2022年电流互感器模型 .pdf

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1、A currenttransformermodelingYann Le FlochCedrat Recherche,Meylan, FranceLaboratoire dElectrotechnique de Grenoble, Saint Martin dHeres,FranceChristophe Gue rinCedrat Recherche,Meylan, FranceDominique BoudaudSchneider Electric, Grenoble, FranceGe rard MeunierLaboratoire dElectrotechnique de Grenoble,

2、 Saint Martin dHeres,France, andXavier BrunotteCedrat Recherche,Meylan, FranceKeywordsElectrical circuits, Transient ?ow, Nonlinearity, Magnetic ?eldsAbstractThis paper presentsthe modelingof a current transformer byvarious methodswith theFLUX3D software. The techniqueusedis basedon the ?nite elemen

3、tmethod coupledwith electriccircuits. A magnetic scalarpotential reducedversusT0formulation eT0f 2 f Ttaking into accounttheelectriccircuits with anair-gap is usedfor this purpose.The air-gap is describedeitherby a thinvolume region or by a surface region.1. IntroductionThe study deals with a curren

4、t transformer used in a low voltage circuitbreaker made by SchneiderElectric (seePlate 1).FLUX3D software allows usto take into account nonlinear transient magnetic problems coupled withelectric circuits. This software enablesto model in an effective way the currenttransformers by introducing a thin

5、 volume air-gap. This solution can be usedwhen modeling simple devices such as the current transformer presented inthis paper. When modeling more complex devices, dif?cultiesdue to thegeometrical description and themeshing of the thin volume air-gaps canoccur.We would then like to model the thin vol

6、ume air-gap in another way by usingshell elements which are surface elements with a thickness. Thus, a newversion which allows us to take into account electric circuits and surface air-gaps hasbeendeveloped.Wewill describethe improvements obtained,thanksto the introduction of a surface air-gap with

7、the electric circuits.The current issue and full text archive of this journal is available athttp:/ currenttransformermodeling505COMPEL: The InternationalJournalfor Computationand MathematicsinElectricaland Electronic Engineering,Vol. 21 No. 4, 2002, pp. 505-511.qMCB UP Limited,0332-1649DOI 10.1108/

8、03321640210437761名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 1 页，共 7 页 - - - - - - - - - 2. Descriptionof the currenttransformerThe transformeris constituted by a magnetic core surrounded by twosecondary coils connected in series. The ?nite element modeling (in ti

9、mestepping and circuit equations) represents 1/8th of the device (seeFigure 1).The simulated curves correspond to a primary sinusoidal excitation eI0?11; 137A and f ? 50HzTand apurely resistive load. The total simulation time(40ms) correspondsto the transient modeof the sensor.3. Formulation:T0f2 fT

10、he present formulation eT0f 2 f T(Biro et al., 1993;Meunier et al., 1998)totreat couplings between electric circuits and magnetic devices is shown inFigure 2.Plate 1.The current transformerused for the modelingFigure1.Description of thecurrent transformerCOMPEL21,4506名师资料总结 - - -精品资料欢迎下载 - - - - - -

11、 - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 2 页，共 7 页 - - - - - - - - - In magnetic circuit (Vt):H ? 2 grad ef TB ? mHIn air and in air-gap (V0)H ?k? 1;mXIkt0k2 grad ef TB ? m0Hwhere m is the number of inductors.t0kis calculatedin theV0region with aunit current in the inductor k,suchas:t0kn ? 0

12、on G? Vt V0With this assumption, the relation between current and voltage is (Piriou andRazek,1992):Uk? RkIktZV0t0k?B?tdVTo compute t0k, we have two solutions. The ?rst solution is to use edgeelements,which is natural in order to take into account the surface conditiont0kn ? 0 on G. The other one is

13、 to compute nodal t0k. For this purpose, wecompute t0kin the air (V0) such ast0k? h0k2 grad e dfkTwhere h0kis the magnetic?eld dueto aunit current in the inductor k, calculatedwith Biot and Savart sformula (nodal value) in the air (V0), dfkthe reduced-total increment (Simkin and Trowbridge, 1979;Luo

14、ng et al., 1996)calculatedwith a unit current in the inductor k such as: grad edfkTn ? h0kn onG ? Vt V0:Figure2.Formulation T0f 2 fcon?gurationA currenttransformermodeling507名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 3 页，共 7 页 - - - - - - - - - Remark on cancella

15、tionerrorsWe do not have cancellation errors becausewe compute and useh0kin the air(V0)and the total scalar potential in the magnetic circuit (Vt). Thus, we canusea non linear material in the magnetic circuit without any cancellation errors.Thus, on G, we respect the conditions: t0kn ? 0 because t0k

16、? h0k2grad edfkTand we compute dfkas follows: h0kn ? grad edfkTnNow, we will seewhich solution we chooseto model our current transformer.4. Modelingair-gapsOne of the dif?cultiesof the current transformer modeling is to take intoaccountthin air-gaps.In our case,for a 40mm long devicethe air-gapthick

17、nessis 50mm. This scale difference makes the device dif?cultto geometricallydescribe it and to mesh it (seeFigure 3).Thus, we would like to model thin volume air-gaps by surface air-gapswith a thickness. For this purpose, we have to use surface elements withpotential jump (shell element). Our experi

18、ence in magnetostatics leads usto use shell elements witha nodal approximation(Guerin et al., 1994).The solutionis then to use the formulationpresented above withthenodal t0kwhich enables us to describe the air gap withshell elements.First, we will present in a short way the shell elements and its l

19、imitationand, in a secondpart, the t0kcomputation.4.1 ShellelementsAs mentioned before, we can model air-gaps with shell elements.Indeed,themagnetic ?eld is mainly normal to the air-gap surface,sothere is ajump of themagnetic scalar potential in thethickness direction. Therefore,the new elementwill

20、be asurface elementin the plane of the air-gap and will have doublenodes(seeFigure 4).Each coupleof double nodeswill have the samecoordinates andFigure3.Surface mesh of theair-gap and the magneticcircuitCOMPEL21,4508名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 4 页，

21、共 7 页 - - - - - - - - - the shell element willbe considered as a conventional prismatic element(Guerin et al., 1994).However, shell elementshave thickness limitations. Theratio betweentheair-gap thickness and the devicelength has to besmaller than1/10 and higher than 1/105.We now usetheseshell eleme

22、ntswith theT0f 2 f formulation with anodalt0kpresentedbelow.4.2 t0kComputation with shellelementsWhen we computedfkfor the inductor k,we impose:dfkib2 dfkit? constant ?1 on shell elements(Notation on Figure 4).This constant is the current in theinductor k (1A) becauseof the Ampe re slaw (Luong et al

23、., 1996).This reduced-totalincrement enablesus to makethe potential jump betweenthe two sides of the air-gap surface (seeFigure 5).Figure4.Prismatic element (a),shell element withpotential jump (b)Figure5.Reduced-totalincrement(dfB 2) calculated with aunit current in theinductor B2 and thesurface me

24、sh of themagnetic circuitA currenttransformermodeling509名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 5 页，共 7 页 - - - - - - - - - 5. The resultsWe have performed two simulations, one with a thin volume air-gap and anedget0k,and another with a surface air-gap and ano

25、dal t0k. We comparethesetwo computations with measurementsgiven by SchneiderElectric. For the thinvolume air-gap and thesurfaceair-gap, the currents obtainedarenot sinusoidaldue to the saturation of themagnetic material (seeFigure 6).The shapesof theresulting waves for both simulations are the same

26、(see Figure 6) and areaccuratein comparison with measurements(lessthan 5per centof variation onthe whole simulation period).The more accuratethe provided B(H)curve of themagnetic material, especially at the saturation bend,the smaller the variationbetweensimulation and measurements.The contribution

27、of the surface air-gap leads to strong improvements interms of computation time which is divided by four (see Table I) withoutmodifying the results (seeFigures 6 and 7).In Figure 7,the isovalues of the Flux density in the air are almost identical,made smoother with the surface air-gap. This differen

28、ce is due to the t0kcalculated with edge elementsused with the volume air-gap and with nodalelementsused with the surface air-gap.MethodDegreesof freedomComputing time (CPU)Volume air-gap187216 h 12min 56sSurfaceair-gap53541 h 25min 55sTableI.Computation timefor the variousmethods(for 80timesteps)wi

29、th PentiumIII 450MHz, 512Moof RAMFigure6.Induced current in thesecondary circuit (B2)COMPEL21,4510名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 6 页，共 7 页 - - - - - - - - - 6. ConclusionFLUX3D software is therefore a powerful tool for modeling and analyzing lowvoltag

30、e current transformers. The dif?cultiesof the current transformermodeling is to take into accountthin air-gaps.To avoid the problems linked toair-gap geometrical descriptions and meshing, a new computation of t0kisintroduced which allows us to take into account both circuit equations andsurface air-

31、gaps with thickness. This contribution strongly improves problemdescription (geometry and mesh of thin volume regions), computation times(four times faster) as well as the smoothnessof the isovalue results.ReferencesBiro, O., Preis, K., Renhart, W., Vrisk, G. and Richter, K.R. (1993),“Computation of

32、 3D currentdriven skin effect problem using a current vector potential”,IEEE Trans. Magn., Vol. 29No. 2, pp. 1325-8.Guerin, C.,Tanneau, G., Meunier, G.,Brunotte, X. and Albertini, J.B.(1994),“Three dimensionalmagnetostatic ?nite elementsfor gaps and iron shells using magnetic scalar potentials”,IEEE

33、 Trans. Magn., Vol. 30 No. 5, pp. 2885-8.Luong, H.T., Mare chal, Y., Labie, P., Guerin, C. and Meunier, G. (1996), “Formulationofmagnetostatic problems in termsof source, reduced and totalscalar potentials”,Proccedings of 3rd InternationalWorshop on Electric AndMagnetic Field, Liege(Belgium), 6-9 Ma

34、y 1996,pp. 321-6.Meunier, G.,Luong, H.T. and Mare chal, Y. (1998),“Computation of coupled problem of 3D eddycurrent and electrical circuit by using T02 T 2 fformulation”,IEEE Trans. Magn.,Vol. 34 No. 5, pp. 3074-7.Piriou, F. and Razek,A. (1992),“A non-linear coupled 3D model for magnetic ?eld and el

35、ectriccircuit equations”,IEEE Trans. Magn., Vol. 28 No. 2, pp. 1295-8.Simkin, J.and Trowbridge, C.W. (1979),“On the used of a total scalar potential in the numericalsolution of ?eld problems in electromagnetics”,Int. J.Num. Meth. Eng.,Vol. 14,pp. 423-40.Figure7.Flux density (Tesla) attime t ? 0:033s withvolume air-gap (a)andwith surface air-gap (b)A currenttransformermodeling511名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 7 页，共 7 页 - - - - - - - - -