France



It is known that the assessment of local magnetization in objects exhibiting hysteresis is crucial to the accurate design and/or performance estimation of a wide range of electromagnetic devices. Among the widely used approaches to carry out the afore-mentioned assessment tasks, the integral equations approach has been widely adopted (See, for instance [1]). In the past, it was demonstrated that an elementary hysteresis operator may be constructed using a two-node Hopfield Neural Network (HNN) having step activation functions and positive inter-node feedback [2]. The approach was further extended in [3] to construct a 2D vector hysteresis operator using a tri-node Hopfield Neural Network (HNN) having orientation-dependent inter-node feedback values. Utilization of this extension was previously partially utilized in 2D field computations by coupling the proposed approach with time consuming iterative integral equation methodologies [4]. The purpose of this paper is to offer a novel 2D field computation approach in media exhibiting hysteresis that solely utilizes HNNs. The approach is based on the novel incorporation of domain-domain interaction in the overall HNN energy formulation. Consequently, 2D field computation may be accomplished by employing typical HNN automated energy minimization algorithms. Consider a HNN comprised of N clusters of tri-node HNNs. The total applied field Ht at any node u in HNN #i having a triangular median given by (x&lt;sub&gt;i&lt;/sub&gt;, y&lt;sub&gt;i&lt;/sub&gt;) may be given by:&lt;br/&gt;Ht&lt;sub&gt;iu&lt;/sub&gt;=(H+∑&lt;sub&gt;j=1,N &amp; j≒i&lt;/sub&gt; h&lt;sub&gt;ji&lt;/sub&gt;)●e&lt;sub&gt;iu&lt;/sub&gt;+∑&lt;sub&gt;v=1,3 &amp; v≒u&lt;/sub&gt; f&lt;sub&gt;iv&lt;/sub&gt;k&lt;sub&gt;copl&lt;/sub&gt;│e&lt;sub&gt;iu&lt;/sub&gt;●e&lt;sub&gt;iv&lt;/sub&gt;│, (1)&lt;br/&gt;h&lt;sub&gt;ij&lt;/sub&gt;=-▽(∑&lt;sub&gt;s=1,3&lt;/sub&gt; (f&lt;sub&gt;js&lt;/sub&gt;e&lt;sub&gt;js&lt;/sub&gt;●R&lt;sub&gt;jp&lt;/sub&gt;)/(4π│R&lt;sub&gt;jp&lt;/sub&gt;│&lt;sup&gt;3&lt;/sup&gt;))│&lt;sub&gt;(xi,yi)&lt;/sub&gt; , R&lt;sub&gt;jp&lt;/sub&gt;=(x-x&lt;sub&gt;j&lt;/sub&gt;)e&lt;sub&gt;x&lt;/sub&gt;+(y-y&lt;sub&gt;j&lt;/sub&gt;)e&lt;sub&gt;y&lt;/sub&gt;, (2)&lt;br/&gt;where H is the external applied field, e&lt;sub&gt;ab&lt;/sub&gt; is the unit vector denoting the orientation of node #b in HNN #a, k&lt;sub&gt;copl&lt;/sub&gt; is the internal tri-node coupling factor using which the switching value of the elementary operator may be tuned, while f represent the HNN step activation function given by:&lt;br/&gt;f(Ht)={+1 if Ht&amp;gt;0, -1 if Ht&amp;lt;0, Unchanged if Ht=0}. (3)&lt;br/&gt;It should be mentioned here that, in HNN terminology, all coupling factors between any two nodes in the overall HNN may be deduced from (1) and (2).&lt;br/&gt;The proposed implementation yields node outputs corresponding to, an automatically achieved, minimized value of the HNN energy E (in line with the magnetostatic energy) given by:&lt;br/&gt;E=-∑&lt;sub&gt;i=1,N&lt;/sub&gt;(∑&lt;sub&gt;u=1,3&lt;/sub&gt;[(H+∑&lt;sub&gt;j=1,N &amp; j≒i&lt;/sub&gt; hji)●f&lt;sub&gt;iu&lt;/sub&gt;e&lt;sub&gt;iu &lt;/sub&gt;+1/2 ∑&lt;sub&gt;v=1,3 &amp; v≒u&lt;/sub&gt; f&lt;sub&gt;iu&lt;/sub&gt;f&lt;sub&gt;iv&lt;/sub&gt;│e&lt;sub&gt;iu&lt;/sub&gt;●e&lt;sub&gt;iv&lt;/sub&gt;│]). (4)&lt;br/&gt;Local magnetization values of any tri-node HNN, denoted by HNN #i, representing a triangular subdivision may be deduced from:&lt;br/&gt;M&lt;sub&gt;i&lt;/sub&gt;(H)=∑&lt;sub&gt;u=1,3&lt;/sub&gt; f&lt;sub&gt;iu&lt;/sub&gt;e&lt;sub&gt;iu&lt;/sub&gt;. (5)&lt;br/&gt;Hence, total output of the overall HNN representing the magnetiation of the magnetized body under consideration may thus be expressed using superposition in the form:&lt;br/&gt;M(H)=∑&lt;sub&gt;i=1,N&lt;/sub&gt;M&lt;sub&gt;i&lt;/sub&gt;(H). (6)&lt;br/&gt;In order to demonstrate the proposed methodology, consider the case of a pair of tri-node HNN (i.e., N=2) as shown in Fig. 1.a. Sample results for this particular case are given in Figs. 1 and 2 for the two tri-node HNNs representing triangular subdivisions whose vertices have the coordinates (1.0,0.5), (-0.5,0.5), (-0.5,-0.5) and (1.0,0.5), (1.0,-0.5), (-0.5,-0.5), respectively. In Figs. 1.b and 1.c results corresponding to x-axis cyclic and rotational applied inputs are demonstrated when the two triangles join to form a rectangular shape. Variations of the x-axis cyclic input-output curves resulting from the change in the vertical spacing Dy and/or horizontal spacing Dx between both triangular subdivisions are further demonstrated in Fig. 2. Results clearly highlight the impact of the incorporation of domain-domain interaction in the introduced field computation approach. It should be mentioned that using such an approach HNN modules, which are widely available in a number of mathematical simulation tools, may be utilized in 2D field computation in media exhibiting hysteresis. More details of the approach for arbitrary 2D objects, demonstration of local magnetization vector plots, simulations and comparisons with finite element analysis tools will be presented in the full paper.

**References:**

[1] A.A. Adly, I.D. Mayergoyz and R.D. Gomez, IEEE Transactions on Magnetics, Vol. 29, p. 2380 (1993).&lt;br/&gt;[2] A.A. Adly and S.K. Abd-El-Hafiz, Journal of Magnetism and Magnetic Materials, Vol. 263, No.3, p. 301 (2003).&lt;br/&gt;[3] A.A. Adly and S.K. Abd-El-Hafiz, Proceedings of ECTICON, Paper No. 1067, p. 1 (2016).&lt;br/&gt;[4] A.A. Adly and S.K. Abd-El-Hafiz, Journal of Magnetism and Magnetic Materials, Vol. 434, p. 151 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e3a633f16b17803e9942186d7f4db975.jpg)
Fig. 1. (a) Two identical tri-node HNNs having different orientations, (b) Superposition of the HNN outputs for  Dx=Dy=0 corresponding to a cyclic input along the x-direction, and (c) Superposition of the HNN outputs for Dx=Dy=0 corresponding to a rotational input.

![](https://assets.underline.io/uploads/markdown_image/1/image/fe05ca9c7dace532cec2ca4a8f6d47c4.jpg)
Fig. 2. Superposition of the HNN outputs corresponding to a cyclic input along the x-direction for; (a) Dx=Dy=infinity (i.e., no interaction), (b) Dx=1.0, Dy=0, and (c) Dx=0.0, Dy=-1.0.

INTERMAG 2021

 Field Computation in Media Exhibiting Hysteresis Using Hopfield Neural Networks

AIAI 2020 (16th International Conference on Artificial Intelligence Applications and Innovations) should be held in Halkidiki, Greece. Due to current Covid-19 situation the conference will be held virtually. Underline will host a repository of video talks and preserve them for the future. Videos will be released on Underline one day after the conference.

AIAI 2020

It is known that the assessment of local magnetization in objects exhibiting hysteresis is crucial to the accurate design and/or performance estimation of a wide range of electromagnetic devices. Among the widely used approaches to carry out the afore-mentioned assessment tasks, the integral equations approach has been widely adopted (See, for instance [1]). In the past, it was demonstrated that an elementary hysteresis operator may be constructed using a two-node Hopfield Neural Network (HNN) having step activation functions and positive inter-node feedback [2]. The approach was further extended in [3] to construct a 2D vector hysteresis operator using a tri-node Hopfield Neural Network (HNN) having orientation-dependent inter-node feedback values. Utilization of this extension was previously partially utilized in 2D field computations by coupling the proposed approach with time consuming iterative integral equation methodologies [4]. The purpose of this paper is to offer a novel 2D field computation approach in media exhibiting hysteresis that solely utilizes HNNs. The approach is based on the novel incorporation of domain-domain interaction in the overall HNN energy formulation. Consequently, 2D field computation may be accomplished by employing typical HNN automated energy minimization algorithms. Consider a HNN comprised of N clusters of tri-node HNNs. The total applied field Ht at any node u in HNN #i having a triangular median given by (xi, yi) may be given by: Htiu=(H+∑j=1,N & j≒i hji)●eiu+∑v=1,3 & v≒u fivkcopl│eiu●eiv│, (1) hij=-▽(∑s=1,3 (fjsejs●Rjp)/(4π│Rjp│3))│(xi,yi) , Rjp=(x-xj)ex+(y-yj)ey, (2) where H is the external applied field, eab is the unit vector denoting the orientation of node #b in HNN #a, kcopl is the internal tri-node coupling factor using which the switching value of the elementary operator may be tuned, while f represent the HNN step activation function given by: f(Ht)={+1 if Ht&gt;0, -1 if Ht&lt;0, Unchanged if Ht=0}. (3) It should be mentioned here that, in HNN terminology, all coupling factors between any two nodes in the overall HNN may be deduced from (1) and (2). The proposed implementation yields node outputs corresponding to, an automatically achieved, minimized value of the HNN energy E (in line with the magnetostatic energy) given by: E=-∑i=1,N(∑u=1,3[(H+∑j=1,N & j≒i hji)●fiueiu +1/2 ∑v=1,3 & v≒u fiufiv│eiu●eiv│]). (4) Local magnetization values of any tri-node HNN, denoted by HNN #i, representing a triangular subdivision may be deduced from: Mi(H)=∑u=1,3 fiueiu. (5) Hence, total output of the overall HNN representing the magnetiation of the magnetized body under consideration may thus be expressed using superposition in the form: M(H)=∑i=1,NMi(H). (6) In order to demonstrate the proposed methodology, consider the case of a pair of tri-node HNN (i.e., N=2) as shown in Fig. 1.a. Sample results for this particular case are given in Figs. 1 and 2 for the two tri-node HNNs representing triangular subdivisions whose vertices have the coordinates (1.0,0.5), (-0.5,0.5), (-0.5,-0.5) and (1.0,0.5), (1.0,-0.5), (-0.5,-0.5), respectively. In Figs. 1.b and 1.c results corresponding to x-axis cyclic and rotational applied inputs are demonstrated when the two triangles join to form a rectangular shape. Variations of the x-axis cyclic input-output curves resulting from the change in the vertical spacing Dy and/or horizontal spacing Dx between both triangular subdivisions are further demonstrated in Fig. 2. Results clearly highlight the impact of the incorporation of domain-domain interaction in the introduced field computation approach. It should be mentioned that using such an approach HNN modules, which are widely available in a number of mathematical simulation tools, may be utilized in 2D field computation in media exhibiting hysteresis. More details of the approach for arbitrary 2D objects, demonstration of local magnetization vector plots, simulations and comparisons with finite element analysis tools will be presented in the full paper.

**References:**

[1] A.A. Adly, I.D. Mayergoyz and R.D. Gomez, IEEE Transactions on Magnetics, Vol. 29, p. 2380 (1993). [2] A.A. Adly and S.K. Abd-El-Hafiz, Journal of Magnetism and Magnetic Materials, Vol. 263, No.3, p. 301 (2003). [3] A.A. Adly and S.K. Abd-El-Hafiz, Proceedings of ECTICON, Paper No. 1067, p. 1 (2016). [4] A.A. Adly and S.K. Abd-El-Hafiz, Journal of Magnetism and Magnetic Materials, Vol. 434, p. 151 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e3a633f16b17803e9942186d7f4db975.jpg)
Fig. 1. (a) Two identical tri-node HNNs having different orientations, (b) Superposition of the HNN outputs for Dx=Dy=0 corresponding to a cyclic input along the x-direction, and (c) Superposition of the HNN outputs for Dx=Dy=0 corresponding to a rotational input.

![](https://assets.underline.io/uploads/markdown_image/1/image/fe05ca9c7dace532cec2ca4a8f6d47c4.jpg)
Fig. 2. Superposition of the HNN outputs corresponding to a cyclic input along the x-direction for; (a) Dx=Dy=infinity (i.e., no interaction), (b) Dx=1.0, Dy=0, and (c) Dx=0.0, Dy=-1.0.

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Transcranial magnetic stimulation (TMS) is a non-invasive and painless treatment for brain disorders. Rodent models of TMS provide a possibility to reveal the mechanism of TMS at cellular and molecular level. However, their application has been hampered by lack of stimulating coils with appropriate focus and intensity. In this study, we analyzed various types of coils and proposed conductive shield and iron core materials to improve the coil performance. The results showed that combination of Halo coil and figure of eight coil with shield and iron core could generate better stimulus focus (29.2 mm2) and intensity (87.6 V/m). Keywords: Transcranial magnetic stimulation; Coils; Focus; Mouse model I. Introduction At present, most commercial animal coils generally cannot meet the requirements of stimulus depth, stimulus intensity and stimulus focus simultaneously. Therefore, it is necessary to further develop a special TMS coil for small animals [1, 2]. II. Method In this study, a mouse model was constructed with three-compartment body-skull-brain model of conductivities 0.3344, 0.0203 and 0.106 S/m. The mouse and coil models were shown in Fig.1. The shield conductor (Cu) with a thickness of 1 mm had a window in the middle. The iron core was made of sheets of silicon steel. All the coils were kept at a distance of 5 mm from the highest point of the mouse head. The stimulus intensity is defined as the max of the electric field (Emax), and the focus was defined as the half power region (HPR). III. Results The results showed that Halo coil, shield and iron core could complement each other well. The combination of Halo coil and figure of eight coil with shield, icon core could get a better stimulus focus (29.2 mm2) and a higher stimulus intensity (87.6 V/m), which had increased about 79% and 54% respectively compared to the general commercial coil, as shown in Fig.2. IV. Conclusion Our results suggested that the combination of Halo coil and figure of eight coil with shield and icon core could improve TMS coil performance. We provided an alternative method to deliver TMS to small animals by constructing a coils with better stimulus intensity and stimulus focus.

**References:**

[1] Meng Q, Jing L, Badjo J P, et al. A novel transcranial magnetic stimulator for focal stimulation of rodent brain [J]. Brain Stimulation, 2018, 11(3): 663-665. [2] Alekseichuk I, Mantell K, Shirinpour S, et al. Comparative modeling of transcranial magnetic and electric stimulation in mouse, monkey, and human [J]. Neuroimage, 2019, 194: 136-148.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c980cf69f9b7ab8e29272aa563f9908a.jpg)
Fig.1 Mouse and coil models of this study

![](https://assets.underline.io/uploads/markdown_image/1/image/ff8235f037f228bd4581f42851cc12cf.jpg)
Fig.2 Electric field induced in the brain of the mouse

 Design of Transcranial Magnetic Stimulation Coils for Mouse with Improved Stimulus Focus and Intensity

Magnetic hysteresis loop measurement [1]-[4] is crucial for the development of advanced magnetic materials. From the magnetization versus magnetizing field (M-H) curve, one can extract the useful magnetic properties, which are important for applications in sensors, transformers, actuators, and miscellaneous electrical devices. The M-H measurements were usually done by using the vibrating sample magnetometer [1] or the SQUID magnetometer [2], for which the magnetizing field are virtually static during the measurement. The induction-coil-based hysteresis loop and susceptibility measurement systems [3], [4] are rapid and low-cost devices capable of determining the alternating-current (AC) magnetic properties, which is important for inferring the frequency-dependent features such as the energy loss due to eddy currents in conducting materials. The convention hysteresis loop measurement device [3] is designed for the magnetic induction versus field (B-H) measurement of toroidal samples. The alternating susceptibility measurement system (AC susceptometer) is capable of determining the frequency dependent initial magnetization and susceptibility. But the induction coil can’t detect the environmental direct-current (DC) field. For the soft magnetic materials, the non-zero environmental magnetic field can significantly alter the measured result, which must be compensated by using a magnetic sensor to detect the unwanted magnetic interference. To achieve higher accuracy in the M-H measurement, we proposed a M-H measurement system combining the signal processing method of B-H measurement and the coil design of the AC susceptometer. The interference from the environmental magnetic field is eliminated by using a giant magnetoresistance (GMR) sensor as the zero field detector, as shown in Fig. 1. The system consists of solenoidal magnetization and pickup coils. The pickup coil consists of reference and sensing coils symmetrically positioned with respect to the excitation coil. The sensing coil detects the magnetization of the sample, whereas the reference coil is wound in the opposite direction to null out the magnetizing signal. The signal output of the integrator is proportional to the magnetization of the sample. The current in the magnetization coil is directly proportional to the magnetizing field strength H. However, the DC level of the magnetizing field at the sample is usually shifted by the environmental field, which can’t be observed by monitoring the current in the coil. To solve the problem, the GMR sensor, which is the GF708 spin-valve sensor [5] from Sensitec GmbH, is used for the zero field detector. The sensor has a very high sensitivity but the dynamic range less than 2 Oe. The output of GMR is used as the trigger for taking the M-H curve to eliminate the disturbance from the environmental DC field, as shown in Fig. 2. The 1-Hz sinusoidal current signal measured by a resistor shows a peak-to-peak amplitude of 0.96 A, corresponding the magnetizing field amplitude of 6.7 mT. When the environmental DC field was completely eliminated in the pickup coil, the observed output signal of the GMR has a zero-field-crossing point defined by the trigger level, shown as in Fig. 2. The trigger point to activate the M-H measurement is at the intersection of the trigger level line and the GMR signal curve for “No external field”. When the DC field (of about 0.77 mT) was applied, the trigger point shifted by 40 ms in comparison with the zero-field curve, while the current waveform in the coil remained unchanged. The new trigger point to activate the M-H measurement is at the intersection between the trigger level line and the GMR signal curve with “External field”. The coercivity and the asymmetry pinning field of the GMR sensor induces a constant time shift in the trigger point, which can be corrected by employing a calibration procedure provided that the sweeping magnetizing field always saturates the output of GMR on both field polarities. With the GMR zero-field detector to set the trigger point, the high-quality M-H curve can be obtained by taking averaging for multiple cycles of the sweeping magnetizing field. In this way, the accuracy of the M-H curve measurement for the soft magnetic materials is successfully improved by eliminating the interference from the environmental field without using a costly magnetic shield chamber. This work is supported by the Ministry of Science and Technology of Taiwan under Grant No. MOST108-2221-E992-083MY2.

**References:**

[1] S. Foner, “Vibrating sample magnetometer”, Review of Scientific Instruments, 27(7), 548-548 (1956). [2] R. Nagendran, N. Thirumurugan, N. Chinnasamy, M. P. Janawadkar, & C. S. Sundar, “Development of high field SQUID magnetometer for magnetization studies up to 7 T and temperatures in the range from 4.2 to 300 K”, Review of Scientific Instruments, 82(1), 015109 (2011). [3] V. Munoz, M. C. Martinez, & V. Such, “Improved systems for the measurement of hysteresis loops: DC and AC characterization”, Journal of Physics E: Scientific Instruments, 20(7), 861 (1987). [4] R. B. Goldfarb & J. V. Minervini, “Calibration of ac susceptometer for cylindrical specimens”, Review of scientific instruments, 55(5), 761-764 (1984). [5] C. Reig, S. Cardoso, & S. C. Mukhopadhyay, “Giant magnetoresistance (GMR) sensors”, In Smart Sensors, Measurement and Instrumentation, 6(1), pp.1-301 (2013).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e67c168e41c249ae72cd97327beac3a7.jpg)
Fig. 1. Schematics of the magnetization measurement system with GMR zero-field detector
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![](https://assets.underline.io/uploads/markdown_image/1/image/53502c98b622d9f578af31fde8a199d5.jpg)
Fig. 2. The trigger points of the GMR zero-field detector for the M-H measurement system
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 Magnetization measurement system with giant magnetoresistance zero-field detector

Many scholars have found that magnetic anomaly signals can reflect the characteristics of magnetic target, such as distance, speed, magnetic moment and other related information [1-3]. This provides a new idea for target recognition by matching magnetic anomaly signals. However, the magnetic anomaly signals in the detection environment are weak and easily disturbed by the strong background noise of the geomagnetic filed, resulting in a low signal-to-noise ratio (SNR) of the signals output by the magnetic sensor [4], which brings great challenges to the traditional signal matching method. In this paper, a method combining OBF (Orthogonal Basis Function) decomposition and EDR (Edit Distance on Real sequence) is proposed to match magnetic anomaly signals under low SNR. The calculation process of OBF-EDR method is divided into three steps (Fig. 1). Step 1: The OBF method is used to decompose the magnetic anomaly signals. According to the characteristics that the background noise is not correlated with the basis function, three sets of discrete basis function coefficient sequences with higher SNR can be obtained [5]. Step 2: The sequences of discrete basis function coefficient are taken as the calculation object, and the EDR method with strong anti-noise ability is used to measure and get their similarity results. Step 3: The matching result of the magnetic anomaly signals is calculated through the similarity results of the three sets of discrete basis function coefficient sequences by the weight method. The simulation shows that the curves of matching accuracy of OBF-EDR and EDR method have the same changing trend with the change of signals’ SNR, but the curve of matching accuracy of OBF-EDR method moves to the direction of lower SNR compared with that of EDR (Fig. 2). It indicates that the OBF-EDR method can match the magnetic anomaly signals correctly at a lower SNR than the EDR method. What’s more, when the matching accuracy of OBF-EDR reaches 0.9, the required signal’s SNR is -2dB, which is 7dB lower than that of EDR method. The above conclusions prove the advantage of the OBF-EDR method to matching magnetic signals under low SNR.

**References:**

 [1] Y. Shen, J. Wang, and J. Shi, J. Magn. Magn. Mater., 484, 164-171, (2019). [2] Y. Shen, D. Hasanyan, and J. Gao, Smart Mater. Struct., 22, 095007, (2013). [3] J. Wang, Y. Shen, and R. Zhao, J. Magn. Magn. Mater., 505, 166761, (2020). [4] O. Faggioni, M. Soldani, and A. Gabellone, J. Appl. Geophys. 72, 46-56, (2010). [5] B. Ginzburg, L. Frumkis, and B. Kaplan, Sens. Actuator A-phys., 102, 67-75, (2002).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/342a1c3ef4df12f64377b274bb525671.jpg)
Fig.1. OBF-EDR method calculation process.

![](https://assets.underline.io/uploads/markdown_image/1/image/5e76e72a07c2b094b80ba3d24bfe86fb.jpg)
Fig.2. Matching accuracy of OBF-EDR and EDR method to match the magnetic anomaly signals under different SNR.

 A Matching Method for Magnetic Anomaly Signals Under Low Signal-to-Noise

Biphasic or monophasic pulsed magnetic fields are commonly used in non-invasive stimulation of biological systems [1]. Biphasic systems bring the advantage of using both the positive and negative pulsed fields, thereby leading to twice the amount of signal obtained by using monophasic pulsed field systems [2]. With a focus on medical applications, biphasic pulses have been used in transcranial magnetic stimulation applications [2, 3] for neuronal excitation and in magnetic resonance applications for inducing proton transitions [4, 5]. Typically, commercial field generator systems with sophisticated data acquisition and processing mechanisms are used for such medical applications. However, with the recent trends towards portable diagnostics, a redesign of the pulsed field generator systems for improved portability is required. In this work, design and challenges of creating biphasic low-pulsed magnetic fields, which may be utilized for portable NMR applications, is discussed. Since portable solutions rely on permanent magnet geometries, the strength of static field is low and correspondingly, the strength of the pulsed field is typically low (less than 0.1T). Prior portable pulsed field generators have been designed to obtain repetitive pulses at both high (up to 0.5T) [2,3] and low-field strengths (0.05T) [4]. However, obtaining high-frequency (1-10MHz) pulsed sinusoidal fields (both monophasic or biphasic) is a challenge for most portable systems. The monophasic pulse generators are useful in obtaining higher pulsed fields and amplitudes with limits on the number of nuclear spins that may be excited during an excitation cycle. Meanwhile, biphasic pulses can provide uniform excitation but have lower field amplitudes. Both the techniques are equally valuable and appropriate pulse sequencing techniques can be designed to make such pulsed field generators valuable additions in pulsed NMR applications. This work highlights the design parameters and challenges of both techniques and details the design of a biphasic portable pulsed field generator operational between 0.5 – 2 MHz. The primary challenge of biphasic pulsed generators is the rapid switching between the different phases. The designed biphasic pulse generator (Figure 1) includes two switching devices and intertwined inductors connected to a common terminal and pulsed with a 180 degree offset. The two switches operate such that only one inductor is active at a particular time. The biphasic generator is designed to obtain a maximum of +/- 15A through the inductor (Figure 2) at an operational frequency of 1MHz. Synchronizing the phase offset between the two switching devices ensures simultaneous operations. In summary, different aspects of the switching device, load, operational frequency and phase synchronization needed to obtain stable biphasic pulsed magnetic fields will be highlighted in this paper.

**References:**

 [1]. N. M. Shupak, F. S. Prato and A. W. Thomas, "Therapeutic uses of pulsed magnetic-field exposure: A review," in URSI Radio Science Bulletin, vol. 2003, no. 307, pp. 9-32, Dec. 2003. [2]. J. Selvaraj, P. Rastogi, N. Prabhu Gaunkar, R. L. Hadimani and M. Mina, "Transcranial Magnetic Stimulation: Design of a Stimulator and a Focused Coil for the Application of Small Animals," in IEEE Transactions on Magnetics, vol. 54, no. 11, pp. 1-5, Nov. 2018, Art no. 5200405. [3]. N. R. Bouda, M. Mina and R. J. Weber, "High-Current Magnetic Field Generator for Transcranial Magnetic Stimulation Applications," in IEEE Transactions on Magnetics, vol. 50, no. 11, pp. 1-4, Nov. 2014, Art no. 5400204. [4]. N. Prabhu Gaunkar, J. Selvaraj, L. Bauer, M. Mina, R. Weber and D. Jiles, "Design and Experimental Implementation of a Low Frequency Pulsed Magnetic Field Generator," in IEEE Transactions on Magnetics, vol. 53, no. 11, pp. 1-4, Nov. 2017, Art no. 6501004. [5]. N. Prabhu Gaunkar, "Developments in static and pulsed magnetic field systems for detection of magnetic resonance in non-uniform magnetic fields" (2018). Graduate Theses and Dissertations, Iowa State University, 17291.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/01008017854c7244c2b8306e8ac882b5.jpg)
Schematic of biphasic pulsed field generator

![](https://assets.underline.io/uploads/markdown_image/1/image/d82b09b7d55f78fed164814ed756efac.jpg)
Current through inductor at a frequency of 1MHz (estimated from current sense resistor)

 Design of stable biphasic pulsed magnetic fields for portable diagnostic applications

Progress in the development of nanomagnetism has forced the scientific community to pay close attention to 3D magnetic nanostructures [1]. Currently, the use of X-ray nanotomography makes it possible to determine temporal evolution of the magnetization [2, 3]. Since both spatial and temporal resolution of experimental methods is limited, it is natural to compare the experimental results with the computer simulation predictions. This approach allows resolve both the fine spatial structure of vortex aggregates and the fast processes of topological rearrangements. In our simulations, we take advantage of instability of a domain wall motion in a magnetic field above a certain threshold value. In continuous films this instability generates various dynamic vortex structures and spin waves [4, 5]. In the case of antidot films, new types of vortex-induced topological rearrangements of magnetization can be observed in the region of the array of holes. We simulate the process of magnetization reversal in a permalloy cuboid with the dimensions X = 800 nm, Y = 200 nm (the film thickness), Z = 400 nm, with periodic boundary conditions along the z axis. This axis is the easy magnetization axis in the film plane and the y axis is perpendicular to the film plane. We have considered a single rectangular hole with size d as well as arrays with three or nine such holes, where d = 25, 50, or 100 nm. Micromagnetic simulations of magnetic dynamics were performed by solving the Landau-Lifshitz-Gilbert (LLG) equation using mumax3 solver [6]. We used a 3D orthogonal uniform mesh of size a = 3.125 nm. To visualize the magnetization topological structure we employed an approach based on the calculation of topological charges of two types [4, 5]. They satisfy the laws of conservation and we have classified permissible transformations of the magnetization distributions at the film boundaries. When a domain wall moves in a sample with holes, various scenarios of transformations of vortex structures are realized. We have specified the initial distribution of magnetization as a specially formed configuration containing a C-shaped (vortex) domain wall (DW) near the boundary of the sample. The axis of the DW (the z axis) is in the plane of the sample. To the right (left) of the DW, the magnetization is directed against (along) the direction of the z axis. After turning on the DC magnetic field H = 200 Oe along the z axis, the DW begins to move, becoming the S–shaped one. Further movement of the DW is accompanied by the generation of vortex structures leading to the complex dynamics of magnetization reversal in the antidot array. (a) In the case of large holes (d = 100 nm), after the DW reaches the antidot array boundary, the vortex structures disappear and the process of magnetization reversal stops. The final steady-state configuration of the magnetization in the three-hole array is as follows: when traversing the contours of the boundaries of the holes clockwise, the magnetization makes a full turn clockwise (counterclockwise) for the middle (lower) hole. This means that the holes are carriers of topological charges and are topologically equivalent to local vortex and antivortex. (b) In the case of medium-sized holes (d = 50 nm), the displacement of the magnetization reversal region is accompanied by the appearance of vortices near the holes. Some vortices connect the lateral boundary surfaces of adjacent holes. (c) In the case of small holes, in the regions between the holes, emerging magnetization distributions look like fragments of DWs. The appearance of vortex filaments similar to the vortex core of C-shaped DWs is typical. Evolution of the 3D magnetization distribution leads to a number of peculiar instant topological configurations. (a) Two vortex filaments connect the boundary surfaces of the film and two other filaments connect the lateral boundaries of the hole with the boundaries of the film (Fig 1a). At the upper boundary of the film, there are two local vortices and one local antivortex (ends of vortex filaments on the surface, Fig. 1b). Additionally, there is an antivortex-like configuration around the hole. At the lower boundary, there is one local vortex, two local antivortices, and a vortex-like configuration around the hole (Fig. 1c). (b) One vortex filament connects the boundary surfaces, the other filament connects the intra-film boundary of the hole and the boundary of the film. There is a local vortex and an antivortex-like hole at the upper boundary, and a vortex-antivortex pair at the lower boundary. (c) A fully antivortex-like hole is combined with a local vortex at the upper boundary and vortex-like hole at the lower boundary.

**References:**

[1] A. Fernández-Pacheco, R. Streubel, O. Fruchart, et al. Nature Comm. Vol. 8, 15756 (2017). [2] C. Donnelly, S. Finizio, S. Gliga, et al. Nat. Nanotechnol. Vol. 15, p. 356 (2020). [3] C. Donnelly, K. L. Metlov, V. Scagnoli, et al. Nat. Physics Lett. https://doi.org/10.1038/s41567-020-01057-3 (2020). [4] V. V. Zverev, E. Zh. Baykenov, I. M. Izmozherov. Phys. Met. Metallogr. Vol. 120, p. 1299 (2019). [5] V. V. Zverev, E. Zh. Baykenov, I. M. Izmozherov. Phys. Solid State, Vol. 61, p. 2041 (2019). [6] A. Vansteenkiste, J. Leliaert, M. Dvornik et al. AIP Advances, Vol. 4, 107133 (2014).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f1b905260d9cf801ccb2bec91c5c2329.png)
Fig. 1. A snapshot of the vortex structure (a) inside the film, (b) at the upper boundary (top), (c) at the lower boundary (bottom).

 Dynamical Rearrangements of 3D Vortex Structures in Moving Domain Walls in Continuous and Antidot Patterned Permalloy Films

Nickel thin films (100 nm) deposited on 128o Y-cut: LiNbO3 exhibit a well-defined magnetic anisotropy upon annealing at 325o C [1]. Simulating the magnetization of these films using a temperature dependent Stoner-Wohlfarth model indicates that the films have a very narrow angular distribution of anisotropy axes [2]. Here we report a direct measurement of the angular distribution of anisotropy axes by the angular remanence technique [3]. The angular range over which the easy axis remanence drop significantly is a measure of the angular distribution of the anisotropy axes. For annealed nickel thin films on 128o Y-cut: LiNbO3 the FWHM of the remanence as a function of angle (α) is approximately 0.44o .

**References:**

[1] S.A. Mathews, N.S. Bingham, R.J. Suess, K.M. Charipar, R.C.Y. Auyeung, H. Kim, and N.A. Charipar, IEEE Trans. Mag., Vol. 55, Issue 2, (2019) [2] S.A. Mathews, A. Ehrlich, and N. A. Charipar, “Hysteresis branch crossing and the Stoner-Wohlfarth model”, Scientific Reports, 10:15141 (2020). [3] H.J. Richter and H. Hibst, Jour. Mag. Mag. Mat., 95, 118-122 (1991).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a8bbb46dcce86d5d88cbc7706792ebd7.jpg)
Figure 1: Absolute value of easy axis remanence as a function of α (angle between the hard axis and the saturating field). Open squares represent measured data. The solid line represents the model, assuming an ensemble of Stoner-Wohlfarth particles with two normal (Gaussian) distributions of anisotropy orientation.

 Angular Remanence and Anisotropy Orientation Distribution in Nickel Films on LiNbO3

In this work, we report transverse susceptibility (TS) measurements performed on nickel thin films, deposited on 128o Y-cut LiNb03 substrates and annealed at low temperature. The films exhibit a strong uniaxial anisotropy [1] and hysteresis branch crossing (HBC) [2]. The results are compared to numeric solutions to the Stoner-Wohlfarth (SW) model [3] using a finite temperature approximation [4] and a distributed anisotropy. The experimentally observed TS-curves, measured with applied field close to the hard axis, are shown to be in excellent agreement with the SW-model, confirming that the magnetization in these samples occurs via coherent rotation. As predicted by Aharoni [5], the TS curves show three peaks or cusps: two reversible peaks associated with anisotropy field, and one irreversible peak associated with the discontinuity in the transverse magnetization.

**References:**

[1] S.A. Mathews, N.S. Bingham, R.J. Suess, K.M. Charipar, R.C.Y. Auyeung, H. Kim, and N.A. Charipar, IEEE Trans. Mag., Vol. 55, Issue 2, (2019) [2] S.A. Mathews, A.C. Ehrlich, and N.A. Charipar, Scientific Reports, 10:15141 (2020) [3] Stoner, E.C. and Wohlfarth, E.P., Phil. Trans. Roy. Soc. of London Series A, 240, no. 826, pp.599-624 (1948). [4] L. Lanci and D. Kent, J. Geophys. Res., Vol. 108, No. B3, 2142 (2003). [5] A. Aharoni, E. M. Frei, S. Shtrikman, and D. Treves, Bull. Res. Counc. Isr., Sect. F 6A, 215 (1957).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3ddf7ccaf7051008650d6c58dc9c0178.jpg)
Figure 1: Measured (open squares) and simulated (solid lines) transverse susceptibility of a nickel thin film on LiNbO3. The field is applied at an angle α=4o from the hard axis. The ascending and descending curves are shown in red and blue, respectively.

 Transverse susceptibility of nickel thin films with uniaxial anisotropy

In this paper, an improved interpolating element-free Galerkin method (IIEFGM) with nonsingular weight function is introduced to calculate the magnetic field distribution of electrical machine. Since the shape function satisfied the property of Kronecker-delta function, the IIEFGM can impose the boundary conditions directly. Meanwhile, the IIEFGM adopts nonsingular weight function, which means the weight function of the conventional element-free Galerkin method (EFGM) can be used. To demonstrate the effectiveness of IIEFGM, the magnetic field distribution of a permanent magnet coreless disc machine is analyzed. The selection of weight function and the nodes distribution are discussed, the comparison with Finite element method (FEM) shows the IIEFGM has high computational accuracy. I. Introduction Meshless methods are the important supplement to the finite element method in solving boundary problems, and element-free Galerkin method is the most widely used method. The mainly difference between the EFGM and the FEM is the construction method of the shape function, however, since the shape function of EFGM does not satisfy the property of Kronecker-delta function, the imposition of boundary conditions required additional methods, such as penalty method. To solve this problem, reference [1] proposed an interpolating element-free Galerkin method (IEFGM), however, the method adopt singular weight, which makes the computation difficult. Reference [2] proposed IIEFGM with nonsingular weight function, which has the advantages of both the FEM and the IEFGM. In this paper,a permanent magnet coreless disc machine (PMCDM) is taken as an example [3] the IIEFGM is introduced to calculate the magnetic field distribution of the PMCDM, and the comparison with the FEM validated the effectiveness of the IIEFGM. II The foundation of the IIEFGM In conventional EFGM, the basis function is defined as: p1=1, p2=x, p3=y (m=3), and the trial function is uh(x)=pT(x)a(x). The coefficients a(x) can be obtained by minimizing the weighted least-square function, then the shape function N(x) can be derived. To make the shape function satisfied the property of the Kronecker-delta function, the IIEFGM modified the basis function. Then the boundary conditions can be imposed directly, and the negative exponential function is selected as the weight function. III Magnetic field calculation of the PMCDM Fig.1 shows the magnetic field distribution of the PMCDM by using IIEFGM, the problem domain is divided into the radial part and the circular part. Fig.2 depicts the main flux density on the airgap center line. It can be seen that the calculated results obtained by IIEFGM are in excellent agreement with the results with the FEM. IV Conclusion This paper introduced an improved interpolating element-free Galerkin method. The Galerkin weak form and the discretized equation are derived, the flow of the algorithm is optimized, and a weight function with higher accuracy is given. Based on IIEFGM, the magnetic field distribution of a PMCDM is analyzed, numerical results verified the effectiveness of the IIEFGM. The study results provide references for the application of IIEFGM.

**References:**

[1] Ren H , Cheng Y ,"The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems," Engineering Analysis with Boundary Elements, Vol. 36(5), p.873-880, 2012. [2]Wang J F , Sun F X , Cheng Y M, "An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems," Chinese Physics B, Vol. 21, No.9, 2012. [3]Gu C L , Wu W , Shao K R, "Magnetic field analysis and optimal design of DC permanent magnet coreless disc machine," IEEE Transactions on Magnetics, Vol. 30(5), p. 3668-3671, 1994.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/16d3efc5f003de67c4a8d3fbbb91d924.jpg)
Fig.1 Magnetic field distribution. (a) circular direction. (b) radial direction

![](https://assets.underline.io/uploads/markdown_image/1/image/4fb8054f5ba97bdbc11fc680df6f9a1b.jpg)
Fig.2 Main flux density on airgap center line. (a) circular direction. (b) radial direction

 Application of an improved interpolating element-free Galerkin method in magnetic field calculation

Introduction Hysteresis of magnetic materials is a complicated physical phenomenon. The appropriate mathematical expression which can describe the hysteresis mechanism is desired in the design of electromagnetic devices, such as transformers and reactors. It could enable researchers to model the hysteresis accurately, and hence improve the performance and efficiency of the devices. This paper modifies the Bouc-Wen model to model the hysteresis of magnetic materials, and accords the parameters of the model with physical meaning. At present, the magnetic hysteresis models are classified as mathematical models and physical models. The Preisach model, as a mathematical model, can describe the hysteresis phenomenon, but it cannot describe the magnetization process of the materials[1]. The domain structure model, which has a clear physical meaning and takes into account of energies related to the growth of the domain, can describe the hysteresis well, but it only studies in microscopic range, and is complicated to be employed in engineering applications[2]. To model the magnetic properties efficiently and accurately, a practical model which can correlate model parameters describing magnetization process with the hysteresis phonemonen responsible to the applied magnetic field is needed. The model proposed in this paper has the feature to simulate the magnetization process in general by adjusting the parameters according to the materials characteristic in applied magnetic fields. Theorem Ferromagnetism reveals that the internal mechanism of hysteresis is very complicated. The magnetization process mainly includes motion of the domain wall and rotation of the spontaneous magnetization of domain [3]. The process of wall motion is very similar to the kinetic process of viscous friction, such as nonlinear hysteretic mechanics Bouc-Wen model, so that the kinetics model is exploited to describe the domain wall motion. However, it cannot characterize the rotation of the domain magnetization[4]. The model is modified by introducing a term which can characterize the rotation process. The improved model includes the four resistances encountered when the domain wall moves and the domain magnetization rotates during the magnetization process: i) the resistance caused by the increasing energy of the domain wall, which is linearly related to the displacement of the domain wall; ii) the resistance caused by the local eddy current, which is proportional to the moving velocity of the domain wall; iii) the resistance caused by the rotation of the domain magnetization; and iv) the hysteretic restoring force caused by material defects, which is proportional to the hysteretic displacement. The hysteretic displacement which describes the inelastic behavior of domain wall motion is calculated by the differential equation of the Bouc-Wen model. The improved model can be used to calculate the magnetic hysteresis under high and low magnetic field strengths. The parameters of the model representing the damping coefficients of each resistance are identified by the particle swarm algorithm. It is found that the damping coefficient is a function of the strength of the applied field under quasi-static magnetization, and the shape parameters of the model are constants. Since eddy current will cause the reactive force during the motion of domain wall and rotation of domain magnetization under dynamic excitation, the model must be modified to take the account of dynamic excitation into damping coefficients. A function of damping coefficients related to applied field strength and frequency is discussed in detail in the full paper. Experiment In order to simulate magnetization process using the improved model, the oriented silicon steel B27R090 with relatively simple domain structure is used. The Single Sheet Test is used as the experimental platform to measure the magnetic properties of the sample under quasi-static and dynamic magnetizations. Fig. 1 and Fig.2 show the measured hysteresis loops and the predicted hysteresis loops under sinusoidal excitation at 5 Hz and 50 Hz. The predicted hysteresis loops calculated by the improved Bouc-Wen model are in good agreement with the measured loops when the flux density is between 0.3 T and 1.4 T. The calculated total losses using the improved Bouc-Wen model is also in good agreement with the the measurement results. The full paper will discuss the results in detail.

**References:**

[1] Mayergoyz I D, IEEE Transactions on Magnetics., Vol. 22, p. 603-608(1986) [2] Sudo M, Mifune T and Matsuo T, IEEE Transactions on Magnetics., Vol. 49, p. 1829-1832 (2013) [3] Bertotti G, IEEE Transactions on Magnetics., Vol. 24, p. 621-630 (2002) [4] Ikhouane F, José Rodellar, Nonlinear Dynamics., Vol. 42, p. 63-78 (2005)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d8bb2736bdfccbddfea7878208a89264.jpg)
Fig.1 Measured (solid lines) and calculated (dashed lines) hysteresis loops under sinusoidal excitation with amplitudes of flux density of 0.3 T, 0.7 T, and 1.4 T at frequencies of 5 Hz

![](https://assets.underline.io/uploads/markdown_image/1/image/11057d74d6a094ebc1633ff25b2967f4.jpg)
Fig.2 Measured (solid lines) and calculated (dashed lines) hysteresis loops under sinusoidal excitation with amplitudes of flux density of 0.3 T, 0.7 T, and 1.4 T at frequencies of 50 Hz

 An Improved Hysteresis Model Based on Bouc-Wen Model Under Quasi-Static and Dynamic Magnetizations

—A circular polarized vortex beam generating based on leaky-wave antenna with substrate integrated waveguide(SIW) is proposed in this paper. Comparing with phased array antenna, the leaky-wave antenna is working without complicated feeding network and phase shifter. Then the unit is designed to realize circular polarization. Finally The loop leaky-wave antenna with SIW is designed to generate vortex beam and works at 14.5 GHz. The simulated gain reaches 11.4 dBi with generating -1 mode orbital angular momentum(OAM) of vortex beam. Vortex electromagnetic wave are electromagnetic waves with orbital angular momentum Wave. The most important characteristic of orbital angular momentum is its orthogonality in different modes. The degree of freedom of electromagnetic wave includes frequency and polarization. It is amazing that the orthogonality of vortex electromagnetic wave provides extra freedom for electromagnetic wave. Therefore, vortex electromagnetic wave has the potential to be used in communication and imaging systems[1]-[3]. In this paper, A circular polarized vortex beam generating based on leaky-wave antenna with substrate integrated waveguide(SIW) is proposed. Because the leaky-wave antenna is working without complicated feeding network and phase shifter. And the unit is designed to realize circular polarization. Then the radiation pattern of loop leaky-wave antenna is derived from the leaky-wave unit. Finally The loop leaky-wave antenna with SIW is proved to radiate -1 mode vortex electromagnetic wave with circular polarization in the simulation of CST Studio. The simulation model is showed as Fig. 1 a), the reason the loop leaky-wave antenna is three-quarter circle is that CST Studio limits the wave port should be put at the coordinate axis. And from [4], we know a part of circle to radiating is enough to radiate vortex beam. The microstrip line is designed for impedance matching to transmit microwave to the loop leaky-wave antenna. The S11 and S21 is showed in Fig. 1 b. It is showed the S11 is lower than -10dB and S21 is also low, because the most energy is radiated to open space. Fig. 2 a) is the phase distribution in far field at 14.5GHz. As we can see, the phase distribution is -1 mode vortex beam form. The radius of loop antenna determines the phase difference between adjacent unit, and in this paper, the radius is 52mm to make the OAM mode is -1. Fig. 2 b) shows the amplitude distribution in far field. The distribution is not uniform, which is caused by deletion of loop leaky-wave antenna. If the loop leaky-wave antenna is complete circle, the distribution will be improved. A circular polarized vortex beam generating based on leaky-wave antenna with substrate integrated waveguide (SIW) is proposed in this paper. Comparing with phased array antenna, the leaky-wave antenna is working without complicated feeding network and phase shifter. We proof linear polarized unit can radiate circular polarized beam but the beam is not clean, so we design unit with circular polarized radiation. Finally, the loop leaky-wave antenna with SIW is designed to transmit vortex beam and works at 14.5 GHz. The simulated gain reaches 11.4 dBi with generating -1 mode vortex beam.

**References:**

[1] Yan, Y. , Xie, G. , Lavery, M. P. J. "High-capacity millimetre-wave communications with orbital angular momentum multiplexing." Nature Communications vol. 5, pp. 4876, 2014. [2] Shilie, Zheng, Xiaonan, et al. "Orbital angular momentum mode-demultiplexing scheme with partial angular receiving aperture." Optics express. vol. 23, no. 9, pp.12251-12257, 2015. [3] Bai, Q. , A. Tennant , and B. Allen . "Experimental circular phased array for generating OAM radio beams." Electronics Letters 50.20:1414-1415, 2014. [4] Xiong, Xiaowen , et al. "Direct Generation of OAM Mode-Group and Its Application in LoS-MIMO System." IEEE Communications Letters PP.99(2020):1-1.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/399fb16ca0487d903477aca5d9a245b8.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/e87535880351c24dc4b35b3f36a74626.jpg)

Field Computation in Media Exhibiting Hysteresis Using Hopfield Neural Networks

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