# Comparing toroidal inductors of different core dimensions

I often see comparisons of toroidal inductors of different core dimensions with all other characteristics (eg turns, core type, frequency) held the same.

There seems an implicit assumption by many that the bigger the core, the larger the inductance. There are several failure in that thinking.

The ‘inductance’ of a toroidal inductor is µ*n^2*a/l where:

• µ is complex permeability, µ0+µr;
• n is the number of turns;
• a is the cross section area; and
• l is the effective magnetic path length.

Note that at RF, permeability may be a complex frequency dependent value, and therefore ‘inductance’ will be a complex value.

Many online calculators incorrectly calculate l from core dimensions using a simplistic formula.

Many online calculators treat permeability as a real number that is not frequency dependent, they use initial permeability (µi).

Nevertheless, it can be seen that the factor in the inductance expression that varies with cores of the same mix is the quotient A/l. Few datasheets give the value, but it can be calculated fairly easily for regular shapes if you ignore the rounding of corners.

Here is a code snippet for a rectangular toroid section:

```a=(od-id)/2*width;
aol=width/2/pi*Math.log(od/id);```

An online expert recently compared an FT240-77 toroid and an FB77-1024 bead expressing surprise at the outcome. Though the FB77-1024 is smaller than the FT240-77, their A/l quotients are 0.00232 vs 0.00109, the smaller core has over twice the A/l quotient of the larger.

There is no general rule that A/l increases or decreases with ‘larger’ cores, all three dimensions play into the result.

‘Larger’ cores tend to have greater volume and surface area, and for those reasons may have greater capacity to absorb and dissipate heat.

Of course other factors also play into the impedance characteristic, even if the same number of turns are wound on each core type.

You could achieve the same result by comparing the ‘Inductance Factor’ Al for two cores as Al is proportional to A/l (for the same mix), though again published values may be unreliable and use different units (nH for 1t is most common) and aware of the common simplistic misuse of Al, some manufacturers do not publish Al for cores seen as ‘suppression products’.