Application of FEA to Image-based Models of Electrical Trees with Uniform Conductivity

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1 IEEE Trnstions on Dieletris nd Eletril Insultion Vol. 22, No. 3; June Applition of FEA to Imge-sed Models of Eletril Trees with Uniform Condutivity Simon M. Rowlnd 1, Roger Shurh 1,2, Mihlis Pttours 1 nd Qi Li 1 1 Shool of Eletril nd Eletroni Engineering, University of Mnhester, Mnhester, UK 2 Deprtment of Eletril Engineering, Federio Snt Mri Tehnil University, Vlpriso, Chile ABSTRACT X-ry omputed tomogrphy nd seril lok-fe SEM hve provided detiled threedimensionl reonstrutions of eletril trees for the first time. The pplition of finite element nlysis (FEA) to the nlysis of eletril fields in n epoxy lok ontining tree is onsidered. Illustrtions re provided y wy of numer of se studies. It is shown tht the limittions of FEA do not rise from the disrete nture of the meshing: rther unertinties re more onerned with mteril properties in high fields on the mirometer sle, the limittions imposed y the pixel size of the imging tehnique, nd the disrete nture of the imge reonstrution tehnique. For dynmi model of tree growth spe hrge dynmis on the sme physil sle need lso to e modelled. A meshing strtegy is used, lirted ginst the hrge simultion method, to ensure urte ut mngele omputtions in ritil prts of tree suh s rnh tips. Exmples of field vlues re given using geometri onstruts nd low-field mteril hrteristis s illustrtive vlues. The field vrition round onduting tree struture, inluding the mximum field diretion s rnh strts to ifurte, is determined s n exmple. These yield vlues in exess of those expeted if spe hrge movement ws onsidered, ut onsistent with nlytil lultions. Index Terms - XCT, x-ry omputed tomogrphy, SBFSEM, Seril lok-fe SEM, FEA, finite element nlysis, eletril tree, field, model, hrge simultion method, CSM, imge-sed modeling. 1 INTRODUCTION ELECTRICAL trees re the result of long-term geing mehnisms in polymeri mterils nd n led to filures in high voltge equipment. Suh trees normlly originte in lotions where voids, impurities nd/or protrusions use high, divergent eletri field. During periods of growth, prtil dishrges re oserved, nd growth is widely onsidered to e result of these dishrges. Sixty yers go Mson showed, for exmple, tht pitting resulted from dishrges within hnnels in polyethylene, nd identified tht roniztion of surfes ourred in some ses nd not in others [1]. Under high AC voltges, the dishrges in the voids eroded their surfe reting non-onduting protrusions into the mteril. As the dishrges ontinued, the protrusions eme ondutive lolly enhning the eletri field nd susequently resulting in the growth of tree. Muh of wht is understood onerning tree growth hs een lernt from diret oservtions of lortory grown trees in trnsprent polymers. Diret optil mesurements of the visul spet of trees, nd ssoited prtil dishrges nd optil emissions [2, 3] hve eome stndrd in their study. Mnusript reeived on 8 Septemer2014, in finl form 7 Jnury 2015, epted 8 Jnury In most lortory experiments, eletril trees re grown in pure polymer resins using needle-plne geometries to rete very high divergent fields, ut in servie trees grow in pprently less onerous onditions over longer times [4]. The use of needle-plne geometries enles high fields to e hieved t reltively low voltges, nd esy oservtion of the tree growth in the trnsprent mterils. Suh trees re oserved to grow from length of 10 µm to ross the insultion, typilly few mm, efore totl rekdown ours. Eletril trees re typilly lssified s one of three types: rnh, ush, or ush-rnh depending on their pprent similrity with otnil tree shpes [5]. Lortory-grown trees re normlly oserved y optil mirosopy. This results in two-dimensionl (2D) projetion. The depth of field of the projetion depends upon the mirosopy system used, nd this influenes the imge, s seen in Figure 1. Eh 2D projetion of tree n e hrterized y frtl dimension, f d [6]. Kudo pointed to the limittions of frtl nlysis of three dimensionl (3D) ojet with 2D imge [7]. Typilly vlues determined from suh imges re 1 < f d < 2 for rnhed nd 2 < f d < 3 for ush trees [8]. The tree shpe nd hrteristis re hevily dependent on the voltge (field) pplied, the mteril (omposition nd glss trnsition temperture) nd mehnil stresses [9]-[12]. Snning eletron mirosopy (SEM) nd Trnsmission eletron mirosopy (TEM) [13] hve given exellent DOI /TDEI This work is liensed under Cretive Commons Attriution 3.0 Liense. For more informtion, see

2 1538 S. M. Rowlnd et l.: Applition of FEA to Imge-sed Models of Eletril Trees with Uniform Condutivity Figure 2. SEM imges of mirotomed surfes giving ross-setions of tree rnhes [14]. Figure 1. A rnhed (left) nd ush type tree (right) reted in lortory onditions. Sle r: 200 µm. lolized imges of individul tree rnhes / tuules. This hs provided diret evidene of the physil nture of suh fetures: onfirmed to e onneted tues of dimeters of the order of mirons. Figure 2 reprodues two exellent imges tken y SEM of tree grown in polyethylene, fter mirotoming nd ething. However the potentil for developing full understnding of the three-dimensionl struture of the trees from this pproh is limited. Reent reserh hs llowed omplete three dimension (3D) replis of trees to e generted using oth X-ry omputed tomogrphy (XCT) nd seril lok-fe snning eletron mirosopy (SBSEM) [15, 16]. The use of these methods enles the mesurement nd lultion of mny quntities whih hrterize the tree: for exmple, prmeters suh s the dimeter of tree hnnels, tree volume nd surfe re, nd 3D frtl dimension [17]. Also, prmeters from the skeleton of the tree n e otined: numer of verties nd segments, segment lengths, rnh ngles nd tortuosity of the hnnels, mongst others. Moreover, the reonstruted 3D models my, for the first time, llow field estimtions to e mde in the dieletri ontining tree, in prtiulr using FEA. However, this egs the question of wht FEA is ple of t suh sles, nd lso wht the true nture of tree is nd how tht n represented y the 3D model. This pper ddresses these questions, not to generte new model of tree growth t this stge, ut to determine whether FEA is n pproprite tool with whih to uild field models. As strting point: time independent, dishrge-free model will e onsidered. In the following setion the reonstruted trees re introdued. This is followed y onsidertions of the physil prmeters used to generte the FEA struture. Finlly the models re generted nd their limittions onsidered in detil. 2 GEOMETRIC THREE-DIMENSIONAL TREE RECONSTRUCTION Three-dimensionl models (or virtul replis ) from rel (rther thn simulted) eletril trees hve een generted using the imging tehniques of X-ry omputed tomogrphy (XCT) nd seril lok-fe snning eletron mirosopy (SBFSEM) [15, 16]. In the pst, optil mirosopy omined with either omputed tomogrphy or seril setioning methods were used to reonstrut the struture of trees [17]. However, optil tomogrphy nnot resolve omplex trees nd optil mirosopy is limited in resolution. In ontrst, XCT nd SBFSEM n provide su-mirometer resolution nd e pplied to optilly-opque mterils [18]. The generl methodology for eletril tree 3D model retion onsists in experimentl smple preprtion, eletril tree formtion, mhining the smple to prepre it for the sn, imge quisition nd post-proessing of the dt whih inludes imge segmenttion nd surfe genertion. XCT is non-destrutive tehnique sed on the mesurement of trnsmission of X-rys through n ojet over rnge of ngles. A series of rdiogrphi projetions re tken s the smple is rotted xilly nd then omined with reonstrution lgorithm to form ross-setionl imges ( slies ) of the volume [19]. In onventionl X-ry imging, ontrst is otined from the ttenution of X-rys s they pss through the ojet. However, when imging eletril trees in polymers, ttenution (sorption) ontrst does not provide suffiient ontrst etween the mteril nd the gs filled tues omprising the tree. As n lterntive, ontrst n e otined from the phse shift in the X-rys s they pss through the ojet. This mode is lled phse-ontrst nd hs een shown y the uthors to e well-suited to eletril tree imging [16]. In this mode, the distortion of X-ry wvefronts (phse shift of the X-ry wves) is due to the different refrtive indies of the different mterils present in the ojet thus, phse ontrst n e oserved even when sorption ontrst is lmost undetetle [20]. In SBFSEM, the 3D set of dt is quired y utomted seril setioning omined with lok-fe imging of the smple, inside hmer of low-vuum SEM [21]. By mintining gseous environment round the smple, this tehnique llows SEM imging of non-onduting surfes, removing the need for onduting oting nd thus, enling utomted seril setioning inside the hmer. The finl output from oth imging tehniques is stk of imges (slies) tht represent ross-setions of the eletril tree ( seleted slie is shown in Figure 3). These slies re in grey levels, where different pixel intensities represent different mterils (XCT nd SBFSEM) or surfe topologil informtion (SBFSEM only). From this stge, the model retion proedure follows the sme route in oth imging tehniques. The pixel size of the imges reltes to the resolution of the sn nd the dimensions to the field of view ptured. Using the gry level informtion, the proedure follows to imge segmenttion, whih is the proess wherey the feture (eletril tree) is extrted nd lelled. In this proess, new stk of imges is generted, where pixels re no longer gry sle levels ut olor-lelled (8 it depth).

3 IEEE Trnstions on Dieletris nd Eletril Insultion Vol. 22, No. 3; June Figure 3. ) Cross-setionl slie in gry levels otined from reonstruted XCT sn; ) Slie fter segmenttion proess; ) Model generted nd the lotion of the slie shown in ) nd ). All sle rs: 10 µm. Eh olor represents different mteril. In the se of eletril trees in unfilled epoxy, imges re omposed of only two olors: eletril tree nd epoxy resin (see the seleted segmented slie in Figure 3). With these segmented slies, numeril quntifition nd surfe genertion (3D model or virtul repli) is possile. The genertion of the surfe of the eletril tree ( losed volume) involves onneting the lelled pixels mong the slies nd omputing tringulr pproximtion of the interfes etween different mterils. This proess ws rried out using Avizo softwre, where different smoothing types nd extent levels re ville. Surfe smoothing is required in order to void rtifiil stirse-like surfes resulting from the onnetion of voxels (3D pixels), ut without modifying the originl shpe of the feture. Despite speil re, this smoothing proess my modify detils of the tree or rete rtifiil fetures. For the se presented here, the lowest smoothing extent tht removed the stirse-like nture of the model surfe ws hosen. An exmple of the surfe generted is shown in Figure 3. Mny elements my hinder the genertion of the est possile (losest to relity) 3D model: First, issues relting to imge qulity of the slies from the sns. Imge qulity is ffeted y resolution, ontrst, noise nd rtefts [22]. Currently, the finl sptil resolution using these 3D imging tehniques is round 0.7 µm for XCT nd in the se of SBFSEM, in prtie round 0.1 µm [16]. The seond hindrne n e misguided deisions out whih gry levels represent tree tues during the imge segmenttion proess. As disussed erlier, in relity there is unlikely to e lerly defined surfe t the su-miron sle. Finlly, surfe genertion from the segmented slies requires deisions out the level of surfe smoothing. This gretly ffets the model for FEA use, through the inlusion or exlusion of shrp points or singulrities. Among the ftors tht ffet the ury of the model generted for FEA, the ones tht re most influentil re: the imge qulity of the slies (minly resolution) nd surfe genertion (seletion of smoothness nd susequent inlusion of singulrities). In the exerise of reting virtul replis of eletril trees to e used in FEA modelling, it is worth sking not only wht the inuries of generting 3D models from the sns re, ut lso, wht is missing from the imging sns. It is not relly known wht fetures re not seen (this is lso true for optil tehniques). Any tue finer thn the sptil resolution of the imges, will not e ptured. The dt used for the 3D model shown in Figure 3 ws quired using XCT nd pixel size of 0.37 µm. The 3D model generted hs the following hrteristis [23]: length of 59 µm, tree hnnel dimeters of round 2 µm, tree volume of µm 3, tree surfe re of µm 2 nd 3D frtl dimension (ox ounting method) of 1.7. The hrteristis of the skeleton of the tree were: 78 verties, segment lengths of round 7 µm, rnh ngles verging 67 nd tortuosity of tree hnnels of MODEL PARAMETERS So tht model n e onstruted, ssumptions onerning the physil nture of the tree need to e expliitly onsidered. Reviewing the literture, this is now onsidered in three prts; the gseous phse (within the tuules), the interfe etween gs nd polymer, nd the ulk polymer. 3.1 THE GAS PHASE In pssive tree in whih no dishrges re tive the gs will hve similr onstitution to ir perhps with some voltiles from the surrounding polymer, nd t tmospheri pressure. During periods in whih prtil dishrges re ourring the gs will e prtly ionized - more so t the dishrge time nd lotion, nd less elsewhere. In ddition, voltile y-produts from the polymer surfe re likely to e present. Pressure is likely to flutute s gseous onstituents re onsumed or generted through onsumption of oxygen initilly nd the low moleulr weight speies eing voltilized from the tree wll surfe [24, 25]. During nd fter prtil dishrge evolution, hnge in ioni density nd ir ondutivity will lso our. It is ler then tht for omplete model of ehvior the time dependent properties of dishrges, nd their intertion with the polymer surfe eomes importnt. This is not onsidered here, nd the hnnels re onsidered to e irfilled with fixed, uniform ondutivity. 3.2 THE INTERFACE BETWEEN GAS AND POLYMER Eletril tree ondutivity plys n importnt role towrds the shpe nd propgtion of the tree, nd s suh their lssifition s onduting or non-onduting hs eome norml prtie. By onduting is normlly ment tht the interfe etween the gs nd dieletri solid phse is onduting nd it is normlly ssoited with roneous lyer. Vughn et l found the differene etween res of tree

4 1540 S. M. Rowlnd et l.: Applition of FEA to Imge-sed Models of Eletril Trees with Uniform Condutivity with nd without ron to e quite distint [26]. The simplest onsidertions of this re tht if the tree is onduting, the field within is negligile nd so dishrges will not our in its rnhes, ut the field t the rnh tips will e enhned nd so dishrges will preferentilly our there. The ility or otherwise of tree to sustin dishrge within its extent is onsidered to e one reson for it to grow s ush- or rnh-type: ondutive tuules generlly eing onsidered to prevent internl dishrges, nd so led to rnh-type trees [27]. It hs lso een speulted tht hnge in ondutivity long tree my lso e ssoited with rnh point [5]. Vlues of tuule ondutivity for ondutive trees were estimted t etween 1 10 Ω/μm in polyethylene y Vughn [28] using resisitivity of 10 3 Ωm nd wll thikness of 0.5 µm. Vlues of nd Ω/50µm were respetively modelled s produing onduting nd nononduting trees respetively y Chmpion nd Dodd [27]. 3.3 THE BULK POLYMER It is likely tht the polymer in whih the tree grows hs time-vrying properties, prtiulrly in regions of tree growth nd high fields [29]. The moility of hrge rriers will vry etween polymers, nd will lso e strong funtion of temperture, prtiulrly round the glss trnsition temperture. The highly divergent field expeted t the needle point nd tree tips is likely to rete lrge vrition in eletril ondutivity. It hs long een epted tht spe hrge injetion t the high fields generted will modify nd moderte the fields otherwise expeted [30]. In the se onsidered here it is ssumed tht no spe hrge enters the polymer, nd tht the ulk polymer is uniform. Therefore the fields lulted in these models do not represent expeted vlues: ut enle explortion of the proess of evlution nd sptil resolution using FEA nd the imging tehniques. In the epoxy resin se onsidered, the following vlues hve een tken: reltive permittivity ε r = 3.6, nd ondutivity σ = 1x10-13 Sm FEA MODELS In generl, the Finite Element Method n e divided into five proedures: determining the geometry of the model meshing ssigning oundry onditions solving post proess nd results nlysis Due to the omplexity of the eletril tree, speifi tehniques re required within eh step to hieve high ury in field lultions. These re now onsidered. 4.1 GEOMETRY MODELING The eletril tree geometril model is generted using Avizo softwre s explined in Setion 2 nd inludes the host epoxy ue of 20 mm sides. The simultion is onerned with the eletri field distriution within the ue only. Figure 4 illustrtes the tree geometry explored. Figure 4. The imported XCT geometry for modelling in FEA. The tree is 59 µm long. 4.2 MESHING TECHNIQUE The model deployed for simultion hs reltively fine detils (tree rnhes) within lrge simultion domin (the epoxy ue). One hllenge of performing suh simultion is forming proper strtegy for mesh genertion. An pproprite meshing sheme is ritil for the suessful simultion of the in terms of ury, given the omputing resoure ville. The sle ftor (µ) is defined s: mx = l (1) l min where l mx nd l min refer to the mximum nd minimum length of feture within the whole simultion. The minimum edge length within the geometry is set y the shrpest tip of eletril tree rnhes whih hs dimeter of 0.2 µm while the mximum size is the edge of the epoxy ue of 20 mm. This retes sle ftor of 100,000 nd requires lrge omputing memory in order to otin n urte result. Figure 5 illustrtes the proess of dptive mesh refinement. To pply it to the eletril tree model, preliminry solution is omputed first. This tenttive model ssumes ondutive tree (the potentil of the tree hnnel is the sme s the voltge pplied on the needle). The eletrostti equtions re pplied to this model. Within this step of study, the mesh is not optimized nd orse mesh is generted to sketh the field distriution. The lol mxim re determined y nlyzing the volume eletri field distriution. As expeted, the lol mxim re loted t the tips of the tree rnhes in the solid dieletri. This is due to the divergent field generted t ll void tips with smll urvture. This divergent field distriution requires finer mesh size t these speifi lotions to ensure the ury of results. To ddress this, lol mesh refinement tehnique of figure 6 is introdued. The res of field mxim re prtitioned into spheril regions s shown. This pproh, leding to meshing illustrted in figure 7, lso hs the following dvntges in oth the mesh ontrol nd post-proesses: sudivided domins llow the lol refinement of mesh

5 IEEE Trnstions on Dieletris nd Eletril Insultion Vol. 22, No. 3; June not only the minimum element size, ut lso the speed of mesh growth n e ontrolled inside the sudivided domin divided surfes of the tree ody llow not only lol mxim ut lso integrted verge eletri fields to e determined onveniently su-divided domins llow integrted volume eletri field nlysis Figure 7. Higher mesh densities re used on the surfes of smll rdii, s seen here t rnh tip. Figure 5. Mesh Sheme designed for Eletril Tree Modeling. A model of sphere-plne gp, hs een uilt to understnd the equivlent errors whih ould exist in the tree model. This simplified model n e solved urtely y the Chrge Simultion Method (CSM) nd offers the opportunity to ompre the FEA results with the nlytil sed method. A perfetly onduting sphere ove n infinite ground plne is shown in Figure 8. The sphere hs rdius, R = 1µm nd is pled t distne 2 mm ove the ground. The rdius of this sphere simultes the smllest tips within eletril tree struture while the distne etween the sphere nd the ground reflets the distne etween tree tips nd the ground. The voltge pplied is 18 kv. The spe outside the sphere is filled with n insultion mteril with reltive permittivity of 3.6, representing the epoxy resin. This model is first solved y CSM (progrmming in MATLAB). A series of fititious point hrges re onstruted distne elow the surfe of the sphere (the lk points within Figure 8) to simulte the tul hrge distriution on the sphere surfe. Aury is determined y the numer of point hrges. The ury is evluted y exmining the devition etween the tul voltge on the sphere surfe (18 kv) nd tht lulted from the fititious point hrges. It is found tht the results stilized t kv/mm when extr fititious point hrges only use devition of less thn 0.1% on the eletri field. This vlue is then utilized s enhmrk ginst whih to evlute the ury of the finite element method. Figure 6. Regions of high field, segregted into spheres for refined meshing. 4.3 FEA ACCURACY The previous setion introdued n dptive mesh sheme for refining the mesh. The stop riterion (in Figure 5) needs to e determined when the lol mesh hs een refined to n eptle level. This setion estlishes enhmrk on how intense the mesh needs to e so tht ertin level of ury n e hieved where the refinement loop shown in Figure 5 n e terminted. Figure 8. A sphere ove n infinite ground plne. Used to verify the FEA results ginst the Chrge Simultion Method.

6 1542 S. M. Rowlnd et l.: Applition of FEA to Imge-sed Models of Eletril Trees with Uniform Condutivity As explined in the previous setion, the ury of FEA relies on how the overll domin hs een meshed. Finer mesh hieves higher ury ut rises the omputing power required. Moreover mesh size smller thn the imge resolution my provide misleding onfidene in the model ury whih is fundmentlly limited y the imported reonstrution. To exmine the ury of the FEA eletri field omputtion, vrious mesh sizes hve een pplied on this simplified sphere-to-ground model. As shown in Figure 9, the numeril residul redued from 40% to elow 1% when the mesh rtio (rtio etween the minimum mesh size nd the minimum urvture) is deresed from 0.5 to If rtio of 0.05 is preserved etween the mesh size nd originl imge resolution, the FEA error will e less thn 1% nd given the unertinty in geometry, this will not e the limiting ftor for the model ury. As stted previously, suh high fields re not expeted in rel mterils, ut this exerise shows the FEA meshing proedure dopted n provide sptil ury for fetures down to the 1 µm sle. =0.5 =0.4 =0.3 =0.2 =0.1 =0.05 Mximum Eletri Field (kv/mm) Mesh Size s Rtio to Minimum Curvture Results from FEA Figure 9. Mximum eletri field predited y FEA s funtion of mesh size. 5 RESULTS AND ANALYSIS In the nture of ny numeril method, finite element methods seek for solutions to prolems fter resonle simplifitions hve een mde. By resonle is ment tht the simplifition is understood, quntified nd only introdues errors in known nd eptle mnner. The development in omputing resoures hs mde the simultion of geometry ontining omplex detils (suh s the tree model developed here) possile. Coupling different physis enles simultion of not only one physil prmeter ut lso y other physil fields. With refully designed simultion shemes, even nonliner prolems n e ddressed with onsiderle ury. This setion presents simultion strtegies s well s their imposed limittions. 5.1 RESULTS FOR THE SIMPLIFIED MODELS As disussed in Setion 3, eletril trees n e lssified s either: insulting (where the tree hnnels n e onsidered s ir filled with no ondutivity) ondutive (onduting surfe 10 3 Ωm, nd wll thikness of 0.5 µm) ondutive with extended ioniztion volume (onduting surfe s ove ut only to within 100 µm of the tree tips) To evlute the sensitivity of vrious prmeters, suh s tree ondutivity, wll thikness, nd extended ioniztion volume, simplified model (Figure 10) is introdued s se study. As shown in Figure 10, needle with 3 µm tip rdius is emedded in n epoxy ue (22 mm x 22 mm x 20 mm). A 60 µm long tree hnnel with 3 µm tip rdius extends from the tip of the needle. 18 kv is pplied t the needle nd the ottom surfe of the epoxy ue 2 mm from the needle tip is grounded. To evlute the effet of tree hnnel ondutivity on the eletri field distriution, finite element model lulting the pitive-resistive field is introdued. Within this model, the ir hs reltive permittivity of 1 nd ondutivity of 1x10-15 S/m, the epoxy hs reltive permittivity of 3.6 nd ondutivity of 1x10-13 S/m. The tree hnnel hs reltive permittivity of 1, ut its ondutivity vries depending on the nture of the tree (n insulting tree hnnel is given homogeneous ondutivity of 1x10-13 S/m while ondutive tree hnnel hs ondutivity of 1x10-5 S/m). A ut plne through the xis of rottionl symmetry is introdued to present the 3D simultion results in 2D plots. The eletri field distriution nd potentil ontour re plotted in Figure 11, nd. The eletri field drop long the tips of needle nd tree hnnel re plotted in Figure 9d s funtion of tree ondutivity. It is oserved from the simultion results tht: The highest eletri field is loted on the needle tip for n insulting tree hnnel The highest eletri field is loted on the tip of tree hnnel for ondutive tree hnnel In the two extreme se the fields re 2000 kv/mm s this is ontrolled y their rdii whih re the sme in this model One limittion of this model is tht the ondutivity of the dieletris re not dependent on the lol eletri field strength nd spe hrge is not onsidered. The 2000 kv/mm is then, whilst higher thn expeted in rel mteril, typil of n nlytil lultion for 3 µm onduting feture enhning Lplin field [30, 31]. Inorporting field-dependent hrteristis will e importnt in studying the mehnism of tree growth. FEA tools n redily ommodte non-liner field dependenies with itertive methods. This requires exponentilly inresed omputing power (oth proessors nd memories) nd imposes limittion on the omplexity of model whih ould e resolved. Moreover it requires urte knowledge of lolized high field ondutivities.

7 IEEE Trnstions on Dieletris nd Eletril Insultion Vol. 22, No. 3; June Figure 10. Geometry of omplete model (left) nd lose-up of needle tip with 3 µm rdius (right). pitive-resistive field. At the tip of the ylindril tree hnnel, it is ssumed tht the min tree hnnel (mother) divided into two su-hnnels (dughter rnhes) s shown in Figure 12. It is lso ssumed tht one of the dughter rnhes follows extly the sme diretion of the mother rnh, the other dughter rnh grows off t ertin ngle: typil ngle of 60 degrees is onsidered here. The eletri field distriution in oth 3D nd 2D re ompred etween single hnnel nd pronged hnnels, in Figure 12. MV/mm MV/mm () () () 15 kv 15 kv 17 kv 13 kv 13 kv 11 kv 11 kv 15 kv 13 kv D2 D1 (d) Needle Tip (x=0 µm) 11 kv Tree Tip (x=62 µm) d f e f d e f Figure 12. Regions of high field s single hnnel strts to ifurte. Figure 11. Eletri field distriution nd voltge ontour for: () insulting tree (C tree =10-13 S/m), () semi-ondutive tree (C tree =10-7 S/m) nd () ondutive tree (C tree =10-5 S/m); (d) the eletri field vlue ginst the distne long the line of symmetry through the needle tip. 5.2 RESULTS FOR BIFERCATING TREE CHANNELS When ondutive tree strts to grow, it is likely to grow from the lotion where eletri field is highest, suh s the tip of tree hnnel. At some stge mother hnnel ifurtes into dughter hnnels. This newly developed geometry modifies the lol eletri field drmtilly. To understnd the mehnism of the proedure of tree growth, good understnding of the eletri field distriution is required. This setion nlyzes how FEA n ssist in determining the mgnitude nd diretion of the eletri field strength s modified y formtive rnhed geometry. A ondutive tree hnnel (with the ondutivity of 10-5 S/m) s desried in the previous setion is onsidered. The lultion of eletri field tkes into onsidertion of From the results, it is oserved tht: From the single hnnel to the pronged hnnels, the mgnitude of the mximum eletri field is redued 10% Due to the effet of the dughter rnh, the diretion of the mximum eletri field shifted 6 degrees from the xis of rottion of the needle If the lowest eletri field round the tip of single hnnel is defined s threshold vlue for tree growth. The re of the tip surfe whih ould grow (hs n eletri field strength higher thn the threshold vlue) on pronged tree hnnels is redued from degree to degree (s mrked in dsh line in Figure 12) 5.3 RESULTS FOR XCT MODEL After the lol mesh refinement, the eletri field distriution surrounding the tree my e omputed. The eletri field is highest t tree tips where divergent field is generted. The lol mxim vlue is not only depending on the ury of X-ry sn ut lso on the numeril pproximtion when CAD tools re used to proess geometries. Within this prt of dt nlysis, the verge nd

8 1544 S. M. Rowlnd et l.: Applition of FEA to Imge-sed Models of Eletril Trees with Uniform Condutivity mximum vlue of oth the spheril surfes nd the volume domins surrounding tips re presented LOCAL MAXIMA AND AVERAGES AROUND TIPS To desrie the eletri field strength t tree tips, lol mxim nd verges re onsidered. Lol mxim refer to the highest eletri field vlue within the 3D volume surrounding of tip. The lol verge vlue is otined y integrting the eletri field within 1 m sphere of the mximum lotion inluding only the solid region (i.e. not the ir/rnh spe). This typilly inludes 100,000 voxels nd so removes issues over errors ssoited with sptil resolution. Figure 13 shows the lotions of some of the rnh tips (for lrity, not ll re identified), nd Figure 14 gives vlues of the lol fields for ll the 56 rnh tips of the tree. It is found tht the highest verge field is loted t the furthest extent of the tree, losest to the ground plne; the highest mximum field is loted t the su-rnh whih is oriented t 45 degrees towrds the ottom nd left hnd side. The lowest mximum nd verge is loted third of the wy up the tree, shielded within the tree struture ELECTRIC FIELD GRADIENT AROUND BRANCH TIPS As disussed in the previous setion, the physil mehnism of tree growth is likely to e not only dependen t on the lol mximum vlue, ut lso the grdient nd divergene surrounding the mximum point. To onsider this spheril surfe is onstruted, with the entre of the sphere t the mximum eletri field position. By inresing the rdius, the verge eletri field on the surfe of the sphere my e deteted. Here the rdius of the sphere inresed from 0.1 m to 2 m with 0.1 m steps. T40 shown in Figure 13 is seleted s n exmple to nlyze oth the field drop nd diretion of highest eletri field. Figure 15 shows tht from 0.1 to 2 m, the eletri field flls from 9000 kv/mm to 1200 kv/mm. Figure 14. The verge nd mximum field determined for eh ondutive rnh tip. Figure 15. The verge eletri field drop over ll tips plotted ginst the rdius of sphere used for otining surfe verge vlue. Figure 13. The lotions of the mximum nd minimum fields. To provide representtive view olours re sled up to 6 MV/mm, ll fields ove tht vlue re red. Pek vlues re given in Figure 14. As the ioniztion proess hppens round tree tips, the distriution of eletri field round tree tips is ritil prmeter to determine in whih diretion the eletril tree will grow. The disdvntge of the pproh of figure 13 is tht the unertinties nd roughness of the surfes impt the results. As n lterntive pproh, Figure 16 shows the eletri field distriution round some tree tips on the equipotentil surfe t 90% of the energiztion voltge. A typil tip, T40, is seleted to nlyze the field distriution round the tree tips. This pproh provides smooth reproduile surfe within the solid dieletri for nlysis.

9 IEEE Trnstions on Dieletris nd Eletril Insultion Vol. 22, No. 3; June relisti model. If less onduting trees re onsidered, is ler tht the ssumption of non-onduting gs is unlikely to e stisftory nd this lso needs further onsidertion. ACKNOWLEDGMENT This work is prt of the EPSRC Supergen HuNet Projet EP/I013636/1, Roger Shurh would like to knowledge the sholrship support of CONICYT (Chilen Reserh Counil). Figure 16. The eletri field distriution over tree tip surfe. The tip lelled T40 in Figure 14 is ritrrily hosen for the lose-up view. 6 CONCLUSIONS FEA hs een shown to e powerful tool with whih to determine eletri field distriutions in nd round eletril trees. FEA s limittions do not rise from the meshing pilities if the models re orretly onstruted. Methods hve een shown to yield highly urte results in simplified ses. Two key ftors do limit the ppliility of this tool: Firstly the unertinties surrounding rel tree geometries. The 3D reonstrutions generted using XCT nd SBFSEM tehniques provide exellent models for the lrger fetures of the trees, ove mirometre. However the surfes nd smller fetures re not distint. Other tehniques suh s SEM nd TEM my e used to determine more struturl detil, nd used in onert with the 3D pprohes, however re is required so tht preprtion tehniques do not lter the nture of the surfes exmined. Two tehniques hve een presented to overome these issues, nd use the genertion of surfes lose to the tree surfe, firstly using geometri spheril onstrution round tree tip, nd seondly using equipotentil surfes. The seond ftor whih requires further study re the mteril properties used in the FEA models. Mesurements for surfe ondutivity vlues re ville for ronised surfes, ut high field ondutivity vlues re lso required for the solid dieletris. Moreover spe hrge in the gs nd solid phses needs etter understnding for inlusion in omprehensive nd relisti model. An illustrtion hs een provided suggesting FEA n e used to predit fields (mgnitude nd diretions) for omplex tree. This suggests these my e used to predit nd model tree development nd growth, inluding topologies. Certinly powerful pltform hs een developed for testing hypothesis onerning mehnisms of field-driven tree growth. The model uilt hs required ssumptions onerning the physil properties of ll mterils onerned. In prtiulr the surfes of the tree tuules re likely to hve mjor effet on the dishrge proess through photo-ioniztion, reomintion nd spe hrge effets. As the proess of dishrge is not eing onsidered here this hs not een onsidered. For simpliity the models presented here hve ssumed onduting tree, whih removes mny of the unknown prmeters. Even so the dieletri properties of the polymer in the high field region t the tree tips re essentil for more REFERENCES [1] J. Mson, The deteriortion nd rekdown of dieletris resulting from internl dishrges, Pro. IEE, Vol. 98, pp , [2] X. Chen, Y. Xu, X. Co, S. J. Dodd nd L. A. Dissdo, Effet of tree hnnel ondutivity on eletril tree shpe nd rekdown in XLPE le insultion smples, IEEE Trns. Dieletr. Eletr. Insul., Vol. 18, pp , [3] N. Shimizu nd C. Lurent, Eletril tree initition, IEEE Trns. Dieletr. Eletr. Insul., Vol. 5, pp , 1998 [4] R. M. Eihhorn, "Treeing in solid extruded eletril insultion", IEEE Trns. Eletr. Insul., Vol. 12, pp. 2-18, [5] L. A. Dissdo nd J. C. Fothergill, Eletril Degrdtion nd Brekdown in Polymers. Peter Peregrinus for IEE, London, UK, [6] L. Niemeyer, L. Pietronero nd H. J. Wiesmnn, Frtl dimension of dieletri rekdown, Phys. Rev. Letters, Vol. 52, No. 12, pp , [7] K. Kudo Frtl nlysis of eletril trees, IEEE Trns. Dieletr. Eletr. Insul., Vol. 5, pp , [8] A. L. Brly, P. J. Sweeney, L. A. Dissdo nd G. Stevens, Stohsti modelling of eletril treeing: frtl nd sttistil hrteristis, J. Phys. D: Appl. Phys., Vol. 23, No. 12, pp , [9] X. Chen, Y. Xu, X. Co, S. J. Dodd nd L. A. Dissdo, Effet of tree hnnel ondutivity on eletril tree shpe nd rekdown in XLPE le insultion smples, IEEE Trns. Dieletr. Eletr. Insul., Vol. 18, pp , [10] J. V. Chmpion, The effet of mteril omposition nd temperture on eletril tree growth in epoxy resins, IET Conf. Dieletr. Mterils, Mesurements nd Applitions, Edinurgh, UK, pp , [11] B. X. Du, Z. L. M, Y. Go nd T. Hn, Effet of mient temperture on eletril treeing hrteristis in silione ruer, IEEE Trns. Dieletr. Eletr. Insul., Vol. 18, pp , [12] H.-Z. Ding nd B. R. Vrlow, Thermodynmi model for eletril tree propgtion kinetis in omined eletril nd mehnil stresses, IEEE Trns. Dieletr. Eletr. Insul., Vol. 12, pp , [13] N. Hozumi, T. Okmoto nd H. Fukgw, TEM oservtion of eletril tree pths nd miro-strutures in polyethylene, Jpnese J. Appl. Phys., Vol. 27, pp , [14] A. S. Vughn, I. L. Hosier, S. J. Dodd, nd S. J. Sutton, On the struture nd hemistry of eletril trees in polyethylene, J. Phys. D. Appl. Phys., Vol. 39, No. 5, pp , [15] R. Shurh, S. M. Rowlnd, R. S. Brdley, nd P. J. Withers, Imging nd Anlysis Tehniques for Eletril Trees using X-ry Computed Tomogrphy, IEEE Trns. Dieletr. Eletr. Insul., Vol. 21, pp , [16] R. Shurh, S. M. Rowlnd, R. S. Brdley, nd P. J. Withers, "Comprison nd Comintion of Imging Tehniques for Three Dimensionl Anlysis of Eletril Trees", IEEE Trns. Dieletr. Eletr. Insul., Vol. 22, No. 2, pp , [17] H. Uehr nd K. Kudo, "Three-Dimensionl Frtl Anlysis of Rel Eletril Trees in Polyethylene", Jp. J. App. Phys., Vol. 38, pp , [18] R. Shurh, S. M. Rowlnd, R. S. Brdley, T. Hshimoto, G. E. Thompson nd P. J. Withers, Three dimensionl imging of eletril trees in miro nd nno-filled epoxy resin, IEEE Conf. Eletr. Insul. Dieletr. Phenomen (CEIDP), Des Moines, Iow, USA, pp [19] A. C. Kk nd M. Slney, Priniples of Computerized Tomogrphi Imging: Soiety for Industril nd Applied Mthemtis, [20] R. Fitzgerld, "Phse-Sensitive X-ry Imging," Phys. Tody, Vol. 53, pp , [21] W. Denk nd H. 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10 1546 S. M. Rowlnd et l.: Applition of FEA to Imge-sed Models of Eletril Trees with Uniform Condutivity [22] J. C. Dinty nd R. Shw, Imge Siene: Priniples, Anlysis nd Evlution of Photogrphi-type Imging Proesses, Ademi Press In., [23] R. Shurh, S. M. Rowlnd, R. S. Brdley, nd P. J. Withers, "Three dimensionl hrteristion of eletril trees", IEEE Conf. Eletr. Insul. Dieletr. Phenomen (CEIDP), Shenzhen, Chin, pp , [24] A. C. Gjerde nd J. Sletk, Influene of prtil dishrges on void gs pressure, Int l. Conf. Prtil Dishrges, Cnterury, UK, pp , [25] C. Myoux nd C. Lurent, Contriution of prtil dishrges to eletril rekdown of solid insulting mterils, IEEE Trns. Dieletr. Eletr. Insul., Vol. 2, pp , 1995 [26] A. S. Vughn, S. J. Dodd, nd S. J. Sutton, A Rmn miroproe study of eletril treeing in polyethylene, J. Mter. Si., Vol. 39, No. 1, pp , [27] J. Chmpion nd S. Dodd, Simultion of prtil dishrges in onduting nd non-onduting eletril tree strutures, J. Phys. D. Appl. Phys., Vol. 34, pp , [28] A. S. Vughn, I. L. Hosier, S. J. Dodd, nd S. J. Sutton, On the struture nd hemistry of eletril trees in polyethylene, J. Phys. D. Appl. Phys., Vol. 39, No. 5, pp , [29] L. A. Dissdo, Understnding eletril trees in solids: From experiment to theory, IEEE Trns. Dieletr. Eletr. Insul., Vol. 9, pp , [30] R. W. Hre nd R. M. Hill, Spe hrge in insultors with needleplne geometry, J. Phys. D. Appl. Phys., Vol. 24, pp , [31] L. Cisse, S. S. Bmji nd A. T. Bulinski, Eletri field lultions for needle-plne geometry nd spe hrge in polyethylene, IEEE Trns. Dieletr. Eletr. Insul., Vol. 10, pp , Simon M. Rowlnd (SM 07) ws orn in London, Englnd. He ompleted the B.S. degree in physis t The University of Est Angli nd the Ph.D. degree t London University. He hs worked for mny yers on dieletris nd their pplitions nd hs lso een Opertions nd Tehnil Diretor within multintionl mnufturing ompnies. He joined The Shool of Eletril nd Eletroni Engineering in The University of Mnhester s Senior Leturer in 2003, nd ws ppointed Professor of Eletril Mterils in He ws President of the IEEE Dieletri nd Eletril Insultion Soiety from He is n Assoite Editor of the IEEE Trnstions on Dieletris nd Eletril Insultion. Roger Shurh (S 11) ws orn in Temuo, Chile. He reeived degree in eletril engineering in 2006 from Federio Snt Mri Tehnil University (UTFSM), Vlpriso, Chile. He ws high voltge equipment nlyst t Trnsele trnsmission ompny in Sntigo, Chile, during He joined the Deprtment of Eletril Engineering UTFSM, s Junior Leturer in 2008, where he lso rried out dieletri tests for mining nd utility ompnies. Sine 2011, he hs een Ph.D. degree student t the Shool of Eletril nd Eletroni Engineering in The University of Mnhester. His reserh projet involves the study of eletril trees nd prtil dishrges in polymeri insultion. He ws wrded IEEE Grdute Fellowship of the Dieletris nd Eletri Insultion Soiety in Mihlis Pttours ws orn in Niosi, Cyprus. He reeived the M.Eng. degree in eletril nd eletroni engineering t The University of Mnhester in Sine then, he hs een Ph.D. degree student t the Shool of Eletril nd Eletroni Engineering in The University of Mnhester. His reserh projet involves the understnding of the role of mteril interfes (joints) on filure proesses in HV equipment. He is studying eletril tree ehvior ross interfes nd wys to inhiit this phenomenon. Qi Li (k Steven) ws orn in Hunn Provine, Chin, in Septemer He ompleted the B.Eng. degree in eletril nd eletronis engineering t oth the University of Birminghm nd Huzhong University of Siene nd Tehnology, in 2007 (s n exhnge student). He reeived the M.S. degree with distintion in eletril power engineering from the University of Mnhester in 2009, nd ompleted the Ph.D. degree in the sme institution in Dr Qi Li is urrently working s reserh ssistnt in the Ntionl Grid High Voltge Reserh Centre in the University of Mnhester, UK. His min reserh interests inlude: surfe potentil grdient lultion for overhed line ondutors, the mehnism of hum noise from trnsmission lines nd udile noise evlution for different types of ondutors.