Low-power and high-sensitivity magnetic sensors and systems 9781630812430, 1630812439

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Low-Power and High-Sensitivity Magnetic Sensors and Systems

For a complete listing of titles in the Artech House Remote Sensing Series, turn to the back of this book.

Low-Power and High-Sensitivity Magnetic Sensors and Systems Eyal Weiss Roger Alimi

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Cover design by John Gomes

ISBN 13: 978-1-63081-243-0

© 2019 ARTECH HOUSE 685 Canton Street Norwood, MA 02062

All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.   All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.

10 9 8 7 6 5 4 3 2 1

This book is a gift to Hervé Abraham Alimi, beloved brother, gone too early but present every day in my memory, —R. A. It is also dedicated to Prof. Eugene Paperno, esteemed mentor, who tragically and prematurely passed away. —E. W.

Contents

Acknowledgments

1

Introduction

1

1.1

Overview

1

1.2 1.2.1 1.2.2

Magnetic Sensors Fluxgates Fluxgate Applications

2 3 6

1.3 1.3.1 1.3.2 1.3.3 1.3.4

Orthogonal Fluxgates Why Focus on Orthogonal Fluxgates Low-Power Parallel Fluxgates Low-Power Orthogonal Fluxgates Summary of the State of the Art

7 9 11 12 13

1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5

Reducing Power Consumption in Fluxgates The Goal Method to Reduce the Power Consumption Core Saturation Discontinuous Excitation Techniques Sampling and Processing

14 14 14 15 17 18

1.5 1.5.1 1.5.2

Magnetic Systems Why Low-Power Consumption Is Important DC Jumps in Low-Power Fluxgate Magnetometers

19 19 19

vii

xiii

viii

Low-Power and High-Sensitivity Magnetic Sensors and Systems

1.5.3 1.5.4 1.6 1.6.1 1.6.2

Power Supply Lines Data Lines Magnetic Data Processing Magnetic Anomaly Detection Localization of Moving Objects References Selected Bibliography

20 20 21 21 22 24 29

2

Magnetic Systems

31

2.1

Overview

31

2.2 2.2.1 2.2.2 2.2.3

Noise from the Natural Environment Clutter from the Magnetic Interactions in the Ionosphere Magnetic Geology Magnetic Hydrodynamics

33

2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5

Internal Sensor Noise and Clutter Internal Sensor Magnetic Noise Alignment Noise Internal Electronic Noise Internal Data Cable Noise Noise from Power Lines

38 38 39 40 41 42

2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5

Environmental Anthropogenic Noise and Clutter Clutter from Moving Ferromagnetic Objects Clutter from Eddy Currents in Conducting Objects External Power Supply Lines External Data Lines Noise and Clutter from Motion of Measurement System

43 43 45 46 46

2.5 2.5.1 2.5.2 2.5.3

Surveillance Systems Detection Schemes Sensor Arrays Generic Surveillance Applications

47 48 51 54

2.6 2.6.1 2.6.2

Survey Systems Unexploded Ordnance Detection Maritime Magnetic Surveys

61 61 63

33 35 37

46



Contents 2.6.3

Low-Power Survey Magnetometers References

ix

71 81

3

Low-Power Fluxgates

87

3.1

Overview

87

3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7

A Tube-Core Orthogonal Fluxgate Operated in the Fundamental Mode Tube Core Fluxgate Experiments Optimal Excitation Parameters Magnetic Noise Suppression Fluxgate Sensitivity Fluxgate Equivalent Magnetic Noise Dominant Origin of the Noise Tube Core Orthogonal Fluxgate Conclusion

88 89 90 90 93 93 94 95

3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.3.8

Excess Magnetic Noise in Orthogonal Fluxgates Employing Discontinuous Excitation Experimental Setup Excitation Magnetic Field and Skin Effect Distribution of the Magnetic Field in the Core Domain Morphology Method for Investigating Fluxgate Noise Results Discussion on Source of Magnetic Noise Conclusion

3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6

Inhomogeneous Core Material Inner Core Neutrality Interference with the Gating Effect Perming Phenomenon High-Power Consumption of Inner Core Thermal Effects of Core Heating Composite Wires Conclusion

3.5 3.5.1 3.5.2 3.5.3

Noise Investigation of the Orthogonal Fluxgate Employing Alternating Direct Current Bias Experimental Setup Measurement Synchronization Excess Noise Suppression

95 96 97 98 101 103 104 104 107 108 109 110 113 113 117 119 120 120 122 123

x

Low-Power and High-Sensitivity Magnetic Sensors and Systems

3.5.4 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.6.5

Alternating DC Bias Conclusion DC Jumps in Low-Power Fluxgate Magnetometers Model of DC Jumps in Parallel Fluxgates Model of DC Jumps in Orthogonal Fluxgates DC Jump Dynamic Model DC Jumps’ Dynamic Model in Amorphous Wire Core Orthogonal Fluxgates DC Jumps Dynamic Model in Parallel Fluxgates References

126 126 127 129 129 132 132 136

4

Low-Power Sampling

141

4.1 4.1.1 4.1.2 4.1.3

Overview Sampling Resolution Sampling Options Sampling of Low-Power Fluxgates

141 141 142 142

4.2 4.2.1 4.2.2 4.2.3 4.2.4

Sampling and Processing of an Orthogonal Fluxgate Output Experimental Setup Digital Selective Bandpass Sampling Technique Discussion Conclusion on Digital Selective Bandpass Sampling

143 143 145 146 147

4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6

Duty Cycle Operation of an Orthogonal Fluxgate Synthesis Method Experimental Setup Experiment Results Conclusion

148 149 150 151 152 155 156

4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5

Concatenation of Discontinuous Operated Orthogonal Fluxgate Noise Measurements and Analysis Excitation Waveforms Noise Measurements Method for Eliminating the Excess Noise Discontinuous Excitation Conclusion

157 157 158 158 161 162

4.5

Conclusion

163

References

163



Contents

xi

5

Magnetic Data Processing

167

5.1 5.1.1 5.1.2 5.1.3

Magnetic Anomaly Detection Orthonormal Basis Functions Representation Minimum Entropy Detection Filter Periodic Anomaly Detection Filter

167 167 169 172

5.2 5.2.1 5.2.2

Magnetic Anomaly Localization The Levenberg-Marquardt Localization Algorithm Genetic Algorithm for Magnetic Dipole Localization References

196 198 209 227



About the Authors

233



Index

235

Acknowledgments Eyal Weiss thanks his wife Lilach for her support, patience, and for giving us the most wonderful family and the best children: Yoav, Tamar, Yonatan and Avigail. He also thanks his parents Sonia and Aharon, for their unconditional support and in lingual editing and care. Roger Alimi would like to thank his wife Odile: without her none of this work would have ever been accomplished. Avital and Noam, Naomi and David, Sarah and Yehuda, Maayan-Chen and Elad, Michael: each of them deserves my gratitude for their supporting presence and encouragement all through the writing task. Special thoughts also go to his late father Sylvain and his beloved mother Julie. Eugene Paperno, who had tragically passed away, for being an advisor, for his creativeness, for his relentlessness, and constant and unyielding strive for perfection,and most of all for his never-ending care and support. The authors express their deep gratitude to the following contributors: Tsuriel Ram-Cohen for the assistance in research on periodic anomaly detection. Asaf Grosz, for his help in the research and implementation of the band pass sampling technique in orthogonal fluxgates. Shai Amrusi, for his never-ending help in the lab and for many fruitful discussions. Elad Fisher, for his help in research of sensors noise and system view. Amir Ivry, for his work on Barkhausen jump detection algorithm.

xiii

xiv

Low-Power and High-Sensitivity Magnetic Sensors and Systems

Alon Shavit for his research on fluxgate calibration methods. Shahar Shoham and Elazar Sarid, Research Management at Soreq NRC, Yavne, Israel. The publishing team at Artech House and especially to Soraya Nair and Aileen Storry for their endless patience and persistence.

1 Introduction 1.1  Overview A magnetometer is a scientific instrument for the measurement of the strength and direction of a magnetic field. Room-temperature magnetometers are used in many applications, especially in geosciences, archeology, space exploration, biomedicine, defense, aerospace, and, recently, mobile devices such as smartphones and tablet computers [1–4]. Each application requires different magnetometer parameters. For example, in magnetometers employed in space exploration [4], the measurement offset, power consumptions, and weight are the dominant requirements, whereas in biomedical applications [5–7], the accuracy and resolution are the dominant factors. Emerging mobile applications [8–11] and defense applications [11, 12] require very low-power consumption, low price, and small size. Fluxgate magnetometers are induction sensors utilizing a ferromagnetic core that is saturated periodically to modulate the measured magnetic field. The modulated field is picked up by a coil. Parallel fluxgate cores are modulated by a bipolar excitation field [13] where fundamental mode orthogonal fluxgates [14] are best modulated by a unipolar field [15, 16]. Fluxgates are commonly used in large sensor arrays for surveillance applications. This is because they are sensitive (typical noise density of around 10 pT/√Hz at 1 Hz), small, and relatively inexpensive. To facilitate a magnetic sensing system, engineering must be involved. However, in magnetic sensing systems, conventional engineering does not suffice, as there appears to be a lot of “black magic” when transferring magnetic 1

2

Low-Power and High-Sensitivity Magnetic Sensors and Systems

research systems to products. The term “black magic” in our case refers to insufficient scientific knowledge of what affects magnetic sensing technology. In fact, many of the conventional engineering practices cannot be used as-is in magnetic systems. This is mainly because of the following aspects of highsensitivity magnetic systems. In many cases, the magnetic sensors cannot be easily shielded from the environment. This is because the measured field is typically in low frequencies (millihertz to kilohertz band with corresponding wavelength 300 to 300 ∙ 106 km) and hence the measured field is not significantly attenuated by the media. It is therefore very difficult to shield the sensor from external magnetic interferences as, in this case, shielding the sensor results in shielding the measured field as well. Additionally, when the measurements take place in a nonshielded environment the Earth magnetic field is the most dominant field. It is a vectorial quasi-DC bias field with magnitude 5 to 6 orders of magnitude larger than that of the measured field. This fact affects the measurements in many ways. It forces the magnetometer to be very stable so as not to rotate in the highly directional field. It also requires monitoring of the environment to distinguish between random changes in the natural magnetic field and those caused by the measured object. Moreover, uncontrolled magnetic fields are emitted from many contraptions that may affect the measuring system. This is because most of the electronic devices are not controlled for low-frequency electromagnetic emissions. Electromagnetic interference (EMI) standards control emissions in much higher frequencies, typically in the range of radio frequencies. In this book, we focus on low-power and high-sensitivity magnetic sensors and systems. We describe some new methods to reduce the power consumption of fluxgates. We then proceed to removing some of the black magic from producing low-sensitivity magnetic system.

1.2  Magnetic Sensors Magnetometers can be divided into two basic types: • Scalar magnetometers, which measure the total strength of the magnetic field; • Vector magnetometers, which measure the component of the magnetic field in a particular direction.



Introduction

3

The use of three orthogonal vector magnetometers enables the magnetic field strength, inclination, and declination to be uniquely defined. Examples of vector magnetometers are fluxgates, superconducting quantum interference devices (SQUIDs), and the atomic Spin-Exchange Relaxation Free (SERF) magnetometer [7, 17]. The measurement band is typically very narrow and low ranging from direct current (DC) to a few hertz. The resolution of a magnetic sensor is universally referenced at 1 Hz because in most low-noise magnetometers the noise is approximately white at frequencies higher than 1 Hz. Because the magnetic field band is near DC, the dipole magnetic field attenuation is barely influenced by the surrounding media. This property is what makes magnetic measurement attractive for many applications. The media are transparent to the magnetic field in low frequencies. Magnetic sensors can measure field variation through water, air, and soil, regardless of weather or visibility. For the same reason, magnetometers are also used as magnetic anomaly detectors in applications such as submarine detection [13, 14]. Magnetometers are used in ground-based electromagnetic geophysical surveys (as magneto-telluric and magnetic surveys) to assist with detecting mineralization and corresponding geological structures [13]. Airborne geophysical surveys use scalar magnetometers that can detect magnetic field variations caused by mineralization. Magnetometers are also used to uncover archaeological sites, shipwrecks [18], and other buried or submerged objects, and in metal detectors they are used to detect metal objects in security screening [2, 15], for example, in a magnetic survey we have performed to locate the wreckage of an airplane buried few meters in seabed at a water depth of 25m (see Figure 1.1) [19]. Another example that we have developed is the use of triaxial fluxgates to localize a ferromagnetic object on an inspected person crossing a security checkpoint [12]. Fluxgates are used in directional drilling for oil or gas to detect the azimuth of the drilling tools near the drill bit. They are most often paired up with accelerometers in drilling tools so that both the inclination and azimuth of the drill bit can be found. A three-axis fluxgate magnetometer was part of the Mariner 2 and Mariner 10 missions and is often used in space explorations [20]. A dual technique magnetometer is part of the Cassini-Huygens mission to explore Saturn [13, 21]. A comprehensive introduction to different types of magnetometers can be found in [2]; at this point, we elaborate on fluxgates. 1.2.1  Fluxgates

Fluxgates are vector induction sensors capable of measuring magnetic fields as large as Earth’s field and as small as a heartbeat [22]. The fluxgate method

4

Low-Power and High-Sensitivity Magnetic Sensors and Systems

Figure 1.1  Three-dimensional (3-D) magnetic map of maritime magnetic survey conducted using scalar magnetometers. Scale is in nano-Tesla and grid is in meters [19].

of operation is based on modulation of the permeability of the soft magnetic core, which creates changes in the DC flux of the pickup coil wound around the sensor core [2, 23]. The output voltage appears on the second harmonics of the excitation frequency because the permeability arrives at its minimum and maximum twice in an excitation cycle (Figure 1.2) [2, 24, 25]. The fluxgate sensor consists of a core of magnetic material surrounded by a pickup coil (Figure 1.3) [26]. The Earth’s field along the core axis, Be, produces a magnetic flux BA in the core of (average) cross-sectional area A. If the permeability µr of the core material is changed, the flux changes and a voltage Vsec is induced in the n turns of the pickup coil, where

V sec = nA dB dt

(1.1)

B is proportional to Be, for small Bex, and the factor of proportionality µeff (the effective permeability) depends on the core material and on the geometrical shape of the core. The effective permeability of the core is given by:



Introduction

5

Figure 1.2  Fluxgate principle of operation [27].

Figure 1.3  The basic fluxgate consists of a ferromagnetic core and a pickup coil. Changes of core permeability cause the core field B to change, thereby inducing a voltage in the pickup coil [26].



µa =

µr

(

)

1 + D µeff − 1

(1.2)

and the fluxgate equation is given by:



V sec = nABex (1 − D )

d µr 1 dt 1 + D ( µ − 1) 2 r  

(1.3)

The fluxgate function is based on time variation of the core permeability. When a magnetic material is saturated, its permeability to further magnetization drops. Figure 1.4 shows the magnetization curve for a ferrite material. It may be seen from Figure 1.4 that the slope µ which is the permeability of the ferromagnetic material of the curve varies when a magnetizing field H is applied. The changing core magnetization induces voltage in the

6

Low-Power and High-Sensitivity Magnetic Sensors and Systems

Figure 1.4  Magnetization curve B against µ0H for a tube of Permax 51 ferrite (Ferroperm, Trerrad, Denmark). The slope dB/µ0dH decreases for large µ0H [26]. (Figure reproduced from [32] with permission.)

pickup coil, but if two opposite magnetized cores are placed inside the same coil, then the two magnetizations cancel, and the only flux change is that caused by the constant external field Bex and the changing differential permeability µ. The effective permeability of the fluxgate core is the governing parameter in the geometrical design of the fluxgate core because it determines how much of the measured field will flow through the core [28]. The effective permeability of rods is determined for cylindrical cores by the length over diameter ratio L/D of the core presented in Figure 1.5. 1.2.2  Fluxgate Applications

Typical fluxgate applications are in de-Gaussing ranges, magnetic signature measurement, surveillance and perimeter security, unattended ground sensors (UGS) [11, 29], active field cancellation, MRI preinstallation site surveys, and geophysical studies including time-domian electromagnetic (TDEM) and magnetotellries (MT) surveys [11, 19, 30]. Fluxgates are commercially available and reach magnetic noise as low as 5 pT/√Hz at 1 Hz with power consumption as low as 5 mW per axis. Their size is on the order of a 100 cm3 and their price is on the order of $1,000 for a sensitive triaxial magnetometer. Integrated circuits fluxgates such as TI DRV425 [31] have lower sensitivity and are used in linear position sensing, current sensing, and so forth. The excitation of the fluxgate core must drive it to deep magnetic saturation to achieve low remanence (perming) [2]. It is also important to drive the entire core to deep saturation because some random parts of the core material may not switch polarity by low excitation field. This is due to core material impurities that may undergo spontaneous Barkhausen jumps, which result in a sudden measurement DC jump. This excitation method consumes most of the power in fluxgates. Commercial low power parallel fluxgates



Introduction

7

Figure 1.5  Relation between apparent permeability µ’ and true permeability µ of cylinders having any given ratio m of length to diameter [32]. ��

are susceptible to these phenomena (remanence, perming, and spontaneous Barkhausen jumps) because the low excitation current is insufficient to cancel them out.

1.3  Orthogonal Fluxgates The orthogonal fluxgate effect in ferromagnetic wires has been known since the 1950s [26]. Some researchers have observed this effect during magnetoimpedance studies, and they use the term “nonlinear magneto-impedance” for the same effect [2]. The main advantages of orthogonal fluxgates are their simplicity and small size compared to parallel fluxgates. A conventional parallel fluxgate usually employs either two separate cores or a single ring core and an excitation coil [26], which is absent in the simplest orthogonal fluxgates.

8

Low-Power and High-Sensitivity Magnetic Sensors and Systems

Types of orthogonal fluxgates are presented in [33] (see Figure 1.6). Orthogonal fluxgates do not require an excitation coil; the sensor can be excited directly by the current flowing through the core (see Figure 1.7). Similarly, as parallel fluxgates, in conventional orthogonal fluxgates, the output appears at the second harmonic of the excitation frequency. The fundamental-mode orthogonal fluxgate [14, 34, 35] uses DCbiased excitation (see Figure 1.8); the output is on the same frequency as the excitation. With this operating mode, the AC bias fundamental, rather than the second harmonic, carries information on the external axial field. It is found that a great enough DC bias practically suppresses magnetic noise generated in the fluxgate core by the AC bias field; the fluxgate resolution becomes limited only by the excess electric noise in the AC bias current [36].

Figure 1.6  Orthogonal fluxgates types: (a) Alldredge and (b) Schonstedt [33].

Figure 1.7  Structure of an orthogonal fluxgate with DC-biased excitation [33].



Introduction

9

Figure 1.8  Excitation with a constant DC bias: the excitation waveform (top) and the fluxgate output (bottom).��

1.3.1  Why Focus on Orthogonal Fluxgates

Mobile and defense applications such as magnetic anomaly detectors are typically employed in very low frequencies in the range of DC to 5 Hz (Figure 1.9). They require a magnetometer with low noise on the order of 5 to 50 pT/√Hz to be able to detect ferromagnetic objects carried by a person in range of few meters. This kind of application is not very sensitive to measurement drift, offset, or accuracy. The important parameters are resolution, size, durability, price, and power consumption. The only commercially available

Figure 1.9  Magnetic band and field for different application fields [2].

10

Low-Power and High-Sensitivity Magnetic Sensors and Systems

magnetometers that can deliver most of those properties are fluxgates (Figure 1.10). However, modern sensitive fluxgates are only partially suitable for those applications. Their size, in the case of ring core fluxgates, is quite large (Figure 1.11) as their construction includes two cumbersome coils as well as a synchronous detector and processing electronics. The price is quite high as the construction is complex and the core materials are expensive and remain secrets of the trade [28]. Their power consumption is quite high as the fluxgate must work continuously and repeatedly to completely saturate the magnetic core. Table 1.1 presents properties of common commercial magnetic sensors. Our experiments show that the power consumption of an orthogonal fluxgate operated in the duty cycle can come close to other extremely

Figure 1.10  Noise spectrum of magnetoresistors and fluxgate sensors. HMC 1001 and 1021 are AMR magnetoresistors, NVE AAxx are GMR magnetoresistors, and NVE SDT is a prototype of a spin-dependent tunneling device [3].

Figure 1.11  Magnetic sensors typical volume versus noise.



Introduction

11

Table 1.1 Properties of Common Commercial Magnetic Sensors Hall with Field Concentrators (Sentron CSAIVG) Linear Range 5 mT Size 6 mm Linearity 0.1 < 0.2% Sensitivity TC Offset at 25°C Offset TC noiseRMS (0.1–10 Hz) Perming, hystereis BW Power Consumption

200 ppm/K 50 µT 600 nT/K 1 µT 1 µT 100 kHz 55 mW

AMR (Philips KMZ 51, Honeywell, HMC1001) 300 µT 6 mm 1%

AMR Flipped + Feedback (KMZ 51) 300 µT 6 mm 40 ppm

Fluxgate Billingsley TFM100 100 µT 15 mm 20 ppm

600 ppm/K < 10 µT 100 nT/K 10 nT(1 nT) 300 nT 100 kHz 30 mW

20 ppm/K < 1 µT 2 nT/K 10 nT 10 nT 100 Hz 100 mW

20 ppm/K 10 nT 0.2 nT/K > r, that is, the clutter distance is much larger than r3the distance between the sensors, we can write that R2 ≈ R1 + dR. After rearranging



µ0  3 ( M ⋅ R ) R  M µ0  3 ( M ⋅ R ) R − M   −   = 4π   R 3 4π   R5 R3    

(2.6)

∆c can now be defined as



  µ0  3 ( M ⋅ R ) R − M 3 ( M ⋅ (R + dR )) (R + dR ) − M  ∆c − 3  4π  R5 R + dR ) (  

(2.7)

However, since the unit vector R + dR is the same as the unit vector R, one can rewrite

54



Low-Power and High-Sensitivity Magnetic Sensors and Systems

  µ0  3 ( M ⋅ R ) R − M 3 ( M ⋅ R ) R − M  ∆c = − = 3  4π  R3 R + dR ) (   µ0 3  ∂ 1  dR  = − c ( M , R ) 3 ( M ⋅ R ) R − M   3  ∂R R  4π R

(2.8)

Meaning that the clutter contribution to the gradiometric signal is not only very small, but also rapidly drops out following the fourth power of its distance from the event. Finally, the last contribution is the noise difference, ∆n. Taking into account the internal noises coming from two independent sensors cannot be expressed by a difference. Rather, we should expect that the resulting noise level would look like a sum, in a vector sense of the term, of the two sources. We have seen then that the gradiometric scheme is not expected to affect the basic signal data and that it will probably lead to an increase of the basic internal noise level. However, it is certainly a small price to pay regarding the high clutter denoising that the scheme can achieve in real-world conditions. We illustrate the scheme and its advantages by the following examples. Figures 2.11 and 2.12 present the signals recorded by two nearby sensors. Plots (a) and (b) present the raw data (three axes magnetic field as a function of time, each sample represent one-tenth of a second). Plot (c) shows the gradiometric signal obtained by subtracting (b) from (a). In Figure 2.11(a, b), noisy background signal with strong peak-to-peak variations (up to 1.2nT swing) is visible in both sensors. In Figure 2.10(c), the high fluctuations have disappeared and the remaining signal show only the internal noise of the sensors. In Figure 2.12, the subplots (a) and (b) show the raw data of a longrange, highly drifted signals generated by a far-moving object. Subplot (c) shows the gradiometric signal in which a pattern that was hardly visible in plot (a) is now clearly detectable: this is the motion of a ferromagnetic object that was close to a sensor but was completely screened by the far and big moving object. Here one can see the true power of the gradiometric scheme.� 2.5.3  Generic Surveillance Applications

The customization of the physical sensor array parameters to a specific application is a masterwork of optimization between multiple components in the sensing system and the environment. We present some generic applications for magnetic surveillance.



Magnetic Systems

55

Figure 2.11  (a, b) Single sensor data and (c) the resulting gradiometric signal. The strong fluctuating variations clearly visible in the raw data completely vanish using a gradiometric scheme.

2.5.3.1  Line Crossing

In line-crossing applications the magnetic sensor array is deployed to detect a person carrying a ferromagnetic mass or a vehicle crossing a virtual line (see Figure 2.13). The virtual line may be overt or covert. The line crossing may be utilized to trigger detection for other surveillance systems or to determine if the crossing object carries a ferromagnetic object. Different environmental clutter and information tiers require sensor array optimization. There are different layers of information available in line crossing application and the requirements. The processed information may be some of the following: • Detection of object crossing the detection line; • Magnetic mass moment estimation (size of object); • Estimation of the crossing point along the detection line; • Estimation of crossing angle (at what angle relative to the line the target crosses).

56

Low-Power and High-Sensitivity Magnetic Sensors and Systems

Figure 2.12  (a, b) Single sensor data and (c) the resulting gradiometric signal. The small moving object visible after the gradiometric scheme was almost completely screened by the far and big distractor that both sensors (a) and (b) measure with the phase and amplitude.

Figure 2.13  Line-crossing scheme.

Conventionally, magnetic line-crossing detection is realized using the same technology applied to detect vehicles on roads. The method includes an induction loop buried shallowly underground (Figure 2.14 [38]) or under the road that measured the loop induction change due to a ferromagnetic mass over it [39, 40]. The ferromagnetic object serves as a core to the induction coil and thus induces current in the coil. The induced current in the coil is



Magnetic Systems

57

Figure 2.14  Line-crossing magnetic detection system using induction loops [38].

measured and analyzed. However, in some cases, more information about the intrusion is required. For each information tier and clutter, the environment different sensor array scheme optimization is required. Where an array of sensors is deployed and the location of all the sensors is known, and at least two sensors measure the object with large enough SNR, the field equations can be solved to obtain the location without knowing the magnetic moment in advance. The more sensors measure the object with sufficient SNR, the higher the accuracy of the object localization. The distance between sensors in the daisy-chain linear array is selected according to the required probability of detection, accuracy of localization, and the expected magnetic moment of the crossing object. For example, for a high-sensitivity sensor array performing a single detection in a rural environment detecting a car, the distance between sensors can be as high as 100m where for accurate localization the required distance would be about half. Oil lines are typically installed in a rural environment and are vulnerable to thieving. There are different ways to protect oil lines, mostly by visual or acoustic systems. Acoustic sensors are used to detect impacts on the metal pipe during an attempt to get access to the oil in the line. However, in many cases detecting the thieving only when the tools hit the pipe is too late for deploying an effective countermeasure. It is therefore possible to employ an array of magnetic sensors to detect an approaching vehicle before it reaches the pipe. This deployment is basically a line-crossing scheme where the distance

58

Low-Power and High-Sensitivity Magnetic Sensors and Systems

between the magnetic sensors is tailored to the magnetic moment of a tanker truck. In this case, the distance between sensors using a high-sensitivity sensor array could be larger than 150m. This is because there is no real need for accurate localization and the environment is typically free of clutter. Processing of the measurements to obtain the detection and localization is elaborated in Chapter 5. 2.5.3.2  Checkpoint

Checkpoints are used to monitor personnel getting into a secure parameter. The requirement is to clear only a person that carries no threat. This typically performed by an active metal detector gate. However, a metal detector is visible, power-consuming, and cumbersome for some applications. In addition, a metal detector is conventionally used where the path of the inspected person is predetermined and exact because the metal detector range is very short, which is not always the case. Furthermore, a metal detector detects all conducting material where in most cases the leading threat derives from ferromagnetic objects. In addition, metal detectors are active and may be hazardous for people with medical implants. The same functionality of screening personnel entering a secure domain may be performed by a passive, low-power consumption magnetic sensor array. The array may be covert and not resemble a gate and detect and localize ferromagnetic object on the inspected person. In cases where the location of the ferromagnetic mass is important, a 2-D array may be used to resemble a gateway (see Figure 2.15) [41]. For this application, the sensor array must be used to both clear the environmental clutter and to localize the ferromagnetic object on the body of the screened person. If located in a quiet environment, the distance between sensors may be as high as several meters to detect small arms. If the checkpoint is placed in a noisier environment, the sensors’ pitch should be reduced as to allow the target signal to be significantly larger than the surrounding clutter and allow an effective gradiometric scheme. 2.5.3.3  Vehicle Detection

Vehicles create magnetic anomaly in the natural Earth field. The anomaly is created by the ferromagnetic content in the structure of the vehicle, electric currents in power components of the vehicle, and rotating parts in the vehicle (shaft, gears, and so forth). Each source generates anomaly in the magnetic field that can be measured by a specialized sensor array. The design requirements may differ according to the level of information on the vehicle. For example, to detect on which lane a vehicle is traveling on a road or to estimate the size class of the traveling vehicle [42], a single axis linear sensor ar-



Magnetic Systems

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Figure 2.15  Schematic view of a magnetic checkpoint [41].

ray operating in the ultralow-frequency band for detecting the ferromagnetic structure of the vehicle is a good fit [43, 44]. If the exact location or course is required, the shape of the anomaly may be required and as a result, a more comprehensive three-axial sensor array operating at larger band is needed [45]. In the following section, we describe some of the sources of magnetic anomalies in vehicles and analyze some generic vehicle detection applications such as traffic lane detection, estimated vehicle type, landmine triggering, and approaching vehicle detection. Source of Magnetic Anomaly in Vehicles

There are different mechanisms that affect the magnetic field in a vehicle. One source is the ferromagnetic mass in the structure of the vehicle. In many vehicles, the chassis of the vehicle is made of large segments made of ferromagnetic steel that may extend to few meters in a passenger’s car and much more in trucks and trailers. The elongated ferromagnetic segments have magnetic moments that create a strong anomaly in the magnetic field. The motion of the vehicle along a sensor array detection line results in a classical magnetic dipole anomaly taking the shape of a combination of the three Anderson functions [46] (see Figure 2.16).

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Figure 2.16  An example of classical magnetic dipole anomaly taking the shape of a combination of the three Anderson functions.

Rotating ferromagnetic mechanisms in the vehicle such as the motor parts, the dry shaft, crank shaft, or clutch shaft in the vehicle also creates an anomaly in the magnetic field [47]. The anomaly shape will be periodic with a smaller magnetic moment because the rotation of those mechanisms is typically along the smaller axis, which produces smaller magnetic moment. As a result, the rotating mechanism will have a periodic and smaller signature (see Figure 2.17) compared to the large Anderson function of the chassis of the vehicle. Electric currents in the vehicle and electric cars also create an anomaly in the magnetic field [48]. The source of the field is by electric currents inherent

Figure 2.17  An example of a rotating moment with environmental clutter. The rotating moment appears from sample 120 to 250.



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to several mechanisms in a vehicle. For example, the ignition system in an Otto cycle engine produces high-current sparks in the coil. The high current is driven along unpaired cables that emit a magnetic field in every ignition cycle of the spark system. The resulting magnetic field will have a periodic spike shape at a multiplication of the engine’s rotating speed. Another source may be from the power alternator mechanism that transforms rotation energy from the engine to voltage by a coil rotor. The current in the coil mechanism generates the magnetic field.

2.6  Survey Systems In this section, we describe survey systems and applications. We first review unexploded ordnance detection, and then we present magnetic maritime survey applications followed by land surveys. 2.6.1  Unexploded Ordnance Detection

One of the most important fields of the application of magnetometers is probably the detection of the unexploded ordnance (UXO) such as underwater mines, landmines, or other bomb or explosive remnants of war. More than 100 million buried landmines are killing or injuring from 15,000 to 20,000 people in a year [49]. Reliable, safe, rapid, and cheap detection systems have been a challenge during the past 3 decades, involving both governments and civil organizations [50]. Reviews or surveys are periodically issued to present cutting-edge technologies that can potentially answer this challenge [51–53]. Although several sensing systems can be employed [54], it is well accepted today that magnetic surveys remain the leading approach to the threat. This comes from the fact that most UXOs contain ferromagnetic components and that magnetometers are insensitive to geological sites and provide one of the most effective technologies in terms of area coverage per day [55]. 2.6.1.1  Land-Buried Targets

Among the different techniques that make use of magnetic field distortion due to the presence of ferrometallic objects, the portable magnetic gradiometer system is of particular interest [56]. It combines a man-portable magnetic tensor gradiometer and a unique method for determining the target position and magnetic moment. According to the authors, the magnetic detector, combined with a noise compensation algorithm and the triangulation and ranging method, should provide motion-noise-resistant magnetic sensing technology for real-time detection, localization, and classification of unexploded ordnance and buried mines.

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Another interesting integration system has been presented by Atya et al. [57]. It combines a ground-penetrating radar (GPR), a fluxgate vertical magnetic gradiometer, a neutron backscattering, and a hydrogen detector, in addition to a distance meter and telemetric transmitter. Despite some limitations, such as the inability to install the magnetic part of the system on a car (or any metallic moving body), the testing experiments on a variety of buried mines have been proven to be successful. Recently, Jane’s International Defence Review reported that a remotecontrolled system that could detect buried or underwater mines during amphibious beach landings had been developed and successfully tested [58]. It was designed to help explosive ordnance disposal teams to quickly find mines and dangerous metal obstacles within coastal surf zones and very shallow water zones. The system is called the Mine Warfare Rapid Assessment Capability (MIW RAC). It comprises a light quadrotor unmanned aircraft system outfitted with an ultrasensitive magnetometer to detect mines and provide real-time data to a handheld Android device. 2.6.1.2  Underwater Targets

Securing seaways from underwater mines is a critical issue in many places in the world. Sonar and acoustic wave propagation techniques cannot always detect buried mines. In these cases, magnetic systems can provide the required solution. A system similar to the one described in [55] was actually developed 2 years earlier for detecting and localizing ferrous mines in shallow water environments [59]. The gradiometer method combined with a powerful processing algorithm has shown promising results both in simulation and real-world testing conditions. A real-time tracking gradiometer (RTG) has been designed and constructed by Kumar et al. [60]. It was aimed to combine the range of a sonar detection system with the capabilities of a four-fluxgate magnetometer device. Field tests were satisfactory even if the experimental conditions were set to be optimal for generating a clean magnetic dipole signal (high SNR). Another interesting direction research is the work performed by Pandya et al. [61]. It comprises a 16-element array of finely spaced microelectromechanical systems (MEMS) hot-wire flow sensors. In addition to the magnetic hardware or more precisely because of this new technology, a dedicated family of algorithms has been developed to process the signal and provide localization of underwater dipole sources. A few years ago, the French team of Yann Yvinec, already known for developing a method for analyzing the detection signal of land-buried UXO,



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adapted a scheme for an underwater survey with a gradiometer [62]. A total field three-axis gradiometer measures the gradient of the field along the three axes. The authors proved that in an ideal situation the inversion problem operates a two-stage linearization that can be solved without imposing any initial estimation of the equation unknowns [63]. 2.6.2  Maritime Magnetic Surveys 2.6.2.1  Introduction to Maritime Surveys

There are several maritime survey methods, utilizing optical, sonar, or magnetic technologies [4]1. Marine magnetic surveys have been used for mapping marine ferrous objects [4] and for mapping archeological structures [6]. The magnetic method was proved to be effective for locating ferrometallic objects masked by sea floor sediments or buried under the seabed. Many modern magnetic surveys are conducted using gradiometers that consist of two or more sensors. There are situations in littoral waters where the survey area complex bottom topography exposes the tow fish to hazardous rocks or debris. Consequently, in littoral water, the survey may be performed from a surface vessel rather than by using a gradiometer fish [64]. Most underwater excavations are labor-intensive and therefore the need to accurately locate artifacts is paramount for the success of the expedition. In this section, we describe a high-resolution marine magnetic survey of shallow water littoral area performed using a towed floating sensor platform [12]. The survey system was optimized for shallow water and high localization accuracy. A catamaran was chosen to serve as the survey vessel for the magnetic survey because of its width, stability, and ability to explore shallow waters and access the shoreline. It is built of nonferrous composite materials and therefore does not interfere with magnetic measurements. The span of the catamaran is roughly 5m wide with the sensors creating a horizontal measurement (see Figure 2.18). The catamaran was towed some 40m behind the tow boat. A ground secondary base station was used to reference the measurements of the rover magnetometer. 2.6.2.2  Survey Goal

The purpose of the survey was to search for the wreck of a T-6 Harvard airplane that crashed in 1960. The T-6 engine, model Pratt and Whitney R-1340 Wasp, contains parts consisting of ferromagnetic materials (see Figure 2.19). Fortunately, a similar engine was available at the Israeli Air Force Museum in Hatzerim.

1. Section reproduced from [12] with permission.

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Figure 2.18  Towed catamaran with sensor array.

2.6.2.3  Survey Magnetometer

An optically pumped potassium magnetometer GSMP-30GS (GEM Systems, Canada) was used. It was selected because of its low noise (1 pT/√Hz at 1 Hz) and low heading error of less than 0.05 nT for 360° full rotation about sensor axis. Because the Earth’s magnetic field inclination is close to 45° in the survey area, orientating the sensors vertically provided their optimal orientation relative to the ambient magnetic field for all bearings of the catamaran (Figure 2.18). A stray magnetic field emitted by electric currents in the electronic circuits had a noticeable influence. This stray field caused a heading error of the measurement system and therefore needed to be compensated for. The compensation was implemented by placing a three-component coil inside the magnetometer electronics box as explained earlier in this section. The three coils were connected in series with a power supply input. The number of turns in each coil was trimmed to compensate for all stray magnetic field components. The position of the survey unit was measured using a submeter precision Differential Global Positioning System (DGPS), model Navcom RT3020M. The catamaran attitude was measured by a digital Honeywell HMR3000 compass module equipped with two tilt meters. The attitude data is important in the case of rough sea conditions to compensate for rolling and wave noises. 2.6.2.4  Reference Magnetometer

To compensate for temporal changes of the environmental magnetic field during the survey, the acquired data was referenced to a stationary reference



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station. The reference station was placed at a magnetically quiet place in the vicinity of the survey area. The rover and reference magnetometers were synchronized by a GPS clock. 2.6.2.5  Survey Objectives: T-6 Wasp Engine

To estimate the anticipated level of the magnetic anomaly produced by the Wasp engine (Figure 2.19), we measured its magnetic moment. We measured the magnetic field produced by the engine relative to the ambient Earth’s magnetic field. Obtained values of engine magnetic moments made up 8.5, 9.0, and 11.0 Am2 for induced magnetism. Remnant magnetic moment components of the measured sample were 5.5, 2.0, and 4.5 Am2. The value and direction of remnant magnetism may vary significantly for different engines. However, the remnant magnetism value is set during the crash impact of the plane. Therefore, remnant magnetism is most likely oriented parallel to the induced one. The maximal depth is ~25m; as a result, the magnetic anomaly intensity on the sea surface produced by the buried engine is estimated to be 0.2 nT. In practice, this figure could be smaller because the aircraft parts may be scattered, forming a magnetic cluster rather than a pure dipole.

Figure 2.19  Harvard T-6 with Wasp engine in the Israeli Air Force Museum.

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2.6.2.6  Survey Area

The survey area is trapeze-shaped, roughly 1.8 km wide and 2 km long (see Figure 2.20). It is about 1 km Northwest of the Atlit Crusader fortress off the Israeli Mediterranean coast at a water depth from 7m to 25m. Rock magnetic properties may have spatial changes depending on concentration of ferrousrich layers. The survey area is located where thousands of years of seafaring have left abundant ancient relics on the sea floor. The survey tracks in the survey area are shown in Figure 2.20. 2.6.2.7  Survey Data Processing

Data postprocessing included diurnal and leg corrections, tie-line, and micro-leveling to remove uncompensated diurnal errors. Data from the survey magnetometer, tilt and compass, reference magnetometer, and reference GPS were processed to create the measurement grid. The corrected magnetic data was gridded with a cell spacing of 0.5m using a Kriging algorithm to generate the total-field magnetic map. The data was processed using the algorithm described in Figure 2.21. The magnetic measurements were plotted on a 3-D map (see Figure 2.3) and a contour map (see Figure 2.22).

Figure 2.20  Survey area with survey lines.

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Figure 2.21  Data-processing algorithm.

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Figure 2.22  Contour map of the survey. Magnetic anomalies generated by ferromagnetic object with magnetic moment near 10 Am2 are marked in geometrical markers.

2.6.2.8  Maritime Survey Results

Analysis of the survey map in Figure 2.22 revealed 59 anomalies in an area of ~3.5 km2. The anomalies were graded according to their similarities to the Wasp engine magnetic moment. The objects’ map was verified by probing some of the objects in selected locations. The ferromagnetic objects were often buried in the seabed and therefore required an underwater dig. The dig was conducted by the Marine Archeology Unit of the Israel Antiquities Authority. After reaching the mapped location with ~0.5-m accuracy, a sinker was dropped to mark the measured position of the anomaly. Scuba divers dove to the measured points to reveal the source of the anomaly.



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Objects located on rocky seabed were visible among the rocks, although they were covered with encrustation. Objects buried in sandy areas required a metal detector for final verification of the target before beginning the excavation. All measured and excavated points contained ferromagnetic objects, although most of them were clutter. Some of the revealed anomaly sources are shown in Figures 2.23 and 2.24. The distance between two anomalies was measured by scuba divers and compared to the magnetic anomaly map. The distance between the objects in Figure 2.23 was found to be 26m, which was in excellent concordance with

Figure 2.23  Two ferrous objects recovered in Figure 2.25. Ferrous object 1 artifact is a fifth-century Byzantine iron anchor, and ferrous object 2 is a thick wall steel tube. The actual distance between the two objects matches the measured distance on the contour map.

Figure 2.24  Magnetic object found at a depth of 17m buried under 1m of sand. The artifact is a dud of aircraft flare.

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27m extracted from the magnetic map (Figure 2.25). In addition, parts similar to T-6 airplane wreckage were found in two of the surveyed target points. 2.6.2.9  Maritime Survey Conclusion

A maritime high resolution magnetic survey system is presented. The system has high capability for detecting and accurate locating of ferromagnetic objects in shallow littoral waters. Excellent performance was confirmed by probing and excavations performed by scuba divers. For each investigated target location, a corresponding ferrometallic item was dug, some of which turned out to be parts of the goal, the crashed airplane. Analyzing the data obtained, one may deduce that in maritime surveys the measurement system should be optimized for low-system internal noise. This is because after optimization of the measurement system (magnetometer and vessel), we were able to measure the seabed clutter and distinguish between localized ferromagnetic objects and large-scale geomagnetic features. The seabed clutter did not affect the system measurement noise because it was relatively far away from the sensors. In the case of this survey, the minimal depth was ~8m, which is far enough to reduce small-size magnetic debris clutter.

Figure 2.25  A zoom on two anomalies in the survey area contour map. The two anomalies were recovered and measured (see Figure 2.23).�



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2.6.3  Low-Power Survey Magnetometers

In ground-level magnetic surveys, the ground clutter is very dominant and the same assumptions cannot be used. Small ferromagnetic objects and localized ground structures create significant clutter. An optimization should be made between getting nearest the measured magnetic object to get larger signal and to reduce ground clutter by getting further from the ground. In that sense, a maritime littoral survey is very similar to a low-altitude, air-borne magnetic survey where there is some stand-off from the survey vessel and the ground clutter. In cases where the survey is performed on the ground level or through bore-holes where the sensor is very close to the ground clutter, a different approach should be adopted. In this case, the sensor resolution is not the dominant source of noise and rather spatial resolution becomes the key feature. This means performing the survey with multiple sensors with lower resolution. This survey sensor array requires a low-power consumption, a low price, and the ability to assemble and operate the sensor’s array in the field. In the next section, we describe a method tailored for this survey types. Magnetometers are commonly used in ground surveys to map anomalies in the natural magnetic field. Magnetic anomalies are caused by contrast of magnetic permeability in the media manifested by objects containing ferromagnetic mass and henceforth used to detect heavy metals [65], archeological [66] and prehistoric tools [67], underground pipes [68], and even wrecks [12]. The location of the anomaly caused by ferromagnetic objects such as UXOs can be estimated [69] by employing magnetic anomaly detection and localization schemes [12]. Magnetic surveys are conventionally performed by scanning a domain with a portable scalar magnetic sensor. The scalar sensor measures only the amplitude of the magnetic field and is not sensitive to the field bearing. This is crucial as the natural magnetic field orientation is highly directional and, as a result, rotation of an uncalibrated vector sensor in the natural Earth magnetic field produces distorted results. Unfortunately, commerical off the shelf (COTS) scalar magnetometers are expensive, power-consuming, and bulky. Furthermore, most scalar sensors also have dead zone orientation that renders them impractical to some applications. In this section, a simple, conventional, low-cost method for performing a magnetic survey is presented. The setup of the survey system is quick and can be performed right before performing the magnetic survey, using the same hardware as in surveillance sensors. A practical approach to overcome the shortcomings of scalar magnetometers is to use vector magnetometers. A fluxgate magnetometer [70] is a low-cost, low-power, and high-resolution sensor [71] and is therefore

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a straightforward candidate for this application. Unfortunately, due to mechanical inaccuracies, the sensors windings are not perfectly perpendicular. In addition, fluxgate magnetic sensors have differences in sensitivity and a constant magnetic offset, formed by ferromagnetic materials and electrical currents from within the sensor unit itself [23, 28]. Typical values of nonorthogonality is 4t = 1 ms. Additional extending of Tr does not further reduce the fluxgate output noise. The second method for suppressing the excess noise is based on adding idle intervals Ti between the DC bias reversals (see Figure 3.31), allowing the domains to relax [17]. Subsequently, the corresponding idle samples are eliminated in the fluxgate output [40]. The idle intervals were increased gradually from Ti = 0 to Ti = 20t. An example of adding an idle interval for the case of Ti = 10t can be seen in Figure 3.27(b), where the added intervals are marked in gray. The fluxgate output voltage after removing the idle intervals and concatenation can be seen in Figure 3.31(c). The sensitivity in this case was measured at the effective excitation frequency fex2:

f ex 2 = f ex ±

T + Ti fa Tr

(3.19)

Figure 3.31  Excitation with idle intervals between the DC bias reversals: (a) the excitation waveform and (b) the fluxgate output. The signal parts shaded by gray (10t segments in this example) were eliminated from the fluxgate output and the remaining signal parts were concatenated (c).

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Figure 3.30 shows that the second method reduces the excess noise almost down to its level in the constant DC bias mode if the suppression time Ti is greater than 9t = 2.25 ms. Hence, one can conclude that it takes 2.25 ms for the magnetic noise to relax. The approximately twofold difference in the noise suppressing times implies that the AC bias reduces the magnetic domain relaxation time. The slight difference in the noise levels between the methods may be attributed to the jump in the fluxgate output following the idle interval that is not trimmed out (see Figure 3.31(c)) in the second method. Whereas in the first method, the DC bias reversal occurrence is removed (see Figure 3.27(c)). Additionally, as the wire core is excited in a noncontinuous mode, less power is consumed by the core [40]. 3.5.4  Alternating DC Bias Conclusion

It has been shown that the fluxgate excess noise related to the magnetic domain dynamics can be avoided either by excluding the parts of the fluxgate output appearing right after the DC bias reversals or by delaying the AC excitation parts with the opposite DC bias. In the processing of the fluxgate output, we concatenate the output parts that do not contain the excess noise. As a result, the excess noise is practically eliminated, and the fluxgate total noise approaches that in the constant DC bias mode. Suppressing the excess noise in the alternating DC bias mode is important because in this mode the fluxgate offset can be eliminated [30, 42, 43, 46]. Thus, it becomes possible to construct an orthogonal fluxgate with both a very small offset and noise.

3.6  DC Jumps in Low-Power Fluxgate Magnetometers Decreasing the power consumption in fluxgates is mainly achieved by decreasing the core excitation magnetic field because it is the principal energy consumer in fluxgates. It is well known that the noise level is generally decreasing with increasing of the excitation field amplitude [7]. The power consumption of conventional parallel fluxgates magnetometers was successfully reduced by more than an order of magnitude by decreasing the excitation current. Unfortunately, decreasing the power consumption in fluxgates comes at a cost. Low-power fluxgates tend to have jumps at their output. The DC jumps are characterized by a random and sudden change in fluxgate output voltage, which its occurrence subside in time (see Figure 3.32). We present a model for the DC jump phenomena in the cores of fluxgates in two types of fluxgates, which differ by the way their core is being excited. We then expand the model to explain why and how the DC jumps subside in time.



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Figure 3.32  An example of magnetic noise in a magnetometer with a DC jump. The DC jump is the negative step visible at ~125 seconds.

In this section, we suggest that because of the low-saturation excitation some domains get temporarily stuck in one magnetization direction by metallurgical imperfections in the lattice [17], while the rest of the core domains continues in periodical rotations. The stuck domains disturb the effective permeability of the core that is translated to a jump in fluxgate sensitivity. This phenomenon is stochastic and the unstable domain may snap back and forth to alignment at random times and causes a sudden offset in sensitivity [19, 21], which, in turn, creates a DC voltage jump in the output of the fluxgate [16]. This DC jump (Figure 3.32) is a source of internal magnetic noise that takes effect mostly in low-power fluxgates. 3.6.1  Model of DC Jumps in Parallel Fluxgates

Parallel fluxgates are operated by employing a bipolar excitation magnetic field to the core (see Figure 3.33). The magnetic domains in the core are periodically rotating by the reversals of the excitation magnetic field direction. In conventional fluxgates, the excitation field required to drive the core to deep saturation is on the order of 20 kA/m [51]. However, in low-power fluxgates, the excitation field is an order of magnitude lower and, as a result the magnetization of the core, becomes less uniform. The magnetization nonuniformity is enhanced by the demagnetizing effect caused by the rod structure of the core [52] and by superstructures in the core, such as material

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Figure 3.33  An illustration of a sine bipolar (bottom curve) and unipolar (top curve) excitation field.

casting imperfections, random metallurgical grains, dislocations, and surface roughness [53] (see Figure 3.34). An additional source of magnetization inhomogeneity is the internal stress imprinted during the wire casting. The stress is inhomogeneous across the core and, as a result, the domains’ switching field is inhomogeneous as well [54].

Figure 3.34  Magnetic domains schematics of a fluxgates core during (a) positive excitation and (b) negative excitation. The core is broken to metallurgical superstructure each containing magnetic domains. The case presented of a shallow bipolar saturation, where the magnetization is not homogeneous across the core.



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We suggest that the core magnetization inhomogeneity caused by the mechanisms described is the source of the DC jumps (see Figure 3.34). This is because the magnetization reversals mechanism in bipolar excitation fluxgates consist of domain walls motion, coherent domain rotations, and nucleation of domains [55]. A superstructure or inhomogeneity in the core is a source of inhibition to the smooth rotations of the domains [56]. The domain reorganization during excitation may be localized in the core by the source of rotation inhibition, where a part of the domain wall or an entire domain may jump to an adjacent stable position [56]. This domain snap results in a DC jump in the fluxgate sensitivity. 3.6.2  Model of DC Jumps in Orthogonal Fluxgates

Orthogonal fluxgates are operated by employing a unipolar excitation magnetic field to the core (see an illustration in Figure 3.33) by driving current through it. The excitation direction is circular across the wire core. The coherent rotation of magnetization [8] is used in a fundamental-mode orthogonal fluxgate [21, 22]. In this method, the excitation field never reaches zero as the entire excitation cycle is unipolar with a DC bias. As a result, the domain walls never flip by the excitation field. However, the magnetization across the core in orthogonal fluxgates is also inherently nonuniform [4]. This is because, the fluxgate core is excited by the field generated by the current flowing through it [14]. In this excitation scheme, the excitation magnetic field increases from the inner core to the periphery because the integral current loops become larger radially. As a result, the periphery of the core is driven to saturation by the current while the inner shells of the wire core magnetization level are weaker. In the neutral center of the wire, the domains are free to nucleate and shift. The natural magnetization direction of the inner core is axial because of its high effective permeability due to its length-to-diameter ratio [52]. As a result, there is a gradient of magnetization level across the core in both amplitude and direction. An unstable spring onion-like boundary layer of magnetic domains separates between the two domain regimes [4] (see Figure 3.12 and 3.15). When this unstable layer transforms, it causes magnetic noise in the core. 3.6.3  DC Jump Dynamic Model 3.6.3.1  Dynamics of a Single DC Jump

In this section, we analyze the DC jumps evolution during the fluxgate operation. As described earlier, DC jumps are caused by the movement of domains or domain walls that got stuck in the core lattice. In Figure 3.35, we present

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Figure 3.35  Average number of DC jumps time during 48 hours of recording.

the measured dynamics of the DC jumps. Nevertheless, the reason why this phenomenon subsides in time remains unexplained. One possible explanation [56] is that the permanent remagnetization of the domains leads to domain walls smoothing after numerous conjoint frictions. During excitation, there may be conditions for self-homogenization due to the rise of temperature, which supplies the necessary energy for liquidation of the material structural impurities. However, according to this explanation the lattice becomes plastically homogenous after some self-healing time. Nevertheless, we have observed that while DC jumps subside in time during continuous operation, after powering down and powering up of the fluxgate, the DC jumps reinitiate. We can therefore conclude that the selfhomogenization of the core is not permanent. Another possible explanation is that the self-homogenization of fluxgates cores is a dynamic process where zones in the core with different magnetization levels interact. A model for the core self-homogenization was presented in [4] for an orthogonal fluxgate operating in the duty cycle. In a duty-cycle operation of an orthogonal fluxgate, idle intervals are introduced between excitation cycles. The idle intervals leave enough time for self-homogenization of the magnetization gradient in the core. Similarly, in parallel fluxgates core, the stuck zones near imperfections are not in equilibrium because they share a boundary with other domains that are pumping out its spin energy in every excitation cycle [19]. When, in time, enough energy is pumped by the spin gradient to equalize the coercive energy required by the domain, it will relax its coercive energy and transform. The



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relaxation occurs in a snap and is a stochastic process, which involves rapid motion of the domain walls. The precession of the magnetization of the ferromagnetic core transfer spins from adjacent, more magnetized core material. This pumping of magnetic spins slows down the precession corresponding to an enhanced Gilbert damping constant in the LLG equation. The magnetized zones or shells magnetize the nonmagnetized zones. This slow rate pumping mechanism gradually increases the magnetization level of magnetic domains in the boundary layer between stuck domains and the lattice. We propose a similarity solution between DC jump phenomenon and mechanical metal fatigue. In the case of metal fatigue, a micro-crack propagates incrementally every strain cycle in proportion to the strain amplitude and the amount of stress concentration in the crack tip. This can be represented as an expansion of the classical nucleation and propagation of the domain wall during magnetization reversal [23]. The crack propagation under cycling load is described in Paris law:

da m = c ( ∆K ) dN

(3.20)

where a is the crack length, growing with cycles N, in terms of the cyclical component ΔK of the stress intensity factor K. m is typically in the range of 3 to 5. Similarly to metal fatigue, the crack length is equivalent to the amount of rotated magnetic spins in a single domain. The incremental propagation of a crack is similar to the amount spin being pumped in or out of a magnetic domain with each excitation reversal. In metal fatigue, the crack propagation accumulates during multiple cycles until a critical crack length is reached and the propagation rate abruptly changes to the speed of sound in the metal. Similarly, in magnetic DC jumps, magnetic spins are being pumped incrementally in every excitation cycle until enough magnetization energy was transferred to the magnetic domain and at this time, the domain superstructure snaps and propagates. 3.6.3.2  Dynamics of Multiple DC Jumps in a Sensor

The relaxation sequence that we describe is based on a gradual increase of the reversal magnetization field in each excitation cycle by pumping of spin energy between adjacent differently magnetized areas in the core. The observed decreasing rate of the DC jumps is due to the distribution of inhibition seeds in relation to the location of the magnetization boundary layer in the core.

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We validate the model in both orthogonal and parallel fluxgates that differ in their magnetization gradient across the core. One of the cores’ inhomogeneity source is the wire drawing production process [54]. The drawing process imprint the core with intrinsic stress field, which is typically up to 150 MPa in the inner core and increases exponentially to ~900 MPa in the periphery of the wire [54]. The stress affects the magnetic field required for domain magnetization reversal [56]. For example, the reversal magnetization field at stress of 900 MPa is 2 to 5 times larger than that required at 150 MPa. Another source of inhomogeneity in the structure of the core is imperfections existing in the core from its rapid casting process. We assume that the outer shell of the wire has the most imperfections per volume unit because it is the surface in contact with the nozzle during the casting process. 3.6.4  DC Jumps’ Dynamic Model in Amorphous Wire Core Orthogonal Fluxgates

As described earlier, in amorphous wire orthogonal fluxgates the magnetic core is excited by driving current through it, creating shells with different magnetization levels. The instable boundary layer between saturated and magnetized shells is where the DC jumps originate. Domains with seeds of rotation impendence are restricted to this boundary shell. Hence, once this layer has turned, no further DC jumps are expected. As a result, we can predict that the amount of DC jumps in wire core orthogonal fluxgates will be small and their occurrence will be random and not progressing. 3.6.5  DC Jumps Dynamic Model in Parallel Fluxgates

In parallel fluxgates, there is no inherent gradient in the magnetization of the core because the core is excited by a solenoid and not by driving current through the core. The dynamics of the DC jumps is therefore determined predominantly by the distribution of the inhibition sources in the core material and the distribution of stress imprinted into the core during casting. We stated above that the wire core drawing process imprints the core with intrinsic stress field, typically up to 150 MPa in the inner core with an exponential increase up to ~900 MPa in the periphery of the wire [54]. We would like now to describe how these facts can explain the long-term behavior of the DC jumps that we saw in Figure 3.35. It was shown experimentally in [59] how stress applied to the wire affects the field that is required to switch the domain magnetization. Such dependence is also shown in [54], together with the change of the stress value



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as a function of the distance from the core center. The combination of these two effects can be performed according to the following steps. Step 1

We first express the internal axial component of the stress as a positive function of the wire radius. It was shown [59] that this component may play a non-negligible role in the switching process of amorphous wire cores. We obtain the curve shown in Figure 3.36. Note in Figure 3.36 that, at half the radius, the stress derivative changes sign. This corresponds to the end of the inner core that consists of a mono-domain cylinder with easy magnetization direction parallel to the wire axis [59]. At this point, the stress value is approximately 150 MPa. Step 2

From a simple analysis of the normalized switching field (NSF) as a function of the internal stress [54], one can remark two important facts. First, the NSF is linear with the stress, and second, the slope changes sign but not its absolute value at a stress of 150 MPa. This means that to express the NSF as a function of the wire radius, two domains must be considered: stress lower or higher than 150 MPa. After this stress threshold, the NSF varies in propor-

Figure 3.36  Internal radial stress as a function of normalized radius of the wire. (After: [60, Figure 7].)

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tion to the stress. At lower stress amplitudes, one needs to take the symmetric value of the stress relative to the line axis S = 150 MPa. This simple exercise produces the graph shown in Figure 3.37. Step 3

We would like now to project the switching field into time units to determine the dynamics of the DC jump’s subsiding rate. This can easily be done if one remembers that the NSF is proportional to the number of rotation cycles that are required to obtain complete magnetization of a single region. This is also directly related to the switching excitation frequency fexc. Therefore, a simple formula can take us from the NSF to the time domain:

t sw =

C ⋅ NSF ⋅ SF (0) f exc

(3.21)

where SF (0) is the switching field value at which the stress is close to 0, and C is a constant that depends on physical and mechanical properties of the specific wire. Step 4

By using (3.21) and Figure 3.37 and inverting the X-Y axes, one can see the correlation between the radial distance from the wire center to the time re-

Figure 3.37  Normalized switching field as a function of wire radius.



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quired for domain switching. For an example, by using the same parameters as obtained in the Figure 3.35 timescale, we use an excitation frequency of 5 kHz, a zero stress SF of 7.5 Am−1 [59], and a dimensionless constant C equals to 2.1∙107. This is shown in Figure 3.38. It is not surprising that the best fit for the graph in Figure 3.38 is also a biexponential function as shown in Figure 3.39, exactly like the fit obtained in our experimental result shown in Figure 3.35. We have presented a straightforward explanation for the biexponential behavior that we measured empirically. There is an implicit relation that takes us from the Y-axis of Figure 3.35 to the Y-axis of Figure 3.39. This relation reflects the fact that the large number of DC jumps at the beginning of the sensor working time has its origin in the inner region of the wire. This region is probably constituted by a very few number of domains with an easy magnetization. This region can be quite large and, since it is easily magnetized, it will produce a large number of DC jumps right on the onset of the fluxgate operation. As time progresses, only regions far from the center will provide more DC jumps. The reason why this occurs, despite the large internal stress present in this part of the wire, is because as we increase the radius of these external shells, the number of impurities increases as well. These impurities provide excellent starting points for the nucleation process, and it is known

Figure 3.38  Switching shells measured from the surface wire (normalized, that is, 1 = surface, 0 = center) as a function of time.

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Figure 3.39  Biexponential fit used to match the result obtained in Figure 3.38.

that such a nucleation process is necessary to switch the magnetization of a domain into its opposite direction, that is, to produce a DC jump.

References [1] Paperno, E., E. Weiss, and A. Plotkin, “A Tube-Core Orthogonal Fluxgate Operated in Fundamental Mode,” IEEE Transactions on Magnetics, Vol. 44, No. 11, November 2008, pp. 4018–4021. [2] Sasada, I. U. T., “A High Performance Orthogonal Fluxgate Magnetometer Based on the Fundamental Mode of Operation,” IEEE 2005 International Magnetics Conference, 2005, p. 485. [3] Weiss, E., A. Grosz, and E. Paperno, “Duty Cycle Operation of an Orthogonal Fluxgate,” IEEE Sens. J., 2014, pp. 1–5. [4] Weiss, E., et al., “Excess Magnetic Noise in Orthogonal Fluxgates Employing Discontinuous Excitation,” IEEE Sens. J., Vol. 14, No. 8, August 2014, pp. 2743–2748. [5] Butta, M., “Orthogonal Fluxgate Magnetometers,” in High Sensitivity Magnetometers, A. Grosz, M. J. Haji-Sheikh, and S. C. Mukhopadhyay, (eds.), New York: Springer, 2017, pp. 63–102. [6] Goleman, K., and I. Sasada, “High Sensitive Orthogonal Fluxgate Magnetometer Using a Metglas Ribbon,” IEEE Transactions on Magnetics, Vol. 42, No. 10, October 2006, pp. 3276–3278.



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[7] Sasada, I., “Symmetric Response Obtained with an Orthogonal Fluxgate Operating in Fundamental Mode,” IEEE Transactions on Magnetics, Vol. 38, No. 5, September 2002, pp. 3377–3379. [8] Paperno, E., “Suppression of Magnetic Noise in the Fundamental-Mode Orthogonal Fluxgate,” Sensors Actuators A Phys., Vol. 116, No. 3, October 2004, pp. 405–409. [9] Plotkin, A., E. Paperno, and A. Samohin, “Compensation of the Thermal Drift in the Sensitivity of Fundamental-Mode Orthogonal Fluxgates,” J. Appl. Phys., Vol. 99, No. 8, April 19, 2006, pp. 8–10. [10] Sasada, I., “Orthogonal Fluxgate Mechanism Operated with DC Biased Excitation,” J. Appl. Phys., Vol. 91, No. 10, 2002, p. 7789. [11] Primdahl, F., “The Fluxgate Magnetometer,” Elements, Vol. 12, No. 4, 1979. [12] Ripka, P., and M. Janosek, “Advances in Magnetic Field Sensors,” IEEE Sens. J., Vol. 10, No. 6, June 2010, pp. 1108–1116. [13] Chiriac, H., “Study of the Noise in Multicore Orthogonal Fluxgate Sensors Based on Ni-Fe/Cu Composite Microwire Arrays,” IEEE Transactions on Magnetics, Vol. 45, No. 10, October 2009, pp. 4451–4454. [14] Ripka, P., Magnetic Sensors and Magnetometers, 1st ed. Norwood, MA: Artech House, 2001. [15] Primdahl, F., “The Fluxgate Magnetometer,” J. Phys. E., Vol. 241, 1979. [16] Bazinet, R., et al., “A Low Noise Fundamental Mode Orthogonal Fluxgate Magnetometer,” IEEE Transactions on Magnetics, Vol. 50, No. 5, 2014, pp. 8–11. [17] Tipek, A., et al., “Excitation and Temperature Stability of PCB Fluxgate Sensor,” IEEE Sens. J., Vol. 5, No. 6, December 2005, pp. 1264–1269. [18] Weiss, E., E. Paperno, and A. Plotkin, “Orthogonal Fluxgate Employing Discontinuous Excitation,” J. Appl. Phys., Vol. 107, No. 9, 2010, p. 09E717. [19] Pommier, J., et al., “Magnetization Reversal in Ultrathin Ferromagnetic Films with Perpendicular Anistropy: Domain Observations,” Phys. Rev. Lett., Vol. 65, No. 16, October 1990, pp. 2054–2057. [20] Bordin, G., et al., “Temperature Dependence of Magnetic Properties of a Co-Based Alloy in Amorphous and Nanostructured Phase,” J. Magn. Magn. Mater., Vol. 195, No. 3, June 1999, pp. 583–587. [21] Kamruzzaman, M., I. Z. Rahman, and M. A. Rahman, “A Review on MagnetoImpedance Effect in Amorphous Magnetic Materials,” J. Mater. Process. Technol., Vol. 119, No. 1–3, December 2001, pp. 312–317. [22] Infante, G., et al., “Locally Induced Domain Wall Damping in a Thin Magnetic Wire,” Appl. Phys. Lett., Vol. 95, No. 1, 2009, p. 012503. [23] Vázquez, M., “Advanced Magnetic Microwires,” in Handbook of Magnetism and Advanced Magnetic Materials, New York: John Wiley & Sons, 2007, pp. 1–34.

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[24] Vázquez, M., and A. -L. Adenot-Engelvin, “Glass-Coated Amorphous Ferromagnetic Microwires at Microwave Frequencies,” J. Magn. Magn. Mater., Vol. 321, No. 14, July 2009, pp. 2066–2073. [25] Vulfovich, P. J., and L. V. Panina, “Magneto-Impedance in Co-Based Amorphous Wires and Circular Domain Dynamics,” Sensors Actuators A Phys., Vol. 81, No. 1–3, April 2000, pp. 111–116. [26] Weiss, E., and E. Paperno, “Noise Investigation of the Orthogonal Fluxgate Employing Alternating Direct Current Bias,” J. Appl. Phys., Vol. 109, No. 7, 2011, p. 07E529. [27] Donoho, D. L., and J. Tanner, “Precise Undersampling Theorems,” Proc. IEEE, Vol. 98, No. 6, June 2010, pp. 913–924. [28] Vaughan, R., N. Scott, and D. White, “The Theory of Bandpass Sampling,” IEEE Transactions on Signal Processing, Vol. 39, No. 9, 1991. [29] Weiss, E., et al., “Orthogonal Fluxgate Employing Digital Selective Bandpass Sampling,” IEEE Transactions on Magnetics, Vol. 48, No. 11, 2012, pp. 4089–4091. [30] Ripka, P., et al., “Sensitivity and Noise of Wire-Core Transverse Fluxgate,” IEEE Transactions on Magnetics, Vol. 46, No. 2, February 2010, pp. 654–657. [31] Ripka, P., and M. Jano, “Advances in Magnetic Field Sensors,” Vol. 10, No. 6, 2010, pp. 1108–1116. [32] Zorlu, O., et al., “A Novel Planar Magnetic Sensor Based on Orthogonal Fluxgate Principle,” Res. Microelectron. Electron., 2005, pp. 237–240. [33] Baglio, S., et al., “Exploiting Nonlinear Dynamics in Novel Measurement Strategies and Devices: From Theory to Experiments and Applications,” IEEE Transactions on Instruments and Measurement, Vol. 60, No. 3, 2011, pp. 667–695. [34] Ando, B., et al., “‘Residence Times Difference’ Fluxgate,” Measurement, Vol. 38, No. 2, September 2005, pp. 89–112. [35] Ripka, P., and W. Hurley, “Excitation Efficiency of Fluxgate Sensors,” Sensors Actuators A Phys., Vol. 129, No. 1–2, 2006, pp. 75–79. [36] Ripka, P., “Excitation of Fluxgate Sensors,” Sensors and Actuators, 2000, pp. 767–770. [37] Ripka, P., et al., “Micro-Fluxgate Sensor with Closed Core,” Sensors and Actuators, Vol. 91, 2001, pp. 65–69. [38] Sasada, I., and S. Harada, “Fundamental Mode Orthogonal Fluxgate Gradiometer,” IEEE Transactions on Magnetics, Vol. 50, No. 11, 2014. [39] Sui, Y., et al., “Compact Fluxgate Magnetic Full-Tensor Gradiometer with Spherical Feedback Coil,” Rev. Sci. Instrum., Vol. 85, No. 1, January 2014, p. 014701. [40] Ripka, P., and S. W. Billingsley, “Crossfield Effect at Fluxgate,” Sensors Actuators, A Phys., Vol. 81, No. 1, 2000, pp. 176–179. [41] Primdahl, F., et al., “The Sensitivity Parameters of the Short-Circuited Fluxgate,” Meas. Sci. Technol., Vol. 2, No. 11, November 1991, pp. 1039–1045.



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[42] Kim, Y. K., et al., “Temperature Dependence of Magnetoimpedance Effect in Amorphous Co66Fe 4NiB14Si15 Ribbon,” J. Appl. Phys., Vol. 83, No. 11, 1998, p. 6575. [43] Buttino, G., A. Cecchetti, and M. Poppi, “Temperature Dependence of Structural and Magnetic Relaxation in Amorphous and Nanocrystalline Co-Based Alloys,” J. Magn. Magn. Mater., Vol. 241, No. 2–3, March 2002, pp. 183–189. [44] Buttino, G., A. Cecchetti, and M. Poppi, “Magnetic Softening and Nanocrystallization in Amorphous Co-Rich Alloys,” J. Magn. Magn. Mater., Vol. 172, No. 1–2, August 1997, pp. 147–152. [45] Ripka, P., X. P. Li, and J. Fan, “Orthogonal Fluxgate Effect in Electroplated Wires,” IEEE Sensors, 2005, pp. 69–72. [46] Vázquez, M., et al., “On the State-of-the-Art in Magnetic Microwires and Expected Trends for Scientific and Technological Studies,” Phys. Status Solidi, Vol. 208, No. 3, March 2011, pp. 493–501. [47] Sinnecker, J. P., et al., “AC Magnetic Transport on Heterogeneous Ferromagnetic Wires and Tubes,” J. Magn. Magn. Mater., Vol. 249, No. 1–2, August 2002, pp. 16–21. [48] Butta, M., et al., “Bi-Metallic Magnetic Wire with Insulating Layer as Core for Orthogonal Fluxgate,” IEEE Transactions on Magnetics, Vol. 45, No. 10, 2009, pp. 4443–4446. [49] Sasada, I., and T. Usui, “Orthogonal Fluxgate Magnetometer Utilizing Bias Switching for Stable Operation,” Proc. IEEE Sensors 2003 (IEEE Cat. No.03CH37498), 2003, pp. 468–471. [50] Sasada, I. K. H., “Simple Design for Orthogonal Fluxgate Magnetometer in Fundamental Mode,” J. Magn. Soc. Japan, Vol. 33, 2009, pp. 43–45. [51] Flohrer, S., et al., “Dynamic Magnetization Process of Nanocrystalline Tape Wound Cores with Transverse Field-Induced Anisotropy,” Acta Mater., Vol. 54, No. 18, 2006, pp. 4693–4698. [52] Bozorth, R. M., “Demagnetizing Factors of Rods,” J. Appl. Phys., Vol. 13, No. 5, 1942, p. 320. [53] Liu, S., “Study on the Low Power Consumption of Racetrack Fluxgate,” Sensors Actuators A Phys., Vol. 130–131, August 14, 2006, pp. 124–128. [54] Squire, P. T., et al., “Amorphous Wires and Their Applications,” J. Magn., Vol. 8853, No. 94, 1994. [55] Liu, Y., D. J. Sellmyer, and D. Shindo, Handbook of Advanced Magnetic Materials: Vol 1. Nanostructural Effects. Vol 2. Characterization and Simulation. Vol 3. Fabrication and Processing. Vol 4. Properties and Applications, New York: Springer Science & Business Media, 2008. [56] Korepanov, V., and A. Marusenkov, “Flux-Gate Magnetometers Design Peculiarities,” Surv. Geophys., Vol. 33, No. 5, 2012, pp. 1059–1079. [57] Chiriac, H., et al., “Magnetic Behavior of Rapidly Quenched Submicron Amorphous Wires,” J. Appl. Phys., Vol. 107, No. 9, 2010, p. 09A301.

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4 Low-Power Sampling 4.1  Overview 4.1.1  Sampling Resolution

The sampling of high-sensitivity magnetometers typically requires low-noise and high-resolution sampling techniques. This is because the sensor’s internal noise power spectrum density (PSD) is on the order of 10 pT-rms /√Hz at 1 Hz and in most practical applications the measurement range is at least 3 orders of magnitude larger. For example, if a measurement is to be made in the natural Earth magnetic field and encompass its range, the largest measured signal is ±60 µT, and the required resolution will thus be 120 µT/1 pT = 120 million. This can be attained by a 27-bit sampling resolution. Decreasing the measurement range will relax those requirements by the factor of the range decrease. Furthermore, the apparent noise of the magnetometer is the sum of all the noise sources of the measurement system. The internal noise of the analog-to-digital converter (ADC) and the sensors’ noise are uncorrelated. Hence, all noise sources are added via their root mean square. The contribution of noise from the ADC is conventionally selected to be roughly an order of magnitude smaller than that of the sensor at the required measurement band and sampling rate.

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4.1.2  Sampling Options

In fluxgate magnetometers, there are at least 3 options to process the sensor output. The conventional approach is to demodulate the output of the sensor by employing an analog synchronous detector to demodulate the output to the baseband and then to sample the demodulated signal [1]. This classical approach requires generation of the demodulated signal to convolute with the output signal. Another approach is to sample the fluxgate output at a sufficiently high rate to perform the demodulation digitally [2]. This method requires a much faster sampling apparatus as the sampling must be at a rate exceeding twice the pickup frequency of the fluxgate. In the case of orthogonal fluxgate, the pickup frequency is the fundamental [3, 4] and in parallel fluxgate it is conventionally the second harmonic [1]. In addition, a more exhaustive digital postprocessing is required. A third approach is to perform selective undersampling to the fluxgate output [5]. This method requires analog filtering before sampling to reduce clutter aliasing into the measurement band. 4.1.3  Sampling of Low-Power Fluxgates

The sampling and processing of orthogonal fluxgate operated in the fundamental mode with discontinuous excitation are different than that of a conventional fluxgate as a phase detector cannot easily be employed. Consequently, different sampling and processing techniques must be developed to optimize the power consumption reduction by the optimized excitation methods. We propose a digital selective bandpass sampling where the fluxgate output is sampled only once, at a single time instance during a number N of excitation cycles. This provides reconstruction of a measured magnetic field with a finite bandwidth. The proposed approach not only improves the fluxgate equivalent magnetic noise, but also simplifies the fluxgate output processing by eliminating the need for analog synchronous detection. We also combine fluxgate digital concatenation and bandpass sampling techniques to attain low magnetic noise and low-power consumption. This method limits the measurement bandwidth. However, in many applications, the measured magnetic fields have a narrow bandwidth. We show that applying the above concept enables us to operate orthogonal fluxgates in duty cycle without decreasing its resolution.



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4.2  Sampling and Processing of an Orthogonal Fluxgate Output In a conventional orthogonal fluxgate, the output signal is collected throughout the entire excitation cycle [1, 6, 7] and therefore the entire excitation cycle is being averaged1. However, the fluxgate equivalent magnetic noise varies significantly within the time in the excitation cycle [8–11]. Therefore, the conventional method does not provide the lowest possible equivalent magnetic noise of the fluxgate. Selective sampling of the output signal at time instances when the equivalent magnetic noise is minimal could improve the fluxgate resolution [5]. In many applications, the required measured magnetic field has a narrow bandwidth [12, 13] because in low-noise environmental magnetic measurements the changes anomalies in the field are progressing relatively slowly. However, the bandwidth of the fluxgate output is much wider, because the measured signal is modulated by the high-frequency excitation. Furthermore, according to the bandpass sampling theory [14] and Shannon’s theorem, the sampling frequency may be lower than the Nyquist frequency. Sampling at frequencies related to the signal bandwidth rather than to the maximum frequency is known as bandpass sampling or undersampling [14]. Bandpass sampling is performed at a predetermined rate to intentionally alias the modulated carrier, fexcitation, into the measurement baseband [14]. Therefore, the measured signal can be sampled only once during a number of excitation cycles and still fulfills the undersampling law. For example, if a measured field has a bandwidth of 5 Hz and the excitation frequency is 4 kHz, the signal can be sampled at 10 Hz, which corresponds to a single sample for every 400 excitation cycles. In this chapter, we show that applying the above concept to an orthogonal fluxgate enables us to improve its equivalent magnetic noise. 4.2.1  Experimental Setup

The fluxgate experimental setup (see Figure 4.1) comprises a Co-based amorphous wire (AC-20 type made by Unitica) of a 120-µm diameter, 30-mm length, and a sensing coil with 3,200 turns of a 127-µm copper wire [5]. The excitation current is applied directly to the fluxgate core, and the output voltage is generated by the sensing coil. The excitation signal is produced by an NI PXI-5421 function generator. The external magnetic field is produced by a calibrated solenoid. The fluxgate is placed inside a three-shell magnetic shield to reduce the ambient magnetic noise. The fluxgate output is amplified by a low-noise AD8605 operational amplifier and sampled with an 1. Section reproduced from [5] with permission.

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Figure 4.1  Schematic view of an orthogonal fluxgate employing an amorphous wire core.

NI PXI-4461, 24-bit, 204.8-ksps data acquisition module. The excitation is synchronized with the acquisition module clock. The fluxgate is operated at a 4-kHz excitation frequency, 36-mA zeroto-peak current, and 36-mA zero-to-peak DC bias. The fluxgate excitation waveform is presented in Figure 4.2(a), and the output is presented in Figure 4.2(b). The chosen excitation waveform and DC bias minimizes our fluxgate equivalent magnetic noise. The fluxgate output is not a sine wave because the core is not on the verge of deep saturation during the entire excitation cycle. This is because the excitation current drops to zero in each excitation cycle. However, the core remains magnetized in the excitation field polarity due to the residual magnetization of the amorphous wire. All the measurements are postprocessed using the LabView software.

Figure 4.2  (a) Fluxgate excitation voltage and (b) the measured output signal in an excitation cycle averaged over multiple cycles [5].



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4.2.2  Digital Selective Bandpass Sampling Technique

To find the time instances when the fluxgate equivalent magnetic noise is minimal, we calculated the equivalent magnetic noise as a function of the sampling instances within the excitation cycle. To this end, we measured the fluxgate output voltage noise and sensitivity and obtained the equivalent magnetic noise (see Figure 4.3), by dividing the output noise by the sensitivity. To measure the fluxgate noise, we sampled its output signal k = 40 times within each excitation cycle, after which we removed out-of-band noise using a digital bandpass filter. To find the noise magnitude for the ith sample in Figure 4.2, we collected the ith samples during M = 40,000 excitation cycles. Then Fourier transform was performed, and the noise magnitude was calculated as the noise spectral density at 1 Hz. This bandpass sampling procedure demodulated the output signal to baseband. The noise was averaged over 20 sets of measurements. To measure the fluxgate sensitivity, we repeated the same procedure with the fluxgate placed inside a solenoid producing a calibrated external field at 1 Hz. The frequency response of the fluxgate from direct current (DC) to 100 Hz was measured and found to be uniform within ±1.5%. Figure 4.3 shows that the equivalent magnetic noise significantly varies within the excitation cycle from 13 to 850 pT/√Hz at 1 Hz. The equivalent magnetic noise curve has minimum noise at samples 9 and 30.

Figure 4.3  Fluxgate output noise (the dashed curve) and sensitivity (the solid curve) as a function of the sampling instances within the excitation cycle [5].

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4.2.3  Discussion

To compare between the new sampling method and the conventional one, we have calculated the fluxgate equivalent magnetic noise by applying a conventional method. To this purpose, we followed the procedure described earlier. Also, to measure the fluxgate output noise and sensitivity in this case, we have performed Fourier transform to all the k = 40 samples in each excitation cycle for all the M = 40,000 excitation cycles. The equivalent magnetic noise of the fluxgate without digital bandpass sampling is the average of the output noise divided by the average of the sensitivity within each cycle (see Figure 4.3). The output noise and sensitivity were measured at a frequency of 1 Hz above the excitation frequency. The calculated equivalent magnetic noise has a magnitude of 18.5 pT/√Hz, which is by 40% higher than the magnitude obtained with the new method. Figure 4.3 shows that the fluxgate output noise is nearly constant within the entire excitation cycle. However, the fluxgate sensitivity varies from 0.5 to 30 µVrms/nT. As a result, the fluxgate equivalent magnetic noise in Figure 4.4 varies from 13 to 850 pT/√Hz at 1 Hz. The minima of the equivalent magnetic noise nearly correspond to the maxima of the fluxgate sensitivity. Figure 4.5 presents the fluxgate equivalent magnetic noise versus frequency for two selected sampling time instances within the excitation cycle

Figure 4.4  Fluxgate equivalent magnetic noise (the solid curve) as a function of the sampling instances within the excitation cycle (the dashed curve) [5].



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Figure 4.5  Noise spectrum of the orthogonal fluxgate with (a) digital selective bandpass sampling with the lowest equivalent magnetic noise (sample #9 in Figure 4.4), (b) sampling point with the highest equivalent magnetic noise (sample #20 in Figure 4.4), and (c) the conventional sampling method [5].

(9 and 20), compared to the conventional method where the entire cycle is being averaged. The equivalent magnetic noise presented in Figure 4.5 demonstrates the improvement gained by selecting the best time instances within the excitation cycle. The variations in the fluxgate sensitivity can be explained by the variations in the slope of the excitation current (see Figure 4.2(a)), according to the orthogonal fluxgate model [15]. The proposed method can be applied to both demodulate and sample the output of a fluxgate. The excitation cycle and the sampling clocks must be synchronized, and the sampling time instances must be set to one of the two optimal points. The exact sampling point can be experimentally found for each fluxgate configuration. 4.2.4  Conclusion on Digital Selective Bandpass Sampling

We show that it is possible to decrease the equivalent magnetic noise of an orthogonal fluxgate by employing digital selective bandpass sampling of its output signal. This is because the sensitivity varies within the excitation cycle, while the output noise is almost constant. As a result, the equivalent magnetic noise reaches a minimum two times during the excitation cycle. Selective digital bandpass sampling at time instances corresponding to the equivalent magnetic noise minima provides a 40% improvement compared to that obtained by the conventional method. Using digital selective bandpass sampling also simplifies the fluxgate output processing. Further simplification can be attained by exploiting the possibility to operate at lower sampling rates. In

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this case, the digital bandpass filter should be replaced with an analog one. This is required to eliminate unwanted aliasing.

4.3  Duty Cycle Operation of an Orthogonal Fluxgate To attain high resolution and to avoid perming [1, 11, 14, 16, 17], the fluxgate core must be fully magnetically saturated by relatively high excitation current2. As a result, the typical power consumption of an orthogonal fluxgate is very high [16]. The high-power consumption limits its operation in battery-operated applications where the power consumption is important, for example, space research and unattended ground sensors (UGS) [12] or for vehicle detection [18–21]. In Chapter 3, we have shown that the power consumption of an orthogonal fluxgate can be significantly reduced by duty cycle operation without sacrificing its resolution [2, 22]. However, although lower duty cycle does not increase spectral noise density, it comes at the price of reduced measurement bandwidth. Furthermore, when employing duty cycle operation, a conventional approach such as the synchronous detection cannot be used to demodulate the fluxgate output. Other demodulation techniques require a high sampling rate and rigorous digital signal processing, which results in high system complexity and increased power consumption. In Section 4.2, digital selective bandpass sampling [23, 24], to the output of an orthogonal fluxgate is presented. It has been shown [5] that the output signal can be demodulated by sampling it only once in every cycle and also that selecting the instance can improve the sensitivity and noise. Both approaches are combined here. We employ the digital selective bandpass sampling technique on a fluxgate operated in duty cycle. Consequently, we achieve a significant decrease in its power consumption [25]. We also show that this method can lower the equivalent magnetic noise and simplify the processing technique. Although this method limits the measurement bandwidth, it remains suited to many applications where the magnetic field of the measured objects has a narrow bandwidth [12, 13, 26, 27]. We also show that by intentionally aliasing harmonics adjacent to the fundamental, the magnetometer equivalent magnetic noise can be improved. By optimizing the bandwidth of the bandpass filter at the fluxgate output, we integrate the signals from a predetermined number of harmonics. Since the magnetic noise in each harmonic is uncorrelated [8], this procedure does improve the signal-to-noise ratio and therefore increases the resolution. 2. Section reproduced from [25] with permission.



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A fluxgate prototype has been built and tested to verify the methods (see Figure 4.1). Our experiments show that the excitation power consumption of an orthogonal fluxgate operated in duty cycle can be lower than 1 mW while maintaining equivalent magnetic noise lower than 6 pT/√Hz at 1 Hz. 4.3.1  Synthesis

In conventional orthogonal fluxgates, most of the power is consumed by the core excitation current given that the excitation power consumption is usually tens of milliwatts, whereas the amplification, demodulation, and sampling circuit consumes only a few milliwatts. To lower the power consumption, we excite the fluxgate in duty cycle (DC) where DC e (0, 1) is the ratio of the duration of the active excitation to the total period of the excitation cycle, and 1/DC is an integer. During the idle intervals, a small DC offset persists to maintain the core in positive magnetization. The excitation cycle is described by:

x (t ) = [I 0 + I 1cos ( 2 πωt + ϕ)] ⋅ g (t )

(4.1)



kT < t ≤ T ⋅ DC ; k = 0,1,..n − 1   f (t ) g (t ) =   ;k = 1,2,..n  I 0 − I 1 T (k + DC ) < t < kT

(4.2)



f (t ) =

τ ∞ 2 +∑ sin (DC ⋅ πn )cos (DC ⋅ 2 πn ωt ) T n =1 πn

(4.3)

where ω = 2πfa, T = 1/fa, fa = 4 kHz, φ = 180°, I0 = 18.5 mA, and I1 = 18 mA. In duty cycle operation, the average excitation power consumption, Pex, is the sum of the active excitation power and the idle offset power, calculated as follows:

 I2 2 Pex = DC ⋅ R ⋅  I 02 + 1  + (1 − DC ) ⋅ R ⋅ (I 0 − I 1 ) 2  

(4.4)

where R is the sum load resistance including the amorphous wire core and the stabilizing series resistor. Hence, choosing DC = 1/128, for example, reduces the excitation power consumption by almost two orders of magnitude.

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However, although duty cycle operation significantly reduces the power consumption, it conventionally results in a reduced sensitivity and an increase in the magnetic noise. The sensitivity is reduced since the core is excited only a fraction of the time and not gating any flux during the idle intervals. Furthermore, since the excitation in duty cycle operation is discontinuous, the core magnetic domain relaxation after the excitation transients results in an additional magnetic noise [2, 28]. Figure 4.6 exhibits or demonstrates the fluxgate output response to discontinuous excitation. In Figure 4.6 (bottom), two excitation cycles are depicted with their corresponding fluxgate outputs in Figure 4.6 (top). Our measurements have shown [28] that excess magnetic noise occurs immediately after the end of an active excitation cycle and subsides within ~175 µs therein after. 4.3.2  Method

The output signal of a discontinuously excited fluxgate cannot be demodulated by conventional synchronous detection techniques since the signal and noise of both excitation and idle intervals will be averaged together (Figure 4.7) leading to increased equivalent magnetic noise. To maintain the same equivalent magnetic noise as in continuous excitation, the idle intervals should be discarded from the acquired signal by, for example, employing digital concatenation [2]. However, it requires sampling

Figure 4.6  Fluxgate output noise within two excitation cycles averaged over 25,000 cycles for a duty cycle of 0.25 measured in a shielded environment without employing a bandpass filter (top) and the fluxgate excitation voltage (bottom) [25].



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Figure 4.7  The sensitivity of an orthogonal fluxgate operated in duty cycle and demodulated conventionally by a synchronous detector (dashed curve), and the sensitivity attained using the duty cycle sampling technique (solid curve).

the fluxgate output at higher rates of ~40·fa to make sure that the concatenation is precise enough to avoid spectral leakage. This results in greater power consumption and increased complexity. The limitations of synchronous detection and digital concatenation can be overcome by employing digital selective bandpass sampling [5] thus simultaneously sampling the signal and demodulating it back to baseband. In digital selective bandpass sampling, the signal is acquired by sampling the modulated output voltage only once in every cycle. The obtained signal does not merely maintain the fluxgate equivalent magnetic noise, but even improves it because the sampling instance is optimally selected. The selection technique is described in detail in the experiment section and in [5]. In Figure 4.7, it is shown that, by employing bandpass sampling (solid curve), the sensitivity remains invariant to the operation duty cycle and surpasses the sensitivity obtained by using synchronous detection. Moreover, bandpass sampling lets us avoid the excess magnetic noise during the idle intervals as it occurs during only small predictable duration. 4.3.3  Experimental Setup

To demonstrate the full potential of the combined methods, we have built and tested an experimental model. Our model (similar to Figures 4.1 and 4.8) comprises an amorphous wire (CoFeSiB based made by the National Institute of R&D for Technical Physics, Romania) of a 120-µm diameter, 50-mm length, and a sensing coil with 4,400 turns of an 80-µm copper wire. The coil resistance and inductance were measured at 4 kHz and found to be Rcoil = 225Ω and Lcoil = 18.1 mH, respectively. The fluxgate wire core is con-

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nected in series to a 10.3Ω resistor and excited by an arbitrary waveform generator of model TTI TG5011 operated as a voltage source. A series resistor is used to improve the stability of the current passing through the core and was found to minimize the fluxgate equivalent magnetic noise. The modulated magnetic field is picked up by the sensing coil, amplified by a low-noise, lowpower OPA377 operational amplifier, filtered by a bandpass filter (see Figure 4.8) and sampled by an NI PXI-4461, 24 bit, 204.8-ksps data acquisition module. The excitation and acquisition are synchronized by a 10-MHz high stability clock. All the measurements were performed in a four-shell magnetic shield to reduce the ambient magnetic noise. To measure the sensitivity, we placed the fluxgate inside a calibrated solenoid producing a well-known magnetic field and measured the amplitude of the output signal. 4.3.4  Experiment

We operate our experimental model in different duty cycles, and examine the equivalent magnetic noise and power consumption for each duty cycle. The voltage output of the orthogonal fluxgate is sampled and demodulated by digital selective bandpass sampling. For every duty cycle and filter, we empiri-

Figure 4.8  Block diagram of the measurement system.



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cally find the optimal sampling instance within each excitation cycle [5]. Figure 4.9 (bottom) presents the excitation voltage throughout an active cycle, whereas Figure 4.9 (top) presents the equivalent magnetic noise acquired in each sampling instance within the cycle. In the example presented in Figure 4.9, the lowest equivalent magnetic noise is attained by sampling point 33 of each cycle. This sampling instance corresponds to 206.25 µs from the start of the excitation cycle. Repeatedly sampling only instance 33 in every cycle results in a string of demodulated measured signals with the optimal equivalent magnetic noise. Bandpass sampling causes out-of-band signals and noise to alias into the measurement band, resulting in spurious signals and increased equivalent magnetic noise. Thus, a bandpass filter is required to make sure that the entire measured band is uncontaminated. In duty cycle operation, the output spectrum becomes more populated because the excitation contains harmonics of both fa and DC∙fa. Figure 4.10 illustrates the comparison between the spectra of the fluxgate operated continuously, DC = 1 (top), and in duty cycle DC = 1/128 (bottom). It may be seen in Figure 4.10 (bottom) that where DC = 1/128 the DC∙fa harmonics appear next to the active excitation frequency fa. The bandwidth gap between the harmonics is DC∙fa. In the example of Figure 4.9, the bandwidth gap is DC∙fa = 1/128·4,000 = 32 Hz, which represent a bandwidth of 16 Hz. The bandwidth gap DC∙fa determines the clean measurement bandwidth. This is

Figure 4.9  Fluxgate equivalent magnetic noise illustrated within a single excitation cycle. The sampling instance within an active excitation cycle is presented in the bottom figure and the resultant equivalent magnetic noise for that instance is presented at the top figure (digitally bandpass filtered and averaged over 25,000 cycles for a duty cycle of 1/8).

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Figure 4.10  The spectrum of the fluxgate output for a duty cycle of 1 (top) and 1/128 (bottom). Only 96-Hz band about the fundamental is presented. A 15-nT external field is applied at 1 Hz.

because overlapping frequencies cannot be separated once they alias one into the other. Sampling a fluxgate operated in a duty cycle only once in every cycle results in aliasing of harmonics adjacent to the fundamental from both higher and lower frequencies. A filtering process must be introduced and carefully designed to control the amount of aliased harmonics. To design the optimal antialiasing filter, both sources of out-of-band noise and interferences must be considered: the magnetometer self-noise and outer magnetic noise and interference sources measured by the fluxgate. In this chapter, we assume that the bandwidth of the measured signal with its noise and interferences is pre-known to be smaller than the bandwidth dictated by the selected duty cycle. Therefore, only the aliased fluxgate self-noise is to be considered. Such measurement environment can be found, for example, by performing the measurements inside a magnetically shielded room or in a rural (or almost sterile) environment.



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In the case of a pre-known signal bandwidth, an antialiasing bandpass filter must be introduced prior to the bandpass sampling. Antialias filtering by the classical approach dictates that the bandwidth of the filter must be equal to or thinner than the signal bandwidth. Fortunately, as demonstrated in [29], gathering measurements from several harmonics may improve the magnetometer equivalent magnetic noise. In a similar manner, we gather measurements from a controlled number of adjacent harmonics by selecting the bandwidth of the antialiasing bandpass filter. The bandwidth of the antialiasing filter that leaves a high enough number of neighboring harmonics that guarantees minimal equivalent magnetic noise is found empirically for each duty cycle. We sample the fluxgate output by a 24-bit sigma-delta ADC with an internal antialiasing filter at a rate of 160 kHz (see Figure 4.8). The internal antialiasing filter guarantees an alias-free bandwidth of 0.4 of the sampling rate. The acquired signal is then digitally bandpass filtered using a third-order windowed elliptic IIR filter with a sideband attenuation of 60 dB and demodulated back to baseband by resampling the signal using the bandpass sampling technique. The bandwidth (highpass and lowpass) design of the filter that results in a minimal equivalent magnetic noise for each duty cycle is given at Table 4.1. Based on the above results, we have designed an analog bandpass filter matched to the requirements of a 1/4 DC operation. The filter was built using a combination of a third-order Sallen-Key highpass filter with a corner frequency of 3 kHz and a third-order Sallen-Key lowpass filter with a corner frequency of 5 kHz based on operational amplifier model OPA333 and a total power consumption of 100 µW. This results in a central frequency of 4 kHz and a bandwidth of 2 kHz. 4.3.5  Results

The equivalent magnetic noise as a function of the duty cycle is presented in Figure 4.11. One can see that for an ideal digital filter the equivalent magnetic noise is roughly 6 pT/√Hz at 1 Hz for all of the duty cycles smaller than 1. While operating the sensor at a duty cycle of 1/4 and filtering the signal Table 4.1 The Digital Filter Parameters Used Duty Cycle Highpass Lowpass

1 1/2 1,000 1,000 7,000 7,000

1/4 1/8 3,000 3,200 5,000 4,800

1/16 3,700 4,300

1/32 3,700 4,300

1/64 3,900 4,100

1/128 3,900 4,100

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Figure 4.11  Equivalent magnetic noise as a function of duty cycle of the orthogonal fluxgate with a wide analog filter (solid curve) and with a flitted digital filter (dashed curve).

by the appropriate analog filter, the resolution is similar to the one obtained using the similar bandwidth digital filter. However, while operating at lower duty cycle and keeping the same analog filter, the resolution degrades as the bandpass filter is passing a bandwidth wider than the optimal. The excitation power consumption is reduced by the duty cycle operation. For example, operating the sensor at a duty cycle of 1/128 reduced the excitation power consumption by a factor of 126 from 19.1 mW to 151 µW. 4.3.6  Conclusion

A method for the reduction of power consumption in orthogonal fluxgates is presented [25]. The method entails duty cycle excitation of the wire core, optimal bandpass filtering of the output, and selective digital bandpass sampling. The excess magnetic noise and the decrease in sensitivity caused by the discontinuous excitation are avoided by omitting the idle intervals that only contain the excessive noise and not the information that comes from the measured external field. The major advantage of our method is to collect only the energy of each individual active interval rather than averaging the power of both active and idle intervals. This way, we do not dilute the active intervals with the idle ones and the sensitivity of the digital signal remains the same regardless of the duty cycle. The obtained equivalent magnetic noise at 1 Hz is comparable to both orthogonal and parallel state-of-the-art fluxgates [17, 30]. Compared to other ultralow-power, high-resolution magnetometers such as search coils and



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magneto-electric sensors [31–34], duty cycle-operated fluxgate exhibits better resolution at low frequencies while consuming similar power consumption. The total power consumption of the fluxgate magnetometer is the sum of the power consumed by the excitation, preamplifier, analog antialiasing filter, and ADC. The power consumption of the preamplifier and antialiasing filters can be reduced by an order of magnitude [31]. Modern 24-bit ADCs offer fast conversion settling time and therefore can be operated in duty cycle (see, for example, AD7767 by Analog Devices) with extremely low average power consumption. We therefore believe that a high-resolution orthogonal fluxgate with power consumption lower than 1 mW can be designed.

4.4  Concatenation of Discontinuous Operated Orthogonal Fluxgate It is known that discontinuities in the excitation increase the fluxgate noise [1]3. However, existing literature [35–39] does not provide thorough quantitative analysis of this phenomenon. In this section, we bridge this gap and analyze the noise level of an orthogonal fluxgate employing discontinuous excitation. We also suggest a method for eliminating the excess noise caused by the excitation discontinuities [2, 28]. We employ fast digital concatenation of the fluxgate output to eliminate the output portions that contribute the least to the fluxgate sensitivity and increase the effective output noise. 4.4.1  Noise Measurements and Analysis

Our experimental model of the fluxgate (Figure 4.12) comprises a U-shape Co-based amorphous wire (AC-20 type made by Unitica) of 120-µm diameter and 50-mm length and a sensing coil with 180 turns of a 45-µm copper wire. The excitation current is applied directly to the fluxgate wire core and the output voltage is measured by the sensing coil. The output voltage is measured at the harmonics that provides highest sensitivity. We use ceramic isolators (see Figure 4.12) to thermally stabilize the fluxgate. We have selected for our experiments the above fluxgate type because its wire core impedance is very low; thus, the excitation current can easily be manipulated. The fluxgate core is excited by a TTI TG4001 arbitrary function generator. The external magnetic field is applied to the fluxgate by a calibrated solenoid. The fluxgate and the solenoid are placed inside a three-shell closed magnetic shield to reduce the ambient magnetic noise. The fluxgate output is sampled by an NI PXI-4461, 24-bit, 204.8-ksps data acquisition module, 3. Section reproduced from [2] with permission.

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Figure 4.12  Orthogonal fluxgate employing a U-shaped amorphous wire [2].

which is connected to the fluxgate output through a low-noise BB INA103KP instrumentation amplifier. The data sampling is synchronized with the PXI data acquisition clock. All the measurements are post processed using NI LABVIEW software. 4.4.2  Excitation Waveforms

Since the fluxgate noise level in the unipolar excitation mode [28, 40] (see Figure 4.13(a)) is lower than in the bipolar mode [27, 41], we analyze the fluxgate noise for the unipolar excitation only [42]. We keep the period of the excitation constant (0.5 ms) and to change the duty cycle, we introduce idle time intervals as shown in Figure 4.13(b). The excitation wave form in the active intervals is the minus cosine plus a DC bias that is equal to the cosine amplitude. In the idle intervals, the excitation is zero. The minus cosine AC part of the excitation has been chosen to smooth the transitions between the active and idle excitation intervals. By reducing the excitation duty cycle from 100% to 6.25%, we increase the effective excitation frequency fx = 1/ta, where ta is the duration of the active excitation intervals from 2 to 32 kHz. 4.4.3  Noise Measurements

To calculate the noise at the fluxgate input, we first measured the fluxgate sensitivity. The sensitivity is measured at the effective excitation frequency fx.



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Figure 4.13  Fluxgate output measurements (solid curves) and excitation current (dashed curves) in different excitation modes: (a) in continuous excitation (100% duty cycle), (b) in discontinuous excitation (50% duty cycle), and (c) in discontinuous excitation (50% duty cycle) and discarding the data related to idle intervals [2].

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The measured sensitivity is described by the dashed curve in Figure 4.14. As one can see from Figure 4.14, the fluxgate sensitivity slowly decreases when decreasing the duty cycle, despite the increasing of fx. This is because the fluxgate output averaged over the entire excitation period decreases when the duty cycle decrease and the excitation idle intervals increase. By dividing the noise at the fluxgate output by the fluxgate sensitivity, we obtain the noise at the fluxgate input. This noise as a function of the excitation duty cycle is represented by the dashed curve in Figure 4.15. It may be seen from this figure that the fluxgate noise significantly increases when the duty cycle decreases. For an example, when the duty cycle decreases from 100% down to 6.25%, the noise increases by a factor of 3.5. To find a possible physical reason for the above excess noise, we note that the only difference between the continuous and discontinuous excitation modes is in the time periods where no excitation field is applied to the amorphous wire (see the excitation waveform in Figure 4.13(a, b)). In the continuous excitation mode, these time periods are infinitely short. As a result, the magnetic domains are always forced (to a greater or lesser extent) to be aligned in a given direction. In the discontinuous mode, the time periods where no excitation field is applied to the material are relatively long. Thus, the magnetic domains have an opportunity to relax [43]. Since the domain relaxation is a stochastic process, it may generate the excess magnetic noise.

Figure 4.14  Fluxgate sensitivity as a function of excitation duty cycle. The dashed curve represents the fluxgate sensitivity without eliminating the data related to the idle excitation intervals (see Figure 4.13(b)). The solid curve represents the fluxgate sensitivity after omitting the data related to the excitation idle intervals (see Figure 4.13(c)).



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Figure 4.15  Fluxgate noise density referred to the fluxgate input as a function of the excitation duty cycle. The dashed curve represents the fluxgate sensitivity without eliminating the data related to the idle excitation intervals (see Figure 4.13(b)). The solid curve represents the fluxgate sensitivity after omitting the data related to the excitation idle intervals (see Figure 4.13(c)) [2].

4.4.4  Method for Eliminating the Excess Noise

To eliminate the excess noise, we suggest discarding the idle intervals from the signal processing of the fluxgate output. To verify this idea, we recalculated the noise at the fluxgate input after omitting the data related to the idle intervals in the excitation (see Figure 4.13(c)). As in the previous case, we measured the fluxgate sensitivity as a function of the duty cycle first (see the solid curve in Figure 4.15). One can see from this figure that, in contrast to the previous case, the fluxgate sensitivity increases when decreasing the duty cycle. This is because the effective excitation frequency fx increases and there are no dead times in the fluxgate output (i.e., no loss in the fluxgate output averaging). By dividing the noise at the fluxgate output in Figure 4.13(c) by the fluxgate sensitivity, we can calculate the noise at the fluxgate input (see the solid curve in Figure 4.15). One can see from Figure 4.15 that at low duty cycles the fluxgate noise remains at almost the same level comparing to the continuous excitation mode (100% duty cycle). The fluxgate noise spectrum is shown in Figure 4.16. This figure illustrates the efficiency of the proposed noise reduction. One can see from Figure 4.16 that the concatenation of the fluxgate output reduces the fluxgate noise (see the upper group of the curves that are measured without concatenation) below the noise level measured at the continuous excitation (see the bottom group of the curves). For example, the fluxgate

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Figure 4.16  The spectrum of fluxgate noise referred to its input [2].

noise at 6.25% duty cycle measured with concatenation is much lower than the fluxgate noise measured at the same duty cycle after the signal without concatenation. There is almost no difference between the fluxgate noise measured at 6.25% duty cycle with concatenation and the noise measured at the continuous excitation. As a result, it is possible to reduce the fluxgate power consumption, by reducing the excitation duty cycle, without decreasing the fluxgate resolution. 4.4.5  Discontinuous Excitation Conclusion

We have found that the power consumption of an orthogonal fluxgate employing an amorphous wire can be reduced significantly without affecting its resolution. The power consumption reduction is obtained by introducing idle intervals into the fluxgate excitation. To eliminate the excess noise caused by the discontinuous excitation, we have discarded the data related to the idle intervals from the signal processing of the fluxgate output. As a result, the fluxgate noise referred to its input remains at almost the same level compared to the continuous excitation mode, despite the increase of the excitation idle intervals. Without the excess noise elimination, the fluxgate noise would increase by a factor of 3.5. Reducing the duty cycle from 100% to 6.25% corresponds to a 16-fold reduction in the power consumption: from 6.4 to 0.4



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mW. Thus, our fluxgate provides a 20 pT/√Hz resolution (measured at 1 Hz) and a power consumption of 0.4 mW.

4.5  Conclusion The sampling of high-sensitivity magnetometers requires special low-noise techniques. This is because the sensor’s internal noise is very low and the measurement range is typically large. Furthermore, the overall measured noise of the magnetometer is the sum of all the noise sources of the measurement system including internal and external ones, which in some cases are difficult to manage. We have presented three options to process the sensor output: (1) the conventional approach by demodulating employing a synchronous detector and then samples the demodulated signal, (2) by sampling the fluxgate output at a high rate and performing the demodulation digitally [2], and (3) by performing selective undersampling.

References [1] Ripka, P., Magnetic Sensors and Magnetometers, 1st ed. Norwood, MA: Artech House, 2001. [2] Weiss, E., E. Paperno, and A. Plotkin, “Orthogonal Fluxgate Employing Discontinuous Excitation,” J. Appl. Phys., Vol. 107, No. 9, 2010, p. 09E717. [3] Primdahl, F., et al., “The Fluxgate Magnetometer,” Elements, Vol. 241, No. 4, 1979. [4] Sasada, I. K. H., “Simple Design for Orthogonal Fluxgate Magnetometer in Fundamental Mode,” J. Magn. Soc. Japan, Vol. 33, 2009, pp. 43–45. [5] Weiss, E., et al., “Orthogonal Fluxgate Employing Digital Selective Bandpass Sampling,” IEEE Transactions on Magnetics, Vol. 48, No. 11, 2012, pp. 4089–4091. [6] Cerman, A., et al., “Digitalization of Highly Precise Fluxgate Magnetometers,” Sensors Actuators A Phys., Vol. 121, No. 2, June 2005, pp. 421–429. [7] Primdahl, F., “The Fluxgate Magnetometer,” J. Phys. E., Vol. 241, 1979. [8] Kubik, J., and P. Ripka, “Noise Spectrum of Pulse Excited Fluxgate Sensor,” Sensors Actuators A Phys., Vol. 132, No. 1, November 2006, pp. 236–240. [9] Fan, J., X. Li, and X. Qian, “Pulse Excitation Approach for Low Power Orthogonal Fluxgate Sensor,” INTERMAG 2006 - IEEE Int. Magn. Conf., May 2006, pp. 877–877. [10] Ubizskii, S. B., and L. P. Pavlyk, “The Pendulum-Like Fluxgate Magnetic Field Sensor,” Sensors Actuators A Phys., Vol. 141, No. 2, February 2008, pp. 440–446.

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[11] Baglio, S., et al., “Exploiting Nonlinear Dynamics in Novel Measurement Strategies and Devices: From Theory to Experiments and Applications,” IEEE Transactions on Instrumentation and Measurement, Vol. 60, No. 3, 2011, pp. 667–695. [12] Alimi, R., et al., “Ferromagnetic Mass Localization in Check Point Configuration Using a Levenberg Marquardt Algorithm,” Sensors, Vol. 9, No. 11, November 2009, pp. 8852–8862. [13] Weiss, E., et al., “High Resolution Marine Magnetic Survey of Shallow Water Littoral Area,” Sensors, Vol. 7, No. 9, September 2007, pp. 1697–1712. [14] Ando, B., et al., “‘Residence Times Difference’ Fluxgate,” Measurement, Vol. 38, No. 2, September 2005, pp. 89–112. [15] Sasada, I., “Symmetric Response Obtained with an Orthogonal Fluxgate Operating In Fundamental Mode,” IEEE Transactions on Magnetics, Vol. 38, No. 5, September 2002, pp. 3377–3379. [16] Ripka, P., and M. Janosek, “Advances in Magnetic Field Sensors,” IEEE Sens. J., Vol. 10, No. 6, June 2010, pp. 1108–1116. [17] Butta, M., and I. Sasada, “Orthogonal Fluxgate with Annealed Wire Core,” IEEE Transactions on Magnetics, Vol. 49, No. 1, January 2013, pp. 62–65. [18] Li, H., et al., “Vehicle Classification with Single Multi-Functional Magnetic Sensor and Optimal MNS-Based CART,” Meas. J. Int. Meas. Confed., Vol. 55, 2014, pp. 142–152. [19] Lan, J., et al., “Vehicle Detection and Classification by Measuring and Processing Magnetic Signal,” Meas. J. Int. Meas. Confed., Vol. 44, No. 1, 2011, pp. 174–180. [20] Paulraj, M. P., et al., “Moving Vehicle Recognition and Classification Based on Time Domain Approach,” Procedia Eng., Vol. 53, 2013, pp. 405–410. [21] He, Y., Y. Du, and L. Sun, “Vehicle Classification Method Based on Single-Point Magnetic Sensor,” Procedia - Soc. Behav. Sci., Vol. 43, 2012, pp. 618–627. [22] Weiss, E., and E. Paperno, “Noise Investigation of the Orthogonal Fluxgate Employing Alternating Direct Current Bias,” J. Appl. Phys., Vol. 109, No. 7, 2011, p. 07E529. [23] Vaughan, R., N. Scott, and D. White, “The Theory of Bandpass Sampling,” IEEE Transactions on Signal Processing, Vol. 39, No. 9, 1991. [24] Donoho, D. L., and J. Tanner, “Precise Undersampling Theorems,” Proc. IEEE, Vol. 98, No. 6, June 2010, pp. 913–924. [25] Weiss, E., A. Grosz, and E. Paperno, “Duty Cycle Operation of an Orthogonal Fluxgate,” IEEE Sens. J., 2014, pp. 1–5. [26] Kaluza, F., A. Gruger, and H. Gruger, “New and Future Applications of Fluxgate Sensors,” Sensors Actuators A Phys., Vol. 106, No. 1–3, September 2003, pp. 48–51. [27] Olsen, N., et al., “Calibration of the Ørsted Vector Magnetometer,” Response, 2003, pp. 11–18. [28] Weiss, E., et al., “Excess Magnetic Noise in Orthogonal Fluxgates Employing Discontinuous Excitation,” IEEE Sens. J., Vol. 14, No. 8, 2014, pp. 2743–2748.



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[29] Jeng, J., J. Chen, and C. Lu, “Enhancement in Sensitivity Using Multiple Harmonics for Miniature Fluxgates,” IEEE Transactions on Magnetics, Vol. 48, No. 11, 2012, pp. 3696–3699. [30] Koch, R. H., and J. R. Rozen, “Low-Noise Flux-Gate Magnetic-Field Sensors Using Ring- and Rod-Core Geometries,” Appl. Phys. Lett., Vol. 78, No. 13, 2001, p. 1897. [31] Grosz, A., et al., “A Three-Axial Search Coil Magnetometer Optimized for Small Size, Low Power, and Low Frequencies,” IEEE Sens. J., Vol. 11, No. 4, 2011, pp. 1088–1094. [32] Xing, Z. P., et al., “Modeling and Detection of Quasi-Static Nanotesla Magnetic Field Variations Using Magnetoelectric Laminate Sensors,” Meas. Sci. Technol., Vol. 19, No. 1, January 2008, p. 15206. [33] Paperno, E., and A. Grosz, “A Miniature and Ultralow Power Search Coil Optimized for a 20 mHz to 2 kHz Frequency Range,” J. Appl. Phys., Vol. 105, No. 7, 2009, p. 07E708. [34] Wang, Y., et al., “An Extremely Low Equivalent Magnetic Noise Magnetoelectric Sensor,” Adv. Mater., Vol. 23, No. 35, September 2011, pp. 4111–4114. [35] Kubik, J., L. Pavel, and P. Ripka, “PCB Racetrack Fluxgate Sensor with Improved Temperature Stability,” Sensors Actuators A Phys., Vol. 130–131, No. December 2005, August 2006, pp. 184–188. [36] Infante, G., et al., “Double Large Barkhausen Jump in Soft/Soft Composite Microwires,” J. Phys. D. Appl. Phys., Vol. 43, No. 34, September 2010, p. 345002. [37] Butta, M., “Orthogonal Fluxgates,” In Tech. [38] Ripka, P., et al., “Micro-Fluxgate Sensor with Closed Core,” Sensors and Actuators, Vol. 91, 2001, pp. 65–69. [39] Butta, M., and P. Ripka, “Linearity of Pulse Excited Coil-Less Fluxgate,” IEEE Transactions on Magnetics, Vol. 45, No. 10, 2009, pp. 4455–4458. [40] Sasada, I., “Orthogonal Fluxgate Mechanism Operated with Dc Biased Excitation,” J. Appl. Phys., Vol. 91, No. 10, 2002, p. 7789. [41] Paperno, E., E. Weiss, and A. Plotkin, “A Tube-Core Orthogonal Fluxgate Operated in Fundamental Mode,” IEEE Transactions on Magnetics, Vol. 44, No. 11, November 2008, pp. 4018–4021. [42] Paperno, E., “Suppression of Magnetic Noise in the Fundamental-Mode Orthogonal Fluxgate,” Sensors Actuators A Phys., Vol. 116, No. 3, October 2004, pp. 405–409. [43] Gurevich, L. E., and E. V. Liverts, “Possibility of Small Barkhausen Jumps in an Ideal Crystal.pdf,” JETP, Vol. 33, 1981, pp. 506–508.

5 Magnetic Data Processing The purpose of this chapter is to describe how magnetic anomalies generated by the presence of moving ferromagnetic objects can be detected and localized using suitable digital signal processing tools. The importance of precise detection and localization schemes is described in Chapter 2. We will first discuss the detection process of two types of motions that are more frequently encountered: constant velocity linear and stationary periodic motions. The first will be brief, as this subject has been extensively reported. A more detailed description will be provided to the second type because of its novelty. In the second part of this chapter, we present two important types of localization algorithms. The first one is a deterministic, straightforward class of inverse problem algorithms, the Levenberg-Marquard algorithm (LMA). We will exemplify cases where LMA solver processing is well suited for magnetic motion tracking. The second type of solver belongs to the family of metaheuristic algorithms and more precisely to the evolutionary algorithm subclass: the Genetic Algorithm (GA). The cases in which LMA types of solver fail while GA programing can be successful will be emphasized. Both LMA and GA applications are presented within realistic background scenarios from a theoretical point of view and in real but sterile experimental conditions.

5.1  Magnetic Anomaly Detection 5.1.1  Orthonormal Basis Functions Representation

This representation was first developed in the late 1940s by J. E. Anderson while working at the Naval Air Development Center in Warminster, Pennsyl167

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vania, and includes the functions known as Anderson functions. Those functions transpire when there is relative linear motion at a constant velocity v of a magnetic dipole relative to a magnetic sensor. To describe those functions, we define ω by (5.1): ω=



νt r0

(5.1)

It is the ratio between the position of the target along the track at time and the closest point from the sensor to the track as r0. A schematic scenario is shown in Figure 5.1. In this case, the signal can be represented as a linear combination of several elementary functions [1]. Further work by Ginzburg et al. [2] and Frumkis et al. [3] decomposed it to three orthonormal basis, so that the functions are normalized and linear independent with some coefficients an: 3

B ( ω) = ∑ αn f n ( ω)



(5.2)

n =1

where fn equals to:



5 1 − ω2 24 128 ω 3 f1 = f2 = 2.5 5 π 1 + ω2 5 π 1 + ω2

(

f3 =

)

(

)

2.5

2

128 ω 3 π 1 + ω2

(

)

2.5

Figure 5.1  Moving magnetic dipole target along straight line trajectory.



(5.3)



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The set of those orthonormal functions is shown in Figure 5.2. The convolution process is performed by computing the projection of a moving window of length N of the input signal on the three orthonormal base functions given by (5.3). After squaring the output of each of the three functions, we define the signal energy by:

2

N  energy (B ) = ∑  ∑B ( j ) f i ( ω, j )  i =1  j =1 3

(5.4)

Detection occurs when the energy value exceeds a predetermined threshold, as shown in the algorithm scheme of Figure 5.3. Since there is no a priori knowledge of the object’s velocity characteristic, a multichannel approach may be adopted for various possible values of w. Practically, when using physical knowledge of the problem, there is typically no need for more than 2 or 3 channels. 5.1.2  Minimum Entropy Detection Filter

The basic function approach described above is aimed to detect a well-known type or shape of signal within a given moving window of streaming data. This is possible when a model of the moving magnetic anomaly is available. If this is not the case, monitoring significant change in the noise behavior can reflect the presence of a ferromagnetic object. This kind of approach is the basic idea of the minimum entropy filter developed by Sheinker et al. [5]. Entropy is a key concept of information theory usually used to measure the lack of order in a given amount of information. High entropy means strongly uncorrelated data while low entropy reveals the presence of a positive autocorrelation of one the properties that differentiate a signal from the surrounding noise.

Figure 5.2  The set of orthonormal functions f1 (ω), f2 (ω), f3 (ω) for presentation of the target field [4].

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Figure 5.3  The orthonormal base functions algorithm general scheme [4].

For a sample xi is the normal probability density function represented by the mean μ and variance σ2:

1

f (xi ) =

2 πσ

2

e

−

(xi − µ)2 2σ2



(5.5)

First, we estimate the mean and variance within the moving window at size using:

ˆ= µ



1 σˆ 2 = N

by:

1 N

i

∑

xj

(5.6)

j =i − N −1

i

∑

j =i − N −1

(x

)

2

j

–µ

(5.7)

The probability of the noise sample taking a discrete value of xi is given p (xi ) =

x i +∆x

∫

xi

f ( x i ) dx ≅ f ( x i ) ∆x

(5.8)



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where ∆x is a quantization step. The entropy filter calculates the entropy in moving window N samples according to:

Entropy ( x i ) = −

i

∑

p ( x n ) log ( x n )

n =i − N +1

(5.9)

Similar to the Anderson function reconstruction technique, detection occurs whenever the entropy value crosses a threshold value. In this case, crossing the threshold is in the negative direction, moving from high to low entropy. For a given signal-to-noise ratio (SNR), the entropy detector performs better than the energy filter. This is shown in Figures 5.4 and 5.5 for a specific case and a general case, respectively. The entropy detector is more sensitive than the energy one since it does not rely on a specific scenario: it can extract from the noise everything that differs from uncorrelated noise. Unfortunately, it will also detect correlated signals that are not the ones for which we are looking. In other words, in terms of false alarm rate (FAR), one expects the entropy to be less effective. The next section presents a novel class of algorithms that combines high PD like the entropy filter and low FARs like the energy filter.

Figure 5.4  Entropy filter compared to orthogonal base function (OBF) detector.

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Figure 5.5  Probability of detection (PD) performance of the entropy and the OBF detector.

5.1.3  Periodic Anomaly Detection Filter

The purpose of the periodic anomaly detection (PAD) filter is to provide a fast and efficient algorithm for detecting magnetic anomalies that exhibit a periodic time pattern. We first present the mathematical model on which the algorithm is based. Then we will discuss the algorithm performance using Monte Carlo simulations. The reason for using simulations is quite obvious: it is the simplest way to generate large data samples for which statistical analysis can be performed. However, since no simulation can fully account for realistic conditions, we have designed and built an experimental setup for testing the algorithm in a true nonsterile environment. The experiment setup and the results analysis will be the subject of the last part of this section. We already know that the internal magnetic noise of the magnetometer performs like a pink noise that follows a 1/f k frequency law. Such a behavior can be better represented by a Gaussian autoregressive (AR) signal of order p [4], the coefficients of which can be estimated from the experimental background recording. We denote the magnetic signal w with length N. The autocorrelation function (ACF) parameters of w are given by:

Rˆ [l ] =

1 N−l

N −1− l

∑ w n + l  w [ n] , n=0

l = 0,1,.., p

(5.10)



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Using the Yule-Walker equations in which we set the ACF parameters Rˆ[l ], we can extract the autocorrelation coefficients aˆ[l ]. We solve the resulting equations using the Levinson recursion algorithm [6]:



      ˆ   ˆ  ˆ ˆ  … R [ p − 1]  aˆ [1] R [1]  R [0 ]  R [1]     Rˆ [1] Rˆ [0]  Rˆ [ p − 2 ]  aˆ [2 ]  = −  Rˆ [2 ]  (5.11)                      … Rˆ [0]  aˆ [ p ] Rˆ [ p − 1] Rˆ [ p − 2 ] Rˆ [ p ] Then the estimated noise variance σˆ 2 is calculated:



σˆ 2 = Rˆ [0] + aˆ [1]Rˆ [1] + …+ aˆ [ p ]Rˆ [ p ]

(5.12)

From which the noise model of w[n] can be formulated: p



w [n ] = − ∑aˆ [k ]v [n − k ] + v [n ] k =1

(5.13)

where v[n] is a white Gaussian noise (WGN) with variance σˆ 2. 5.1.3.1  Signal Modeling

We proceed in two steps: first, we propose a simple physical and mathematical representation of the oscillating magnetic dipole (OMD), and second, using a gravity pendulum model, we present a realistic dynamic model of the physical system. 5.1.3.2  A Simplified Model for OMD

The magnetic moment M can be represented by a rotating vector over time. Figure 5.6 depicts the rotation axis K around which the moment Mrot is rotating around. The time-varying angle is notated by θ. We use the Rodrigues rotation algorithm [7] for computing Mrot directly. The vector form of the Rodrigues formula is given by:

(

)

(

)

M rot = M cos θ + Kˆ × M sin θ + Kˆ Kˆ ⋅ M (1 − cos θ )

(5.14)

where Kˆ is the unit vector of the fixed rotation axis and q is the rotation angle.

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Figure 5.6  A simplified model illustrating the OMD.

If M and K are defined as column vectors and [K]x is defined as the cross-product matrix, we get:



K 1   0    K × M = [K ]× M = K 2  × M =  K 3  −K 2 K 3 

−K 3 0 K1

K2  −K 1  M (5.15) 0 

Leading to: M rot =



(

)

(

)

M cos θ + [K ]× M sin θ + K K T M (1 − cos θ ) = M cos θ + [K ]× M sin θ + KK T M (1 − cos θ )



(5.16)

= I cos θ + [K ]× sin θ + KK T (1 − cos θ ) M

The rotation angle θ can be written as a time-dependent function of the rotation frequency f:

θ = ωt = 2 πft

(5.17)

Using the dipole equation, we can finally formulate the simplified model of the OMD as:





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B (r , M , K ,t , f ) =

175

µ0 −5  T 2 3 rr − r I r   4π

Icos (2 πft ) + [K ]× sin (2ð ft ) + KK T (1 − cos(2 πft ))] M 



(5.18)

5.1.3.3  OMD Generalization to Equation of Motion of Gravity Pendulum

A more realistic description of the OMD motion can be represented by a nonlinear gravity pendulum as shown in Figure 5.7. The swinging angle is described as a fractional periodical rotation:

θt = θ0cos (2 πft )



(5.19)

where θ0is the angle swing away from the vertical of the moment. Again, using the dipole equation motion, we get: B (r , M , K ,t , f , θ0 ) =



(

)

µ0 r 4π

−5

3 rrT − r 2 I ⋅  

(

I cos θ0 cos (2 πft ) + [K ]× sin θ0 cos (2 πft )   + KK T − cos θ0 cos (2 πft ) 

(

(

))

)  



(5.20)

M

5.1.3.4  Fourier Series Representation

We may simplify the model by using a Fourier series representation. The signal model (per axis) can be written as:

Figure 5.7  Magnetic moment swing around fixed axis as a pendulum motion.

176



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(5.21)

n = 0,1,…, N − 1 where fs and f0 are the sampling rate and the fundamental frequency, respectively. We take into account M – 1 harmonics (2f0, 3f0, …) and ak, bk are the amplitudes of each harmony, respectively. 5.1.3.5  Overall Data Model for Three-Axis Magnetometer

The whole streaming data B is usually represented by the addition of a pure signal s and noise n: B[n] = s[n] + w[n]. The matrix form can be written as follows:



M   ∑αx k cos  2 πk  k =1 Bx [n ]     M B y [n ] =  ∑α y k cos  2 πk    k =1 Bz [n ]  M    ∑αz k cos  2 πk  k =1

 f0  M f  n  + ∑ bx k sin  2 πk 0 n   f s  k =1 fs      f0  M f  n  + ∑ b y k sin  2 πk 0 n   f s  k =1 fs      f0  M f0   n + ∑ bz sin  2 πk n   f s  k =1 k fs   

   ∑a x [k ]v x [n − k ] + v x [n ] k =1   p  +  ∑a y [k ]v y [n − k ] + v y [n ] n = 0,1,…, N − 1 k =1   p   a k v n −k +v n  ] z [ ] ∑ z [ ] z [  k =1 

(5.22)

p

where αk and βk are the Fourier series coefficients of eachaxis, f0 is the fundamental frequency of the oscillation within the range f 0min ≤ f 0 ≤ f 0max with M – 1 harmonics,fs is the sampling rate, a[k] are the AR noise coefficients of order p, and v[n] is the WGN with variance σ2 and zero mean. We also assume that each axis is independent except for the dipole frequency f0. 5.1.3.6  Signal Model Verification

The next step is to check the model by comparing the signals that it generates to real experimental data. For this purpose, we developed a MATLAB



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simulation code in which the model was implemented. We test the complete exact formulation rather than the Fourier series representation. The latter will be used at the algorithm development stage. We consider only gradiometric signals (see Chapter 2) since gradiometric data tend to produce a higher SNR value than a single sensor data. The ACF and Yule-Walker equations (YWE) provided simulated noise added to the background recording. The best (minimal) AR order fit to the noise spectrum was found to be equal to 10. It is presented in Figure 5.8. The experimental setup and results are described in detail in Section 5.1.3.9. The simulated and experimental signals exhibit very similar noise floor, the same fundamental and harmonics frequencies, and nearly the same SNR. Let us consider, for example, the experiment with the highest SNR value (experiment No. 1). Figure 5.9 compares the gradiometric channel (S4S2) spectrum to the simulation. The second comparison is experiment No. 2 (medium SNR), which compared gradiometric channel S4-S2 to the simulation, as shown in Figure 5.10. The final comparison is experiment No. 5 (low SNR), is shown in Figure 5.11. It appears that the simulated signal has the same noise floor, the same fundamental frequencies with its harmonics, and nearly the same SNR. The good agreement between the simulation and the real data gives us confidence that as long as the detection algorithm works in the frequency domain, we can use simulated data rather real-world measurements to develop the optimal detection scheme.

Figure 5.8  Background recording PSD versus AR (p = 10) estimation signal.

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Figure 5.9  Spectral comparison of experiment No. 1 (high SNR): gradiometric channel S4-S2 versus simulation.�

Figure 5.10  Spectral comparison of experiment No. 2 (medium SNR): gradiometric sensors S4-S2 versus simulation.�



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Figure 5.11  Spectral comparison of experiment No. 5 (low SNR): gradiometric sensors S4-S2 versus simulation.

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5.1.3.7  The Detection Algorithm Noise Whitening

Before applying the detection algorithm, the data is first preprocessed by whitening the signal to flatten the spectrum. Only then the multiple harmonics peaks search can be applied, based on a periodic characterization of the signal. When fomin approaches direct current (DC), it is generally advised to perform noise whitening to obtain an efficient peak detection. This is usually done using a linear finite input response (FIR) filter with a transmission function (Z spectrum) given by:

Bˆ ( Z ) =

σˆ

∑

p aˆ k =0

[k ] Z −k



(5.23)

when σˆ and aˆ are prior noise estimated parameters. The output signal of the AR signal is a white Gaussian noise with variance of 1. Figure 5.12 depicts the effect of whitening on real signal spectrums: DC and linear trend are correctly removed and the whole SNR has also improved.

Figure 5.12  Spectral comparison between signals before and after the whitening filter.



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Harmonics Peak Search

Since each sinusoidal component of the signal has an unknown amplitude, frequency, and phase, we can make use of the comb filter presented in [8]. First, the power spectral density (PSD) is computed to determine harmonics peaks with an integer multiplier that lies over the flattened spectrum P(f). The peak search of the fundamental frequency f0 made by the maximum summation of all harmonics is performed using a detector T value given by:

Tdetector =

max

M

∑P (nf 0 ) > γ

f min ≤ f 0 ≤ f max n =1



(5.24)

where Tdetector exceeds a predetermined threshold γ, a detection flag is raised. An overall scheme of the detection algorithm using a gradiometric triaxial magnetometer is shown in Figure 5.13. Examples of the algorithm output of the experimental gradiometric data with low SNR = −3.91 dB with clutter noise (temporal wideband noise that usually comes from moving vehicles) and experimental gradiometric data with medium SNR = −1.36 dB with clutter noise are shown in Figures 5.14 and 5.15, respectively. Algorithm Evaluation

Performing a very large number of controlled experiments was not practical. Instead, the algorithm evaluation was determined using Monte Carlo computer simulation evaluations (using MATLAB software). A large number of data sets (noise alone, signal and noise) with various parameters were generated. The results are processed to provide the performance plots. Threshold Determination

The Neyman-Pearson (NP) criterion was used to determine the threshold value to achieve maximal detection probability under a constraint on the false alarm rate. This criterion is useful when no information is known about the target. In our situation, a statistical test of 100,000 gradiometric magnetic noise signals, using the noise model according to (5.13), was generated. Using A preliminary background recording, we estimated the noise parameters’ values. They are listed in Table 5.1. The data sets were evaluated by the proposed algorithm to find the threshold g as a function of the FAR. The output curve of this function is shown in Figure 5.16. The horizontal axis of the graph was plotted using logarithmic axis. This scale helps to find, in a convenient way, the corresponding threshold for

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Figure 5.13  Block diagram of the proposed detection algorithm.

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Figure 5.14  Algorithm output of experimental gradiometric data with low SNR = 3.91 dB and clutter noise.

Figure 5.15  Algorithm output of experimental gradiometric data with medium SNR = −1.36 dB and clutter noise.

desirable FAR. As expected, smaller values of desirable FAR require higher threshold values.

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Low-Power and High-Sensitivity Magnetic Sensors and Systems Table 5.1 Noise Parameter Values Used in Simulation Parameter Value N 300 10 [Hz] fs p 10 2 2.246e-04 [nT2] σˆ x

σˆ y2 σˆ 2z aˆx [k ], k = 1, 2, …p aˆy [k ] aˆz [k ]

Description Window size Frequency of sampling AR order Estimated noise variance of X 2.176e-04 [nT2] Estimated noise variance of Y 2.451e-04 [nT2] Estimated noise variance of Z Estimated AR [−0.456,−0.034, −0.078, −0.011, −0.061, −0.054, −0.039, −0.039, −0.059, −0.0793] coefficients of X [−0.491, −0.003, −0.078, −0.005, −0.020, Estimated AR coefficients of Y −0.050, −0.009, −0.034, −0.016, −0.069] [−0.470, −0.020, −0.043, −0.0193,0.021, −0.024, −0.030, −0.066, −0.042, −0.102]

Estimated AR coefficients of Z

Figure 5.16  FAR of the proposed algorithm as a function of threshold value when the input is simulated data.

For selected FAR, the corresponding PD can be calculated. To find this value, we generated an additional 100,000 random data sets of OMD signals, according to the signal model in (5.20). The signal parameter values used in the data sets are listed in Table 5.2.



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Table 5.2 Magnetic Signals Parameter Values Used in Simulation Parameter N n fs

Value 300 1, 2, …, N 10 [Hz]

r M

[~N(0, 5 ),~N(0, 5 ),~N(0, 5 )] Dipole’s location vector Magnetic moment vector [~N(0,1),~N(0,1),~N(0,1)]

K

Kˆ

[~N(0,1),~N(0,1),~N(0,1)] K K

t f0

n/fs [s] ~U[0.5, 1.5] [Hz]

θ0

π  ~ U  , π 3 

Rotation angle

[0, 0, 0] [m] [0, 8, 0] [m]

Sensor 1 location Sensor 2 location

S1 S1

Description Window size Sample vector Frequency of sampling

Dipole’s rotation axis Normalized rotation axis Time vector Rotation frequency

For a constant FAR, the PD depends only on the SNR. The SNR value is computed from the noise energy and the signal energy affected by the model parameters. However, since the model is relatively complex, we cannot control the signal energy by changing those parameters directly. Therefore, after generating each signal, we multiply it by a constant number to scale it to the required signal energy level. Then we add random noise. This completes the data model with a given SNR. The tested SNR values are [−5 dB, −10 dB, −15 dB, −20 dB]. For each value a receiver operating characteristic (ROC) curve was generated, in which plots PD versus FAR are calculated. This is shown in Figure 5.17. Logarithmic scale was used for the abscissa. Given a FAR value and a constrained SNR, the log scale helps to conveniently find the corresponding PD. As expected, higher values of desirable PD required higher FAR values. Higher values of SNR result in a higher ROC response. Another method to evaluate at the detection performance is to plot the PD versus the SNR. This graph indicates the SNR required to meet the specification of a given PD for a fixed FAR. The tested FAR values are [1e-1, 1e-2, 1e-3, 1e-4]. For each value, a curve was generated in which plots SNR versus PD are shown in Figure 5.18. As shown, PD of almost 100% is achieved for the various FAR values, even at FAR as low as 1e-5, when the SNR is greater than 0 dB. When the SNR value decreases to −5 dB, the differences of the PD value of each FAR are less than 2%. Once the SNR is less than −10 dB,

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Figure 5.17  ROC curve for the complete simulated model.�

Figure 5.18  SNR versus PD for the complete simulated model.

the differences between the PD of the high and low FAR values increased to 20% but remain almost stable. Another remarkable parameter is the dependence of the PD on the window size of the algorithm input. The importance of window size is its direct effect on the computation cost of the algorithm. In this test, the values of N



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are in the range of 10 to 1,000. The values of FAR that were tested are the same as before: [1e-1, 1e-2, 1e-3, 1e-4]. For each value a curve was generated, in which plots N versus PD are shown in Figure 5.19. 5.1.3.8  Comparison of Detection Algorithms

The performance of the detection algorithm is now compared to known and competitive detection algorithms: the minimum entropy detection (MED) and the energy detector. As we have shown before, MED has a higher PD than the orthonormal base functions (OBFs) detector. This is true even when the object is moving along a line [5]. The energy detector is a simple detector, very easy to implement. The detector provides an alert if the overall energy of the data increases relative to that expected for noise only. No other information about the signal is assumed to be available [9]. The detector decides a signal is present if:

Tenergy (x [n ]) =

N −1

∑ x 2[n ] > γ

n =0

(5.25)

where x[n] is the input signal, N is the window size, and γ is a predetermined threshold. We employed the example from [5] in which the OBF and MED are compared. Figure 5.20 considers the case of a FAR less than 4%. As we see, our algorithm performs better at an SNR of −10 dB with 90% PD than the MED that has only a PD of 50%. As expected, the energy detector has the weakest performance with PD of 40%.

Figure 5.19  N versus PD for the complete simulated model.

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Figure 5.20  The PD of the proposed algorithm versus the MED and energy detector.

5.1.3.9  Experimental

Monte Carlo simulations give insight regarding the algorithm performance because they allow significant statistical analysis. However, the highest confidence level is achieved by testing it under realistic operational conditions. The test included real-world noise signals, as well as natural background noise. Apparatus

An array of Bartington Mag634 three-axial fluxgate magnetometers were used as a synchronized magnetic sensor network. To generate the signals, we used a gimbaled electromagnet system that allows us to control the orientation, frequency and intensity of the OMDs. These signals are used to develop and test the algorithm proposed in this study. The Mag634 includes an internal bandpass filter that compensates for the Earth’s magnetic DC field. The sensor is characterized by internal noise of the order of 10 pT/√Hz at. The magnetometer filtered output with a dynamic range of ±5V was sampled by a 16-bit analog-to-digital converter (ADC), with a sampling rate of. All the sensors were orientated in the same direction. The distance between the sensors was set to 4m. The sensors can be taken into account as stand-alone magnetometers or as gradiometer pairs to cancel out background noise. The data was acquired and then decimated by a factor of 100, resulting in a 10-Hz sampling frequency. The data of each sensor axis was sent serially



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and saved separately on a remote PC. The entire experimental setup was ferromagnetic-free, a minimum magnetic disturbance that might interfere with the magnetometer measurement; the setup scheme is shown in Figure 5.21. Acquiring Preliminary Data

To research the OMD algorithm, several experiments were performed with different SNRs. The experiments were carried out with the following settings: • One-hour recording of background noise was made to characterize the magnetic noise. • Fifty-nine SNR scenarios were recorded. • Each scenario was 120 seconds long where the first 30 seconds of each signal recording is background noise and the remaining 90 seconds contains the OMD signal. • Seven hundred background noise scenarios were measured where the total recording time of each noise experiment is 30 seconds. Tables 5.3 and 5.4 list the nongradiometric and gradiometric experiments conducted, and present the corresponding SNR values (in decibels) for each sensor. The differences between gradiometric and nongradiometric signals in experiments #1 and #5 are presented in Figures 5.22 and 5.23, respectively. To analyze the noise characteristics, we recorded the background noise for nearly 1 hour. Figure 5.24 presents 600-second recording of gradiometric and nongradiometric background noise. Figure 5.25 presents the PSD of the same noise signal. It is clear that the nongradiometric signal is much noisier with respect to peak to peak, low frequencies, and noise floor. As we see from the collected data, the gradiometric measurement shows a 10-dB to 14-dB

Figure 5.21  The experimental setup.

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Low-Power and High-Sensitivity Magnetic Sensors and Systems Table 5.3 The Nongradiometric Experiments and Their SNRs Sensor Exp. No. 1 2 3 4 5 Sensor Exp. No. 1 2 3 4 5

S1 SNR [dB] TF Bx By Bz –22.03 –25.11 –21.25 –11.77 –26.29 –29.86 –27.55 –12.31 –24.16 –26.75 –28.46 –15.35 –21.16 –24.61 –24.53 –11.69 –23.96 –28.06 –24.99 –12.26 S3 SNR [dB] TF Bx By Bz –17.25 –22.02 –19.67 –7.66 –18.98 –23.50 –21.17 –9.82 –20.50 –23.35 –22.90 –11.48 –18.76 –23.83 –18.19 –8.14 –22.44 –27.22 –24.58 –10.41

S2 SNR [dB] TF Bx By Bz –25.68 –29.55 –23.62 –10.99 –25.19 –31.49 –26.55 –11.15 –24.53 –27.76 –28.84 –13.02 –21.32 –26.91 –26.24 –9.79 –24.05 –29.40 –26.04 –11.08 S3 SNR [dB] TF Bx By Bz –4.27 –9.95 –8.09 0.43 –8.30 –15.95 –13.58 –3.14 –13.14 –19.97 –18.32 –6.98 –17.78 –24.93 –22.06 –7.69 –17.90 –26.13 –23.97 –8.94

�Table 5.4 The Gradiometric Experiments and Their SNRs Sensor S3–S1 SNR [dB] Exp. No. TF Bx By 1 –1.96 –3.38 –1.40 2 –4.93 –6.48 –3.13 3 –6.21 –6.14 –4.82 4 –3.39 –5.40 0.46 5 –8.19 –9.47 –7.72

Bz –0.25 –4.22 –5.61 –2.29 –8.45

S4–S2 SNR [dB] TF Bx By 10.13 12.82 11.61 2.05 6.80 5.79 –0.02 3.04 1.01 –1.36 –2.01 –1.70 –3.91 –4.35 –5.24

Bz 13.43 9.23 5.82 3.35 0.29

SNR improvement compared to the single sensor measurement; hence, the former was selected for preprocessing. Once the data was collected, a detection threshold was determined from the noise data sets by running the algorithm over the noisy dataset and calculating the FAR for each threshold. In Figure 5.26, we can see the FAR with the corresponding threshold value. Because we are limited to 1,400 noise data sets, a reasonable value of FAR can be taken as 1% compatible with a threshold value of 0.017. After determining the threshold value, the 118 signal datasets were tested using the proposed algorithm. For each SNR value, we found the number of datasets that surpass the threshold value. Those data points were fitted to an approximate curve using a simple least square method. This curve was compared to the performance of simulated data as shown in Figure 5.27. It



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Figure 5.22  Exp. #1: Nongradiometric signal (S4) with SNR = −4.27 dB (left) and gradiometric signal (S4-S2) with SNR = 10.13 dB (right).

can be seen from the graph that the SNR differences stand only on less than 2.5 dB and PD differences are 10% to 15%. 5.1.3.10  OMD Detection Algorithm: Summary and Discussion

This section presented a solution for the problem of characterization of OMD signals, and the development of a suitable detector. The resulting outcomes

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Figure 5.23  Exp. #5: Nongradiometric signal (S4) with SNR = −17.9 dB (left) and gradiometric signal (S4-S2) with SNR = −3.9 dB (right).

are the following. (1) A comprehensive model of the noise and the signal based on a nonlinear gravity pendulum model is developed. This model was compared and verified against real-world magnetic signals, as well as simulated ones. (2) A detection algorithm utilizes this model by whitening the noise and seeking periodic features at the signal.



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Figure 5.24  Background noise of nongradiometric magnetic signal (left) and gradiometric magnetic signal (right).

The algorithm has a high noise immunity and high detection probabilities at SNR as low as −10 dB. Compared to benchmark detectors, this detector offers improved performance of 5 to 10 dB. Compared to benchmark detection algorithms, this detector performs significantly better. For instance, for FAR = 0.04 and SNR = −8 dB, the PD of an energy detector is 0.5, the PD of the MED is 0.75, and the PD of

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Figure 5.25  PSD of background noise of nongradiometric magnetic signal (left) and gradiometric magnetic signal (right).

the proposed detector is >0.95. This finding enables extending the detection range of magnetic measurement systems. Furthermore, in cases of OMD signals, this difference may grow substantially due to the low resilience of other detectors to temporary signals that affect the signal fidelity.



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Figure 5.26  Threshold value as a function of FAR of the proposed algorithm when the input is field data.

Figure 5.27  Detection performance: comparison of simulated data and field data.

However, the computing resources required for each detection window are significantly greater than the one required for energy and MED detectors, because spectral transformations are applied in each detection window.

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5.2  Magnetic Anomaly Localization Assuming the dipole approximation for describing the magnetic field measured by the magnetometers, the basic localization problem can be formulated by solving a system of equations with 6 unknowns: the 3 components of the position and of the magnetic moment vectors of the ferromagnetic object. In all relevant cases under investigation, two or more triaxial sensors measure the field as a function of time. As a result, recovering the position and the moment is basically an inverse problem. Several methods and approaches have been developed for that purpose since the pioneer work of Wynn [1]. These methods can be roughly classified to three categories: direct analytic or semianalytics algorithms [10–26], pure heuristic [18], [27–36] or statistical schemes [27, 37–39] and less common but interesting hybrid or combination of the two first types [27, 40, 41]. The frequent approach is the direct approach. It can be either analytical or numerical but it is always deterministic. In this category, the LMA solver has been proven to be the most precise and most efficient compared to other nonlinear optimization schemes. For this reason, we will describe in detail an example of a dedicated LMA method applied to the localization of a moving target by an array of sensors arranged according to a checkpoint configuration setup. On the opposite side, one finds the pure heuristic schemes, mainly the genetic algorithm (GA) [30, 31, 34, 42, 43] or the particle swarm optimization (PSO) [29]. Some works also reported the use of neural network (NN) [44] or simulated annealing [45]. The rationale behind these approaches is the highly nonlinearity of the dipole field expressions and the fact that the SNR value may be rather low in some real-world situations. This explains why deterministic solvers are often unsuitable for solving the equation with a good level of confidence in real, nonsterile field configurations. Since the GA approach is well established, we will dedicate the last part of this section to the description of a modified GA that solves in pseudo-real-time the dipole equations of a ferromagnetic object that moves within a large yet bounded domain. Intermediate approach based on statistic/probabilistic methods have also been tested. For instance, a numerical approximation to the recursive Bayesian filtering has been successfully performed for tracking magnetic dipole targets [37, 38]. Another interesting study compares the random complex algorithm (RCA) to the classical LMA scheme and came to the conclusion that RCA provides more accurate and more stable results with a better denoising capacity [39].



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Finally, few attempts to combine direct and heuristic approach (mainly the PSO scheme together with LMA) argue that such a hybrid scheme can take advantages of both methods. That is, utilize the flexibility of the heuristics to find, with good confidence, the first-order solution, which, in turn, serves as a guess point for the more robust and low CPU-cost deterministic solver [40]. Another field of investigation where the magnetic localization provided a most promising method to a solution is the localization of wireless capsule endoscopy. The endoscopic capsule appeared in 2000 and has since attracted unquestionable interest among gastroenterologists. Constituting a miniature endoscope that fits into a capsule swallowed by the patient, the endoscopic capsule progresses freely in the digestive tract and does not have the limitations of a traditional endoscope that can only explore part of the small intestine. The video-endoscopic M2A capsule was developed by Given Imaging (Yoqneam, Israel) and presented for the first time at the AGA Congress in May 2000. The principle was described in nature [46] and consists of a miniature endoscope comprising an optical system and an electronic chip capable of recording images according to a principle comparable to that of the charge coupled device (CCD) of video-electronic endoscopes. It also contains a light source and a transmission system that sends the images to sensors placed on the patient’s skin. The entire device is powered by two watch batteries. Obviously, the correct analysis and interpretation of the images sent by the capsule during the examination depends on the possibility to localize precisely the position and the orientation of the capsule as a function of time. In 2017, an extensive literature survey was conducted in which several different technologies were reviewed: electromagnetic localization, magnetic localization, video localization, and other localization techniques that include reflected marker, X-ray image-based, gamma ray-based, and other hybrid approaches [46]. The authors finally came to the following conclusion [46]: Magnetic localization is the most employed localization technique in the field of Wireless Capsule Endoscopy. It is more practical and reliable with less estimated error compared with other localization methods. Most of the studies mainly focus on the required number, arrangement, position, and array of the magnet and magnetic sensors. However, some challenging issues arise in this technique, such as interference between two or more magnetic sources and the influence of magnetic actuation while charging the capsule through the induction process.

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5.2.1  The Levenberg-Marquardt Localization Algorithm

This section presents a two-stage LMA particularly designed for a surveillance checkpoint configuration (see Chapter 2)1. This application requires precise (on-body) localization of ferromagnetic mass moving in a controlled passageway. The sensors can either be located on ground level or on a vertical mount (sand bags, doorways, and so forth). Obviously, response time is of great importance while on-body localization precision requirements are less stringent. Target location includes height and body flank, left or right, at which the object is carried. The section is outlined as follows: after a brief recall of the magnetic anomaly characterization theory, we describe the two-stage localization algorithm. The experimental setup is then presented together with a detailed event analysis. Finally, a summary of the results for various configurations of the checkpoint geometry is given. 5.2.1.1  The LMA in Checkpoint Application

The detection trigger is given by an appropriate detection algorithm as presented in Section 5.1. After detection, the localization algorithm is initiated. For the sake of clarity and self-content, we recall here that the magnetic field B created by a ferromagnetic target with a moment m at a distance r is given by (5.26):

B (m ,r ) =

3 µ0  (m ⋅ r )r m  − 3 4 π  R 5 R 

(5.26)

where m = (Mx, My, Mz) and R = |r|. In our sensors array the distance r is a function of the (fixed) sensors positions. Therefore, for each sensor, r = rsource – rsensor. By applying a simple rotation-translation transformation, a unique reference frame is applied to all calculations. It is shown in Figure 5.28. It is commonly accepted that if the distance between the source center and the sensor is at least 3 times larger than the largest dimension of the source, then the source may be considered as a magnetic dipole. Since our sources are approximately 100 to 150 mm long, it is easily shown from Figure 5.28 that our setup indeed fulfills this condition. The three components of the field can be expressed by (5.27):

1. Section reproduced from [10] with permission.



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Figure 5.28  Schematic view of the checkpoint. The relative distance between the sensors are indicated.



3x 2 − R 2  Bx    = µ0 3 yx  B y  4 πR 5  B  3zx z

3xy 2

3y − R 3zy

 Mx    3 yz  M y  3z 2 − R 2  M z  3xz

2

(5.27)

The Earth magnetic field DC is filtered out by an analog highpass filter embedded in the sensor itself. The problem of localization may be expressed in terms of a classical inverse problem. Given the magnetic field measured by a well-positioned sensor at time t, one wishes to find the source position and moment that generates the measured field. Strictly speaking, from a mathematical point of view, six unknowns (x, y, z, Mx, My, Mz) require at least six equations to solve the problem. The equations are provided by the data measured by two three-axial magnetic sensors. Nevertheless, for the high nonlinearity of the equations and SNR values that can be quite low, there are ambiguities that demand processing additional data. These data are obtained by adding sensors to the grid. This leads to a system of equations that become overdetermined. The least squares method may be efficiently applied to get approximate solutions to an overdetermined system. In our case, the nonlinearity of the magnetic dipole equations suggests the use of nonlinear least squares. This method often utilizes iterative procedures for the functional error minimiza-

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tion. For the algorithm to converge efficiently and provide a precise solution, it must suit the mathematical and physical characteristics of the problem. The simplest optimization algorithm is the gradient descent or steepest gradient method, whose principle is to start from a random point and then move in the direction of the steepest slope. Suppose we have a function f that depends on a vector of variables p. By applying a number of iterations, the algorithm converges to a solution that is a local minimum of a function f. An alternative way to decrease the number of iterations of an optimization algorithm is to use the second derivatives of f. Even if the gradient gives a direction towards which to move to find the minimum, it does not provide us with the step size. In the classical gradient descent, this step is a fixed coefficient, and in the adaptive variant, it can vary at each iteration. Since the second derivative is related to the radius of curvature of the function, it allows us to determine this step more finely. Let us show how. Assume that f has a quadratic form, formally expressed by:

f ( p ) = a + bT p + pT Cp

(5.28)

where a and b are column vectors of coefficients, xT is the transpose of the vector x and C is a symmetric matrix. We can find the extremum of the function: it is simply the point at which the derivative of f vanishes:

∇f = 0 ↔ b + 2Cp = 0

(5.29)

p = −2C −1b

(5.30)

that is:

provided that C is invertible. It is possible to locally approximate any function f at a point pi by a quadratic function by a Taylor series, using its first and second derivatives, and with (5.30) to determine the vector for the next iteration, by using an optimization algorithm more advanced than the gradient descent. However, the calculation of second derivatives can be very time-consuming, first because the number of second derivatives is the square of that of the first derivatives, and also because the expression of the second derivative of f can be quite complex. Actually, many algorithms use an approximation of the second derivatives computed from the first derivatives. Still, they keep the advantage of requiring much fewer iterations than a gradient descent. The LMA is one of these algorithms, and it applies to the particular case where f is a mean squared error. We can express it in the form:



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f ( p ) = g (x , p ) − y



201

)

2



(5.31)

where g denotes a function of two vectors x and p and · denotes the average calculated over a set of pairs (x, y). We assume here that g is a scalar function to simplify the notation, but the same steps can be made if is g a vector function. In the rest of this section, all the derivatives are a function of the vector p. This is because only this vector has to be varied to find the minimum of f. It is assumed that we have reached iteration number i, and that one tries to compute a new vector pi as a function of pi–1, such f (pi) as is closer to a local minimum of f. For this, we compute a quadratic approximation fˆ of f from a linear approximation gˆ of g around the point pi–1. By determining the point p at which the gradient of fˆ vanishes, we obtain: p = pi −1 − H −1d



(5.32)

where

d = g (x , pi −1 ) − y ∇g (x , pi −1 )

(

)

(5.33)



H = ∇g (x , pi −1 ) ∇g (x , pi −1 )

(5.34)

T

provided that H is invertible. The matrix H is an approximation of the Hessian of f, calculated from the gradient of g. The previous equation could be used in an optimization algorithm, which allows one to compute pi from pi–1. However, this is effective in practice only if g is actually close to a straight line around the point pi–1. In this case, (5.32) describes the Gauss-Newton (GN) algorithm formula for a nonlinear least square problem. Otherwise, this algorithm gives very poor results. Levenberg’s idea is therefore to use the quadratic GN approach in the areas where g is quasi-linear, and a steepest gradient descent in the other cases. The step of an iteration of this algorithm is calculated as follows:

pi = pi −1 − (H + λI ) d −1

(5.35)

When λ is small, this equation is equivalent to (5.32) and the new parameter vector is simply determined with the quadratic approximation of f. When λ is large, this equation is equivalent to:

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pi = pi −1 − λ−1d



(

)

= pi −1 − λ−1 g (x , pi −1 ) − y ∇g (x , pi −1 ) = pi −1 − (2 λ)

−1

f (x , pi −1 )

(5.36)

which indeed corresponds to a gradient descent expression. For intermediate values of λ, the algorithm is a mixture between the steepest descent and the quadratic approach based on the linear approximation of g. The coefficient λ is modified at each iteration, like in the adaptive steepest gradient descent method. If f (pi) decreases during the iteration, λis decreased (by dividing it by 10, for example), and one thus approaches the quadratic limit. On the contrary, if f (pi) increases, it means that we are in a region in which g is not very linear, and therefore we will increase λ (by multiplying by 10, for example) to go back closer to the gradient descent approach. This algorithm was further improved by Marquardt: the step of the iteration being now defined by:

pi = pi −1 − (H + λdiag (H)) d −1

(5.37)

The identity matrix has been replaced by the diagonal of H. The goal here is to modify the behavior of the algorithm in cases where λ is large, that is, when one is close to a gradient descent. This replacement allows us to move faster towards the directions where the gradient is more reliable to avoid spending many iterations on a plateau. This is called the LMA [47, 48]. It appears to be especially well adapted to our physical system [15, 23, 26, 28, 38, 49]. 5.2.1.2  The Tracking Algorithm

After detection trigger has been given by the detection algorithm, a relatively short portion of the signal is filtered and cleaned from bias and trends. The next step is to isolate the part of the path that corresponds to the sensor network crossing event contained by the entire data. If a distracting target interferes, it may then be truncated out of the analysis. Only the cleaned data will be processed for localization. Local false minimum is avoided by applying a simple annealing procedure together with the LMA. A first LMA cycle computes the target trajectory for the full six components vector (x, y, z, Mx, My, Mz). After that, a robust statistics analysis is applied to extract an average value of the moment components, assuming that these values remain constant during the crossing. Although this assumption is probably correct for the total magnetic moment, it not necessarily true for the component values (Mx, My, Mz). However, it is fair to assume that, during the



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few seconds that the path is processed, the ferrometallic object does not rotate significantly and each moment component remains constant. Then a second LMA cycle follows solving only the three path variables using the average values of the moment as constant numbers. Figure 5.28 presents the trajectory; X-coordinate represents the path progression, Y-coordinate represents the height at which the metallic object is carried and the sign of the Y-coordinate indicates the side on the body where the object is placed. Y and Z must be found with high precision to fit the application requirements. 5.2.1.3  Experimental Setups

The experimental setup simulates a checkpoint passageway by stacking sand bags and carefully placing three-axis sensitive magnetic fluxgates sensors (Bartington Mag634). Eight sensors were installed in the checkpoint to cover all relevant sensors configurations. Figure 5.28 presents a schematic description. The reference system places the axis origin (0, 0, 0) on the ground at equal distance between S2 and S6. Directions are shown in the figure. The checkpoint was designed using an excess of magnetometers allowing selection of certain sensors data according to a given required check point configuration. Two main configurations are considered: a vertical one, in which the sensors are located on a vertical mount as in a doorway, and a horizontal one, where all the sensors are positioned at the ground level. For each configuration the number of participating sensors can be dynamically selected according to the configuration. The sensors’ location and their symbols used in this work are shown. In the vertical setup, sandbags provide a doorway through which the inspected person is walking. The doorway width is 1.7m. This configuration utilizes six sensors [S2, S3, S4, S6, S7, and S8] located at positions shown in Figure 5.28. The localization algorithm processes the data of 2 to 6 sensors among the 6 available. We shall see that the performance varies with the number of sensors actually used. In the horizontal setup, the inspected person is passing through the sensors network while all sensors are on ground level. The sensors are placed on the ground creating a passage through which the person under test is walking. For this configuration, we consider four sensors, namely, S1, S2, S5, and S6 (see Figure 5.28). In this case, the localization algorithm uses from 2 to 4 sensors out of the 4 available. The experiment procedure is as follows: the inspected person carries a ferromagnetic object at a specific location on his body. The person stands 5m away from the gate. He then starts to walk towards the gate at an approximate velocity of 1 m/s and crosses the checkpoint. He stops 10m after

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the gate. During this time, the signals measured by the entire sensor network are recorded and stored on a PC. Although the data can be processed online, at this stage of the algorithm, development, and optimization, it was more convenient to analyze the signals outside the experiment field. 5.2.1.4  Tracking Demonstration

As a typical example, we show the detailed analysis of the localization of a small metallic object carried at the left side breast of the inspected person (about 1.3m above ground). Figure 5.29 presents the raw data measured by sensors S1, S2, S5, and S6. Units for the X-axis are sample points, at a sampling rate of 10 Hz, and the Y-axis units are nano-Teslas (nT). During the crossing event, a distracting target (a person carrying a large ferromagnetic object simulating an armed guard) is in motion 5m from the sensors array. The distracting target signal is clearly visible between 100 and 200 sample points particularly in Bx, and By in all sensor data. The first step is to clean the data from high-frequency noise (see Bx in sensor S2) straightforward searching procedure since we know the current position of the crossing person (by using online camera synchronized with the data acquisition system). The clean and isolated signals are presented in Figure 5.30.

Figure 5.29  Magnetic field raw data of the crossing of a small metallic object in the left armpit. Units for the x-axis are sample points, and units for the y-axis units are nano-Teslas (nT).



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Figure 5.30  Isolated relevant data of the object crossing as measured by sensors 1, 2, 5, and 6.

Only 26 data points were selected for the localization procedure to be processed by the LMA. Figure 5.31 shows the results of the first LMA round for the (x, y, z) coordinates and Figure 5.32 presents the calculated magnetic moment components at the same time. The crossing event occurs between 15 and 20 sample points and the average value of the moments is extracted within this interval. The results of the first iteration are then loaded by the second iteration procedure at which only position vector is being calculated. The modified vector can be seen in Figure 5.33 depicting straight and stable trajectories for the three coordinates. The X-axis starts about 2m North, crosses the gate amid 15 to 20 sample points, and terminates 1.5m South. A constant negative offset from the center of the path is clearly visible in the Y-coordinate, indicating that the ferromagnetic mass is probably located in the left flank of the body. Finally, the Z-coordinate shows an almost constant value of 1.27m above ground, which indeed corresponds to the chest height of the crossing person. 5.2.1.5  Algorithm Performance Analysis

A series of controlled tests were conducted to estimate the algorithm performance. Over 100 trajectories were recorded in which several parameters were varied: three different objects (overall mass of 1 kg, 0.7 kg, and 0.5 kg,

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Figure 5.31  Calculated trajectory after the first round of LMA.�

Figure 5.32  The calculated magnetic moment after the first round of LMA.



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Figure 5.33  The calculated trajectory after the second round of LMA.

respectively) were carried at three different heights, at the flank of the person crossing the checkpoint. In most cases, controlled distracting objects were moving around during the experiment. Results show that these distractors had very little effect on the algorithm performance. For the horizontal configuration, two subsystems were considered: two sensors (S1 and S5) and four sensors (S1, S2, S5, and S6). For the vertical configuration, two subsystems were designed: three sensors (S2, S3, and S6) and four sensors (S2, S3, S6, and S8). For each configuration, the cumulative distribution function (CDF) of the trajectory height (Z-coordinate) and flank (Y-coordinate) were calculated using the data provided by the localization algorithm. The height parameter results are shown in Table 5.5. The calculated height shows a deviation of about 250 mm from the true value while the flank position and 50-mm error for the flank position. A result of P% indicates that in P% of the cases the object was localized at a position that falls within 250 mm (for Z and 5 cm for Y) of the true position. The vertical configuration leads to better results than the horizontal. This is not surprising since the ferromagnetic object is crossing along the Zaxis, which is not well sampled by the horizontal configuration. Nevertheless, by using 4 sensors in the horizontal setup, one still obtains quite satisfactory estimation of the true object position.

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4 sensors 90% 85% 75%

Vertical Setup 3 sensors 100% 90% 100%

4 sensors 100% 100% 100%

The flank of the object was also evaluated by the localization algorithm. The probability distribution for finding the true side is shown in Table 5.6. Contrary to intuition, we notice that, for both configurations, the lower the number of sensors, the better the results. We suggest the following explanation. In the horizontal setup, adding two more sensors to the array does not change the symmetry relative to relevant axis (Y-axis). However, in the vertical setup, the 3 and 4 sensors setups exhibit a different symmetry relative to Y-axis. This could improve the results accuracy, specifically in the 3-sensor geometry. This hypothesis should be scrutinized in further testing. The magnetic moments provided by the algorithm are shown in Table 5.7. The object magnetic moment has been correctly estimated by the

�Table 5.6 The Flank Localization Probability Target Horizontal Setup 2 sensors Large 90% Medium 85% Small 70%

4 sensors 80% 80% 65%

Vertical Setup 3 sensors 95% 100% 95%

4 sensors 90% 100% 80%

Table 5.7 The Mean and Standard Deviation Calculated Values of the Moment for All Configurations Horizontal Setup

Vertical Setup

2 Sensors

4 Sensors

3 Sensors

4 Sensors

Large

6.4 ±1.8

6.9 ±1.5

6.3 ±1.3

6.1 ±1.3

Medium

2.6 ±2.0

2.0 ± 1.7

2.7 ±0.9

2.4 ±0.7

Small 1.1 ± 0.8

0. 8 ± 0.4

0.7 ± 0.4

0.7 ± 0.2

Units are

100*Am2.



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algorithm and can be classified in a large (>0.05 Am2), medium (>0.02 Am2), or small size (