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Time Domain and Statistical Modeling and of the Barkhausen Effect Used As a Magnetic Sensor

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Time Domain and Statistical Modeling and of the Barkhausen Effect Used As a Magnetic Sensor Time Domain and Statistical Modeling and of the Barkhausen Effect used as a Magnetic Sensor Dr. Theodore R. Anderson NavalUndersea Warfare Center Division Newport 1176Howell Street Newport , RI 02841 Abstract: This researchpresents a time domain model of the by sensing the motion of a device with a ferromagnetic Barkhauseneffect as a magnetic field sensor. This type of materialaffixed to the device. magneticfield sensorwill supplementmagnetic field sensors alreadyin useand should have wide applicability in the EMC MODEL community. When a silicon steel sheetis placedat the center The energy expendedduring the realignment of a magnetic of a coil, an external magnetic field from a moving magnet domainin the Barkhauseneffect is modeledusing principals in will causethe magneticdomain walls to snapand click as they solid state physics [l]. This model is then connectedto suddenlyorient themselves. This suddenorientation of the macroscopic physical quanities that are measured in a magnetic domains causes suddenchanges in magnetization laboratory.The model is basedon the energyexpended in the which produceimpulses of emf which can be heardas distinct Bloch wall in an iron crystal that is the transition layer that clicks on a loudspeakerconnected to an amplifier and with the separatesadjacent regions (domains) magnitiied in different amplifier connectedto the coil. This suddenorientation of the directions.The entire changein spin direction takes place in a magneticdomain walls in the presenceof an externalmagnetic Bloch wall over many atomic planes. The total exchange field is called the Barkhauseneffect. In this paper a time energyof a line of Ncl atoms is: domainmodel of the Barkhauseneffect as a magneticsensor is presented. Equations are also developed which model Barkhauseneffect as a motion sensor by sensing magnetic NW,, = JS2n2! N (1) fields in a moving referenceframe. modeled. The sudden orientation of the magnetic domain described by the where w, is the exchangeenergy per pair of neighboring Barkhauseneffect is mathematicallymodeled by calculatingthe atoms [I]. The quantity J is the exchangeintegral, and the energyneeded to changethe direction of the Bloch wall or the quantity S is the spin in Heisengerg’s model. Following transition layer that separatesadjacent domains magnetized in Kittel, S= 1, and above the Curie temperatureJ= 11.9 MEV. different directions.The spectrumof the Barkhauseneffect was The total the number of atoms N in the Bloch wall is theoreticallyderived. approximateley300 [I]. Henceit is easy to calculatethe total exchangeenergy through a Bloch wall. Our model treats this INTRODUCTION energyas the energyof domain realignmentin the Barkhasen This paperdevelops the time domain model of the Barkhausen noise effect. We will asume that this model for domain effect and mathematicallypredicts it ’s use as a magneticfield realignmentis a stochastic event in Barkhausennoise and a sensor.Magnetic sensorsin use include the Hall effect sensor, powerspectral density for Barkhausennoise can be computed. the fluxgate sensor, magnetostrictive, magnetoresistive. Treating our domain realigmnnet as an event in Barkhausen magnetoinductive,and squid sensors. These magnetic field noise, we will assumethat this Barkhausenevent utimateley sensorshave a wide applicability, but noneuse the Barkhausen appearsas a measurableinduced emf E. We will assumethe effect to sensea magneticfield. The Barkhauseneffect sensesa size of a magneticdomain is on the order of the size of a Bloch magnetic field by aligning small atomic size magnetic wail. The size of the Bloch wall accordingto Boswoth is about domainsin the direction of an externalmagnetic field and by 500 angstumsthick for iron [2]. We will model the domain as Faraday’slaw causing inductive kicks which can be measured a spherewith a diameterof 500 angstrums.We will asume when amplified and even connectedto a loudspeaker.The that the exchangeenergy is uniformly distributed over the primary advantagesin using the Barkhauseneffect lies in its domainand obtain the energyper unit volume by dividing the simplicity and the fact it is a passivedevice. The Barkhausen exchangeenergy by the domain volume. We will then connect effect usedas a magneticsensor can be usedas a motion sensor the microscopic quantum mechanical process. to the macroscopicmeasuring process by equating this energy per -i”dt unit volume to the differentiatedformula for the energy stored (7) in a magneticfield as a changein energy due to the changein the magnetic field from a Barkhausenevent. We use the formula for energy storedin a magneticmaterial ( here iron): Carrying out the integration yields: ’ B2 (2) AJS2n2 -ioJr- ‘3 U=r &w, z) = [ (8) and diiferentiateit: 4niVNBAt e Au = 1zBAB (3) Multiplying the above quantity by its complex conjugate, we Y get: where the differential of the magnetic field representsa change in the magneticenergy due to a Barkhausenevent. This causes p-p4q (9) a changein the energy which we equateto the exchageenergy 16@2nV2N2B2At2 per unit volume of a domain due to a Barkhausenevent, assuming that the hysterisis curve is narrow enough (such as Assuming that the correlation time and the frequency are in silicon steel) so all of the quantum mechanicalexchange independentvariables, we can write the Fourier transform of a energygoes into the magnetic energy. Dividing both sides of Barkhausenemf event as: the differentiatedmagnetic equation by a differential unit of time which representsthe time duration of a Barkhausenevent we get: (10) AU The statistical distribution of the Barkhausen events will -=1&i! (4) assumeto be Poissondistributed [3], [4]. This is a probability At At p distibution that works well for Johnson noise arki hydrodynamic bursting events [3], [4]. Our probability Now we evokeFaraday ’slaw: distribution for Barkhausenevents is : (3 where A is taken as the cross sectionalarea of a domain which we assumeto be a rectangularsquare of 500 angstomsby 500 angsturns. The time duration of a Barkhausenevent can be To =At (12) estimatedform a plot in Bozorth’s book where he shows that a Which is the mean time duration of a Barkhausenevent. We typical Barkhausenevent causesa changein magnetic field of are now ready to average the spectral density over the about one Gauss [2]. Becausethe cross sectional area of a correlation time [5], [6], [7]. This yields: typical domain is known and the techniquefor calculating the a cahangein magnetic field from a Barkhausenevent has been B(w)=jg(z)lE(w, z)(‘ dr (13) dlscribed,we can experimentally measurethe inducedemf due 0 to a Barkhausenevent and use Faraday’slaw to determinethe Upon doing the integration we get: time duration of a Barkhausenevent. We are now in a position to combine formulae to obtain an expressionfor the induced 2A2 J2S4z4 emf due to a Barkhausen event in terms of microscopic IE(w)12= (14) quantummechniacal physical quantities: 4n2V2N2B2(1 + dr;) This is the equationfor the spectraldensity of Barkhausen A@S2n ’ noise - a potential EM1 problem. E=- (6) VNBAt CONCLUSIONS BARKHAUSEN NOISE SPECTRA DERIVATION In this paper , applications of the Barkhauseneffect used as a Taking the Fourier Transform of the induced emf of a magneticand motion sensorhave been discussed.This type of Barkhausenevent: magnetic field sensor will supplement magnetic field sensors alreadyin useand should have wide applicability in the EMC community. The author along with Ed Javor of the same 362 workplace performedan experiment at the 1997 IEEE EMC conferenceon Barkhausennoise. In that experiment,we placed a silicon steel sheetat the centerof a coil, and used an external magnetic field from a moving magnet to cause the magnetic domain walls to snap and click as they suddenly oriented themselves.This suddenorientation of the magnetic domains caused sudden changes in magnetization which produced impulses of emf which were heard as distinct clicks on a loudspeakerconnected to an amplifier which was connectedto the coil. This suddenorientation of the magnetic domain walls in the presenceof an external magnetic field is called the Barkhauseneffect. In this paper a time domain and statistical model of the Barkhausen effect as a magnetic sensor is presented . Equations are also developed which model Barkhauseneffect as a motion sensor by sensing magnetic fields in a moving referenceframe. modeled . The sudden orientation of the magnetic domain described by the Barkhauseneffect is mathematicallymodeled by calculatingthe energyneeded to changethe direction of the Bloch wall or the transition layer that separatesadjacent domains magnetizedin different directions The spectrum of the Barkhauseneffect is theoreticaIly derived in this paper. The Barkhausen effect becomesan EM1 preoblem when the magnetic domain size becomesof the order of the magnetic material used. This problem occurs in the recording industry when Barkhausen noise can be significant. This research formulates the Barkhausen effect as stochastic noise by considering a Barkhausenemf responseas a event in a stochasticprocess and obtaining a spectral density of this noise. This becomes a quantitative expressionfor Barkhausennoise as EM1 noise. REFERENCES [I] Kittel , C. Introduction to Solid State Physics , 5th edition, John Wiley Jr Sons 1976 [2] Bozorth, Richard M. ,Ferromagnetism , IEEE Press , New York , 1951 [3] Yariv , Amnon , Optical ElectTonics in M&em Communications , 5th edition , New York , 1997 [4] Anderson,T., Structure of Turbulence and Drag Reduction, IUTAM Symposium, Springer Verlag , 1989 [5] Bennett, W.R. Methods of Solving Noise Problems ,Proc. IRE 44:609 [6] Rice , S. 0. , Mathematical Analysis of Random Noise , Bell Syst. Technical Journal 23:282, 1944;24;46,1945 [7] Johnson , J.B. , Thermal Agitation of Electricity in Conductors , Phys. Rev. 32;97,1928 363 .
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