In some cases it is necessary that core of inductor should have a gap, unlike with transformer core. I understand the reason with the voltage transformer core; there is nothing to worry about core saturation and we want to keep the winding inductance as high as possible.
The formula for the inductance is:
$$
L = N^2A_L = N^2\dfrac{1}{R} = \dfrac{N^2}{\dfrac{\ell_c}{\mu_cA_c} + \dfrac{\ell}{\mu_0A_c}} = \dfrac{N^2A_c}{\dfrac{\ell_c}{\mu_c} + \dfrac{\ell}{\mu_0}}
$$
And, the formula for the magnetic flux density:
$$
B = \dfrac{\mu N I}{\ell} = \dfrac{N I}{\dfrac{\ell}{\mu}} = \dfrac{N I}{\dfrac{\ell_c}{\mu_c} + \dfrac{\ell_g}{\mu_0}}
$$
Where,
\$N\$: Number of turns
\$R\$: Total core reluctance
\$A_L\$: The \$A_L\$ factor
\$I\$: Current through the wire
\$\mu_c\$: Permeability of the core
\$\ell_c\$: Mean magnetic path of the core
\$\ell_g\$: Length of the gap
\$A_c\$: Cross-section area of the core
\$L\$: Inductance
\$B\$: Magnetic flux density
What I understand from these two formulas is, the length of the gap affects both the magnetic flux density and inductance with the same proportion. When designing inductor, we would like to keep magnetic flux density low, so that the core wouldn't saturate and core loss stay low. People say that they leave the gap in order to keep the reluctance high, so that there are less flux flowing in the core, and the core stays away from the saturation region. However, doing so will reduce the inductance as well. By leaving the gap, we reduce magnetic flux density and inductance with the same coefficient. Then, instead of leaving the gap, we can also decrease the number of turns in the winding as well.
The only reason to leave gap that makes sense to is to increase the number of design parameters to obtain a closer resulting inductance value at the end. I can't find any other reason to leave gap.
What makes leaving the gap an inevitable action while designing an inductor?
Best Answer
And...
There is a major reason and it's clear from the formulas you quote: -
What saturates an inductor is too much current and too many turns for a given core geometry and core material. However, by adding a gap we might halve the permeability of the core and this means that we could double the amps (or double the turns) to obtain the same level of saturation we had before but, the inductance will have halved when we halved the permeability.
Fortunately, when we halve the core permeability, in order to restore the original value of inductance, we only need to increase the number of turns by \$\sqrt2\$ so, if we have halved the permeability with a gap, the potential for avoiding saturation has improved by \$\frac{2}{\sqrt2}\$ = \$\sqrt2\$.
This means that you get the same inductance but now you can have an operating current that is \$\sqrt2\$ higher for the same level of core saturation when the core was not gapped.
And...
No; look at your 1st formula - it tells you inductance is proportional to turns squared whilst in your 2nd formula, flux is proportional to turns (no square term) so no, they don't alter with the same proportion or coefficient.
If a gap causes permability to halve, flux density also halves for the same operating current but, to return inductance to what it was previously, turns must increase by \$\sqrt2\$ hence the bottom line is that flux density has gone down by \$\sqrt2\$ for the same operating current. This is a benefit and a big one.