**9.6. The Drop Yoga (\*binduyoga) and the Subtle Yoga (\*suks ¯ .mayoga)**

→ *D. ak¯ arn ¯ . ava* 15.280–283b.

de nas rang gi snying ga'i hu¯m. yongs su gyur pa las rdo rje sngon po rtse lnga pa lte ba la nyi ma la gnas pa'i hum¯ . gis byin gyis brlab po//

Now,2374 in his own heart [he should meditate on the letter] *hu¯m.* transformed into a vajra, [which is] dark blue [in color], five-pronged, and blessed (marked) with *hu¯m.* placed on a sun [disk] at the center.

de nas hu¯m. las 'od zer thogs **(D 34r3)** pa med pas sems can thams cad mkha' 'gro rgya mtshor byas pa khyer nas 'ongs tshur 'dus te/2375 hu¯m. las zhugs pas/ hu¯m. yig yongs su gyur pa las thig ler gyur te/ de'i nang du skad cig gis 'khor lo bcu gnyis kyi lha tshogs rten gyi dkyil 'khor dang bcas pa bsgoms te/ de'i gtso bo'i thugs **(D 34r4)** ka'i sa bon gyi 'od zer gyis dkyil 'khor gsal bar bsgom zhing/ dmigs pa 'di ni thig le'i rnal 'byor ro//

Subsequently, all-penetrating rays from the *hu¯m.* change all sentient beings into D. ak¯ ar¯ n. avas, take [them], carry [them], gather together, and come back into the *hu¯m.* . The *hu¯m.* is transformed and becomes a drop (*thig le*). Having visualized in that [drop] the deities of the twelve circles with the base man. d. ala (*rten gyi dkyil 'khor*) (viz., the whole man. d. ala), in an instant, he sees and meditates on the man. d. ala clearly with rays from the seed [letter *hu¯m.* ] in the heart of the Lord [residing] in that [same man. d. ala]. This is the Drop Yoga (\**binduyoga*).

2373 rang rang gi ] D; rang rang gis P ♦ nam mkha'i ] D; namkha'i P.

2374 Jayasena's interpretation of the discourse of the Drop and Subtle Yogas in the *D. ak¯ ar¯ n. ava* is largely different from Padmavajra's interpretation of it in his *Bohita¯*. For the version in the *D. ak¯ ar¯ n. ava* and Padmavajra's interpretation, see the *D. ak¯ ar¯ n. ava* (15.280–283b) and footnotes 940–942 in this monograph.

2375 thogs pa med pas ] D; thogs pa med pa P ♦ tshur 'dus ] D; chur 'dus P.

de nas yang thig le'i rnal 'byor gyis gtso bo'i thugs ka'i sa bon las 'od byung ste/ 'thon pa rnams kyis de bzhin gshegs pa thams cad la reg ste zhal gyi sgo nas zhugs te/ zhu **(D 34r5)** ba rdo rje'i bum par lhung ba las rdo rje nas kyi 'bru tsam gyi lte bar dkyil 'khor rdzogs pa thig le'i rnal 'byor gyi bar du lhag par mos par byas te/2376 sems brtan par ma gyur gyi bar du bsgom par bya'o//

Next, again by the Drop Yoga, [he] emits rays from the seed [*hu¯m.* ] in the heart of the Lord [of the man. d. ala in his heart]. Those emitted [rays] touch all tathagatas [in ¯ the external world] and enter [their bodies] through [their] mouths as gates. From a fluid, [which] falls into an adamantine vessel, a vajra of the size of a kernel of barley [emerges, and then] in the middle of [the vajra], a man. d. ala perfectly manifests.2377 He should have faith in [that man. d. ala] during the Drop Yoga. He should meditate on [that] as long as [his] mind is not firm.

de nas snying ga'i rdo rjer drangs te de la bstims la gzhag go// de dag gi sems rim **(D 34r6)** gyis brtan pa'i rtags su du ba la sogs pa mthong bar 'gyur te/2378 'di ni phra mo'i rnal 'byor ro//

Subsequently, [that man. d. ala] is attracted toward the vajra in the heart, is absorbed [into it], and remains [in it]. He will see smoke and so on2379 as signs, [showing that his] meditation on them is gradually becoming firm. This is the Subtle Yoga (\**suks ¯ .mayoga*).

## **9.7. Mantra Recitation**

de nas sngags kyi bzlas pa bya ste/ rkang pa bzhi bcu rtsa brgyad pa dang bcas pa'i rtsa ba'i sngags gtso bo'i hu¯m. yig las byung ste/ rang gi zhal nas yum gyi zhal du zhugs te/ rdo rje'i lam gyi yum gyi **(D 34r7)** pa dma nas 'khor ba'i rim pas bzlas pa'i skyon rnams dang bral bas dung chen gyi phreng ba bzung ste/2380 skabs yod na lha ma lus pa'i sngags kyang bzlas par bya'o//

→ *D. ak¯ arn ¯ . ava*, 15.283c–284a.

2376 byas te ] D; byas ste P.

2377 It is obscure how the rays and the fluid are connected. I speculate that the rays, which have entered the bodies of all tathagatas, draw out drops of bodily fluid from their bodies. The "adamantine vessel" ¯ is also unclear. Perhaps a practitioner visualizes a vessel in the sky, into which the fluid from the bodies of all tathagatas falls. ¯

2378 rim gyis ] D; rims kyis P.

2379 "Smoke and so on" (*du ba la sogs pa*) seems to indicate the meditative steps that start with smoking to ignite the inner fire at the center of the navel area. This is widely taught in the Buddhist Subtle Yoga (or Hat.ha yoga) systems.

2380 bzung ste ] D; gzung ste P.

Now, he should perform the recitation of mantra. The fundamental mantra accompanied by the forty-eight parts2381 emerges from the *hu¯m.* [in the heart] of the Lord. From his mouth, [the mantra] enters the mouth of [his] female consort, and then [comes] into the adamantine channel [of the Lord through his penis] from the lotus (womb) of the female consort;2382 with [this] ritual sequence of circulating [the mantra between the bodies of the Lord and his female consort], avoiding mistakes in recitation, he keeps the circle of great conch shell.2383 When time permits, he should also recite the mantras of all goddesses.


**4. Discussion**

Statistics in the OD matrix of residents' travels at di fferent hours show that, most of the spatial units follow the pattern of minimum tra ffic at 9:00, but there are also a few plots, such as Plot 27, that do not match the major pattern of travel numbers. Situated in the South Lake area and serving mainly residential functions, Plot 27 is one of the most congested areas in Wuhan during rush hours. A possible explanation is that the residents of the area choose to delay their departure time to 9:00 in order to avoid the tra ffic congestion period from 7:00 to 8:00, or they are stalled in tra ffic for too long and are considered as having not departed from the area in the statistics. This result, to some extent, verifies the significance of OD statistics and simulation by each hour: when the travel patterns of each space unit during the four hours change, tra ffic pressure at each intersection may also vary, and the causes behind these changes demand further analysis with the support of simulations. This is also why this study divides the unit time span of commuting travels into a one-hour basis.

In order to test the simulation results of the model for di fferent schemes, and combined with the analysis of the causes of congestion, the roads in the study area are optimized according to the Wuhan master plan. As a measure of optimization, a waterfront north–south road along the Yangtze River and the road to the South Lake area are planned (Figure 6b). The planned and optimized road network is simulated in the model and compared with the original one. In this simulation, it is assumed that the population in this area remains unchanged and so do the places of residence and work. Comparing the simulation results of the two schemes (Figure 9), it can be seen that the optimized scheme is evidently better than the scheme before optimization: first, tra ffic has been distributed to multiple road intersections instead of being concentrated at an intersection before the optimization. Second, duration of tra ffic congestion is significantly shortened, which means congestions can be alleviated quickly even if they do occur.

According to Gaode's Tra ffic Report on major cities in China, 81% of them su ffer from congestions during rush hours of residents' commuting [49]. Therefore, studying residents' commuting behavior as a starting point to address the wider problem of urban tra ffic congestion bears practical significance, not only for China but also for the world at large. The era of big data is coming. When it is less di fficult to acquire data, how to use them in urban research becomes an issue that calls for deliberation [40]. Previous studies prove that mobile phone call data can more accurately reflect the commuting features of urban residents. However, most studies focus on the overall analysis of cities on a macro scale and

the visual representations. Few studies are found on the micro scale dynamic analysis of residents' commuting behaviors, or on how commuting relates urban traffic.

However, as the sole data source used in the present study to understand residents' mobility, CDR data still has limitations because it is a relatively sparse data in recording the travel trajectory of residents. It is difficult to obtain the traffic mode (or speed) of residents' travel through statistical analysis. Thus, the specific correlations between commuter vehicles and mobile phone users are not discussed in the present study. Therefore, in follow-up studies, additional data sources such as bus card and traffic cameras at road intersections may facilitate the cross-examination of our research results or the setting rules of residents' commuting at a finer time-scale. Of course, these rely on the availability of data, which remain difficult to collect compared with other sources at present.

**Figure 9.** Comparison of traffic conditions before and after the optimization of the road network. (**a**) The traffic conditions of road intersections before optimization; (**b**) The traffic conditions of road intersections after optimization.


**Dr. Nawaz Mohamudally**

**1**

Section 1

Introduction

Department of Industrial Systems and Engineering, School of Innovative Technologies and Engineering, University of Technology, Mauritius

Section 1 Introduction

**3**

**Chapter 1**

**1. Introduction**

Introductory Chapter: Rising

*Daniel Almeida Marinho and Henrique Pereira Neiva*

Sport became a product of interest in different domains of the current society. Beyond the competition or leisure practice of physical activity, there is a large variety of interests regarding the sports contexts, specifically those with social and economic purposes. Using physical activity and sport, we can develop social values and those that today are linked with emerging market economies. These are nowadays mainly based on the notions of success, progress, and external conformity. There is a clear need for being properly informed and aware of this set of values for the optimal development life in society, benefiting from a clear involvement in sport. The business world has found in sports a new investment path, either through participating in scientific and technological development or by taking

The investment in technological development by specialized companies, in partnership with the sports institutions, ensures continuous development of athletes through the optimization of performance but also ensures more information to those who are spectators. Furthermore, this development also benefits non-competitive activities, with people now being able to follow and monitor their physical activities in real-time. The progress of technological devices leads to the optimization of sports performance, increases the efficiency of the training process, and creates reliable monitoring and increased motivation to practice for the "common" person. The user benefits either by increasing performance or by improved health status. Being seen as a potential of an investment, large companies are looking closely at these areas associated with sports and health, to guaranty better quality of life for all people. Also, this technological development allows for different experiences and contributes to what is called sports tourism. The search for new sensations, different events, and the visit to places that provide different experiences for users, increase the investment of cities and local towns in sports events and the different range of activities. The purpose is to increase tourist interest and

Technology is a resource in constant evolution, which increasingly provides greater utility in almost all activities developed by humans. Sport is no exception. Technology is closely associated with sports and physical activity, offering both new possibilities and improvements in existing activities. The technology allows us to obtain better results in sports performances, optimize training programs, and deepen the knowledge through a reliable and more precise output. It provides a

Interests in Sports Sciences

advantage of the profitability of the activity itself.

investment in local development.

**2. The technological development**


**Advanced Signal Processing in Wearable Sensors for Health Monitoring**

Editors

**Maysam Abbod Jiann-Shing Shieh**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Editors* Maysam Abbod Brunel University London UK

Jiann-Shing Shieh Yuan Ze University Taiwan

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Sensors* (ISSN 1424-8220) (available at: https://www.mdpi.com/journal/sensors/special issues/ signal processing wearable sensors).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

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**ISBN 978-3-0365-3887-7 (Hbk) ISBN 978-3-0365-3888-4 (PDF)**

© 2022 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.

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**4. Results and discussions**

To understand the effect that the PHLC would have on the flow downstream, a simulation was carried out considering two scenarios: a historical condition and the PHLC for dry and rainy seasons.

#### **4.1 PHLC functioning during the dry season**

**Table 2** presents the discharges observed in the lagoon system without and with dam during the dry season in the sections identified along the river. The negative signs in the average discharge indicate that the direction of the flow is toward the sea.

When the average flow rate is 0 m3 s−1, it indicates that during the analyzed period, the volume of water that entered the system is the same as the one that left. If this is different from 0 with a positive sign, it indicates that, during the studied period, the volume of water that enters is greater than the one that leaves and, on the contrary, if the sign is negative, it indicates that more water leaves than the one that enters the system. In this way, it is observed that with the operation of the PHLC, when the discharge in the San Pedro river increases, the volume of water that leaves the system toward the sea increases (**Table 3**).

Also, it was observed that when the levels in the lagoon system increase due to the operation of the PHLC during the dry season, the difference in levels between the system and the sea is modified by: (a) in high tide conditions the gradient is lower; therefore, the flow of water from the sea to the system decreases, a situation that is reflected in the reduction of the discharges that enter through the sea toward the lagoons; and (b) low tide condition, the gradient is greater; therefore, the flow of water toward the sea increases as well as in the flows that leave the system. In particular, in the Camichín mouth (SEC25), there is a major exchange of water between the sea and the lagoon system, leaving the Palapares mouth with a reduced exchange (SEC11). This water exchange defines the maximum, average, and minimum water levels at each site with respect to the two established conditions (without and with the PHLC) as shown in **Table 4**.

As highlighted in **Table 4**, in the four sites, the water level impact due to the operation of the dam is not significant. This is confirmed in **Table 5**, which shows the maximum, average, and minimum variations reached for each site.

The water level variation associated with the operation of the PHLC during the dry season showed that at the Camichín mouth there are no differences, but in the Palapares mouth and in both lagoons (Mexcaltitán and Toluca), there is an increase in the average level of the order of 1 cm. Thus, the average water level for the four sites is 0.18 m, 0.14 m, 1.0 m, and 1.6 m, respectively. In addition to the discharge and water

#### **Table 2.**

*Maximum (toward the system), reflux (toward the sea), and average discharges in the lagoon system, without and with dam during the dry season.*

*Interconnection among River Flow Levels, Sediments Loads and Tides Conditions and Its Effect… DOI: http://dx.doi.org/10.5772/intechopen.109175*

**Table 3.**

*Discharges variation for the operation of the PHLC in the dry season.*

#### **Table 4.**

*Maximum, average, and minimum water levels without and with dam during the dry season.*

#### **Table 5.**

*Variation of levels due to the operation of the P. H. Las Cruces (PHLC) during the flash flood season.*

levels data, other aspects to be considered in the operation of the lagoon system are velocity and salinity.

Reviewing velocities for the four sites, one can find that the main changes were observed in both mouths of Camichín and Palapares. In the first, the average flow velocity for both scenarios (without and with a dam) is around 0.933 m s−1. However, the maximum velocity was 2.097 m s−1 for current conditions and 2.099 m s−1 for the project conditions, so there would be an increase of the order of 0.002 m s−1, which is not significant. At the Palapares mouth, an average velocity of 0.840 m s−1 for the current condition, and 0.842 m s−1 for the project condition were observed, which shows an increase of the order of 0.002 m s−1, which is also not significant. The maximum velocity registered was 1.807 m s−1 for the current conditions and 1.808 m s−1 with a dam; thus, there is an increment due to the operation of the project of 0.001 m s−1.

In 2016, the Institute of Marine Sciences and Limnology (ICMyL) of the UNAM established an initial salinity of the system and concluded that there is an important change from almost 0 in the river to close to 30 PSU in the lagoons, and then a gradual increase from the lagoon to the sea, as shown in **Figure 6**. Salinity is measured in practical salinity units (PSU) or ppt (parts per thousand).

To estimate the salinity of the system under PHLC operation, simulations were carried out considering that the San Pedro river has fresh water, while the lagoon

**Figure 6.** *Salinity measured in the system in May 2014.*

system presents high salinity concentrations (29–32 ppt), and close to the sea the salinity is 33 ppt. Thus, for the dry season, the average salinity of 32.9 ppt was obtained in the mouths of Camichín and Palapares, as well as in the lagoons of Toluca and Mexcaltitán in current conditions, and with the dam, the salinity decreased by only 0.1 ppt. The latter is due to the PHLC operation, where it is expected a greater contribution of freshwater coming from the San Pedro river; thus, the salinity concentration decreases mainly in the Mexcaltitán and Toluca Lagoons, with average values of 11 and 3 ppt, respectively. Similarly, in the mouths of Camichín and Palapares, the salinity decreases on average by 0.1 and 0.8 ppt, respectively.


**3.1 Classification**

*Essential Oils - Bioactive Compounds, New Perspectives and Applications*

investigation on terpenoids.

**3. Terpenes and terpenoids**

techniques [102, 118–120]. The major volatile constituents may be classified into two main categories: terpenoids and polypropanoids [121–123]. We will focus our

Terpenes are defined as secondary metabolites with molecular structures containing carbon backbones of isoprene (2-methylbuta-1,3-diene) units [124]. Terpenes are synthetized in the cytoplasm of plant cells through the mevalonic acid pathway. Biochemical modification such as oxidation or rearrangement of terpenes produces the related terpenoids. Terpenoids are then oxygenated derivatives of hydrocarbon terpenes such as aldehydes, ketones, alcohols, acids, ethers, and esters [34]. Terpenoids are the largest classes of plants' natural products accounting for more than 40,000 individual compounds of both primary and secondary metabolisms been identified; to date, new terpenoids are being discovered every year [12, 124].

In general, terpenoids can be divided into at least four groups of compounds that

These types of natural lipids can be found in every class of living things, mainly in plants as constituents of EOs, and are therefore considered as the largest and structurally diverse group of natural products [125]. In general, only the hemiterpenoids, the monoterpenoids, and sesquiterpenoids are sufficiently volatile to be components of EOs. As widely acknowledged, the composition of EOs is mainly represented by mono-, sesqui-, and even diterpene hydrocarbons and their respec-

include true terpenes, steroids, saponins, and cardiac glycosides.

tive oxygenated derivatives [30, 126–128].

**88**

**Figure 14.**

**Figure 13.**

*Structure of few isolated hemiterpenes and hemiterpenoids.*

*Biosynthesis pathways of monoterpenes, sesquiterpenes, and diterpenes.*

Structurally, EO constituents typically have low molecular weights, which contribute to their high volatility. Terpenes are the most common constituents found in EOs [128]. They are made from isoprene units (several five carbon base units). Each group of terpenes arises from the head-to-tail condensation of a variable number of isoprene units. Variations in the number of isoprene unit repetitions, cyclisation reactions, and rearrangements are primarily responsible for their chemical and structural diversity. EOs consist of mainly monoterpenes (C10) and sesquiterpenes (C15) but also have diterpenes (C20), triterpenes (C30), and tetraterpenes (C40) at very low concentration with their oxygenated derivatives, respectively (**Figure 13**) [15, 102, 130].


**Induction Magnetometers Principle, Modeling and Ways of Improvement**

Christophe Coillot and Paul Leroy *LPP Laboratory of Plasma Physics France*

### **1. Introduction**

44 Magnetic Sensors – Principles and Applications

Zorlu, O.; Kejik, P. & Popovic, R.S. (2006). An orthogonal fluxgate-type magnetic

Kraus, L.; Butta, M. & Ripka P. (2010). Magnetic anisotropy and giant magnetoimpedance in NiFe electroplated on Cu wires. *Proceedings of EMSA 2010,* Bodrum, July 2010 Butta, M; Ripka, P.; Vazquez, M.; Infante, G. & L. Kraus (2010c). Microwire electroplated

Vol. 135, Iss. 1, (March 2007), pp. 43-49, ISSN 0924-4247

**Papers in conference proceedings**

July 2010

microsensor with electroplated Permalloy core. *Sensors and Actuators A-Physical*,

under torsion as core for coil-less fluxgate. Proceedings of EMSA 2010, Bodrum,

Induction sensors (also known as search coils), because of their measuring principle, are dedicated to varying magnetic field measurement. Despite the disadvantage of their size, induction magnetometer remains indispensable in numerous fields due to their sensitivity and robustness whether for natural electromagnetic waves analysis on Earth Lichtenberger et al. (2008), geophysics studies using electromagnetic toolsHayakawa (2007); Pfaffling (2007), biomedical applications Ripka (2008) or space physic investigations Roux et al. (2008). The knowledge of physical phenomena related to induction magnetometers (induction, magnetic amplification and low noise amplification) constitutes a strong background to address design of other types of magnetometers and their application.

Let us describe the magnetic field measurement in the context of space plasma physics. AC and DC magnetic fields are among the basic measurements you have to perform when you talk about space plasma physics, jointly with electric fields measurements and particle measurements . The magnetic field tells us about the wave properties of the plasma, as the electric field also does. At the current time, several kinds of magnetometers are used onboard space plasma physics missions: most often you find a fluxgate to measure DC fields and a searchcoil to measure AC fields. The searchcoil is better than the fluxgate above coarsely 1 Hz. In-situ measurements of plasma wave in Earth environment have been performed since many years by dedicated missions (ESA/CLUSTER (2002), NASA/THEMIS (2007)) and will continue with NASA/MMS mission, a 4 satellites fleet, which will be launched in 2014 with induction magnetometer (see Fig. 1) onboard each spacecraft. Earth is not the only planet with a magnetic field in the solar system. Search coils are for space plasmas physics and how their development still is a challenge.

### **2. Induction sensor basis**

Induction sensor principle derives directly from the Faraday's law:

$$e = -\frac{d\Phi}{dt} \tag{1}$$

where Φ = ' (*S*) −→*B* −→*dS* is the magnetic flux through a coil over a surface (*S*).

Fig. 2. Induction sensor using ferromagnetic core.

In this chapter, we will start with the description of magnetic amplification provided by a ferromagnetic core. Our description will rely on a simplified modelling of demagnetizing field (Chen et al. (2006); Osborn (1945)). Demagnetizing field energy modelling is still of great importance for micromagnetism studies of magnetic sensors. The search coil demagnetizing field effect study has the advantage to be a pedagogic application and to give magnitude scales to the designer through the apparent permeability concept which is at the source of the magnetic amplification of a ferromagnetic core but which also a way to produce welcome magnetic amplification Popovic et al. (2001). This last point remains a common denominator

Induction Magnetometers Principle, Modeling and Ways of Improvement 47

When a magnetic field is applied on a ferromagnetic material, this one becomes magnetised. This magnetization, linked to the magnetic field as expressed by eq. 3, implies an increase of

−→*M* = *χ*

The magnetic susceptibility *χ* can vary from unity up to several tens of thousands for certain ferromagnetic materials. When magnetic field line exit from the ferromagnetic core, a magnetic interaction appears Aharoni (1998) which is opposite to the magnetic field. This interaction, designed as demagnetizing field, is related to the shape of the core through the

By combining, eq. 3, 4 and 5 we can express the ratio between flux density outside of the ferromagnetic body (*Bn*) and inside (*Bext*). This ratio, designed as apparent permeability (*μapp*, cf. eq. 6), depends on the relative permeability of the ferromagnetic core (*μr*) and

Under the apparent simplicity of the previous formulas is hidden the difficulty to get the demagnetizing field factor. Some empirical formulas and abacus are given for rods in

= *μr*

−→*H* + −→*M*

−→*B* = *μ*0

demagnetizing coefficient tensor ||*N*|| and magnetization (cf. eq. 5).

the demagnetizing field factor (*Nx*,*y*,*z*) in the considered direction (*x*, *y*or *z*).

*μapp* = *Bn*

*Bext*

−→*H* (3)

−→*Hd* = −||*N*||−→*M* (5)

1 + *Nz*(*μr* − 1) (6)

(4)

**2.1 Ferromagnetic core**

of many magnetic sensors.

flux density (eq. 4).

Fig. 1. tri-axis induction magnetometer designed for a NASA mission

The voltage is proportionnal to the time derivative of the flux, thus, by principle, DC magnetic field can not be measured with a static coil. Higher will be the frequency, higher will be the ouput voltage (in the limit of the resonance frequency of the coil).

For (*n*) coils of section (*S*), into an homogenous induction magnetic field (*B*) equation 1 becomes:

$$e = -nS\frac{dB}{dt} \tag{2}$$

This single n turns coils, is designed as air coil induction sensor. As reminded in Tumanski (2007), an increase in sensitivity of air coil induction sensor can be obtained by increasing number of turns (*n*) or coil surface (*S*). In applications where the size, mass and performances of the sensor are not too stringent the air-coil induction sensor is an efficient way to get magnetic field variation measurement. Low costs and independancy from temperature variations are two other advantages of the air coil induction sensor.

An important way to improve the sensitivity of an induction sensor consists in using a ferromagnetic core. In that configuration, the ferromagnetic core acts as a magnetic amplifier and coils is wounded around the ferromagnetic core (Fig. 2)

Unfortunately, things are not so easy as ferromagnetic cores have some drawbacks, among them: non-linearity and saturation of the magnetic material, complexity of the design, tricky machining of the ferromagnetic core, whatever the material (ferrite, mu-metal) and the extra costs induced by this part in the design. But the ferromagnetic core still is worth being used as it dramatically increases the sensitivity of the sensor. Thereafter we will consider induction sensors using ferromagnetic cores even if most of our discussion is applicable to the air-coils too.

Fig. 2. Induction sensor using ferromagnetic core.

### **2.1 Ferromagnetic core**

2 Will-be-set-by-IN-TECH

Fig. 1. tri-axis induction magnetometer designed for a NASA mission

ouput voltage (in the limit of the resonance frequency of the coil).

variations are two other advantages of the air coil induction sensor.

and coils is wounded around the ferromagnetic core (Fig. 2)

becomes:

too.

The voltage is proportionnal to the time derivative of the flux, thus, by principle, DC magnetic field can not be measured with a static coil. Higher will be the frequency, higher will be the

For (*n*) coils of section (*S*), into an homogenous induction magnetic field (*B*) equation 1

*e* = −*nS dB*

This single n turns coils, is designed as air coil induction sensor. As reminded in Tumanski (2007), an increase in sensitivity of air coil induction sensor can be obtained by increasing number of turns (*n*) or coil surface (*S*). In applications where the size, mass and performances of the sensor are not too stringent the air-coil induction sensor is an efficient way to get magnetic field variation measurement. Low costs and independancy from temperature

An important way to improve the sensitivity of an induction sensor consists in using a ferromagnetic core. In that configuration, the ferromagnetic core acts as a magnetic amplifier

Unfortunately, things are not so easy as ferromagnetic cores have some drawbacks, among them: non-linearity and saturation of the magnetic material, complexity of the design, tricky machining of the ferromagnetic core, whatever the material (ferrite, mu-metal) and the extra costs induced by this part in the design. But the ferromagnetic core still is worth being used as it dramatically increases the sensitivity of the sensor. Thereafter we will consider induction sensors using ferromagnetic cores even if most of our discussion is applicable to the air-coils

*dt* (2)

In this chapter, we will start with the description of magnetic amplification provided by a ferromagnetic core. Our description will rely on a simplified modelling of demagnetizing field (Chen et al. (2006); Osborn (1945)). Demagnetizing field energy modelling is still of great importance for micromagnetism studies of magnetic sensors. The search coil demagnetizing field effect study has the advantage to be a pedagogic application and to give magnitude scales to the designer through the apparent permeability concept which is at the source of the magnetic amplification of a ferromagnetic core but which also a way to produce welcome magnetic amplification Popovic et al. (2001). This last point remains a common denominator of many magnetic sensors.

When a magnetic field is applied on a ferromagnetic material, this one becomes magnetised. This magnetization, linked to the magnetic field as expressed by eq. 3, implies an increase of flux density (eq. 4).

$$
\overrightarrow{M} = \chi \overrightarrow{H} \tag{3}
$$

$$
\overrightarrow{B} = \mu\_0 \left( \overrightarrow{H} + \overrightarrow{M} \right) \tag{4}
$$

The magnetic susceptibility *χ* can vary from unity up to several tens of thousands for certain ferromagnetic materials. When magnetic field line exit from the ferromagnetic core, a magnetic interaction appears Aharoni (1998) which is opposite to the magnetic field. This interaction, designed as demagnetizing field, is related to the shape of the core through the demagnetizing coefficient tensor ||*N*|| and magnetization (cf. eq. 5).

$$
\overrightarrow{H\_d} = -||N||\overrightarrow{M}\tag{5}$$

By combining, eq. 3, 4 and 5 we can express the ratio between flux density outside of the ferromagnetic body (*Bn*) and inside (*Bext*). This ratio, designed as apparent permeability (*μapp*, cf. eq. 6), depends on the relative permeability of the ferromagnetic core (*μr*) and the demagnetizing field factor (*Nx*,*y*,*z*) in the considered direction (*x*, *y*or *z*).

$$
\mu\_{app} = \frac{B\_{\rm n}}{B\_{\rm ext}} = \frac{\mu\_{\rm r}}{1 + N\_{\rm z}(\mu\_{\rm r} - 1)}\tag{6}
$$

Under the apparent simplicity of the previous formulas is hidden the difficulty to get the demagnetizing field factor. Some empirical formulas and abacus are given for rods in

will be reduced. To take advantage of this improvment, design of the sensor by means of

Induction Magnetometers Principle, Modeling and Ways of Improvement 49

Assuming a coil of N turns wounded on single or multi layers and assuming negligible abundance coefficient. The coil exhibits a resistance which can be computed with eq. 9.

*d* + *N* (*dw* + *t*)

where *ρ*is the electrical resistivity, *dw* is the wire diameter, *t* is the thickness of wire insulation,

This coil exhibits a self inductance which can be expressed 10 in case of an induction sensor using ferromagnetic core and using Nagaoka formulas in case of an air-coil induction sensor

where (*S*)is the ferromagnetic core section, *μo*is the vacuum permeability and *λ* = (*l*/*lw*)

is a correction factor proposed in (Lukoschus (1979)). Designers should notice that practically relative permeability is complex and apparent permeability too, the imaginary part corresponds to loss in the ferromagnetic core. At low field, main mechanisms involved in the imaginary part of the permeability, will be damping of magnetization and domain wall motion in ferrite ferromagnetic coreLebourgeois et al. (1996) while eddy current will be significant in electrically conductive ferromagnetic material like NiFe type. The imaginary part of permeability should be taken into account in the inductance modelling, and also in the following of the modelling detailed in this chapter, since resonance frequency becomes higher

The difference electric potential between each turn of the coil implies an electrostatic field and as a consequence a capacitance because of the stored electric energy between the turns of the coil. For a single layer winding, the capacitance between conductors should be considered. For a multi layer winding the capacitance between layer will be preponderant. In that case and assuming a discontinuous winding (presented on Fig. 5) equation 11 allows to estimate the capacitance of the coil. The capacitance of the continuous winding (see Fig. 5) will not be derived here. Because of his bad performances (explained at the end of this paragraph) we don't recommend its use for induction sensor design. The computation of winding capacitances in continuous and discontinuous windins is detailed pages 254-258 of Ferrieux

where *εo*and *εr* are respectively the vacuum permittivity and the relative permittivity (of the wire insulator), (*nl*)is the number of layers and the other parameters were defined before. Now, for a given sensor, the elements of the induction sensor electrokinetic modelling (Fig. 4)

*N*2*μ*0*μappS*

*d*2 *w* 2 /*Lw*

*l* (10)

*t*(*nl* − 1) (*d* + 2*nl*(*dw* + *t*)) (11)

(9)

2/5

mathematical optimization (Coillot et al. (2007)) is recommended.

*R* = *ρN*

*d* is the diameter on which coil is wounded and *Lw*is the length of the coil.

*L* = *λ*

**2.2 Electrokinetic's representation**

( (n.d.a)).

than few kHz.

can be fully determined.

& Forest (1999) which is unfortunaley in French.

*C* = *πεoεrlw*

Fig. 3. Ferromagnetic core using flux concentrators.

Bozorth & Chapin (1942) while analytic formulas for general ellipsoids, where demagnetizing coefficient is homogeneous into the volume, are computed in Osborn (1945). However in common shapes of ferromagnetic cores demagnetizing coefficients are inhomogenenous and numerical simulation could be helpfull to guide the design. As explained in Chen et al. (2006) one should distinguish between fluxmetric and magnetometric demagnetizing coefficient. The apparent permeability of concerns should be the one along the coil. To take advantage of the maximum flux, the coil should be localized into 70-90% of the total length depending on the length to diameter ratio of the core Bozorth & Chapin (1942). In the case of rods or cylinders with high aspect ratio (i.e m=length/diameter>>1), the formulas for ellipsoid given in Osborn (1945) allows to get a good estimate of demagnetizing factor in the longitudinal direction (*z*) :

$$N\_z = \frac{1}{m^2} \left(Ln\left(2m\right) - 1\right) \tag{7}$$

#### **2.1.1 Increase of magnetic amplification using flux concentrator**

A way to increase magnetic amplification is to used magnetic concentrators at the ends of the ferromagnetic core (3). Let us consider a ferromagnetic core using magnetic concentrators of length (*L*), center diameter (*d*) and ends diameter (*D*). The classical formula of apparent permeability 6 becomes eq. 8. For a hollow core an extension of this formulas is given in Grosz et al. (2010).

$$
\mu\_{app} = \frac{B\_{\rm n}}{B\_{\rm ext}} = \frac{\mu\_{\rm r}}{1 + \mathcal{N}\_{\rm z} \left( {\rm L/D} \right) \frac{d^2}{D^2} (\mu\_{\rm r} - 1)}\tag{8}
$$

For a given set of length, diameter and magnetic material, an increase of magnetic concentrators diameter will lead to a significant increase of apparent permeability (we report increase of apparent permeability higher than 50% in (Coillot et al. (2007)). This increase allows to reduce the number of turns of the winding, all things being equal. Thus the mass of the winding and as a consequence the thermal noise due to the resistance of the winding will be reduced. To take advantage of this improvment, design of the sensor by means of mathematical optimization (Coillot et al. (2007)) is recommended.

### **2.2 Electrokinetic's representation**

4 Will-be-set-by-IN-TECH

Bozorth & Chapin (1942) while analytic formulas for general ellipsoids, where demagnetizing coefficient is homogeneous into the volume, are computed in Osborn (1945). However in common shapes of ferromagnetic cores demagnetizing coefficients are inhomogenenous and numerical simulation could be helpfull to guide the design. As explained in Chen et al. (2006) one should distinguish between fluxmetric and magnetometric demagnetizing coefficient. The apparent permeability of concerns should be the one along the coil. To take advantage of the maximum flux, the coil should be localized into 70-90% of the total length depending on the length to diameter ratio of the core Bozorth & Chapin (1942). In the case of rods or cylinders with high aspect ratio (i.e m=length/diameter>>1), the formulas for ellipsoid given in Osborn (1945) allows to get a good estimate of demagnetizing factor in the longitudinal direction (*z*) :

A way to increase magnetic amplification is to used magnetic concentrators at the ends of the ferromagnetic core (3). Let us consider a ferromagnetic core using magnetic concentrators of length (*L*), center diameter (*d*) and ends diameter (*D*). The classical formula of apparent permeability 6 becomes eq. 8. For a hollow core an extension of this formulas is given in

> = *μr* 1 + *Nz* (*L*/*D*) *d*�2

For a given set of length, diameter and magnetic material, an increase of magnetic concentrators diameter will lead to a significant increase of apparent permeability (we report increase of apparent permeability higher than 50% in (Coillot et al. (2007)). This increase allows to reduce the number of turns of the winding, all things being equal. Thus the mass of the winding and as a consequence the thermal noise due to the resistance of the winding

*m*2 (*Ln* (2*m*) − 1) (7)

*D*2 (*μr* − 1) (8)

*Nz* = 1

**2.1.1 Increase of magnetic amplification using flux concentrator**

*μapp* = *Bn*

*Bext*

Grosz et al. (2010).

Fig. 3. Ferromagnetic core using flux concentrators.

Assuming a coil of N turns wounded on single or multi layers and assuming negligible abundance coefficient. The coil exhibits a resistance which can be computed with eq. 9.

$$R = \rho N \frac{\left(d + N \left(d\_w + t\right)^2 / L\_w\right)}{d\_w^2} \tag{9}$$

where *ρ*is the electrical resistivity, *dw* is the wire diameter, *t* is the thickness of wire insulation, *d* is the diameter on which coil is wounded and *Lw*is the length of the coil.

This coil exhibits a self inductance which can be expressed 10 in case of an induction sensor using ferromagnetic core and using Nagaoka formulas in case of an air-coil induction sensor ( (n.d.a)).

$$L = \lambda \frac{N^2 \mu\_0 \mu\_{app} S}{l} \tag{10}$$

where (*S*)is the ferromagnetic core section, *μo*is the vacuum permeability and *λ* = (*l*/*lw*) 2/5 is a correction factor proposed in (Lukoschus (1979)). Designers should notice that practically relative permeability is complex and apparent permeability too, the imaginary part corresponds to loss in the ferromagnetic core. At low field, main mechanisms involved in the imaginary part of the permeability, will be damping of magnetization and domain wall motion in ferrite ferromagnetic coreLebourgeois et al. (1996) while eddy current will be significant in electrically conductive ferromagnetic material like NiFe type. The imaginary part of permeability should be taken into account in the inductance modelling, and also in the following of the modelling detailed in this chapter, since resonance frequency becomes higher than few kHz.

The difference electric potential between each turn of the coil implies an electrostatic field and as a consequence a capacitance because of the stored electric energy between the turns of the coil. For a single layer winding, the capacitance between conductors should be considered. For a multi layer winding the capacitance between layer will be preponderant. In that case and assuming a discontinuous winding (presented on Fig. 5) equation 11 allows to estimate the capacitance of the coil. The capacitance of the continuous winding (see Fig. 5) will not be derived here. Because of his bad performances (explained at the end of this paragraph) we don't recommend its use for induction sensor design. The computation of winding capacitances in continuous and discontinuous windins is detailed pages 254-258 of Ferrieux & Forest (1999) which is unfortunaley in French.

$$\mathcal{C} = \frac{\pi \varepsilon\_o \varepsilon\_r l\_w}{t(n\_l - 1)} \left( d + 2n\_l (d\_w + t) \right) \tag{11}$$

where *εo*and *εr* are respectively the vacuum permittivity and the relative permittivity (of the wire insulator), (*nl*)is the number of layers and the other parameters were defined before.

Now, for a given sensor, the elements of the induction sensor electrokinetic modelling (Fig. 4) can be fully determined.

Fig. 6. Principle & bloc-diagram of induction sensor using feedback flux

transfer function of induction sensors.

how the transfer function is flattended.

according to 14.

**2.3.2 Induction sensor using current amplifier**

**2.3.1 Induction sensor using feedback flux**

*T*(*jω*) = *Vout*

be presented. They give an efficient way to suppress the first resonance and to flatten the

Induction Magnetometers Principle, Modeling and Ways of Improvement 51

With a feedback flux added to the induction sensor, as presented on figure 6, the resonance of the induction sensor can be suppressed. A representation of the sensor using bloc diagram makes easy the computation of the transmittance of the feedback flux induction sensor:

> *B* = −*jωNSμappG* (1 − *LCω*2) + *jω*

where *M* is the mutual inductance between the measurement winding and the feedback winding, *Rf b*is the feedback resistance and *G* = (1 + *R*2/*R*1) is the gain of the amplifier.

Transfer function of induction sensor using feedback flux is illustrated in 8. It demonstrates

Transmittance of the current amplifier (also designed as transimpedance) is expressed

As current amplifier have a propensation to oscillate a capacitance in parallel is needed to

1 + *jRf Cf ω*

stabilize it (n.d.b). In such case, the transmittance of the current amplifier becomes :

*Vout* = − *Rf*

 *RC* +

*GM*

*Vout* = −*Rf Iin* (14)

*Iin* (15)

*Rf b* (13)

Fig. 5. Induction sensor transfer function for continuous and discontinuous winding (figure extracted from Moutoussamy (2009))

Then, the transfer function between the output of the seameasurable voltage and flux density can be expressed using equation 12.

$$\frac{V\_{out}}{\underline{B}} = \frac{-j\omega N S \mu\_{app}}{(1 - L\mathcal{C}\omega^2) + j\mathcal{R}\mathcal{C}\omega} \tag{12}$$

It shows that induced voltage will increase with frequency until the occurence of the resonance of the induction sensor ( √ 1 (*LC*)). Even if the gain at the resonance can be extremely high it is considered as a drawback since resonance can saturate the output of the electronic conditionner. Moreover the output voltage decreases after the resonance (5). It can be noticed that, in case of continuous winding, multiple resonances appear beyond the main resonance frequency (Seran & Fergeau (2005) & Fig. 5). These multiple resonances do not appear when discontinuous winding strategy Moutoussamy (2009) is implemented. The absence of multiple resonance using discontinuous winding could be due to the homogeneous distribution of the electric field between the layers of the winding.

The methods presented in the next chapter (feedback flux and current amplifier) give an efficient way to suppress the first resonance and to flatten the transfer function of induction sensors.

#### **2.3 Electronic conditioning of induction magnetometers**

In this section we will give some details about the classical electronic conditioning associated to induction sensorsTumanski (2007). The feedback flux design and the current amplifier will 6 Will-be-set-by-IN-TECH

Fig. 5. Induction sensor transfer function for continuous and discontinuous winding (figure

Then, the transfer function between the output of the seameasurable voltage and flux density

*B* = −*jωNSμapp*

It shows that induced voltage will increase with frequency until the occurence of the resonance

it is considered as a drawback since resonance can saturate the output of the electronic conditionner. Moreover the output voltage decreases after the resonance (5). It can be noticed that, in case of continuous winding, multiple resonances appear beyond the main resonance frequency (Seran & Fergeau (2005) & Fig. 5). These multiple resonances do not appear when discontinuous winding strategy Moutoussamy (2009) is implemented. The absence of multiple resonance using discontinuous winding could be due to the homogeneous

The methods presented in the next chapter (feedback flux and current amplifier) give an efficient way to suppress the first resonance and to flatten the transfer function of induction

In this section we will give some details about the classical electronic conditioning associated to induction sensorsTumanski (2007). The feedback flux design and the current amplifier will

(1 − *LCω*2) + *jRCω*

(*LC*)). Even if the gain at the resonance can be extremely high

(12)

*Vout*

distribution of the electric field between the layers of the winding.

1

**2.3 Electronic conditioning of induction magnetometers**

Fig. 4. Représentation électrocinétique du fluxmètre

extracted from Moutoussamy (2009))

can be expressed using equation 12.

of the induction sensor ( √

sensors.

Fig. 6. Principle & bloc-diagram of induction sensor using feedback flux

be presented. They give an efficient way to suppress the first resonance and to flatten the transfer function of induction sensors.

### **2.3.1 Induction sensor using feedback flux**

With a feedback flux added to the induction sensor, as presented on figure 6, the resonance of the induction sensor can be suppressed. A representation of the sensor using bloc diagram makes easy the computation of the transmittance of the feedback flux induction sensor:

$$\underline{T}(j\omega) = \frac{V\_{out}}{\underline{B}} = \frac{-j\omega N \text{S} \mu\_{app} G}{(1 - L\text{C}\omega^2) + j\omega \left(R\text{C} + \frac{GM}{R\_{fb}}\right)}\tag{13}$$

where *M* is the mutual inductance between the measurement winding and the feedback winding, *Rf b*is the feedback resistance and *G* = (1 + *R*2/*R*1) is the gain of the amplifier.

Transfer function of induction sensor using feedback flux is illustrated in 8. It demonstrates how the transfer function is flattended.

#### **2.3.2 Induction sensor using current amplifier**

Transmittance of the current amplifier (also designed as transimpedance) is expressed according to 14.

$$V\_{\rm out} = -R\_f I\_{\rm in} \tag{14}$$

As current amplifier have a propensation to oscillate a capacitance in parallel is needed to stabilize it (n.d.b). In such case, the transmittance of the current amplifier becomes :

$$V\_{out} = -\frac{R\_f}{1 + jR\_fC\_f\omega}I\_{in} \tag{15}$$

Characteristics Value Ferromagnetic core length 12cm Ferromagnetic core diameter 4mm Diabolo ends diameter 12mm Relative permeability 3500 Turns number 2350 Inductance 0.306H Resistance 48 Capacitance (incl. 20cm cable) 150pF

Induction Magnetometers Principle, Modeling and Ways of Improvement 53

Fig. 8. Transmittance of induction sensor using either a transimpedance or a feedback flux

**2.4 Noise Equivalent Magnetic Induction (NEMI) of induction magnetometers**

the transmitances reported for the two types of electronic conditionner reported, i.e. the feedback amplifier on one side (Coillot et al. (2010); Seran & Fergeau (2005); Tumanski (2007)) and transimpedande amplifier on the other side (Prance et al. (2000)). The shape of the transmittance can be a reason to have a preference for transimpedance amplifier in some applications especially as the flatness of the transmittance can be increased up to 6 decades by using a compensation network (Prance et al. (2000)), which is simply an integration of the

√*Hz* is defined as the output noise

Table 1. induction sensor design example

(analytic modelling versus Pspice simulation).

Noise equivalent magnetic induction, expressed in *T*/

related to the transfer function of the induction magnetometer 19.

induction signal.

Fig. 7. Principle & bloc-diagram of induction sensor using transimpedance amplifier

In the case of an induction sensor the relation between the induced voltage and the current flowing througth the induction sensor is expressed following:

$$\underline{\varepsilon} = (\mathcal{R} + jL\omega)\underline{I\_{\text{in}}} \tag{16}$$

From the previous equation, is is obvious to obtain a block diagram representation of the induction magnetometer using transimpedance amplifier (7).

Finally, using the bloc-diagram representation 7 the transmittance (*T*(*jω*)) of the induction magnetometer is expressed:

$$\frac{\frac{\mathcal{V}\_{\text{out}}}{\underline{\mathcal{B}}}}{\underline{\mathcal{B}}} = \frac{j \text{NS} \mu\_{app} \omega}{\mathcal{R} + j \text{L}\_1 \omega} \times \frac{\mathcal{R}\_f}{1 + j \mathcal{R}\_f \mathcal{C}\_f \omega} I\_{\text{in}} \tag{17}$$

which finally can be written:

$$\underline{T}(j\omega) = \frac{V\_{out}}{\underline{B}} = \frac{R\_f}{R} \frac{j\omega \text{NS}\mu\_{app}}{1 + \left(\frac{L + R\_f \mathbf{C}\_f}{R}\right) j\omega - \frac{R\_f}{R} \text{LC}\_f \omega^2} \tag{18}$$

The response of the transimpedance amplifier can be either computed using the transmittance equation or computed by simulation software. The comparison of equation 18 with a pspice simulator gives a very good agreement even if the real tansmittance of the operational amplifier modifies sligthly the cut-off frequency. Let us consider an induction sensor designed for VLF measurements, using a 12cm length ferromagnetic core (ferrite with a shape similar to the diabolo juggling prop) and with 2350 turns of 140μm copper wire. The parameters of the electrokinetics can thus be obtained. The parameters of the sensor are summarised in Table 1.

With Rf=470k and Cf=3.3pF for the transimpedance amplifier and Go=100 & Rfb=4,7k for the feedback flux amplifier. We assume the bandwidth of the amplifiers are higher than needed. The transmittance of the transimpedance amplifier exhibits a nice flat bandwidth on 4 decades while the one of the feedback flux is flat on less than 2 decades (Fig. 8).

The comparison between analytic modelling formulaes and pspice simulations validates the analytic formulas proposed respectively for feedback flux induction sensor and transimpedance induction sensor. The analytic formulas are in good agreements with

Table 1. induction sensor design example

8 Will-be-set-by-IN-TECH

Fig. 7. Principle & bloc-diagram of induction sensor using transimpedance amplifier

flowing througth the induction sensor is expressed following:

induction magnetometer using transimpedance amplifier (7).

*Vout*

*T*(*jω*) = *Vout*

*B* = *jNSμappω R* + *jL*1*ω*

1 +

4 decades while the one of the feedback flux is flat on less than 2 decades (Fig. 8).

*B* = *Rf R*

magnetometer is expressed:

which finally can be written:

In the case of an induction sensor the relation between the induced voltage and the current

From the previous equation, is is obvious to obtain a block diagram representation of the

Finally, using the bloc-diagram representation 7 the transmittance (*T*(*jω*)) of the induction

×

 *L* + *Rf Cf R*

The response of the transimpedance amplifier can be either computed using the transmittance equation or computed by simulation software. The comparison of equation 18 with a pspice simulator gives a very good agreement even if the real tansmittance of the operational amplifier modifies sligthly the cut-off frequency. Let us consider an induction sensor designed for VLF measurements, using a 12cm length ferromagnetic core (ferrite with a shape similar to the diabolo juggling prop) and with 2350 turns of 140μm copper wire. The parameters of the electrokinetics can thus be obtained. The parameters of the sensor are summarised in Table 1. With Rf=470k and Cf=3.3pF for the transimpedance amplifier and Go=100 & Rfb=4,7k for the feedback flux amplifier. We assume the bandwidth of the amplifiers are higher than needed. The transmittance of the transimpedance amplifier exhibits a nice flat bandwidth on

The comparison between analytic modelling formulaes and pspice simulations validates the analytic formulas proposed respectively for feedback flux induction sensor and transimpedance induction sensor. The analytic formulas are in good agreements with

*Rf* 1 + *jRf Cf ω*

*jωNSμapp*

*jω* − *Rf*

*R LCf ω*2

*e* = (*R* + *jLω*)*Iin* (16)

*Iin* (17)

(18)

Fig. 8. Transmittance of induction sensor using either a transimpedance or a feedback flux (analytic modelling versus Pspice simulation).

the transmitances reported for the two types of electronic conditionner reported, i.e. the feedback amplifier on one side (Coillot et al. (2010); Seran & Fergeau (2005); Tumanski (2007)) and transimpedande amplifier on the other side (Prance et al. (2000)). The shape of the transmittance can be a reason to have a preference for transimpedance amplifier in some applications especially as the flatness of the transmittance can be increased up to 6 decades by using a compensation network (Prance et al. (2000)), which is simply an integration of the induction signal.

### **2.4 Noise Equivalent Magnetic Induction (NEMI) of induction magnetometers**

Noise equivalent magnetic induction, expressed in *T*/ √*Hz* is defined as the output noise related to the transfer function of the induction magnetometer 19.

Fig. 9. Feedback flux induction sensor and noise sources.

*PSDePA* = *e*

*PSDiPA* (*V*2/*Hz*) = (|*Z*|*iPA*)

where |*Z*| is the impedance modulus of induction sensor :

(while equation seems coming from sky in some papers).

By considering the noise contribution of the impedance of the feedback:

preamplifier is expressed:

2 *PA* *G*2

(1 − *LCω*2)

2

 *R*2

(1 − *LCω*2)

Finally we express the total output noise contribution as a sum of the different contributions:

We can notice that the total ouput noise expressions presented here have only few differences to the one reported in Seran & Fergeau (2005). The objectives of this chapter being double: get the analytic modelling and expose, in a pedagogic way, the method of total noise computation

1 − *LCω*22 + (*RCω*)

*RCω* +

2 +

2 + (*RCω*)

*PSDout*(*f*) = *PSDR* + *PSDePA* + *PSDiPA* + *PSDRf* (26)

2 +

Thirdly, the contribution to the PSD of input current noise (converted in noise voltage) of the

Induction Magnetometers Principle, Modeling and Ways of Improvement 55

*G*2

(1 − *LCω*2)

+ (*Lω*) 2  2

*Rf b* 2 (22)

*Rf b* 2 (23)

2

2 (24)

*GMω*

*GMω*

1 − *LCω*22 + (*RCω*)

*RCω* +

*PSDRf* = 4*kTRf* (25)

$$NEMI(f) = \sqrt{\frac{PSD\_{out}(f)}{T(j\omega)^2}}\tag{19}$$

NEMI is the key parameter for induction magnetometer performances. In most of the design reported the NEMI is computed at a given frequency (often at a low frequency to simplify the problem) while the computation of NEMI on the whole spectrum could permit to adjust the design in case of wide band measurments. The way to compute the NEMI consists in adding different relevant noise sources on the magnetometer schematic. Then, we must consider the transmittance seen by each noise contribution and compute the output PSD. Finally the resulting output PSD will be divided by the transmittance to get the NEMI 19.

For this study we assume that equivalent input noises (both for voltage and current) at positive and negative inputs of the operationnal amplifier are identical and that transmittances of positive and negative inputs signals are closed. Thus, we consider an equivalent voltage input noise (*ePA*) and an equivalent current input noise (*iPA*). Since induction sensors are high impedance, amplifier with low input current noise should be preferred, this is the reason why this parameter is often neglected in the modelling. For detailed considerations, use of simulation software is recommended. The purpose of the analytic modelling being to have a friendly tool to help designer.

#### **2.4.1 NEMI of induction magnetometer using feedback flux amplifier**

Let us consider first the feedback flux induction magnetometer. Noise sources coming from the sensor and from the preamplifier are reported on the schematic presented on Fig. 9. Using the block diagram representation we can easily get the transmittance "seen" by each noise contribution. Main noise contribution from the induction sensor is the thermal resistance of the winding. Noises sources coming from big volume ferromagnetic core can be neglected at low resonance frequency (Seran & Fergeau (2005)) while Barkhausen noise coming from the magnetic domain displacment can be neglected until reversible magnetization is considered (typ. few mT). We neglect noise contribution from R1//R2 equivalent resistor since this noise can be reduced by design (choosing a small enough R1 resistance).

From the bloc diagram (9), we can easily express the output noise contribution of each noise source, because of the relation between input and output power spectrum density (PSD) through a system characterized by its transmittance (*T*(*jω*)):

$$PSD\_{out} = \underline{|T(j\omega)^2|} PSD\_{in} \tag{20}$$

First, the PSD of the noise coming from the sensor comes essentially from the resistance of the sensor and can be expressed directly as:

$$PSD\_R = 4kTR \xrightarrow[\left(1 - L\text{C}\omega^2\right)^2 + \left(RC\omega + \frac{GM\omega}{R\_{fb}}\right)^2\right] \tag{21}$$

Secondly, the contribution to the PSD of the input voltage noise of the preamplifier (*ePA*) is:

10 Will-be-set-by-IN-TECH

NEMI is the key parameter for induction magnetometer performances. In most of the design reported the NEMI is computed at a given frequency (often at a low frequency to simplify the problem) while the computation of NEMI on the whole spectrum could permit to adjust the design in case of wide band measurments. The way to compute the NEMI consists in adding different relevant noise sources on the magnetometer schematic. Then, we must consider the transmittance seen by each noise contribution and compute the output PSD. Finally the

For this study we assume that equivalent input noises (both for voltage and current) at positive and negative inputs of the operationnal amplifier are identical and that transmittances of positive and negative inputs signals are closed. Thus, we consider an equivalent voltage input noise (*ePA*) and an equivalent current input noise (*iPA*). Since induction sensors are high impedance, amplifier with low input current noise should be preferred, this is the reason why this parameter is often neglected in the modelling. For detailed considerations, use of simulation software is recommended. The purpose of the

Let us consider first the feedback flux induction magnetometer. Noise sources coming from the sensor and from the preamplifier are reported on the schematic presented on Fig. 9. Using the block diagram representation we can easily get the transmittance "seen" by each noise contribution. Main noise contribution from the induction sensor is the thermal resistance of the winding. Noises sources coming from big volume ferromagnetic core can be neglected at low resonance frequency (Seran & Fergeau (2005)) while Barkhausen noise coming from the magnetic domain displacment can be neglected until reversible magnetization is considered (typ. few mT). We neglect noise contribution from R1//R2 equivalent resistor since this noise

From the bloc diagram (9), we can easily express the output noise contribution of each noise source, because of the relation between input and output power spectrum density (PSD)

First, the PSD of the noise coming from the sensor comes essentially from the resistance of the

2 +

*RCω* +

*PSDR* = 4*kTR G*2

(1 − *LCω*2)

Secondly, the contribution to the PSD of the input voltage noise of the preamplifier (*ePA*) is:

*PSDout* = |*T*(*jω*)2|*PSDin* (20)

*GMω Rf b*

2 (21)

*PSDout*(*f*)

*T*(*jω*)2 (19)

*NEMI*(*f*) =

resulting output PSD will be divided by the transmittance to get the NEMI 19.

analytic modelling being to have a friendly tool to help designer.

**2.4.1 NEMI of induction magnetometer using feedback flux amplifier**

can be reduced by design (choosing a small enough R1 resistance).

through a system characterized by its transmittance (*T*(*jω*)):

sensor and can be expressed directly as:

Fig. 9. Feedback flux induction sensor and noise sources.

$$PSD\_{\varepsilon\_{\text{PA}}} = \varepsilon\_{\text{PA}}^2 \frac{G^2 \left( \left( 1 - LC\omega^2 \right)^2 + \left( RC\omega \right)^2 \right)}{\left( 1 - LC\omega^2 \right)^2 + \left( RC\omega + \frac{GM\omega}{R\_{fb}} \right)^2} \tag{22}$$

Thirdly, the contribution to the PSD of input current noise (converted in noise voltage) of the preamplifier is expressed:

$$\text{PSD}\_{\text{ip}\_{\text{PA}}}(V^2/Hz) = \left(|Z|i\_{\text{PA}}\right)^2 \frac{G^2\left(\left(1 - L\mathcal{C}\omega^2\right)^2 + \left(RC\omega\right)^2\right)}{\left(1 - LC\omega^2\right)^2 + \left(RC\omega + \frac{GM\omega}{R\_{fb}}\right)^2} \tag{23}$$

where |*Z*| is the impedance modulus of induction sensor :

$$|\underline{Z}| = \sqrt{\frac{\left(\text{R}^2 + \left(L\omega\right)^2\right)}{\left(1 - LC\omega^2\right)^2 + \left(RC\omega\right)^2}}\tag{24}$$

By considering the noise contribution of the impedance of the feedback:

$$PSD\_{R\_f} = 4kTR\_f \tag{25}$$

Finally we express the total output noise contribution as a sum of the different contributions:

$$PSD\_{out}(f) = PSD\_R + PSD\_{\ell p\_A} + PSD\_{l p\_A} + PSD\_{R\_f} \tag{26}$$

We can notice that the total ouput noise expressions presented here have only few differences to the one reported in Seran & Fergeau (2005). The objectives of this chapter being double: get the analytic modelling and expose, in a pedagogic way, the method of total noise computation (while equation seems coming from sky in some papers).

Fig. 10. Induction sensor using transimpedance electronic amplifier: noise sources.

Induction Magnetometers Principle, Modeling and Ways of Improvement 57

Fig. 11. NEMI comparison: transimpedance amplifier versus feedback flux

Finally, the NEMI curve can be plotted in both case.

filtering. The plots have been superimposed on figure below.

is limited by the input noise voltage.

a transimpedance preamplifier with same characteristics as the one used in the model and with a sensor identical to the one described in 1 confirm the accuracy of the modelling.

With this design we can get NEMI as low as few fT/sqrt(Hz) using short sensors (12cm in this case). We can notice the minimum of the NEMI around a few 10kHz is (in the design reported here) limited essentially by the input current noise while the NEMI at frequency below 10kHz

We found a good agreement with the NEMI curves of feedback flux induction magnetometer reported in Seran & Fergeau (2005) for an induction magnetometer built for space physics. The increase of NEMI measured above a few 30kHz is the presence of 7th order low-pass

By combining 18 with total PSD of noise 31, we can expressed the NEMI defined by 19.

#### **2.4.2 NEMI of induction magnetometer using transimpedance amplifier**

Let us now consider the induction sensor with a transimpedance electronic conditionner. The same method as the one presented above is applied to get the NEMI.

In this case, we refer to block diagram of 10. First, the power spectrum densitiy (PSD) of the noise coming from the sensor resistance can be expressed as27

$$PSD\_R = 4kTR \times \frac{1}{R^2 + \left(L\_1 \omega\right)^2} \times \frac{\left(R\_f\right)^2}{1 + \left(R\_f C\_f \omega\right)^2} \tag{27}$$

Secondly the contribution to the PSD of the input voltage noise of the preamplifier (*ePA*) converted in current noise is simply amplified by the transmittance amplifier.

$$PSD\_{\varepsilon\_{\text{PA}}} = \frac{\varepsilon\_{\text{PA}}^2}{R^2 + \left(L\_1 \omega\right)^2} \times \frac{\left(R\_f\right)^2}{1 + \left(R\_f \mathbb{C}\_f \omega\right)^2} \tag{28}$$

Thirdly, the contribution to the PSD of the input current noise of the preamplifier is expressed:

$$PSD\_{i\_{\rm PA}} = i\_{\rm PA}^2 \frac{\left(R\_f\right)^2}{1 + \left(R\_f \mathcal{C}\_f \omega\right)^2} \tag{29}$$

By considering the noise contribution of the impedance of the transimpedance feedback:

$$PSD\_{i\_{\rm pA}} = 4kTR\_f \frac{1}{1 + \left(R\_f C\_f \omega\right)^2} \tag{30}$$

Finally we express the total output noise contribution by means of PSD summ:

$$PSD\_{out}(f) = PSD\_R + PSD\_{\varepsilon\_{PA}} + PSD\_{i\_{PA}} + PSD\_{R\_f} \tag{31}$$

By combining 18 with total PSD of noise 31, we can expressed the NEMI defined by 19. Finally, the NEMI, which is the first requirement of magnetometers, can easily be determinedd for a given design over the frequency range.

#### **2.4.3 NEMI awards: feedback flux versus transimpedance**

Let us consider the example of sensor design presented on 1. We assume the input voltage noise of the preamplifier is about 3*nV*/ √*Hz* while the input current noise is weak 200 *f A*/ √*Hz* and that the frequency range is beyond the 1/f noise of the preamplifier. The comparison on Figure 11between the NEMI computed in both cases (transimpedance and feedback flux) exhibit quite close performances in terms of NEMI. Measurment achieved with 12 Will-be-set-by-IN-TECH

Let us now consider the induction sensor with a transimpedance electronic conditionner. The

In this case, we refer to block diagram of 10. First, the power spectrum densitiy (PSD) of the

1 *R*2 + (*L*1*ω*)

Secondly the contribution to the PSD of the input voltage noise of the preamplifier (*ePA*)

Thirdly, the contribution to the PSD of the input current noise of the preamplifier is expressed:

1 + *Rf Cf ω*

> 1 + *Rf Cf ω*

By combining 18 with total PSD of noise 31, we can expressed the NEMI defined by 19. Finally, the NEMI, which is the first requirement of magnetometers, can easily be determinedd for a

Let us consider the example of sensor design presented on 1. We assume the input

comparison on Figure 11between the NEMI computed in both cases (transimpedance and feedback flux) exhibit quite close performances in terms of NEMI. Measurment achieved with

√*Hz* and that the frequency range is beyond the 1/f noise of the preamplifier. The

2 ×

 *Rf* 2

*PA R*2 + (*L*1*ω*)

> 2 *PA*

By considering the noise contribution of the impedance of the transimpedance feedback:

converted in current noise is simply amplified by the transmittance amplifier.

*PSDiPA* = *i*

*PSDiPA* = 4*kTRf*

Finally we express the total output noise contribution by means of PSD summ:

given design over the frequency range.

200 *f A*/

**2.4.3 NEMI awards: feedback flux versus transimpedance**

voltage noise of the preamplifier is about 3*nV*/

*PSDePA* = *e*2

2 ×

 *Rf* 2

2 (27)

2 (28)

2 (29)

2 (30)

√*Hz* while the input current noise is weak

1 + *Rf Cf ω*

 *Rf* 2

1 + *Rf Cf ω*

1

*PSDout*(*f*) = *PSDR* + *PSDePA* + *PSDiPA* + *PSDRf* (31)

By combining 18 with total PSD of noise 31, we can expressed the NEMI defined by 19.

**2.4.2 NEMI of induction magnetometer using transimpedance amplifier**

same method as the one presented above is applied to get the NEMI.

noise coming from the sensor resistance can be expressed as27

*PSDR* = 4*kTR* ×

Fig. 10. Induction sensor using transimpedance electronic amplifier: noise sources.

Fig. 11. NEMI comparison: transimpedance amplifier versus feedback flux

a transimpedance preamplifier with same characteristics as the one used in the model and with a sensor identical to the one described in 1 confirm the accuracy of the modelling.

Finally, the NEMI curve can be plotted in both case.

With this design we can get NEMI as low as few fT/sqrt(Hz) using short sensors (12cm in this case). We can notice the minimum of the NEMI around a few 10kHz is (in the design reported here) limited essentially by the input current noise while the NEMI at frequency below 10kHz is limited by the input noise voltage.

We found a good agreement with the NEMI curves of feedback flux induction magnetometer reported in Seran & Fergeau (2005) for an induction magnetometer built for space physics. The increase of NEMI measured above a few 30kHz is the presence of 7th order low-pass filtering. The plots have been superimposed on figure below.

Fig. 13. Tri-axis induction sensor inside a thermal blanket & electrostatic shielding (space

Induction Magnetometers Principle, Modeling and Ways of Improvement 59

As mentionned previously, the signal from induction sensor decreases after the resonance frequency while NEMI increases whatever the electronic conditioning is. To bypass this drawback a mutual reducer made of ferromagnetic core is used between two windings Coillot et al. (2010) designed for contiguous frequency bandwidth. It allows to extend the frequency band of measurement using a single sensor. Adjusting a such dual band sensor is not an easy task and its use has sense mainly for applications where mass constraints are stringent.

An interseting way to reduce inductance of induction sensor and thus increase frequency resonance is presented in Dupuis (2005). It consists in implementing induction sensors on the

In such configuration, each axis is constituted in 4 inductions sensors connected in "serie". Let us consider an induction sensor coil requiring a number of turns *N*. Each induction sensor of the cubic configuration will have *N*� = *N*/4 turns and, by connecting the single induction

the turns are mounted on the same core. Thus, the inductance will be 4 times lower. Since authors claim the capacitance value in the classical configuration and their cubic configuration is the same, the resonance frequency of the cubic induction sensor will be 2 times higher than for classical induction sensor allowing a desirable extend of the frequency range of use.

)2 instead of (4*N*�

)2when

sensor in "serie", the total inductance will be proportionnal to 4 ∗ (*N*�

experiment CLUSTER).

**3.1.1 Dualband search-coil**

**3.1.2 Cubic search-coil**

edges of a cube 14.

**3. Last and future developments**

**3.1 Induction sensor bandwidth extension**

Fig. 12. NEMI of feedback flux amplifier: comparisons between analytic modelling & NEMI reported in Seran & Fergeau (2005)

It would be interesting to compare with other designs, as the one presented in (Grosz et al. (2010); Prance et al. (2000)) but this would require the recollection of datas describing the designs in the mentionned papers.

### **2.5 Calibration**

When designers get their induction magnetometer. Some features of this magnetometer could be useful to know clearly what is measured. First of all, the transfer function and output noise will permit to know the dynamic and the noise equivalent magnetic induction. Next, the directionality of the magnetometer can be a key parameter in applications where the direction of the electromagnetic field must be determined accurately. Another key parameter, which is never mentionned, even if it is of great importance is the sensitivity to electric field. Induction sensors can be sensitive both to magnetic field and electric field and users must care about this last one to avoid to get wrong informations. The electric field sensitivity mechanism is symmetric to the principle of the induction sensor itself. In that case, the electric field will create a current through the wires of the sensor which will be amplify by the amplifier. An electrostatic shielding should surround the sensor and the cable until the preamplifier. This shielding must be refer to a potential (usually the ground), in the same time, this shielding (made of conductive surface) must avoid to allow circulation of eddy current (which would expel the magnetic field at frequency where the skin depth becomes smaller than the thickness of the conductive material). Finally the measurement of the electric field transfer function of the sensor could be an helpfull way to ensure the quality of the electrostatic shielding. As an example, we illustrate, on Fig. 13, a space induction magnetometer where the thermal blanket ensures also electrostatic shielding function.

Fig. 13. Tri-axis induction sensor inside a thermal blanket & electrostatic shielding (space experiment CLUSTER).


**4. Globalization and society**

This section explores the interaction between globalization and society in the context of social sustainability. Three aspects of society, namely quality of life, urbanization, and equality, are considered.

#### **4.1 Quality of life**

In low-income nations, life expectancy has been positively correlated with globalization [46]. Likewise, quality of life measured in terms of human development index has been found to increase with globalization in several Asian countries [47]. At global scale, aggregate positive effect from globalization-related trade has been reported while also acknowledging disproportionate distributional concerns. The disproportionate concern is that globalization-related trade can lead to unemployment, reduced wages, and slower decline in poverty in some geographical regions and commercial sectors [48].

The overall effect of globalization on high-income, as well as low-income, nations is reported to be improved quality of life. Positive impacts include harmonization of the labor market, development of digital society [49], and reduced macroeconomic volatility due to diversification of risks.

#### **4.2 Urbanization**

Urbanization is primarily happening in low- and middle-income countries due to economic growth and globalization [50]. Benefits of urbanization are increase in employment opportunities, abundance of unskilled labor, and expansion of industrial, residential, entertainment, and commercial areas. Urbanization and trade openness can also reduce income inequality [51, 52], particularly when coupled with high institutional quality and democracy. However, a study in Africa has found that urbanization alone may not significantly impact equality in the absence of democratic reforms and institutional quality [52]. Institutional quality here refers to corruption control, law and order, and sound bureaucracy. On the flip side, disadvantages of urbanization include increased crime and land use change [50]. Farmland loss, ecological degradation, and decrease in biodiversity are some of the negative consequences of urbanization [53].

### **4.3 Equality**

The relationship between equality and globalization is very complex. Generally, globalization increases inequality within a nation but decreases inequality between nations [54]. It can therefore be argued that the benefits of globalization have not reached everyone equally. There are also mixed relationships between income inequality and globalization with some countries seeing the benefit of globalization while others have not [55]. Statistical analyses have shown that social spending may not necessarily cushion the impact of inequality brought about by globalization [55]. Overall, better approach to globalization is needed to ensure that inequality does not rise due to globalization and outweigh its positive effects.