duality of the electric covering fieldmill and the fluxgate magnetometer

2306 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 4, JULY 1998

Duality of the Electric CoveringFieldmill and the Fluxgate Magnetometer

Ben-Zion Kaplan,Senior Member, IEEE, and Uri Suissa

Abstract—This work treats two dc field sensors. The firstsensor is the “covering fieldmill” that is employed to measuredc electric fields. The second sensor is the “fluxgate magnetome-ter” that is utilized to measure dc magnetic fields. An unusualapproach is suggested to deal with the devices’ evaluation. Itis shown that both devices can be associated with related acfield sensors—electric and magnetic. This association leads tomodeling the dc sensors by Thevenin circuit models. It is thenshown that in both cases the coupling between the sensors outputcircuitry and the measured field sources is due to series time-varied reactances. This enables a similar analytical treatment toboth devices. Furthermore, it is also suggested that both sensorsare related to the large family of antennas. As a result, an unusualway is found to relate the sensors’ output signal to the measuredfield. It is well known that the signal in electric antennas is relatedto the measured field times the effective antenna length. It istherefore relatively easily acceptable that also in the dc electricsensor case a similar relationship holds. The work, however,suggests that this relationship (that now relates the output signalto the measured dc magnetic field times the effective magneticstructure length) also holds in the fluxgate magnetometer case.An extension of this result is useful when attempting to measurenonuniform dc magnetic fields and is treated toward the end ofthe paper.

Index Terms—Duality, electric field sensor, fieldmill, fluxgate,magnetic field sensor.

I. INTRODUCTION

T HE present work treats certain dc magnetic field andelectric field sensors. Such sensors are usually discussed

in the literature separately. The present work deals withthe subject from a unified viewpoint. It is argued that themagnetic and the electric sensors are closely similar. It is evendemonstrated that devices of one family are strongly related tothe devices of the other family via the rules of electromagneticduality. Duality considerations appear important in enablinga better insight to devices of one family by considering thedevices of the other family. One of the main dc magnetic fieldsensors is the fluxgate magnetometer, which can be sometimesinterpreted as if it alternately chops the flux that originatesin the measured field sources [1]–[7]. Similarly, there existsan electric dc field sensor, the “covering fieldmill,” whoseoperation is interpreted as if it acts as a chopper that affects theelectric flux produced by the measured electric field sources

Manuscript received July 9, 1997; revised February 18, 1998. This paperwas supported by the Paul Ivanier Center for Robotics Research and Produc-tion Management.

The authors are with the Department of Electrical and Computer Engineer-ing, Ben-Gurion University of the Negev, Beer-Sheva 84105 Israel (e-mail:[email protected]).

Publisher Item Identifier S 0018-9464(98)03446-3.

[8]–[16]. The two latter sensors are the sensors treated inthe present work. They are employed mainly in geophysicaland space research [17]–[19]. Fluxgate magnetometers havebeen assembled on most of the spacecrafts [17]. The “cover-ing fieldmill” is installed aboard balloons for measuring theatmospheric electric dc field [10]. Both dc field sensors areemployed also for geophysical investigations on the Earth’ssurface [20]. Fluxgates, for example, are utilized for mineralprospecting and for oil exploration [20]. It is interestingthat fluxgates are employed also for military and securityapplications [20]. The present work approach differs from theusual tradition of seeing the devices as completely separateentities and treats them by employing a unified viewpoint. Theviewpoint is based on ideas borrowed from antenna theory,where the operation of antennas is sometimes explained byemploying circuit models. (Such circuit models are treated inSections II and III.) This viewpoint is not merely helpful inshowing that the dc electric field sensor and the dc magneticfield sensor are related. It is also helpful in demonstratingthat field sensors can be regarded as related to the largefamily of antennas. The circuit model approach employed forthe treatment of the present work devices is also helpful indeparting from the traditional way of seeing the devices asquasi-choppers. In discussing the electric sensor, we show thatthe main item that enables coupling between the sensor andthe external measured field sources is the series capacitance. Itis known from antenna theory that the series capacitance thatappears in the circuit model is due to the external capacitancebetween the two limbs of the antenna (the reader is askedto consider, for example, a short electric dipole) and not theinternal parasitic capacitance that appears between the limbsin the middle [18], [21], [22]. As a result, in the coveringfieldmill case, the coupling to the measured dc field is dueto the periodic variations in time of the capacitance. (This isdiscussed in Section II.) The latter capacitance is associatedwith the external field between the moving vanes. Hence,the present work explanation of the dc electric field sensoroperation differs from the more traditional chopper approachwhich can be interpreted as regarding the internal capacitanceas the main item in the sensor operation [8]–[16]. The dualityexisting between the magnetic and electric dc field sensorssuggests that the main item in the fluxgate magnetometeroperation is similarly due to a series of inductance variations.(This is discussed in Section III.) We demonstrate that themagnetometer sensing coil inductance that is made time variedby the saturating effects of the excitation coils current is theseries inductance that is responsible for the coupling between

0018–9464/98$10.00 1998 IEEE

KAPLAN AND SUISSA: DUALITY OF ELECTRIC COVERING FIELDMILL AND FLUXGATE MAGNETOMETER 2307

Fig. 1. The basic configuration of a short electric dipole structure (in areceiving mode of operation.)

the magnetometer output circuitry and the field sources. Theassociation of the dc field sensors with similar circuit modelsdemonstrates further the close affinity that relates the sensorsand even enables to treat them via the strict rules of electromagnetic (EM) duality. (This is discussed in Section IV.) It isdemonstrated that the duality that relates to the circuit modelsis also valid when treating the devices via their geometricstructural details. This leads, for example, to interpretingthe fluxgate operation as being related to a “magnetic opencircuit voltage” obtained by integrating the measured magneticfield along the magnetic core structure as is explained inSection V. The latter seems to bring forth a new concept intreating magnetometers whose operation is usually evaluatedby integrating the external field flux density in the systemeffective cross sectional area.

II. A D YNAMIC MODEL FOR A DC ELECTRIC FIELD

SENSOR (THE ROTATING COVERING FIELDMILL )

The performance of a dc field sensor that is commonlyentitled as a fieldmill can be interpreted by considering at firstan extremely low frequency (ELF) field sensor. ELF sensorsare usually constructed as small dipoles (Fig. 1) [17], [18],[21]–[23]. Such small dipoles are employed in space research[17], [22] and for detecting electromagnetic interference [21].They are usually much shorter than the wave length and, asa result, their radiation impedance is almost solely reactive(capacitive) and the small radiation resistance can be omitted[17], [18], [21]–[23]. It is also known from antenna theory thatthe radiation reactance mentioned previously is the one thatcouples the sensor output terminals to the field sources [18],[21], [22]. This explanation gives rise to the sensor circuit

Fig. 2. Electric circuit model for the ELF electric dipole sensor. (It alsorepresents the electric dipole in Fig. 1.)

model that is well known from the literature on antennas. Thiscircuit model is shown in Fig. 2 [17], [18], [21]–[23]. This is,in fact, the Thevenin model of the sensor when it is regardedas a voltage source. Antenna theory tells us also that the sourceopen circuit voltage is [21], [24]

(1)

where is the field component parallel to the sensor direction(we assume that the field in the vicinity of the sensor isuniform.) is the dipole length. It also seems obviouswhen considering antenna theory literature that the capacitiveradiation reactance mentioned previously is the reactancemeasured between the antenna terminals in the external field.It is related merely to the external capacitance of the structure,while the small capacitance between the terminals in the centercan be regarded as part of the sensor load. It is interesting torelate the short circuit current in Fig. 2 to the opencircuit voltage as follows:

(2)

This is the relationship existing when assuming sinusoidalsteady state. The difficulty of measuring dc field can bedemonstrated by further considering the performance of thepreviously described ac sensor. Let us suggest that the fre-quency of the ac field has been reduced to an extremely smallone. It appears obvious that in such circumstances the seriescapacitive reactance becomes relatively large, and the fieldsources are not coupled effectively to the measuring systemthat is connected to the terminals. It appears, therefore, thata dc field sensor should perform effectively if its externalcapacitance was made time varied. The dc field sensors whoseexternal capacitance is modulated are well known in thetechnical literature [8]–[16]. One of the latter sensors is therotating covering fieldmill (Fig. 3). This structure is associatedwith a periodically time varied external capacitance which isdue to the rotation of one of the electrodes usually entitledin the literature as the rotating shutter or cover electrode.The other electrode is entitled as a test electrode and isusually stationary. This electrode can be described as if itwere alternately exposed to and shielded from the field linesof force by a moving earthen cover [8]. Some of the coveringfieldmills include a test electrode and a cover electrode, both


Fig. 3. The rotating covering fieldmill.

Fig. 4. The circuit model describing the rotating covering fieldmill system.

consisting of several (say eight) wings that are cut sectoriallyfrom originally round discs. This enables a rapid modulationof the external capacitance, and as a result, the sensor canrespond to relatively rapid variations of the external fields.

The stationary electrode is connected to the input terminal ofthe measuring amplifier. In order to obtain a simple circuitthat conforms with antenna theory, we suggest to model thesystem by assuming that it is completely electrically floatingin relation to the earth. Furthermore, the output is not assumedas connected to an elaborate amplifier, but it is connectedto a simple resistive load. This leads to the following circuitmodel in Fig. 4. This circuit is based on the established methodof describing an antenna through its Thevenin representation.As a result, the dc voltage represents the open circuitvoltage associated with the measured field. (Its value is thedistance between the electrodes times the field strength.)It is based on common methods of describing antennas in the

Fig. 5. A practical construction of an ELF magnetic field sensor.

literature [23]. (The related antenna is entitled in the literatureas capacitor-plate antenna or top-hat-loaded dipole.)

III. A D YNAMIC MODEL FOR A DC MAGNETIC

FIELD SENSOR (THE FLUX-GATE MAGNETOMETER)

In the present magnetic case, the development of the sensormodel is similar to the development of the electric counterpartand it is also assisted by considering a related ac sensor atfirst. A well-known ac magnetic field sensor is of the searchcoil type [18], [20], [24]. Such search coils are employed asmagnetometers in geophysical research and were consideredfor ELF communications [17]–[19]. They are also installedaboard many of the deep space probes [17]. They are utilizedfor recording the ELF and even ultra low frequency (ULF)magnetic fields in space and in the vicinity of the planets [17],[19]. Some of the search coil magnetometers are constructedas air cored coils of relatively large diameter (a meter or so)and they consist of relatively many turns (say ten thousand)[18], [20], [24]. Portable search coil magnetometers, however,are usually constructed with a ferromagnetic core of largepermeability, which enables an overall reduced size and weightof the device [19]. The basic configuration of the device isshown in Fig. 5. The usual way of modeling such devices isthrough their conventional electric circuit, which is given inFig. 6. is the inductance of the search coil, and the voltageinduced in the coil is represented by a series voltage source

(3)


Fig. 6. The circuit model describing the ELF magnetic field sensor in theliterature.

The related amplitude when the field is harmonic of angularfrequency is

(4)

is the amplitude of the flux passing through the coil. Ouraim is to employ a model which is on one hand similar tothat of Fig. 6 and, on the other hand, we would like thismodel with small modifications to represent magnetic dc fieldsensors (fluxgate magnetometers) as well. The idea is to utilizea method similar to that in the previous section (in the electriccase), which enabled us to model both the ac sensor and the dcsensor by closely related models. The model of Fig. 6 does notappear as capable of such comprehensive description of boththe ac and dc sensors. It appears, however, that by borrowingideas from antenna theory, and by relying on the modelsdeveloped previously for the electric case, one is capable ofproposing a comprehensive method for treating both the ac andthe dc cases. The first step in obtaining the comprehensivemodel is in associating the sensor output with the intensityof the field sources in a way similar to that in Fig. 1. Theopen circuit voltage there is directly related to the field. Themagnetic structure (Fig. 5) is also a long structure similarto that in Fig. 1. Furthermore, it acts as an electrode in themagnetic sense. This is due to its large permeability whichcauses it to gather and concentrate the magnetic field lines ina way similar to that of the electric structure in Fig. 1. Theelectric structure behaves as a real electrode due to its largeelectric conductivity. The magnetic system does not possessa real conductivity (since there are no magnetic monopolesand charges in existence), but due to its large permeabilityit can be regarded as a quasi-electrode system. The magneticsystem can become even more closely related to its electriccounterpart by initially removing the coil and by gapping thestructure in the middle. This is shown in Fig. 7. isregarded as the magneto motive force that appears across thegap as a result of an externally measured magnetic field. Dueto the magnetic electrode analogy mentioned previously, theopen circuit “voltage” in the magnetic case should besimilar to (1), namely the field intensity times half the corelength (see Fig. 7) [25]. Hence

(5)

The ferromagnetic core, however, in a real sensor case, shownin Fig. 5, behaves as if it were short circuited. (It is, unlikethe electric antenna previously, not gapped in the middle andalso the coil wound around it is open.) The system dealt with

Fig. 7. A short receiving dipole (magnetic.) The magnetomotive force thatappears across the gap between the two ferromagnetic limbs is analogous tothe terminal voltage (open circuit voltage) that is measured in the similarelectric case previously (Fig. 1). As a result, it is denoted byVo:c:m:. Itsunits are, however, amperes and not volts as is case in Fig. 1. Furthermore,the gap is ferromagnetically bridged in the practical magnetic antenna. Theoutput signal is then obtained as a real voltage across a coil wound aroundthe central region of the bridged ferromagnetic core. The practical magneticantenna (Fig. 5) can be regarded as being magnetically short circuited, if thecoil around it was open circuited.

now in Fig. 5, however, is a loop antenna consisting of a coilwound around the central region of a long ferromagnetic core.It behaves as an analog of the electric antenna previously ifthe electric antenna terminals were short circuited and if thepresent case coil was open circuited. The coil voltage can beevaluated by relying on the flux in the core. The flux amplitude

(when the coil is open circuited) can be evaluated throughthe structure reluctance. Hence

(6)

The reluctance, however, can be expressed in terms of thestructure inductance, namely

(7)

As a result

(8)

Equations (6)–(8) are presented in terms of conventionalmagnetic circuits. Our objective, however, is to describe themodel of the magnetic sensor in a way similar to that ofFig. 2 where the electric ac sensor model is shown.can indeed serve as a source in such a model since it can beregarded as the dual of previously. There is, however,a need to define also a magnetic dual of the electric current


Fig. 8. Magnetic circuit model for the ELF magnetic dipole sensor. (It is a“magnetic circuit” in the special magnetic sense that has been developed inthe present work.)

in order to obtain an appropriate model for the magnetic case.The current in the electric structure of Fig. 1 is simply itsconductance current. The identity of the magnetic counterpartis less obvious. A current of charges (monopoles) does notexist in the magnetic case. Furthermore, it is well known thatthe electric flux time derivative is the displacement current,which replaces in many situations real electric currents ofcharged particles. As a result, it is argued that the magneticentity in Fig. 7, that should be regarded as the analog of theelectric current in the structure of Fig. 1, is the time derivativeof the magnetic flux through the core [25], [26]. Ferromagneticmaterials with large permeability force the flux flow to besimilar to that of currents in a conductor. However, as a resultof the previously mentioned analogy, the magnetic entity thatreplaces the electric current is the time derivative of the fluxwhose flow is confined by the highly permeable ferromagneticmaterial. Modeling magnetic circuits in such a manner wherethe time derivative of the flux is regarded as the currentof the magnetic case is completely consistent with similartreatments in discussing higher frequency antennas [23], [27].A further support for this choice of the magnetic current isdue to dimensional analysis [26]. The traditional magneticcircuit employs magneto-motive force as an analog of theelectro-motive force in the electric circuit. It employs the fluxflowing in the magnetic circuit as an analog of the currentin the electric circuit. As a result, one may erroneously tendto regard the product of magneto-motive force times the fluxas if it were power. The units, however, of this product areAmper Weber Amper Volt sec Watt sec Joule. Thelatter represents Work and not Power as one would have liked.If, however, the flux derivative in time was employed as theanalog of the electric current in the magnetic circuit, then theproblem would be alleviated. As a result, (8) is replaced bythe following relationship:

(9)

This has been obtained from (8) by taking the time derivativeof the flux and by assuming sinusoidal steady state. in(9) is regarded as the magnetic sensor short circuit current.The assumption of a short circuit is due to the fact that thecore in Fig. 5 is not gapped in the middle and the coil woundaround it is assumed open in the electric sense. As a result,we obtain the following circuit (it is a circuit in the magneticsense, similar to the electric circuit in Fig. 2), which is shownin Fig. 8. in (9) plays the same role as in (2).This leads to proposing as the appropriate impedance

Fig. 9. Magnetic circuit model for the fluxgate magnetometer. (It is a“magnetic circuit” in the special magnetic sense that has been developedin the present work.)

(in the magnetic sense) that should be employed in evaluationsrelying on the nonconventional magnetic circuit in Fig. 8. Theproblem of sensing dc magnetic fields can now be shownas being related to the problem of sensing dc electric fieldsdiscussed in Section II. The series magnetic impedance (it isan impedance in the magnetic sense) is extremelylarge when the frequency is reduced, and the sensor load(which is due to the measuring system) becomes decoupledfrom the field sources. This problem is usually alleviatedby employing the well-known fluxgate magnetometer [1]–[7].One of the interpretations for explaining the performance offluxgates is in describing them as sort of search coils whosecore relative permeability is being modulated by employingexcitation fluxes that result from special auxiliary excitationcoils [28]–[30]. Other special fluxgates are interpreted aspossessing a core whose cross sectional area is effectivelymodulated [31]. One can therefore propose that in both cases,a magnetometer for dc magnetic fields is obtained when theinductance of a search coil is appropriately modulated in time.This time varying inductance plays the same role as thatof the time varying capacitor in Fig. 4. The related circuitmodel (it is a circuit in the magnetic sense) of the dc fieldmagnetometer is shown in Fig. 9. Other authors also foundthat the performance of fluxgate magnetometers can be madeexplicable by relying on a time varying inductance that servesto couple the sensor to the field sources. Russel and Narod[29], [30] attribute the performance of a fluxgate magnetometerto time varying inductance. The model of Fig. 9 is applicablefor describing most of the fluxgate magnetometers. The actualgeometry, however, of most fluxgate magnetometers is notrelated directly to that of a long core as is the case of thesearch coil geometry. There exist, nevertheless, some fluxgatemagnetometers that can still be described as a search coilwhose long core magnetic properties are modulated. Suchmagnetometers are mentioned by [28], [32], and [33]. Thestructural model related to Afanasyev’s magnetometer [32]is shown in Fig. 10. The ring in the center in the model inFig. 10 is saturated by the excitation coil current twice percycle of the excitation frequency. As a result, the inductanceof the so-called search coil wound on both sides of thering is periodically modulated in time. Many of the practicalfluxgates possess a somewhat different geometry where thewhole ferromagnetic core consists of a single ring. Both theexcitation coils and the sensing coils are then wound on thesame ring [1], [7], [20]. This geometry is usually suggestedbecause of practical reasons. It enables, for example, the


Fig. 10. A special configuration of the fluxgate magnetometer. (This config-uration is due to Afanasyev [32].)

whole magnetic structure to be alternatively saturated. Asa result, difficulties associated with hysteresis and remnanteffects are prevented. The long core geometry (in Fig. 10) isimportant because of some of its advantages in magnetometricapplications [32], [33], and also because it lends itself toa relatively precise analytical treatment in evaluating theresponsitivity of the sensor output voltage to the measuredfield. This is due to the fact that we are now able to evaluatethe performance of magnetometers by relying on the theoryof electric antennas of a relatively long structure. The latteridea is due to the previously mentioned evaluation ofin (5) and is described in the sequel. The “magnetic circuit”model in Fig. 9 deserves further explanations. This is achievedby relating it to the details of the fluxgate magnetometer inFig. 10. The role of , the open circuit magnetic voltage,is related to the external magnetic field in the same way as themagnetic open circuit voltage in the structure of Fig. 7. Asimilar voltage can be assumed to exist on both sides of thering in Fig. 10. The excitation current causes the central ringto be alternatively saturated. As a result, it causes the loadof the measuring system that is connected to the main coil tobe reflected to the magnetic structure in a way similar to thatby which a transformer load is reflected in the ferromagneticcore. Hence, variation of the electric load resistance appears asa sort of conductance in the “magnetic circuit” of Fig. 9.

The treatment of the role of in this circuit is somewhatsimplified now by assuming the number of turns to bein (7). is inversely related to the time variations of thestructure reluctance that are due to the effects of the excitationcurrent that alternately saturates the central ring. inFig. 9 is a voltage in a “magnetic sense” and it is measured inamperes. Its value can be shown to be identical to that of theelectric current in the output coil in Fig. 10. (We have alreadyassumed that the coil consists of only one turn. We should alsoassume that the coil is wound very tightly on the core.) Theelectric current is in fact inferred through the electric voltageon the measuring system equivalent load—in the case of one turn situation.

IV. THE DUALITY THAT RELATES ELECTRIC DCFIELD SENSORS ANDMAGNETIC DC FIELD SENSORS

Ideas related to duality considerations have already beenemployed in this work when choosing the flux time derivativein the ferromagnetic structure as if it were the “magneticcurrent.” This “current” is regarded as the dual of the electriccurrent in the structure associated with the electric field sensorin Section II. Another duality relationship that has alreadybeen employed in this work is due to the evaluation of ,the open circuit voltage (it is a voltage in the magnetic dualsense. It is measured in amperes and can also be entitled as thestructure magneto motive force) in the magnetic case.has been evaluated by relating it to the external magnetic fieldin a way similar to the one that is employed when ,namely, the open circuit electric voltage is evaluated in theelectric counterpart case. The duality is valid from an overallfunctional viewpoint. However, the geometry of the practicalelectric dc sensor is not the same as that of the practical dcmagnetometer. As a result, we do not assess here the dualityrelations in a detailed manner in relation to the EM fields andto the details of the geometry. Instead, we attempt to assess theduality relations from the viewpoint of circuit theory models.The circuit model of the electric dc field sensor is shownin Fig. 4. The theory of this circuit is similar to that of the“condenser microphone” [34] and is relatively well-known.The differential equation is as follows:

(10)

is the time varying capacitance that is due to therotating covering fieldmill operation.

(11)

It is of course assumed that . As a result, we evaluatethe steady state solution by dealing with a small signaltheory for variations about the average quiescent values. Inaddition, we deal with an ac case where merely result atthe fundamental frequency is taken into account. Severaldifficulties are alleviated in this way. One of them is avoidingthe problems associated with the fact that the system is a circuitwith time varying parameters. Due to the latter assumptions,


we obtain the following steady state:

(12)

is the amplitude of the fundamental of the output voltagein steady state. The magnetic dc sensor, the fluxgate mag-netometer model, is shown in Fig. 10. Its “magnetic circuit”model is shown in Fig. 9. It should be again emphasized thatthe circuit is a circuit in a “magnetic sense.” However, it is nota conventional magnetic circuit that is usually employed forevaluating the flux in ferromagnetic materials. This circuit isa magnetic counterpart of the circuit in Fig. 4. Its significancehas been explained in Section III. It has been shown therethat the role of here is that of a “magnetic capacitor” inthe circuit. Furthermore, the variations of in time in thepresent case serve to couple the structure to the external dcmagnetic field sources. This is similar to the role of the externalstructure time varying capacitance of the fieldmill that servesto couple the measuring system to the field sources. As a result,the differential equation representing the magnetic case is asfollows:

(13)

is the time varying inductance that is due to thefluxgate variable operation that has been explained in theprevious section.

(14)

The procedure here is similar to that related to the capacitivesensor previously. This becomes clear by considering theaffinity between the “magnetic circuit model” in Fig. 9 and theelectric sensor circuit in Fig. 4. Hence, a steady-state treatmentfor attaining the output signal amplitude at the fundamentalfrequency is similar to that in the capacitive case previously.As a result

(15)

It should be re-emphasized that is measured in Siemensand that the “voltages” in (15) are measured in amperes. Theaffinity between the electric Thevenin model representing therotating covering fieldmill and the “magnetic Thevenin model”representing the fluxgate magnetometer has been demon-strated. Our present objective, however, is to demonstratethat the affinity possesses a mathematical rigor since it canbe shown to be related to duality principles in circuit theory[35]–[37]. This is achieved by finding the dual of the circuitin Fig. 4. The dual is shown in Fig. 11. The current sourcein the dual circuit is directly related to the voltage source inFig. 4. is in fact, according to circuit theory, the dualof previously. Similarly is the dual of in Fig. 4.Furthermore, according to rules of networks duality, hereis directly related to in the electric case in Fig. 4. Hence,

Fig. 11. A circuit dual to the electric circuit in Fig. 4. (The duality has beenobtained according to circuit theory rules.) It is interesting to notice that theequation representing this circuit is identical to the equation representing the“magnetic circuit” in Fig. 9.

the amplitude of the output current here at the fundamentalfrequency is

(16)

Equation (16) is an equation in a conventional electric sense,while (15) is an equation of a circuit in a “magnetic sense.”The close relationship between the equations is clearly demon-strated. It is interesting that such a close relationship is attainedin spite of the fact that one of the circuits is a true dualcircuit of the electric sensor Thevenin model, while the other ismerely a quasi circuit in a “magnetic sense.” This close relationis an evidence for the idea that the fluxgate magnetometer isnot only similar to the rotating covering fieldmill. The systemscan be regarded as being dual one to the other.

V. THE MAGNETIC STRUCTURE GAP MMF WHEN THE

STRUCTURE IS IMMERSED IN A NONUNIFORM FIELD

In dealing with the open circuit voltage (electric and mag-netic) in the devices Thevenin models, one should also con-sider the possibility that the measured field is not uniform. Thedemonstrated validity of the duality principles in the previoussections suggests that the rules that are applicable for electricantennas immersed in nonuniform fields can be extended alsoto similar situations occurring in the magnetic case. This isexplained in the sequel. Equation (1) results from a generalizedmethod for evaluating relatively long wire antennas [38]

Integrated along the whole

length of the antenna wires. (17)

is the length coordinate along the antenna wire. is theelectric field intensity projection along the antenna location(when the antenna itself is assumed missing.) is theprofile of the antenna current when a unit current is injectedinto the antenna terminals. This profile is triangular when thelength of the structure is much shorter than a wavelength[21]. Equation (17) is now regarded as a starting point forfinding out the influence of nonuniform fields on . Weneed, however, to rely on a magnetic counterpart of (17).It is relatively easily appreciated that in (17) shouldnow be replaced by the magnetic field . in (17)should be replaced by the magnetic current weighting function

. One should notice that due to physical and geometricconsiderations in the present case also possesses atriangular shape similar to that of in (17). The resulted


Fig. 12. Experimental system for measuring the gap MMF in the nonuniformfield case.

gap MMF is given now by (the meaning of the gap MMF isexplained in Section III and in Fig. 7)

Integrated along the whole

length of the magnetic structure. (18)

The initial experiments are due to the immersion of a devicesimilar to that in Fig. 7 in a nonuniform dc magnetic fieldand to the measurement of the resulted magnetic fieldin the middle of the gap whose length is . The pair offerromagnetic limbs in the experiment is constructed of longradio-metal laminations glued together in a manner that resultsa square cross section of 15 15 mm . The length of eachlimb is 0.6 m and its effective relative permeability is 20 000.The nonuniform field is obtained by a turns ofa m diameter short coil. The system is shownschematically in Fig. 12. The gapped magnetic structure isnow located on the axis of the latter coil at various distances.We rely on the fact that the gap MMF should be

(19)

The evaluation of is due now on one hand to the well-known magnetic field equation on the axis of very short coils[27]

(20)

(It is of course assumed now that the origin of the coordi-nates is at the middle of the coil.) On the other hand, thecurrent weighting function should now be modeledmathematically as follows:

(21)

is the distance between the coil center and the gap center.is the current in the magnetic field generating coil.in the

present experiments has always been A. is the areacircumscribed by the coil ( .) Equation (18) can be

now rewritten as follows:

(22)

The solution of (22) is

(23)

where

Most of the present experiments have been performed atdistances of [m] between the field generatingcoil and the measuring system gap. The gap MMF has beenmeasured for severally different distances. The integral

was also evaluated for the same distances. The resultsare shown in Fig. 13. We can therefore see that the deviationbetween the measured and evaluated results has never beenlarger than 15%.

We have thus shown experimentally that the open circuitvoltage concept is applicable in the magnetic case even whenthe surrounding field is nonuniform. The rules that enable theevaluation of this magnetic voltage are the same as those usedfor small antennas that are immersed in a nonuniform field.The device in Fig. 11 is not regarded as a practical sensor sincethere has been a need to introduce another magnetic sensor inthe middle of the gap in order to pick up the signal. The sensorbecomes practical by converting it to a fluxgate magnetometersimilar to that in Fig. 10 which includes a provision forsensing the flux that is generated by the magnetic open circuitvoltage. A similar fluxgate magnetometer has been constructed


Fig. 13. Comparison between the measured gap MMF (upper curve) and the evaluated gap MMF (lower curve) in the nonuniform field case. The gapMMF’s are traced versus the distanceS0 of the magnetic dipole center from the field generating coil.

by us and is described elsewhere [39]. Furthermore, theoutput signal of this magnetometer has been directly relatedto the obtained from (15). (An experiment similarto the one summarized in Fig. 13 has also been conductedin association with the practical fluxgate and it supports thepresent paper viewpoint—that long fluxgates are helpful inmeasuring nonuniform magnetic fields.)

VI. CONCLUSIONS

The present work deals with the similarity between two dcfield sensors, the electric covering fieldmill, and the fluxgatemagnetometer. It appears that this similarity has not beenexploited by other authors, while we have shown that thesimilarity leads to better understanding of the devices and topractical results. It has been shown that the systems are notmerely similar, but they are even related by the strict rulesof EM duality. We even show that the duality is not validonly at dc, but the relations hold also for associating magneticac field sensors with electric ac field sensors. The associationof the dc sensors with ac sensors is also connected to theassociation of the sensors with the vast field of antennas. Someof the ideas obtained through this channel are mentioned in thesequel. However, a related sidepath is due to showing that theduality between the sensors is also related to the expositionof the systems via their Thevenin circuit models. The lattermodels are strongly connected to methods used in modelinghigh frequency antennas by representative circuit models. Thecircuit model of the electric dc field sensor suggests that thecoupling of the output measuring system to the field sourcesis due to changes of the series capacitor. It should be noticedthat the variations of theexternal capacitanceof the sensorare responsible for the coupling and not the capacitance inbetween the fieldmill vanes. This leads to the suggestion thatin the similar fluxgate magnetometer, the coupling is due to theseries inductance variations. This interpretation of the fluxgateoperation appears new and not related to the conventional flux-gate treatment in the literature [1]–[7]. It provides an insight

for better appreciation of the sensor performance. We wouldlike to emphasize that the main innovation in the manuscriptis in discovering that the principle of operation of the twoinvestigated devices is different from the one that is tradition-ally mentioned in the literature. The usual explanation in theliterature treats the devices as if they were choppers. Namely,the fluxgate magnetometer periodically gates the magnetic fluxpassing through the device and enables the generation of an acsignal related to the dc field. Similarly, the rotating coveringfieldmill is traditionally interpreted as causing the flux of thedisplacement vector passing through the internal capacitanceof the device to be periodically chopped. Hence, the choppedflux can be according to the traditional investigators convertedto an ac current that can serve as an output signal. We on theother hand intend to demonstrate that the devices are coupledto the fields sources in a different manner. Our work shows thatthe present sensors like ELF and ULF antennas are coupledto the field sources by a series capacitance in the electriccase and by a series inductance in the magnetic case. Thecase of dc sensors, however, is different from that of lowfrequency antennas in that the reactive elements in the sensorscase are time varied. Due to this variation in time, the devicesbecome coupled to the external world even at dc. This newapproach enables several innovations. For example, new typesof dc electric field sensors can be proposed. A sensor whoseoperation cannot be interpreted as due to the chopping of thedisplacement vector flux has been built by us. It relies on adipole that possesses two arms and whose length is periodi-cally changed. As a result, the external capacitance changesperiodically and an output signal directly related to the externalfield is generated without relying on a chopped displacementflux. Nevertheless, we have not developed a new magneticsensor that employs this approach. The present approach(which implies that the modulated external reactance is theone responsible for coupling the sensor to the field sources),however, provides proof [40]. As a result, the approach enables


a new treatment in considering the balance of power in fluxgatemagnetometers. Another additional related factor that becomesevident by the present treatment is also due to the performanceof the sensors as antennas. The open circuit voltage in theelectric sensor case is known to be related to its geometricallength times the externally measured electric field intensity.Similarly, we show that also in the magnetic case there existsa sort of “magnetic open circuit voltage” that is also related tothe sensor length times the externally measured magnetic field.In some cases, like the fluxgate model in Fig. 10, the relatedgeometrical length of the sensor can be easily recognized inthe structure. In other cases, however, the fluxgate geometryis complicated and an equivalent length should be identified.It appears that the present work treatment of the antenna-likefeatures of the magnetic sensor is relatively new. Some hintsthat may lead to similar conclusions were mentioned by [29],[30]. They, however, did not obtain the results by systematictreatment. Their approach is intuitive. It appears, therefore,that the present reasoning that is based on employing theduality between electric sensors and magnetic sensors is avalid contribution. Furthermore, the present concept possessespractical results as well. It leads to proposing a methodfor measuring nonuniform magnetic fields. The “magneticopen circuit voltage” can then be evaluated as an integralof the field along the magnetometer structure. It should beappreciated that such sensors are employed inside laboratorieswhere measurement at the proximity of the field sources iscommon. As a result, the present suggestion to exploit thesimilarity between magnetic sensors and long electric sensorsis of value. Most of the fluxgate investigators interpret itsperformance in a different manner in suggesting that it actsas if it modulates the external flux passing through the devicesensing coil [2], [3], [7]. The latter approach is probably lesshelpful when measuring nonuniform fields. This is supportedby the discussion in Section V.

ACKNOWLEDGMENT

The authors would like to thank the donors of the Abra-hams–Curiel Chair in electronic instrumentation for their kindsupport.

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Ben-Zion Kaplan (M’76–SM’82), for biography, see this issue, p. 2305.

Uri Suissa, for biography, see this issue, p. 2305.

duality of the electric covering fieldmill and the fluxgate magnetometer

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