# current transformers 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)
(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

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