NONLINEARITY ESTIMATION BY USING THE HARMONIC DISTORTION METHOD OF SOUND

WAVES PROPAGATION IN A FERROMAGNETIC SAMPLE

Nicolae CREŢU, Mihail POP

Physics Department , Transilvania University Brasov

Nicolae Creţu: cretu.c@unitbv.ro, Mihail Pop: mihailp@unitbv.ro

SISOM 2007andHOMAGIAL SESSION OF THE COMMISSION OF ACOUSTICSBUCHAREST, 29-31 May

About the nonlinearity

Classical nonlinear acoustics

2

22 2

1( ) ( )

3

2( ) ( )

2 3

( ) 0

( , )

i k lik

k i l

vv v p v v

t

v v vST v S v k T

t x x x

vt

p p S

Obtaining the nonlinear acoustics equations:

•

0

20

2

0

22

2222

2 2 2 20 0 0

0

1' ....

2

' '

' '

' 1 ''

2

1 1 1

2

o

c

Bdescribe the nonlinearity leve

p p p

p c

p A

lAv

B

t A tc t c c

B

For cubic crystalline elastic media

• For a longitudinal wave which propagate in the direction(100)

• For a transversal wave which propagate in the direction(100) :

• For a transversal wave propagating in (110) direction:

• The density of the elastic energy of the cubic crystall:

11l

Cv

44t

Cv

11 121( )2

t

C Cv

2 2 2 2 2 211 44 12

1 1( ) ( ) ( )

2 2xx yy zz yx xy zx yy zz zz xx xx yyW C e e e C e e e C e e e e e e

The influence of the magnetocrystalline anisotropy

for a ferromagnetic crystall

• The stable value of the strain and of the energy will be:

2 2 2 2 2 21 1 2 1 3 2 3( )mW K

2 20

2 1(sin )

3 2W B C

22 2 20

0 0

200

1 1 3 3(1 sin ) (1 sin )

2 2 2 2

3(1 sin )

2

2

3

W CC

C

B

C

magnetostriction

Magnetostriction behavior

• Magnetostriction shows a nonlinear behavior vs magnetic field

• Magnetostriction has two regions with strong nonlinearity vs magnetic field

• The propagation of the elastic waves in ferromagnetic samples becomes a nonlinear acoustic propagation especially in these regions

Experimental setup

• Calibrated signal generator sine-wave

• Linear power amplifier

• Laser Velocity Transducer

• Preamplifier

• USB External data acquisition board

• Computer for Signal Analysis

• Magnetic coils

• Calibrated current source

• Magnetic fluxmeter

• Magnetization and magnetostriction measurements were made in CNRS Lab Magnetisme, Grenoble.

Matrix method

• Fourier coefficients of the implied waves:

• Transfer matrix

dCtLU

dBtU

dAtU

e

e

e

ti

dtransferre

ti

rejected

ti

injected

2

1,

2

1,0

2

1,0

A

BT Z Z

CR L

, ,

0

Obtaining of transfer matrix

• For the ferromagnetic sample:

T Z Z D Z Z P k a D Z Z

T Z Z D Z Z P k a D Z Z

R L j j j jj

N

L R N j N j N j N jj

N

, , , ,

, , , ,

0 1 10

1

1 1 10

1

1 0

0

0

1

1

Za

Z

Va

V

transfer matrix is a nonlinear matrix

Nonlinear considerations

• Because the ferromagnetic medium has a nonlinear behavior, the matrix is also nonlinear. The process of the sound propagation in such media is very complicated and was solved only in some particular cases [4].

• Is possible to estimate the nonlinearity from the sound propagation measurements, by considering the ferromagnetic sample as a nonlinear system. From the theory of the nonlinear systems we know that if we inject a monochromatic signal in such sample, the output signal will contain a number of harmonics and the more the nonlinearity means the higher number of harmonics.

Total Harmonic Distortion Method

• Total Harmonic Distorsion Analysis (THD) is an analysis in the frequency domain. To determine the amount of nonlinear distorsion we need to measure the amplitude and the frequency of the fundamental and of the harmonics when in the sample is injected a monochromatic signal . By definition :

with An the corresponding amplitudes of the output signal:

2 2 2 21 2 3

1

...(%) 100nA A A A

THDA

• THD vs magnetic field

• An extension of the THD Method , based on the energy conservation. If the input energy is distributed on the harmonics, the decresing of the first amplitude is an indication of the dispersal of the energy on the others

Conlusions

• By spectral analysis of Fourier components, using THD and first frequency amplitude decreasing, we find that also THD such as first frequency tend to have a minimum value in two regions of the intensity of the electric current from the magnetization coil.

• By inspection of the value of the magnetic field B(T) from the magnetization coil in which appear the minimum of the THD, is possible to connect these values with the regions where magnetostriction has strong nonlinearity on his variation versus magnetic field.

References• 1. LINES, M. E., Elastic properties of magnetic materials, Phys. Rep. 55, 2, pp 133-

181, 1979• 2. SMIT, J.,Magnetic properties of materials, Mc. Graw-Hill Book Comp., 1971• 3. CRETU, N., NITA, G., Pulse propagation in finite elastic inhomogeneous

media,Computational Materials Science, Elsevier, • 31, pp.329-33, 2004• 4.NAYFEH, H., A.,The general problem of elastic wave propagation in multilayered

anisotropic media, J.Acoust.Soc.Am.,89, pp 1521-1531, 1991• 5. KNABNER, P., ANGERMANN, L., Numerical Methods for Elliptic and Parabolic

Partial Differential Equations,Springer New York,Berlin, Heidelberg …, 2003.

• 6. CRETU, N., NITA, G., Acoustic behavior of finite ferromagnetic samples, modeling and simulation, Proc. ICUltrasonics, Wien, 8-13 April

2007 • 7.Ostrovski, I.,Nonlinear acoustics, Cambridge University Press, ISBN 0-521-39080-X

Hardback, 1998