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Modelling of Barkhausen envelop for the characterization of magnetic materials08 October 2020

Patrick FAGAN

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Plan

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Introduction: theoretical basis for the Barkhausen Noise Simulation: Jiles-Atherton-Sablik and multi-scale model Experimental setup and results

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Introduction: the Barkhausen noise

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At micrometric scale, magnetic domains surrounded by domain wallsUnder external magnetizing fields, the domain walls move in such a way to enlarge domains whose internal magnetization M is aligned with the external field H

Faraday Law

M

𝑽𝑽𝑩𝑩𝑩𝑩 𝒕𝒕 ∝ 𝒅𝒅𝒅𝒅𝒅𝒅𝒕𝒕→ 𝑽𝑽𝑩𝑩𝑩𝑩 𝒕𝒕 ∝ 𝒅𝒅𝒅𝒅

𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒕𝒕

(𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅

spatial derivative of M, 0 inside a domain and supposed constant in domain walls)

In this case,𝑽𝑽𝑩𝑩𝑩𝑩 𝒕𝒕 ∝ 𝒅𝒅𝒅𝒅𝒅𝒅𝒕𝒕→ induced voltage is proportional to domain wall speed

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Introduction: the Barkhausen noise

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Proposed setup: use a coil to detect the induced Barkhausen noise voltage

Sum at macroscopic scale: 𝑽𝑽𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒕𝒕 = ∑𝑽𝑽𝑽𝑽𝑩𝑩𝑩𝑩(𝒕𝒕,𝒅𝒅,𝒚𝒚, 𝒛𝒛)Constructive and destructive superposition between induced noises from all the domain wall movements considered→ RMS, and more recently MBNE:

𝒅𝒅𝑩𝑩𝑴𝑴𝑴𝑴 𝒕𝒕 = �𝟎𝟎

𝒕𝒕𝒔𝒔𝒄𝒄𝒔𝒔𝒔𝒔

𝒅𝒅𝒅𝒅𝒅𝒅𝒕𝒕

𝑽𝑽𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝟐𝟐 𝒕𝒕 𝒅𝒅𝒕𝒕

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

From the Barkhausen noise to the MBNE(H)

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4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Simulation: the Jiles-Atherton-Sablik model

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Input: external magnetization field H, strain 𝝈𝝈, magnetostriction λ, 5 parameters and an anhysteretic function 𝒅𝒅𝒂𝒂𝒔𝒔 𝒅𝒅Output: differential equation of M in function of H

𝑑𝑑𝑑𝑑𝑑𝑑𝐻𝐻𝑒𝑒

= 𝑑𝑑𝑑𝑑𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝐻𝐻𝑒𝑒

+ 𝑑𝑑𝑑𝑑𝑖𝑖𝑟𝑟𝑟𝑟𝑑𝑑𝐻𝐻𝑒𝑒

𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅

=𝟏𝟏

𝟏𝟏 + 𝒄𝒄𝒅𝒅𝒂𝒂𝒔𝒔 −𝒅𝒅𝒄𝒄𝒊𝒊𝒊𝒊

𝜹𝜹𝒌𝒌 − 𝜶𝜶(𝒅𝒅𝒂𝒂𝒔𝒔 −𝒅𝒅𝒄𝒄𝒊𝒊𝒊𝒊)+

𝒄𝒄𝟏𝟏 + 𝒄𝒄

𝒅𝒅𝒅𝒅𝒂𝒂𝒔𝒔

𝒅𝒅𝒅𝒅𝒆𝒆

𝐻𝐻𝑒𝑒 = 𝐻𝐻 + 𝛼𝛼𝑀𝑀𝑖𝑖𝑖𝑖𝑖𝑖 + 𝐻𝐻𝜎𝜎𝑀𝑀𝑖𝑖𝑒𝑒𝑟𝑟 = 𝑐𝑐 𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀𝑖𝑖𝑖𝑖𝑖𝑖

𝑀𝑀𝑎𝑎𝑎𝑎 = 𝑀𝑀𝑠𝑠 𝑓𝑓𝐻𝐻𝑒𝑒𝑎𝑎

𝑑𝑑𝑀𝑀𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝐻𝐻𝑒𝑒

=𝑀𝑀𝑎𝑎𝑎𝑎 −𝑀𝑀𝑖𝑖𝑖𝑖𝑖𝑖

𝛿𝛿𝑘𝑘 − 𝛼𝛼(𝑀𝑀𝑎𝑎𝑎𝑎 −𝑀𝑀𝑖𝑖𝑖𝑖𝑖𝑖)

𝛿𝛿 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑑𝑑𝐻𝐻𝑑𝑑𝑑𝑑

𝐻𝐻𝜎𝜎 =32𝜆𝜆𝜇𝜇0

𝜕𝜕𝜆𝜆𝜕𝜕𝑀𝑀

(cos2 𝜃𝜃 − 𝜈𝜈 sin2 𝜃𝜃) 𝒅𝒅𝒅𝒅𝒅𝒅𝝈𝝈

=𝟏𝟏𝝐𝝐𝟐𝟐𝝈𝝈(𝒅𝒅𝒂𝒂𝒔𝒔 −𝒅𝒅) + 𝒄𝒄

𝒅𝒅𝒅𝒅𝒂𝒂𝒔𝒔

𝒅𝒅𝝈𝝈

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Simulation: the Jiles-Atherton-Sablik model

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Advantages Easy to integrate Parameters linked to material characteristics Separation between reversible and irreversible componentsProblems Unsatisfying reconstruction with narrow hysteresis loops Verification of convergence, given a set of parameters, not available

𝑑𝑑𝑀𝑀𝑑𝑑𝐻𝐻

=1

1 + 𝑐𝑐𝑀𝑀𝑎𝑎𝑎𝑎 −𝑀𝑀𝑖𝑖𝑖𝑖𝑖𝑖

𝛿𝛿𝑘𝑘 − 𝛼𝛼(𝑀𝑀𝑎𝑎𝑎𝑎 −𝑀𝑀𝑖𝑖𝑖𝑖𝑖𝑖)+

𝑐𝑐1 + 𝑐𝑐

𝑑𝑑𝑀𝑀𝑎𝑎𝑎𝑎

𝑑𝑑𝐻𝐻𝑒𝑒𝑀𝑀𝑎𝑎𝑎𝑎 = 𝑀𝑀𝑠𝑠 𝑓𝑓

𝐻𝐻𝑒𝑒𝑎𝑎

Parameter Name

a (A / m) Domain walls density

𝛼𝛼 Interdomain coupling coefficient

c Reversibility coefficient

k (A / m) Average pinning energy

𝑀𝑀𝑠𝑠 (A / m) Saturation magnetization

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Simulation: the multiscale model

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Idea: find M by simulating the material at different scales: magnetic domain, monocrystal, polycrystal

Magnetic domainUniform strain and magnetization

MonocrystalUniform strain, variable magnetizations

PolycrystalVariable strains and magnetizations

Proportion of each orientation of M found by solving a minimum energy problem

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Simulation: the multiscale model

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PolycrystalHypothesis: 𝒅𝒅 = 𝒅𝒅𝒔𝒔 for each domain, spherical coordinate system

𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑𝑎𝑎𝑖𝑖𝑎𝑎 = 𝐸𝐸𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑎𝑎𝑎𝑎𝑒𝑒𝑒𝑒 + 𝐸𝐸𝑑𝑑𝑎𝑎𝑒𝑒𝑎𝑎𝑒𝑒𝑚𝑚𝑑𝑑𝑠𝑠𝑚𝑚𝑎𝑎𝑚𝑚𝑖𝑖𝑒𝑒 + 𝐸𝐸𝑑𝑑𝑎𝑎𝑒𝑒𝑎𝑎𝑒𝑒𝑚𝑚𝑑𝑑𝑒𝑒𝑚𝑚𝑎𝑎𝑠𝑠𝑚𝑚𝑖𝑖𝑒𝑒 + 𝐸𝐸𝑑𝑑𝑎𝑎𝑒𝑒𝑎𝑎𝑒𝑒𝑚𝑚𝑑𝑑𝑒𝑒𝑖𝑖𝑦𝑦𝑠𝑠𝑚𝑚𝑎𝑎𝑚𝑚𝑚𝑚𝑖𝑖𝑎𝑎𝑒𝑒 𝑎𝑎𝑎𝑎𝑖𝑖𝑠𝑠𝑑𝑑𝑚𝑚𝑖𝑖𝑑𝑑𝑎𝑎𝑦𝑦

𝐻𝐻 = 𝐻𝐻𝑠𝑠𝑥𝑥𝐻𝐻𝑦𝑦𝐻𝐻𝑧𝑧𝐻𝐻

,𝑀𝑀 = 𝑀𝑀𝑠𝑠

𝑥𝑥𝑑𝑑𝑦𝑦𝑑𝑑𝑧𝑧𝑑𝑑

, 𝑥𝑥2+𝑦𝑦2 + 𝑧𝑧2 = 1

𝐸𝐸𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑎𝑎𝑎𝑎𝑒𝑒𝑒𝑒 = 𝐴𝐴 𝑠𝑠𝑔𝑔𝑎𝑎𝑑𝑑 𝑀𝑀2

(0 inside a magnetic domain)

𝐸𝐸𝑑𝑑𝑎𝑎𝑒𝑒𝑎𝑎𝑒𝑒𝑚𝑚𝑑𝑑𝑠𝑠𝑚𝑚𝑎𝑎𝑚𝑚𝑖𝑖𝑒𝑒 = −𝜇𝜇0𝐻𝐻 ⋅ 𝑀𝑀𝐸𝐸𝑑𝑑𝑎𝑎𝑒𝑒𝑎𝑎𝑒𝑒𝑚𝑚𝑑𝑑𝑒𝑒𝑚𝑚𝑎𝑎𝑠𝑠𝑚𝑚𝑖𝑖𝑒𝑒 = −𝜎𝜎0 ∶ 𝜖𝜖 + 1

2𝜖𝜖 ∶ ℂ ∶ 𝜖𝜖

𝐸𝐸𝑑𝑑𝑎𝑎𝑒𝑒𝑎𝑎𝑒𝑒𝑚𝑚𝑑𝑑𝑒𝑒𝑖𝑖𝑦𝑦𝑠𝑠𝑚𝑚𝑎𝑎𝑚𝑚𝑚𝑚𝑖𝑖𝑎𝑎𝑒𝑒 𝑎𝑎𝑎𝑎𝑖𝑖𝑠𝑠𝑑𝑑𝑚𝑚𝑖𝑖𝑑𝑑𝑎𝑎𝑦𝑦 = 𝐾𝐾1 𝑥𝑥2𝑦𝑦2 + 𝑦𝑦2𝑧𝑧2 + 𝑥𝑥2𝑧𝑧2 + 𝐾𝐾2 𝑥𝑥2𝑦𝑦2𝑧𝑧2 (cubic crystal material)

𝛼𝛼 𝜃𝜃,𝜙𝜙 =𝑒𝑒−𝐴𝐴𝑠𝑠 𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑑𝑑 (𝜃𝜃,𝜙𝜙)

∑𝑎𝑎𝑚𝑚𝑚𝑚 𝑑𝑑𝑖𝑖𝑖𝑖𝑒𝑒𝑒𝑒𝑚𝑚𝑖𝑖𝑑𝑑𝑎𝑎𝑠𝑠 𝑒𝑒−𝐴𝐴𝑠𝑠 𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑑𝑑𝑀𝑀 = ∑𝑎𝑎𝑚𝑚𝑚𝑚 𝑑𝑑𝑖𝑖𝑖𝑖𝑒𝑒𝑒𝑒𝑚𝑚𝑖𝑖𝑑𝑑𝑎𝑎𝑠𝑠 𝛼𝛼 𝜃𝜃,𝜙𝜙 𝑀𝑀 𝜃𝜃,𝜙𝜙

𝑀𝑀 𝜃𝜃,𝜙𝜙 =𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃𝑐𝑐𝑠𝑠𝑠𝑠𝜙𝜙𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠𝜙𝜙𝑐𝑐𝑠𝑠𝑠𝑠𝜃𝜃

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Experimental Setup

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Sampling at 500 kHz (current, Hall signal, Barkhausen noise, analogical square and MBNE, dB/dt)Barkhausen signal filtered with a 4th order band-pass filter (between 1 and 10 kHz)

Physical experiment

Signal treatment and acquisitionSimulation

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Experimental results

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Comparison B(H) - MBNE

MBNE renormalized to have the same value at saturation as the inflexion elbow of B(H) (begin of saturation)

Strong anisotropy materials (FeSi) -> rotation negligible comparing to domain wall movements -> strong similarity between B(H) and MBNE almost equals

Weak anisotropy materials (FeCo) -> rotation not negligible, even at saturation -> B(H) et MBNE different, especially after saturation

Strong similarity B(H) – MBNE around Hc

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

Experimental results

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B(H) simulation with Jiles-Atherton-Sablik

Same Jiles-Atherton parameter for bothB(H) and MBNE, anhysteretic functionobtained thanks to the multiscale model

Similar results -> inverse reconstructionfeasible? (from MBNE to B(H))

4 novembre 2020Commissariat à l’énergie atomique et aux énergies alternatives Auteur

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