What you need to know is the turns ratio (in your parlance, the ratio of
This most basic detail is the point I am trying to get across. I agree the turns ratio is the key factor and what you need to use to compute impedances and gain. Then you go on to mention "voltage in to voltage out" which represents the gain and this number includes the effects of the source and load impedance and the gain is not used to calculate anything impedance related. Turns ratio is a simple definable constant for a device. Determining gain requires info unique to the specific situation.
In the attempt to simplify things and make it easy for the end user, the process of specifying and labelling SUTs has made a simple topic clear as mud. What is needed is the use of accepted math and known well defined terms. The introduction of the yet to be defined X-factor simply makes no sense when proper well defined terms exist.
dave

