A very common design of a n:1 transformer for EFHW antennas uses a 2t primary on and FT240-43 (or even smaller) ferrite core.
In a process of designing a transformer, we often start with an approximate low frequency equivalent circuit. “Low frequency” is a relative term, it means at frequencies where each winding current phase is uniform, and the effects of distributed capacitance are insignificant.
Above is a commonly used low frequency equivalent of a transformer. Z1 and Z2 represent leakage impedances (ie the effect of magnetic flux leakage) and winding conductor loss. Zm is the magnetising impedance and represents the self inductance of the primary winding and core losses (hysteresis and eddy current losses).
For broadband RF transformers, Z1 and Z2 need to be small as they tend to be quite inductive and since inductive reactance is proportional to frequency, they tend to spoil broadband performance.
Zm shunts the input, so it spoils nominal impedance transformation (Zin=Zload/n^2) if it is relatively low. For powdered iron cores Zm is mainly inductive; and for ferrite cores Zm is a combination of inductive reactance and resistance depending on frequency and ferrite type.
Where the transformer secondary turns and load (antenna) are adjusted for a near perfect 50Ω input match, we can estimate the approximate core efficiency as 1 minus 50 divided by Rmp (the parallel resistance equivalent component of Zm), 1-50/Rmp.
If we set a design criteria of at least 80% core efficiency, we can calculate a critical value for Rmp>50/(1-0.8)>250Ω.
We can measure Rmp with out trusty nanovna.
The measurement fixture has been OSL calibrated, and a two turn winding applied to an FT240-43 core and plugged into the fixture.
The sweep is controlled from nanovna MOD v3 and plotted.
Above is the measurement from 1-31MHz, and it can be seen that Rmp falls below 250Ω from 1.6-11.5MHz. Two turns is not sufficient for more than 80% core efficiency from 1.6-11.5MHz.
At 3.5MHz, Rmp=171Ω and so core efficiency is 1-50/171=71%, or -1.5dB.
An exercise for the reader is to try three turns, and to try smaller cores such as the popular FT114-43.