Research of Magnetic Shielding on the Closing Gap of Optical Fiber Current Sensor
Reflective optical fiber current sensors have been widely used in electric power and metallurgy. However, the closing gaps between the wave plate and the reflector leads to a strong background current when they are used in strong magnetic fields. In this paper, a mathematical model of the background current was analyzed, and the relationship between the external magnetic field and the background current was identified through experiments. Therefore, a method to reduce the background current by magnetic shielding was proposed. A magnetic ring to shield the magnetic field effectively was designed, and the effects of the structure and relative permeability of the magnetic ring on its shielding rate were established by experiments and numerical simulations. The results showed that the shielding rate exceeded 95% when the length, thickness, and inner diameter of the magnetic ring were 50 mm, 2 mm, and 13 mm, respectively. Overall, this work provided a comprehensive framework that is useful for the analysis and optimization of magnetic shielding and improved the measurement accuracy of optical fiber current sensors in strong magnetic fields.
The electric current is the most basic process parameter in various industrial applications, such as high-voltage direct current (HVDC) projects [1, 2] and nonferrous metallurgy [3–6]. Accurately measuring the current is an important task. Electromagnetic sensors are used traditionally to measure current, but their large volumes and fixed installation requirements restrict their application in narrow spaces and harsh environments and for collecting numerous measurements.
In the electrolysis production process of metals, such as copper, zinc, and aluminum, dozens of cathodes or anodes are placed in a single cell. The distance between the electrodes and that between the electrode and cell sidewall are too small to use a traditional electromagnetic sensor. Thus, some production workshops have no choice but to use simple pressure drop method to imprecisely measure and even give up the measurement of electrode current in most cases, resulting in the loss of key data, limiting the improvement of production level.
Compared with traditional electromagnetic sensors, optical fiber current sensors (OFCSs) are smaller, lighter, and more flexible, with stronger immunity against electromagnetic interference and wider dynamic range. Therefore, OFCSs have been widely used in many fields, such as HVDC projects, new generation intelligent substations [7–11], and metallurgy .
In recent years, our team has used OFCSs to measure the electrode currents of aluminum reduction cells  and electrolytic troughs of copper , the environment of which was very extreme and harsh. Compared with the voltage drop method, real-time electrode current data with much higher accuracy have been obtained, and we determined the current variation characteristics caused by electrolytic cell conditions in metal electrolysis processes. We proposed that the failure and status in the production process could be diagnosed based on the characteristics of the current change, resulting in fine management during the production process.
These measurement practice shows that the OFCS has broad application potential in the field of electrolytic metallurgy, and it has been gradually promoted and applied in China’s aluminum electrolysis industry, leading to more international reports on its application . Predictably, with further popularizing of OFCSs and further development of semiconductor technology, the cost of the photoelectric signal processing system will significantly reduce, making OFCSs can be installed onto each anode and cathode, measuring electrode current real timely, so that key information can be provided for the next generation of cell control system, which will become an important support for the intelligent manufacturing of electrolytic metallurgy.
The structure of an OFCS is shown in Figure 1, see the next section for the detailed description and measurement principle. Briefly, when the wave plate and the reflector are close enough, forming a closed loop with the sensing fiber, the integration of the external magnetic induction intensity along the loop is zero in theory. Thus, the sensor will only measure the current in the electrode conductive rod surrounded by the sensing loop and completely shield the external magnetic field. Therefore, an OFCS should have strong immunity against electromagnetic interference.
In a real electrolytic metallurgical workshop, the number of cells can usually reach to 200~800, while there are nearly 100 cathodes and anodes placed on each cell. The number of electrodes is so large that it is unrealistic to install OFCSs onto every electrode at present. Therefore, portable OFCSs are developed and promoted, the wave plate and the reflector of which can be separated, in order to wrap around the conductive rod of the measured target electrode, then close the sensing fiber ring again to measure. As shown in Figure 2, the sensing fiber, wave plate, and reflector of the OFCS required package protection using a protective cover with a size of Ø6~10 mm. In this working condition, the wave plate and the reflector can only be lapped together instead of sticking tightly, resulting in a closing gap with a distance of 6 ~10 mm, which destroys the integrity of the closed loop and reduces the immunity against electromagnetic interference. As a result, the obvious effect of external magnetic field on the measured value has been observed during our measurement process. Especially when using the portable OFCSs near an electrolytic cell, even if no conductive rod was wrapped into the sensing loop, the strong external magnetic field still caused non-ignorable background current, sometimes up to tens of amperes, which has nothing to do with the current under measurement in fact.
Thus, in order to further popularizing of the portable OFCSs, the background current problem caused by the closing gap have to be solved. It is necessary to research the influence law of the background magnetic field and the method to shield the closing gap such as by using a magnetic ring. Wera et al.  experimentally studied several geometries of magnetic shields composed of YBa2Cu3O7-coated conductors from SuperPower. Sun and He  proposed and designed a direct current (DC) magnetic concentrator and omnidirectional cascaded cloak using only one or two homogeneous anisotropic materials with positive permeability. Bidinosti and Martin  studied the effect of passive magnetic shielding on the direct current (DC) magnetic field gradients imposed by both external and internal sources for two idealized shield models: concentric spherical models and infinitely long cylindrical shell models made from linear materials. Gutierrez et al.  proposed a novel noncontact DC electromagnetic propulsion concept that was validated numerically and experimentally. The articles discussed above reported the application of the magnetic shielding principle in different fields and found that the external DC magnetic fields of the target area were reduced significantly by using magnetic shielding. However, the research of magnetic shielding on the closing gap of OFCS was not mentioned before. Therefore, in this paper, a method for reducing the influence of the external magnetic field on the measurement of an OFCS using magnetic shielding was studied based on the same magnetic shielding principle presented in References [17–19].
In this study, a method to reduce the background currents’ OFCSs using magnetic shielding was examined. The rest of the paper is organized as follows. The measurement principle of the OFCS and the mathematical model of the background current are discussed in Section 2. Based on the mathematical model derived in Section 2, the relationship between the background current and the external magnetic induction intensity, a method to decrease the background current based on experiments, as well as the influence of the magnetic ring structure on the shielding effect, are studied in Section 3. The influence of the magnetic ring structure on the shielding rate cannot be studied systematically using experiments, so numerical simulations were used to examine the effects of the structure and relative permeability of the magnetic ring on its shielding rate in Section 5. Conclusions are given in Section 5.
2. Mathematical Principle
The system structure of the OFCS, as shown in Figure 1, is composed of a sensing fiber ring, a photoelectric signal module, and the optical fiber between them. The sensing fiber ring consists of a wave plate, sensing fiber, and reflector. During the measurement process, the photoelectric signal module will emit a linearly polarized light beam, which will in consequence become circularly polarized through the wave plate. The circularly polarized light will reach the reflector along the sensing fiber and turn back, during which the Faraday effect will occur twice in total. Then, the light will pass through the wave plate again, changing back to linearly polarized light and transmitting to the photoelectric signal module. When there is an electric current passing through the sensing fiber ring, the change of the light wave phase can be measured in the photoelectric signal module. Thus, the current intensity can be detected by coherent detection and digital closed-loop feedback technology.
The basic principles of the OFCS are a Faraday magnetooptic effect and Ampère’s circuit rule. The integral form of the Faraday magnetooptic effect is as follows : where is the Verdet constant of the optical medium, is the propagation distance of the light in the medium, is the magnetic field intensity, and is the magnetic declination of the polarization plane, i.e., the magnetic rotation angle.
The magnetic field intensity and the magnetic induction intensity have the following relation in a homogeneous medium: where is the relative permeability of the medium .
Consequently, Equation (1) can be rewritten as follows:
When the sensing fiber, wave plate, and reflector form a closed loop, the integration of the magnetic induction intensity generated by the current-carrying conductor along the loop is equal to the current passing through the loop :
Equation (3) can be written as follows:
The total current of the conductor around the fiber ring is , so the current for the measurement can be written as follows:
According to Equation (6), the measured value of the current is only dependent on the magnetic rotation angle, which is affected by the magnetic induction intensity along the whole fiber closed loop. In theory, the integration of the external magnetic induction intensity along the closed fiber loop is zero if the loop is a complete circuit, and the external magnetic field does not reduce the measurement accuracy of the OFCS. However, the fiber loop cannot be closed completely in practical use because of the packaging protection of the wave plate and the reflector, leading to the appearance of a closing gap for the fiber loop. Therefore, the difference of the magnetic rotation angle between theory and practice, which results in a nonzero value of the integral in Equation (4), will result in error in the current measurement by the OFCS to some extent.
Figure 3 shows the diagram of the practical fiber loop. In Figure 3, and are the length of the sensing fiber and that of the unclosed part (closing gap), respectively, so the actual length of the fiber loop is (). The magnetic induction intensity generated by the measured current-carrying conductor along the fiber loop is , and the magnetic induction intensity generated by the external magnetic field at the closing gap is . The actual current of the current-carrying conductor inside the fiber loop is , and the current value measured by the OFCS is . The following can be obtained on the basis of Ampère’s circuit rule:
In practical measurements of the current, the reason for the change of the magnetic rotation angle is not only the magnetic field generated by the current-carrying conductor along the loop but also the external magnetic field at the closing gap, expressed as follows:
is defined as the measurement error:
Equation (10) indicates the difference between the measured value of the OFCS and the actual value. can be reduced effectively both by reducing the magnetic induction intensity () caused by the external magnetic field at the closing gap and reducing the length of the closing gap of the OFCS (). In addition, Equation (10) also shows that the elimination of the closing gap of the OFCS is necessary to ensure the ability to resist the external magnetic field interference. However, it is impossible to eliminate the closing gap in an actual manufacturing and use processes.
3. Experimental Measurements
The experiments were divided into two parts in this paper. The first part examined the effect of the magnetic induction intensity at the closing gap on the background current. The second part examined the magnetic shielding experiments by using high-permeability materials to wind the closing gap.
3.1. Effect of Magnetic Induction Intensity at Closing Gap
The portable FS207-2kA-F-BFG optical fiber current sensor used in the experiments was manufactured by Beijing SWT Intelligent Optics Technology Co., Ltd. in China. The sensing fiber, wave plate, and reflector were encapsulated in a Ø5-mm protective cover, which resulted in a minimum closing gap of approximately 5 mm during the measurements.
A CHROMA DC power supply 20 kA was used to output a 15 kA current in the experiments, and a background magnetic field was formed around an aluminum bus with a cross-section. As shown in Figure 4, all the measurement points were located on a line perpendicular to the bus centerline. The dependence of the magnetic induction intensity on the distance between the measurement points and the bus was measured by a CH-Hall Model 1600 Gauss meter.
As shown in Figure 5(a), the magnetic induction intensity generated by the current bus decreased with the increase in the distance between the measurement points and the bus. As shown in Figure 4, the closing gap of the OFCS was placed at the same position to measure the background current, and the results are shown in Figure 5(b). No current was introduced into the fiber ring during the measurements, so the measured current value was the background current generated by the external magnetic field at the closing gap. The background current corresponded to the current error given by Equation (10).
Figure 5(c) shows the relationship between the background current and the magnetic induction intensity. The background current measured by the OFCS increased linearly with the increase in the magnetic induction intensity at the closing gap, which agreed with the result obtained using Equation (10). When the external magnetic induction intensity reached 0.014 T, the background current was as high as 66 A. Therefore, the presence of the closing gap in the external strong magnetic field led to a significant measurement error for the OFCS.
3.2. Magnetic Shielding Experiments
Equation (10) and the experimental results mentioned above showed that the decrease in the magnetic induction intensity at the closing gap could reduce the background current caused by the external magnetic field. Therefore, a magnetic shielding method using the permalloy winding rings and the permalloy magnetic rings was used to reduce the external magnetic field at the closing gap in this paper. The shielding effect on the external magnetic field was the worst when the axes of both the winding ring and magnetic ring were parallel to the direction of the magnetic line. Therefore, all the winding ring and magnetic ring axes in the experiments were set to be parallel to the magnetic line to simulate the worst-case-scenario shielding effect.
In the preliminary experiments, 1J50 permalloy belts with 25.2 mm widths, 0.2 mm thicknesses, and relative magnetic permeabilities of 30000–50000 N/A2 were used to wind around the closing part of the wave plate and the reflector, causing most of the magnetic lines at this position to pass through the permalloy material. Thus, the magnetic lines inside the closing gap would be reduced greatly to achieve the magnetic shielding effect.
To determine the influence of the number of layers of the permalloy belts on the magnetic shielding effect, the effects of different numbers of winding layers of the permalloy belts (1, 3, and 6 layers) and the distance between the measurement points and the current bus on the background current are shown in Figure 6. It shows that winding the permalloy belts at the closing gap could reduce the background current of the OFCS. The background current decreased with the increase in the number of winding layers of the permalloy belts, which corresponded to a better antimagnetic interference effect.
To describe the antimagnetic interference effect of the permalloy more intuitively, the shielding rate was used to illustrate the shielding effect of the alloy on the external magnetic field, and was defined as follows: where is the background current measured without the permalloy belt and is the current measured with the permalloy belt. The shielding rates to the external magnetic field for different numbers of winding layers of the permalloy belts are shown in Figure 7. The shielding effect increased with the increase in the number of winding layers of the permalloy belts, and the shielding rate exceeded 90% with six winding layers.
Based on the experimental results, the permalloy belts had a good shielding effect on the external magnetic field. The shielding rate was over 90%, but it was significantly affected by the number of winding layers of the belt. In the winding process, it is not easy to precisely control the winding thickness because the gaps between the layers are difficult to eliminate. Therefore, 16 kinds of permalloy magnetic rings were designed to replace the permalloy belts with different winding layers. The effects of magnetic rings with thicknesses of 2, 3, 4, and 5 mm (corresponding to the number of layers of permalloy belt) and lengths of 20, 30, 40, and 50 mm (corresponding to different widths of the permalloy belt) on the shielding rates were studied. All the magnetic rings were constructed from the 1J50 permalloy with inner diameters of Ø13 mm, and the distance between the measurement point and the current bus was set as 4 cm.
Figure 8 shows the influence of the length and thickness of the magnetic ring on the shielding rate of the external magnetic field. As shown in Figure 8, most of the magnetic rings shielded the external magnetic field significantly. The minimum shielding rate of the magnetic ring was about 80%, and the maximum was as high as 97%. The thickness of the magnetic ring in the range of 2–5 mm had little influence on the shielding effect. The main factor affecting the shielding effect of the external magnetic field was the length of the magnetic ring. The shielding effect was improved with the increase in the length of the magnetic ring. However, the increase in the shielding rate slowed when the length exceeded 40 mm. Further experiments were conducted to increase the length of the magnetic ring to 100 mm; the shielding rate remained at about 97%, with little change.
4. Numerical Simulations
The external interference magnetic field can be shielded effectively by winding permalloy belts and magnetic rings that have high magnetic permeabilities around the closing gap of the OFCS, as shown by the experiments described above. The shielding rate was related to the number of layers of the permalloy belts and the lengths of the magnetic rings. Further analyses showed that the shielding rate might be associated with various factors, including the relative magnetic permeability of the magnetic material and other structural parameters. Therefore, the effects of these factors on the shielding rate were studied using numerical simulations. The magnetic field for the measurement was simulated by the COMSOL Multiphysics field simulation software, and the locations of the magnetic rings were set to be the same as those in the experiments. The effects of the relative magnetic permeability and structural parameters of the magnetic rings on the shielding rates of the external magnetic field were studied by simulations.
The physical fields in the simulation model were magnetic and electric fields under the alternating current (AC)/DC model, and the simulation was conducted at steady state. The nature of the simulation was to solve Maxwell’s equation under the condition of the following quasistatic approximation: where is the current density, is the conductivity, is the electric field intensity, is the velocity of the conductor, is the magnetic induction intensity, and is an externally generated current density.
In this study, the conductivity and the parameters of the electric field were determined by the materials and the boundary conditions. The magnetic induction intensity was then calculated from Equation (12).
Figure 9 is schematic diagram of simulation model, which set the conditions as the same as those in the experiments described above, the axes of all the magnetic rings were set to be parallel to the magnetic line of the external magnetic field to simulate the worst-case-scenario shielding effect. The distance between the magnetic rings and the current bus was set to 4 cm, and the bus current was set to 15000 A. Figure 10 shows the magnetic induction intensity distribution around and inside the magnetic ring, and Figure 11 shows the variations of the magnetic induction intensity inside the magnetic ring.
Figures 10 and 11 show that the magnetic ring was magnetized by the current bus, and its magnetic induction intensity was large. The maximum value could reach 0.476 T, which was much higher than the magnetic induction intensity (0.012 T) without a magnetic ring at this position. Because of this kind of magnetic congregation effect, the magnetic induction intensity inside the magnetic ring became quite small, only several ten-thousandth parts of 1 T. The magnetic induction intensity on the center point along the axis was lower than 0.0001 T, indicating that the magnetic ring with a highly magnetically conductive material had a significant shielding effect on the external magnetic field.
4.1. Effect of Relative Magnetic Permeability on Shielding Rate
Figure 12 shows the effect of the relative magnetic permeability of the magnetic ring on the shielding rate to the external magnetic field. The results showed that the shielding rate increased with the increase in the relative magnetic permeability of the magnetic ring when it was below 10000 N/A2, while the shielding rate to the external magnetic field could reach up to 99% when the relative magnetic permeability was over 10000 N/A2. However, the shielding rate could not continue to increase significantly with the increase in the relative magnetic permeability. Therefore, the relative magnetic permeability of the magnetic ring was set to 10000 N/A2 in the subsequent calculations.
4.2. Effect of Magnetic Ring Structure on Shielding Rate
Figures 10 and 11 show that the magnetic ring with a high magnetic permeability material had a significant shielding effect on the external magnetic field, but the magnetic induction intensity at both ends of magnetic ring increased significantly, and the shielding region was usually present in the center of the magnetic ring. According to the COMSOL simulation results in Figure 11, the magnetic induction intensity at a distance of 4 cm from the current bus was 0.012 T. The shielding rate reached 96% when the magnetic induction intensity inside the magnetic ring was 0.0005 T. Therefore, the region below 0.0005 T inside the magnetic ring was defined as the low magnetic region.
There are three main structural parameters of the magnetic ring that should be considered: length, thickness, and inner diameter. The effects of these parameters on the length of the shielding region were analyzed herein using numerical simulations.
Figure 13 shows the dependence of the length of the low magnetic region on the magnetic ring with different lengths. The length of the low magnetic region inside the magnetic ring increased with the increase in the magnetic ring length, which means that the longer the magnetic ring was, the greater the shielding effect was. Figure 14 shows the dependence of the length of the low magnetic region on the magnetic ring with different thicknesses. The thickness of the magnetic ring had little influence on the low magnetic region length inside the magnetic ring, which agreed with the results of Section 3.2. When the magnetic ring length was 50 mm, the length of the low magnetic region was 25.9 mm. Considering the length of the closing gap of OFCS is 20 mm, this length can satisfy the operational requirements in practical applications and effectively shield the external magnetic field at the closing gap. Figure 15 shows the dependence of the length of the low magnetic region on the magnetic ring with different inner diameters. As the inner diameter of the magnetic ring increased, the length of the low magnetic region inside the ring decreased, and the shielding effect of the magnetic ring on the external magnetic field also decreased.
Therefore, increasing the length and thickness of the magnetic ring and reducing the inner diameter can improve the shielding effect of the magnetic ring on the external magnetic field.
In this paper, the background current caused by the external magnetic field at the closing gap of an OFCS was analyzed, and a method to decrease the background current was studied based on the magnetic shielding principle.
The relationship between the magnetic induction intensity at the closing gap and the background current was clarified by experiments. Results showed that the background current increased with the increase in the magnetic induction intensity at the closing gap. The effects of the lengths and thicknesses of the magnetic rings made from permalloy on the shielding effect were tested as well. The results showed that the magnetic ring effectively reduced the background current caused by the external magnetic field. The shielding rate reached a maximum of 97% when the magnetic ring had a length of 50 mm, a thickness of 2 mm, and an inner diameter of 13 mm.
The shielding phenomenon achieved by a magnetic ring made from a material with a high relative magnetic permeability was simulated using COMSOL Multiphysics field simulation software, and the effects of the relative magnetic permeability and structure of the magnetic ring on the shielding rate were analyzed. The results showed that the shielding effect of the magnetic ring resulted from a magnetic congregation effect of the high-permeability material. The shielding effect of magnetic ring on the external magnetic field could be improved by selecting a higher magnetic permeability material, increasing the length of the magnetic ring, and decreasing its inner diameter to produce the magnetic ring.
The (data type) data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Chun Li did the methodology, formal analysis, validation and project administration. Shuihua Zhang did the data curation, and writing original draft. Yi Meng did the validation and writing—review and editing. Jun Tie did the conceptualization and supervision. Rentao Zhao acquired software. Qingyu Zeng provided experimental assistance. Hao Xiao acquired resources. Dongwei Liu acquired resources.
The authors would like to acknowledge support from the National Natural Science Foundation of China (21978004).
J. Sun, S. Yamauchi, and M. Sagano, “Critical current measurements for design of superconducting DC transmission power cable,” Physica C-Superconductivity and its Applications, vol. 471, no. 21-22, pp. 1313–1316, 2011.View at: Google Scholar
T. Nguyen, J. Ely, and G. Szatkowski, “A fiber-optic current sensor for lightning measurement applications,” SPIE Sensing Technology + Applications, vol. 9480, 2015.View at: Google Scholar
G. Duncan, B. Michael, and N. Lauri, “Measurement of electric current in an individual electrode in an electrolysis system,” 2017.View at: Google Scholar
F. Prado, “Device for monitoring current distribution in interconnected electrolytic cells,” 2017.View at: Google Scholar
C. H. Yang, S. J. Deng, and Y. G. Li, “Optimal control for zinc electrowinning process with current switching,” IEEE Access, vol. 5, pp. 24688–24697, 2017.View at: Google Scholar
Y. C. Yao, C. Y. Cheung, and J. Bao, “Detection of local cell conditions based on individual anodecurrent measurements,” Light Metals 2016, 2016.View at: Google Scholar
J. H. Haywood, I. M. Bassett, and M. Matar, “Application of the NIMI technique to the 3×3 Sagnac fiber optic current sensor-experimental results,” Optical Fiber Sensors Conference Technical Digest. OFS 2002, vol. 15, 2015.View at: Google Scholar
E. C. Xin and H. W. Yuan, “Development of a sensor for corona current measurement under high-voltage direct-current transmission lines,” International Journal of Distributed Sensor Networks, vol. 12, 2016.View at: Google Scholar
P. A. Nicati and P. Robert, “Stabilised current sensor using Sagnac interferometer,” Journal of Physics E: Scientific Instruments, vol. 21, no. 8, 1988.View at: Google Scholar
D. Tzelepis, A. Dysko, and G. Fusiek, “Single-ended differential protection in MTDC networks using optical sensors,” IEEE Transactions on Power Delivery, vol. 32, pp. 1605–1615, 2016.View at: Google Scholar
M. Takahashi, K. Sasaki, and K. Terai, “Optical current sensor for DC measurement,” in EEE/PES Transmission and Distribution Conference and Exhibition, Yokohama, Japan, 2002.View at: Google Scholar
K. Bohnert, P. Gabus, and J. Nehring, “Fiber-optic current sensor for electrowinning of metals,” Journal of Lightwave Technology, vol. 25, pp. 3602–3609, 2007.View at: Google Scholar
Y. L. Wang, J. Tie, and S. C. Sun, “Testing and characterization of anode current in aluminum reduction cells,” Metallurgical and Materials Transactions B, vol. 47, pp. 1986–1998, 2016.View at: Google Scholar
Q. Y. Zeng, C. Li, and Y. Meng, “Analysis of interelectrode short-circuit current in industrial copper electrorefining cells,” Measurement, vol. 164, 2020.View at: Google Scholar
V. Potocnik, A. Arkhipov, N. Ahli, and A. Alzarooni, “Measurement of DC busbar currents in aluminium smelters,” in Proceedings of 35 th International ICSOBA Conference, Hamburg, Germany, Travaux, 2017.View at: Google Scholar
L. Wera, J. F. Fagnard, and G. A. Levin, “Magnetic shielding with YBCO coated conductors: influence of the geometry on its performances,” IEEE Transactions on Applied Superconductivity, vol. 23, no. 3, 2013.View at: Google Scholar
F. Sun and S. L. He, “DC magnetic concentrator and omnidirectional cascaded cloak by using only one or two homogeneous anisotropic materials of positive permeability,” Progress in Electromagnetics Research, vol. 142, pp. 683–699, 2013.View at: Google Scholar
C. P. Bidinosti and J. W. Martin, “Passive magnetic shielding in gradient fields,” AIP Advances, vol. 4, no. 4, 2013.View at: Google Scholar
H. Gutierrez, R. Meinke, and T. Fernando, “Non-contact DC electromagnetic propulsion by multipole transversal field: numerical and experimental validation,” IEEE Transactions on Magnetics, vol. 52, no. 8, 2016.View at: Google Scholar
H. D. Young and R. A. Freedman, University Physics with Modern Physics Global Edition, 2015.