Ferrite cored inductors at HF – flux, loss and saturation

I see online experts opine that small signal characteristics (eg complex permeability curves) of ferrite toroids are not valid for applications such as RF common mode choke in transmitting antennas.

Others opine that saturation is a practical design limit, and for example that Bs/2 is a safe / appropriate design target.

Let us consider a ferrite cored inductor at 7MHz. The inductor comprises 11t on a 11t on Fair-rite 5943003801 (FT240-43) toroid. This is a medium to high permeability ferrite material, and for that reason, has significant loss at HF. Higher and lower permeability materials are fashionable at different times, the higher permeability #31 mix is fashionable at this time.

I will work in MKS units.

Above is Fair-rite's B-H curves for #43 material. Let's take saturation flux density Bs to be 1500gauss or 0.15T.

Small signal analysis

Most inductance calculators I see are based on Al, here is popular one, toroids.info.

It calculates an inductance of 130µH and impedance of 0+j5710Ω, both are nonsense. You might be forgiven to thinking the calculated inductance is at 7MHz, it is actually for 10kHz.

We can calculate the expected impedance of the inductor with a better calculator.

The inductor has an impedance of  5490+j1320Ω. Also calculated is the equivalent inductance of the series reactance, 30.1µH.

This size core can safely dissipate around 10W continuous, lets use that figure.

The effective area of the core from the datasheet is 1.58e-4m^2.

The current to cause 10W of dissipation is given by \(I=\sqrt\frac{P}{R}=\sqrt\frac{10}{5490}=42.7 \; mA\).

Recalling that \(L=n\frac{\phi}{i}\) we can calculate peak flux as \(\phi=\frac{L \sqrt{2} I}{n}=\frac{3e-5 \cdot \sqrt2 \cdot 0.0427}{11}=1.65e-7 \; Wb\) and \(B=\frac{\phi}{A}=\frac{1.65e-7}{1.58 e-4}  \frac{3e-5 \cdot 0.0427}{11}=1.04 \; mT\)

So, the core is dissipating 10W at 0.7% of Bs… it is nowhere near saturation.

Unsurprisingly, measurements of real devices reconciles will with small signal based models.

To illustrate how absurd the Bs/2 target is for this scenario, if the material was sufficiently linear to Bs/2, the impressed voltage would be 17kV and it would dissipate 52kW. Of course the core would quickly reach Curie temperature at which point losses become dramatically lower.

None of this is to say that saturation isn't relevant to design, but it will turn out that for many if not most HF RF applications, dissipation is a more constraining factor than saturation.

Flux density analysis

Let's use VK1SV's flux density calculator (B), one of many.

A common mistake is to apply a formula that the user doesn't really understand.

Above is a calculation that overestimates B because it is for a lossless inductor.

When the inductor's loss resistance if factored into the calculation, the voltage across the pure inductance is substantially lower, and B is lower at 10.4 gauss or 1.04 mT which reconciles with the earlier calculation.

This worked example shows that naive / mindless use of online calculators can give significantly wrong results.

Note that if there is a DC component to current through an inductor, that must be considered in flux and saturation calculations.