Publi Journal P 11479

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INSTITUTE OF PHYSICS PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 15 (2004) 365–370 PII: S0957-0233(04)65307-0

Circular magnetization and susceptibilityof an ideal soft ferromagnetic wire

D-X Chen1, A Hernando2 and L Pascual2

1 ICREA and Grup d’Electromagnetisme, Departament de Fısica, Universitat Autonoma

Barcelona, 08193 Bellaterra, Barcelona, Spain2 Instituto Magnetismo Aplicado, UCM-RENFE-CSIC, PO Box 155, 28230 Las Rozas,

Madrid, Spain

Received 20 June 2003, in final form 20 October 2003, accepted forpublication 26 November 2003Published 11 December 2003Online at stacks.iop.org/MST/15/365 (DOI: 10.1088/0957-0233/15/2/008)

AbstractThe voltage induced by an alternating current is derived for an idealferromagnetic cylinder with a linear local magnetization curve ending by asudden saturation, based on which practical formulae relevant to inductivemeasurements of the circular hysteresis loop and the susceptibility of softmagnetic wires are recommended. The anomalously large circularsusceptibility discovered in amorphous soft magnetic wires is furtherstudied using the present results.

Keywords: circular susceptibility, circular magnetization, soft magnetic wire,inductive magnetic measurement

1. Introduction

When an alternating current flows through a conducting wire,

it produces a circular ac field inside the wire based on theAmpere law so that the wire is magnetized in the circular

direction. According to the Faraday law, the circular acflux corresponding to the circular field and magnetizationwill induce emf surrounding its circular path so that an

induced ac voltage can be measured along the wire and theac magnetization obtained. This is the basic principle behind

the inductive measurement of the circular magnetic propertiesof conducting magnetic wires.

Although the relation between the ac impedance Z and

the constant circular susceptibility χc of a conducting cylinderwas rigorously derived many years ago [1, 2], considerable

attention has only been recently devoted to techniques forcircular magnetic measurements after possible applicationsof the magneto-inductive effect of soft magnetic wires were

predicted [3]. For example, in studies of nearly non-magnetostrictive amorphous wires, circular hysteresis loops

were measured in [4–6] using a technique proposed in [7].However, there has been a major complexity that discouragesthe use of such a kind of technique; the circular field produced

by the alternating current is nonuniform and it is difficult togive correct average values of the field and magnetization.Althoughconsiderableefforthas been applied to thederivation

of the relation between the circular magnetization and induced

voltage [4, 7], most experimentalists still regard circularmagnetic measurements as qualitative or relative ones innature,since the mentionedderivationswere based on intuitiveor less sound grounds. In this work, we will give a rigorousderivationof theac voltage induced by thetransportalternating

current for a simple model that characterizes the main featuresof ferromagnetshaving a linear local magnetization curvewitha sudden saturation, following the approach proposed in [8].As a result, we will provide experimentalists with simple andoptimized formulae to use in measurements of the circular

hysteresis loop and susceptibility. A sister paper studying aferromagnetic strip has recently been published [9].

Impedance measurements are often easier than ordinary

inductive magnetic measurements, since the former involveonly the current and voltage of the sample itself butfor the latter separate magnetization and pick-up coilsare necessary. With the formulae derived in this work,we can unambiguously obtain circular magnetic properties

from impedance measurements, which are an importantcomplement of longitudinal properties measured by ordinarymagnetic measurements. As a result, the basic magnetic

properties, magnetization process and magnetic structureof ferromagnetic wires can be studied by measuring their

impedance as a function of current amplitude and frequency,temperature, tension, twist and so on. Such areas of researchare especially convenient and useful for experimentalists whoare studying magneto-impedance, since they already have all

0957-0233/04/020365+06$30.00 © 2004 IOP Publishing Ltd Printed in the UK 365





Circular magnetization and susceptibility of an ideal soft ferromagnetic wire

Figure 1. Comparison between the averaged and local (a) M c/ M c,s

versus H c/ H c,s, (b) M c,m,1/ M c,s versus H c,m/ H c,s, and (c) χc,1/χc

versus H c,m/ H c,s for an ideal ferromagnetic cylinder having a localcircular magnetization curve as denoted by the dashed line in (a).

where

C 2 = 3/4. (22)

Comparing with the above derivation, it can be found thatmany results appearing in [4, 7] are correct and this work puts

them on a sound physical basis.

The averaged and local magnetization curves are plotted

in figure 1(a).

2.4. Fundamental circular susceptibility

With the lock-in technique, induced voltage is measured as

its fundamental component V ind,1(r 0, t ), so that fundamental

circular susceptibility isdetermined. It is identical to V ind(r 0, t )

when hm < 0, but when hm 0, Fourier analysis on

equations (11) and (16) is necessary. This analysis results

in the fundamental induced voltage

V ind,1(r 0, t ) = r 20

2 AB cos ωt , (23)

where

B = 1

π[2 arcsinh−1

m + sin(2 arcsinh−1m )]

+ 4

3πcos arcsin h−1

m (h−1m − h−3

m ). (24)

From equations (11) and (23) we see that the rms value of the

fundamental voltage

V ind,1 =√

2a2 A/4 (hm < 1), (25)

V ind,1 =√

2a2 AB/4 (hm 1). (26)

Since at low I , the fundamental inductance

L1 = V ind,1/ω I = µ0lχc/8π, (27)

we define the averaged fundamental circular susceptibility

χavc,1 = 8π L1/µ0l, (28)

and thus,

χ avc,1 = χc (hm < 1), (29)

χ avc,1 = χc B (hm 1). (30)

2.5. The fundamental circular magnetization curve

Consistent with equations(21),(22), (29)and (30)the averaged

fundamentalcircularmagnetization curve, M avc,m,1 versus H av

c,m,

may be calculated using

H avc,m = 3 H c,m(r 0)/4, (31)

M avc,m,1

=χ av

c,1 H avc,m. (32)

It coincides at low and high fields with the local fundamental

circular magnetization curve as obtained from equations (4)

and (5) by Fourier analysis:

M locc,m,1

M c,s

= 2

π

H loc

c,m

H c,s

arcsin H c,s

H locc,m

+ cosarcsin H c,s

H locc,m

. (33)

Thus, we have the normalized averagedand local M c,m,1/ M c,s,

M avc,m,1/ M c,s = M loc

c,m,1/ M c,s = 4/π ≈ 1.273 at saturation.

A comparison between both is shown in figure 1(b).

Correspondingly, χ avc,1 as a function of H av

c,m is also consistent

with the local one,

χ locc,1 = M loc

c,m,1/ H locc,m, (34)

as seen in figure 1(c).

3. Circular magnetic measurements for wires

3.1. Formulae for circular magnetic measurements

Since the circular demagnetizing factor is zero, a wire can be a

standardsample shapefor circular softmagneticmeasurements

just as a ring for longitudinal soft magnetic measurements.

Another advantageof such circular measurements is that there

is no need to wind the magnetizing and search coils, since the

magnetizing field is produced by a current flowing through the

wire itself and the induced voltage is detected on the surface

of the wire itself.

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D-X Chen et al

Consistent with the above results, the averaged hysteresis

loop of the magnetic wires should be calculated using

H avc (t ) = 3 I (t )/8πr 0, (35)

M avc (t ) − M av

c (0) = 3

µ0lr 0

t

0

V ind(r 0, t ) dt , (36)

if thewaveforms of transportcurrent, I (t ), andinduced voltage

across l lengthof thewire, V ind(r 0, t ), arerecorded. It shouldbe

noted that when measuring V ind(r 0, t ), the contribution of the

voltage dropped on the dc resistance and the voltage induced

by the magnetic field has to be removed using, for example, a

bridge technique [4, 7].

The averaged fundamental circular susceptibility χ avc,1

and the amplitudes of the averaged fundamental field and

magnetization corresponding to this χ avc,1 should be calculated

using

χavc,1 =

8π L1

µ0l= 8π V ind,1

µ0lω I 1, (37)

H avc,m,1 = 3

√ 2 I 1/16πr 0, (38)

M avc,m,1 = χ av

c,1 H avc,m,1, (39)

if the rms values of the fundamental current and induced

voltage, I 1 and V ind,1, are measured. Equation (37)hasalready

been given in [8]. From equation (37) we see that χ avc,1 is very

easy to measure for magnetic wires; not requiring the value of

radius, χ avc,1 can be measured easily and accurately by a lock-in

techniqueeven for wires with micro size, and only a correction

with respect to external inductance is necessary [10, 12].

3.2. Measuring frequency

It should be emphasized that all the above derivations are

performed at so low a frequency that equation (3) holds. For

higher frequencies, eddy-current effects will make χc,1 take

a complex value, and the study of its frequency dependence

will be interesting for understanding the domain structure and

magnetization process as discussed in [8].

Defining a reduced frequency

θ 2 = r 20 µ0χcω/ρ = 2r 20 /δ2s , (40)

where ρ and δs are in turn the resistivity and the classical

skin depth of the wire, we have the eddy-current loss factor

in the classical case and when considering the transverse wall

displacements of large domains,

tan δclaseddy = θ 2/24, (41)

tan δeddy = (4.4c/r 0) tan δclaseddy, (42)

respectively, where 2c is the averaged circular domain

width [8]. Since low frequencies require tan δeddy 1, we

may use the criterion of θ = 1 to estimate the classical

upper-limit frequency f max ≈ 400 Hz, if typical values of

ρ = 1.3× 10−6 m, r 0 = 60 µm and χc = 105 are assumed.

When considering the possible contribution of domain wall

displacements to the circular magnetization process, this f max

can be reduced in proportion to r 0/c.

In equations (37)–(39) we use subscript 1 not only for χ

and M but also H and I which stands for the fundamental

component, because the latter are not required to be modelled

sinusoidally in real measurements. However, using an ac

power supply with sinusoidal output, the sinusoidal condition

for I (t ) has already been met when the low-frequency

requirement is met, since when tan δeddy is small the possible

harmonic reactance is always much smaller than the total

constant resistance of the measuring circuit so that I (t ) isproportional to thesinusoidal outputemfof thepowdersupply.

Nevertheless, the situation is more complex if the frequency is

high.

3.3. Upper limit of the circular field

There is an upper limit of the circular field on the order of a

few 102 A m−1, above which the sample temperaturemay rise

considerably. Therefore, such measurements are especially

suited to very soft ferromagnets. However, this upper limit

canbe greatly increasedif themeasuringcircuit is immersed in

non-conducting liquidor themeasurements arecarriedoutvery

quickly to reduce the temperature rise caused by the heatingof the transport current.

4. Parasitic circular soft ferromagnetism

Being the variation of the ac impedance Z with the axial

dc field H z, the magnetoimpedance (MI) of ferromagnetic

wires and ribbons has become a topic of interest in recent

years. Since MI is determined by the skin effect and the H zdependent circular susceptibility, very soft magnetic wires are

thebest candidates to show a giant MI. Therefore, studying the

impedance of wires [13], we have proposed a formula for the

circular susceptibility of an ideal soft magnetic wire,

χc,ideal = M s/| H z|, (43)

which should be valid for | M s H z| |K eff |, K eff denoting the

effective anisotropy constant in the wire.

However, the measured χ avc,1 of very soft nearly non-

magnetostrictive amorphous or nanocrystalline wires has been

found to be many times greater than χc,ideal at certain values of

H z [10, 11, 14, 15], and this anomalous phenomenon has been

explained by theexistenceof magnetic inclusions embeddedin

a soft magnetic matrix and the formation of exchange-coupled

disc-like soft magnetic structures [15]. Using the formulae

given above to analyse some experimental data, we can showa new feature of this phenomenon, which is the occurrence of

a parasitic circular soft ferromagnetism.

The studied sample was an in-rotating-water quenched,

nearly non-magnetostrictive amorphous CoFeSiB wire of

diameter 0.14 mm and length 15 cm after 475 ◦C annealing

for 30 min, which was actually sample S1 presented in [15].

Low-frequency fundamental inductance L1 was measured by

the lock-in technique for a middle segment of l = 6 cm [15].

The measurement set-up is simple, as shown in figure 2.

The output of the internal oscillator of the lock-in amplifier

provides alternating current to the series connection of a

standard resistor Rs and the middle segment of the sample S,

which is placed east–west in a magnetizing solenoid fedby a dc power supply. At a given alternating current and

dc field H z, the voltages across the resistor and across the

connection of the resistor and the sample are measured by

368



Circular magnetization and susceptibility of an ideal soft ferromagnetic wire

Figure 2. Schematic of the set-up used for circular magneticmeasurements using a lock-in amplifier.

Figure 3. The circular susceptibility χavc,1 of a nearly

non-magnetostrictive CoFeSiB amorphous wire measured at H z = ±1430, ±4470 and ±14 300 A m−1 as a function of H c,m .Dashed curves are the fits by χ av∗

c,1 calculated using equations (24),(29), (30), (43) and (44).

the lock-in amplifier as V 11 and V 21 + jV 22, respectively. The

fundamental alternating current and the inductance across the

middle segment of the sample are calculated by I 1 = V 11/ Rs

and L 1 = RsV 22/ωV 11. The circular susceptibility χ avc,1 was

calculated using equation (28). As explained in [15], by

decreasing H z from 9 × 104 to −9 × 104 A m−1, we have

over a large range of 10 A m−1 <

| H z

|< 104 A m−1 that

(i) χ avc,1( H z = H z1 > 0) > χ av

c,1( H z = − H z1),

(ii) χ avc,1( H z) measured at low H c,m is appreciably greater than

χc,ideal( H z), and

(iii) χ avc,1 measured at a fixed H z decreases with increasing

H c,m(r 0).

It should be emphasized that χ avc,1/χc,ideal can be as large as 55

at H z ≈ 1400 A m−1 and H c,m(r 0) = 2.3 A m−1.

We now showin figure3 the variation of the measured χ avc,1

asa functionof H c,m(r 0) when H z isdecreased to14300,4470,

1430, −1430,−4470, and −14 300 A m−1, respectively. We

see that feature (iii) occurs for all cases except at the largest

| H z|, where χ avc,1 remains constant.Feature (iii) indicates that the circular magnetization

obeys a general rule in ferromagnetism; it has a high initial

susceptibility χc and a saturation magnetization M c,s. We

model the local circular magnetization as above, so that

χavc,1[ H c,m(r 0)] can be calculated using equations (24), (29)

and (30). Assuming the total χ av∗c,1 to be the sum of this

susceptibility and χc,ideal, we use

χav∗c,1 [ H z, H c,m(r 0)] = χ av

c,1[ H c,m(r 0)] + χc,ideal( H z) (44)

to fit the measured χ avc,1[ H z, H c,m(r 0)]. The results are shownin figure 3 by dashed curves. In the fitting, parameters χc and

H c,s are chosenas 18000and 2.7 A m−1, 7800and 4.7 A m−1,

3100 and 3.5 A m−1, and 440 and 14 A m−1 for H z = 1430,

−1430, 4470 and −4470 A m−1, respectively.

We see that the model fits are generally good. There

are two kinds of departure of the fitting curve from the

experimental data: its turning down is too sharp for three

cases with lower χc and it is too high at the maximum value of

H c,m(r 0) for two cases with higher χc. The former is obviously

due to the simplification in the model with only two constant

valuesofd M c/d H c andthe lattermay indicate that χc is smaller

at the axial region of a few micrometres diameter.The good fits imply that the difference between χ avc,1 and

χc,ideal results froma very soft circular ferromagnetism, whose

M c,s decreases with increasing | H z|, being greater at H z > 0

than at H z < 0, like the situation of χc itself. Normalized to

M s, we have M c,s/ M s = 0.085, 0.064, 0.019 and 0.011 for

the above four cases. Since this soft circular ferromagnetism

is parasitized upon an ordinary ferromagnet that has a large

saturation magnetization M s and very small longitudinal and

circular differential susceptibilities (at sufficiently high H z),

and it is not predicted by the conventional theory of domain

magnetization rotations and domain wall displacements, we

name it parasitic circular soft ferromagnetism.

If the saturation magnetization for the parasiticferromagnetism is M s, then the ratio M c,s/ M s should be the

volume fraction occupied by the parasitic ferromagnetism.

This will lead to a very large low-field circular susceptibility

for theparasitic ferromagnetism itself, being 5×104 ∼ 2×105

calculated from this ratio and themeasured χavc,1 = 600–18 000

for the four upper curves in figure3. Such a high susceptibility

cannot be shown if the parasitic ferromagnetism occurswithin

separatedparticlesonly, since the strongdemagnetizing effects

will make the total circular susceptibility very small. For

example, the shape susceptibility of an isolate sphere, which

is determined by demagnetizing effects and sets the upper

limit of the external susceptibility of a ferromagnetic sphere,is merely 3. Therefore, it must occur in a dispersive and

multiconnectedway having thesame entire volumeas the wire

andan averagedsaturation magnetization M c,s. In other words,

thereis a sponge-likeparasitic ferromagnetismcoexistingwith

the ordinary ferromagnetic matrix.

We have used the above-derived formulae to research a

particular topic. Since the phenomenon discovered is new,

interesting and complex, its study is still in progress, and

therefore no definite explanations can be given at present.

However, it is necessary and possible to give a brief review

regarding this phenomenon, since the existing presentation in

the literature is not so logical owing to the absence of one of

the key papers, which has only been published very recentlyafter a long delay [16]. We noticed the anomalous, over-

large circular susceptibility in nearly non-magnetostrictive

amorphous wires after calculating the magnetoimpedance of

369



D-X Chen et al

ferromagnetic wires in 1998 [13], where equation (43) was

presented. Since the anomaly occurred at H z much greater

than the coercive force, it should belong to the problem of

the approach to saturation. In contrast to classical studies

which emphasized the longitudinal magnetization, transverse

magnetization had to be calculated using the Stoner–Wahlfarth

model in various cases. Doing this [13], we confirmed thatthe phenomenon could not be explained by classical technical

magnetization theories, as later stated briefly in [17], and

consequently submitted two papers to announce this fact,

including its asymmetry [10, 11]. Since it was intimately

connected to another phenomenon, the hysteresis loop shift

in annealed amorphous ferromagnetic wires, another paper

was written in the same period [18]. Trying to further

explain the phenomenon, a more detailed study was published

in [15], where a model of micro magnetic structures centred at

magnetic inclusions was proposed. According to this model,

in the wire there are intrinsic and annealing-induced soft

and hard magnetic inclusions embedded in the soft magnetic

matrix; when the spontaneous magnetization of the inclusionsis greater than that of the matrix pointing in the same axial

direction, exchange-coupleddisc-like soft magnetic structures

formand they may hugelyenhance the circular magnetization.

The soft magnetic matrix condition was also justified in an

extended study on nanocrystalline wires; when the magnetic

softness of an amorphous wire was greatly improved by

nanocrystallization, its circular susceptibility changed from

the classical one into the anomalously enhanced one [14]. The

essentialdifferencebetweenanomalouscircularmagnetization

and the classical one was shown in a new way by comparing

the circular susceptibility calculated from the inductance

with that calculated from the second harmonical longitudinalmagnetization [19]. In this work, the modelled soft magnetic

structures are further characterized as a sponge-like parasitic

ferromagnetism.

5. Conclusion

We have performed a rigorous derivation of the alternating

current induced voltage across an ideal soft ferromagnetic

cylinder, which has a local linear magnetization curve with

a sudden saturation. Being consistent with their local

values, the optimum averaged circular field, magnetization

and susceptibility are recommended to be calculated using

equations (35)–(39) in the inductive measurements of soft

magnetic wires. Based on this work, circular magnetic

measurement techniques may be further developed and

standardized to become a powerful tool in materials research.An example is given for the applications of some derived

formulae in materials research.

References

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[3] Mohri K, Kohzawa T, Kawashima K, Yoshida H andPanina L V 1992 IEEE Trans. Magn. 28 3150

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[15] Chen D-X and Pascual L 2002 Phil. Mag. 82 1315[16] Chen D-X and Pascual L 2003 IEEE Trans. Magn. 39 510[17] Chen D-X, Pascual L and Hernando A 2000 Appl. Phys. Lett.

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