The calculator Calculate ferrite cored inductor – rectangular cross section has until now assumed that the toroid has sharp corners. The corner treatment varies across commercial products, some are burnished which removes very little material, some have a chamfer or bevel, some are radiused. All of these treatments give rise to a very small error in calculated ΣA/l.

The calculator has been revised to include 45° chamfers of a specified length on all four corners. If the chamfer angle differs, the error is very small in the range 30-60°. If the corners are radiused, use the radius as the chamfer length, the error is very small.

We do not need to obsess over these errors as they will usually be dwarfed by manufacturing tolerances.

The calculation of ΣA/l for the sharp corner model is fairly simple.

\(\int _{ir}^{or}\frac{w}{2 \pi r}dr\)

\(=\frac{w}{2 \pi }\left(\ln \left(or\right)-\ln \left(ir\right)\right)\)

To implement the chamfer adjustment, the ΣA/l component of the missing material is calculated.

Firstly the inner chamfers (which are simpler).

\(\int _{ir}^{ir+cr}\:\frac{r-ir}{2 \pi r}dr\)

\(=\frac{1}{2\pi}\left(cr-ir\left(\ln \left(ir+cr\right)-\ln \left(ir\right)\right)\right)\)

Then the outer chamfers are calculated.

\(\int _{or-cr}^{or} \frac{\left(r-\left(or-cr\right)\right)}{2 \pi r}dr\)

\(=\frac{1}{2\pi }\left(cr\left(\ln \left(or\right)-\ln \left(-cr+or\right)\right)-or\left(\ln \left(or\right)-\ln \left(-cr+or\right)\right)+cr\right)\)

The final ΣA/l is the first quantity less the two missing components.

Implemented in javascript in the calculator…

aol=width/2/pi*Math.log(od/id)*1e-3; aol=aol-(cr-ir*(Math.log(ir+cr)-Math.log(ir)))/(2*pi)*1e-3*2; aol=aol-(cr*(Math.log(or)-Math.log(-cr+or))-or*(Math.log(or)-Math.log(-cr+or))+cr)/(2*pi)*1e-3*2;

Let's look at some examples.

Above is a calculation for the popular FT240-43. Without the chamfer adjustment, ΣA/l would be 0.001091 (which is the value given by Fair-rite in the datasheet), adjusting for chamfer the reduction is 2.5%.

Above is an example calculation of ΣA/l and Al. the calculated ΣA/l is less than 1% less than if the chamfer were ignored. The difference may be greater on some cores, especially very small cores.

Above is an example calculation for a very small core with radiused corners. The chamfer approximation reduces ΣA/l and Al by about 2%… again the manufacturing tolerances dwarf the adjustment.

Sizing the adjustment is an accurate way to determine if it is significant or not, and 2% accuracy does not have a lot of application to ferrites with 20% tolerance… but it does become more important when trying to characterise the ferrite material based on core dimensions.