France



Strain gauges are sensors that measure a strain generated by a stress applied to an object, metal and semiconductor gauges are widely used. The sensitivity of strain gauges is expressed using the gauge factor, which is about 2 for metal gauges and about 200 for semiconductor gauges. Recently, the SI (stress-impedance) sensor using an amorphous alloy with a gauge factor reaching 1260 has been developed, but it has not been put to practical use due to problems of handling and the influence of external magnetic fields [Shen et al., 1997]. In this paper, we report on the development of a flexible SI sensor element with a magnetic shield to suppress the influence of external magnetic fields.&lt;br/&gt;We use the SI element, which is a negative magnetostrictive amorphous wire (FeCoSiB) with a wire length of 2 mm and a diameter of 13 μm, and a flexible film substrate. The amorphous wire is placed on the film and is connected to the electrodes via lead wires from both ends of the wire using gold deposition technique. We manufacture the SI device with a magnetic shield. The shielded SI element uses permalloy plating around the amorphous wire.&lt;br/&gt;Figure 1 shows the results of measuring the impedance change by applying a stress to the element. The 18% impedance change was obtained when a tensile stress of 45 MPa was applied, which corresponds to a gage factor of 354.&lt;br/&gt;As shown in Figure 2, the effect of the magnetic shield was affected by an external magnetic field of 20 μT without the shield, but with the shield, the rate of change of the impedance of the SI element was reduced to less than 2% even at 100 μT.&lt;br/&gt;In this study, the SI element was miniaturized and mass-produced by placing a 2 mm long amorphous wire on a flexible film for use as a contact sensor. In addition, we have developed a SI device that can reduce the influence of the Earth&#39;s magnetic field by applying a magnetic shield. In the future, we plan to investigate the response to stress.

**References:**

K. Mohri, T. Kohsawa, K. Kawashima, H. Yoshida and L.V. Panina, Magneto-inductive effect (MI effect) in amorphous wires. IEEE Transactions on Magnetics, 28(5), pp.3150-3152, (1992).&lt;br/&gt;&lt;br/&gt;L.P. Shen, T. Uchiyama, K. Mohri, E. Kita and K. Bushida, Sensitive stress-impedance micro sensor using amorphous magnetostrictive wire, IEEE Transactions on Magnetics, 33(5), pp. 3355-3357, doi: 10.1109 / 20.617942 (1997).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a413c180d4ced40ef4e3085c07b5727d.jpg)
Fig. 1: Effect of stress on impedance change.

![](https://assets.underline.io/uploads/markdown_image/1/image/5e541388252910e1f469dafdde8036c6.jpg)
Fig. 2: Shielding effect.


INTERMAG 2021

 Development of High-Sensitivity SI Sensors Using Flexible SI Devices

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2022 Joint MMM-INTERMAG

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Strain gauges are sensors that measure a strain generated by a stress applied to an object, metal and semiconductor gauges are widely used. The sensitivity of strain gauges is expressed using the gauge factor, which is about 2 for metal gauges and about 200 for semiconductor gauges. Recently, the SI (stress-impedance) sensor using an amorphous alloy with a gauge factor reaching 1260 has been developed, but it has not been put to practical use due to problems of handling and the influence of external magnetic fields [Shen et al., 1997]. In this paper, we report on the development of a flexible SI sensor element with a magnetic shield to suppress the influence of external magnetic fields. We use the SI element, which is a negative magnetostrictive amorphous wire (FeCoSiB) with a wire length of 2 mm and a diameter of 13 μm, and a flexible film substrate. The amorphous wire is placed on the film and is connected to the electrodes via lead wires from both ends of the wire using gold deposition technique. We manufacture the SI device with a magnetic shield. The shielded SI element uses permalloy plating around the amorphous wire. Figure 1 shows the results of measuring the impedance change by applying a stress to the element. The 18% impedance change was obtained when a tensile stress of 45 MPa was applied, which corresponds to a gage factor of 354. As shown in Figure 2, the effect of the magnetic shield was affected by an external magnetic field of 20 μT without the shield, but with the shield, the rate of change of the impedance of the SI element was reduced to less than 2% even at 100 μT. In this study, the SI element was miniaturized and mass-produced by placing a 2 mm long amorphous wire on a flexible film for use as a contact sensor. In addition, we have developed a SI device that can reduce the influence of the Earth's magnetic field by applying a magnetic shield. In the future, we plan to investigate the response to stress.

**References:**

K. Mohri, T. Kohsawa, K. Kawashima, H. Yoshida and L.V. Panina, Magneto-inductive effect (MI effect) in amorphous wires. IEEE Transactions on Magnetics, 28(5), pp.3150-3152, (1992). L.P. Shen, T. Uchiyama, K. Mohri, E. Kita and K. Bushida, Sensitive stress-impedance micro sensor using amorphous magnetostrictive wire, IEEE Transactions on Magnetics, 33(5), pp. 3355-3357, doi: 10.1109 / 20.617942 (1997).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a413c180d4ced40ef4e3085c07b5727d.jpg)
Fig. 1: Effect of stress on impedance change.

![](https://assets.underline.io/uploads/markdown_image/1/image/5e541388252910e1f469dafdde8036c6.jpg)
Fig. 2: Shielding effect.

technical paper

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Introduction When an electromagnetic sensor composed of a static excitation coil and a detection coil is moved in a steel tube at high speed, an eddy current is generated in the steel tube. If a defect is present in the steel tube, the defect may be detected by the change in the eddy current. Generally, the heat transfer tubes with ferromagnetism in power plants or oil plants are inspected for defects by moving an internally inserted electromagnetic sensor at a high speed of about 1 m/s [1]. In this research, an inner-type electromagnetic sensor using the static magnetic field that moves inside these steel tubes at high speed and detects on the outer side defect in the steel tubes is proposed. The generated eddy current inside the steel tube by constant velocity movement of the static magnetic field is analyzed by the 3-D nonlinear eddy current FEM analysis using step-by-step method. In addition, verification experiment is also conducted. Inspection Model and Electromagnetic Conditions Fig.1 shows the proposed electromagnetic sensor using only the static magnetic field and an inspection steel tube (STB340SC). This sensor is composed of two static excitation coils in series, a pair of differential detection coils, and a magnetic yoke of permendur material. The direct current of 1 A is passed through two excitation coils in series so that the plus z-direction of the sensor is the S-pole and the minus z-direction is the N-pole. The permendur material is an alloy of iron and cobalt. The two excitation coils and the differential detection coil are 220 turns × 2 and 100 turns × 2, respectively. The outer diameter, length, and thickness of this inspection steel tube are f 19 mm, 460 mm, and 2 mm, respectively. The proposed sensor is inserted in the steel tube at a speed of 1 m/s for minus z-direction. The distance (lift-off: Lo) between the sensor and the steel tube is equal to 0.5 mm. Analysis and Experimental Results Fig.2 shows the distribution of only one layer in the y-direction of eddy current density inside the steel tube when the sensor is moved in the minus z-direction of the steel tube without defect at 1m/s. This figure denotes that the directions of the eddy currents generated on the upper side and the lower side of the steel tube are opposite. Since the lower part of the sensor in the moving direction is the N-pole, the eddy current is generated inside the steel tube near the lower part of the sensor in the direction of canceling the N-pole magnetic field. On the other hand, since the upper part of this sensor is the S pole, an eddy current is generated inside the steel tube near it in the direction of canceling the S-pole magnetic field. In addition, Fig.2 denotes that both the upper and lower eddy currents in the thickness of the steel tube are concentrated on the inner surface of the steel tube by skin effect. Fig.3 shows the distribution of the flux density inside the steel tube when the sensor is moved in the minus z-direction of the steel tube without defect at 1m/s. This figure denotes the flux density inside the steel tube near the lower side of the sensor is distributed in the plus z-direction, since the eddy current is generated inside the steel tube in the direction of canceling the N-pole magnetic field. On the other hand, the flux density on the inner surface of the steel tube near the upper side of the sensor is offset because the eddy current is generated in the opposite direction to when the N-pole of lower side of the sensor passed. However, the distribution of flux density in the steel tube near the central region of the sensor is almost uniform. Fig.4 shows comparison results of analysis and experiment of the steel tube with an outer slit defect in the circumferential direction. The width for the z-direction and the depth for the x-direction of the defect are 10 mm and 0.5 mm, respectively. The horizontal axis of the figure indicates the moving position of the sensor, and the vertical axis indicates the differential voltage output by the pair of detection coils in the sensor. This figure shows that an electromagnetic sensor using only a static magnetic field is detected the outer side defect on the steel tube. Moreover, this figure denotes that the calculated result is in agreement with measurement.

**References:**

[1] M. Zec, R. P. Uhlig, M. Ziolkowski, and H. Brauer, “Finite Element Analysis of Nondestructive Testing Eddy Current Problems with Moving Parts”, IEEE Transactions on Magnetics vol.44, no.8, pp.4785-4794 (2013).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/98aefef80aa754ec6f73137b1cfdd52b.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/c8a51b13b5da909ba1b6aa71e5ae87a4.jpg)

 Examination of insertion type electromagnetic inspection for outer side defect on ferromagnetic steel tube by speed effect using only static magnetic field

Linear variable differential transformer (LVDT) sensors offers many advantages in low power consumption, high sensitivity, low hysteresis, etc. These features allow them to be used in wide range of possible applications in critical environments [1]. However, they suffer from some disadvantages like temperature dependence of output, sensitivity to stray magnetic fields and others. Most of the disadvantages arises from the fact that LVDT sensors sense variation of permeability of magnetic core to external stimuli like magnetic field or temperature. Some of the problems can be solved using glass-coated magnetically bistable microwire as a variable core for linear position sensor. Glass-coated amorphous microwires are composite materials having metallic nucleus covered by glass-coating [2]. They are produced by Taylor-Ulitovski method applying drawing and quenching of molten metal that fills glassy capillary. Having positive magnetostriction, they show magnetic bistability (only two oppose saturation magnetic states are allowed in static state) and switching between the two states appears through the single Barkhaussen jump. This gives us possibility to separate contribution of external magnetic field, temperature, stress or position [3]. Apart from the above mentioned feature, they offer advantages in small size, glass-coating that provides insulation from chemically aggressive environment, or simple, fast and cheap production of large amount of microwire (up to kilometre of wire can be produced from few grams of master alloy). In the given contribution, we show the linear position sensors similar to LVDT based on magnetically bistable microwire. It consists of two coils (excitation and sensing) having single microwire coaxially placed on linearly moving piston. In contrary to classical LVDT sensors, where position defines the amplitude of the signal measured on the sensing coil, we detect the switching field that is proportional to the position. Application of triangular shape of excitation field allows us to measure switching time that is proportional to the switching field. Using fluxgate like sensing system and measuring the positive and negative switching time, we are able to avoid stray fields dependence of linear position piston sensing. We show, how temperature independence of sensing can be solved by proper chemical composition. Such a sensors show no hysteresis and requires quite small power consumption. Proper design allows to obtain sensitivity down to 20 µm within the range 0-6 cm (see fig. 2). However, sensitivity and range can be adjusted by properly selecting the shape of excitation coil. This work was partially supported by Slovak Grant Agencies VEGA 1/0053/19 and APVV-16-0079.

**References:**

[1] D. S. Nyce, Linear Position Sensors: Theory and Application, John Wiley & Sons, Inc. (2004) [2] M. Vázquez: Advanced magnetic microwires. In: Kronmüller, H., Parkin, S. S. P. (eds.) Handbook of Magnetism and Advanced Magnetic Materials, Wiley, Chichester, West Sussex, England (2007) [3] R. Jurc, L. Frolova, D. Kozejova et al. Sensoric application of glass-coated magnetic microwires in Magnetic Nano- and Microwires, Second edition, ed. by M. Vázquez, Woodhead Publishing, (2020), ISBN: 9780081028322.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/805a4234dba066457bf55254449c52f5.jpg)
SEM figure of glass-coated microwire (left), Schematic view of linear position sensor with bistable microwire (right)

![](https://assets.underline.io/uploads/markdown_image/1/image/7112c598e9108dbb47fe8ef47ee2420f.jpg)
Dependence of magnetic response of linear position sensor on piston position

 Linear position sensor using magnetically bistable microwire

A Wiegand sensor is composed of a strip of Wiegand wire and a pickup coil. From literature. the Wiegand wire was initially made of NiFe alloy [1-2]. However, it was reported that the wire made of FeCoV alloy offered the optimum performance in generating the Wiegand effect [3]. Diameter of the Wiegand wire is about 0.25 mm. Heat treatment such as quenching and annealing are applied to the wire in order to change grain size of the alloy. After heat-treatment, cold-treatment such as stretching and twisting are then introduced to the wire to make exhibition of different coercivity in its surface region and center region, respectively [1-6]. When external magnetic field passing through axial direction of the wire with different coercivity reaches a certain threshold, the “soft layer” with relatively low coercivity of the wire can fast reverse its direction of magnetization. The fast magnetization reversal of the soft layer leads to generation of large Barkhausen jump, resulting in changes of magnetic flux in the pick-up coil such that the pulse voltage is induced with 10 µs pulse width and up to 8 V peak voltage [5]. This pulse voltage is also called Wiegand pulse. The peak voltage and pulse width of the pulse is reported to be independent of driving frequency, so it can be used as a zero-speed sensor. Reported in [3], the Wiegand sensor was able to generate electrical power of 600 nJ, which could be used for energy harvesting driving small power components. The material of the Wiegand sensor is stable and cheap, and the pulse energy did not reduce after million times of switching. Advantaged by the above features, the Wiegand sensors have been used in rotary speed measurement [7], Batteryless hall sensors [3], and multi-turn absolute encoders [8]. In this study, we apply Wiegand sensor to linear positioning system so that Wiegand impulse can be used as battery-less limiting switch or reference mark. The focus of this work is placed on the design of the system’s magnetic circuits so that the Wiegand pulse can be triggered effectively. In our experiments, data acquisition NI USB-6216 with up to 400 Ks/s sampling rate and 16-bit resolution to measure Wiegand pulse in different magnetic circuits are employed. The relationship between different magnetic circuits and pulse peak were investigated and obtained in this study. To design the magnetic circuits for the Wiegand sensor, generation of Wiegand pulse if of primary importance. In previous work [1][6], it was reported that to be able to repeatedly generate Weigand pulse, external magnetic field should pass through the wire’s axial axis. To be able to reach this objective, appropriate strength of external magnetic field should be first applied through the axial axis of Wiegand wire so as to magnetize the wire, followed by decreasing external magnetic field to let Wiegand wire have remanence (Br). Lastly, opposite direction of external magnetic field should be applied to the Wiegand wire so that the wire’s soft layer can suddenly reverse its polarity to yield Wiegand pulse. Based on the Wiegand effect mentioned above, we use single magnet to produce external magnetic field. The direction of magnetic field around both sides of the magnet is opposite to that around center of the magnet. The placements of the Wiegand sensor and the magnet are shown and depicted in Fig. 1(a) and Fig. 1(b), respectively. The linear motor carrying the magnet is set to move back and forth at a speed of 30 mm/s along Y-axis, from which the mean value of the pulse peak is computed every 50 runs. The relationship between the mean value of peak voltage and the Flying height (H) is shown in Fig. 2(a), whereas the relationship between pulse signal and time is shown in Fig. 2(b). From measurement results, it was observed that when H is lower than 3.5 mm, the triggering of the pulse signal was limited. As the linear motor moved from left to right, it was observed that the Weigand sensor first generated small pulse signal under 0.1 V first, followed by generation of relatively large pulse signal with magnitude over 0.3 V. This phenomenon was investigated through finite-element simulation. When the magnet moving from left to right, the simulated magnetic flux density in the soft layer of the wire is depicted in Fig. 2(c) showing unsymmetrical hysteresis loop according to the change of magnetic flux density. Based on this unsymmetrical hysteresis loop, one can see that the Br values at state 2 and state 4 are significant different, leading to generation of relatively small pulse signal first at state 2 then relatively large pulse signal at state 4. From Fig. 2(b), it was found that when H is below 3.5 mm, the Wiegand pulse could not be triggered. As illustrated in the comparison in Fig. 2(d), when H=1 mm, the whole soft layer can’t perform fast magnetization reversal at the same time as the case when H=5 mm. Through this study, we conclude that should one need to effectively trigger Wiegand pulse, appropriate strength of external magnetic field should be applied and whole soft layer of Wiegand wire not fast reversing at the same time should be avoided.

**References:**

[1] J. R. Wiegand, “Switchable magnetic device,” U.S. Patent No 4,247,601, 1981. [2] J. R. Wiegand, “Method of manufacturing bistable magnetic device,” U.S. Patent No 3,892,118, 1975. [3] Y. Takemura et al., “Batteryless hall sensor operated by energy harvesting from a single Wiegand pulse,” IEEE Transactions on Magnetics, 53.11: 1-6, 2017. [4] S. Abe, A. Matsushita, and M. Naoe, “Annealing and torsion stress effect on magnetic anisotropy and magnetostriction of Vicalloy fine wire,” IEEE Transactions on Magnetics,33.5: 3916-3918, 1997. [5] S. Abe, A. Matsushita, and M. Naoe, “Dependence of large Barkhausen jump on length of a vicalloy fine wire with torsion stress.” IEEE Transactions on Magnetics, 34.4: 1318-1320, 1998. [6] X. Sun, T. Yamada, and Y. Takemura, “Output characteristics and circuit modeling of Wiegand sensor,” Sensors, 19.13: 2991, 2019. [7] E. Zabler and P. Hauler, “Level amplitude output rotary speed transducer,” U.S. Patent No 4,150,314, 1979. [8] W. Mehnert and T. Theil, “Absolute high resolution segment or revolution counter,” U.S. Patent No 8,655,615, 2014.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c5adaac1a38b3f932e9ecb59aa21c950.jpg)
Fig. 1 (a) isometric and (b) top views of placement of Wiegand sensor and magnet.

![](https://assets.underline.io/uploads/markdown_image/1/image/e1c623ee2fbd72fa918ebee60a5a8c04.jpg)
Fig. 2 Measured and simulated results of (a) pulse voltage obtained from the Wiegand sensor, (b) mean value of peak voltage in different H. Magnetic flux density in different location and hysteresis loop of Wiegand wire is shown in (c) exampled by magnetization direction of soft layer when H=1 mm and 5 mm, respectively in (d).

 Effect of Magnetic Circuit on Pulse Voltage Generated by Wiegand Sensor in a Linear Positioning System

Linear and rotary types of positioning systems have been research and development focus in particular to facilitate precision manufacturing with machine tools. Encoders inside the positioning systems are commonly classified into two types, namely optical and magnetic types. The optical type uses light source reflection to target for position in between, whereas the magnetic type uses a magnetic field to measure position of the target. Although optical encoders may have a greater resolution, magnetic encoders have better performance in harsh environments and dust resistance. Moreover, comparing with the same level of product specifications, magnetic encoders have lower cost and fewer components. These advantages make magnetic encoders more commonly used in machine tools [1]. A magnetic encoder usually consists of a sensor head and a magnetic medium, where the medium is composed of a substrate, and a ferromagnetic material covering on its surface, to generate different alternating polarity pattern by magnetizing the ferrite-magnetic medium under an external magnetic field. Even when the external magnetic field is removed, the magnetization polarity could be still maintained [2]. These patterns are not only used in linear positioning systems, but are also used to obtained spindle shaft angular position by assembling a rotary magnetic encoder in robotics joints or rotating machines. Because of its accuracy performance directly affects the workpiece appearance, factors affecting accuracy of the rotating machines play a significant role in the machining process. In the process, an inevitable thermal expansion cannot be ignored under high-speed machining, particularly for the axial elongation of the machine’s spindle shaft. This axial elongation can lead to catastrophic failure in the rotating machine if it is not adequately monitored. Therefore, the magnetizer in this paper aims to present a different pattern arrangement, which was conceived by Shih and Hsiao [3] to verify elongation of the spindle shaft in axial direction. In the present study, the aforementioned alternating polarity pattern is magnetized on CoMnP ferromagnetic material on the magnetic ring in the way as illustrate in Fig.1(a). A NdFeB35 permanent magnet serves as an external magnetization source, and special magnetization head shape are utilized to concentrate the magnetic flux to create the designed magnetic pattern. The magnetic ring is concentrically attached to the spindle shaft accompanied with a magnetoresistive (MR) sensor to establish a measurement system along the shaft’s axial direction as shown in Fig.1(a). This MR sensor is composed of two Wheatstone bridges, each bridge consisting solely of 4 equally arranged MR elements placed above the magnetic ring. The MR elements can read out a sinusoidal or cosinusoidal voltage signal caused by changes in magnetic field when relative displacement occurs between the medium ring and the sensor. However, the angle between the magnetization vector and the element’s current can be altered as the magnetic field signals are affected subsequently due to assembly errors between the sensor and the ring. These problems result in a high position error. In this paper, finite element studies and experiments are adopted to investigate the effect of assembly errors, including flying height, pitch, yaw, and roll errors between the MR sensor head and the magnetic patterns on the ring as illustrated in Fig.1(b) [4]. We aim to reduce position error by optimizing the sensor’s installation. From the studies, the optimized flying height is found to be 0.6 to 0.7 mm from the MR element or equivalently 0.1 to 0.2 mm when package dimensions of the senor head are considered. It is observed that with this optimal flying height, an optimal signal performance can be achieved in which frequency multiplication can be reduced in the measured magnetic signals. As depicted in Fig.2(a), when the sensor head rotates in pitch and yaw directions from 0° to 1.16° with 0.29° and 0.2° steps, respectively, the error percentage is strictly increased by 81.4% and 29.6%, respectively. Moreover, in error percentage of the sensor increases only 3.8% when it rolls from 0° to 1°, implying that sensor assembly in the roll direction possesses more mounting tolerance than other directions. On the other hand, in order to make the measurement system reach to application level for the spindle shaft axial thermal elongation measurement in a high-speed motor (24,000 rpm), a system with a built-in field-programmable gate array to improve firmware sampling rate to 2 kHz is instrumented so that the measured magnetic signals can accurately reflect spindle shaft’s actual elongation from unreal or distorted signal waveform due to insufficient sampling rates. As shown in Fig.2(b), along with temperature measurement, the spindle shaft’s axial displacements measured by the aforementioned system are compared with commercially available eddy current sensor when the spindle rotates from 1,000 to 24,000 rpm. It can be seen that not only the trends are the same, the difference between the two are neglectable. This suggests that the proposed method and system can instantly measure the spindle’s axial thermal elongation in response to the change of temperature in the machine tools.

**References:**

[1] K. Miyashita, T. Takahashi and M. Yamanaka, “Features of a magnetic rotary encoder.” IEEE Transactions on Magnetics, vol. 23, no. 5 pp. 2182-2184, Sep.1987. [2] Y.-J. Luo, E.-T. Hwang, and S.-M. Huang, “Multi-pole magnetization of high resolution magnetic encoder.” Proceedings of Electrical/Electronics Insulation Conference. IEEE, Oct.1993. [3] S.-W. Shih, H.-S. Hsiao, J.-Y. Chang, “The magnetic encoder apparatus and measuring the displacement of the rotation axis.” Taiwan Patent I662255, July. 2018. [4] K.-Y. Peng and J.-Y. Chang, “Effects of assembly errors on axial positioning accuracy for rotating machinery with magnetoresistance-based magnetic encoders” Microsystem Technologies, January, 2021.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/98c95c7d3c2e06a55caa9dbf7f85bfd7.jpg)
Fig. 1 Schematics of (a) magnetic ring with alternative polarities pattern attached to spindle shaft with (b) sensor assembly errors.

![](https://assets.underline.io/uploads/markdown_image/1/image/be5e7d23f770fbe381f60df660b97f80.jpg)
Fig. 2 (a) error percentage at different angles and (b) comparison of the shaft’s measured axial thermal elongation.

 A Magnetization System for Spindle Axial Thermal Elongation Measurements

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

The sensitivity and automation capabilities of modern superconducting quantum interference device magnetometers are currently unmatched [1]. The measured moment values are, however, prone to deviations from their actual value due to geometric effects, namely sample size, shape, and radial offset [2,3]. A knowledgeable operator will correct measured moment values taking these effects into account. The current procedure for the MPMS3 magnetometer is based on an available simulation tool, valid for both SQUID-VSM and DC-scan methods [4]. Still, determining the correction factor requires samples with well-defined geometric shapes together with accurate sample dimensions and the usually difficult to determine radial offset [5]. There is no current solution to correct geometry effects of irregular shaped samples. Via an extensive use of the available Sample Geometry Simulator tool [4], in all available geometries, with sample dimensions and radial offset values limited only by the sample chamber volume, we find a systematic relation between the difference of SQUID-VSM and DC-scan measurements and the corresponding geometric correction factors of the MPMS3 SQUID-VSM device. This relation follows a clear trend, independent of sample size, shape, or radial offset, for a given pair of DC-scan length and SQUID-VSM amplitude values. Exploiting this trend, a geometry-independent correction method is presented and validated by measurements of two different geometries of the same Al piece, two different radial offsets of a Ni square, and metallic Fe powder using a far from optimal sample mounting. In this last case, we improved the initial maximum observed deviations from about 35% to better than 3%, by following the proposed correction method, as shown in Table 1. While the presented methodology requires both DC-scan and SQUID-VSM data, its ease of use on correcting magnetization measurements of irregular shaped samples and unknown radial offset values, can be of wide application by the magnetism/ magnetometry community.

**References:**

[1] M. Buchner, K. Höfler, B. Henne, V. Ney, A. Ney, J. Appl. Phys. Vol. 124, p. 161101 (2018) [2] P. Stamenov, and J. M. D. Coey, Rev. Sci. Instrum. Vol. 77, p. 015106 (2006) [3] Quantum Design MPMS Application Note 1500-015 (2010) [4] Quantum Design MPMS Application Note 1500-020 (2014) [5] G. Morrison, and H-C zur Loye, J. Solid State Chem. Vol. 221, p. 334 (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ddcfcf8354cffc5511d7f0e84ac2a256.jpg)
Simulated correction factors as a function of the relative difference x = (MVSM −MDC) /MVSM for three different pairs of DC-scan lengths and SQUID-VSM amplitudes.

![](https://assets.underline.io/uploads/markdown_image/1/image/b22be110a92d165fa1ffc58f3f5f794f.jpg)
Measured saturation magnetization and respective corrected values of several Fe powder samples. The measurements were performed using a 1mm SQUID-VSM amplitude and a 30mm DC-scan length.

 A geometry-independent moment correction method for the MPMS3 SQUID-VSM magnetometer

MAGNETIQUE is an itinerant scientific exhibition for the general public with three different levels of reading, presented at the Palais de la Découverte in Paris in 2019-20. Through experiments, manipulations and observations, MAGNETIQUE invites the public to discover principles and effects of this omnipresent but unknown phenomenon. Its goal is to share "science in the making" (cf. Jean Perrin) and the scientific process with public: it brings visitors into the researcher's creativity and imagination. MAGNETIQUE includes some sixty interactive experiments specially designed for this exhibition and completed by many videos on touchscreens. Pedagogical experiments are shown to explain the origin of magnetic properties and everyday-life applications. They are essential to explain some of the recent research results. Do-it-yourself experiments invite visitors to be curious and answer many questions about the use of magnetism in computers and data storage. The exhibition is thus organized in a logical and progressive way around 5 thematic blocks: Block 1: Magnetism, where can it be observed? Visitors can test attraction - repulsion between two magnets. They also discover different kind of magnets, some of their fundamental properties, and the principles of the compass. Some experiments with iron powder or ferrofluids are exhibited to visualize the magnetic field produced by a permanent magnet. Visitors can then observe the various behavior of matters under magnetic field, or discover the magnetic field produced by an electric current and test its applications to waste-sorting. This block ends with the exploration of Earth magnetism. Block 2: Magnetism, how can it be explained? This block is devoted to the explanation of the origin of magnetism. Different experiments show characteristics of various magnetic states (diamagnetism, paramagnetism, hard or soft ferromagnetism). Block 3: Magnetism, how can it be useful? This block shows interactions between current and magnetic field. Visitors discover Laplace’s force, and its use in loudspeakers or electric motors. Later, they observe induction’s phenomenon and its applications to produce electricity in bicycle lighting systems or wind turbines. Two experiments demonstrate the existence of eddy currents and their use in truck breaks. The other experiments concern alternative currents: one illustrates the principle of electromagnetic levitation and its application for ultra-fast trains, other ones shed light on the application for induction heating, electrical transformer, Witricity used in wireless chargers, as well as RFID system used in anti-theft devices. Block 4: Magnetism, where to find it in computer? This block only deals with computers: its goal is to show how omnipresent magnetism is in data storage. After having learnt about binary coding, visitors discover how to write and read a byte on a magnetic media thanks to a giant experiment illustrating the principle of data storage. Then, visitors discover magnetic codes hidden in a credit card or a metro ticket, as well as Halbach networks hidden in a magnet. The end of this block is devoted to the explanation of the giant magnetoresistance phenomenon, and its applications in hard disk read heads. Recent research carried out to develop denser, faster and less energy-intensive information storage are discussed through two examples: writing by ultrashort laser pulses and coding in four states. Block 5: Magnetism, what research is being done? This block shows the experimental universe of a research laboratory, the Jean Lamour Institute (IJL), whose facilities allow the development of new nanomaterials with new magnetic properties. Visitors can therefore observe a 3D model of the Depositing and Analyzing nanomaterials under Ultra-high Vacuum (DAUM) platform. An old spectroscopic analysis chamber used in research is also exhibited. Models show crystal growth under ultra-high vacuum, structural and magnetic characterizations of samples and lithography to make three-dimensional nanometric objects. This bloc ends with experiments illustrating applied research topics: innovative sensors with giant magnetoresistance used in ABS, and magnetography as a secure printing technique useful for legal and banking services. The 5th bock is completed by a virtual and interactive immersive approach of the IJL’s DAUM platform developed with the collaboration of the compagny Com par l'image. The objective of this application is to invite public inside a virtual laboratory to share a growth layers deposition experience in order to produce materials with new properties. The scenario takes the sequence of real experiments in the laboratory: atomic deposition under ultra-high vacuum, characterization of in-situ surfaces, lithography … Visitors are then invited to realize by themselves a magnetoresistive virtual sensor used as position sensor in drones. MAGNETIQUE and the VIRTUAL DAUM are two achievements resulting from the commitment of a research laboratory to support out-reach activities in order to establish the difficult link between fundamental research and the general public. Why get involved in this type of mission? Is it a mission for a research laboratory? What about the mission of researchers in the public life? These are some of the questions that will be addressed in this presentation. MAGNETIQUE is available for rental. Contact: helene.fischer@univ-lorraine.fr

**References:**

- Hélène Fischer, Reflets de la Physique, Vol. 57, p. 32 (2018). - Hélène Fischer, Gazette du vide, Vol. 36, p. 04 (2020). - Hélène Fischer, Stéphane Mangin, Découverte, Vol. 430, p. 44 (2020). - Hélène Fischer, Stéphane Mangin, Découvrte, Vol 431, p. 24 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b68fe7a88b7d45bf13cfa0839e512268.png)
Fig.1 : MAGNETIQUE at the Palais de la découverte, copyright Ph. Levy-EPPDCSI

![](https://assets.underline.io/uploads/markdown_image/1/image/17370546461698490bddcdccd19a770a.jpg)
The Virtual DAUM

 MAGNETIQUE and the Virtual Daum, two out-reach activities carried out in a research laboratory: what about researchers’ engagement in knowledge sharing? INVITED

Composite multiferroic materials represent a novel class of functional materials where multiple types of ferroic ordering coexist. Further, coupling between different types of ferroic materials can result in additional ordering, which can be leveraged to fabricate new multifunctional devices.1, 2 One-dimensional composite magnetoelectric nanofibers, free of substrate-based constraints provide an ideal platform for achieving magnetoelectric coefficients that are orders of magnitude greater than those in ideal thin films.3 Here, we will present the synthesis of one-dimensional magnetoelectric nanofibers via electrospinning followed by device assembly. Electrospinning can be used to fabricate composite nanowires in a range of configurations by varying the orientation of the piezoelectric and magnetoelectric phases, including random, core-shell, or Janus. Here, we will focus on the synthesis and assembly of bi-phasic magnetoelectric Janus nanofibers, this structure retains external access to each phase as well as to the interface, making them well suited to form modular structures (Figure 1). As-electrospun fibers are amorphous and form a continuous fiber network, requiring subsequent processing steps to both crystallize the nanofibers while also taking advantage of thermal stresses to break the nanofibers into shorter nanowires for device applications. The final nanowire length can be controlled by the electrospinning parameters (e.g., applied voltage) as well as the calcination ramp rate. Shorter nanowires (&lt;10 mm) result from as-spun nanofibers with a smaller initial diameter, and/or faster calcination ramp rates. Aligned arrays of the nanowires can then be assembled into device architectures via dielectrophoresis (Figure 1b-d). The efficiency of nanowire assembly via dielectrophoresis can be enhanced solvent choice as well as the frequency of the applied voltage. We found optimum assembly using butanol and an applied voltage of 42 volts peak to peak at a frequency of 5kHz. Additionally, prior to assembly it may be necessary to perform an acid rinse of the nanofibers to remove any undesired carbonate phases that may hinder assembly. This method results in a linear density of assembled nanowires of ~20 nanowires/mm. The assembled arrays of magnetoelectric nanowires can then be used for a variety of devices, including zero-power magnetic field and current sensors.4 Figure 2 shows the magnetoeletric coefficients of BaTiO3/CoFe2O4 nanowire arrays. These nanowire arrays have a magnetoelectric coefficient of 514+/- 27 mV cm-1 Oe-1 at 1 kHz, considerably larger than bulk values 104 mV cm-1 Oe-1. Figure 2 also shows corresponding device sensitivities. Magnetoelectric nanowires can be assembled into arrays for low power sensors, including magnetic field and current sensors and they can also be developed for biomedical applications where the remote delivery of electric fields is desired, such as for neural modulation.5 Here, we will demonstrate the versatility of magnetoelectric nanowires, their synthesis, and applications. This work opens new opportunities and applications for magnetoelectric materials. This work was support by the NSF I/UCRC on Multi-functional Integrated System Technology (MIST Center) IIP-1439644. The authors acknowledge the use of the Major Analytical Instrumentation Center (MAIC) and Nanoscale Research Facility (NRF) at the University of Florida.

**References:**

1. Spaldin, N. A., Proc. R. Soc. A 2020, 476 (2233), 20190542. 2. Nan, C.-W.; Dong, S.; Viehland, D.; Srinivasan, G., Journal of Applied Physics 2008, 103 (3), 031101. 3. Xie, S.-H.; Liu, Y.-Y.; Li, J.-Y., Front. Phys. 2012, 7 (4), 399-407. 4. Bauer, M. J.; Wen, X.; Tiwari, P.; Arnold, D. P.; Andrew, J. S., Microsystems & Nanoengineering 2018, 4 (1), 37. 5. Ferson, N. D.; Uhl, A. M.; Andrew, J. S., P. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 2020.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5dbf229eb0b210860d8fcd24301da24c.jpg)
Figure 1. (a) scanning electron microscope image of a biphasic magnetoelectric nanowire suspended across an electrode gap (b) optical microscope image of an electrical assembly (c) a schematic magnetoelectric voltage source consisting of composite magnetoelectric nanowires and (d) a schematic of a possible architecture suitable if the density of the assembled nanowires is found to be generally uniform.

![](https://assets.underline.io/uploads/markdown_image/1/image/3c329a99c16139ffe78fbd2b64a3eabc.jpg)
Figure 2. Magnetoelectric coefficients and corresponding device sensitivities of BaTiO3/CoFe2O4 nanowire arrays as a function of frequency measured at angles of approximately no inductive effects, the angle of maximum constructive induction, and maximum destructive induction.

 Magnetoelectric Nanowires and Their Applications INVITED

Giant MagnetoImpedance (GMI) is a magnetic phenomenon present in materials or structures with high magnetic permeability - like soft ferromagnetic conductors - that provides a large change of their impedance Z=R+jX when submitted to an external magnetic field H [1]. This effect is under investigation from many years moving promptly from fundamental studies to applicative domains. Indeed, more and more sensors based on this effect, such as magnetic or current sensors are largely deployed. The number and plurality of applications never stop growing up like target detection, high-density information storage, automotive and so on [2]. Up to now, it seems that GMI has never been used for cyber security purposes even though it is a major challenge to be taken up in the coming years, as the number of connected objects is constantly sprouting up [3]. Security and privacy are threatened by the increasing use of insufficiently protected electronic devices that manage, store and share personal or confidential data [4]. Indeed, software protections can be bypassed by attacks aimed at recovering a secret key by physical means (fig. 1) such as fault injection or side channel [5]. Countermeasures [6-7] must be necessarily implemented to prevent hackers from accessing or modifying these components; they must also provide an intrusion alert so that the integrated circuit (IC) can take appropriate action to protect sensitive information. The use of GMI as a hardware-based security offers several advantages in terms of technology (low additional cost, simple process flow, etc.) and cyber security (high sensitivity to the electromagnetic environment, significant Z impedance variation, etc). While the GMI sensor appears to have some very interesting cyber security properties, it also presents some challenges such as the negative impact of film’s width reduction [8]. For all these reasons, GMI structure as an active countermeasure (antiprobing layer embedded inside the packaging) will be evaluated in this article. A set of magnetic striplines with different widths from 2µm to 20µm were fabricated on 200 mm silicon wafers. The soft ferromagnetic/conductor Ni80Fe20/Al/Ni80Fe20 tri-layer with thicknesses of 100nm/200nm/100nm was deposited by dynamic sputtering under a linear magnetic field and patterned. Each device was characterized directly on wafer for both transversal (T) and longitudinal (L) deposition anisotropy between 1 MHz and 10 GHz by using a vector network analyzer. An internal DC current that generates a transversal magnetic field on the studied tri-layer was used as a polarization bias. The experimental procedure reveals that the 6 µm wide film with transversal (T) deposition anisotropy corresponds to the best tradeoff between GMI response and cybersecurity requisites. The film exhibits magneto-inductive effect higher than 100% on a wide frequency range (5MHz-700MHz). At 60MHz, a significant change of inductance of more than 300 % was observed providing a strong DC current variation sensitivity of 2nH/mA between 4mA and 9mA (fig 2). The resistive variation appears with increasing frequency of the AC current in the film with respect to the skin effect and ferromagnetic resonance. Moreover, the experiment highlights a strong influence of shape anisotropy on GMI’s answer for the films with transversal deposition anisotropy. This effect has to be taken into account in further studies. Nevertheless, the addition of the internal DC current polarization method and the outstanding magneto impedance effect of the 6µm wide GMI film makes it a suitable candidate as an antiprobing layer material. In case of an intrusion, the hacker will inevitably modify the design of at least one GMI mesh but also the general magnetic field distribution in the secured packaging due to the unusual polarization method. These first observations are encouraging for the development of a GMI countermeasure against physical attacks.

**References:**

 [1] M. Knobel and K. Pirota, Journal of magnetism and magnetic materials, Vol. 242-245, p.33 (2002) [2] M. H. Phan and H. X. Peng, Progress in Materials Science, Vol.53, p.323 (2008) [3] M. Semeria, IEEE 62nd International Electron Devices Meeting, p.9 (2016) [4] S. Bhunia, S. Ray and S. Sur-Kolay, “Fundamentals of IP and SoC Security” (2016) [5] M Tanjidur Rahman, Q.Shi and S.Tajik, IEEE 3rd International Verification and Security Workshop, p.93 (2018) [6] A. Lecavelier, M. Brizoux and A. Grivon, IMAPS MiNaPAD (2017) [7] S. Borel, L. Duperrex and E. Deschaseaux, Proceedings of IEEE 68th Electronic Components and Technology Conference, p.515 (2018) [8] L.V. Panina, D.P. Makhnovskiy, and D.J.Mapps, Journal of Applied Physics, Vol.89, p.7221 (2001)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b4675bafa48138c42aeb33f9aa4e4f18.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/79e3835a210cbdcb45305aa80fe4e35f.jpg)

 Evaluation of Magnetoimpedance in Narrow NiFe/Al/NiFe Thin Films for Secured Packaging

Magnetic sensors play an important role in modern life. They are actively used in many areas of science and technology from navigation systems and high-density data recording to security systems and biomedicine. The magnetoimpedance effect - the dependence of the material’s impedance on the applied magnetic field – became the working principle basis of somesensors. For advanced efficiency, large values of permeability were required. The discovery of the giant magnetoimpedance effect in amorphous materials [1] allowed the design of the new type of highly sensitive field detectors. Their pronounced soft magnetic properties make it possible to build magnetic field sensors for geomagnetic applications [2-3], as well as mechanical stress and temperature [4-5]. The GMI effect also allows the detection of bioelectrical activity [6]. The modified construction of the GMI sensor may help to avoid the appearance of parasitic effects caused by the intrinsic magnetic field of the detector. In this work, we consider the planar structure with a radial current distribution with GMI effect for sensor applications. The considered structure consists of two identical ferromagnetic disks separated by a dielectric layer and connected at the periphery by a continuous contact (Fig. 1), the contribution to the inductance of which can be neglected. This system is excited with alternating current I = I0exp (-iωt) from the outer plane of the disks. The contact area between the disc and the driving current lead can also be neglected, since it is small compared to the disc area. To determine theoretically the impedance of the disk, it is necessary to consider the distribution of fields in it. The cylindrical coordinate system (ρ, φ, z) was used for the impedance calculation. The expression for the impedance of the disk with radius a was obtained from the surface impedance which depends on the current frequency. Both real and imaginary part of the impedance were taken into account with the expression Z=Z'+iZ''. The impedance of the system was considered as the impedance of the oscillatory circuit, in which the inductance L = 2Ld with resistance R = 2Rd (factor 2 takes into account two disks) and the capacitor formed by the interlayer between the disks are connected in parallel. The component of impedance of such system Z'(f) and Z''(f) were calculated for the different permeability (Figure 2, theory). The experimental investigations were carried out using the samples, which prepared fromFe70Co24,5 Si2,9B3,1, and Co72Ni12,2Fe5,7Si6,5B3,6 amorphous ribbonswith thicknesses 0.025 and 0.02 mm, respectively. Two disks were cut from every ribbon. Copper leading wires were soldered to the center of the disk. The polyethylene film was used as a dielectric interlayer. Periphery parts of the disks were glued with a silver-based conductive glue. Impedance measurements were carried out using vector network analyzer Agilent FieldFox 9923A in the frequency range 2 MHz-1 GHz. The frequency dependences of real and imaginary impedance part obtained via calculations and in the experiment show similar resonant behavior. The described structure demonstrates resonant frequency dependences of the impedance and thus provides high sensitivity of the sensor. Using sputtering techniques the small-scaled structures can be manufactured for implementation into complex devices.

**References:**

[1] L.V. Panina et al. Appl. Phys. – 1994. - v. 76, N 10. - pp. 6198-6203. [2] A. Zhukov et al. Journal of Alloys and Compounds. – 2014. – Vol.586. - S279–S286. [3] J. Olivera et al. Sensors Actuators A. Phys. – 2011. - Vol.168. - pp.90–94. [4] M. Nowicki et al. Materials. – 2019. – Vol.12. – 2110. [5] J. Nabias et al. Sensors and Actuators, A: Physical. - 2019. – Vol. 289. – pp.50-56. [6] Z. Wang et al. IEEE ICMA 2016, 7558909, pp.2209-2214.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/841e8485db02592d4a245be290445b82.jpg)
Planar structure with radial current distribution – section by plane perpendicular to the sensor surface (left), radial current distribution (right).

![](https://assets.underline.io/uploads/markdown_image/1/image/c096006b08da509dec337ba120d3edce.jpg)
Theoretical and experimental frequency dependences of the real and imaginary impedance parts.

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