Magnetizing Inductance

The inductance of a magnetic is a parameter that depends on the total reluctance of the magnetic core, and the number of turns wound around it. As the number of turns is usually a known value, this documentation will focus on the calculation of the reluctance of the magnetic core.

The reluctance of a magnetic circuit is a measure of how reluctant the magnetic flux lines are to circulate through a volume, equivalently to the resistance for electricity. The higher the reluctance of a volume compared to its surroundings; the smaller number of magnetic flux lines will go through that volume. 

The total reluctance of most magnetic components is calculated by the series connection of the reluctance of the magnetic core with the reluctance of the air gap(s). The reluctance for more complex magnetic components, like coupled inductors or mergences, can be obtained by applying circuit solving methods to the magnetic circuit, but they are out of the scope of this article. 

The reluctance of the magnetic core depends on its geometrical parameters, especially the length and the cross-sectional area, and on the relative permeability of the material; while the reluctance of an air gap depends on the perimeter and shape of the magnetic core that defines the gap, the gap length, and the distance to the closest perpendicular surface of the core (in most cases this is equal to half the height of the winding window). 

For both reluctance calculations it is vital to use the correct geometrical values of each shape. It is a common mistake to use the effective area of the magnetic core to calculate the reluctance of the air gap in a central column, when the geometric area should be used. This is especially important when modelling a gap in all the legs of the core, because, except the case of common Es and Us, the lateral legs perimeters and areas are completely unrelated with their counter values in the central column. The different lengths and areas of each magnetic shape are calculated according to EN 60205. 

As mentioned above, the permeability of the magnetic material es vital in order to calculate the reluctance of the ungapped core. This permeability has a dependence on the working temperature, switching frequency and DC bias of the magnetizing current. To obtain it we use a simple multidimensional interpolation from data measured with a Power Choke Tester DPG10 B. This interpolation is used iteratively in design loop, in order to ascertain that the correct permeability is used at each operation point, as the permeability can change the operation point itself.  

Continuing with the other reluctance of our magnetic circuit, in order to obtain the reluctance of the air gap, we use the model proposed by Zhang [1]. He proposes that the reluctance of the gap can be calculated by transforming the air-gap fringing magnetic field into a current source that produces the equivalent magnetic field, which can be easily solved and calculated. Alternatively, other air gap reluctance models [2, 3, 4] were implemented and evaluated for both cases, gap in the central column and gaps in all legs, but Zhang's model was found to have the lowest error when compared with our measured validation data.  

Finally, to obtain the final reluctance of our magnetic core, we add the reluctance of the core together with the reluctance of the air gap(s), for the case of the gap existing only in the central column. For the case of gaps in all legs, as is common in prototyping by using spacers, to the previous addition must be added the parallel calculation of the reluctance in the lateral legs, as the magnetic flux divides and runs in parallel for any lateral legs 

[1] X. Zhang, F. Xiao, R. Wang, X. Fan and H. Wang, "Improved Calculation Method for Inductance Value of the Air-Gap Inductor," 2020 IEEE 1st China International Youth Conference on Electrical Engineering (CIYCEE), 2020, pp. 1-6, doi: 10.1109/CIYCEE49808.2020.9332553.). The other evaluated models were: 

[2] J. Muhlethaler, J. W. Kolar and A. Ecklebe, "A novel approach for 3d air gap reluctance calculations," 8th International Conference on Power Electronics - ECCE Asia, 2011, pp. 446-452, doi: 10.1109/ICPE.2011.5944575. 

[3] E. Stenglein and M. Albach, "The reluctance of large air gaps in ferrite cores," 2016 18th European Conference on Power Electronics and Applications (EPE'16 ECCE Europe), 2016, pp. 1-8, doi: 10.1109/EPE.2016.7695271. 

[4] McLyman C. Transformer and inductor design handbook (Fourth ed.), CRC Press (2011)