A simplified 2D equivalent model for magnetic wire array

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The calculation of demagnetization factor and apparent permeability of magnetic core is essential in induction and fluxgate sensors design [1].The calculations and analyses of demagnetization factors for ellipsoidal and non-ellipsoidal shapes of single element were studied in detail in various publications [1]-[8]. For instance, demagnetization factor for sphere can be analytically calculated and it is 1/3 [2]. Demagnetization factor for single ellipsoidal has closed-form equation [1], but the apparent permeability depends on material permeability (only for very high permeability it depends only on geometry). Demagnetization factor for non-ellipsoidal shape of single element, for example solid cylindrical wire and hollow cylinder, cannot be described in single closed-form formula, however approximations using curve fitting were used [1], [4] and [6]. Magnetic permeability of wire has high impact on the demagnetization [7]-[8]. Using finite element method (FEM) or complex analytical modeling are common methods to take into account magnetic permeability effects on the demagnetization. However, effects of finite relative magnetic permeability can be taken into account using curve fitting [4] and [6] in the closed form equation of the demagnetization factor. The demagnetization factors are categorized to two cases: Fluxmetric (ballistic or central) and magnetometric; magnetometric demagnetization factor is of main interest for sensors applications. we have calculated demagnetization factor and corresponding apparent permeability for multiwire arrays using magnetostatic 3D FEM in [1]. The effect of distance between magnetic wires on the demagnetization factor and apparent magnetic permeability was studied at different relative magnetic permeability. We also compared the simulations with experimental results up to 91 wires.Unfortunatelly the complexity of the 3D model is increasing fast with the number of wires and 3D FEM simulations of large wire arrays are impossible. This is why we propose novel simplified equivalent 2D model for wire arrays in this paper. The simplified model consists of hollow cylinders and it is axisymmetrical.The basic geometry of the 2D equivalent model is shown in Fig. 1. The model was derived for hexagonal arrays, but similar models can be made for other lattices. The mean diameter of each hollow cylinder is calculated by the average of the circumdiameter (maximal diameter) and the innerdiameter (minimal diameter) of the hexagon.The thickness of each hollow cylinder is obtained assuming the same volume as the volume of wires on the circumference of the corresponding hexagon. The validity and the accuracy of our model is tested by the calculations shown in Fig. 2, with relative material permeability of mr-max= 500. The first calculation shows the magnetometric (using volume integral of the flux density B) and fluxmetric (using surface integral of B in the midplane) apparent permeability of the central wire in the array of n wires as a function of n. The results of our simplified 2D model fit very well the full 3D FEM simulations.We have studied apparent permeability and magnetometric demagnetization factors of arrays up to 2000 wires as a function of wire aspect ratio, wire distances and material properties.Two different hexagonal and square arrangements for wires will be considered in the full paper. Increasing wire distance increases apparent permeability and decreases demagnetization factor, which is similar to comparison between solid cylinder and hollow cylinder for demagnetization factor [3]. We will also show the results of the verification measurements on array of Permalloy wires.The simulation results are also valid for the arrays of microwires and even large arrays of nanowires, as for the practically achievable distances the wire interaction is mainly magnetostatic.This study was supported by the Grant Agency of the Czech Republic within the Nanofluxgate project (GACR GA20-27150S). **