Demagnetizing correction in permeability measurements of cylinders

When an ellipsoid is made of material with constant susceptibility χ and is placed in a uniform applied field H_a along a principal axis, its magnetization M and the demagnetizing field H_d produced by surface poles are both uniform and parallel to H_a. The demagnetizing factor N is defined by H_d = −NM and calculated analytically [1]. This definition of the demagnetizing factor cannot be used for bodies of other shapes, since uniform H_a leads to non-uniform M and/or H_d in general. However, if the volume or midplane averaged M and H_d are both parallel to H_a, a magnetometric or fluxmetric demagnetizing factor may be defined as N_{m, f} ≡ −〈H_d〉_{vol, mid}/〈M〉_{vol, mid}, where 〈H_d〉_{vol, mid} and 〈M〉_{vol, mid} are the components of volume or midplane averaged H_d and M, respectively, on the H_a axis. Since the volume or midplane averaged field 〈H〉_{vol, mid} = H_a − N_{m, f}〈M〉_{vol, mid}, χ may be corrected from the measured external susceptibility χ_ext ≡ 〈M〉_{vol, mid}/H_a by

$$\begin{equation} 1/\chi =1/\chi _{\rm ext}-N_{m,f}. \end{equation} \tag{ 1 }$$

In this work, we study cylinders of length 2L_s and radius R_s, whose N_{m, f} are functions of the aspect ratio γ ≡ L_s/R_s and χ. To use equation (1), a large number of values of N_{m, f} over a large range of γ and χ have to be calculated accurately. Before the 1990s, there were complete analytical or accurate numerical calculations of N_{m, f} only for cylinders of χ → 0 or → ∞ [2–4] and a few accurate but complicated numerical calculations of N_m for χ = −1 and ∞ and γ from 0.25 to 4 [5, 6]. Accurate calculations of N_{m, f} at many values of γ and χ became possible only when a numerical technique with a vital correction was developed in [7]. After a conjugate relation was studied, optimal data points of χ could be chosen, so that the number of calculated N_{m, f} points could be greatly minimized and the final results of N_{m, f}(γ, χ) of accuracy 0.1% are tabulated in [8].

Equation (1) for N_m is relevant to magnetometric measurements, by which the magnetic moment of the entire sample is obtained directly. It has been explained in [8] how to make interpolations from the calculated data points to obtain N_m for any values of γ and χ and how to use equation (1), or more explicitly to use

$$\begin{equation} \chi =\chi _{\rm ext}/[1-\chi _{\rm ext}N_m(\gamma ,\chi )], \end{equation} \tag{ 2 }$$

to obtain χ and N_m iteratively from the directly measured χ_ext and γ. Equation (2) has been compared with experimental data for magnetometric measurements of superconducting and soft magnetic cylinders with the field applied along the axis and a diameter, respectively [9, 10].

Magnetometric measurements are usually carried out for small samples by using instruments such as dc magnetometers and ac susceptometers commercially available. For relatively large samples, fluxmetric measurements are often performed, for which the directly measured quantity is a certain kind of average flux density B but not M so that equation (1) cannot be used. In this work, we develop a formula for the fluxmetric case and use it for demagnetizing correction.

Cylindrical samples of radius R_s = 0.005 m (cross-sectional area A_s = πR²_s) and different lengths were cut from a high-resistivity iron alloy rod. Low-field ac susceptibility was measured with a lock-in amplifier at f = 40 Hz and at maximum applied field less than 20 A m⁻¹. After a B-measuring coil of number of turns N_B = 10 was tightly wound around the midplane with a wire of thickness 0.2 mm, the cylindrical sample was inserted in the center of a solenoid of length L_H = 0.392 m, radius 0.0106 m and number of turns N_H = 300, whose winding was connected to a resistance R₀ = 10 Ω and then to the internal oscillator of a lock-in amplifier, to be magnetized. The voltage across the resistance, V_H, and the in-phase and out-of-phase (with respect to V_H) voltages across the B-measuring coil, V_B1 and V_B2, were measured by the lock-in amplifier to obtain the applied field H_a and the cross-sectional out-of-phase and in-phase flux densities, B'' and B', based on the Ampère law and Faraday law, respectively. The in-phase and out-of-phase external permeabilities are calculated by

$$\begin{eqnarray} \mu _{\rm ext}^\prime \mu _0&\equiv &{B^\prime \over H_a}={R_0L_HV_{B2}\over 2\pi fN_HN_BA_sV_H}, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \mu _{\rm ext}^{\prime \prime }\mu _0&\equiv &{B^{\prime \prime }\over H_a}={R_0L_HV_{B1}\over 2\pi fN_HN_BA_sV_H}. \end{eqnarray} \tag{ 4 }$$

Since the measured low-field B''/B' at 40 Hz for all samples is smaller than 0.02, the obtained μ'_ext is regarded as its dc limit μ_ext.

The measured μ_ext for five cylindrical samples of γ between 3.15 and 7.20 is listed in table 1 and shown in figure 1 by open circles. The external permeability is defined, equivalent to equation (3), as μ_extμ₀ ≡ Φ/A_sH_a, where Φ is the flux through one turn of the B-measuring coil. Since Φ/μ₀ = 〈M〉_midA_s + [H_a − N*_f(γ, χ)〈M〉_mid]A_c, where A_c is the average area of the coil and N*_f is the fluxmetric demagnetizing factor when the midplane average of M and H_d is made within A_s and A_c, respectively, we have

$$\begin{equation} \chi _{\rm ext,mid}\equiv {\langle M\rangle _{\rm mid}\over H_a}={\mu _{\rm ext}-A_c/A_s\over 1-N_f^*(\gamma ,\chi )A_c/A_s}. \end{equation} \tag{ 5 }$$

Using equation (5), equation (1) is written in the fluxmetric case as

$$\begin{equation} \chi ={\mu _{\rm ext}-A_c/A_s\over 1-(\mu _{\rm ext}-A_c/A_s)N_f(\gamma ,\chi )-N_f^*(\gamma ,\chi )A_c/A_s}. \end{equation} \tag{ 6 }$$

N*_f(γ, χ) may be replaced by N_f(γ, χ) when A_c is not much larger than A_s as in the present case, so that equation (6) may be simplified as

$$\begin{equation} \chi =(\mu _{\rm ext}-A_c/A_s)/[1-\mu _{\rm ext}N_f(\gamma ,\chi )]. \end{equation} \tag{ 7 }$$

The measured μ_ext may be iteratively corrected to χ by using this equation and the calculated N_f(γ, χ).

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The procedure is described in figure 2 for the case of γ = 4.16. Continuous cubic-spline N_f versus γ curves for χ = 1.5, 9, 99 and 999 are plotted in figure 2(a) on log–log scales based on table 1 in [8]. N_f values at γ = 4.16 and χ = 1.5, 9, 99 and 999 are obtained from the crossing points of these curves and the vertical γ = 4.16 line, and the continuous cubic-spline N_f(γ = 4.16, χ) curve is drawn in figure 2(b) on log-linear scales. For iterations, equation (7) is rewritten as

$$\begin{equation} \chi ^{(i+1)}=(\mu _{\rm ext}-A_c/A_s)/[1-\mu _{\rm ext}N_f(\gamma ,\chi ^{(i)})]. \end{equation} \tag{ 8 }$$

Iteration starts from i = 1 with χ⁽¹⁾ = μ_ext and then increases to i = 2, 3, ... until the difference between χ^{(i + 1)} and χ⁽ⁱ⁾ becomes smaller than 1%. The iterated results for [N_f(γ, χ⁽ⁱ⁾), χ^{(i + 1)}] are shown by the crossing points between the curve and the vertical lines in figure 2(b) for the case of γ = 4.16. The corrected χ as a function of γ is shown in figure 1 by open squares. We observe that while the measured log μ_ext increases linearly with log γ, the corrected χ becomes constant at χ = 74.7 ± 3%.

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As a reference, the interpolated N_f as a function of χ is listed in table 2 and the iterated results of [N_f(γ, χ⁽ⁱ⁾), χ^{(i + 1)}] are listed in table 1 for all the samples with different values of γ.

If N_f is replaced by N_m, equation (7) may also be used for volume-averaged μ_ext when the B-measuring coil uniformly covers the entire length of the cylinder. However, additional error due to possible non-uniformity of the winding has to be considered.

The results here and in [8] are valid for materials with constant χ. For a ferromagnetic cylinder as in the present case, the constant χ is realized at a small applied field, below which χ does not change, and at low frequency, where the eddy-current shielding effect is weak, so that μ''_exp/μ_exp' ≪ 1 and μ'_exp is regarded as the initial dc permeability. If μ''_exp is not negligible, ac hysteresis has to be considered and equations (2) and (7) should be used for complex ac susceptibility or permeability expressed by χ = χ' − jχ'', χ_ext = χ'_ext − jχ''_ext and μ_ext = μ'_ext − jμ''_ext. The problem of demagnetizing correction when considering hysteresis has to be studied for different situations and purposes.

For the ac susceptibility of conducting or superconducting cylinders, where the magnetic moment arises from circulating eddy currents or supercurrents rather than local material χ, N_{f, m} for χ = −1 should be used with γ > 5 for studying the frequency or field dependence, as shown by direct calculations in [8]. When the frequency dependence of low-field susceptibility of a soft ferromagnetic cylinder is studied, N_{f, m} for χ equal to dc μ_ext should be used, and γ should be large enough so that 1/N_{f, m} be comparable with μ_ext and low-frequency demagnetizing correction may be performed with adequate accuracy.

Comparing equation (2) with equation (7), we see that even for an ideal winding with A_c/A_s = 1, for which equation (7) is exactly valid, χ_ext in equation (2) is replaced by μ_ext − 1 in one place of equation (7) and by μ_ext in another place. This is a consequence of the validity of χ_{ext, mid} = (μ_ext − 1)/(1 − N_f) rather than χ_{ext, mid} = μ_ext − 1.

If A_c/A_s > 1, then N*_f < N_f, since the demagnetizing field outside the cylinder is smaller than that inside (see the appendix). In this case, equation (6) should be used. Comparing equation (6) with equation (2), we see an extra term N*_f(γ, χ)A_c/A_s in equation (6). This feature is of great importance in measurements of weakly magnetic samples, for which N*_f(γ, χ)A_c/A_s may be much larger than (μ_ext − A_c/A_s)N_f(γ, χ) if χ ≪ 1 and A_c/A_s ≫ 1 (for increasing the measuring sensitivity, a large N_B requires the B-measuring coil to be wound multi-layered with a large A_c), so that the demagnetizing correction may be much larger than what is intuitively thought without considering the extra term. This has led to a major revision of the fluxmetric method in an ASTM standard [11], avoiding the error owing to incorrect demagnetizing correction, which could reach 30%. The revision was based on a paper written in Chinese [12], whose main results are reviewed and improved in the appendix.

When χ is large, the above feature of equation (6) becomes less important, and the accurate determination of χ requires sufficiently large γ, as in the situation of equation (2) for the magnetometric case. Although ring samples with circular magnetization (so without demagnetizing effects) are standard for the case of high χ, direct measurements of midplane and volume-averaged μ_ext on a long cylinder to study χ and its uniformity are especially important for certain magnetic devices, such as the magnetic core of a self-inductance or a magnetic torquer rod for spacecraft attitude control [13]. In such devices, large values of material χ are required throughout the entire core or rod and are realized by optimizing material composition and heat treatment. The χ check using the proposed technique may provide important data for quality improvement, which could be vital for a very long torquer rod.

All the above statements for cylinders are basically valid for rectangular prisms, for which the values of N_{m, f} as functions of aspect ratios and χ are accurately calculated and tabulated in [14]. We note that exact formulas for N_{m, f} of rectangular prisms at χ → 0 are derived in [15, 16].

Demagnetizing correction is used for converting the measured magnetic permeability of a sample with a finite length along the applied field to the intrinsic susceptibility of the sample material. Such a correction may be made straightforwardly when the sample susceptibility is obtained by magnetometric measurements, but extra work has to be done if the directly measured property is permeability. We have derived the correction formula, equation (7), for the latter case, and its applicability for cylindrical samples is shown experimentally. The importance and applications of such a correction are also discussed.

This work was financially supported by the National Natural Science Foundation of China (no 61171003), the Beijing Municipal Natural Science Foundation (no 1102024) and the New-Star of Science and Technology supported by Beijing Metropolis (no 2008A020).

The fluxmetric method in ASTM designation A342, standard test methods for permeability of feebly magnetic materials, originally published as A342-49 and followed till A342-84 was revised to A342-95 in 1995, based on a paper [12] written in Chinese. Some changes made in the revision are explained here for facilitating more readers.

A342-84 signifies the sample cross-sectional area A_s ⩾ 0.2 cm², sample length 2L_s ⩾ 10 cm, equivalent dimension ratio of the sample γ ⩾ 4, 20 and 30 for χ < 0.1, 0.1–1 and 1–3, respectively, the length of the test coil L_c ⩽ 2.54 cm and the outer cross-sectional area of the test coil A_c2 ⩽ 10A_s. A_c2 is allowed to be larger than 10A_s when χ ⩽ 0.05. The measuring error of χ (written erroneously as permeability μ_r = 1 + χ) is assigned less than 2%.

Since both the measuring errors of the flux Φ and applied field H_a should be around 1%, only the positive demagnetizing error with respect to the field H,

$$\begin{eqnarray} &&\fl \delta _h=H_a/\langle H\rangle _{\rm mid}-1=-\langle H_d\rangle _{\rm mid}/\langle H\rangle _{\rm mid}=\chi N_f(\gamma ,\chi ), \nonumber\\ \end{eqnarray} \tag{ A.1 }$$

is considered. It may be estimated by using the accurate values of N_f(γ, χ) for cylinders given in [8] that δ_h ⩽ 0.29%, 0.13% and 0.19%, for χ < 0.1, 0.1–1 and 1–3, respectively. All these values are negligible compared with the assigned error 2%.

However, A342-84 has ignored a much larger negative demagnetizing error with respect to magnetization M,

$$\begin{equation} \delta _m=\langle H_d\rangle _{\rm mid}^*A_c/(\langle M\rangle _{\rm mid}A_s)=-N_f^*(\gamma ,\chi )A_c/A_s. \end{equation} \tag{ A.2 }$$

Since the test coil is multi-layered, N*_f(γ, χ) and A_c are the average of those defined in section 3, considering the total flux of demagnetizing field linked to an N_B-turn infinitely short test coil uniformly wound on the midplane with inner and outer radii R₁ = α₁R_s and R₂ = α₂R_s, respectively.

For a cylindrical sample of radius R_s and length 2L_s = 2γR_s at χ = 0 and a test coil described above, N*_f(γ, χ) is calculated by

$$\begin{eqnarray} &&\fl N_f^*={3\over \alpha _2^2-\alpha _1^2}\sum _{l=0}^{\infty}{(-1)^l(2l+1)!\big(\alpha _2^{2l+3}-\alpha _1^{2l+3}\big)C_{2l+1}\over 2^{2l}(l!)^2(l+1)(2l+3)\lambda ^{2l}}, \nonumber\\ \end{eqnarray} \tag{ A.3 }$$

where

$$\begin{eqnarray} &&\fl C_{2l+1}={1\over 2^{2l}}\sum _{m=0}^l{(-1)^m(4l-2m)![1-(1+\lambda ^{-2})^{m-2l-1/2}]\over m!(2l-m)!(2l-2m+1)!}. \nonumber\\ \end{eqnarray} \tag{ A.4 }$$

Correspondingly, A_c is calculated by

$$\begin{equation} A_c={1\over R_2-R_1}\int _{R_1}^{R_2}\pi \rho ^2{\rm d}\rho =A_s{\alpha _2^3-\alpha _1^3\over 3(\alpha _2-\alpha _1)}. \end{equation} \tag{ A.5 }$$

N_f and N*_f at χ = 0 calculated using equations (A.3) and (A.4) as functions of γ are shown in figure 3. We see that N*_f < N_f at any given value of γ, and N*_f decreases with increasing α₂ at fixed γ and α₁.

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Using N*_f(γ, χ = 0) shown in figure 3 and assuming N*_f(γ, χ)/N_f*(γ, 0) = N_f(γ, χ)/N_f(γ, 0), where N_f(γ, χ) is given in [8], we obtain approximate N*_f(γ, χ), so that δ_m may be calculated using equations (A.2) and (A.5). The results relevant to A342-84 are that −δ_m ⩽ 13.4–13.7%, 0.70–0.73% and 0.32–0.35% for χ < 0.1, 0.1–1 and 1–3, respectively, when (α₁ = 1.5, α₂ = 10^1/2) is used, and if (α₁ = 1.5, α₂ = 20^1/2) is used for χ ⩽ 0.05, −δ_m can reach 20%. In these calculations, the test coil is assumed infinitely short. A larger δ_m is expected for a coil with finite L_c, so that the maximum error of χ may reach around 30% if L_c/(2L_s) = 0.254 is assumed.

The easiest way to solve this problem is to assign γ > 30 for all materials of χ ⩽ 3 to obtain both negligible δ_h and δ_m. However, this is not realistic or economic for materials with very small χ, for which R_s should be large to increase M-measuring sensitivity. We propose two requirements for assigning the standard dimensions of the sample and test coil: (i) δ_h be negligibly small; (ii) −δ_m be relatively small and fixed. These requirements may be satisfied by γ = 10, 15 and 30 for χ < 0.5, 0.5–1 and 1–3, respectively. In this case, δ_h is negligibly small: δ_h ⩽ 0.26%, 0.24% and 0.19% for χ < 0.5, 0.5–1 and 1–3, respectively. −δ_m = 2.69–2.85%, 1.28–1.33% and 0.32–0.35% at (α₁ = 1.5, α₂ = 10^1/2) for χ < 0.5, 0.5–1 and 1–3, respectively. Since the difference between the lower and upper limits for each χ range is small, one may use their approximate average, −δ_m = 2.8%, 1.3% and 0.33%, to make demagnetizing correction. After an additional correction with respect to finite coil length, whose experimental procedure is introduced in [12], the error of χ measurements can be less than 2%.

Based on the above calculations, the main changes made in A342-95 are as follows: γ ⩾ 10, 15 and 30 for χ < 0.5, 0.5–1 and 1–3, respectively; L_c/(2L_s) ⩽ 0.2; A_c2/A_s ⩽ 10 independent of χ. In this case, without mentioning the total error in χ, the largest negative error in χ due to demagnetizing effects is stated as −3% for χ < 0.5.