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Journal of Magnetism and Magnetic Materials 320 (2008) e583–e588

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Measurement of the stress sensitivity of magnetostriction in electricalsteels under distorted waveform conditions

Philip Anderson�

Wolfson Centre for Magnetics, School of Engineering, Cardiff University, Cardiff, UK

Available online 6 April 2008

Abstract

Electrical steel laminations used in the construction of transformer cores are subject to stresses introduced during their construction

and analysis of the effect of this on the magnetostriction of the lamination has been investigated previously. It has been shown

that higher harmonics of magnetostriction are of greater importance than the fundamental when considering transformer noise.

Whereas previous studies have concentrated on the magnetostriction harmonics generated by sinusoidal magnetization, this

investigation seeks to understand the relationship between harmonics present in the magnetization waveform and those in the

magnetostriction waveform. A measurement system has been designed based on a similar principle to one previously described.

In this case, a single Labview Virtual Instrument (VI) is used for the control of the applied stress, controlled magnetization

and measurement of magnetostriction together with other magnetic parameters such as specific total loss, specific apparent

power, permeability, coercivity and remanence. An adaptive digital feedback algorithm is utilized for control of arbitrary

waveform which may be constructed from discrete harmonics or read from an input waveform. As well as measuring peak

magnetostriction the software utilizes an FFT to calculate the harmonics of magnetostriction at each stress point. The effect of

harmonics introduced into the magnetization waveform on the magnetostriction harmonics will be shown at various applied stresses.

A harmonic, HarmB in the flux density waveform is shown to have the effect of producing a dominant harmonic in the magnetostriction

given by (HarmB+1)/2.

r 2008 Elsevier B.V. All rights reserved.

PACS: 07.55.�w; 75.50.Bb; 75.80.+q

Keywords: Magnetostriction; Electrical steel

1. Introduction

Electrical steel laminations used in the construction oftransformer cores are subject to stresses introduced duringtheir construction and analysis of the effect of this on themagnetostriction of the lamination has been investigatedpreviously [1].

Due to factors including the response of the humanear, higher harmonics of magnetostriction are of greaterimportance than the fundamental when consideringtransformer noise so appreciation of how harmonics inthe magnetizing waveform affect the harmonics of magne-tostriction is important.

- see front matter r 2008 Elsevier B.V. All rights reserved.

/j.jmmm.2008.04.014

44 29 20876729; fax: +44 29 20879538.

ddress: andersonpi1@cf.ac.uk

2. System design

A measurement system (Fig. 1) has been designed basedon a similar principle to one previously described by thisauthor [1].Magnetostriction is measured by accelerometer and

tensile and compressive stress is applied by a pneumaticstressing system. The new system uses a Labview VirtualInstrument (VI) is used for the control of the applied stress,controlled magnetization and measurement of magnetos-triction together with other magnetic parameters such asspecific total loss, specific apparent power, permeability,coercivity and remanence.An adaptive digital feedback algorithm described by

Zurek andMarketos [2] is utilized for control of an arbitrarymagnetization waveform, which may be constructed from

ARTICLE IN PRESSP. Anderson / Journal of Magnetism and Magnetic Materials 320 (2008) e583–e588e2

discrete harmonics or read from an input file. This wave-form is used to magnetize the test sample, in the form of anEpstein strip, at several predefined flux densities across arange of user selected stress values. Fundamental frequen-cies of 25Hz up to 5 kHz may be employed although thiscan be limited by the 5 g maximum acceleration measurableby the accelerometers, particularly for waveforms contain-ing higher harmonics. The sampling rate for the B, H andmagnetostriction waveforms (which are all simultaneouslysampled) is 1MS/s. The stress produced by the pneumaticcylinder is controlled by electro-pneumatic regulators whichare driven by a simple proportional control algorithm usinga load cell for feedback. Two accelerometers are used for themeasurement of magnetostriction. One is built into the

AIR Lc

Power Amp

Fig. 1. Schematic of the magneto

-5

0

5

10

15

20

-15

Applied Stress (M

m

S

-10 -5 0

Fig. 2. Typical peak magnetostriction, specific total loss and relative permeab

50Hz.

clamping system at the free end of the sample whilst theother is mounted in the opposite direction at the fixed end ofthe sample. The outputs of these are summed and doubleintegrated, using a numerical integration algorithm, to givedisplacement from which the strain waveform is calculated.The process is completely automated allowing a typical testwith 15 stress points and 2 flux densities to be completed inapproximately 3min. Output data from the system is in theform of a tabulated text file (for easy importing tospreadsheet software) and includes the following for eachstress value: zero to peak magnetostriction, peak to peakmagnetostriction, ‘A’ weighted magnetostriction [3], specifictotal loss (calculated by integrating h � dB/dt over a cycle),specific apparent power, relative permeability, coercivity,

Coupler

FixedEnd

Windings

oadell

striction measurement system.

Pa)

Peakagnetostriction(microstrain)

pecific total loss(x10 W/kg)

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Rel

ativ

e pe

rmea

bilit

y

Peak magnetostrictionSpecific total lossPermeability

15105

ility versus applied stress for a conventional grain-oriented steel at 1.5 T,

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-5.0

0.0

5.0

10.0

15.0

20.0

2000

No. of samples

Mag

neto

stric

tion

(mic

rost

rain

)

Sin30% 3rd10% 9th

0 200 400 600 800 1000 1200 1400 1600 1800

Fig. 4. Magnetostriction waveforms under the flux density waveforms in Fig. 3 in a conventional grain-oriented steel at 1.5T, 50Hz.

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0

No. of samples

Mag

netic

flux

den

sity

(T)

Sin30% 3rd10% 9th

200 400 600 800 1000 1200 1400 1600 1800 2000

Fig. 3. Example magnetic flux density waveforms applied to a conventional grain-oriented steel at 1.5T, 50Hz.

-5

0

5

10

15

20

-2.0

Magnetic flux density (T)

Mag

neto

stric

tion

(mic

rost

rain

)

sin30% 3rd10% 9th

-1.5 -1.0 -0.5 0.0 0.5 2.01.51.0

Fig. 5. Butterfly loops for the flux density waveforms in Fig. 3 in a conventional grain-oriented steel at 1.5 T, 50Hz.

P. Anderson / Journal of Magnetism and Magnetic Materials 320 (2008) e583–e588 e3

ARTICLE IN PRESSP. Anderson / Journal of Magnetism and Magnetic Materials 320 (2008) e583–e588e4

remanence, peak magnetic field strength and the first tenharmonics of magnetostriction.

3. Typical results

Typical results for zero to peak magnetostriction (to bereferred to as peak magnetostriction from this point),specific total loss and relative permeability versus appliedstress in a conventional grain-oriented electrical steel areshown in Fig. 2 for magnetization at 1.5 T, 50Hz.

Figs. 3–5 show the effect of adding harmonics into the B

waveform. Fig. 3 shows the controlled B waveforms, Fig. 4the measured magnetostriction waveforms and Fig. 5 thecharacteristic ‘‘butterfly loops’’ of flux density versusmagnetostriction, when adding 30% 3rd harmonic and10% 9th harmonic at a compressive stress of 10MPa.

-5

0

5

10

15

20

-10

Applied Stress (MPa)

Pea

k m

agne

tost

rictio

n(m

icro

stra

in)

Sin10% 3rd30% 3rd50% 3rd

-8 -6 -4 -2 0 2 4 6 8 10

Fig. 6. Peak magnetostriction versus applied stress in a conventional

grain-oriented silicon steel at 1.5T, 50Hz with increasing 3rd harmonic

content in the flux density waveform.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

sin% 3rd harmo

Pea

k m

agne

tost

rictio

n (m

icro

stra

in)

10 20

Fig. 7. Harmonics of peak magnetostriction in a conventional grain-oriented

harmonic content in the flux density waveform.

4. Effect of harmonics in flux density waveform on

harmonics of magnetostriction

Fig. 6 shows the stress sensitivity of peak magnetostric-tion in a conventional grain-oriented electrical steel at1.5 T, 50Hz for magnetization waveforms containing anincreasing amount of 3rd harmonic (the 3rd harmonic isalways in phase with the fundamental). The 3rd harmonichas been chosen in this case as it is very common to see asignificant percentage of 3rd harmonic in the magnetiza-tion waveform of electrical machines. Excluding measure-ments at very high compressive stress there is very littledifference between the measured characteristics suggestingthe acoustic noise generated under each of these conditionswould be similar. An analysis of the harmonics ofmagnetostriction reveals significant differences.Figs. 7 and 8 show the first 3 harmonics in the

magnetostriction waveform (i.e. 100, 200, 300Hz) forincreasing 3rd harmonic content of B at 5 and 10MPacompression, respectively. The higher harmonics are notshown here but are insignificant compared to those shown.In both cases a similar trend can be seen whereby

increasing 3rd harmonic in the flux density waveform leadsto decreasing amplitude of the fundamental of themagnetostriction waveform, which is almost mirrored byan increase in the second harmonic amplitude of themagnetostriction waveform. Interestingly, in both cases theamplitude of the third harmonic remains relatively small.Conventional thinking would lead to the conclusion that athird harmonic of 150Hz in the flux density waveformwould lead to a commensurate increase in the magnetos-triction waveform of a component at twice this frequency(300Hz) and therefore a dominant 3rd harmonic.Furthermore, Fig. 9 shows that increasing the percentage

of 5th harmonic in the flux density waveform results inboth the 2nd and 3rd harmonic of magnetostrictionincreasing but the 3rd harmonic becoming dominant.

nic in B waveform

fund2nd3rd

30 40 50

silicon steel at 1.5T, 50Hz under 5MPa compression with increasing 3rd

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0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

sin% 3rd harmonic in B waveform

Pea

k m

agne

tost

rictio

n (m

icro

stra

in)

fund2nd3rd

10 20 30 40 50

Fig. 8. Harmonics of peak magnetostriction in a conventional grain-oriented silicon steel at 1.5T, 50Hz under 10MPa compression with increasing 3rd

harmonic content in the flux density waveform.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

sin% 3rd harmonic in B waveform

Pea

k m

agne

tost

rictio

n (m

icro

stra

in)

fund2nd3rd4th5th

10 20 30 40 50

Fig. 9. Harmonics of peak magnetostriction in a conventional grain-oriented silicon steel at 1.5 T, 50Hz under 10MPa compression with increasing 5th

harmonic content in the flux density waveform.

-5

0

5

10

15

20

Mag

neto

stric

tion

(mic

rost

rain

)

Fig. 10. Single cycle of magnetostriction waveform under sinusoidal, 30%

3rd harmonic and 30% 5th harmonic flux density waveforms.

P. Anderson / Journal of Magnetism and Magnetic Materials 320 (2008) e583–e588 e5

This can be explained if we consider the magnetostric-tion waveform. A full cycle of the magnetostrictionwaveform occurs in one half cycle of the flux densitywaveform and is repeated identically in the second half.A single cycle of the magnetostriction waveform when theflux density is sinusoidal, with 30% 3rd harmonic and with30% 5th harmonic is shown in Fig. 10.

It can be seen that when in-phase odd harmonics aresuperimposed onto the B waveform they generate magne-tostriction harmonics such that the dominant magnetos-triction harmonic is given by

Harmmag ¼HarmB þ 1

2(1)

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where Harmmag is the dominant magnetostriction harmo-nic HarmB is the harmonic in the flux density waveform.

Using Eq. (1) it can be calculated that for a 7th harmonicin the flux density waveform the dominant harmonic in themagnetostriction waveform is the 4th harmonic and 9thharmonic in B generates 5th harmonic in magnetostriction.Experimentation has shown this to be the case.

5. Conclusions

An automated system for the measurement of magne-tostriction along with several other parameters underapplied stress and digitally controlled operational magne-tization waveforms has been described.

The effect of in-phase harmonics in the flux densitywaveform on the magnetostriction waveform has been

demonstrated. A harmonic, HarmB in the flux densitywaveform will have the effect of producing a dominantharmonic in the magnetostriction given by (HarmB+1)/2.

Acknowledgements

The author wishes to thank Cogent Power Ltd. for theirsupport of this work.

References

[1] P. Anderson, A.J. Moses, H.J. Stanbury, J. Magn. Magn. Mater.

215–216 (2000) 714.

[2] S. Zurek, P. Marketos, T. Meydan, A.J. Moses, IEEE Trans. Magn. 41

(2005) 4242.

[3] M. Ishida, S. Okabe, K. Sat, Kawasaki Steel Giho. 29 (4) (1997) 164.