A published design for an EFHW matching device from 80-10m uses the following circuit.

Like almost all such ‘designs', they are published without supporting measurements or simulations.

The transformer is intended to be used with a load such that the input impedance Zin is approximately 50+j0Ω, Gin=0.02S.

Analysis of a simple model of the transformer with a load such that input impedance is 50+j0Ω gives insight into likely core losses.

Let us calculate the magnetising admittance of the 3t primary at 3.6MHz. The core is a FT82-43 ferrite toroid.

Gcore is the real part of Y, 0.00691S.

If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.00691/0.02=65.4%, core loss is 1.87dB.

Now as to whether 65% efficiency is acceptable is a question for the user. This is intended for QRP use, so 5W SSB telephony input is not like to damage it, and you could think that an inefficient antenna system doubles the benefit of QRP, QRP^2 if you like.

## Can it be improved?

If an efficiency target for the transformer is set at say 90% or better, it takes an 6t primary to achieve that on 3.6MHz.

Gcore is the real part of Y, 0.00173S.

If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.00173/0.02=91.3%, core loss is 0.39dB.

Of course twice the turns are needed on the other winding, leakage inductance is increased, and the compensation capacitor will need review. It is likely that even with compensation, less VSWR bandwidth is achieved.

Another option is to stack two or three of the cores, or use a different core better suited to the application.

## Where do these designs come from?

It seems 2t or 3t primary windings on #43 cores are very common which might suggest ‘designers' have simply changed the core dimensions.

The geometry of the core varies from size to size, so just as inductance and impedance are very dependent on magnetic properties of the material (complex permeability), the also depend on cross section area and path length. The calculator shots above show a metric, ΣA/l, which captures the geometry, the larger it is, the fewer turns for same inductance / impedance. ΣA/l for and FT240-43 core is 0.00106 whereas for the FT82-43 core discussed in this article, it is less than half of that at 0.000468 and so drives a need for more turns.

Note that the next size core up or down in a series could have greater or lesser ΣA/l (m), there is no general rule that going to a smaller or a larger core requires more or less turns. Some datasheets show ΣA/l or the inverse Σl/A .

## Does it matter?

Well, in ham radio, everything works. But systems that work better increases the prospects of contacts.

## References

- Find |Z|,R,|X| from VSWR,|Z|,R,Ro
- A new impedance calculator for RF inductors on ferrite cores
- Calculate ferrite cored inductor (from Al)
- Calculate VSWR and Return Loss from Zload (or Yload) and Zo
- Duffy, O. 2015. A method for estimating the impedance of a ferrite cored toroidal inductor at RF. https://owenduffy.net/files/EstimateZFerriteToroidInductor.pdf.
- ———. 2006. A method for estimating the impedance of a ferrite cored toroidal inductor at RF. VK1OD.net (offline).