Recovery of Valuable Incompletely-Recorded Return- Stroke Current Derivative Signals

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1 Rcovry of Valuabl Icomplly-Rcordd Rur- Srok Curr Drivaiv Sigals Lakmii Prra Elcrical ad Compur Egirig Dparm Ryrso Uivrsiy Toroo, Caada Ali M. Hussi Elcrical ad Compur Egirig Dparm Ryrso Uivrsiy Toroo, Caada Absrac Rsarchrs hav sudid a umbr of simulaig fucios for modllig CN Towr lighig rur-srok curr. Thy foud ou ha Hidlr ad h Puls fucios ovrcom crai limiaios, icludig im-drivaiv discoiuiis. O h ohr had, icomplly-rcordd curr drivaiv sigals rprsd aohr challg. Th papr proposs a doubl-rm Puls fucio ha is ivsigad ad compard wih h doubl-rm Hidlr fucio for modllig h lighig rur-srok curr. Th doubl-rm simulaig fucio, usd o rcovr a larg icomplly-rcordd rursrok curr drivaiv sigal masurd o Ju 0, 996. Kywords Hidlr fucio, Puls fucio, lighig rursok curr, sigal rcovry I. INTRODUCTION Lighig has b o of h mos iriguig phomo kow o ma. Bjami Frakli sard ivsigaig lcriciy i 746. I 75, durig a hudrsorm, Frakli coducd his famous xprim wh h flw a ki wih a coducig srig ad a ky id o h boom of h ki. Durig a hudrsorm, h obsrvd sparks flyig from h ky id o h coducig ki srig ad oo his kuckls. This xprim provd ha hudrclouds ar lcrically chargd ad lighig is lcrical [Uma, 97]. Dspi is spcacular aur, lighig has a log rcord of caasrophic damags, spcially for all objcs. Sudy of lighig has hlpd i procig all srucurs, forss ad powrlis [Hussi al., 995; Jaischwskyj al., 997]. Th Caadia Naioal CN) Towr locad i Toroo, 553m i high, has b pivoal o h mrgc of lighig sudis a h Uivrsiy of Toroo ad Ryrso Uivrsiy. I provids a suiabl objc for rcordig all-srucur lighig paramrs. Alhough h lighig flash dsiy i Toroo is abou.5/km /yar, h CN Towr ormally rcivs dozs of dirc sriks yarly [Jaischwskyj al., 997]. I h pas, rsarchrs did ivsiga a umbr of simulaig fucios for modllig CN Towr lighig rur-srok curr, icludig h doublxpoial ad Jos modifid doubl-xpoial fucios. Howvr, hs fucios wr foud o hav probl du o hir im-drivaiv discoiuiis [Hidlr ad Cvic, 00; Jos, 977]. O h ohr had, Hidlr ad h Puls fucios wr usd o ovrcom hs limiaios [Elrodsly ad Hussi, 0; Yazhou al., 00]. Ohr challgs wr arly od du o rflcios from CN Towr s srucural discoiuiis [Hussi, 009; Hussi al., 04; Rahimia ad Hussi, 05] ad h irfrig Lora-C Sigal [Liaos ad Hussi, 005; Ndjah, al., 00]. Furhrmor, h icomplly rcordd curr drivaiv sigals, which xcdd h maximum sigal s lvl, rprsd aohr challg [Rahimia ad Hussi, 05]. I fac hs icomplly rcordd curr drivaiv sigals, whos paks ar wll abov h ois lvl, provd o b qui valuabl for modllig purpos. Th papr proposs a doubl-rm Puls fucio ha is ivsigad ad compard wih h doubl-rm Hidlr fucio for modllig h lighig rur-srok curr. I h proposd papr, a compl larg rur-srok curr drivaiv sigal rcordd o Ju 0, 996, is arificially cu. Th, ach of h chos doubl-rm simulaig fucios is usd o ry o rcovr h origial arificially cu sigal for valuaig h proposd algorihm bfor applyig i o sigals ha wr icomplly rcordd. Th qualiy of fiig of h masurd sigal ad h rcovrd sigal ar valuad wih R fiig facor. Furhrmor, h simulaig fucios ar agai usd o rcovr icomplly-rcordd rur-srok curr drivaiv sigals, which wr masurd o Ju 0, 996. Th mai objciv of his sudy is o succssfully rcovr h valuabl icomplly-rcordd sigals, which ar o b usd for valuaio of all-srucur lighig modls by comparig h simulad lcric ad magic filds wih hos masurd [Rahimia ad Hussi, 05].

2 II. SIMULATING FUNCTIONS Th simulaig fucios, Hidlr ad h Puls fucios, ach cosiss of a ris fucio x) ad a dcay fucio y). Ris ad dcay fucios hav a dcouplig rlaioship wih h codiio ha durig h ris y) ad durig h dcay x) [Hidlr ad Cvic, 00]. Th gral sigl-rm curr wavform of ach simulaio fucio is dfid as: i) = I max x) y) ) Mahmaically, doubl-rm Hidlr curr fucio ad is drivaiv ar dfid as: [Milwski al, 008; Rahimia al., 05; Yazhou al., 00] di) d i) = I ) + ) τ + ) τ I + ) τ = i ) + i ) = i ) [ τ i ) [ τ τ + ) ] + ) + ) ] 3) τ Th proposd doubl-rm Puls fucio is dfid i 4) ad is drivaiv is giv i 5). i) = I ) τ + 4) I = i ) + i ) di) = i d ) [ τ i ) [ τ τ τ ) ) ) τ τ ] + τ ] To obai h bs fi for h masurd curr drivaiv wavform, h im drivaiv of Hidlr ad h Puls fucios ar usd hr for h simulaio. Th curr fucios ar h obaid by igraig h simulad curr drivaivs. Som cosrais ar iroducd o improv h fiig of ach curr drivaiv simulaio as i [Milwski al, 008; Rahimia al., 05]. Cosrais forc h aalyical paramrs, τ,,, τ, ad τ o rach hir opimal valus, which yilds a br fi of h simulad curr. 5) d Forcig h maximum spss cosrai i d = 0, = whr is h im a which h maximum spss of h curr or h im a which h maximum ampliud of h curr drivaiv occurs. This cosrai miod abov provids h bs fi [Elrodsly al., 0; Rahimia al., 05; Yazhou al., 00]. Howvr, i was o possibl o dircly apply h maximum spss cosrai o h doubl-rm Hidlr fucio ad obai a closd xprssio for ad τ. Thrfor, h maximum spss cosrai is applid oo quaio ) o driv h followig xprssios. f h ) = i ) [ ) τ g h ) = τ ) + τ ) τ [ τ 3 + τ ) + τ ) + τ ] 6) + τ ) + τ ] 7) I = f h ) g h ) 8) Th, h xprssio for I is subsiud io 3) o acquir h fial xprssio 9), which is usd i h fiig procss. di) d = i ) [ τ f h ) g h ) [ τ + ) ] + + ) ] 9) τ For h doubl-rm Puls fucio, h sam procdur is applid oo quaio 4) o driv h followig xprssios. f p ) = i ) [ τ ) + τ )) ) )] 0)

3 g p ) = [ + τ ) )) τ τ τ ) )] I = f p ) g p ) ) ) ) STARTING POINT -0.μS ZOOMED STARTING POINT -0.μS Fig.. 6 h curr drivaiv wavform. Zoomd-i viw illusras h sarig poi a which di)/d 0 Equaio ) is subsiud io 5) o driv h fial xprssio 3), which is o b usd i h fiig procss. di) = i d ) [ f p ) g p ) [ τ ) τ τ τ ] + ) τ ] 3) III. METHODOLOGY O of h muli-srok CN owr flashs, rcordd o Ju 0, 996, is uilizd i his papr for modllig. Th 6 h ad 7 h rur-srok curr drivaiv sigals ar usd for modllig du o hir high curr drivaiv paks. 7 h rur-srok curr drivaiv sigal is icomplly-rcordd, which is o b usd for h rcovry procss. A. Machig Wavfor For h fiig procss, h sarig poi of h curr drivaiv wavform is chos from h masurd lighig rur-srok curr wavform [Milwski, 008]. Fig. illusras h 6 h rur-srok curr drivaiv sigal. Th iiial poi is assumd o b a = 0. μs for modllig purpos. Th machig wavfor ar h dividd io hr imwidows. Th dcouplig rlaioship of Hidlr ad Puls fucios allow h sigal o b asily dividd io scios [Elrodsly al., 0]. Th hr im widows ar:. Th iiial impuls bfor arrival of rflcios Fig.). Th rflcios ad arly dcay par of h wavform 3. Th dcay porio is basd o h curr wavform Fig. 3) Fig.. Th firs im widow of h 6 h curr drivaiv machig sigal Figur 3 illusras h hird im widows of h 6 h rursrok curr wavform obaid by umrically igraig is curr drivaiv machig wavform. This im widow is usd o sima τ ad τ, as xplaid i scio B. B. Esimaig τ ad τ Firs sp i approximaig h aalyical paramrs is o sima h dcay im cosas τ ad τ. Durig h dcay porio, boh simulaig fucios ar simplifid o: i) = I τ + I τ 4) τ ad τ ca b simad by fiig h dcay fucio dscribd by 4) io a chos hird im widow, usig MATLAB. For h 6 h rur-srok curr wavform, usig h hird-im widow dscribd i Fig. 3, i was foud ha τ = μs, τ = 0 μs, I = ka, ad I = ka wih R fiig of 0.935, idicaig a good fi. Th fiig procss is illusrad i Fig. 3. Figur illusras h firs im widow of h 6 h curr drivaiv machig sigal. Th duraio of h firs im widow is = μs bcaus h rflcio from h spac dck is visibl afr = 0.6 μs [Rahimia al., 05]. 3

4 IV. RESULTS Th drivaiv of Hidlr ad h Puls fucios dscribd by 9) ad 3), rspcivly, ar usd i fiig h 6 h curr drivaiv machig wavform. Th fiig is coducd usig MATLAB Fiig Toolbox o sima h ukows, I,, τ,,, τ ad τ. Th fiigs of Hidlr ad Puls fucios obaid for h 6 h curr drivaiv machig wavform ad is curr wavform ar illusrad i Figs. 4 ad 5, rspcivly. Th aalyical paramrs obaid for h wavform is summarizd i Tabl I alog wih h R fiig facors. I Fig. 4, of h curr drivaiv of Hidlr fucio machs of h 6 h curr drivaiv machig wavform, which was aaid usig h maximum spss cosrai, whras h maximum drivaiv of h Puls fucio is shifd o h righ by 0.005μs. Th locaio of h curr drivaiv pak basd o Hidlr fucio is foud o b closr o h masurd pak i compariso wih ha basd o h Puls fucio. Tabl I idicas ha h simulaio basd o h drivaiv of h Puls ad Hidlr fucios producd xcll fiigs of R = ad R = 0.998, rspcivly. Th simulaio basd o h Hidlr fucio producd a slighly br fi ha ha basd o h Puls fucio. Furhrmor, h locaio of h maximum curr drivaiv pak basd o Hidlr fucio simulaio is much closr ha ha usig h Puls fucio Fig. 4). Th maximum ampliud of h 6 h curr drivaiv machig wavform is ka/μs, whras ha simad from h drivaiv of Hidlr fucio is 38kA/μs. Howvr, h drivaiv of h Puls fucio is simad o b 37.45kA/μs. Thus, h maximum curr drivaiv pak basd o h Puls fucio simulaio is closr o h masurm ha ha usig Hidlr fucio Fig. 4). Figur 5 idicas ha h pak of h 6 h rur-srok curr wavform was o rachd by Hidlr ad h Puls fucios. Bu h Hidlr curr fucio has a mior ovrshoo compard wih ha of h Puls fucio. Obsrvig Fig. 5, shows ha h Puls fucio rachs h curr pak slowr i compariso wih h Hidlr fucio. Boh Hidlr ad h Puls fucios rasoably simulad h masurm. O of h biggs challgs facd by may rsarchrs is h icomplly-rcordd rur-srok curr drivaiv sigals, which xcdd h maximum sigal s lvls [Rahimia, ad Hussi, 05]. Ths icomplly-rcordd rur-srok curr drivaiv sigals, such as h 7 h rursrok curr drivaiv sigal, which will b furhr ivsigad i his papr, whos paks ar wll abov h ois lvl, provd o b qui valuabl for modllig purpos. Prior o sarig h rcovry of h icomplly-rcordd 7 h rur-srok curr drivaiv sigal, a complly rcordd sigal wih gligibl amou of ois, h 6 h rur-srok curr drivaiv sigal is arificially cu ad h rcovry procss is applid o i Fig. 6). Th masurd daa of h 6 h rur-srok curr drivaiv wihi h irval 0.35μs < < 0.55μs is rmovd o obai a wavform ha is arificially cu a h 50% lvl from h pak, as illusrad i Fig. 6. Th sam mhodology dscribd usig h 6 h rur-srok curr drivaiv machig wavform is followd o fi h simulaig fucios for h arificially-cu 6 h curr drivaiv wavform. Th fiig rsuls obaid for h arificially-cu 6 h curr drivaiv wavform ad is curr wavform ar illusrad i Figs. 6 ad 7, rspcivly. Th aalyical paramrs obaid for his wavform is summarizd i Tabl II alog wih h R fiig facors. Th rcovry of h arificially-cu 6 h curr drivaiv wavform is obaid by usig h curr drivaiv of h Puls fucio. I is foud o b mor succssful i h rcovry of h arificially-cu 6 h curr drivaiv wavform compard wih h curr drivaiv of Hidlr fucio. I Fig. 6, of h curr drivaiv of h Puls wavform, which was aaid usig h maximum spss cosrai, is shifd o h righ by 0.0μs. Tabl I idicas ha h simulaio basd o h drivaiv of h Puls fucio producd a xcll fiig of R = Furhrmor, h im locaio of h curr drivaiv pak, basd o h Puls fucio Fig.6), is clos o h locaio of h masurd curr drivaiv pak. Furhrmor, h maximum ampliud of h 6 h curr drivaiv masurd wavform is ka/μs ad h rcovry usig h drivaiv of h Puls fucio simad pak is 36.5kA/μs. Thus, h maximum curr drivaiv pak basd o h Puls fucio simulaio is provd o b clos o h masurd daa. Figur 7 idicas ha h pak of h 6 h rur-srok curr wavform was o rachd by h Puls fucio. I has a mior ovrshoo. Howvr, h Puls fucio rasoably rcovrd h masurm daa. Th sam mhodology dscribd usig h arificially-cu 6 h rur-srok curr drivaiv machig wavform is followd o fi h simulaig fucios for 7 h rur-srok curr drivaiv wavform, h icomplly-rcordd sigal. Th rcovry of icomplly-rcordd sigal is obaid usig h curr drivaiv of Hidlr fucio. I is foud o b mor succssful i h rcovry procss compard wih h curr drivaiv of h Puls fucio. Th fiig rsuls obaid for h 7 h rur-srok curr drivaiv wavform is illusrad i Fig. 8. Th aalyical paramrs obaid for his wavform is summarizd i Tabl II alog wih h R fiig facors. I Fig. 8, of h curr drivaiv of Hidlr wavform mach of h 7 h curr drivaiv machig wavform, which was aaid usig h maximum spss cosrai. Tabl II idicas ha h simulaio basd o h drivaiv of Hidlr fucio producd a xcll fiig of R = Th maximum ampliud of h 7 h curr drivaiv wavform simad by Hidlr fucio is 40.kA/μs. Thus, h Hidlr fucio simulaio is vry clos o h masurd daa ad rasoably rcovrd missig daa. 4

5 Fig. 3. Third im widow of h 6 h rur-srok curr wavform ad h fiig of h dcay fucio dscribd by ), which is usd o sima τ ad τ. Fig. 6. Th fiig of h Puls drivaiv fucio of h arificially cu 6 h curr drivaiv wavform. Fig. 4. Th fiig of Hidlr ad Puls drivaiv fucios of h 6 h curr drivaiv machig wavform. Fig. 7. Th fiig of h Puls fucio of h arificially cu 6 h curr wavform obaid by umrically igraig). Fig. 5. Th fiig of Hidlr ad Puls fucios of h 6 h curr wavform obaid by umrically igraig h 6 h curr drivaiv machig wavform). Fig. 8. Th fiig of Hidlr drivaiv fucio of h 7 h curr drivaiv wavform icomplly rcordd wavform). 5

6 V. CONCLUSION Followig umrous ivsigaios, rsarchrs foud ou ha Hidlr ad h Puls fucios ovrcom a lo of limiaios xpricd by h doubl-xpoial ad Jos modifid doubl-xpoial fucios. Th papr prsd a compariso bw Hidlr ad h Puls fucios for modllig h lighig rur-srok curr, usig h 6 h rur-srok curr drivaiv wavform masurd o Ju 0h 996 a h CN owr. A compl larg rur-srok curr drivaiv sigal is arificially-cu a h 50% lvl from h pak. Th, ach of h chos doubl-rm simulaig fucios ar usd o ry o rcovr h origial arificially-cu sigal for valuaig h proposd algorihm bfor applyig i o 7 h rur-srok curr drivaiv wavform, was o a complly rcordd sigal. Wih h 6 h curr drivaiv machig wavform, h drivaiv of h Puls ad Hidlr fucios producd xcpioal fiigs of R = ad R = 0.998, rspcivly. Wih h arificially-cu 6 h curr drivaiv wavform, h drivaiv of h Puls fucio foud o b succssful i h rcovry procss ad producd R = Usig h 7 h rursrok curr drivaiv wavform, h drivaiv of Hidlr fucio is foud o b mor succssful i h rcovry procss ad producd R = Boh Hidlr ad Puls simulaio fucios ca succssfully simula all-srucur lighig rur-srok currs ad rcovr valuabl missig daa. ACKNOWLEDGMENT This work was fudd i par by h Naural Scics ad Egirig Rsarch Coucil of Caada, NSERC Discovry Gra 0-07), which is grafully ackowldgd. Th auhors also ackowldg Ryrso Uivrsiy iral fudig ad faciliis. REFERENCES Elrodsly, K., ad A.M.Hussi 0), C Towr Lighig Rur-Srok Curr Simulaio, Joural of Lighig Rsarch, 4, pp Hidlr, F., ad J. Cvic 00), A Class of Aalyical Fucios o Sudy h Lighig Effcs Associad Wih h Curr Fro, Europa Trasacios o Elcrical Powr, ), pp Hussi, A.M. 009), Curr wavform characrisics of gaiv ad posiiv lighig o h CN Towr, Procdigs X Iraioal Symposium o Lighig Procio, SPIDA, Curiiba, Brazil, pp Hussi, A.M., S. Kazazi, M. Awar, M. Yusouf ad P. Liaos 04), CN Towr Lighig Characrisics Basd o Curr-Rcordd Flashs, Procdigs Iraioal Cofrc o Lighig Procio, ICLP, Shaghai, Chia, pp Hussi, A.M., W. Jaischwskyj, J.S. Chag, V. Shosak, W.A. Chisholm, P. Dzurvych, ad Z.I. Kawasaki 995), Simulaous masurm of lighig paramrs for sroks o h Toroo Caadia Naioal Towr, Joural of Gophysical Rsarch, 005), pp Jaischwskyj, W., A.M. Hussi, V. Shosak, I. Rusa, J.X. Li, ad J.S. Chag 997), Saisics of Lighig Sriks o h Toroo Caadia Naioal Towr ), IEEE Trasacio o Powr Dlivry, 3), pp. 0-. Jos, R.D. 977), O h Us of Tailord Rur-Srok Curr Rprsaios o Simplify h Aalysis of Lighig Effcs o Sys, IEEE Trasacios o Elcromagic Compaibiliy, 9), pp Liaos, P., ad A.M. Hussi 005), Characrizaio of Nois i h Lighig Curr Drivaiv Sigals Masurd a h CN Towr, IEEE Trasacio o Elcromagic Compaibiliy, 474), pp Milwski, M., ad A. M. Hussi 008), Lighig rur-srok rasmissio li modl basd o CN owr lighig daa ad drivaiv of hidlr fucio,caadia Cofrc o Elcrical ad Compur Egirig, CCECE, Niagara Falls, Caada, pp Ndjah, O., A.M. Hussi, S. Krisha ad R. Soudh 00), Comparaiv sudy of adapiv d-oisig chiqus for lighig curr drivaiv sigals, Europa Associaio for Sigal Procssig EURASIP) Joural, 0), pp Rahimia, M.S., ad A.M. Hussi 05), ATP Modlig of Tall-Srucur Lighig Curr: Esimaio of Rur-Srok Vlociy Variaio ad Upward-Cocig Ladr Lgh, IEEE Trasacio o Elcromagic Compaibiliy, 576), pp Rakov, V.A., ad M.A. Uma 003), Lighig: Physics ad Effcs, Cambridg Uiv. Prss, Nw York. Uma, M.A. 97), All abou lighig, Dovr Publicaios, Nw York. Yazhou, C., L. Shagh, W. Xiaorog, ad Z. Fizhou 00), A Nw Kid of Lighig Chal-bas Curr Fucio, Procdigs 3rd Iraioal symposium o Elcromagic Compaibiliy, Bijig, Chia, pp