Further to Amidon's method of rating ferrite inductors and transformers, this article discusses some interesting measurements of ferrite toroids by Manfred Mornhinweg (Mornhinweg 2019).

Above are his measurements of a FB-31-6873 sleeve. Essentially there are two measurements at each frequency, and the expected flux density B is in the ratio of approximately 2:1. He has fitted a straight line on a log/log graph to the measurements at each frequency. The similarity of the slopes is not unexpected, and is a tribute to his experiment design, execution and calculations.

The first thing to consider is the slope of the the fitted line.

If the material was linear in its characteristic, we might expect the core heating to increase as the square of B. The blue line has such a slope and can be used to compare the slope of the red lines. By eye, they appear similar in slope, and no conclusions can be drawn about linearity as there are only two points for each frequency.

The other interesting this is B relative to the critical numbers given in Amidon's advice. The table is in gauss, and 1 tesla (1T)=10000gauss.

The sparse points are a bit limiting… so lets explore whether there is a likely curve fit, whether they appear based on a simple relationship.

When plotted on log/log axes, the points fall roughly on a straight line. Considering that the table numbers are rounded, we cannot do better than to say Bcrit=f^2*15.1 where f is in MHz.

So, lets plot them on the chart.

Above, the magenta X is at Amidons Bcrit value. It can be seen that both measurements at 1.8MHz were above Bcrit, and one of the measurements at 3.5MHz was well above Bcrit. Nevertheless each pair of measurements appear to be on approximately the same slope as the blue line, linear material. There is little evidence of saturation.

Nor should there be, 23mT (230gauss) @ 1.8MHz is way below saturation.

## Core loss

Mornhinweg performed thermal analysis to estimate the core loss, and at 1.8MHz, the graph indicates the two data points were (0.0114,0.216), (0.023,1.000).

Lets estimate the core loss of a 1 turn inductor using complex permeability.

We should keep in mind that suppression sleeves are not controlled for permeability in manufacture (they are controlled for Z at some frequency), so predictions based on published permeability curves have additional uncertainty.

The key figure is the real part of Y, G which is 0.0109S.

To be able to calculate the core loss, we need to know the voltage impressed on the winding at each of the test flux densities using the expression \(E=\frac{4.44 B A_e N F}{1e8}\).

Power dissipated is given by \(P=E^2 G_m\).

We can then divide that by the core volume to obtain the core loss in W/cm^3 to compare to the plotted measurements.

The calculated result is (0.0114,0.242), (0.023,1.010) which reconciles very well considering the tolerances of ferrite cores and the uncertainty of measurements.

The core loss could also be calculating by finding the magnitude of magnetising current \(Im=\frac{V}{Z_m)\) and calculating \(P=I^2R_m\). It reconciles with \(P=E^2 G_m\).

## References

- Mornhinweg, M. 2019. Ferrite core loss in HF power applications.

- Ferrite permeability interpolations
- Inductance of RF cored inductors and transformers
- Calculate ferrite cored inductor – rectangular cross section
- Calculate ferrite cored inductor – circular cross section
- Calculate ferrite cored inductor (from Al)
- Calculate ferrite cored inductor – ΣA/l or Σl/A
- Ferrite permeability interpolations