Estimate inductance of core with roof shaped gap

[Miyuki]

Hi folks,
is there any simple way of determining inductance roll off for a core with roof shaped gap (as in the picture)?
Or do I need to make a complete simulation or an entire physical measurement?

[T3sl4co1l]

This might be helpful,
https://www.pes-publications.ee.ethz.ch/uploads/tx_ethpublications/10_A_Novel_Approach_ECCEAsia2011_01.pdf
but you'll need a different transform for the geometry shown, if it really is like that.

Generally speaking, effective area goes up with gap.  You can imagine adding the gap to the diameter of the center peg, and using the corresponding area.  This implies flux spreads out in the area of the gap, which is true, but the total effect is more complex.  For a single winding inductor filling the winding area, some of the winding intercepts the reduced magnetization within the fringe (that is, it's no longer looped around the full N*I seen by the core, but a fraction of it).  Thus two things happen: 1. those turns are effectively not as closely coupled to the rest of the winding, and 2. the flux density seems to go down, or Ae goes up.

Note that I must be careful describing this, because the total field seen by a turn of wire near the gap is incredibly dense and complex: it is subject to the full skin, proximity and fringing effects of the region.  There is no more strenuous location in the coil.  The actual magnitude of field strength might not even be too crazy, but it changes rapidly with position i.e. is very inhomogeneous and thus induces eddy currents in even very fine strands (which is how proximity and fringing effects manifest).  I refer to N*I above, to invoke the equivalent magnetic circuit; effectively some turns are on a parallel path, partly decoupled from the rest; this isn't to say the field strength at the turns/strands is necessarily more or less than it is elsewhere.

So, as a very basic / rough approximation, you could say Ae goes up by the apparent ratio of areas, by adding gap to the radius of the center peg; this gap dimension will be further modified by the geometry of the non-planar gap; and further still, it'll all depend on turn position within the window, which is to say A_L apparently varies by location, or the turns have some coupling factor k slightly less than 1 between them, or L = A_L N^(2-x) for some small x > 0, or etc.

If you need to design more accurately than say 20% of actual saturation, you'll need a simulation or measurement.

Tim

[Conrad Hoffman]

Maybe try student edition of Quickfield- https://quickfield.com/