Electronic – transformer has continuity but no voltage output

The winding ratio of a transformer does not translate to a simple voltage step-up / step-down ratio.

The actual transfer function is affected by the transformer core material, the frequency of the signal used, winding quality, core design, environmental factors, and possible several other parameters as well.

In your specific experiment, here are the likely sources of significant variation from expectation:

• Frequency of signal applied to the primary (which in this context is whichever side you applied a signal to). Soft ferrous core transformers transfer well at line frequencies (e.g. 50Hz), ferrite cores are good for intermediate frequencies, air core is good for radio frequency but pretty much useless at line frequency.
• Waveform of signal applied: For anything but a pure sine wave, the transfer function would be different for the different harmonics making up the waveform, so output signal would have a complex relationship to the input, certainly not linear as you expect.
• Method of measurement of signal: A multimeter or AC voltmeter is not ideal, an oscilloscope would be recommended.

If you could amend your question with more detail on how you conducted your experiment, a better response could be formed.

For "concentric" wound transformers, I find both ways of them to be used, and the factors influencing it are most likely (may not all apply to your case):

• Cost of the material involved. Higher current needs thicker wires, but the same number of turns (and even thicker if its on the outside because it is longer there). Just calculate what is needed for the number of turns required in either inner or outer layer, and then do a comparison. It seems to me the higher the step down ratio, the more beneficial it is to keep the high current ones inside.
• Tapping. A lot of transformers have multiple tappings. Precise positioning and space for them is a lot easier on the outside than it is on the inside.
• Serviceability and failure modes. Bigger transformers are actually so expensive that it might be feasible to repair them. Depending on the expected failure modes putting one or the other winding on the outside is more useful.
• Rectangular cores might need different material strenghts and properties since internal forces are higher than in circular cores. Aluminum might be better suited here than the softer copper. This applies of course only at rather high currents (that are distributed unevenly in the conductor). Metal fatigue considerations however might drive you away from aluminum here.

Note that I don't call them secondary/primary but low/high voltage sides, I think those factors are more influential than the direction they are used in.