Influence of microstructure and magnetizing mechanisms on magnetic complex permeability (imaginary part) of a Cu-doped Ni–Zn polycrystalline ferrite

Frequency dispersion model of the complex permeability of soft ferrites in the microwave frequency range

“…(ii) 𝑏 ′′ values, which were previously obtained by the authors for the four angular frequencies studied (10 6 , 10 7 , 10 8 , and 10 9 Hz), 11 have now been empirically…”

“…This experimental value can be estimated by plotting relative density versus magnetic permeability and extrapolating magnetic permeability to zero. 2,11 For the studied polycrystalline ferrite, a 𝜙 lim value of 0.65 has been reported. 11 Authors have satisfactorily tested Equation ( 9) at four specific angular frequencies (10 6 , 10 7 , 10 8 , and 10 9 Hz), determining for each of the angular frequencies studied the experimental values of the parameters 𝜇 ′′ 𝑆 , 𝜇 ′′ DW , 𝑏 ′′ , and 𝑑 𝑊 .…”

“…Particularly, average grain size and relative density play an important role in the magnetization process of polycrystalline ferrites: complex magnetic permeability increases with the average grain size and relative density. 2,3,8,9,11,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]39 However, very few models relating magnetic permeability to microstructure have been developed in the past, and all of them have been uniquely tested at a specific constant angular frequency. In this regard, it is worth mentioning the models proposed by Globus, 36 Rikukawa, 38 Johnson and Visser, 25 and Pankert.…”

“…40 Authors have tested these models in polycrystalline ferrites and proven that they are not able to predict the influence of average grain size and porosity on magnetic permeability. Hence, models were modified, and the following relationship, derived from Pankert model, was validated with a good accuracy with the experimental imaginary part of complex permeability 2,11 :…”

“…In Equations ( 9) and (10), 𝜇 ′′ 𝑆 is the imaginary partcomplex magnetic permeability spin rotation contribution, 𝜇 ′′ 𝐷𝑊 is the imaginary part-complex magnetic permeability domain-wall motion contribution, 𝑑 𝑊 is the domainwall width, 𝐺 is the average grain size, and 𝜓 is the densification defined by Equation (11). Densification 𝜓 is an effective concept when comparing systems of different initial porosities, 41 and 𝜙 lim stands for the relative density below which the value of the magnetic permeability is virtually zero.…”