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Modeling High-Order Ferromagnetic Hysteretic Minor Loops and Spirals Using a Generalized Positive-Feedback Theory

Figure 1

Figure 1
Two theoretical demagnetization spirals calculated using the G-PFB algorithm. Reversible segments are shown in thick line, irreversible segments in thin line. Odd-order and even-order segments are shown in dark and light gray respectively (blue and green online). Normalized parameters are given in the inset.

Figure 2

Figure 2
Theoretical asymmetric demagnetization spirals (continuous curves). (a) A low-order spiral in which NORC Formula$\Gamma _{n}$ has both lower-reversible Formula$({\rm L}_{n})$ and ascender-irreversible Formula$({\rm A}_{n})$ segments and is therefore Type-LA. The major loop Formula$\Gamma _{0}$ is indicated in dashed line. (b) Part of a higher-order spiral in which NORC Formula$\Gamma _{n}$ has only a lower-reversible Formula$({\rm L}_{n})$ segment and is therefore Type-L. In (a) and (b), Formula$n$ is an odd integer. In this figure, Formula${\rm P}_{n}$ are reversal points, Formula$X_{n}$ are reversal fields, Formula$Y_{n}$ are reversal magnetizations, and Formula$Y_{Bn}$ are bifurcation magnetizations [2]. Reversible segments are shown in bold line, irreversible segments as light arrows, reversal points as open circles, and bifurcation points as diamonds. The “connectors,” due to the RPM effect, are shown as dotted lines with arrows.

Figure 3

Figure 3
Simplified flow chart for calculating demagnetization spirals and minor loops of arbitrary order Formula$n$.

Figure 4

Figure 4
Summary of the G-PFB algorithm depicted in the flow chart of Fig. 3.

Figure 5

Figure 5
Measured third-order asymmetric demagnetizing spiral (dotted curve) in Isomax [6] compared with theoretical results (continuous curves). Reversible curve segments are shown in heavy line, irreversible segments in light line.

Figure 6

Figure 6
A measured eighth-order demagnetization spiral in Formula$\gamma$-Fe2O3 (dotted line) compared with theory (gray, blue, green online). The 1ORC is almost coincident with the major loop. Reversible segments are in bold line, irreversible segments in light line. Measured data are courtesy of I. Mayergoyz [7].

Figure 7

Figure 7
(a) A theoretical major loop with a quasi-symmetric demagnetization spiral of order Formula$n = 111$, computed for Isomax using the G-PFB theory (odd/even-order NORCs are blue/green online). Open circles are reversal points, diamonds are bifurcation points. (b) The vertex curve obtained by plotting the calculated demagnetization reversal points (circles).

Figure 8

Figure 8
Enlargement of central region of Fig. 7(a), showing in bold line the lenticular Rayleigh quasi-loops that characterize the low-field reversible region. The rest of the spiral is shown in light gray line.

Figure 9

Figure 9
Enlargement of the theoretical first-quadrant “vertex” curve (open circles) of Fig. 7(b), compared with the experimental initial-magnetization (IM) curve for Isomax (red online), from [6]. The portion of the theoretical IM curve associated with low-field reversibility is shown as black dots.

Figure 10

Figure 10
Three closed asymmetric minor loops in Isomax. (a) In this instance, each minor loop is reached via a 1ORC (blue online), and consists of a 2ORC (green online) followed by a 3ORC (red online). In the case of the uppermost loop, the 3ORC overlaps the 1ORC. (b) Comparison between the theoretical curves (gray, 1ORCs suppressed) and measured data (dots) [6]. Reversible segments are shown in thick line, irreversible segments in thin line.

Figure 11

Figure 11
A (normalized) scenario for calculating a symmetrical closed minor loop in Isomax. The 1ORC Formula$\Gamma _{1}$, the 2ORC Formula$\Gamma _{2}$, and the 3ORC Formula$\Gamma _{3}$ are in color online. Thick lines indicate reversibility, thin lines irreversibility The theoretical initial magnetization curve (IMC) is shown in dotted line (magenta online).

Figure 12

Figure 12
Comparison between calculated (continuous line) and measured data (dots) for four nested symmetric closed minor loops in a soft 6.5% Si-Fe material. As in other figures, thick lines indicate reversibility, thin lines irreversibility. Measured data are courtesy of V. Basso [46].

Figure 13

Figure 13
Closed minor loops in Sm18Fe11Co71. Measured data due to Cornejo et al. [10], shown in dotted line, is compared with continuous curves calculated using the G-PFB theory.

Figure 14

Figure 14
(a) CPU times to calculate up to 11 reversal branches. (b) The CPU times to calculate up to 111 branches.