Apparently it really isn't that simple.
I got an output transformer and measured a voltage transfer ratio of 24.35 @ 1kHz.
The square root of the primary to secondary impedance ratio should be the same as the voltage ratio
I then connected various load resistors to the secondary and measured the impedance at the primary side with an LCR meter, also @ 1kHz, obtaining the following results:
Code:
Load Resistance Primary Impedance Square Root of impedance ratio
(measured)
1Ω 1190Ω 34.50
2Ω 1642Ω 28.65
4Ω 2854Ω 26.71
8Ω 4842Ω 24.60
16Ω 8068Ω 22.46
32Ω 12215Ω 19.54
64Ω 15710Ω 15.67
128Ω 17896Ω 11.82
The measured results aren't following the simple rule you gave, so apparently there's more to it than the simple rule.