentl nd norml feld component. The lst set of Fgs s relted to the lghtnng rod protectng the rdr ntenn system of length 7.5m, rdus 0.5m excted by the unt current source t frequency fmh. Fgs 9 nd 0 show the norml nd xl feld component, respectvely. WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

7 Boundry lements nd Other Mesh Reducton Methods XXIX 85 Fgure 3: Rel, mgnry nd bsolute vlue of the nduced current (0m, 0.09m, fkh. Fgure 4: Rel, mgnry nd bsolute vlue of the nduced chrge (0m, 0.09m, fkh. Fgure 5: Absolute vlue of the tngentl nd norml feld components (0m, 0.09m, fkh. WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

8 86 Boundry lements nd Other Mesh Reducton Methods XXIX Fgure 6: Rel, mgnry nd bsolute vlue of the nduced current (0m, 0.09m, fmh. Fgure 7: Rel, mgnry nd bsolute vlue of the nduced chrge (0m, 0.09m, fmh. Fgure 8: Absolute vlue of the tngentl nd norml feld components (0m, 0.09m, fmh. WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

9 Boundry lements nd Other Mesh Reducton Methods XXIX 87 Fgure 9: Absolute vlue of the tngentl feld component for dfferent dstnces from the rod (7.5m, 0.5m, fmh. Fgure 0: Absolute vlue of the norml feld component for dfferent dstnces from the rod (7.5m, 0.5m, fmh. 4 Concludng remrks The frequency domn nlyss of the sngle lghtnng rod representng the smple lghtnng protecton system (PS s undertken n ths work. The full wve model s bsed on the wre ntenn theory. The lghtnng nduced current long the rod s obtned s the BM soluton of the correspondng homogeneous ntegro-dfferentl equton of the Pocklngton type. Hvng obtned the current dstrbuton t s possble to compute the nduced chrge long the rod nd the relted rrdted electrc feld. Further extenson of the present nlyss wll nvolve the tretment of complex lghtnng protecton systems consstng of conductors n vertcl nd horontl rrngement. A prtculr feture of the future work wll be relted to the drect tme domn nlyss of the problem. WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

10 88 Boundry lements nd Other Mesh Reducton Methods XXIX Appendx A: BM soluton of homogeneous Pocklngton equton The Pocklngton ntegro-dfferentl equton (3 s numerclly hndled by mens of the Glerkn-Bubnov scheme of the ndrect Boundry lement Method [6]. The opertor form of equton (3, cn be symbolclly wrtten s: K ( I 0 (A where K s lner opertor nd I s the unknown current to be determned. The unknown current s expressed by the sum of n lnerly ndependent bss functons {f } wth unknown complex coeffcents I,.e.: n I I n I f (A Applyng the weghted resdul pproch nd choosng the test functons to be the sme s bss functons (Glerkn-Bubnov procedure the opertor equton (A s trnsformed nto system of lgebrc equtons: n ( f f d 0 j,,..., n I K j (A3 0 Performng certn mthemtcl mnpultons the followng mtrx equton s obtned: M [ Z] j{ I} 0, j,,..., M (A4 where the vector {I} contns the unknown coeffcents, nd M s the totl number of boundry elements. The mutul mpednce mtrx [Z] j representng the ntercton of the -th source boundry element wth the j-th observton element s gven by: [ Z ] j j j 4πωε + k j+ + T { D} j { D} j+ + T { f } j { f } j g (, ' d' d + g (, ' d' d (A5 WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

11 Boundry lements nd Other Mesh Reducton Methods XXIX 89 where, +, j nd j+ re the coordntes of -th nd j-th wre segment, respectvely. A lner pproxmton over ech boundry element hs been used n ths pper s ths choce hd lredy been shown to provde ccurte nd stble results [6, 7]. Mtrces {f} nd {f, } contn the lner shpe functons: + - f ( - f ( (A6 + whle {D} nd {D, } contn ther dervtves: ( ' I I+ I (A7 Appendx B: BM evluton of the feld ntegrls The rdl feld component, defned by the ntegrl: ρ ( ' g(, ', ρ I d' j4πω (B ε 0 ' ρ - cn be evluted usng the BM formlsm,.e. the current long the segment cn be wrtten s: nd, therefore, t follows: I ( I f ( + I f ( (B + + x j4πωε M 0 I I + + ( x,, ' g x d' (B3 Furthermore, to overcome the qussngulrty problem, the kernel s pproxmted v centrl fnte dfference formul: f ( xy, f( x+ xy, f( x xy, x x (B4 nd the fnl expresson for the rdl electrc feld s then: ρ j4πωε M 0 + I+ I [ G(, ', ρ + ρ G(, ', ρ ] d' (B5 ρ where ρ denotes the fnte dfference step. WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press

12 90 Boundry lements nd Other Mesh Reducton Methods XXIX The tngentl feld component s gven by: j4πω ε 0 - ( G(, ', ρ I ' ' d' + k - I ( ' G(, ', ρ d' (B6 Usng lner nterpolton for current over the segment yelds: (, ', ρ + I G + I d ' + M j4 + πωε 0 k I f( + I+ f+ ( G(, ', ρ d' (B7 nd pproxmtng the kernel wth fnte dfferences the fnl formul for the xl feld component s: + G( + /, ', ρ G( /, ', ρ d' M I I j4πωε k I f( + I+ f+ ( G(, ', ρ d' (B8 References [] F.C. Yng, K.S.H. ee, Nturl Frequences of Rod wth ghtnng Return Stroke, pp , n ghtnng lectromgnetcs, R. Grdner, (d CRC Press 990. [] D. Poljk, B. Jjc, "ghtnng Induced Current on Metllc Rod- Frequency Domn Anlyss " ICCOM 97, pp Dubrovnk, Crot, Oct [3] D. Poljk, B. Jjc, "On the Use of Monopole Antenn Model n ghtnng Protecton System Anlyss " MC 98 ROMA, pp , Rom, Itly, Sept [4] K. Anserowt, "Methods of Creton of ghtnng Protecton Zones Ner Tll Telecommuncton Structures Accordng to Dfferent Ntonl Stndrds ", TCST 00, vv- Slvsko, Ukrne, Feb 8-3, 00. [5] B. Jjc, D. Poljk, N. Kovc "Boundry lement Modellng of the Metllc Rod Protecton Zone", BM XXV, Southmpton, UK, Boston, nd USA: WIT Press, 003. [6] D. Poljk, C.A. Brebb, "Boundry lement methods for lectrcl ngneers ", WIT Press, Southmpton-Boston, 005. [7] D. Poljk, V. Dorc, S. Antonjevc "Computer ded Desgn of Wre Structures: Frequency nd Tme Domn Anlyss", WIT Press, Southmpton-Boston, 007. WIT Trnsctons on Modellng nd Smulton, Vol 44, ISSN X (on-lne 007 WIT Press