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The complex initial reluctivity, permeability and susceptibility spectra of magnetic materials
Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
The complex initial reluctivity, permeability and susceptibility spectra of magnetic materials N.C. Hamilton North View, Chawston Lane, Chawston, Bedford MK44 3BH, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 24 February 2014 Received in revised form 13 September 2014 Accepted 17 October 2014 Available online 22 October 2014
The HF complex permeability spectrum of a magnetic material is deduced from the measured impedance spectrum, which is then normalized to a series permeability spectrum. However, this series permeability spectrum has previously been shown to correspond to a parallel magnetic circuit, which is not appropriate. Some of the implications of this truth are examined. This electric/magnetic duality has frustrated efforts to interpret the shape of the complex magnetic permeability spectra of materials, and has hindered the application of impedance spectroscopy to magnetic materials. In the presence of magnetic loss, the relationship between the relative magnetic permeability and the magnetic susceptibility is called into question. The use of reluctivity spectra for expressing magnetic material properties is advocated. The relative loss factor, tanδm/μi is shown to be an approximation for the imaginary part of the reluctivity. A single relaxation model for the initial reluctivity spectra of magnetic materials is presented, and its principles are applied to measurements of a high permeability ferrite. The results are presented as contour plots of the spectra as a function of temperature. & 2014 Elsevier B.V. All rights reserved.
Keywords: Complex permeability spectrum Reluctivity Magnetic susceptibility Soft ferrite
1. Introduction 1947 was very significant in the history of ferrite and of magnetism in general. Snoek′s 1947 book [1] described the preparation of MnZn and NiZn ferrites; these are still the chemical basis of most of the soft ferrites in use today. Snoek′s 1947 letter [2] proposed that the complex series susceptibility spectrum of ferrite was due to gyro-magnetic resonance. Also in 1947, Macfadyen [3] gave a detailed description of the use of permeability and reluctivity as complex quantities. In the earlier part of the century, it had become firm practice to use permeability rather than reluctivity, and as long as these were viewed as scalar quantities, this caused no difficulties. Permeability was probably chosen because the values were of a convenient size when using the CGS system of units, and because it seems more natural for a magnetic field strength to cause a flux density. But by 1950, the use of the complex series permeability spectra μr,s(f)¼ μ′(f)-jμ″(f) in SI units for small-signal material characterization was becoming standard practice. It remains so today, in spite of successive observations [4–7] that the complex permeability spectra of ferrites at HF are simpler when expressed in parallel terms, rather than in series terms. Snelling [8] and Wijn [9] state that it was Snoek and Six who established an alternative measure for characterizing materials: E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmmm.2014.10.061 0304-8853/& 2014 Elsevier B.V. All rights reserved.
the relative loss factor tanδm/μi (or its reciprocal, μiQm, probably due to Owens [10]), where μi is the initial permeability, a scalar value measured when both frequency and flux density are very small. The relative loss factor is widely used by manufacturers of soft ferrites, especially in Northern America. Since 1970s, work has progressed from material characterization to the interpretation and simulation of the permeability spectra. In the author's view, the exclusive use of complex series permeability is hindering progress on the subject. This article reviews the basis of permeability spectra with the intention of encouraging the use of complex reluctivity or parallel permeability.
2. Theory 2.1. Reluctivity and permeability Consider the magnetic circuit in a ring core. The components of energy storage and of energy loss in the core are homogenously distributed throughout the material, so all the flux change in the core is subject to both energy storage and dissipation. If the core's distributed energy storage were to be lumped in one sector of the core, and the magnetic losses in the other sector, then the lumped element small-signal magnetic circuit must consist of these two material types in series. Macfadyen [3] observed "It is generally best to work with permeability when there are magnetic paths in parallel and with
N.C. Hamilton / Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
497
the loss term of the series magnetic circuit, but they conceal this, the origin of their usefulness.
2.2. Susceptibility and permeability The magneto-static relationship between these quantities as dimensionless scalars is, by definition, μr ¼ χ þ 1, where χ is the (rationalized magnetic volume) susceptibility. Assume the simplest susceptibility spectrum, a single relaxation given by a Debye type equation [14] for the complex magnetic susceptibility χ(ω):
χ (ω) =
Fig. 1. Reciprocals (x 1) relate the various forms of complex relative permeability μr and reluctivity rr and their commonly used approximations. The series components of reluctivity sr,s are shown for completeness, but are not used.
reluctivity when the magnetic paths are in series…". Macfadyen then showed that the complex reluctivity gives a parallel equivalent electrical circuit. This view was amplified by Cherry [11], who showed that the electric and magnetic circuits were duals of each other. Cherry then showed explicit equivalent circuits of a lossy magnetic material as a series magnetic circuit of real and imaginary parts of reluctance, modelled by an electrical circuit consisting of an admittance: an inductor and a resistor connected in parallel. Fig. 1 shows the relationships between the series and parallel forms of both permeability and reluctance, where the sub-scripts denoting series and parallel forms are based on IEC 62044-2 2005. Although characterization of a material's series magnetic energy storage and magnetic loss components is best expressed in terms of complex reluctance, Fig. 1 shows that the same data may be expressed as a so-called parallel permeability, the components of which in fact represent a series magnetic circuit. It is regrettable that common usage differentiates between the series and parallel forms of these magnetic quantities in terms of the series and parallel dual electric circuit. As Schlicke observed [12] of basic discontinuous material structures: "What seems magnetically a series arrangement is electrically a parallel circuit and vice versa." and that the equivalent electric circuit is "… contrary to what the eye seems to see so clearly in the magnetic structure". From Fig. 1, it can be deduced that the ratios between the real and imaginary parts of the permeability and reluctivity are constant, so:
″ μr,p ′ μr,p
=
″ μ′ −σr,s σr′ 1 = r = = = Qm ′ σr″ σr,s − μr″ tanδm
χ0 1 + jωτ
(2)
where χ0 is the static susceptibility, ω is the angular frequency, τ is the time constant of relaxation and j is the complex operator. There are two possible methods of proceeding [15]. 2.2.1. Series susceptibility conversion Add 1 to the series real component of χ(ω), thereby making the assumption that μr(ω) ¼ χ(ω) þ1. Resolve χ(ω) into real and imaginary parts:
χ ′(ω) =
χ0 1 + (ωτ)2
and χ ″(ω) =
−χ0 ωτ 1 + (ωτ)2
(3a) and (3b)
and add 1 to the real part, giving:
μr′(ω) = 1 +
χ0 1 + (ωτ)2
(4a)
and
μr″(ω) =
−χ0 ωτ 1 + (ωτ)2
(4b)
Unfortunately, Eqs. (4a) and (4b) do not easily convert to an expression for complex reluctivity. Fig. 2 shows a graph of the ‘series’ permeability spectra resulting from values of χ0 and τ appropriate to a ferrite for use in HF/VHF inductors. At low frequencies, Qm falls as the frequency increases: this is expected. However, the frequency axis of Fig. 2 is extended to show that, above 800 MHz, Qm starts to rise again. This is not observed in practice, and is an artefact of the series conversion of susceptibility to permeability, which is unlikely to be correct. 2.2.2. Parallel susceptibility conversion Add 1 to the parallel real component of χ(ω). Using the relations given in Fig. 1, convert series susceptibility to its reciprocal. (Magnetic immunity has been suggested [16] as a name for
100 (1)
where Qm is the material's magnetic quality factor, and tanδm is the loss factor. Now, as Macfadyen observed [3]: "Reluctivity possesses the advantage that (σ′r and hence μ′r , p ) is constant over a wide range of frequency…". This is confirmed by examination of the series and parallel permeability spectra of the 22 ferrite grades presented by Snelling [13]. See also the measurements presented in Fig. 5. It follows that the initial permeability μi ≈ μr′, p , and this is shown in Fig. 1. From this, and the relationships given in Eq. (1), the approximations for the relative loss factor tanδm/μi and its reciprocal μiQm follow. These two factors are approximations for
10 1 0.1 1M
10M
100M 1G frequency (Hz)
10G
100G
Fig. 2. Series relative permeability spectrum real (solid) and imaginary (dashed) parts modelled for χ0 ¼ 79 and τ¼ 2 ns, if μr(ω) ¼ χ(ω) þ 1. Also shown (dot dashed) is Qm, which is unrealistic above 800 MHz.
498
N.C. Hamilton / Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
In the 1960s, Olsen wrote a book [17] which gave a careful and plainly stated re-appraisal of the subject. A similar re-appraisal is needed now.
100 10 1
2.3. Snoek's factor
0.1 0.01 0.001 1M
10M
100M 1G frequency (Hz)
10G
100G
Fig. 3. Series relative permeability spectrum real (solid) and imaginary (dashed) parts modelled for χ0 ¼79 and τ ¼2 ns, by adding 1 to the parallel real component of χ(ω). In contrast to Fig. 2, Qm (dot dashed) remains realistic throughout the frequency range.
reciprocal susceptibility):
1 1 + jωτ = χ (ω) χ0
(5)
separating real and imaginary parts and taking their reciprocals gives [16]:
χp′ = χ0 and χp″ (ω) =
(Hz)
(9)
For a single relaxation, S is independent of frequency. Where the material's complex reluctivity spectrum is more complex, S is a weak function of frequency. S(fr)¼ χ0fr is widely known as Snoek's product [18]. But, as S is a function of frequency, Snoek's factor may be a more appropriate name. Substituting Eq. (9) into Eq. (8), and converting from angular frequency ω (rad/s) to frequency f (Hz) gives
1 f +j χ0 + 1 S
(10)
2.4. Hysteresis constant
χ0 ωτ
(7a) and (7b)
From which the complex reluctivity can be assembled:
ωτ 1 +j χ0 + 1 χ0
χ0 2πτ
(6a) and (6b)
ωτ
Add 1 to the parallel real component of susceptibility to give the parallel permeabilities:
σr(ω) =
S=
σ (f ) =
χ0
μ p′ = χ0 + 1 and μ p″(ω) =
In Section 2.2, the time constant of relaxation τ has been used as the variable that characterizes the loss element of the permeability frequency spectrum. This is the approach taken by Debye, and has been taken ever since. However, 1/(2πτ) defines the relaxation frequency (fr) at which −μr″ peaks; this does not characterize a core material. For example, if the core is provided with an air-gap, then τ will change, even though the material's performance does not change. Instead, define a new variable S:
(8)
There is no compact expression for Eq. (8) in terms of 'series' permeabilities, but Fig. 3 shows a graph of the permeability spectrum of this version, using the same parameters as Fig. 2. 2.2.3. Summary-susceptibility and permeability The conversion between susceptibility and permeability causes a problem, the origins of which lie in the series-parallel duality of magnetic and electric circuits. A single relaxation gives a parallel inductor/resistor (LR) model of susceptibility. Converting susceptibility to permeability requires an increase of the model's inductive component. Section 2.2.1 shows the series approach and Section 2.2.2 the parallel approach. Both approaches preserve the χ ¼ μr-1 scalar relationship. The series conversion adds a series inductance to the parallel LR circuit (Eq. (4a)). This is often used, but has an unrealistic high frequency response. The parallel approach gives a realistic high frequency response, because it takes the parallel LR model of susceptibility, and increases the value of L (Eq. (7a)). So, a material's complex relative permeability is found by adding 1 to the parallel real component of magnetic susceptibility. Furthermore, it is convenient to do this, because the parallel approach results in a simple expression for complex reluctivity, and Section 2.1 has argued that the reluctivity is the measure to use. This conclusion applies to the deduction of complex permeability and susceptibility from impedance data. It is not the end of the matter. For example, it may be that these considerations do not apply to free space permeability measurement techniques.
Eq. (10) has a shortcoming: it predicts that, as the frequency approaches zero, the material's quality factor Qm approaches infinity. In practice, as the frequency approaches zero, Qm approaches a limiting value Qmax set by the small-signal hysteresis loss. A typical value for a ferrite core with no air gap might be 200. Introducing an extra term for this gives
σ (f ) =
⎛f ⎞ 1 + j⎜ + σi″⎟ ⎝ ⎠ χ0 + 1 S
(11)
100k 10k 1k 100 10 1 100m 10m 1m 100n 10n 10k
100k
1M 10M 100M frequency (Hz)
1G
10G
Fig. 4. The modelled permeability, reluctivity ‘parallel’ permeability and Qm for the data in Fig. 3, (single relaxation χ0 ¼ 79 S¼ 6.3 GHz) but with Qmax ¼ 200. Note the change in frequency scale from Fig. 3.
N.C. Hamilton / Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
1M
2.5. Ferromagnetic resonance
measurement bandwidth (Hz) 10 30 100 300
Fig. 4 shows the same data as Fig. 3, but using Eqs. (11) and (12) with Qmax ¼200, and showing the curves of the alternative formats for the complex permeability/reluctivity data. The peak in the imaginary part of the 'series' permeability spectrum has no corresponding peak in the reluctivity or ‘parallel' permeability spectra, as the peak is an artefact caused by analysing a series magnetic circuit in parallel terms. It is not due to the resonance effect suggested by Snoek [2]; others have given this peak the name ‘ferromagnetic resonance'. It does not exist. Instead, the aim should be to examine the deviations of the spectra of real ferrites from the single relaxation model of Fig. 4. A ′ (f ) (as permeability has convenient way of doing this is to graph μr,p been used for 100 years, it is hard to avoid) and S(f). For a single relaxation, these will be constants. The cause of the deviations must then be sought.
100k 10k 1k 100 10 1 10k
100k
1M 10M frequency (Hz)
100M
1G 3. Experimental technique
Fig. 5. Permeability and ‘parallel’ permeability spectra deduced from impedance measurements of 1 turn on 2.5 mm diameter T38 (MnZn ferrite) ring core at 20 °C.
where σ″ i is the imaginary part of the initial reluctivity of the material. Its value may be found from
σi″ =
499
tanδi 1 = (χ0 + 1)Q max μi
(12)
where tanδi is the initial loss factor. Note that, in the electrical model, the extra term σ″ i results in a frequency dependent resistor; it is difficult to model this circuit element in a conventional SPICE type simulator. It can only be approximated by a ladder network, used over a limited frequency range.
The aim here is to measure a ferrite's HF impedance spectrum over a range of temperatures, and apply the theory to the pre′ (f ) and S(f) were simultaneously sentation of the results. So that μr,p available over the broadest frequency range, a high permeability MnZn ferrite (T38, μi ¼ 10,000 nominal) was selected. This material had the additional advantage that impedance measurements at the ferrite's Curie temperature needed no special heat resisting equipment. The upper frequency limit of the measurement was set by the sample's dimensional resonance [19]; this undesirable effect was minimised by measuring a small sample, a ring core [20] with an outer diameter of 2.5 mm, held in a single turn coaxial mounting. However, these test conditions were not optimum for the measurement of either Qmax, or of the small permeability remaining above the Curie temperature due to paramagnetism. To improve the characterisation of Qmax, high sample impedance
f t
Fig. 6. Impedance spectra measurements of a 2.5 mm diameter ring core of T38 ferrite at various temperatures. Data presented here as contours of μ′r , p (f , t) , the real part of parallel permeability, which gives rise to the parallel reactance. Permeability contours are in 1000 s; the nominal μi of the ferrite is 10,000. At each measurement temperature, a small circle marks the 'relaxation frequency' fr, the measurement frequency corresponding to the greatest value of the series imaginary part of permeability.
500
N.C. Hamilton / Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
S ft
Fig. 7. Impedance spectra measurements of a 2.5 mm diameter ring core of T38 ferrite at various temperatures. Data presented here as contours of Snoek's factor, S (f , t) = μ″r , p (f , t). f , which gives rise to the parallel resistance. Snoek's factor contours are in GHz. As in Fig. 6, small circles mark the ‘relaxation frequency’.
would be required at low frequencies, implying measurement of a large core with multiple turns. Above the Curie temperature, a large core with a single coaxial turn would be required. As shown in Fig. 5, measurement accuracy was improved by using a narrow IF bandwidth at LF, increased at higher RF, thereby achieving a reasonable sweep time of 8.5 s by the HP 4195A network analyzer, which measured at 83 logarithmically spaced frequencies between 40 kHz and 500 MHz. The HP 87512A resistive signal divider gave fair accuracy for the required low impedance measurements. The available power at the ring core was 23 dBm, which was found to be just below the onset of reduction of Qmax at 20 °C. The ring core was mounted in an SMA connector that was fitted with a bead thermistor for temperature measurement. The SMA connection was OSL calibrated and short circuit compensated. The SMA core mount was immersed in a bath of alcohol at 60 °C or oil heated to 145 °C, and the temperature of the core was allowed to relax back to ambient, while taking complex impedance measurements every 5 °C; this required a short analyzer sweep time to avoid temperature drift during measurement.
resonance frequency rises (Fig. 6). It can also be seen in Fig. 7 that the frequency of minimum Snoek's factor is a function of temperature, but is always close to fr, so these are probably related in some way. The small discontinuity in Fig. 7 between 20 °C and 25 °C is due to the approach to 20 °C from cold, and to 25 °C from hot, and is probably due to the thermal hysteresis [21] of the ferrite's magnetic properties. Fig. 7 shows that Snoek's factor is small near the Curie temperature. In consequence, the 40 kHz lower limit of the measurement frequency range was not quite low enough to register values of initial permeability at temperatures above 120 °C. The data of Figs. 6 and 7 could have been plotted in the more usual terms of series permeabilities. Such graphs would emphasize the dominant ‘relaxation frequency’. By plotting Figs. 6 and 7 as shown, a single relaxation has been eliminated, allowing other, more subtle, effects to be shown.
5. Conclusions 4. Experimental results Fig. 5 shows the permeability spectra at 20 °C deduced from the analyzer's impedance measurements, and using the manufacturer's published value of core factor Σl/A¼ 12.3 mm 1. Fig. 5 shows dimensional resonance above 200 MHz, and Snoek's factor varied between 4.5 and 9 GHz, with a shallow minimum near 0.4 MHz. This occurred near the dominant ‘relaxation frequency’ fr, this is taken to be the measurement frequency corresponding to the maximum value of −μ″r (f). To conveniently display the information in Fig. 5 at 39 different temperatures, the data are given ′ (f , t) and S(f,t) in Figs. 6 and 7. Small circles as contour plots of μr,p indicate fr at each temperature. Figs. 6 and 7 show that the ferrimagnetic properties of core were absent at 135 °C, but had returned by 130 °C. As the ferrite's temperature reduces below the Curie temperature, its high frequency performance improves (Fig. 7), and the core's dimensional
The following conclusions apply to coupled electric and magnetic circuits, as in an inductor with a magnetic core. The conclusions do not necessarily apply to other applications. 1. The characterization of a material's series magnetic energy storage and magnetic loss components is best expressed in terms of a complex reluctance. 2. Because of long usage, it may be preferable to express the complex reluctance in terms of parallel permeability. 3. Parallel permeability is an inappropriate name, as it characterizes a series magnetic circuit. Further clarification and formalization will be required to resolve the matter. 4. The commonly accepted relationship between permeability and susceptibility is suspect, and the conversion is best done in ‘parallel’ terms. 5. These considerations lead to a two element model for a simple relaxation.
N.C. Hamilton / Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
6. The deviations from the simple two element model have been illustrated by measurements of a MnZn ferrite. These deviations, and others that will doubtless be found in other soft magnetic materials, remain to be fully characterized and explained. 7. A third element can be introduced to model the hysteresis loss, as shown in Eq. (11). 8. Impedance Spectroscopy has been widely used for the analysis of dielectric materials, but has been little used on magnetic materials. To progress with this technique, it is essential to acknowledge, and to make allowance for, the series-parallel circuit transformation that is caused by the duality of the magnetic and electric circuits.
References [1] J.L. Snoek New, Developments in Ferromagnetic Materials, 2nd Ed, Elsevier, 1949 (1947). [2] J.L. Snoek, Gyromagnetic resonance in ferrites, Nature 160 (1947) 90. http://dx. doi.org/10.1038/160090a0; Later expanded in J.L. Snoek, Dispersion and absorption in magnetic ferrites at frequencies above one Mc/s Physica 14 (4) (1948) 207–217. http://dx.doi.org/ 10.1016/0031-8914(48)90038-X. [3] K.A. Macfadyen, Vector permeability, J. IEEE Part III Radio Commun. Eng. .94 (.32) (1947) 407–414. http://dx.doi.org/10.1049/ji-3-2.1947.0079. [4] W. Six, Some applications of Ferroxcube, Philips Tech. Rev. 13 (11) (1952) 301–336. [5] P.M. Prache, B. Chiron, Complex susceptibility of high-resistivity ferrites, Proc. IEEE Part B 104 (5 Part S) (1957) 209–212. http://dx.doi.org/10.1049/pi-b1.1957.0030. [6] Y. Naito, Inverse Cole–Cole plot as applied to the ferrimagnetic spectrum in VHF through the UHF Region, in: Proceedings of the International Conference on Ferrites (ICF1-1970), University Park Press, 1971, pp. 558-560. [7] J.K. Watson, S. Amoni, Using parallel complex permeability for ferrite characterisation, IEEE Trans Magn. 25 (5) (1989) 4224–4226. http://dx.doi.org/ 10.1109/20.42576.
501
[8] E.C. Snelling, Soft ferrites, 2nd Ed, Butterworths, London (1988) p1. [9] H.P.J. Wijn, Some remarks on the history of ferrite research in Europe, in: Proceedings of the International Conference on Ferrites (ICF1-1970), University Park Press, 1971, pp. xix–xxi. [10] C.D. Owens, Analysis of Measurements on Magnetic Ferrites, in: Proceedings of the IRE, March 1953, pp. 359–365. http://dx.doi.org/10.1109/JRPROC.1953. 274382. [11] E.C. Cherry, The duality between interlinked electric and magnetic circuits and the formation of transformer equivalent circuits, Proc. Phys. Soc. B 62 (2) (1949) 101–111 http://dx.doi.org/10.1088/0370-1301/62/2/303. [12] H.M. Schlicke, Dielectromagnetic Engineering, John Wiley, New York, 1961 (pp. 8, 9 and 14). [13] E.C. Snelling, Soft Ferrites, 2nd Ed, Butterworths, London (1988) 92–98. [14] J. Smit, H.P.J. Wijn, Ferrites, John Wiley, New York (1959) p290. [15] N.C. Hamilton, Electrical circuit models of the HF initial permeability spectra of soft ferrite lead to revised definition of fundamental material properties, in: Proceedings of the IET 1st Annual Active and Passive RF Devices Seminar, Glasgow, 29 Oct 2013, pp. 59–62. http://dx.doi.org/10.1049/ic.2013.0239. [16] N.C. Hamilton, The small-signal frequency response of ferrites, High Freq. Electron. Mag. 10 (6) (2011) 36–52 http://www.highfrequencyelectronics.com/ index.php?option=com_flippingbook&view=book&id=79:2011-06-june&catid= 8:2011-high-frequency-electronics&Itemid=149. [17] E. Olsen, Applied Magnetism: A Study in Quantities, Philips Technical Library, Eindhoven, 1966. [18] T. Nakamura, Snoek′s limit in high-frequency permeability of polycrystalline Ni–Zn, Mg–Zn, and Ni–Zn-Cu spinel ferrites, J. Appl. Phys. 88 (1) (2000) 348–353 http://dx.doi.org/10.1063/1.373666. [19] F.G. Brockman, P.H. Dowling, W.G. Steneck, Dimensional effects resulting from a high dielectric constant found in a ferromagnetic ferrite, Phys. Rev. .77 (.1) (1950) 85–93 http://dx.doi.org/10.1103/PhysRev.77.85. [20] B64290J35X38 core manufactured by Siemens AG. Now replaced by part B64290P0035X038 from EPCOS AG. [21] A. Globus, Some physical considerations about the domain wall size theory of magnetization mechanisms, J. de Phys. C1-38 (C1) (1977) 1–15. http://hal.ar chives-ouvertes.fr/docs/00/21/69/60/PDF/ajp-jphyscol197738C101.pdf.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
The complex initial reluctivity, permeability and susceptibility spectra of magnetic materials N.C. Hamilton North View, Chawston Lane, Chawston, Bedford MK44 3BH, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 24 February 2014 Received in revised form 13 September 2014 Accepted 17 October 2014 Available online 22 October 2014
The HF complex permeability spectrum of a magnetic material is deduced from the measured impedance spectrum, which is then normalized to a series permeability spectrum. However, this series permeability spectrum has previously been shown to correspond to a parallel magnetic circuit, which is not appropriate. Some of the implications of this truth are examined. This electric/magnetic duality has frustrated efforts to interpret the shape of the complex magnetic permeability spectra of materials, and has hindered the application of impedance spectroscopy to magnetic materials. In the presence of magnetic loss, the relationship between the relative magnetic permeability and the magnetic susceptibility is called into question. The use of reluctivity spectra for expressing magnetic material properties is advocated. The relative loss factor, tanδm/μi is shown to be an approximation for the imaginary part of the reluctivity. A single relaxation model for the initial reluctivity spectra of magnetic materials is presented, and its principles are applied to measurements of a high permeability ferrite. The results are presented as contour plots of the spectra as a function of temperature. & 2014 Elsevier B.V. All rights reserved.
Keywords: Complex permeability spectrum Reluctivity Magnetic susceptibility Soft ferrite
1. Introduction 1947 was very significant in the history of ferrite and of magnetism in general. Snoek′s 1947 book [1] described the preparation of MnZn and NiZn ferrites; these are still the chemical basis of most of the soft ferrites in use today. Snoek′s 1947 letter [2] proposed that the complex series susceptibility spectrum of ferrite was due to gyro-magnetic resonance. Also in 1947, Macfadyen [3] gave a detailed description of the use of permeability and reluctivity as complex quantities. In the earlier part of the century, it had become firm practice to use permeability rather than reluctivity, and as long as these were viewed as scalar quantities, this caused no difficulties. Permeability was probably chosen because the values were of a convenient size when using the CGS system of units, and because it seems more natural for a magnetic field strength to cause a flux density. But by 1950, the use of the complex series permeability spectra μr,s(f)¼ μ′(f)-jμ″(f) in SI units for small-signal material characterization was becoming standard practice. It remains so today, in spite of successive observations [4–7] that the complex permeability spectra of ferrites at HF are simpler when expressed in parallel terms, rather than in series terms. Snelling [8] and Wijn [9] state that it was Snoek and Six who established an alternative measure for characterizing materials: E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmmm.2014.10.061 0304-8853/& 2014 Elsevier B.V. All rights reserved.
the relative loss factor tanδm/μi (or its reciprocal, μiQm, probably due to Owens [10]), where μi is the initial permeability, a scalar value measured when both frequency and flux density are very small. The relative loss factor is widely used by manufacturers of soft ferrites, especially in Northern America. Since 1970s, work has progressed from material characterization to the interpretation and simulation of the permeability spectra. In the author's view, the exclusive use of complex series permeability is hindering progress on the subject. This article reviews the basis of permeability spectra with the intention of encouraging the use of complex reluctivity or parallel permeability.
2. Theory 2.1. Reluctivity and permeability Consider the magnetic circuit in a ring core. The components of energy storage and of energy loss in the core are homogenously distributed throughout the material, so all the flux change in the core is subject to both energy storage and dissipation. If the core's distributed energy storage were to be lumped in one sector of the core, and the magnetic losses in the other sector, then the lumped element small-signal magnetic circuit must consist of these two material types in series. Macfadyen [3] observed "It is generally best to work with permeability when there are magnetic paths in parallel and with
N.C. Hamilton / Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
497
the loss term of the series magnetic circuit, but they conceal this, the origin of their usefulness.
2.2. Susceptibility and permeability The magneto-static relationship between these quantities as dimensionless scalars is, by definition, μr ¼ χ þ 1, where χ is the (rationalized magnetic volume) susceptibility. Assume the simplest susceptibility spectrum, a single relaxation given by a Debye type equation [14] for the complex magnetic susceptibility χ(ω):
χ (ω) =
Fig. 1. Reciprocals (x 1) relate the various forms of complex relative permeability μr and reluctivity rr and their commonly used approximations. The series components of reluctivity sr,s are shown for completeness, but are not used.
reluctivity when the magnetic paths are in series…". Macfadyen then showed that the complex reluctivity gives a parallel equivalent electrical circuit. This view was amplified by Cherry [11], who showed that the electric and magnetic circuits were duals of each other. Cherry then showed explicit equivalent circuits of a lossy magnetic material as a series magnetic circuit of real and imaginary parts of reluctance, modelled by an electrical circuit consisting of an admittance: an inductor and a resistor connected in parallel. Fig. 1 shows the relationships between the series and parallel forms of both permeability and reluctance, where the sub-scripts denoting series and parallel forms are based on IEC 62044-2 2005. Although characterization of a material's series magnetic energy storage and magnetic loss components is best expressed in terms of complex reluctance, Fig. 1 shows that the same data may be expressed as a so-called parallel permeability, the components of which in fact represent a series magnetic circuit. It is regrettable that common usage differentiates between the series and parallel forms of these magnetic quantities in terms of the series and parallel dual electric circuit. As Schlicke observed [12] of basic discontinuous material structures: "What seems magnetically a series arrangement is electrically a parallel circuit and vice versa." and that the equivalent electric circuit is "… contrary to what the eye seems to see so clearly in the magnetic structure". From Fig. 1, it can be deduced that the ratios between the real and imaginary parts of the permeability and reluctivity are constant, so:
″ μr,p ′ μr,p
=
″ μ′ −σr,s σr′ 1 = r = = = Qm ′ σr″ σr,s − μr″ tanδm
χ0 1 + jωτ
(2)
where χ0 is the static susceptibility, ω is the angular frequency, τ is the time constant of relaxation and j is the complex operator. There are two possible methods of proceeding [15]. 2.2.1. Series susceptibility conversion Add 1 to the series real component of χ(ω), thereby making the assumption that μr(ω) ¼ χ(ω) þ1. Resolve χ(ω) into real and imaginary parts:
χ ′(ω) =
χ0 1 + (ωτ)2
and χ ″(ω) =
−χ0 ωτ 1 + (ωτ)2
(3a) and (3b)
and add 1 to the real part, giving:
μr′(ω) = 1 +
χ0 1 + (ωτ)2
(4a)
and
μr″(ω) =
−χ0 ωτ 1 + (ωτ)2
(4b)
Unfortunately, Eqs. (4a) and (4b) do not easily convert to an expression for complex reluctivity. Fig. 2 shows a graph of the ‘series’ permeability spectra resulting from values of χ0 and τ appropriate to a ferrite for use in HF/VHF inductors. At low frequencies, Qm falls as the frequency increases: this is expected. However, the frequency axis of Fig. 2 is extended to show that, above 800 MHz, Qm starts to rise again. This is not observed in practice, and is an artefact of the series conversion of susceptibility to permeability, which is unlikely to be correct. 2.2.2. Parallel susceptibility conversion Add 1 to the parallel real component of χ(ω). Using the relations given in Fig. 1, convert series susceptibility to its reciprocal. (Magnetic immunity has been suggested [16] as a name for
100 (1)
where Qm is the material's magnetic quality factor, and tanδm is the loss factor. Now, as Macfadyen observed [3]: "Reluctivity possesses the advantage that (σ′r and hence μ′r , p ) is constant over a wide range of frequency…". This is confirmed by examination of the series and parallel permeability spectra of the 22 ferrite grades presented by Snelling [13]. See also the measurements presented in Fig. 5. It follows that the initial permeability μi ≈ μr′, p , and this is shown in Fig. 1. From this, and the relationships given in Eq. (1), the approximations for the relative loss factor tanδm/μi and its reciprocal μiQm follow. These two factors are approximations for
10 1 0.1 1M
10M
100M 1G frequency (Hz)
10G
100G
Fig. 2. Series relative permeability spectrum real (solid) and imaginary (dashed) parts modelled for χ0 ¼ 79 and τ¼ 2 ns, if μr(ω) ¼ χ(ω) þ 1. Also shown (dot dashed) is Qm, which is unrealistic above 800 MHz.
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In the 1960s, Olsen wrote a book [17] which gave a careful and plainly stated re-appraisal of the subject. A similar re-appraisal is needed now.
100 10 1
2.3. Snoek's factor
0.1 0.01 0.001 1M
10M
100M 1G frequency (Hz)
10G
100G
Fig. 3. Series relative permeability spectrum real (solid) and imaginary (dashed) parts modelled for χ0 ¼79 and τ ¼2 ns, by adding 1 to the parallel real component of χ(ω). In contrast to Fig. 2, Qm (dot dashed) remains realistic throughout the frequency range.
reciprocal susceptibility):
1 1 + jωτ = χ (ω) χ0
(5)
separating real and imaginary parts and taking their reciprocals gives [16]:
χp′ = χ0 and χp″ (ω) =
(Hz)
(9)
For a single relaxation, S is independent of frequency. Where the material's complex reluctivity spectrum is more complex, S is a weak function of frequency. S(fr)¼ χ0fr is widely known as Snoek's product [18]. But, as S is a function of frequency, Snoek's factor may be a more appropriate name. Substituting Eq. (9) into Eq. (8), and converting from angular frequency ω (rad/s) to frequency f (Hz) gives
1 f +j χ0 + 1 S
(10)
2.4. Hysteresis constant
χ0 ωτ
(7a) and (7b)
From which the complex reluctivity can be assembled:
ωτ 1 +j χ0 + 1 χ0
χ0 2πτ
(6a) and (6b)
ωτ
Add 1 to the parallel real component of susceptibility to give the parallel permeabilities:
σr(ω) =
S=
σ (f ) =
χ0
μ p′ = χ0 + 1 and μ p″(ω) =
In Section 2.2, the time constant of relaxation τ has been used as the variable that characterizes the loss element of the permeability frequency spectrum. This is the approach taken by Debye, and has been taken ever since. However, 1/(2πτ) defines the relaxation frequency (fr) at which −μr″ peaks; this does not characterize a core material. For example, if the core is provided with an air-gap, then τ will change, even though the material's performance does not change. Instead, define a new variable S:
(8)
There is no compact expression for Eq. (8) in terms of 'series' permeabilities, but Fig. 3 shows a graph of the permeability spectrum of this version, using the same parameters as Fig. 2. 2.2.3. Summary-susceptibility and permeability The conversion between susceptibility and permeability causes a problem, the origins of which lie in the series-parallel duality of magnetic and electric circuits. A single relaxation gives a parallel inductor/resistor (LR) model of susceptibility. Converting susceptibility to permeability requires an increase of the model's inductive component. Section 2.2.1 shows the series approach and Section 2.2.2 the parallel approach. Both approaches preserve the χ ¼ μr-1 scalar relationship. The series conversion adds a series inductance to the parallel LR circuit (Eq. (4a)). This is often used, but has an unrealistic high frequency response. The parallel approach gives a realistic high frequency response, because it takes the parallel LR model of susceptibility, and increases the value of L (Eq. (7a)). So, a material's complex relative permeability is found by adding 1 to the parallel real component of magnetic susceptibility. Furthermore, it is convenient to do this, because the parallel approach results in a simple expression for complex reluctivity, and Section 2.1 has argued that the reluctivity is the measure to use. This conclusion applies to the deduction of complex permeability and susceptibility from impedance data. It is not the end of the matter. For example, it may be that these considerations do not apply to free space permeability measurement techniques.
Eq. (10) has a shortcoming: it predicts that, as the frequency approaches zero, the material's quality factor Qm approaches infinity. In practice, as the frequency approaches zero, Qm approaches a limiting value Qmax set by the small-signal hysteresis loss. A typical value for a ferrite core with no air gap might be 200. Introducing an extra term for this gives
σ (f ) =
⎛f ⎞ 1 + j⎜ + σi″⎟ ⎝ ⎠ χ0 + 1 S
(11)
100k 10k 1k 100 10 1 100m 10m 1m 100n 10n 10k
100k
1M 10M 100M frequency (Hz)
1G
10G
Fig. 4. The modelled permeability, reluctivity ‘parallel’ permeability and Qm for the data in Fig. 3, (single relaxation χ0 ¼ 79 S¼ 6.3 GHz) but with Qmax ¼ 200. Note the change in frequency scale from Fig. 3.
N.C. Hamilton / Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
1M
2.5. Ferromagnetic resonance
measurement bandwidth (Hz) 10 30 100 300
Fig. 4 shows the same data as Fig. 3, but using Eqs. (11) and (12) with Qmax ¼200, and showing the curves of the alternative formats for the complex permeability/reluctivity data. The peak in the imaginary part of the 'series' permeability spectrum has no corresponding peak in the reluctivity or ‘parallel' permeability spectra, as the peak is an artefact caused by analysing a series magnetic circuit in parallel terms. It is not due to the resonance effect suggested by Snoek [2]; others have given this peak the name ‘ferromagnetic resonance'. It does not exist. Instead, the aim should be to examine the deviations of the spectra of real ferrites from the single relaxation model of Fig. 4. A ′ (f ) (as permeability has convenient way of doing this is to graph μr,p been used for 100 years, it is hard to avoid) and S(f). For a single relaxation, these will be constants. The cause of the deviations must then be sought.
100k 10k 1k 100 10 1 10k
100k
1M 10M frequency (Hz)
100M
1G 3. Experimental technique
Fig. 5. Permeability and ‘parallel’ permeability spectra deduced from impedance measurements of 1 turn on 2.5 mm diameter T38 (MnZn ferrite) ring core at 20 °C.
where σ″ i is the imaginary part of the initial reluctivity of the material. Its value may be found from
σi″ =
499
tanδi 1 = (χ0 + 1)Q max μi
(12)
where tanδi is the initial loss factor. Note that, in the electrical model, the extra term σ″ i results in a frequency dependent resistor; it is difficult to model this circuit element in a conventional SPICE type simulator. It can only be approximated by a ladder network, used over a limited frequency range.
The aim here is to measure a ferrite's HF impedance spectrum over a range of temperatures, and apply the theory to the pre′ (f ) and S(f) were simultaneously sentation of the results. So that μr,p available over the broadest frequency range, a high permeability MnZn ferrite (T38, μi ¼ 10,000 nominal) was selected. This material had the additional advantage that impedance measurements at the ferrite's Curie temperature needed no special heat resisting equipment. The upper frequency limit of the measurement was set by the sample's dimensional resonance [19]; this undesirable effect was minimised by measuring a small sample, a ring core [20] with an outer diameter of 2.5 mm, held in a single turn coaxial mounting. However, these test conditions were not optimum for the measurement of either Qmax, or of the small permeability remaining above the Curie temperature due to paramagnetism. To improve the characterisation of Qmax, high sample impedance
f t
Fig. 6. Impedance spectra measurements of a 2.5 mm diameter ring core of T38 ferrite at various temperatures. Data presented here as contours of μ′r , p (f , t) , the real part of parallel permeability, which gives rise to the parallel reactance. Permeability contours are in 1000 s; the nominal μi of the ferrite is 10,000. At each measurement temperature, a small circle marks the 'relaxation frequency' fr, the measurement frequency corresponding to the greatest value of the series imaginary part of permeability.
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S ft
Fig. 7. Impedance spectra measurements of a 2.5 mm diameter ring core of T38 ferrite at various temperatures. Data presented here as contours of Snoek's factor, S (f , t) = μ″r , p (f , t). f , which gives rise to the parallel resistance. Snoek's factor contours are in GHz. As in Fig. 6, small circles mark the ‘relaxation frequency’.
would be required at low frequencies, implying measurement of a large core with multiple turns. Above the Curie temperature, a large core with a single coaxial turn would be required. As shown in Fig. 5, measurement accuracy was improved by using a narrow IF bandwidth at LF, increased at higher RF, thereby achieving a reasonable sweep time of 8.5 s by the HP 4195A network analyzer, which measured at 83 logarithmically spaced frequencies between 40 kHz and 500 MHz. The HP 87512A resistive signal divider gave fair accuracy for the required low impedance measurements. The available power at the ring core was 23 dBm, which was found to be just below the onset of reduction of Qmax at 20 °C. The ring core was mounted in an SMA connector that was fitted with a bead thermistor for temperature measurement. The SMA connection was OSL calibrated and short circuit compensated. The SMA core mount was immersed in a bath of alcohol at 60 °C or oil heated to 145 °C, and the temperature of the core was allowed to relax back to ambient, while taking complex impedance measurements every 5 °C; this required a short analyzer sweep time to avoid temperature drift during measurement.
resonance frequency rises (Fig. 6). It can also be seen in Fig. 7 that the frequency of minimum Snoek's factor is a function of temperature, but is always close to fr, so these are probably related in some way. The small discontinuity in Fig. 7 between 20 °C and 25 °C is due to the approach to 20 °C from cold, and to 25 °C from hot, and is probably due to the thermal hysteresis [21] of the ferrite's magnetic properties. Fig. 7 shows that Snoek's factor is small near the Curie temperature. In consequence, the 40 kHz lower limit of the measurement frequency range was not quite low enough to register values of initial permeability at temperatures above 120 °C. The data of Figs. 6 and 7 could have been plotted in the more usual terms of series permeabilities. Such graphs would emphasize the dominant ‘relaxation frequency’. By plotting Figs. 6 and 7 as shown, a single relaxation has been eliminated, allowing other, more subtle, effects to be shown.
5. Conclusions 4. Experimental results Fig. 5 shows the permeability spectra at 20 °C deduced from the analyzer's impedance measurements, and using the manufacturer's published value of core factor Σl/A¼ 12.3 mm 1. Fig. 5 shows dimensional resonance above 200 MHz, and Snoek's factor varied between 4.5 and 9 GHz, with a shallow minimum near 0.4 MHz. This occurred near the dominant ‘relaxation frequency’ fr, this is taken to be the measurement frequency corresponding to the maximum value of −μ″r (f). To conveniently display the information in Fig. 5 at 39 different temperatures, the data are given ′ (f , t) and S(f,t) in Figs. 6 and 7. Small circles as contour plots of μr,p indicate fr at each temperature. Figs. 6 and 7 show that the ferrimagnetic properties of core were absent at 135 °C, but had returned by 130 °C. As the ferrite's temperature reduces below the Curie temperature, its high frequency performance improves (Fig. 7), and the core's dimensional
The following conclusions apply to coupled electric and magnetic circuits, as in an inductor with a magnetic core. The conclusions do not necessarily apply to other applications. 1. The characterization of a material's series magnetic energy storage and magnetic loss components is best expressed in terms of a complex reluctance. 2. Because of long usage, it may be preferable to express the complex reluctance in terms of parallel permeability. 3. Parallel permeability is an inappropriate name, as it characterizes a series magnetic circuit. Further clarification and formalization will be required to resolve the matter. 4. The commonly accepted relationship between permeability and susceptibility is suspect, and the conversion is best done in ‘parallel’ terms. 5. These considerations lead to a two element model for a simple relaxation.
N.C. Hamilton / Journal of Magnetism and Magnetic Materials 377 (2015) 496–501
6. The deviations from the simple two element model have been illustrated by measurements of a MnZn ferrite. These deviations, and others that will doubtless be found in other soft magnetic materials, remain to be fully characterized and explained. 7. A third element can be introduced to model the hysteresis loss, as shown in Eq. (11). 8. Impedance Spectroscopy has been widely used for the analysis of dielectric materials, but has been little used on magnetic materials. To progress with this technique, it is essential to acknowledge, and to make allowance for, the series-parallel circuit transformation that is caused by the duality of the magnetic and electric circuits.
References [1] J.L. Snoek New, Developments in Ferromagnetic Materials, 2nd Ed, Elsevier, 1949 (1947). [2] J.L. Snoek, Gyromagnetic resonance in ferrites, Nature 160 (1947) 90. http://dx. doi.org/10.1038/160090a0; Later expanded in J.L. Snoek, Dispersion and absorption in magnetic ferrites at frequencies above one Mc/s Physica 14 (4) (1948) 207–217. http://dx.doi.org/ 10.1016/0031-8914(48)90038-X. [3] K.A. Macfadyen, Vector permeability, J. IEEE Part III Radio Commun. Eng. .94 (.32) (1947) 407–414. http://dx.doi.org/10.1049/ji-3-2.1947.0079. [4] W. Six, Some applications of Ferroxcube, Philips Tech. Rev. 13 (11) (1952) 301–336. [5] P.M. Prache, B. Chiron, Complex susceptibility of high-resistivity ferrites, Proc. IEEE Part B 104 (5 Part S) (1957) 209–212. http://dx.doi.org/10.1049/pi-b1.1957.0030. [6] Y. Naito, Inverse Cole–Cole plot as applied to the ferrimagnetic spectrum in VHF through the UHF Region, in: Proceedings of the International Conference on Ferrites (ICF1-1970), University Park Press, 1971, pp. 558-560. [7] J.K. Watson, S. Amoni, Using parallel complex permeability for ferrite characterisation, IEEE Trans Magn. 25 (5) (1989) 4224–4226. http://dx.doi.org/ 10.1109/20.42576.
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[8] E.C. Snelling, Soft ferrites, 2nd Ed, Butterworths, London (1988) p1. [9] H.P.J. Wijn, Some remarks on the history of ferrite research in Europe, in: Proceedings of the International Conference on Ferrites (ICF1-1970), University Park Press, 1971, pp. xix–xxi. [10] C.D. Owens, Analysis of Measurements on Magnetic Ferrites, in: Proceedings of the IRE, March 1953, pp. 359–365. http://dx.doi.org/10.1109/JRPROC.1953. 274382. [11] E.C. Cherry, The duality between interlinked electric and magnetic circuits and the formation of transformer equivalent circuits, Proc. Phys. Soc. B 62 (2) (1949) 101–111 http://dx.doi.org/10.1088/0370-1301/62/2/303. [12] H.M. Schlicke, Dielectromagnetic Engineering, John Wiley, New York, 1961 (pp. 8, 9 and 14). [13] E.C. Snelling, Soft Ferrites, 2nd Ed, Butterworths, London (1988) 92–98. [14] J. Smit, H.P.J. Wijn, Ferrites, John Wiley, New York (1959) p290. [15] N.C. Hamilton, Electrical circuit models of the HF initial permeability spectra of soft ferrite lead to revised definition of fundamental material properties, in: Proceedings of the IET 1st Annual Active and Passive RF Devices Seminar, Glasgow, 29 Oct 2013, pp. 59–62. http://dx.doi.org/10.1049/ic.2013.0239. [16] N.C. Hamilton, The small-signal frequency response of ferrites, High Freq. Electron. Mag. 10 (6) (2011) 36–52 http://www.highfrequencyelectronics.com/ index.php?option=com_flippingbook&view=book&id=79:2011-06-june&catid= 8:2011-high-frequency-electronics&Itemid=149. [17] E. Olsen, Applied Magnetism: A Study in Quantities, Philips Technical Library, Eindhoven, 1966. [18] T. Nakamura, Snoek′s limit in high-frequency permeability of polycrystalline Ni–Zn, Mg–Zn, and Ni–Zn-Cu spinel ferrites, J. Appl. Phys. 88 (1) (2000) 348–353 http://dx.doi.org/10.1063/1.373666. [19] F.G. Brockman, P.H. Dowling, W.G. Steneck, Dimensional effects resulting from a high dielectric constant found in a ferromagnetic ferrite, Phys. Rev. .77 (.1) (1950) 85–93 http://dx.doi.org/10.1103/PhysRev.77.85. [20] B64290J35X38 core manufactured by Siemens AG. Now replaced by part B64290P0035X038 from EPCOS AG. [21] A. Globus, Some physical considerations about the domain wall size theory of magnetization mechanisms, J. de Phys. C1-38 (C1) (1977) 1–15. http://hal.ar chives-ouvertes.fr/docs/00/21/69/60/PDF/ajp-jphyscol197738C101.pdf.