The fringing flux is an effect that happens around air gaps when the magnetic cores are excited. The magnetic flux lines swell because the magnetic lines repel each other when passing through a nonmagnetic material, causing a decrease in the flux density.
Many models have been developed along the years. In this app note six models are studied and compared against measurements, plotting the values and their relative error.
Effective Air Gap Cross-Sectional Area
Effective Length of Air Gap
This method ignores the existence of fringing flux, and simply calculates the inductance based on the series connection of core and air gap reluctances.
This method assumes the cross-sectional area is increased by a length of the gap on each side in the core leg, calculating a factor (>1) that raises the classical inductance.
This method assumes than the effective length of the magnetic flux in the gap is increased by the expansion in the cross-sectional area, adjusting the factor in the previous model.
As specified in Marian Kazimierczuk's book "High-Frequency Magnetic Components", this model modifies the factors of previous models with a cross-sectional - length ratio calculated empirically.
As specified in Colonel Wn. McLyman's book "Transformer andinductor designhandbook", this model calculates an inductance factor (>1) dependant on the gap, cross-sectional area, and winding height.
Model presented by J. Mühlethaler based on Schwarz-Christoffel Transformation in his paper of 2011, where air gap reluctances are calculated for basic geometries and a generic 3D model is built upon them.
Values for different turns and gaps
Errors for different turns and gaps
Since fringing flux is unavoidable from the moment there is a gap, calculating and understanding its effects is necessary in order to design high-quality magnetics. As can be seen in the charts, the models behave generally good when the gap is small and there is one stack, especially Mühlethaler's model, though its complexity might discourage designers, while McLyman's can be a good compromise between complexity and accuracy. For larger gaps, or more than one stack, the models start to lose accuracy and an intelligence (human or artificial) is needed to correct the models based on its experience from the real world.