# Calculation Methodologies of Complex Permeability for Various Magnetic Materials

^{1}

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## Abstract

**:**

_{e}, current I

_{e}, and phase difference θ

_{e}, which can be obtained simply by an oscilloscope and a function generator, the real and imaginary permeability can be calculated w.r.t. operating frequency by the suggested calculation methodologies. Such information for the real and imaginary permeability is important to determine the size of the magnetic components and to analyze the core loss. To identify the superiority of the high-frequency magnetic materials, three prototypes for a ferrite core, amorphous core, and nanocrystalline core have been built and verified by experiment. As a result, the ferrite core is superior to the other cores for core loss, and the nanocrystalline core is recommended for compact transformer applications. The proposed calculation for the complex (i.e., real and imaginary) permeability, which has not been revealed in the datasheets, provides a way to easily determine the parameters useful for industrial electronics engineers.

## 1. Introduction

_{sat}(≈1.5 T) and can reduce the eddy loss due to ultra-thin-laminated Fe-based plates [1,4]. In order to miniaturize the magnetic materials, Nanocrystalline alloy, which is usually composed of Fe, Si, B with the other chemical combinations, can be utilized over 1 kHz–100 kHz of operating frequency applications [5]. Such magnetic materials mentioned above have their own merits and demerits for various applications. Thus, by referring to inherent magnetic characteristics of the magnetic materials, the most appropriate magnetic materials should be selected, according to the operating frequency of its applications.

^{3}], etc., should be analyzed. As one of the important factors, the permeability is the baseline design parameter to determine the size of the magnetic components and analyze the core loss [1,2,3,4,5,6,7]. Although the major characteristics of the selected magnetic cores were already described in datasheets, the complex permeability characteristics w.r.t. the operating frequency need to be further investigated, if the permeability is not revealed in detail. Especially, imaginary permeability, which implies the core loss, is not usually described in datasheets; hence, investigating the imaginary permeability becomes important for estimating the core loss and managing the heat dissipation of the magnetic core. As the typical methods to investigate the complex permeability, high resolution based LCR meters for kHz ranges [7], microwave generators for MHz-GHz [8], vector network analyzers (VNA) for GHz ranges [9,10,11], and millimeter-wave spectrometers for GHz ranges [12] could be utilized to derive the complex permeability as well as inductance and resistance. Although several special measurement methods to investigate the complex permeability have been presented for various purposes, e.g., ultra-wideband permeability measurements [8], transmission bridge method [9], and using rod samples inside a rectangular waveguide [10], they are limited to be used for millimeter-wave frequency ranges. Because such measurement devices cannot provide appropriate AC test current levels [7,8,9,10,11,12,13], the measurement results of the complex permeability may not be accurate for the specified frequency ranges; hence, they are not suitable for the general magnetic components applied in power applications due to extremely low AC test currents in magnetic components. Furthermore, the previous literatures mentioned above require complicated experimental environments, which make the power engineers difficult to investigate the major characteristics of the magnetic materials. For this reason, an appropriate AC power supply based complex permeability measurement is required so that appropriate AC test current is applied to the magnetic components [14].

_{r}″ can be specified w.r.t. the operating frequency. To evaluate a comparative analysis of different magnetic materials, a ferrite core, an amorphous core, and a nanocrystalline core have been selected and verified by experiments. By virtue of the proposed calculation methodology for the complex permeability, the major characteristics of the magnetic materials for the design of the magnetic components can be found for a wide operating frequency range.

## 2. Proposed Calculation Methodology of the Complex Permeability

_{s}. However, the imaginary permeability, which is related to the core loss resistance, is not described below f

_{s}= 250 kHz from the PM11 datasheet [15]. For this reason, in order to exactly find the value of imaginary permeability of PM11 for f

_{s}< 250 kHz, a simple inductor model with a ferrite core has been built, as shown in Figure 1a. A simplified equivalent circuit of Figure 1a is shown in Figure 1b, where the voltage source is connected to the inductor model in series to evaluate the characteristics of the inductor model. For simplicity of analysis, all the magnetic field is assumed to be in the magnetic core in this paper. In Figure 1b, the complex impedance of the inductor model

**Z**can be defined as follows:

_{e}_{s}is the operating frequency, and ω

_{s}is the angular switching frequency. Bold characters mean a complex value.

**Z**and

_{e}**L**are the complex values for the impedance and inductance, and r

_{e}_{cp}is the copper loss-term resistance including the skin effect and eddy effect determined by frequency. From (1b), the complex inductance

**L**includes core loss resistance r

_{e}_{co}because r

_{co}depends on the magnetic characteristics [6].

**L**can be expressed by the complex permeability as follows:

_{e}_{e}and l

_{e}are effective magnetic area and effective magnetic path length, respectively.

_{e}and the imaginary permeability ${\mu}_{r}^{\u2033}$ is proportional to the core loss resistance r

_{co}.

**V**is used, as shown in Figure 1b. Depending on the electrical and magnetic characteristics of the inductor model, test current

_{e}**I**and phase difference between the voltage and current θ

_{e}_{e}are different as follows:

_{e}and θ

_{e}can be measured by an oscilloscope, and r

_{cp}can be found by measuring only the copper wire resistance without the ferrite core.

_{e}, I

_{e}and θ

_{e}, the values of L

_{e}and r

_{cp}can be obtained as follows:

## 3. Experimental Verification

_{e}, l

_{e}, and N

_{e}in Figure 1 were arbitrarily selected based on a piece of magnetic core, commercially available in markets: A

_{e}is magnetic flux area, l

_{e}is the length of effective magnetic path, and N

_{e}is the number of winding turns. The physical parameters are summarized in Table 1. A PM11 magnetic material from TODAISU has been selected for Mn-Zn-type ferrite core [15]. The AMLB-8320 amorphous core from AMOGREENTECH, which is formed by layers of laminated 40-µm-thin amorphous-metal-film, has been selected for the amorphous core, as shown in Figure 2b. A VITROPERM 500F core from VACUUMSCHMELZE has been selected for the Nanocrystalline core, as shown in Figure 2c [16]. Considering the applicable operating frequency of kHz ~ several MHz for three major magnetic materials, e.g., ferrite core, Amorphous core, and Nanocrystalline core in this paper, 1 kHz to 1 MHz frequency range has been specified.

**V**,

_{e}**I**, and θ

_{e}_{e}in (4) by a function generator (KEYSIGHT 33522A) and an oscilloscope (Lecroy 104Xi-A oscilloscope), the inductance and resistance in (5) w.r.t. operating frequency can be calculated, as shown in Figure 3 and Figure 4, where LCR measurement results have been obtained by a KEYSIGHT network analyzer E5071C. In case of LCR measurement used in this paper, uncertainty below a 10 MHz frequency range is highly less than 0.1(dB) [17,18]. A little discrepancy between calculation and LCR measurement, i.e., lower than 8%, comes from leakage magnetic flux, which is leaking out to the air, not circulating through the test prototypes of the magnetic materials in Figure 2. Because the proposed idea to calculate the inductance and resistance in (1)–(5) is assumed to be ideal inductor having no leakage magnetic flux, this inherent leakage magnetic component is not reflected to the calculated results of Figure 3 and Figure 4. Unfortunately, this leakage component cannot be analytically modeled due to the unpredictable radiation of the magnetic field and the non-linear property of the ferrite core [3]. This leakage magnetic flux could be minimized if a manufactured piece of the magnetic core in Figure 2 is small enough so that the generated magnetic flux from the test currents is all passing through the magnetic cores, not leaking to the air. Because the manufactured pieces of the magnetic cores in Figure 2 are the commercialized products in markets, and the size of magnetic pieces are already fixed, this small discrepancy is inevitable in this case. Nevertheless, the confirmation of the tendency for the permeability w.r.t. operating frequency is highly similar, and the relative comparison w.r.t. operating frequency can be obtained by the proposed idea. The real permeability of the ferrite core increases w.r.t. frequency, whereas that of the amorphous core and nanocrystalline core decreases. The imaginary permeability always increases for three core cases due to the increment of the core loss, which is usually proportional to the operating frequency.

_{s}> 250 kHz, which were already described in the datasheet [15], are matched well with the calculation and measurement results. According to the datasheet [15], unfortunately, the information for the imaginary permeability has not been revealed for the f

_{s}< 250 kHz range. From the matching tendency between datasheet and calculation results, it can be said that the values of real and imaginary permeability for f

_{s}< 250 kHz is reliable. Although the magnetic characteristic of the amorphous core is not revealed, the real and imaginary permeability can be found by the proposed methodology, as shown in Figure 5b. In case of the nanocrystalline core, the calculation results are also in good agreement with the datasheet that has described only the tendency of the permeability w.r.t. the operating frequency [16]; hence, the numerical values for the permeability can be specified. The test current level I

_{e}and the calculated magnetic flux density inside the ferrite core B

_{e}w.r.t. the operating frequency is shown in Figure 6. The magnetic field inside the magnetic cores B

_{e}can be calculated from (2), and the cases of amorphous core and nanocrystalline core are omitted due to the same tendency of Figure 6. It is noteworthy that the test currents and magnetic flux density can be increased until B

_{e}< B

_{sat}≈ 0.3 T by the other AC power supply devices, so that the exact complex permeability can be obtained under rated operating condition, although the function generator, which can provide a convenient frequency modulation function, has been used in this paper.

_{μ}is defined as μ″/μ′, which represents the magnetic core loss and can be usefully utilized for a finite-element-method (FEM) simulation analysis [3]. Magnetic loss tangent δ

_{μ}in Table 2 is essential to specify the core loss resistance modeling from the 3D FEM simulation analysis [2,3]. Based on the core loss resistance obtained by magnetic loss tangent δ

_{μ}, the static core loss resistance can be obtained to analyze the core loss of the magnetic components for power converters and wireless power transfer systems. According to the results of tan δ

_{μ}in Table 2, the ferrite core is superior to the amorphous core and nanocrystalline core for coil loss term. The complex permeability of the amorphous core is relatively lower than the others. The nanocrystalline core has very high real permeability, i.e., the core can be highly minimized. However, the imaginary permeability is also very high, i.e., the nanocrystalline core is of inferior core efficiency, although the magnetic component can be minimized by the nanocrystalline core. From the measured results in Table 2, the core loss of amorphous core may be almost 2.7 times larger than that of ferrite core at f

_{s}= 10 kHz under the same physical and operating conditions. If the operating frequency increases, the loss gap significantly increases. Therefore, in terms of the low core loss and high-frequency characteristics, the ferrite core is recommended. On the other hand, the nanocrystalline core is recommended for compact magnetic core. According to the magnetic results obtained by the proposed calculation methodology, the optimal magnetic core can be appropriately selected at the specified operating frequency condition.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

μ_{r}′ | Relative real permeability |

μ_{r}″ | Relative imaginary permeability |

V_{e} | AC test voltage |

I_{e} | AC test current |

θ_{e} | Phase difference between V_{e} and I_{e} |

Z_{e} | Input impedance |

L_{e} | Calculated inductance |

r_{co} | Calculated core loss resistance |

r_{cp} | Calculated copper loss resistance |

A_{e} | Magnetic flux area |

l_{e} | Magnetic flux path |

N_{e} | Winding turns |

## References

- Azuma, D.; Hasegawa, R. Audible Noise From Amorphous Metal and Silicon Steel-Based Transformer Core. IEEE Trans. Magn.
**2008**, 44, 4104–4106. [Google Scholar] [CrossRef] - Zhang, J.; Liu, J.; Yang, J.; Zhao, N.; Wang, Y.; Zheng, T.Q. A Modified DC Power Electronic Transformer Based on Series Connection of Full-Bridge Converters. IEEE Trans. Power Electron.
**2019**, 34, 2119–2133. [Google Scholar] [CrossRef] - Lee, E.S.; Choi, B.G.; Choi, J.S.; Nguyen, D.T.; Rim, C.T. Wide-Range Adaptive IPT Using Dipole-Coils with a Reflector by Variable Switched Capacitance. IEEE Trans. Power Electron.
**2016**, 32, 8054–8070. [Google Scholar] [CrossRef] - Kurita, N.; Nishimizu, A.; Kobayashi, C.; Tanaka, Y.; Yamagishi, A.; Ogi, M.; Takahashi, K.; Kuwabara, M. Magnetic Properties of Simultaneously Excited Amorphous and Silicon Steel Hybrid Cores for Higher Efficiency Distribution Transformers. IEEE Trans. Magn.
**2018**, 54, 1–4. [Google Scholar] [CrossRef] - Ohta, M.; Hasegawa, R. Soft Magnetic properties of magnetic cores assembled with a high Bs Fe-based Nanocrystalline alloy. IEEE Trans. Magn.
**2017**, 53, 1–5. [Google Scholar] [CrossRef] - Ayachit, A.; Kazimierczuk, M.K. Sensitivity of effective relative permeability for gapped magnetic cores with fringing effect. IET Circuits Devices Syst.
**2017**, 11, 209–215. [Google Scholar] [CrossRef] - Shin, K.H.; Kim, Y.; Lim, S.H. AC Permeability of Fe-Co-Ge/WC/Phenol magnetostrictive composites. IEEE Trans. Magn.
**2005**, 41, 2784–2786. [Google Scholar] [CrossRef] - Chen, Y.; Wang, X.; Chen, H.; Gao, Y.; Sun, N.X. Novel Ultra-Wide Band (10 MHz–26 GHz) Permeability Measurements for Magnetic Films. IEEE Trans. Magn.
**2018**, 54, 1–4. [Google Scholar] [CrossRef] - Dimri, M.; Tripathi, V.; Kashyap, S.; Jeong, Y.; Dube, D. Theoretical Modeling and Experimental Verification of the Permeability Measurements of Thick Films at Microwave Frequencies. IEEE Trans. Instrum. Meas.
**2009**, 58, 2911–2915. [Google Scholar] [CrossRef] - Roussy, G.; Chaabane, H.; Esteban, H. Permittivity and Permeability Measurement of Microwave Packaging Materials. IEEE Trans. Microw. Theory Tech.
**2004**, 52, 903–907. [Google Scholar] [CrossRef] - Afsar, M.N.; Korolev, K.A.; Namai, A.; Ohkoshi, S.I. Measurements of complex magnetic permeability of Nano-size ε -Alx Fe
_{2}−x O_{3}powder materials at microwave and millimeter wavelengths. IEEE Trans. Magn.**2012**, 48, 2769–2772. [Google Scholar] [CrossRef] - Yabukami, S.; Nozawa, K.; Tonthat, L.; Okita, K.; Sai, R. Impact of Complex Permeability Measurements Up to Millimeter-Wave Frequency Range. IEEE Trans. Magn.
**2021**, 57, 1–5. [Google Scholar] [CrossRef] - Yang, K.; Lu, J.; Ding, M.; Zhao, J.; Ma, D.; Li, Y.; Xing, B.; Han, B.; Fang, J. Improved Measurement of the Low-Frequency Complex Permeability of Ferrite Annulus for Low-Noise Magnetic Shielding. IEEE Access
**2019**, 7, 126059–126065. [Google Scholar] [CrossRef] - ANSI. Cores Made of Soft Magnetic Materials-Measuring Methods-Part 2: Magnetic Properties at Low Excitation Level; IEC 62044-2:2005; ANSI: Washington, DC, USA, 2021. [Google Scholar]
- Toda-isu, Ferrite Core Datasheet. Available online: http://forum.cxem.net/applications/core/interface/file/attachment.php?id=318271, (accessed on 9 August 2021).
- VACUUMSCHMELZE, Datasheet. Available online: https://www.vacuumschmelze.com/Assets-Web/en%20NanocrystallineVITROPERM-EMC-Products-2016_01.pdf (accessed on 9 August 2021).
- Szewczyk, R.; Nowicki, M.; Ostaszewska-Liżewska, A.; Bieńkowski, A.; Nowak, P.; Malinen, M. Accuracy of frame-shaped samples based measurements of magnetoelastic characteristics of soft magnetic materials. Measurement
**2020**, 162, 107899. [Google Scholar] [CrossRef] - E5071C ENA Vector Network Analyzer, Datasheet. Available online: https://www.keysight.com/us/en/assets/7018-01424/data-sheets/5989-5479.pdf (accessed on 9 August 2021).

**Figure 3.**Calculated and LCR measurement results of the inductance for three different core materials.

**Figure 4.**Calculated and LCR measurement results of the resistance for three different core materials.

**Figure 5.**Calculated and LCR measurement results of the inductance for three different core materials.

**Figure 6.**Calculated B

_{e}and measured I

_{e}w.r.t. the frequency for the test condition. (the ferrite core PM11 case).

Classification | A_{e} (mm^{2}) | l_{e} (mm) | N_{e} (Turns) |
---|---|---|---|

Ferrite core (PM11) | 576 | 600 | 10 |

Amorphous core (AMOGREENTECH) | 600 | 480 | 10 |

Nanocrystalline core (VITROPERM) | 250 | 280 | 10 |

Classification | f_{s} = 10 kHz | f_{s} = 100 kHz | f_{s} = 500 kHz | ||||||
---|---|---|---|---|---|---|---|---|---|

μ′ | μ″ | tan δ_{μ} | μ′ | μ″ | tan δ_{μ} | μ′ | μ″ | tan δ_{μ} | |

Ferrite core (PM11) | 2148 | 2.1 | 0.001 | 2186 | 6.5 | 0.0029 | 2301 | 61.5 | 0.0267 |

Amorphous core (AMOGREENTECH) | 601 | 5.8 | 0.009 | 602 | 83.3 | 0.1383 | 464 | 217.7 | 0.4692 |

Nanocrystalline core (VITROPERM) | 75,281 | 25,743 | 0.3419 | 28,743 | 22,018 | 0.7660 | 11,064 | 9063 | 0.8191 |

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Lee, E.S.; Choi, B.G.
Calculation Methodologies of Complex Permeability for Various Magnetic Materials. *Electronics* **2021**, *10*, 2167.
https://doi.org/10.3390/electronics10172167

**AMA Style**

Lee ES, Choi BG.
Calculation Methodologies of Complex Permeability for Various Magnetic Materials. *Electronics*. 2021; 10(17):2167.
https://doi.org/10.3390/electronics10172167

**Chicago/Turabian Style**

Lee, Eun S., and Byeong Guk Choi.
2021. "Calculation Methodologies of Complex Permeability for Various Magnetic Materials" *Electronics* 10, no. 17: 2167.
https://doi.org/10.3390/electronics10172167