Frequency dependence of magnetostrictive phenomenon of as-cast 2605SA1 ribbon was studied. We applied a sinusoidal sweeping field (H), with a fixed frequency (f), along length (L) of the ribbon, and simultaneously recorded the longitudinal magnetostriction (λ_{∥}) and the transverse magnetostriction (λ_{⊥}) as a function of time (t), respectively. f was varied from 0.07 to 122 Hz. In the low-f case (f =0.07 Hz), we observed the frequency-doubling (FD) feature in λ_{∥}(t) and λ_{⊥}(t) curves; i.e., only even harmonic magnetostrictive signals showed up. In the high-f case (f = 122 Hz), we observed the no-frequency-doubling (NFD) feature; i.e., both odd and even harmonic magnetostrictive signals showed up. A theory, based on the balance among various torques acting on magnetization, is developed to explain the f dependence of the magnetostriction phenomenon observed. From this theory, we conclude that only when the reflection symmetry of the system is reserved, i.e., when the equivalent easy axis (EEA) is perpendicular to L, will λ_{∥}(t) and λ_{⊥}(t) have the true-frequency-doubling (TFD) feature. However, for the as-cast 2605SA1 ribbon, EEA is not perpendicular to L. Thus, strictly speaking, we should observe the NFD feature only. Nevertheless, in the low-f limit, we can show that the FD feature is *somewhat* allowed under the condition, b/α being close to 1, where b and α are the two parameters used in the theory. From experimental data, this condition is met for as-cast 2605SA1. To make a distinction from TFD, this low-f feature is called close-frequency-doubling (CFD) in this paper. In general, the theory explains all the experimental results fairly well.

## I. INTRODUCTION

The “humming” noise from a transformer is a well-known phenomenon, which always raises an environmental issue.^{1} Although we have already understood that the origin of this noise is associated with the magnetostrictive vibrations of the ferromagnetic cores (or sheets) in the transformer, the exact mechanism is still a little complicated, as disclosed later, and requires further analysis. For example, when the magnetization (M) of the core is driven by a magnetic field (H) from the exciting coil with a fundamental frequency (f_{o}), due to the fact that magnetostriction (λ) is an even function of H (or M), one would expect the frequency-doubling (FD) effect always appear in the vibration signal. That is, since the core sheet extends and contracts twice during a full cycle of magnetization, the noise signal should be composed of harmonics with frequencies (fs) of even multiples of f_{o} only. However, from a simple analysis, such as the fast Fourier transformation (FFT) spectrum of the noise signal from a (60 Hz) power transformer, we immediately find that the statement above is not true, because the spectrum shows not only the even harmonics, but also the odd harmonics. This phenomenon shall be referred to as the no-frequency-doubling (NFD) effect in later discussion.

In this study, we used the Metglas^{®} 2605SA1 ribbon as the test sample, because it is often been taken as the core material in a power transformer due to economic reasons.^{2} Strain gauges were employed to monitor the real-time magnetostrictive signals from the ribbon, when it was driven by a longitudinal field H with f = f_{o}. In particular, we like to focus our attention on the analysis of the frequency dependence of the longitudinal magnetostriction (λ_{∥}) and the transverse magnetostriction (λ_{⊥}) signals. Moreover, a simple mathematical model is set up to explain those observed magnetostrictive phenomena.

## II. EXPERIMENTS

A spool of as-cast 2605SA1 ribbon, with thickness t = 17.8 μm, was purchased from Metglas^{®}, Inc.. A rectangular sample, with length L = 25 mm and width w = 8 mm, was cut out from the spool. A pair of tee-rosette type strain gauges, WK-06-060WT-120, bought from Vishay Intertechnology, Inc., were glued on the sample surface with M-Bond 200 adhesive. From Eq. (4) of Ref. 3, the relation between the true stain (λ) and the measured stain (ε), by using the metal strain gauge on the 2605SA1 ribbon sample, is λ = ε[1 + (2A_{g}E_{g})/(A_{m}E_{m})], where A and E are the cross-section area and Young’s modulus, and the subscript g and m denote gauge and 2605SA1. Since in our situation, A_{g} = 0.21 × 0.024 mm^{2} (for 6 grid lines), E_{g} = 195 GPa (for the K-alloy), A_{m} = 8 × 0.018 mm^{2}, and E_{m} = 110 GPa, the correction factor in the bracket is as small as 0.12 < < 1. Also, even if the ΔE/E effect, which is about 20 to 30% for 2605SA1, is considered, the above conclusion is not affected. Thus, the metal gauge, used by us, measures λ of the 2605SA1 ribbon within an error about 12% only.

As shown in Fig. 1, the orientation of the longitudinal (or “blue”) gauge was parallel to L (or the x-axis), and that of the transverse (“red”) gauge was parallel to w (or the y-axis). Then, the ribbon sample, with gauge on the bottom surface, was glued to the free end of each flexible Cu strip with Deuco Cement (Fig. 1). The purpose of using this cement is that we like to hold the ribbon sample in the right place (the homogenous field region), and at the same time let it expand or contract as freely as possible. The top-ends of the strips were fixed on the two sides of a sample holder, respectively. With this sample arrangement, we could also avoid twisting of the ribbon plane, when the saturation magnetization $ M s \u21c0 $ was not parallel to $ H \u21c0 $.

When performing the magnetostriction experiment, we placed the sample at the symmetric center of a pair of Helmholtz coils, which could provide a uniform in-plane longitudinal field, i.e., $ H \u21c0 \u2225L\u2225 x \u02c6 $. Since we wish to study the frequency dependence, the coils were excited by a set of signal generator and power amplifier operating at f = 0.07, 0.14, 0.55, 1.5, 5.6, 12, 32, 61, 92, 112, and 122 Hz, respectively. The waveform of the H-signal was sinusoidal, and the maximum amplitude of H was H_{m} = 60 Oe. From an independent measurement by using the same alloy based gauge, we found that the saturation magnetization (λ_{S}) of the as-cast 2605SA1 (stacked and/or un-stacked) is 41 ppm (@ H = 4.8 × 10^{5} A/m or 6 KOe). Other auxiliary data of 2605SA1 (from our studies) include the Young’s modulus E = 110 GPa, density D = 7.18 × 10^{3} Kg/m^{3}, electrical resistivity ρ = 130 μΩcm, magnetic permeability μ = 4.5 × 10^{4}, saturation magnetization 4πM_{S} = 1.57 T.

## III. THEORY

Since the structure of the as-cast 2605SA1 ribbon is amorphous, the isotropic magnetostriction condition is satisfied, i.e., λ_{100} = λ_{111} = λ_{S}, where λ_{100} and λ_{111} are the [100] and [111] elongations, when $ M s \u21c0 $ is along the [100] and [111] directions, respectively. Then, it is easy to show that due to the magnetization rotation the magnetostrictive strain (λ) is expressed as,^{4,5}

where θ is the angle between $ M \u21c0 s $ and the direction of the strain gauge. At the demagnetized state, when H =0, we should assume $ M s \u21c0 $ is along the equivalent easy axis (EEA), which is an average of the local easy axes in as-cast ribbon. Notice that this *uniaxial* EEA must exist, because we have used Eq. (1) only to describe the magnetostriction phenomena. For simplicity, the magnetostriction contribution due to the 90-degree domain wall (DW) motion is not considered here, and the 180-degree DW motion makes no contribution at all.^{6} As shown in Fig. 1, the EEA of the ribbon is in general deviated from the $ x \u02c6 $-axis by an angle δ. Hence, according to Eq. (1), when H =0, the strain sensed by the longitudinal (L) gauge is $ \lambda \u2225 \delta = 3 2 \lambda s [ cos 2 \delta \u2212 1 3 ] $, and that by the transverse (T) gauge is $ \lambda \u22a5 \delta = 3 2 \lambda s [ sin 2 \delta \u2212 1 3 ] $. Next, |H| = H_{m} = 4.8 × 10^{3} A/m or 60 Oe used in this study is not large enough to saturate the sample completely, due to the large demagnetizing field H_{D} near edges or corners of the sample as discussed later. In other words, with H_{m} = 4.8 × 10^{3} A/m , we could only magnetize the central region of the sample, i.e., each of our tests was done only on a *minor* hysteresis loops, not the major loop. As a result, $ M \u21c0 s $ was rotated symmetrically about EEA with a small angle (or amplitude) ξ. The total change of strain indicated by the L gauge or the peak amplitude of λ_{∥} is defined as $\Delta \lambda \u2225 P = \lambda \u2225 P \u2212 \lambda \u2225 \delta $ and the peak amplitude of λ_{⊥} is $\Delta \lambda \u22a5 P = \lambda \u22a5 P \u2212 \lambda \u22a5 \delta $. Finally, we can show that

As discussed later, in the quasi-static case, when f =0.07 Hz, we find that $ \Delta \lambda // P =-8\u2009ppmand\u2009\Delta \lambda \u22a5 P =+5\u2009ppm$. Thus, from Eq. (2), it is easy to show that δ = 48^{o} and ξ = 8^{o}. Clearly, since we did not have the complete saturation (where $ M s \u21c0 \u2009\u2225 x \u02c6 $-axis), the amplitude of swing angle, ξ, should be much smaller than 48^{o}.

As mentioned previously, the time varying field H is expressed as H = H_{m}sinωt, where ω = 2π f. It causes $ M \u21c0 s $ to oscillate around EEA back and forth with a period T_{θ} = 1/f. In turn, the balance of the various torques on $ M \u21c0 s $ imposes the condition,

where $ \tau \u21c0 D $ is the driving torque due to the internal field H_{i} = H − H_{D}, H_{D} = D_{L}M_{S}cosθ_{L} is the demagnetizing field, and D_{L} is the demagnetization factor along L (see Fig. 1), $ \tau \u21c0 K $ is the dissipating torque due to the eddy-current mechanism, and $ \tau \u21c0 K $ is the restoring torque due to the uniaxial anisotropy energy E. Then, a simple estimation, based on Eq. (2.13) of Ref. 6 or Eq. (2.22), i.e., D_{L}≒(2/π)(t/L)(2w/L)^{1/2}, of Ref. 7 with our sample’s dimensions, shows that at or near the center region, D_{L}≒5.2 × 10^{−4} or 3.6 × 10^{−4}. Notice that since our sample plane is rectangular, D_{L} is not a constant. In fact, near the edges or corners of the sample, H_{D} may diverge logarithmically: we can hardly turn the local magnetization in the corners. In other words, complete saturation by finite H is almost impossible for a ribbon sample. Thus, in this study, we only consider the magnetization state in or near the center region of the ribbon sample. Moreover, from Fig. 1, it is easy to show that

Notice that $ z \u02c6 $ is a unit vector pointing into the ribbon plane. Further, according to the Lenz’s rule, any rotation of $ M \u21c0 s $, as driven by $ H \u21c0 $, would induce eddy-current flowing, which, in turn, dampens the rotation al motion. Thus, $ \tau \u21c0 E $ is expressed as

where β is a positive constant, and M_{x} = M_{S}cos[θ_{L}(t)]. Finally, based on Fig. 1, the uniaxial anisotropy E can be written as E = − K_{u}cos^{2}[δ − θ_{L}]. Then, $ \tau \u21c0 K $ is

Due to smallness of D_{L}M_{S} = 52 A/m or 0.65 Oe, on the right hand side of Eq. (7), the D_{L}M_{S}[cotθ_{L}(t)] term is negligible compared with the H_{m}sinωt term for most of the time, except at or very near the time satisfying ωt = pπ, where p =0 or 1, during one period. Thus, Eq. (7) is approximated as,

Similarly, since θ_{L}+θ_{T} = 90^{o}, we can also write

Obviously, Eqs. (8) and (9) are two coupled non-linear differential equations. In order to solve θ_{L}(t), we need to assume that the perturbed solution θ_{L}(t), resulted from H_{m}≠0, will eventually oscillate around the equilibrium value δ, i.e.,

where ψ_{L} is the phase angle (or lag) of θ_{L} due to the anisotropy and eddy-current terms. For simplicity, Eq. (8) is re-written as

where $F \theta =\u2212 [ sin 2 \delta \u2212 \theta ] sin \theta $. Notice that an equilibrium condition, θ_{L } = δ, of the system is a particular solution with (dδ/dt) =0 and F(δ) = 0 under H_{m} = 0. Mathematically speaking, there may exist many equilibria for Eq.(11), but θ_{L} = δ is the equilibrium most interest to us. Thus, we shall linearize F(θ) around δ by the Taylor expansion

where α = [K_{u}/(M_{S}β)](dF/dθ)_{δ}, (dF/dθ)_{δ} = 2/sin(δ) > 0, and b = H_{m}/β. This perturbation equation has the general solution,

where $cos \psi L =\u2212\alpha / \alpha 2 + \omega 2 $ and $sin \psi L =\u2212\omega / \alpha 2 + \omega 2 $. Thus, the phase lag of θ_{L} is

Also, from Eq.(14), since α > 0 and α is usually large, the decaying transient term, Ae^{−αt}, will soon die out after finite time, and θ_{L} eventually vibrates around the stable equilibrium δ. As a result, we simply write

where $\eta \u2261b/ \alpha 2 + \omega 2 $. From Eq. (9), we can also derive

where ψ_{L} = ψ_{T} + 180^{∘}, i.e., when ψ_{T} is in the first quadrant, ψ_{L} is in the third quadrant. Next, from Eq. (1), we should find that λ(t) ∝ cos[2θ(t)]. Thus , in principle, we may simulate the λ_{∥}(t) and/or λ_{⊥}(t), based on the following two equations

where h ≡ 2δ and n ≡ 2η. However, because it is difficult to get values of β and K_{u}, it is impossible to do the simulation for λ_{∥}(t) or λ_{⊥}(t) exactly. Nevertheless, we still can raise the following discussion to explain our experimental data satisfactorily. First, if h = 2δ = 180^{o}, the second term of Eq. (18A) should disappear, which means that no matter what values n and f are, λ_{∥}(t) is *always* sinusoidal with a period T= 1/(2f). For example, Fig. 2(a) is plotted with δ = 90^{o}, n =π, ψ_{L} = 0^{o}, and f =0.1 Hz in Eq. (18A). Thus, clearly from Fig. 2(a) λ(t) has the true-frequency-doubling (TFD) feature, because δ = 90^{o}. This is understandable, since when δ = 90^{o} or the EEA is parallel to w, the reflection symmetry (or the even condition of the change of magnetization ΔM) of the system is fulfilled. As a result, θ_{L}(t) or θ_{T}(t) oscillates symmetrically around EEA. Second, if h = 2δ = 96^{o}, the symmetry is broken. Hence, λ(t) in Fig. 2(b) or 2(c) cannot exhibit the TFD feature in general. Fig. 2(b) is plotted by using parameters; f = 10 Hz (the medium-f case), n =π/2, δ = 48^{o}, and ψ_{L} = 0^{o} in Eq. (18A). In Fig. 2(b), there are two unequal valleys during a full period (T=1/f) of the λ _{∥}(t) plot. Because of this two-valley feature, we call it quasi-frequency-doubling (QFD). Thirdly, in the special case of the low-f limit (i.e., ω → 0) and t is finite, we shall show that the FD feature in λ _{∥}(t) or λ_{⊥}(t) is *somewhat* allowed, even when h = 96^{o}≠180^{o}. From Eqs. (14) – (15), in the near static case, we have,

Since we were at time t > (1/α), when the decaying in Eq. (14) has gone, we can write,

Further, from definitions for b and α, we have (b/α) = [H_{m}M_{S}sin(δ)]/(2K_{u}) = [H_{m}sin(δ)]/H_{K} = 0.94, where H_{K} = (2K_{u})/M_{S} ≈ 3.8 × 10^{3} A/m or 47 Oe is the anisotropy field. Thus, we can show that approximately speaking,

which indicates a close-frequency-doubling (CFD) feature in the low-f case, even for δ≠90^{o}. Besides, we also conclude that in the δ≠90^{o} case, the CFD feature means that the two valleys are almost equal, while the QFD feature means that two valleys are largely un-equal. Fourthly, Fig. 2(c) is plotted by using f = 100 Hz (the high-f case), n =π/10, δ = 48^{o}, and ψ_{L} = 0^{o} in Eg. (18A). As expected, there is no-frequency-doubling (NFD) for λ_{∥}(t) in Fig. 2(c). Fifthly, Fig. 2(d) is plotted by using f = 100 Hz, n =π/10, δ = 48^{o}, and ψ_{T} = 180^{o} in Eq. (18B). Obviously, λ_{⊥}(t) in Fig. 2(d) also shows the NFD feature. Moreover, by comparing Figs. 2(c) and 2(d), we find that in the high-f case, the phase difference, Δϕ≡ϕ_{L}−ϕ_{T}, where ϕ_{L} and ϕ_{T} are the phases of λ _{∥}(t) and λ⊥(t), respectively, is also close to 180^{o}.

Finally, all the theoretical discussion above is based on the assumption that H(t), generated by the signal generator and power amplify system, contains only one drive frequency (f) and no other higher harmonics (f_{k} = kf with k = 2, 3, 4, 5, - - -). As shown later, this does not reflect the real situation. Thus, the analysis of the experimental results is a little more complicated.

## IV. RESULTS AND DISCUSSION

Fig. 3(a) shows the H(t), λ _{∥}(t) and λ_{⊥}(t) plots in the low-f case (f = 0.07 Hz). At t = t_{o}, the green (or H) curve crosses zero from positive to negative. It serves as a good reference point. Correspondingly, when t = t_{o}, λ_{∥} should decrease from $ \lambda \u2225 P $ to $ \lambda \u2225 \delta $, and λ_{⊥} should increase from $ \lambda \u22a5 P $ to $ \lambda \u22a5 \delta $. The former event is characterized when t = t_{∥} at the valley of the λ_{∥} (or blue) curve, and the latter event when t = t_{⊥} at the peak of the λ_{⊥} (or red) curve. Then, in Fig. 3(a), we can measure Δt_{∥} = t_{∥} − t_{⊥} and Δt_{⊥} = t_{⊥} − t_{0}. Hence, the phase ϕ_{i} is set equal to 2πfΔt_{i}, where i = ∥ and ⊥. From Fig. 3(a), the CFD features appear in the λ_{∥}(t) or λ_{⊥}(t) curve; e. g., Ref. 1 the averaged period of each curve is T_{λ} = 7.123 sec ≒ 0.997×(T/2), where T = 14.286 sec; and Ref. 2 The neighboring amplitudes of the λ(t) curve are almost, but not exactly, equal to each other. These results agree with our previous theoretical discussion (#3) in Sec. III that in the low-f limit, both λ _{∥}(t) and λ_{⊥}(t) should exhibit the CFD feature with b/α = 0.94 ≒1, even in the asymmetric case, such as δ = 48^{o}. Moreover, in Fig. 3(a), the λ_{∥}(t) and λ_{⊥}(t) curves are square-wave-like, instead of sinusoidal-like. This is because in theory or Eqs. (7)- (9), we did not introduce the rotation-pinning mechanism. In reality, $ M s \u21c0 $ cannot start rotating, until H is larger than a critical pinning field (H_{RC}). Thus, the incipient magnetostrictive response was abrupt, rather than smooth. It explains why the curves are more square-wave-like, at least in the low-f case.

Fig. 3(b) shows H(t), λ_{∥}(t) and λ_{⊥}(t) in the medium-f case (f =32 Hz). One distinct feature for λ_{∥}(t) is that over a full period (T =0.0313 sec.) there are two un-equal valleys, and for λ_{⊥}(t) the two un-equal peaks. They exactly describe the QFD feature found in the simulated result (e.g., Fig. 2(b)) for the medium-f case. Using the same criteria, t_{∥} should correspond to the lowest valley, while t_{⊥} to the highest peak, as shown in Fig. 3(b).

Fig. 3(c) shows H(t), λ_{∥}(t), and λ_{⊥}(t) in the high-f case (f = 122 Hz). Obviously, both λ_{∥}(t) and λ_{⊥}(t) exhibit the NFD feature. In addition, according to t_{o}, t_{∥}, and t_{⊥} in this figure, we find that Δϕ = -180^{o}. Notice that since λ_{∥} and λ_{⊥} are periodic with a 2π period in Fig. 3(c), Δϕ = -180^{o} is equivalent to Δϕ = 180^{o}. Thus, the experimental findings above are in agreement with the simulated results in Figs. 2(c) and 2(d). Moreover, as f increases, the λ_{∥}(t) and λ_{⊥}(t) curves are more distorted, Δλ_{pp}, defined as $\Delta \lambda p p \u2261 \lambda \u2225 P \u2212 \lambda \u22a5 P $ increases, and Δϕ increases from 0^{o} to 180^{o}.

From Fig. 2(a), the f =0.07 Hz case, we can plot the λ_{∥} vs. H, λ_{⊥} vs. H, and λ_{∥} vs. λ_{⊥} curves as shown in Figs. 4(a) and 4(b). In Fig. 4(a), both the λ_{∥} vs. H and λ_{⊥} vs. H curves show the typical feature of the “butterfly” pattern.^{8} The error due to noise/drift in these two plots is estimated to be about 15%. The pinning field H_{C} and the saturation field H_{S} can also be determined from the λ_{∥} vs. H or λ_{⊥} vs. H curve. There is little hysteresis in the two butterfly loops. But, as f increases, the degree of hysteresis increases, and the loops are more distorted, which make the determinations of H_{C} and H_{S} virtually impossible in the high-f case. In Fig. 4(b), the λ_{∥} vs. λ_{⊥} plot is linear, and located in the second quadrant. The linear behavior confirms that the phase difference (Δϕ) between λ_{∥} and λ_{⊥} is zero in the low-f case. A little degree of hysteresis is expected in Fig. 4(b). When f increases, the λ_{∥} vs. λ_{⊥} plot is rotated from the second to third quadrant, and the enclosed area (or degree of hysteresis) of the plot increases significantly. The former indicates Δϕ changes from 0^{o} to 180^{o}, and the latter indicates the enhancement of the eddy-current (EC) effect, as f increases. From auxiliary data, the EC skin depth δ_{m,} defined as δ_{m} = [(2ρ)/(μω)]^{1/2}, for 2605SA1 decreases from 10 mm to 240 μm, as f increases from 0.07 to 122 Hz.

In general, as shown in Fig. 5(a), both H_{C} and H_{S} increase with increasing f. All these facts also indicate that the EC effect becomes more important at higher frequencies.

According to the sample arrangement in Fig. 1, we may consider that both ends (along L) of the ribbon vibrate quasi-freely, when magnetized by H. Then, we have another mechanical problem, and its equation of motion, without damping, is described as,

where x, m, and *A _{R}* are the vibration amplitude, mass, and the cross-section area of the ribbon, respectively. Hence, from Eq. (22) the natural resonance frequency (f

_{R}) of the mechanical system is expressed as,

It is noticed that in Refs. 9 and 10, their sample was clamped at one end to form a cantilever. But, in this study both ends were quasi-free. Thus, their resonance driving f_{n} of H(t) should be equal to (n/2)f_{R} here, and this is true only if the TFD criterion is strictly satisfied. Using Eq. (23), we find that f_{R} = 14.4 KHz. Thus, it is expected that in our case, as f increases from 0 to f_{R}, the waveforms of λ _{∥}(t) and λ_{⊥}(t) are more distorted, and Δλ_{pp} should increase monotonically, which are partially confirmed in Figs. 3(b), 3(c), and 5(b). In addition, as shown in Appendix, there may exist another resonance, called domain-wall moment of inertia resonance, occurring at f_{r}, which is much higher than f_{R}.

The H(t) curves in Figs. 3(a)- 3(c) look purely sinusoidal, and seem containing one driving f only. In order to check, we did the following studies. Fig. 6(a) show the FFT spectrum of H(t) in Fig. 3(a). As shown in Fig. 6(a), besides the drive signal at f, there are other higher-harmonic signals at f_{k} = kf, respectively. Here, H, at f =0.07 Hz, is the main signal, 4.88×10^{3} A/m. H_{k} at higher f_{k} is small, and decreases with the increasing k-th order. As H_{k}≦ H_{C} = 37.6 A/m in this case, we think that H_{k} should have little effect to cause any significant λ_{∥} or λ_{⊥} signal at f_{k}. Fig. 6(b) and 6(c) show the FFT spectrums of the λ_{∥}(t) and λ_{⊥}(t) in Fig. 3(b) and 3(c) (the low-f case), respectively. In Fig. 6(b) or 6 (c), we can only find the even-harmonic signals at f_{2k} = 2(k-1)f. This result again confirms our previous conclusion that Ref. 1 in the low-f case, the λ _{∥}(t) and/or λ_{⊥}(t) should exhibit the CFD feature, and Ref. 2, the odd-harmonic magnetostriction signals do not show up at all, because the H_{k}, though existing, should have no effect on magnetostrictions.

When f = 122 Hz, the FFT spectrum of H(t), in Fig. 3(c), is shown as in Fig. 7(a). Besides the main H signal at f, there are also higher-harmonic signals (H_{k}) at f_{k}, respectively. Since H_{k} < < H_{C} ≅ 4.4 × 10^{3} A/m (estimated), we believe that only the main driving H (at f) can enable λ_{∥} and λ_{⊥} signals in their FFT spectrums. In Fig. 7(b) or 7(c), we find the main λ_{∥} or λ_{⊥} signal (at f), as well as all the higher-harmonics signals (at f_{k} = kf), respectively. This result confirms our previous conclusion that in the high-f case, the λ_{∥}(t) and/or λ_{⊥}(t) should exhibit the NFD feature.

Fig. 8 shows Δϕ plotted as a function of f. Notice that Δϕ was defined from t_{o}, t_{∥}, and t_{⊥} in Fig. 3, as discussed previously. Thus, if the TFD or CFD feature applies, Δϕ = 0 is equivalent to Δϕ = π. Similarly, if the NFD feature applies, Δϕ = π is equivalent to Δϕ = − π. From Figs. 3(a)- 3(c), we can make the following classifications: (1) when 0 < f ≦ 1.5 Hz, the CFD feature is observed; (2) when 1.5 < f≦60 Hz, the QFD feature; and (3) when f > 60 Hz, the NFD feature. Thus, although in Fig. 8, we observe such trends that Δϕ = 0 in the range 0 < f≦5.6 Hz, Δϕ = π in the range 12≦f≦112 Hz, and Δϕ = − π at f = 122 HZ, we can actually consider Δϕ is independent of f and Δϕ = π always. This agrees with our previous conclusion that ϕ_{L} = ϕ_{T} + π or Δϕ = π. Notice that since ϕ_{T} = (tan^{−1}ω/α), both ϕ_{L} and ϕ_{T} are frequency dependent, but Δϕ is not.

## V. CONCLUSION

We have studied the frequency (f) dependence of λ_{∥} or λ_{⊥} in the as-cast Metglas^{®} 2605SA1 ribbon far below f_{R}. A theoretical model was proposed to explain all the experimental results. Following conclusions are summarized. First, the true-frequency-doubling (TFD) feature occurs in the magnetostriction signal, only when the symmetry of the system is reserved; i.e., δ = π/2. If δ≠π/2, for example δ = 48^{o} (our case), the symmetry is broken, and only the close-frequency-doubling (CFD), or quasi-frequency-doubling (QFD), or no-frequency-doubling (NFD) feature is possible. Second, when it is in the low-f limit, the λ signals show the CFD feature, when in the medium-f case, the QFD feature, and in the high-f case, the NFD feature. Thirdly, the eddy-current effect becomes more prominent, as f increases. It is supported by the facts that with increasing f the degree of the λ-hysteresis (or the enclosed area within the λ_{∥} vs.λ_{⊥} loop ) increases, and H_{C} and H_{S} also increases. Fourthly, as the natural mechanical resonance of the ribbon is approached from below, Δλ_{pp} increases monotonically. Fifthly, the phase difference Δϕ between ϕ_{∥} and ϕ_{L}, as determined from the _{ }λ_{∥}(t) and λ_{⊥}(t) curves, is independent of f, and Δϕ is always equal to π.

## ACKNOWLEDGEMENT

This work was supported in part by IOP-AS and in part by Grant No. NSC 100-2112-M-001-017-MY3.

### APPENDIX

If the domain-wall moment of inertia term, i.e., the $ ( I DW ) d 2 \theta d t 2 $ term, has been included in Eq. (7), the general solution for θ(t) becomes,

where ω_{r} = 2πf_{r} = (2K_{u})/[I_{DW}M_{S}(sinδ)] is the resonance frequency of the DW rotation, and tan $ ( \psi DW )= \beta \omega ( \omega r 2 \u2212 \omega 2 ) I DW $. Since I_{DM} is usually a very small quantity, we expect f_{r} >>f_{R}. Both f_{R} and f_{r} are out of the frequency range used in this study.