One of the most important parts of a fluxgate sensor is the magnetic core. This work is dedicated to the study of fluxgate sensors based on the nanocrystalline wire as a core
and comparing them to the performance of amorphous wire with chemical composition (Co94Fe6)75Si15B10 [1]. Both wires were manufactured by the Institute of Technical Physics, Iasi,
Romania. Compared to the previous studies on nanocrystalline materials, we demonstrate that proper processing of nanocrystalline alloys can bring sensors with properties comparable to
the traditionally used amorphous alloys, but with better stability at elevated temperatures.
The amorphous Fe73,5Si13,5B9Cu1Nb3 master alloy wires 0.5m in length, and 0.12 mm in diameter were annealed by DC pulse to achieve the nanocrystalline structure. Contrary to other
researchers we found problems with surface oxidation in air, so our annealing was performed in Argon. During the annealing and cooling, we applied mechanical stress to achieve
transverse anisotropy. The character of changing Hc varies depending on tension as shown in Fig.1. Higher tension increases anisotropy and reduces the longitudinal permeability (Fig. 2).
Best results were shown by wires annealed under tension 9 g and 13 g with the current in the range 485 mA up to 510 mA. In these cases, coercivity was in the range of 7 A/m up to 17
A/m at 1 kHz. Annealed wires were used as a core for fluxgate sensors of the Förster and Vacquier types and with ring core.
As an example, Förster type sensor with amorphous (Hc = 20 A/m) wire as a core had max. sensitivity of 0.88 V/mT and min. the noise of 164pT/√Hz at 1 Hz.
By using nanocrystalline wire with Hc = 17A/m we achieved lower sensitivity (0.7 V/mT) but lower noise (111 pT/√Hz). The nanocrystalline core with lower Hc (9 A/m) had higher
sensitivity (1.1 V/mT) but higher noise (204 pT/√Hz) compared to the amorphous core. And the core with Hc = 13A/m had almost the same results as the amorphous wire.
This study was supported by the Grant Agency of the Czech Republic within the Nanofluxgate project (GACR GA20-27150S). 

**References:** 

[1] P. Ripka, D. Hrakova, “Multiwire Parallel Fluxgate Sensors”, IEEE Transactions on Magnetics. 2022, 58(2), 1-5. ISSN 0018-9464. DOI 10.1109/TMAG.2021.3093017 
[2] Y.-F. Li, M. Vazquez, D.-X. Chen, “Circular magnetization process of nanocrystalline wires as deduced from impedance measurements”, Journal of Applied Physics 97, 124311 (2005) 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/dc0c8aa0034b27a0fcea032449a5d896.png)

Creep induced Magnetic Anisotropy of Nanocrystalline Wire as a Core of Fluxgate Sensor

Arrays of magnetically soft nanowires and microwires have potential sensor applications [1, 2]. We have reported nanowire array manufactured by electrodeposition of permalloy into polycarbonate membranes [3]. In this paper we report permalloy nanowire and microwire arrays produced using alumina and silicon membranes with hexagonally ordered pores with constant pitch. These membranes allow to optimize the nanowire array properties such as coercivity, apparent permeability and amplification factor. Finite element modelling of nanowire arrays is demanding due to the computational complexity. In [4] we calculated apparent permeability and demagnetization factor of single wire and small wire arrays. We have also proposed simplified 2D equivalent model allowing to calculate properties of wire arrays very efficiently [5] and verified the validity of this model by full 3D simulations and also experimentally on arrays of up to 91 permalloy microwires. In this paper we apply this model to manufactured arrays of 3 mm diameter containing 100 000 to 40 million wires with diameters between 180 nm and 2.5 μm and compare simulation results to measurements by SQUID. Figs. 1 and 2 show examples of the measured hysteresis loops with red line showing the slope corresponding to apparent amplitude permeability. The values calculated by FEM using constant permeability model show significant deviations especially at low density arrays. We will show that this difference is caused by the nonlinear shielding effect: the field in the outermost wires is much stronger and thus they are saturated first. Because of that the saturation process propagates fast from the periphery towards inside array which can be shown on non-linear model.

![](https://assets.underline.io/uploads/markdown_image/1/image/4818309d16f32532f9a688524ccc7d9a.jpg)

Fig. 1 Hysteresis loop of the array of 180 nm diameter 100 μm long wires with 480 nm pitch. Measured apparent amplitude permeability μa = 45, calculated incremental μai = 8.6.

![](https://assets.underline.io/uploads/markdown_image/1/image/7178e6971765f6f4dd7a495b3982dd0b.jpg)

Fig. 2 Hysteresis loop of the array of 2.5 μm diameter 200 μm long wires with 4.2 μm nominal pitch. Measured apparent amplitude permeability μa = 5.5, calculated incremental μai = 3.8.

Apparent permeability of ordered magnetically soft nanowire and microwire arrays

Some of the early parallel fluxgate sensors had cores made of magnetic wires and this design is still used in many devices, including gradiometers and low-cost sensor arrays in magnetoelastic torque meters [1]. Wire and also multiwire cores are also being used in transverse fluxgates [2]. Despite their larger noise and lower temperature stability compared to ring-core fluxgates, wire-core parallel fluxgates have several advantages [3]: 1. their sensing direction is precisely defined by the direction of the sensor core 2. they have excellent spatial selectivity which is important for the detection of magnetic particles [4] 3. High sensitivity allows decreasing the core length 4. The large shape anisotropy gives them low crossfield error [5]. The sensitivity of ring-core and race-track fluxgate sensors is well understood and described in the literature [6,7], while such study on wire-cored sensors is missing. We offer it in this paper for the case of voltage output. We calculate the sensitivity as a function of core and pickup coil geometry and configuration of wire core by Finite-element modeling and compare these results with the measurements. We analyze both existing configurations (single-core, Foerster and Vacquier) and propose multiwire fluxgates of two types: with collective excitation and pickup coils and with individual pickup coils. This paper is based on model cores made of 36 mm long, 0.2 mm diameter Permalloy wires with chemical composition Ni78Fe15Cu4Mo3. The wires were annealed for 3 hours in a dry hydrogen atmosphere at 1080 °C to obtain near-zero magnetostriction. Our simulation models were based on constant permeability and also on the magnetization curve estimated from the measurement on the wire. Sensitivity and noise properties were measured for sensors containing 2 to 12 wires with different distances between the wires. The model can be upscaled for magnetic rods and downscaled for bundles of microwires and arrays of nanowires. However, the results strongly depend on the aspect ratio of the used wires. Fig. 1 shows examples of the measured fluxgates. The multiwire type has a 3D printed honeycomb bobbin which allows insertion of different configurations of wires in glass or ceramic capillaries. For each configuration, we found the optimum working point (excitation frequency and amplitude) to maximize the sensitivity and minimize the noise. In order to simplify the problem, we did not use the parametric amplification of the output voltage. We studied the effect of pickup coil geometry by FEM modeling and verified the results experimentally: the optimum coil length is 80 % of the core length and the coil diameter should be minimized as the sensitivity is inversely proportional to the pickup coil diameter as shown in Fig.2. In this regard’s fluxgate behaves differently from induction sensors without excitation, which shows the increase of sensitivity with the pickup coil diameter. This dependence does not change with the wire permeability. This decrease of the sensitivity is the reason, why multiwire fluxgate with large common pickup coil has larger noise (0.7 to 14 nT/√Hz at 1 Hz, depending on configuration) than single-core (180 pT/√Hz@1Hz) and Vacquier type (51 pT/√Hz@1Hz) sensors. The effective solution to this problem is to use an individual pickup coil for each wire instead of a collective coil common for all wires. The sensitivity of the multiwire fluxgate is increasing with the wire pitch and the noise is inversely proportional to the wire pitch. This is caused by two factors: 1. With increasing wire distance their magnetostatic interaction is decreasing, which results in lower demagnetization 2. With increasing distance the noise correlation between wires is decreasing and the total noise is reduced by the averaging effect, which is stronger for independent noise sources This study was limited to the initial selection of optimum geometry of the wire core and pickup coil of the voltage-output parallel fluxgate. Further increase of sensitivity and noise reduction can be achieved by tuning the pickup coil by the parallel capacitor to the mode of parametric amplification at second harmonics. This study was supported by the Grant Agency of the Czech Republic within the Nanofluxgate project (GACR GA20-27150S).

**References:**

[1] P. Ripka, Magnetic sensors and magnetometers, Artech House, 2001 [2] M. Butta, M. Vazquez, R. P. del Real et al., AIP Advances Vol. 10, (2020) [3] M. Janosek, "Parallel fluxgate magnetometers," in High Sensitivity magnetometers, Springer, 2017 [4] J. Lei, T. Wang, C. Lei et al. Applied Phys. Lett. Vol. 102, 022413 (2013) [5] P.Ripka, W. Billingsley, Sens. Act. A Vol. 81, p. 176-179 (2000) [6] C. Hinnrichs, J. Stahl, K. Kuchenbrandt et al., IEEE Trans. Magn. Vol. 37, p. 1983 (2001) [7] L. Perez, I. Lucas, C. Aroca et al., Sens. Act. A Vol. 130, p. 142 (2006)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/197886eeec5ca839cbd0ec0c6da03e29.jpg)
Figure 1 Experimental fluxgate sensors: a) single core, b) Vacquier, c) multiwire

![](https://assets.underline.io/uploads/markdown_image/1/image/b8f674df538b8b10c95c207e3fa16986.jpg)
Figure 2 Relative flux of the 36 mm long pickup coil with diameter D as a function of its relative diameter D/d – values calculated by 2D FEM for 36 mm long, d = 0.2 mm diameter magnetic wire core

 Sensitivity and Noise of Multiwire Parallel Fluxgate Sensors

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).

**References:**

[1] P. Ripka, V. Grim, M. Mirzaei, acc. for publ. in JMMM, 2021 [2] M. Stafl, Electrodynamics of Electrical Machines, Publisher: Iliffe , 1st March 1968 [3] S. G. Sandomirskii, Russian Electrical Engineering, vol. 80, pp. 109–112, 2009 [4] S. G. Sandomirskii, Russian Electr. Eng. 84, pp. 376-381, 2013 [5] V. F. Matyuk, A. A. Osipov, and A. V. Strelyukhin, Russian Journal of Nondestructive Testing, vol. 43, no. 3, pp. 154–162, 2007 [6] M. Nirei, S. Suzuki, K. Tashiro, and H. Wakiwaka, Journal of the Japan Society of Applied Electromagnetics and Mechanics, vol. 17, no. 3, pp. 449-452, 2009 [7] D.-X. Chen, J. A. Brug, and R. B. Goldfarb,”, IEEE Trans. on Magn.. 27, , pp. 3601-3619, 1991 [8] D.-X. Chen, E. Pardo, A. Sanchez, “Fluxmetric and magnetometric demagnetizing factors for cylinders,”JMMM 306, pp. 135–146, 2006

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/83e685bd809a1efe045b10918b3686dc.jpg)
Fig. 1 The wire array and its equivalent 2D model based on the hollow cylinders

![](https://assets.underline.io/uploads/markdown_image/1/image/94cfaa14032d2e088b2f7c7b1141e354.jpg)
Fig. 2 Apparent permeability (magnetometric using volume integral and fluxmetric using surface integral) of the wire array as a function of the number of wires, n. Calculated by 3D FEM and by the 2D equivalent model

 A simplified 2D equivalent model for magnetic wire array

Introduction Rogowski coils (RC) are used for the measurement of large alternating current, particularly as a short circuit detector in electric substations. Another application is in electricity meters. A regular, air-core RC cannot be saturated and therefore its dynamic range is limited only by noise voltage and supply voltage of the accompanying integrator or ADC [1]. An RC used for precise current measurement should have a consistent cross-sectional area and turn-to-turn spacing, limiting the winding arrangement to a single layer. The sensitivity of such coils is poor, causing a low signal-to-noise ratio [2]. Our proposed design adds a low-permeability ferromagnetic core to improve the sensitivity and thus reduce the required number of turns. Ferromagnetic core has been already tested for different types of RC, e.g. self-integrating RCs which are used for measurement of short current pulses. In this case increasing the core permeability decreases the low frequency limit of self-integrating operation [3]. Contrary to that, our target application is precise current sensing at the mains frequency and its harmonics. Such a sensor should be able to work with high accuracy (typical requirement is 0.5% or 0.2%) in the nominal range, as well as above this range with reduced accuracy with partly saturated core as a fault current detector [4]. Because the coil does not require a load resistor on its winding, there is no concern about overheating during overload conditions.However, there are some open questions, how will the ferromagnetic core affect the frequency characteristics of the RC, its ability to suppress external currents and resistance against the position deviations of the measured (primary) conductor. Several different materials, either in the form of a finished toroidal core or a 3-D printer filament filled with magnetic particles were tested, both theoretically by a FEM simulation and practically by measurement. FEM Simulation We have made a series of 3D FEM simulations to evaluate the effect of the core permeability on rejection of external currents and position of the measured conductor. In both cases the perfectly homogenous coil would be absolutely immune against both these influences. Practical coil always has uneven winding density, in particular between the first and last turn [5]. Also the cross-sectional area of individual turns varies slightly. We simulated the role of these imperfections in crosstalk from an external conductor as a function of its position and core permeability. In general, the error is diminishing with increasing permeability due to demagnetization effect. However, for practical permeability of 20 the error reduction is only very small (from 0.23 % to 0.15 %). The permeability effect on the resistance against the position deviations of the measured (primary) conductor is even lower and for permeability of 20 it is negligible. The effect of nonuniformity of the core permeability along its circumference was found to be insignificant as well. Samples used for measurement The first batch of test specimens (labeled A, B, C) are toroidal powder cores with an outer diameter of 79 mm and rated relative permeability of 26 (Kool Mju and Sendust materials) or 14 (Sendust). The winding is electrostatically shielded by a copper adhesive tape, care was taken to avoid the creation of a short-circuited turn. We also tested 3D printed cores from two types of plastic filament filled with magnetic particles and for the full paper we plan to test resin-based cores of various sizes. Measurement procedure Two important sensor characteristics were measured: linearity and frequency response. Test current was provided by a Techron 7548 power amplifier with an additional step-down transformer. Reference current measurement was done by a LEM Ultrastab current sensor (IT60S for frequency response, IT1000S for linearity). Two different measurement approaches were tested: either integrating the induced voltage with a Lakeshore 480 fluxmeter in AC mode, or measuring the voltage directly with a HF2LI lock-in amplifier. Both methods give similar results in amplitude, but the phase can only be measured by a lock-in amplifier. A primary conductor with a loop length of 10 meters was passed once through the measured coil to minimize the influence of the return current. Fig. 1 shows the frequency dependence: there is no observable sensitivity drop due to frequency dependence of the permeability. The sensitivity increase at high frequencies is due to the resonance with parasitic coil capacitance and its amplitude could be reduced by a parallel damping resistor [6] at the expense of heat being generated in the associated circuitry. Fig. 2 shows the measured linearity error at 93 Hz, which is in a good agreement with the catalogue values of permeability vs. flux density in the core. A nominal flux line length of 0.196 m was used to convert flux density to primary current. Conclusion We have demonstrated that adding a ferromagnetic core to a RC of the classical dI/dt type is beneficial for its sensitivity, which would allow the design of very precise RCs for lower currents. The RC linearity is decreased, but remains acceptable for most applications. The amplitude error caused by self-resonance can be reduced by standard methods such as a damping resistor, or by using a different material with high core loss. Acknowledgement This study was supported by the Grant Agency of the Czech Republic within the Nanofluxgate project (GACR GA20-27150S).

**References:**

[1] S. Tumanski, Meas. Sci. Technol. 18, p. R31, 2007 [2] J. D. Ramboz, IEEE Trans. Instrum. Meas. 45, p. 515, 1996. [3] J. D. Zhu, Q. G. Zhang, J. B. Jia et al., Plasma Sci. Tech. 8, p. 457l, 2006. [4] L. A. Kojovic, M. T. Bishop, and D. Sharma, IEEE Trans. Ind. Appl. 49, pp. 8, 2013. [5] L. Ferkovic, D. Ilic, and R. Malaric, IEEE Trans. Instr. Meas. 58, p. 122, 2009. [6] M. H. Samimi, A. Mahari, M. A. Farahnakian et al., IEEE Sens. J. 15, p. 651 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b316cd355ace0c3db4ef22e26ec8fa2a.jpg)
Fig. 1 Frequency response

![](https://assets.underline.io/uploads/markdown_image/1/image/fd59bddadd4847af0a850bbd099da4d6.jpg)
Fig. 2 Linearity

 Rogowski Coil with Ferromagnetic Core

OOB: Magnetic Sensors and Devices II

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Presentations

Creep induced Magnetic Anisotropy of Nanocrystalline Wire as a Core of Fluxgate Sensor

Apparent permeability of ordered magnetically soft nanowire and microwire arrays

Sensitivity and Noise of Multiwire Parallel Fluxgate Sensors

A simplified 2D equivalent model for magnetic wire array

Rogowski Coil with Ferromagnetic Core

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