Barkhausen Noise and Magnetic Amplifiers. I. Theory of Magnetic AmplifiersJ. A. Krumhansl and R. T. Beyer Citation: Journal of Applied Physics 20, 432 (1949); doi: 10.1063/1.1698399 View online: http://dx.doi.org/10.1063/1.1698399 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/20/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Barkhausen noise and size effects in magnetic microstructures J. Appl. Phys. 90, 2416 (2001); 10.1063/1.1388023 Magnetic Barkhausen noise analysis of residual stress and carburization AIP Conf. Proc. 557, 1732 (2001); 10.1063/1.1373962 Plastic deformation effects on magnetic Barkhausen noise AIP Conf. Proc. 509, 1541 (2000); 10.1063/1.1306217 Power Spectrum of the Barkhausen Noise of Various Magnetic Materials J. Appl. Phys. 34, 3223 (1963); 10.1063/1.1729168 Barkhausen Noise and Magnetic Amplifiers. II. Analysis of the Noise J. Appl. Phys. 20, 582 (1949); 10.1063/1.1698430

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Barkhausen Noise and Magnetic Amp1i1iers. I. Theory of Magnetic Amplifiers'" J. A. KRUMHANSL** AND R. T. BEYER

Brown University, Pruvidence, Rhode Island (Received July 27, 1948)

An analysis is made of the operation of magnetic amplifiers or non-linear transformers. Open circuit output voltages are calculated for transformers driven by a sinusoidal primary current, and also for sinusoidal primary current plus a direct current bias. The case of a loaded secondary is also considered and it is shown that there exist conditions for instability or resonance. A subsequent paper will consider Barkhausen noise as a limiting condition on the sensitivity of these devices.

I. INTRODUCTION

T HE term magnetic amplifier or non-linear transformer is customarily applied to a trans­

former containing ferromagnetic core material which, under the operating conditions, undergoes such magnetization that the flux density, B,l is not a linear function of the magnetic field intensity H. In the course of some experimental work previously reported,2 it was noticed that the background noise which ultimately limited the sensitivity of the de­vices could not be entirely accounted for by the as­sociated circuits. It seemed that this noise might be due, at least in part, to Barkhausen effect in the core materials.

In order to discuss this subject this paper (1) presents a theory of operation of these magnetic amplifiers.3 A subsequent paper (II) will present an analysis of the Barkhausen noise in terms of a simplified physical model.

n. METHOD AND ASSUMPTIONS

The usual mode of operation of the magnetic amplifiers can be understood with the aid of Fig. 1. Two identical ferromagnetic transformers have their primary windings (N 1) connected in series. The transformers are excited by an audio oscillator which drives the cores into the region of saturation during each cycle, thus generating a potential across each secondary winding which contains the fundamental frequency plus odd harmonics. If a given secondary winding (N2) on each transformer is connected in series opposing (relative to the primary), these signals will cancel out. However, if a direct current is entered on one of the secondaries (Na) , even harmonics will be produced across each secondary.

* Part of the research reported in this paper was performed under contract NOrd 508. The authors wish to express their appreciation to Professor T. B. Brown for his encouragement of the work done under this contract.

U Now at Cornell University, Ithaca, New York. 1 A table of chief symbols used in this paper is included in

Appendix B. 2 Sack, Beyer, Miller, and Trischka, Proc. I.R.E. 35, 1375

(1947). 'The authors are indebted to the Bell Telephone Labora­

tories, whose unpublished work leading to results similar to those of Part I was made available in the course of the work mentioned in reference 2.

432

Because of the method of connection, the contribu­tion from each transformer will add, so that the voltage across the No.2 windings will contain only even harmonics.

The analysis in this paper will be based on one such transformer. In the analysis it is assumed that the relation B = F(H) is known, that there is no flux leakage, and that B is uniform over any cross section of the core. In most of the discussion, the effect of hysteresis is neglected. Finally, only the steady state is considered.

m. EXPRESSIONS FOR INDUCED VOLTAGES

If the constant cross-sectional area of the core is A, the magnetic flux through this area is BA. Around this core there may be a number of windings (W). Let the jth of these have N j turns. If the cur­rent in the jth winding is I j , the total mmf, M, is

w M = L O.41fNjI;. (1)

i-I

where the summation is taken over all windings. Also, H = Mil, where 1 is the length of the core and H is the magnetic field intensity.

The voltage appearing across the terminals of the jth coil is

Vj=NjX lo-sdq,ldt, (2) so

Vi=AN;Xl0~8(dB) ~(M). (3) dH H=M11dt 1

If in each mesh there is a generator with e.m.f. E j

and linear impedance Z;(p) (where p is the operator dldt), then the jth equation may be written

w Ej=Zj(p)Ii+Njg(lt, h ... , Iw) L Nkph (4)

.10-1

where

O.41fA (dB) g(Il,I2, ••• , I w)=--Xl0-8 -

1 dH H-M/I.

All voltages and currents may be expressed as harmonics of some basic frequency w. Accordingly

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one can express any of these in terms of a Fourier series. Substitution in Eq. (4) and collection of coefficients for each harmonic would yield 2W simultaneous non-linear algebraic equations for these Fourier coefficients. Because of the tedious nature of such an operation, the discussion is limited to certain special cases, for which the equations can be "linearized.'"

Under the assumption that 11» Ij ""1, the function g(ft, h ... , Iw) can be expanded about the pointP: II, 0, 0, .... That is,

In the above expression, g(Il)P and each coefficient in the sum are functions of time. 5

Then

w g(ft, 12, ••• , Iw) E N kpIk=g(Il)pN1pIl

k-l

+(N1pI1) ~ (~) I k+g(Il)p ~ NkpIk k-2 iJh P k-2

+ higher order terms. (6)

The first step in approximating solutions to Eq. (4) is to solve

Ej=Zj(p)Ij+NINjg(Il)pPft. (7)

In this case, one can prescribe either the E j or the I j

and solve for the other. Since Eq. (7) gives Ej ex­plicitly, but I j only implicitly, the I j are prescribed.

IV. FIRST APPROXIMATIONS FOR INDUCED VOLTAGES

It shall be assumed throughout this analysis that II = 110 sinwt. 6 Also, the analysis is carried out for one possible approximation to the average mag­netization characteristiC of the form

FIG. 1. Schematic diagram of windings on a pair of non-linear transformers.

The non-linear term in Eq. (7) is now expanded in a Fourier series. After some calculation, it can be shown that the coefficients G .. (cos) and H .. (sin) are given by

where

G.. 2 f T/2 cos 'Y coswtdt =- (nwt) ,

Hn T -T/2 sin 1 +a2D2 sin 2wt

'Y O.41rAJlO ---x 10-8,

1

0.41r D=--Ntflo.

1

(9)

(10)

The solution of integrals of this type IS given In

Appendix A. The results here are

G .. =~ exp( -n sinh-1_1_), n odd

aD aD

= 0, n even or zero

H .. =O, all n, (11)

2S (1rJlO) 2S B=-tan-1 - H=-tan- 1(aH),

and the final expression for the open circuit voltage (8) appearing across the jth winding is

1r 2S 1r

S is the saturation value of Band Jlo is the initial permeabili ty.

4 It is often possible to carry the analysis from this point through a Fourier calculation (such as that done in the next section) without the linearizing approximation. This is done in the work cited in reference 3.

& Although the present discussion is concerned principally with sinusoidal excitation current, this same approximation is equally useful for other types of operating conditions. For many B-H characteristics, (ag/aIj)p is a sharply peaked function of II in the neighborhood of certain values of II and small elsewhere, which can simplify approximate evaluations of Fourier coefficien ts.

8 While this is not the usual experimental arrangement, it serves as a useful first approximation to what is otherwise a very involved analysis.

VOLUME 20, MAY, 1949

xexp( -n sinh-la~) cosnwt. (12)

It can easily be seen that as D-'>O (the linear case), the higher harmonic voltages vanish. For j = 1, the coefficient of 110 coswt becomes the linear inductance of the winding.

The relative magnitudes of the various harmonics of the open-circuit voltage appearing across the jth winding are plotted in Fig. 2. This graph indicates

433

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~ I"OJ = £..n .m"',fo «0

0(0

FIG. 2. Relative amplitudes of odd harmonics produced in non­linear transformers.

that there is an optimum driving condition for the production of any given harmonic.

V. FURTHER APPROXIMATIONS FOR . INDUCED VOLTAGES

To consider the effects produced by currents other than h Eqs. (4) and (6) are combined to give

Ej= Z;(p )Ij+ NjNlg (II) ppIl + Nig(Il)P

(13)

When the same form for the magnetization curve and primary current are used as in the preceding section, it can be shown that·

(14)

0.46

n08 '1'10(0)

10 15 0<0

FIG. 3. Relative amplitudes of even harmonics produced in non-linear transformers.

434

Then Eq. (13) can be rewritten as

Ei= Z;(p )1;+ NjNlg(Il)ppIl + Nig(Il)p

x r: NkPlk+(PI1)(~) ~ Nklk • k=2 aI; p k=2

(15)

The various terms of this expression are then de­veloped in Fourier Series. If

<Xl

I;=!P;o+ 1: (P;n cosnwt+Q;n sinnwt) , (16) n=l

and if the linear impedances at each harmonic are given by R; .. , X j .. , then

<Xl

Ei=!R;oP;o+ 1: [(RjnP; .. +X;nQ;n) cosnw~ n=l

<Xl

+(RjnQ;n-XinPjn) sinnwtJ+N;Nl 1: n=1. 3. 5, •••

2w,¥ ( 1 ) X-Ilo exp -n sinh-l- cosnwt ,aD aD

+Q •• co,,,,,,t)]+ N 1--,F, ...

2nw,¥ exp ( -n Sinh-la~). 1 X smnwt

(1 +a2D2)i

X[~ f. Nk(!PkO+Pkn cosnwt k=2 .. -l

+Qk .. sinnwt) 1 (17)

For one particularly simple special case, it is assumed that R10=O, i.e., no d.c. in primary; 12=C=!P2o (d.c. on No.2 winding, P2,,=Q2 .. =O. n~O); W=2 (only two windings). In this case, substitution inEq. (17) leads to an expression for the even harmonic voltage appearing across the No.

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B

H

FIG. 4. Idealized B-H curve used in estimating effect of hysteresis.

2 winding:

e2n=[-2N22'Ynw exp ( -n sinh-I~) (1+a 2D2)i

x,;n"",,]c, n~2, 4, 6, •... (18)

The amplitude of the e.mJ., e2n, is thus directly proportional to the input direct current (for small C) as has been observed experimentally.2 The de­pendence of these voltages on aD is shown in Fig. 3.

The effect of hysteresis can be studied by as­suming a hysteresis loop as shown in Fig. 4. In such a case, dB/dH is represented by a delta-function. If the analysis is carried out as above, the first ap­proximation indicates that the only effect of hysteresis is to introduce a phase shift in the output, without diminIshing the amplitudes of the various harmonics. These amplitudes are affected by second-order terms, i.e., large direct currents.

In the next important case it is supposed that II is prescribed but that 12 depends on the loading conditions for No.2 winding. Asbefore, II =IIosinwt, RIO=O, and W=2. Under these conditions Eq. (13) reduces to

Ej=Z;(p)Ii+ NjNIgppII +Nj N 2P(gpI2). (19)

It should be noted that gP is an even function of II and pII is an odd function of h In order to elimi­nate the effects of this large second term in Eq. (19), it is customary to connect the secondaries of two identical transformers in series opposing (relative to the primaries). The total voltage across the jth set of secondaries is then

VOLUME 20, MAY, 1949

x

R

FIG. 5. Region of instability for tuned secondary on non­linear transformers.

If the only appreciable currents in the No.2 sec­ondary are the d.c. and the mth harmonic, then

E2 = R 20P 20 + (Q2mR 2m - P 2mX 2m) sinmwt

+ (Q2mX2m+P2mR2m) cosmwt+2N22mw

x[-P2m'Y{ exp ( -2m sinh-I~) +1}

(1+a 2D2)!

xsinmwt+2N22mw[ -Q2m'Y (1+a2D2)!

x{exp ( -2msinh-la~)-1}]cosmwt. (21)

If the coefficients of sinmwt, cosmwt, and d.c. term are equated, and if the notation

_ 2N22mw'Y ( 1 ) X m = exp -2m sinh-I- ,

(1 +a2D2)! aD

2N22mw'Y

(1+a2D2)!'

is used, further calculation leads to

e2Om=R2mQ2m-(Xm+X2m+XmO)P2"., 0= (X2m - Xm+Xmo)Q2m+R2P2m.

(22)

(23)

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y

-.". ·.".

FIG. 6. Integration contour.

The current amplitudes are then

where

1 Sirh"'_I_

8

.:l =R2m2- Xm2+(X2m+ XmO)2.

(24)

Since .:l vanishes for certain values of the parame­ters, it follows that there exists a region of insta­bility, the locus of which is the semicircle drawn in Fig.5.

One should note that apparently the behavior in the vicinity of the mth harmonic is analogous tQ a voltage generator which supplies the "open circuit" voltage e20m to a four-terminal network whose transfer characteristics are described by Eq. (21).

APPENDIX A

The various Fourier expansions involve integrals of the form

f.. eiKtdz

I(K, (j) = . . _ .. 1 +{j2 sin2z

(25)

In the corresponding contour integral, the poles are given by

1 z= ±i sinh-L+mr, n=O, ±1, ±2, . ... (26)

(j

The contour for positive K is shown in Fig. 6. Since the integrand -+0 as z-+x+i 00 , the upper side of the contour does not contribute to the integral. Also, the opposite sides differ in value of z by 211" so that

436

their contributions cancel. Hence the value of the contour integral is the same as that of Eq. (25).

The residue at z=i sinh-11/{j is found to be

exp ( - K sinh-~) Ro=--------

2(1+{j2)i

while that at z=i sinh-11/{j+1I" is

exp( -K sinh-l~) Ro=(-l)K------

2(1+{j2)i

The value of the integral is then

__ 2_11"_ exp ( _ K sinh-l~), (1 +(j2) 1 (j

Keven

I(K, (j)

(27)

(28)

=0 K odd. (29)

For negative K, the integration is performed in the lower half plane and Eq. (29) is again obtained, ex­cept that IKI replaces K .

B H M S <P A 1 W Ni Ij Ej Zj(P)

P Rj ..

Xi"

w p.o a c

APPENDIX B

Table of Symbols

Flux density. Magnetic field intensity. Magnetomotive force. Saturation value of flux density. Flux. Cross-sectional area of core. Length of core. Number of windings on core. Number of turns on jth winding. Current in jth winding. E.mJ. in jth mesh. Linear impedance in jth mesh. d/dt. Resistance in jth mesh at nth harmonic. Reactance of load impedance of jth winding

at nth harmonic. Angular frequency of driving oscillator. Initial permeability of core. 7rp.o/2S. Direct current input in No.2 winding. Even harmonic voltages produced in No.2

winding. 110 Amplitude of current in No.1 winding. g(lt, It. ... , Iw) 'Y(1+a2D2 sin2wt)-I. g(I.)p gp(lt, 0, ···,0). 'Y (0.47rAjl)p.oXlO-8•

D (O.47r/l)Ntllo. Pi'" Qj" Fourier coefficients of expansion for Ij. G", H.. Fourier coefficients of expansion for

g (II 0 sinwt) coswt.

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