On measurement and modelling of 2D magnetization and magnetostriction of SiFe sheets Anders Lundgren Royal Institute of Technology Electric Power Engineering Stockholm 1999 Anders Lundgren On measurement and modelling of 2D magnetization and magnetostriction of SiFe sheets TRITA-EEA-9901 ISSN 1100-1593 Department of Electric Power Engineering Royal Institute of Technology SE-100 44 Stockholm SWEDEN On measurement and modelling of 2D magnetization and magnetostriction of SiFe sheets Anders Lundgren Royal Institute of Technology Electric Power Engineering Stockholm 1999 Akademiskavhandlingsommedtillst(cid:23)andavKunglTekniskaH(cid:127)ogskolanframla(cid:127)gges till o(cid:11)entlig granskning f(cid:127)or avla(cid:127)ggande av teknisk doktorsexamen m(cid:23)andagen den 21 juni 1999 kl 14.00i Kollegiesalen, Administrationsbyggnaden, Kungl Tekniska H(cid:127)ogskolan, Valhallav(cid:127)agen 79, Stockholm. TRITA-EEA-9901 ISSN 1100-1593 cAnders Lundgren, 1999 (cid:13) KTH Reprocentral, Stockholm 1999 Abstract The development and technological aspects of a 2D magnetization and magne- tostriction measurement setup are documented and described. Local magnetic in- tensity and (cid:13)ux density are measured with Rogowskiand material encircling coils. In-plane strain is measured with a homodyne laser interferometer. Measured and processed time-domain signals, hysteresis plots and signature data such as loss are presentedbyane(cid:14)cientandcommunicativeinterface. Measurementsonquadratic siliconironsheetsamplesareincluded. Materialtypestestedonthesetuparewith non-oriented and oriented textures. Possible excitations include uniaxial alternat- ingmagnetic(cid:12)eldintherollingandtransversedirectionsbetween10and300Hzat least. Rotational excitations are possible at least for the non-oriented and conven- tional grain-orientedtypes. The value of the setup lies in the possibility of using it for routine measurements on samples. The interplay between mathematical modelling and physical experimenting is de- scribed. Investigationsby algebraicandnumericalmethods aredone to(cid:12)nd a pos- sible way to parameterize material behaviour and include this behaviour in (cid:12)nite element programs. On the basis of a proposed one-dimensional nonlinear model, algorithms are devised to compute magnetostrictive responses to uniaxially alter- nating magnetic (cid:12)elds. An experimental FEM program to calculate strain (cid:12)elds frominhomogeneousmagnetizationisdeveloped. Itsuseforinvestigationofsample behaviour during the operation of the setup is shown. The value of the proposed modelling methodology lies in the study of possibilities of loweringthe production of magnetostrictive vibration in transformer,motor and generatorcores. IEEE index terms: Magnetostriction, silicon steel, magnetic cores, strain, inter- ferometry, magnetic anisotropy, magnetic (cid:12)elds, magnetic measurements, magne- toelasticity, nonlinear magnetics, power transformers, power distribution acoustic noise, (cid:12)nite element methods. TRITA-EEA-9901 ISSN 1100-1593 Acknowledgements I would like to thank the members of the reference committee, Jan Anger (ABB Transformers), Thomas Edstro(cid:127)m (ABB Corporate Research) and Birger Nilsson (ABBCorporateResearch)andElektraprogrammemanagerStenBergman(Elforsk AB) for their work in supporting this project. On the department side I owe thanks to the project manager G(cid:127)oran Engdahl for energizingtheproject,applying forfunding andproofreading. Headofdepartment RolandErikssonisthankedforemployingmeandforadministeringthe(cid:12)nancesand agreements. I especially wish to thank former research associate Anders Bergqvist for many stimulating discussions and collaborations. I thank Olle Br(cid:127)annvall, Go(cid:127)te Bergh and Yngve Eriksson for making parts to the experimental setup and trans- porting it. I send greetings to friendly department colleagues Eckart Nipp, Niklas Magnusson, Fredrik Stillesj(cid:127)o, Mats Kvarngren and Anders Helgesson. I (cid:12)nally express heartily thanks to my girlfriend Cecilia H(cid:127)aggmark for her encour- agement, proofreading and general support. Anders Lundgren i Contents 1 Introduction 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivation and goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 Magnetic hysteresis models . . . . . . . . . . . . . . . . . . . 5 1.4.2 Magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.3 Stress dependence . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.4 Measurement methods . . . . . . . . . . . . . . . . . . . . . . 12 1.4.5 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.6 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Measurement system 17 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Drawing and design system . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Data acquisition programs . . . . . . . . . . . . . . . . . . . . . . . . 21 ii 2.5 Magnetic circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Excitation frequency limits . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Voltage or current sti(cid:11) ampli(cid:12)er . . . . . . . . . . . . . . . . . . . . 28 2.8 B-coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.9 Calibration of the H-coil . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.10 Measurement table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.10.1 Support placement . . . . . . . . . . . . . . . . . . . . . . . . 33 2.10.2 Optic component placement . . . . . . . . . . . . . . . . . . . 33 2.11 Vibration of material . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.12 Digital control issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.13 Strain measurement by interferometry . . . . . . . . . . . . . . . . . 36 2.14 Stress in(cid:13)uence, frame e(cid:11)ect . . . . . . . . . . . . . . . . . . . . . . . 37 2.15 Yoke design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.16 Magnetic sensor design . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.17 Temperature drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.18 Signal conditioning and Nyquist limit . . . . . . . . . . . . . . . . . 39 2.19 Signal bu(cid:11)ering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.20 Measurement coil misalignments . . . . . . . . . . . . . . . . . . . . 43 2.21 Using the measurement system . . . . . . . . . . . . . . . . . . . . . 43 2.21.1 Magnetic measurements . . . . . . . . . . . . . . . . . . . . . 43 2.21.2 Peak (cid:13)ux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.21.3 Measurement procedure . . . . . . . . . . . . . . . . . . . . . 44 3 Interferometer 46 iii 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Homodyne interferometry . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Heterodyne interferometry . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Interferometer alignment . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Doppler e(cid:11)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 Motion of measurement table . . . . . . . . . . . . . . . . . . . . . . 52 3.7 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.8 The acousto-opticmodulator . . . . . . . . . . . . . . . . . . . . . . 60 3.9 Beam splitters and prisms . . . . . . . . . . . . . . . . . . . . . . . . 62 3.10 Interference (cid:12)lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.11 Photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.12 Demodulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.13 Interferometer type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.14 Re(cid:13)ector placements and properties . . . . . . . . . . . . . . . . . . 65 4 Strain analysis 68 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 De(cid:12)nitions of observables . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.1 2D strain measurement analysis. . . . . . . . . . . . . . . . . 73 4.2.2 Deformation of volume elements . . . . . . . . . . . . . . . . 76 4.3 Stress and 3D elastic material relations. . . . . . . . . . . . . . . . . 77 4.4 2D elastic material modelling . . . . . . . . . . . . . . . . . . . . . . 80 4.4.1 Magnetostriction components and constitutive relations . . . 80 4.4.2 Elasticity and compliance matrices . . . . . . . . . . . . . . . 85 iv