David, VK3IL posted EFHW matching unit in which he describes a ferrite cored transformer matching unit that is of a common / popular style.
Above is David's pic of his implementation. It is a FT140-43 toroid with 3 and 24t windings and note the 150pF capacitor in shunt with the coax connector.
The popular belief is that these are a broadband impedance transformer with impedance ratio equal to the square of the turns ratio, 64 in this case and therefore a broad band match from 3200Ω to 50Ω.
To his credit, David took some measurements of several different variations and reported them in his article.
Above are David's measurements of the subject transformer.
Lets explore the matching detail for the case of a 3.3kΩ load at 22MHz, and using the 150pF shunt cap.
Superficially, you might convince yourself that this is explained by the turns squared story, but the 150pF doesn't reconcile with that story, nor does the variation with frequency, eg the rapid change above 22MHz.
There are several imperfections that impact behaviour, and some might offset others to some extent at some frequencies. Let's list the most significant ones:
- permeability of the core is frequency dependent and complex (ie has real and imaginary parts);
- some finite current must flow in the primary winding to magnetise the core, and the equivalent magnetising impedance is complex;
- not all flux created by one winding cuts the other, the transformer has flux leakage; and
- each winding exhibits self resonance.
Factors regarded as less significant and omitted from the analysis:
- loss of the 150pF capacitance (as the capacitor characteristic is unknown, though even if its Q was as low as 50 at 22MHz, the loss will not impact the network significantly);
- self resonance of the primary winding; and
- conductor loss.
Above, a calculation of the expected magnetising impedance.
Above is a graphic of the model and Smith chart solution. Key points:
- T1 is an ideal transformer with turns ratio 3:24 (n^2 ratio 64);
- L2 models flux leakage, it is 393nH;
- L1 and R1 model the magnetising impedance (including core loss);
- C2 models the self resonance of the secondary, it is 2.1pF.
This model is at a single frequency, note that L1, L2, L3 and R2 are dependent on permeability which is frequency dependent.
David's measurement above without C1 shows the extent to which the basic transformer is frequency dependent. Increasing leakage reactance as permeability decreases with increasing frequency is the main reason for the behaviour above 7MHz, and the 150pF C1 provides some level of compensation that makes the transformer usable over a considerably wider range.
It is possible to calculate the core loss from R1, but it has not been done here at 22MHz as core loss at lower frequencies is typically the most limiting value (it would be around 15% at 3.6MHz).
The common turns squared explanation for the principle of operation of these transformers is superficially naive, and traditional linear circuit analysis does provide a rational explanation for their behaviour.
Foonote: Some revisions have been made to the topology of the equivalent circuit, but they are all similar, the changes have been about the referral of values to primary or secondary side of the ideal transformer, or representation of the magnetics as a T circuit or L circuit.