Nordc Insulaton Symposum - Nord-IS - Trondhem, Norway, June 9-2, 2 Dpl.-Ing. D. Geßler (danel.gessler@kt.edu), Prof. Dr.-Ing. T. Lebfred Insttute of Electrc Energy Systems and Hgh Voltage Technology Karlsruhe Insttute of Technology (KIT), Karlsruhe, Federal Republc of Germany The short crcut wthstand capablty of power transformers descrbes the ablty of the wndngs to wthstand electromagnetc forces resultng from current densty n the conductors and magnetc stray felds. Especally operatonally aged transformers often have a reduced short crcut strength what can lead to rreversble deformaton of wndngs f hgh fault currents occur. Short crcut strength s to some degree dependent on the mechancal strength of the conductors. Manly the cellulosc nsulaton materals experence degradaton durng operaton tme due to hgh temperatures and mosture. Hence a reducton n short crcut strength can be correlated to some degree wth the agng of the cellulose. In ths paper the nfluence of paper nsulaton around Contnuously Transposed Conductors (CTC) on ther mechancal strength wll be analyed by a smplfed analytcal model as well as by a FEA based mechancal smulaton model. CTC delver an mportant contrbuton for the ncrease of power transformers effcency compared to the conventonal paper-nsulated sold conductors. CTC based wndngs manly reduce eddy current losses through ther dvson n sngle strands. Moreover loop voltages as a result of radal varyng magnetc stray felds are compensated by transposton of the strands. Also the wndng space factor can be ncreased as a result of the thn nsulaton of the sngle strands. Hence the temperature dstrbuton throughout the whole wndng s more homogeneous []. Electromagnetc forces can be dvded nto a radal and axal drecton of acton. In ths paper only radal force components are consdered as the rgdty of CTC especally n radal drecton s much lower compared to a sold conductor wth dentcal outer dmensons. The reason here for s that n axal drecton the cross-secton of CTC compared to a sold conductor s only dvded nto two stacks, whle radal drecton s dvded nto usually to 4 sngle strands. Radal forces actng on the nner wndng are drected nwards resultng n a bendng stress. Outward drected forces on the outer wndng result n hoop stress. Thus nward drected forces are more crtcal, as they can lead to bucklng of CTC between to axal spacers whle outward drected forces only generate a stran of the conductor n amuthal drecton. The frst part of ths paper deals wth the nfluence of paper nsulaton around CTC on the radal bendng stffness wth respect of lnear materal propertes: A constant modulus of elastcty for paper and copper and some further smplfyng assumptons enable the use of smple equatons of bendng theory. The more complex nonlnear case wll then be analyed n the second part by the use of a FEA based model, whch descrbes a CTC segment between two spacers under the nfluence of radal actng forces. The age condton of paper nsulaton s respected by varyng the correspondng stress stran characterstcs n the FEA. 2. CTC are made of an odd number of rectangular, enamel coated copper strands, whch are arranged n two stacks. These two stacks are solated by a pressboard separator. The number of strands s odd, as always one strand transposes ts radal poston after a fxed dstance n turn drecton, called transposton ptch. The whole assembly s solated by the use of several layers of paper (Fg. ). Enamelled strands Paper nsulaton Transposton The CTC geometrc parameters used n ths paper are shown n Fg. 2. The strands are descrbed by the axal heght and the radal wdth. Paper nsulaton thckness s represented by. The axal and radal CTC dmensons CTC and CTC can be calculated as shown n formula (.) and (.2). Here descrbes the number of strands n radal drecton. The separator thckness s neglected. CTC Pap EL Pap (.) CTC Pap R EL Pap (.2)
Nordc Insulaton Symposum - Nord-IS - Trondhem, Norway, June 9-2, 2 axal wndng drecton radal wndng drecton t EL T=N R t EL T CTC 2.2 Bendng stffness s a quantty that descrbes the ablty of a bar-shaped geometry to wthstand bendng. If the curvature of a CTC segment between two axal spacers s neglected the lnear bendng theory for a straght beam can be used to descrbe dsplacement dependent on appled load and beam geometry. As an approach the electromagnetc force that s dstrbuted over the whole conductor volume wll be replaced by a lne load q actng on ts surface. Fg. shows a unformly loaded beam wth smply supported ends. Maxmum deflecton n the mddle of the beam can be calculated by (.), [2]. The bendng stffness as product of modulus of elastcty E and second moment of nerta s the only quantty that contans parameters of the geometry and materal of cross-secton. As s the bendng axs, can be calculated for a massve cross-secton after (.4). d Pap sngle strand paper nsulaton separator paper-layers aganst each other (Fg. ). Ths avods movement of the sngle layers aganst each other under the nfluence of bendng and allows the smplfcaton of rectangular tubes. Furthermore the transposton s neglected. The second moment of nerta for a sngle strand EL, can be calculated by the use of (.6). EL, EL EL 2 (.6) The second moments of nerta for the ndvdual paper layers Pap,, can be calculated as shown below: Pap,, 2 Pap,,2 2 Pap,, n 2 I T 2d H 2d T H I T 4d H 4d T 2d H 2d I T 2nd H 2nd t EL T d Layer n Layer 2 Layer q L q H T 2 n d H 2 n d y The effectve second moment of nerta for n nsulaton layers can be summed up accordng to (.7). It s shown that paper layers can be summared to a homogeneous nsulatng layer wth a total thckness Pap =. y max 5 q L 5 84 E I 84 4 4 q L S (.) (.4) To regard nfluence of paper nsulaton on the radal mechancal strength t thus s suffcent to focus on the bendng stffness of CTC. The effectve bendng stffness, for a composte cross-secton can be calculated accordng (.5) f the sngle cross-sectons are not bonded. In the case of a CTC that means the sngle strands can move frctonless aganst each other or aganst paper nsulaton. S E I (.5),eff, Fg. 4 shows the composte cross-secton of a CTC where sngle paper layers are approxmated by rectangular tubes, as the spral wrappng of nsulaton paper strpes cannot be respected. Thus n a layer wndng wthout radal spacers the wndng s axal clampng force wll press all top and bottom CTC I Pap,,eff Pap,, n 2 2 I T 2nd H 2nd T H T 2dPap H 2dPap T H (.7) So the effectve bendng stffness for a CTC wth 2 R strands and paper layers can be calculated by (.8). S S S CTC,,eff Cu,,eff Pap,,eff E I E I Cu Cu,,eff Pap Pap,,eff E 2N I E I Cu R EL, Pap Pap,,eff (.8) 2. To vsuale the contrbuton of paper nsulaton stffness to effectve CTC bendng stffness the quotent Z accordng was defned, (.9). CTC,,eff Cu,,eff Pap,,eff Cu,,eff (.9) 4
Nordc Insulaton Symposum - Nord-IS - Trondhem, Norway, June 9-2, 2 4 dfferent CTC geometres wth varyng radal wdth T accordng to Table were examned. Fg. 5 shows the quotent Z n a D surface plot for these CTC geometres. Therefore paper thckness Pap and radal number of strands per stack r s vared. It can be seen that especally for a hgh number of strands n radal drecton the nfluence of paper nsulaton on effectve bendng stffness ncreases: Z reaches values up to 8. The reason here for s the smaller radal wdth EL of strands what leads to a sgnfcant decrease n radal bendng stffness Cu,,eff. It s clear that wth the ncrease n total paper thckness Pap the paper bendng stffness rses and hence Z rses. So the smplfed analytc consderaton proofs an nfluence of paper on radal bendng stffness of CTC. But the smplfcatons that were necessary are responsble for a manly qualtatve character of the analytc results. complexty (Fg. 6). The (, )-plane dvdes the CTC n the mddle between the two axal stcks. The (, )- plane les between the two stacks of strands. As only radal forces and thus deformatons wll be smulated t s suffcent to regard one stack. Table 2 shows geometrc parameters for the FEA CTC model (Fg. 7). h EL (mm) 7.65 axal strand heght t EL (mm).22 radal strand wdth N r strands per stack L ( ) 5 angle of CTC segment between 2 axal stcks b L (mm) 2 wdth of axal spacer n amuthal drecton t L (mm) 5 radal wdth of axal spacer D K (mm) 2 core outer dameter N Pap number of paper layers d Pap (mm).6 total paper thckness L (r,)-plane (r, )-plane T (mm) 2 4 6 N r 5 5 2 5 4 5 6 H (mm) 5 N 2 d Pap (mm),5 2 E Cu (GPa) E Pap (GPa) 5 As the analytc consderatons only respect lnear materal propertes and a straght CTC a more detaled model was bult by the use of a FEA based mechancal smulaton program. But also here some smplfyng assumptons have to be accepted: Transposton wll stll reman neglected to smplfy D geometry. They are supposed to have no sgnfcant nfluence on radal bendng stffness of CTC. The spral paper wrappng s approxmated by rectangular tubes of paper as already shown n Fg. 4. Frcton and dynamc effects lke nerta are neglected.. The FEA model respects a CTC segment between two axal stcks. To speed up computaton tme two symmetry planes were ntroduced to reduce the model s.2 The model conssts of dfferent materals: The copper strands, the nsulatng paper and pressboard for the axal spacers. For copper and paper nonlnear stress stran curves were mplemented to reale the nonlnear materal characterstc under the nfluence of bendng. The spacer s mplemented wth a lnear modulus of elastcty, as t only has the functon to support the CTC and no plastc compresson wll occur. To descrbe a nonlnear materal the FEA Software needs the plastc part of correspondng stress stran characterstc whch can be calculated from the data of a tensle test as shown n (.) and (.) wth mechancal stress, the stran, and the lnear modulus of elastcty. (.) (.) Two dfferent sorts of copper were used for smulaton wth both havng a modulus of elastcty of E = GPa but dfferent plastc stran stress data (Fg. 8 left). Copper9 has a yeld stress of 9 MPa, whle Copper5 has a yeld stress of 5 MPa. To regard the nfluence of paper agng state dfferent plastc stress stran curves were used n smulaton (Fg. 5
Nordc Insulaton Symposum - Nord-IS - Trondhem, Norway, June 9-2, 2 8 rght). Hereby aged paper has half of the maxmum tensle stress compared to the new paper, whle aged2 paper has the same plastc data but a bsected lnear modulus of elastcty. 4 2 copper9 copper5,,,2,,4 plast. stran [mm/mm]. In FEA Software the CTC model had to be fxed by approprate boundary condtons to gan reasonable smulaton results. Thus deformaton of the nner face of the axal stck was restrcted n all spatal drectons what relates to the rgd constrant of the transformer core. All cross sectons at the symmetry cut planes had to be restrcted n normal respectvely amuthal drecton. As FEA Software only allows exctaton by force or pressure on sngle faces, the radal Lorent force densty actng n the complete volume of all copper strands (.2) has to be approxmated. Here for a pressure EL actng n nward radal drecton on the sngle strands was chosen. To smulate plastc materal behavor wth FEA the pressure EL has to be appled n the form of a ramp (.). Smulaton tme was set to s as n a statc structural smulaton ths tme has no effect on results. f J B (.2) r,,4,8,2 plast. stran [mm/mm] N m rn tel t t ˆ N EL r EL 2 m r s s n p t f dr p (.) Dscretaton of CTC model was done by dfferent types of mesh elements: For the strands quadratc D Sold-Elements (Sold86) were used wth 2 elements n radal drecton. The paper layers were meshed wth Sold-Shell Elements (Solsh9) usng element n paper thckness drecton, Fg.. Shell elements are well suted to model thn structures where one dmenson s much smaller than the other two dmensons []. = 8 6 4 2 new paper (E=5.4 GPa) paper aged (E=5.4 GPa) paper aged2 (E=2.7GPa) = = = = Smulatons were done for a CTC wth geometrc parameters accordng to Table 2. For both copper speces Cu9 and Cu5 the materal property of paper nsulaton was vared: Frst smulaton was done wthout paper to have a reference of the behavor of the blank CTC. The followng smulatons were done wth the plastc stress stran curves for the paper nsulaton accordng to the rght part of Fg. 8. Fg. shows a D plot of the deformed CTC model colored accordng to total deformaton. To compare effect of dfferent paper speces on radal deformaton a deformaton probe on the nnermost strand was placed, Fg.. In Fg. 2 the maxmum deformaton n nwards radal drecton of ths probe s shown. Due to logarthmc scale of the y-axs the lnear elastc range s logarthmcal shaped. If pressure EL s ncreased the CTC suddenly collapses what results n a dramatc ncrease of deformaton. Due to FEA convergence problems after runnng over ths pont of nstablty smulaton was stopped. However the smulaton results are suffcent to evaluate the nfluence of dfferent paper agng states on radal bendng stffness. The crtcal pressure EL,crt a the pont of nstablty s summared n Table : For both copper speces the new paper results n an ncrease by about 45% compared to a CTC wthout paper wrappng. The use of aged paper wth ts lower plastc stran curve but the same modulus of elastcty as the new paper has no effect on crtcal pont. The plot of maxmum equvalent stress n all paper layers (Fg. ) reveals that both new paper and aged paper have the same curve. So the aged paper wll experence the pont of damage earler as ts plastc stress stran curve ends wth a lower breakng stress of about 4 MPa compared to the new paper wth about 8 MPa. The aged2 paper wth bsected modulus of elastcty leads to a decrease of EL,crt compared to the aged paper. So manly the lnear part of paper stress stran curve s responsble for rgdty of the CTC model. The plot maxmum equvalent stress of all strands n Fg. 4 shows how all three paper models delay the sudden ncrease of mechancal stress n the strands. Pont of nstablty s shfted to hgher pressure EL what s equal to an ncrease n radal bendng stffness of CTC. 6
Nordc Insulaton Symposum - Nord-IS - Trondhem, Norway, June 9-2, 2 EL,crt deformaton probe EL,crt EL,crt Cu9 no 58 new 8 4, aged 8 4, aged2 72 24, Cu5 no 74 new 9 47, aged 8 45,9 aged2 9 25,6 bottom vew top vew, Cu9_no_paper Cu9_paper_new Cu9_paper_aged Cu9_paper_aged2, Cu5_no_paper Cu5_paper_new Cu5_paper_aged Cu5_paper_aged2 E- 2 4 6 8 2 45 4 5 25 Cu9_paper_new Cu9_paper_aged Cu9_paper_aged2 Cu5_paper_new Cu5_paper_aged Cu5_paper_aged2 Two methods of evaluatng the nfluence of paper nsulaton on the radal bendng stffness of CTC were proposed. The frst analytc approach only respects lnear materal propertes for a CTC segment between to axal spacers. Curvature s neglected as well as the spral paper wrappng. It was shown that the nfluence of paper ncreases wth number of sngle strands n radal drecton and total paper thckness. The second FEA based approach respects nonlnear materal behavor and the curvature of the CTC segment. Spral paper wrappng stll remans neglected. Smulaton results show that pont of nstablty s ncreased for the CTC by the paper nsulaton. Manly the lnear modulus of elastcty of paper s responsble for CTC rgdty f thckness and number of layers reman constant. An aged paper model wth decreased plastc stress stran data shows no sgnfcant dfference compared to the new paper model. A drect comparson of results from analytc and FEA approach s not possble. The analytc approach enables a quck estmaton of nfluence from geometrc dmensons whle the FEA model addtonally respects nonlnear materal behavor and thus provdes more detaled results. 2 5 5 2 4 6 8 2 4 2 8 Cu9_no_paper Cu9_paper_new Cu9_paper_aged Cu9_paper_aged2 Cu5_no_paper Cu5_paper_new Cu5_paper_aged Cu5_paper_aged2 [] M. Fard, V. Nabae, S.A. Mousav, and M. Mohammad, Modelng of Contnuously Transposed Cable n power transformer for fault analyss based on FEM, Proc. Int. Conf. Electrcal Machnes and Systems, ICEMS, 29, pp. -4. [2] S. Tmoshenko, Strength of Materals, Part and Part 2, D. Van Nostrand Co., Inc., Toronto New York London, 2nd Edton, 94. [] ANSYS, Inc., ANSYS Mechancal APDL Element Reference, SAS IP Inc., USA, Release 4., November 2. 6 4 2 2 4 6 8 2 7