This may not be important anymore, since I am getting results that work for what I am trying to do... But I am struggling to find how my version of derivations has equivalence to your derivations algebraically. Numerically they agree. I might have to play around some more with the algebra to really understand the differences there.
I start with this:
zis=1/y11, zos=1/y22, zio=z11, zoo=z22
Keeping in mind that your two-ports are reciprocal.
Using the conversion from Z to Y parameters, we can get:
Y11 = Z22/(Z11 Z22 - Z12 Z21)
Y22 = Z11/(Z11 Z22 - Z12 Z21)
Using this, we get Y11/Y22 = Z22/Z11
and then (Z11 Y11)/Y22 = Z22
Using the equivalences from the beginning, this becomes:
(Zio Zos)/Zis = Zoo
Now starting with your formula Z12 = Z21 = Sqrt(Z11*Z22 - Z22/Y11)
Leaving the Sqrt off until the end, we can convert Z11*Z22 - Z22/Y11 to Zio*Zoo - Zis*Zoo
and then Zio*Zoo - Zis*Zoo to Zoo(Zio-Zis)
Now using (Zio Zos)/Zis = Zoo, we replace Zoo in Zoo(Zio-Zis) to finally get Z12 = Z21 = Sqrt((Zio Zos*(Zio-Zis))/Zis)
And since the C element of the ABCD parameters is 1/Z21, the just above expression for Z21 can be reciprocated to give:
C = 1/Z21 = 1/Sqrt((Zio Zos*(Zio-Zis))/Zis) which is the formula for C in these formulas:
Those are the second set of formulas I posted in reply #48