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Theory and Practice – Making use of the Barkhausen Effect
David C. Jiles Anson Marston Distinguished Professor Palmer Endowed Chair Department of Electrical & Computer Engineering Iowa State University
Workshop on Large Fluctuations and Collective Phenomena in Disordered Materials May 18, 2011 Summary
• Stochastic/deterministic Barkhausen model – Equations describing the phenomenon – Comparison of theory and experiment • Variation of emissions with other factors such as stress – Variation of Barkhausen signal amplitude – Theoretical predictions – inverse law • Variation of signal with frequency and distance – Possibilities for depth profiling of properties – Challenges
2 www.ece.iastate.edu learn invent impact The variation of magnetization with magnetic field looks deceptively simple
Yet behind it lies a large number of very complicated interrelated mechanisms which are hard to model 1 2 3 4 A further problem is that magnetic behavior is inhomogeneous and does not “scale” easily as dimensions change The result has been a number of different models which do not fit together very well
www.ece.iastate.edu learn invent impact Langevin-Weiss Model
• An array of magnetic moments at temperature T and in a magnetic field H will distribute themselves among the available energy states according to the probability distribution P
P = P0 exp (− µ0mH / kBT ) • Integrating over all possible angles and normalizing gives
M = Nm [coth (µ0mH / kBT )− (kBT / µ0mH )] • Weiss extended this to include internal coupling proportional to the magnetization He = aM with the result
M = Nm [coth (µ0m(H +αM )/ kBT )− (kBT / µ0m(H +αM ))] which decribes the anhysteretic magnetization curve for ferromagnets www.ece.iastate.edu learn invent impact Anhysteretic magnetization curves with different temperatures and different anisotropies M M Low temperature Ms + Ms 1.0
High temperature -15 -10 -5 5 10 15 H H
3D 2D
-1.0 1D
"Superparamagnetic magnetization equation in two dimensions," D.C. Jiles, S.J. Lee, J. Kenkel and K. Metlov. Applied Physics Letters , 77, 1029, 2000.
www.ece.iastate.edu learn invent impact Energy Dissipation and Hysteresis
• Energy is only dissipated by irreversible magnetization processes
• A model with dissipation proportional to change in M gives good results M M n επ Epin ()M = ∫dM irr =µok∫dM irr 2m 0 0
• Energy output (change in magnetostatic energy) must equal energy input minus energy dissipated due to any losses such as hysteresis
µ Mirr dH e = µ Man dH e − µ k dM 0 ∫ 0 ∫0 ∫ irr
www.ece.iastate.edu learn invent impact Irreversible Changes in Magnetization
• Differentiating the energy equation lead to a differential for the rate of change of magnetization
dM irr 1 = ()M an − M irr dH e k
• This can be written in the form of a differential with respect to the applied field H
dM irr 1 = ()Man −Mirr dH k −α()Man −Mirr
www.ece.iastate.edu learn invent impact Isotropic Model of Hysteresis
• The irreversible component of magnetization varies according to the differential equation
dM irr 1 = ()Man −Mirr dH k −α()Man −Mirr • The reversible component of magnetization varies as dM dM dM rev = c an − irr dH dH dH • Therefore hysteresis can be represented in terms of Ms, a, α, k, and c using the equation dM 1 (M − M ) c dM = an + an dH ()1+ c k −α ()M an − M ()1+ c dH
"Determination of theoretical parameters for modelling bulk magnetic hysteresis properties using the theory of ferromagnetic hysteresis," D.C. Jiles, J.B. Thoelke and M.K. Devine. IEEE Trans. Mag. www.ece.iastate.edu learn invent impact Comparison of Model and Experimental Measurement
• Comparison of measured and modelled hysteresis curves of cobalt modified gamma iron oxide material
After P. Andrei and A. Stancu, “Hysteresis in particulate recording media. Experiment and simulation with Preisach and Jiles-Atherton models,” Journal of Magnetism and Magnetic Materials , 206 , 160-164, 1999.
www.ece.iastate.edu learn invent impact Other magnetic properties - Barkhausen effect
1.5 H Virr 1.0 0.5 V 0.0 -0.5 Voltage Max/ Voltage -1.0 Barkhausen -1.5 0Time (s) 0.006
Barkhausen signal shown as voltage against time. The time dependence of magnetic field H, voltage in flux coil V, and Changes in Barkhausen emissions in envelope of irreversible component of voltage in flux coil V irr carbon steel as a result of applied are also shown on the same time scale. tensile stress.
www.ece.iastate.edu learn invent impact Stochastic Process Barkhausen model
Model assumes Barkhausen activity is proportional to the rate of irreversible change in magnetization
• •
Avalanches: Number of Barkhausen events N(t n) in a given time interval is correlated with number of events in the previous time interval tn-1
N(tn) = N(tn−1)+∆N(tn−1)
δrand : Random number ∆N(tn−1) =δrand N(tn−1) (-1.47 < δrand < 1.47) • • ' ' ' MBE (tn) =< Mdisc > χrr HN (tn−1)+δrand N (tn−1) www.ece.iastate.edu learn invent impact Comparison of model with experiment
www.ece.iastate.edu learn invent impact Other Factors – Effects of Stress
• Applied stress can be treated in most respects like an effective magnetic field which changes the anisotropy of the material 3 σ ∂λ Hσ = 2 µo ∂M T • Later this was extended to cover the case of a uniaxial stress at an arbitrary direction to the applied magnetic field
3 σσσ 2 2 ∂λ∂λ∂λ Hσσσ()θθθ = ()cos θθθ − νννsin θθθ 2µµµo ∂∂∂M T σσσ is the stress, θθθ is the angle between the stress axis and the
direction of H σσσ, and ν is Poisson’s ratio.
www.ece.iastate.edu learn invent impact Modelled Effects of Stress on Magnetic Moment Orientation
Positive magnetostriction stress applied along the y axis
www.ece.iastate.edu learn invent impact Detection of Stress
Specimens calibration tested under load in Instron Tensile Test machine
BN sensor in the middle of gauge length of sample
Typical Magnetising and Analysis Conditions :
Magnetising frequency : f = 300 Hz Magnetising voltage : V = 12 volt No of bursts : 20 Analysing bandwidth frequency : fa = 20-1250 KHz (about nearest 100 µm) Smoothing parameter: sp = 100 Sampling frequency: fs =2.5 MHz
15 www.ece.iastate.edu learn invent impact Measured Barkhausen Results
Core Surface
Strip #3#20#36 - -Depth -Depth Depth = = 22.9mm =11.2mm 0.2mm 1.5 RMSRMS = =0.187= 0.0100.029 volt volt volt
1.0
0.5
0.0 0.0
-0.5 -0.5 BarkhausenBarkhausen effecteffect signalsignal (V) (V) -1.0 Barkhausen effect signal (V) -1.0
-1.5 0.00 0.02 0.04 0.06 0.08 0.10 -1.5 0.00 0.02Time 0.04 (sec) 0.06 0.08 0.10 Time (sec)
16 www.ece.iastate.edu learn invent impact Effects of Stress on Barkhausen Emissions Applied stress causes a change in anisotropy energy of magnetic moments 3 E = − σλ( cos 2 θ − v sin 2 θ ) σ 2 This can be expressed as an equivalent field
1 ∂Eσ 3 σ ∂λ Hσ = − = µ0 ∂M 2 µ0 ∂M σ The total field is then the sum of magnetic field, exchange field and
“stress-equivalent field” H σ 3λ σ H H H M H s M M e = + σ +α = + 2 + α µoMs The magnetization is then an anisotropic function of this total field H + Hσ +αM a M an ()H,σ = M s coth − a H + Hσ +αM www.ece.iastate.edu learn invent impact Effects of Stress on Barkhausen Emissions
dM 1 1 ′() an () χ σ = H,σ ≅ Ms = Ms dH 3a − ()ασ +α Ms 3λσ a s M 3 − 2 +α s µoMs
This predicts how the rate of change of magnetization with field depends on stress and so can be used to calculate stress. However there is a much easier way... 3λsσ 3a−α + 2 Ms 1 3a−()α +α M µoMs = σ s = χ′(σ) Ms Ms
1 1 3λsσ which predicts a straight line graph = − 2 χ′(σ) χ′ )0( µoMs of 1/χ’ against σ
18 www.ece.iastate.edu learn invent impact Detection of Tensile Stress - test results
Applying tensile stress with monotonically increasing load 0 M P a 1 0 0 0 50 M P a 100 M P a 9 0 0 150 M P a 200 M P a 8 0 0 250 M P a 300 M P a 7 0 0 350 M P a 400 M P a 450 M P a 6 0 0 500 M P a 550 M P a 5 0 0 600 M P a 650 M P a 4 0 0 700 M P a 750 M P a 3 0 0 800 M P a BN peak amplitudepeak(mp) BN 850 M P a 2 0 0 900 M P a 950 M P a 1 0 0 Yield strength = 993MPa
0 -100 -80 -60 -40 -20 0 20 40 60 80 100 Magnetic Field Strength H (in % )
19 www.ece.iastate.edu learn invent impact Dependence of Barkhausen emissions on stress
Fig. 1. Envelope curves of the rectified Fig. 2. MBN Peak Amplitude for carburized MBN bursts for carburized SAE 9310 SAE 9310 specimen as a function of applied specimen for different amplitudes of applied stress. stress using a tensile test machine (residual stress value from XRD measured before stress application was -805MPa). 20 www.ece.iastate.edu learn invent impact Effects of Stress on Differential Susceptibility and Barkhausen Emissions
• Barkhausen voltage V MBE is known from previous work* to be proportional to the differential susceptibility
• Therefore, a similar linear expression should hold for reciprocal Barkhausen voltage as a function of stress as for the reciprocal differential susceptibility 1 1 3b′σ = − VMBE (σ) VMBE )0( µo
• This suggests that the most useful calibration curve for Barkhausen effect as a function of stress is the reciprocal plot
www.ece.iastate.edu learn invent impact Comparison of Model equation with Measurements
www.ece.iastate.edu learn invent impact Conclusions
• Magnetic properties, including permeability and Barkhausen effect depend on other external factors such as stress.
• A phenomenological/stochastic model has been developed to describe Barkhausen and these effects
• Development of models is essential for understanding and interpretation of measurement results and for predicting changes, such as with stress.
• Measurements of these properties can be used for determination of stress and even its variation with depth.
23 www.ece.iastate.edu learn invent impact www.ece.iastate.edu learn invent impact Depth Profiling using Barkhausen Effect
There are three key equations • The change in flux density that propagates to the surface is ω2 xmax x Bmeas ,0( xmax ,ω1,ω2) = ∫ ∫ Borigin (x,ω) exp − dx dω ω1 0 δ • The voltage measured in an induction coil on the surface is
ω2 xmax dB (x,ω,σ ) dH (x) x V ,0( x ,ω ,ω ) = NA origin exp − dx dω meas max 1 2 ∫ ∫ dH dt δ ω1 0 • The component of the measured signal coming from a particular depth is −4 d ′ () V x,( ω1,ω2) = 3 Vmeas ,0 xmax ,ω1,ω2 x µoµrσ dω
www.ece.iastate.edu learn invent impact SAE 9310 CEMENTATO (CARBURIZED) 0,70 mm Comparison with X-ray results SN 107 A1 Unground 0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0,2 Ground initial 600 500 side surface surface TOP BOTTOM 400 300 200 BN PEAK VALUE 939,9 106,0 1416,6 1035,4 100 0 -100 -200 BN PEAK POSITION 59,15 48,6 45,4 46,2 -300
Tensione Residua - -400
Residual Stress (MPa) -500 -600 profondità - depth (mm) Envelope curves of the rectified bursts for107A1 sample
1500 BN RMS vs distance (side surface)
800 1200
700
900 600
500
600 400
300 BN RMS [mp] BN PEAK VALUE [mp] 300 200
100
0 0 0 10 20 30 40 50 60 70 80 90 1001 2 3
magnetising current I[%] distance
bottom surface side surface initial surface top surface 90 180 270 360
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