ARTICLE IN PRESS
Physica B 372 (2006) 269–272 www.elsevier.com/locate/physb
Hysteresis in conducting ferromagnets Carl S. Schneider, Stephen D. Winchell Physics Department, US Naval Academy, Annapolis, MD 21402, USA
Abstract Maxwell’s magnetic diffusion equation is solved for conducting ferromagnetic cylinders to predict a magnetic wave velocity, a time delay for flux penetration and an eddy current field, one of five fields in the linear unified field model of hysteresis. Measured Faraday voltages for a thin steel toroid are shown to be proportional to magnetic field step amplitude and decrease exponentially in time due to maximum rather than average permeability. Dynamic permeabilities are a field convolution of quasistatic permeability and the delay function from which we derive and observe square root dependence of coercivity on rate of field change. Published by Elsevier B.V. Keywords: Ferromagnetic; Hysteresis; Conductivity; Eddy currents
1. Introduction and theory Williams et al. [1] and others [2,3] have shown that excess eddy current increases in ferromagnetic coercive field are due to microstructure in ferromagnets in concert with Brown’s [4] micromagnetic perspective. Weiss’ [5] domain model of ferromagnetism, confirmed electronically by Barkhausen [6] and optically by Bitter [7] has dominated the perspective of ferromagnetism for a century. The recently published macromagnetic model [8] of ferromagnetic hysteresis, based on magnetic anisotropy and cooperation between magnetized domains, accurately predicts both the virgin curve of magnetization, M, due to effective magnetic field, Heff, in Eq. (1) and hysteretic reversals from any magnetic state in Eq. (2) for all multidomain ferromagnetic materials. Reversible susceptibility, denoted Xrev, is a single-valued function of magnetization. Eqs. (1) and (2) are single valued if both magnetization and field are measured in the direction of field change, subject to the wiping out [2, p. 463] or memory rule of hysteresis. We denote initial susceptibility by Xi and coercive field by Hc. M ¼ H eff ðX rev þ MX 2rev =X i H c Þ,
(1)
dM ¼ X rev ðMÞeDM=2X i H c dH eff .
(2)
Corresponding author. 1047 Little Magothy View, Annapolis, MD 21401, USA. Fax: +410 293 3729. E-mail address:
[email protected] (C.S. Schneider).
0921-4526/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.physb.2005.10.064
In the cooperative anisotropic theory, differential susceptibility rises exponentially with change in magnetization, M, from reversals due to domain cooperation until it decreases to saturation due to magnetic anisotropy expressed through reversible susceptibility, Xrev, following a differential extension of virgin susceptibility in Eq. (1). Magnetic anisotropy can also be imposed on ferromagnetic materials through fields due to external currents, HI, eddy currents, Ho, thermal viscosity, HT, stress, Hs, and demagnetizing field, HD, which may be superimposed in a linear unified field theory, with an effective field in Eq. (3): H eff ¼ H I þ H o þ H T þ H s þ H D .
(3)
Directionality of hysteresis represents the irreversibility of ferromagnetism, while Eqs. (1) and (2) represent its nonlinearity. Linear field dependence follows from magnetic properties as a function of magnetization, a major change from the anthropocentric view of magnetic properties as a function of applied field. Eddy current field, Ho, and viscous field, HT, cause time dependence while stress and demagnetizing fields cause anisotropy and inhomogeneity. The effects of stress have been described [9] and we refer readers to the work of Kronmuller [10] on the viscous field. Radial penetration of magnetic flux into a thin toroidal sample in time is described by an electromagnetic wave in cylindrical coordinates. The magnetic field is polarized
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along the cylindrical z-axis, independent of z and independent of azimuthal angle f. Delayed magnetic flux penetration, resulting in skin depth for AC fields, is derived from the diffusion equation, Eq. (4), which comes from Faraday’s and Ampere’s laws, excluding Maxwell’s displacement current which is negligible below 1 Hz used here: qB . (4) qt We write Eq. (4) in cylindrical coordinates in Eq. (5) to accurately describe a ferromagnetic cylinder wave:
r2 B ¼ sm
qB qB ¼ sm . (5) r qr qt A wave of magnetic induction B moves inward at a velocity given in Eq. (6) which varies with conductivity, s, permeability, m, and radius, r: v¼
qr 1 ¼ . qt smr
(6)
The time for flux to completely penetrate a solid cylinder or thin toroid is the integral over the minor radius, r, of the inverse velocity: t ¼ 12 smr2 .
(7)
Following Stoll [11], magnetic flux change after a discontinuous change in applied magnetic field, DH, penetrates with the magnetic field and the net flux sensed by secondary Faraday turns around the entire sample is given in Eq. (8): DB ¼ m DHð1 et=t Þ.
(8)
The time rate of change of magnetic induction in a toroid can now be derived: dB 2DH t=t ¼ e , dt sr2 and the resulting Faraday voltage is given by
(9)
dB 2pN DH t=t ¼ e . (10) dt s For small changes in applied magnetic field relative to the coercive field, DH, permeability changes little and the voltage across N secondary turns is proportional to DH, as observed by Sixtus and Tonks [12] in 1931. For Cartesian samples this voltage is enhanced by factor pw=2d, where w is the sheet width and d is its thickness. Eq. (10) can be used to measure both the conductivity and susceptibility of a ferromagnetic cylinder using high-speed data acquisition systems such as LabView DAQ. Eq. (10) is integrated over differential changes in applied field to give an exact expression for delayed permeability: Z H 2 dH 0 0 m¼ 2 . (11) eðtt Þ=t sr H i dH=dt
V ¼ Npr2
From the cooperative anisotropic theory of ferromagnetic hysteresis in Eq. (2), quasistatic susceptibility, mq, and penetration time constant are functions of magnetization
change, not applied field, during integration of Eq. (11) which is rewritten as a convolution in Eq. (12) through the approximation of linear magnetic field increases, H ¼ t dH=dt normalized by the classical eddy current field, Ho in Eq. (13). Permeability is thus delayed and broadened with a maximum on the saturate curve equal to the change in B, roughly 2Brem, divided by Ho, also linear in permeability, and solved with Eq. (7) to give square root dependence in Eq. (14): Z H 0 mðHÞ ¼ mq ðMÞeðHH Þ=H o dH 0 =H o ; (12) Hi
H o ¼ t dH=dt,
2DB mmax ðoÞ ¼ 2 sr dH=dt
(13) 1=2 .
(14)
Ferromagnetic hysteresis curves, B(H), for conducting samples can be computed by integrating the delayed permeability over the applied field. Maximum permeability decreases as the square root of rate of field change, and coercivity must similarly increase to achieve the same change in induction while frequencies are low enough that induction penetrates past remanence. 2. Experiment and conclusions No rod or toroidal sample has magnetic field homogeneous in both axial and radial directions, and measured susceptibilities result from convolution over either demagnetizing field HD for rods or Amperian field for toroids. Radial variations of magnetic field around the toroidal circuit cause B(H) curves to be smeared, which decreases maximum permeability by an amount dependent upon the line shape X(H), given in Eqs. (1) and (2). Axial flux path variations in area or permeability cause changes in effective field in rods similar to moving poles. We have used a thin toroidal pipe section with no demagnetizing field. Vibrational stress fields, H s ; were avoided by adding roughly 1 mF capacitance across the primary power supply which shifts the LC resonant frequency band away from the mechanical resonance over the range of permeabilities. Viscous field, HT, and thermal instabilities were minimized and the conductivities of both sample and primary turns were increased by immersing the sample in a bath of liquid nitrogen. This work then involves only applied Amperian and eddy current fields. The overall experimental design has been described before [13]. The initial Faraday voltage proportional to magnetic field and flux step, predicted in Eq. (10), requires a primary voltage that must not exceed the 20 V of the power supply. A field step of 58 A/m induces about thirty microvolts per turn which is low enough that the field steps much faster than the flux penetrates. The voltage intercept in Fig. 1 is about three millivolts which agrees well with Eq. (10) and a conductivity of 10 MS/m, measured and reported in Bozorth [14]. Thermal and 60 Hz noise in these turns was
ARTICLE IN PRESS C.S. Schneider, S.D. Winchell / Physica B 372 (2006) 269–272
0.01 0
5
10 time (s)
15
271
B(T) for Steel Toroid dH/dt = 11, 27, 84, 268, 823 2738 A/ms
20
1.5
V(X,t) = NΦd /dt X = 810, 1373, 3383, 7550, 9850 ∆H = 58 A/m, T = 78K
1.0 0.5
0.001
H (A/m)
0.0 -8000
-4000
0
4000
8000
-0.5 -1.0 -1.5 -2.0 0.0001 Fig. 1. Faraday voltage due to a constant field step from five initial fields and susceptibilities, sensed by 100 turns around our steel toroid, decreases from a constant predicted by Eq. (10) for 10 MS/m conductivity. Slope of exponential decreases slightly with permeability.
Fig. 3. Ferromagnetic hysteresis curves of a thin steel toroid show increasing coercive field and decreasing slope for triangular excitation field at rates, dH/dt, from 11 A/ms for the narrow curve to 2738 A/ms for the broad curve.
Eddy Current Time Constants of Steel Toroid
10000
Susceptibility of a Mild Steel Toroid
9000
3.5
X(H, dH/dt) = dM/dH T = 78 K r = 4.5 mm ___ Classical Delay
3.0 2.5
8000
dH/dt = 11, 27, 84
7000
268, 823, 2738 A/ms
6000
2.0
5000
1.5
4000 3000
1.0
2000 0.5
1000 2738 A/ms
0.0 0
2000 4000 6000 8000 Average Susceptibility X = dM/dH
0
10000
Fig. 2. Magnetic penetration time constants in Fig. 1 are due to maximum susceptibility over a field step and toroid radius and exceed their classical values in Eq. (7) due to average susceptibility, causing ‘‘excess’’ losses.
about one millivolt but with up to 5000 data points per second noise was reduced to about 20 mV by averaging. The 16 bit A/D converter resolved 0.2 mV which was reduced significantly by averaging. The time constant for exponential decrease of this voltage in Eq. (7) is consistent with observations in Fig. 2 if we use susceptibility five times greater than average. Broadening of B(H) curves is due to convolutions with 60 Hz noise, thermal fluctuations, magnetoelastic resonance, toroidal thickness, conductive shielding and spatial inhomogeneities while the eddy current field is due to maximum or root mean square values of Ho since magnetic waves cannot move faster than the most slow permeable regions of the sample. The cooperative model predicts the fractional change in susceptibility, X, to be Xr/XiR and this reaches five over
-1000
0
1000 2000 3000 Magnetic Field H(A/m)
4000
Fig. 4. The susceptibility of dynamic hysteresis curves is the convolution of the quasistatic curve and an exponential delay function of the rate of magnetic field change, whose normalized shape is invariant at high rates.
the magnetic field band in the toroid of minor radius r and major radius R. Digitally integrated magnetic induction plotted against applied field in Fig. 3 shows an increase in coercivity over its quasistatic value due to the delay in penetration of field and thus flux. Classical eddy current power loss is proportional to the increase in area within the B(H) curve, and to the eddy current field, which is linear in frequency from Eq. (13). At temperature 78 K the steel has a coercive field of 392 A/m, a saturation induction of 2.1 T, initial susceptibility of 120. Maximum susceptibilities for homogeneous flux changing quasistatically exceed 20,000 for mild steel, but sample geometry, stress and temperature cause an initial convolution of susceptibility X(H) over the range of effective field variation. The effect of sample
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Acknowledgements
Scaled Coercivity and Susceptibility Hc/H0c
We are grateful to Mary Wintersgill for providing liquid nitrogen, John Ertel for suggestions on digital convolutions, Jeff Walbert for sample machining, Mark Lewis of National Instruments for writing the LabView virtual instrument software for experiment control and data acquisition, and the Naval Academy for time to prepare this work and funding to present it.
Xmax/X0max ________ Theory
100
10
Hω /H0c (A/m)
1 0.1
1
10
100
0.1
0.01 Fig. 5. Maximum susceptibility and coercive field normalized to their quasistatic values vary nearly as the square root of the rate of field change as derived in Eq. (14) are and shown as solid lines.
conductivity is an additional delayed convolution over the eddy current field. Susceptibilites of dynamic hysteresis curves in Fig. 3 are shown as functions of magnetic field in Fig. 4, initially increase with field and then decrease with time, and vary as the square root of frequency. Peak susceptibilities and their fields normalized by the quasistatic susceptibility, Xq, are shown in Fig. 5. The cooperative theory of hysteresis has explained both internal sources of ‘‘excess’’ losses and coercive fields and their nonlinear increase with frequency.
References [1] H.J. Williams, W. Shockley, C. Kittel, Phys. Rev. 80 (1950) 1090. [2] G. Bertotti, Hysteresis in Magnetism, Academic Press, New York, 1998 (Chapter 12). [3] D.C. Jiles, J. Appl. Phys. 70 (1994) 5849. [4] W.F. Brown Jr., Magnetostatic Principles in Ferromagnetism, Interscience, New York, 1962. [5] P. Weiss, J. Phys. 6 (1907) 661. [6] H. Barkhausen, Phys. Z. 20 (1919) 401. [7] F. Bitter, Phys. Rev. 41 (1932) 507. [8] C.S. Schneider, Physica B 343 (2004) 65. [9] C.S. Schneider, J. Appl. Phys. 97 (2005) 10E503. [10] H. Kronmuller, Nachwirkung in Ferromagnetika, Springer, Berlin, 1968. [11] R.L. Stoll, The Analysis of Eddy Currents, Clarendon Press, Oxford, 1974, p. 44. [12] K.J. Sixtus, L. Tonks, Phys. Rev. 37 (1931) 930. [13] C.S. Schneider, J. Appl. Phys. 89 (2001) 1282. [14] R.M. Bozorth, Ferromagnetism, Van Nostrand, New York, 1951, p. 107.
