Current Transformer Modelling.

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Current Transformer Modelling.

Yann Le Floch(1)(2), Christophe Guérin(1), Dominique Boudaud(3), Gérard Meunier(2), Xavier Brunotte(1) Cedrat Technologies, Meylan, France ; (2) Laboratoire d'Electrotechnique de Grenoble, UMR 5529 INPG/UJF - CNRS, ENSIEG, France; (3) Schneider Electric, Grenoble, France.

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Keywords - Air gap, Electric circuit, Transient, Nonlinear material, Nodal element, Shell element, Time stepping, Reduced magnetic scalar potential. Abstract - This paper presents the modelling of a current transformer by various methods with the FLUX3D software. The technique used is based on the Finite Element Method coupled with electric circuits. A magnetic scalar potential reduced versus T0 formulation (T0φ -φ) taking into account the electric circuits with an air-gap is used for this purpose. The air-gap is described either by a thin volume region or by a surface region.

I. Introduction

With this assumption, the relation between current and voltage is [3]:

Figure 2: Description of the current transformer.

II. Description of the current transformer

The transformer is constituted by a magnetic core surrounded by two secondary coils connected in series. The finite element modelling (in time stepping and circuit equations) represents 1/8th of the device (see figure 2). The simulated curves correspond to a primary sinusoidal excitation (I0 = 11.137 A and f = 50 Hz) and a purely resistive load. The total simulation time (40ms) corresponds to the transient mode of the sensor.

The study deals with a current transformer used in a low voltage circuit breaker made by Schneider Electric (see figure 1). FLUX3D software allows to take into account nonlinear transient magnetic problems coupled with electric circuits. This software enables to model in an effective way the current transformers by introducing a thin volume air-gap. This solution can be used when modelling simple devices such as the current transformer presented in this paper. When modelling more complex devices, difficulties due to the geometrical description and the meshing of the thin volume air-gaps can occur. We would like then to model the thin volume air-gap in. another way by using shell elements which are surface elements with a thickness. Thus, a new version which allows to take into account electric circuits and surface air-gaps has been developped. We will describe the improvements obtained thanks to the introduction of a surface air-gap with the electric circuits.

The present formulation (T0φ - φ) [1] [2] to treat coupling between electric circuits and magnetic devices is : In magnetic circuit (Ωt) : φ) H = -grad(φ B = µH

Figure 1: Photo of the current transformer used for the modelling.

Figure 3: Formulation T 0 φ - φ configuration.

III. Formulation: T0 φ - φ

In air and in air-gap (Ω0) : B = µo H With : m the number of inductors. t0k is calculated in the Ω0 region with a unit current in the inductor k, such as: t0k x n = 0 on Γ = Ωt ∩ Ω0

To compute t 0k , we have two solutions. The first solution is to use edge elements, which is natural in order to take into account the surface condition t0k x n = 0 on Γ. The other one is to compute nodal t0k. For this purpose, we compute t0k in the air (Ω0) such as : t0k = h0k - grad(δφk) Where : - h0k is the magnetic field due to a unit current in the inductor k, calculated with Biot and Savart's formula (nodal value) in the air (Ω0). - δφk is the reduced-total increment [4] [5] calculated with a unit current in the inductor k such as : grad(δφk) x n = h0k x n on Γ = Ωt ∩ Ω0. Thus, on Γ, we respect the conditions: tok x n = 0 because t0k = h0k - grad(δφk) and we compute δφk as follow : h0k x n = grad(δφk) x n Now, we will see which solution we choose to model our current transformer.

IV. Modelling air-gaps

One of the difficulties of the current transformer modelling is to take into account thin air-gaps. In our case, for a 40 mm long device the air-gap thickness is 50 µm. This scale difference makes the device difficult to geometrically describe it and to mesh it (see figure 4). Thus, we would like to model thin volume air-gaps by surface air-gaps 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 elements. Firstly, we will present in (continued on page 6)

Number 38 - January 2002 - CEDRAT - CEDRAT TECHNOLOGIES - MAGSOFT

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Yann Le Floch , Christophe Guérin , Dominique Boudaud , Gérard Meunier , Xavier Brunotte(1) Cedrat Technologies, Meylan, France ; (2) Laboratoire d'Electrotechnique de Grenoble, UMR 5529 INPG/UJF - CNRS, ENSIEG, France; (3) Schneider Electric, Grenoble, France. (1)(2)

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a short way the shell elements and its limitation and, in a second part, the t 0k computation. A. Shell elements As mentioned before, we can model air-gaps with shell elements. Indeed, the magnetic field is mainly normal to the airgap surface, so there is a jump of the magnetic scalar potential in the thickness direction. Therefore, the new element will be a surface element in the plane of the air gap and will have double nodes (see figure 5). Each couple of double nodes will have the same coordinates and the shell element

Figure 4: Surface mesh of the airgap and the magnetic circuit. We use now these shell elements with the T0φ -φ formulation with a nodal t0k presented below. B. t0k Computation with shell elements When we compute δφ k for the inductor k, we impose: δφkib - δφkit = constant = 1 on shell elements (Notation on figure 5). This constant is the current in the inductor k (1A) because of the Ampère's law [5].

Figure 5: Prismatic element (a), Shell element with potential jump (b). will be considered as a conventional prismatic element [6]. However, shell elements have tickness limitations. The ratio between the air-gap tickness and the device length has to be smaller than 1/10 and higher than 1/105.

This reduced-total increment enables to make the potential jump between the two sides of the air-gap surface.

V. The results

We have performed two simulations, one with a thin volume air-gap and an edge t 0k , and another with a surface air-gap and a nodal t0k. We compare these two computations with measurements given by Schneider Electric.

For the thin volume air-gap and the surface air-gap, the currents obtained are not sinusoidal due to the saturation of the magnetic material (see figure7). The shapes of the resulting waves for both simulations are the same (see figure 7) and are accurate in comparison with measurements (less than 5% of variation on the whole simulation period). The more accurate is the provided B(H) curve of the magnetic material, especially at the saturation bend, the smaller is the variation between simulation and measurements. The contribution of the surface airgap leads to strong improvements in term of computation time which is divided by 4 (see table I) without modifying the results (see figures 7 and 8). On figure 8, the isovalues of the flux density in the air are almost identical, made smoother with the surface air-gap. This difference is due to the t0k calculated with edge elements used with the volume airgap and with nodal elements used with the surface air-gap.

VI. Conclusion

FLUX3D software is therefore a powerful tool for modelling and analyzing low voltage current transformers. The difficulties of the current transformer modelling is to take into account thin air-gaps. (continued on page 8)

Figure 7 : Induced current in the secondar y circuit (B2).

Figure 6: Reduced-total increment (δφ B2 ) calculated with a unit current in the inductor B2 and the surface mesh of the magnetic circuit.

Table 1: Computation time for the various methods (for 80 time steps) with Pentium II 450 MHz, 512Mo of RAM.


New

& Axial Field Electric Machine for Energy Storage.

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Olivier Gergaud, Bernard Multon, Hamid Ben Ahmed, LÉSiR - Antenne de Bretagne de l'ENS de Cachan.

Conclusion

The 3D finite element computation allowed us to evaluate the 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 element computation allowed us to evaluate the efforts of the reluctant 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,

- An effort of the radial moment of the order of magnitude of milliNewton-meter for Laplace efforts,

- An axial stiffness of 5 N/mm for reluctant efforts.

Figure 6: Normal component of the magnetic flux density under the disk of average radius, at different heights.

Figure 7: Normal component of the magnetic flux density under the disk.

Figure 8: Orthoradial component of the magnetic flux density in the air-gap.

Figure 9: Radial component of the magnetic flux density in the air-gap.

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Yann Le Floch , Christophe Guérin , Dominique Boudaud , Gérard Meunier , Xavier Brunotte(1) Cedrat Technologies, Meylan, France ; (2) Laboratoire d'Electrotechnique de Grenoble, UMR 5529 INPG/UJF - CNRS, ENSIEG, France; (3) Schneider Electric, Grenoble, France. (1)(2)

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To avoid the problems linked to airgap geometrical descriptions and meshing, a new computation of t0k is introduced which allows to take into account both circuit equations and surface air-gaps with thickness. This contribution strongly improves problem description (geometry and mesh of thin volume regions), computation times (4 times faster) as well as the smoothness of the isovalue results. References

[1] O. Biro, K. Preis, W. Renhart, G. Vrisk, K.R. Richter, Computation of 3D Current Driven Skin Effect Problem Using a Current Vector Potential , IEEE Trans. Magn., vol. 29 n°2 (1993), [2] G. Meunier, H.T. Luong, Y. Maréchal, Computation of Coupled Problem of 3D Eddy Current and Electrical Circuit by using T0 - T φ Formulation, IEEE Trans. Magn., vol. 34 n°5 (1998),

[3] F. Piriou and A. Razek, A Non-linear Coupled 3D Model for Magnetic Field and Electric Circuit Equations, IEEE Trans. Magn., vol. 28 n°2 (1992), [4] J. Simkin and C.W. Trowbridge, On the used of a total scalar potential in the numerical solution of field problems in electromagnetics, Int. J. Num. Meth. Eng., Vol. 14 (1979), [5] H.T. Luong, Y. Maréchal, P. Labie, C. Guerin and G. Meunier , Formulation of magnetostatic problems in terms of source, reduced and total scalar potentials, Proccedings of 3rd International Worshop on Electric And Magnetic Field, Liege (Belgium), 6-9 May 1996, [6] C. Guerin, G. Tanneau, G. Meunier, X. Brunotte, J.B. Albertini, Three dimensinal magnetostatic finite elements for gaps and iron shells using magnetic scalar potentials, IEEE Trans. Magn., vol. 30 n°5 (1994).

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Figure 8: Flux density (Tesla) at time t=0.033s with volume air-gap (a) and with surface air-gap (b).
