BINGO -  Tnx Rik, thats it, convincing.  Sorted

 

So my thing  IS 1:4 turns ratio, (1:16 Z ratio)  just as the calculations said it needed to be, and is indeed, effectively, an auto transformer.  Otherwise known as a bootstrapped coaxial transformer, but just configured with different reference points to confuse everyone. Your second reference, the Semelab paper, (UKuW see below)  gives the answer.  

 

Look at page 10,   the 1:9 Ruthroff   UnUn .   With just two bits of coax, which are then bootstrapped up on top of the input to give a 3:1 ratio (for 1:9 impedance).   Add another turn, so three windings are  bootstrapped on top of the primary, rearrange the grounding points and there, in all its glory, is my mystery.   

 

The reference / grounding points can be shifted as the mosfet PA  has  two separate, non magnetically linked cores.  This is is just normal push-pull HF PA practice, and being so normal in its concept, completely hides hides a big real advantage - it enables the designer to float just whatever ternainals need to be floated for balanced / unbalanced operation, and even to provide DC isolation,   Any conductor passing through the core allows one end to be completely Rf decoupled from the other end, so that's how balanced operation is permitted with the basic Ruthrof design shown in the paper.

 

On the same page, a single length of coax is shown giving 1:2 transformation by the same bootstrapping arrangement, and when the transmision line is shown as a twisted pair rather than coax, then wrapped on a torroid, it becomes the 1:4 balun beloved of HF antenna constructors.   A natural progression that helps illustrate the concept  of  a loop of coax giving one turn more than it appears to.

 

It all falls out. 

... life is worth living again ...

... can retire happy ...

 

Andy
www.g4jnt.com

... now, next job, how to sort out the world financial crisis ...