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# Current Transformer Modelling. Yann Le Floch(1)(2), Christophe Guérin(1), Dominique Boudaud(3), Gérard Meunier(2), Xavier Brunotte(1) (1) Cedrat Technologies, Meylan, France ; (2) Laboratoire d'Electrotechnique de Grenoble, UMR 5529 INPG/UJF - CNRS, ENSIEG, France; (3) Schneider Electric, Grenoble, France. Keywords - Air gap, Electric circuit, With this assumption, the relation Transient, Nonlinear material, Nodal between current and voltage is [3]: element, Shell element, Time stepping, Reduced magnetic scalar potential. Abstract - This paper presents the To compute t 0k , we have two modelling of a current transformer by solutions. various methods with the FLUX3D The first solution is to use edge software. The technique used is elements, which is natural in order based on the Finite Element Method to take into account the surface coupled with electric circuits. A Figure 2: Description of the current condition t0k x n = 0 on Γ. magnetic scalar potential reduced transformer. The other one is to compute nodal versus T0 formulation (T0φ -φ) taking into account the electric circuits with t0k. For this purpose, we compute an air-gap is used for this purpose. t0k in the air (Ω0) such as : The air-gap is described either by a II. Description of the current t0k = h0k - grad(δφk) thin volume region or by a surface transformer region. The transformer is constituted by a Where : magnetic core surrounded by two - h0k is the magnetic field due to a I. Introduction secondary coils connected in series. unit current in the inductor k, The study deals with a current The finite element modelling (in time calculated with Biot and Savart's transformer used in a low voltage stepping and circuit equations) formula (nodal value) in the air (Ω0). circuit breaker made by Schneider represents 1/8th of the device (see - δφk is the reduced-total increment Electric (see figure 1). FLUX3D figure 2). The simulated curves [4] [5] calculated with a unit current software allows to take into account correspond to a primary sinusoidal in the inductor k such as : nonlinear transient magnetic excitation (I0 = 11.137 A and f = 50 grad(δφk) x n = h0k x n on Γ = Ωt ∩ Ω0. problems coupled with electric Hz) and a purely resistive load. The circuits. This software enables to total simulation time (40ms) Thus, on Γ, we respect the model in an effective way the corresponds to the transient mode conditions: tok x n = 0 current transformers by introducing of the sensor. because t0k = h0k - grad(δφk) and we a thin volume air-gap. This solution compute δφk as follow : can be used when modelling simple III. Formulation: T0 φ - φ h0k x n = grad(δφk) x n devices such as the current The present formulation (T0φ - φ) [1] transformer presented in this paper. [2] to treat coupling between Now, we will see which solution we When modelling more complex electric circuits and magnetic choose to model our current devices, difficulties due to the devices is : transformer. geometrical description and the In magnetic circuit (Ωt) : meshing of the thin volume air-gaps H = -grad(φ) φ B = µH IV. Modelling air-gaps can occur. We would like then to One of the difficulties of the current model the thin volume air-gap in . In air and in air-gap (Ω0) : transformer modelling is to take another way by using shell into account thin air-gaps. In our B = µo H elements which are surface case, for a 40 mm long device the elements with a thickness. Thus, a air-gap thickness is 50 µm. This new version which allows to take With : m the number of inductors. scale difference makes the device into account electric circuits and t0k is calculated in the Ω0 region with difficult to geometrically describe it surface air-gaps has been a unit current in the inductor k, such and to mesh it (see figure 4). developped. We will describe the as: t0k x n = 0 on Γ = Ωt ∩ Ω0 improvements obtained thanks to Thus, we would like to model thin the introduction of a surface air-gap volume air-gaps by surface air-gaps with the electric circuits. with a tickness. For this purpose, we have to use surface elements with potential jump (shell element). Our experience in magneto-statics leads to use shell elements with a nodal approximation [6]. The solution is then to use the formulation presented above with the nodal t 0k which enables to describe the air gap with shell Figure 1: Photo of the current Figure 3: Formulation T 0 φ - φ elements. Firstly, we will present in transformer used for the modelling. configuration. (continued on page 6) Number 38 - January 2002 - CEDRAT - CEDRAT TECHNOLOGIES - MAGSOFT $ Current Transformer Modelling. Yann Le Floch , Christophe Guérin , Dominique Boudaud , Gérard Meunier , Xavier Brunotte(1) (1)(2) (1) (3) (2) (continue) (1) Cedrat Technologies, Meylan, France ; (2) Laboratoire d'Electrotechnique de Grenoble, UMR 5529 INPG/UJF - CNRS, ENSIEG, France; (3) Schneider Electric, Grenoble, France. a short way the shell For the thin volume air-gap and the elements and surface air-gap, the currents its limitation obtained are not sinusoidal due to and, in a second the saturation of the magnetic part, the t 0k material (see figure7). The shapes of computation. the resulting waves for both simulations are the same (see A. Shell elements figure 7) and are accurate in comparison with measurements As mentioned before, we (less than 5% of variation on the can model air-gaps with shell Figure 4: Surface mesh of the air- whole simulation period). The more elements. Indeed, the magnetic gap and the magnetic circuit. accurate is the provided B(H) curve field is mainly normal to the air- of the magnetic material, especially gap surface, so there is a jump of We use now these shell elements at the saturation bend, the smaller the magnetic scalar potential in the with the T0φ -φ formulation with a is the variation between simulation thickness direction. Therefore, the nodal t0k presented below. and measurements. new element will be a surface element in the plane of the air gap B. t0k Computation with shell elements The contribution of the surface air- and will have double nodes (see gap leads to strong improvements figure 5). Each couple of double When we compute δφ k for the in term of computation time which nodes will have the same inductor k, we impose: δφkib - δφkit = is divided by 4 (see table I) without coordinates and the shell element constant = 1 on shell elements modifying the results (see figures 7 (Notation on figure 5). This and 8). constant is the current in the inductor k (1A) On figure 8, the isovalues of the flux because of the Ampère's density in the air are almost law [5]. identical, made smoother with the surface air-gap. This difference is This reduced-total increment due to the t0k calculated with edge enables to make the potential elements used with the volume air- jump between the two sides of gap and with nodal elements used Figure 5: Prismatic element (a), the air-gap surface. with the surface air-gap. Shell element with potential jump (b). V. The results VI. Conclusion will be considered as a conventional We have performed two FLUX3D software is therefore a prismatic element [6]. However, simulations, one with a thin volume powerful tool for modelling and shell elements have tickness air-gap and an edge t 0k , and analyzing low voltage current limitations. The ratio between the another with a surface air-gap and transformers. The difficulties of the air-gap tickness and the device a nodal t0k. We compare these two current transformer modelling is to length has to be smaller than 1/10 computations with measurements take into account thin air-gaps. and higher than 1/105. given by Schneider Electric. (continued on page 8) Figure 7 : Induced current in the secondar y circuit (B2). Table 1: Figure 6: Reduced-total increment Computation (δφ B2 ) calculated with a unit current time for the in the inductor B2 and the surface various mesh of the magnetic circuit. methods (for 80 time steps) with Pentium II 450 MHz, 512Mo of RAM. Number 38 - January 2002 - CEDRAT - CEDRAT TECHNOLOGIES - MAGSOFT & New Axial Field Electric Machine for Energy Storage. (continue) Olivier Gergaud, Bernard Multon, Hamid Ben Ahmed, LÉSiR - Antenne de Bretagne de l'ENS de Cachan. Conclusion - An effort of the radial moment of - An axial stiffness of 5 N/mm for The 3D finite element computation the order of magnitude of milli- reluctant efforts. allowed us to evaluate the Newton-meter for Laplace efforts, distribution of the magnetic flux density in the air-gap, due to the inductor coil. The interaction between the inductor magnetic field and the induced currents, allowed an appropriate evaluation of the parasitic efforts that are exerted on the magnetic suspension that operates perfectly centered (vertical, axial and angular), but also in the case of various non - alignment. Finally, a 2D finite Figure 6: Normal component of the Figure 7: Normal component of the element computation allowed us to magnetic flux density under the disk of magnetic flux density under the disk. evaluate the efforts of the reluctant average radius, at different heights. type. When applied to the validation mock up (0.1 Nm at 10,000 rpm), the computations showed that at rating operating conditions of the machine, the magnetic bearings should be dimensioned in order to support: - A radial and axial stiffness of the order of magnitude of Newton per millimeter, Figure 8: Orthoradial component of the Figure 9: Radial component of the magnetic flux density in the air-gap. magnetic flux density in the air-gap. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Current Transformer Modelling. Yann Le Floch , Christophe Guérin , Dominique Boudaud , Gérard Meunier , Xavier Brunotte(1) (1)(2) (1) (3) (2) (continue) (1) Cedrat Technologies, Meylan, France ; (2) Laboratoire d'Electrotechnique de Grenoble, UMR 5529 INPG/UJF - CNRS, ENSIEG, France; (3) Schneider Electric, Grenoble, France. To avoid the problems linked to air- [3] F. Piriou and A. Razek, A Non-linear gap geometrical descriptions and Coupled 3D Model for Magnetic Field and meshing, a new computation of t0k Electric Circuit Equations, IEEE Trans. Magn., is introduced which allows to take vol. 28 n°2 (1992), into account both circuit equations (a) [4] J. Simkin and C.W. Trowbridge, On the and surface air-gaps with thickness. used of a total scalar potential in the This contribution strongly improves numerical solution of field problems in problem description (geometry and electromagnetics, Int. J. Num. Meth. Eng., mesh of thin volume regions), Vol. 14 (1979), computation times (4 times faster) [5] H.T. Luong, Y. Maréchal, P. Labie, C. Guerin as well as the smoothness of the and G. Meunier , Formulation of magnetostatic isovalue results. problems in terms of source, reduced and total scalar potentials, Proccedings of 3rd References International Worshop on Electric And Magnetic (b) [1] O. Biro, K. Preis, W. Renhart, G. Vrisk, Field, Liege (Belgium), 6-9 May 1996, K.R. Richter, Computation of 3D Current Driven Skin Effect Problem Using a Current [6] C. Guerin, G. Tanneau, G. Meunier, X. Vector Potential , IEEE Trans. Magn., vol. 29 Brunotte, J.B. Albertini, Three dimensinal n°2 (1993), magnetostatic finite elements for gaps and iron shells using magnetic scalar potentials, [2] G. Meunier, H.T. Luong, Y. Maréchal, IEEE Trans. Magn., vol. 30 n°5 (1994). Figure 8: Flux density (Tesla) at time Computation of Coupled Problem of 3D Eddy t=0.033s with volume air-gap (a) and Current and Electrical Circuit by using T0 - T - with surface air-gap (b). φ Formulation, IEEE Trans. Magn., vol. 34 n°5 (1998), Number 38 - January 2002 - CEDRAT - CEDRAT TECHNOLOGIES - MAGSOFT