The transducer needs to be driven by a matching transformer. But how do we calculate the windings ratio on this, and what is this transformer made of ?

After some searching I found an article about building a sonar that has a decent explanation on how to calculate the transformer, based on the known capacitance of the transducer:

The inductance of the secondary winding L2b must be such that the resonant frequency of the circuit forming the L2b, the distributed transmit/receive transmittance and C22 are identical with the same transmit/receive element frequency.

This frequency is given by the relation f = 1 / 2π x L x C, where f is the resonance frequency in Hz, L is the inductance at H and C the total capacity in F.

Reversing the terms, L = 1 / 4π² x f² x C, and for f = 200kHz, C = 3n2 we have L2b = 198 μH.The corresponding number of turns N is calculated from the relation N = L2b / Ls. where Ls is the specific inductance of the ferrite core. If, for example, Ls = 250nH, the number of turns becomes 28.If the coil ratio is 1:9, L2a must have 3 turns.

If a ferrite core with a different special inductance value is used, the above calculations should of course be re-generated. The turns ratio can be held at 1:9. Correspondingly, if a different transmit/receive element is used, L2 inductance must be recalculated.

Also, if the frequency is not 200 kHz, C14 must be recalculated by: C14 = 1 / 4π² x f² x L1, where f is the new frequency and L1 = 630μH

Now this article is clearly a copy of a muchh older one, as I found something with exactly the same text and illustrations in a Dutch magazine Elektuur from 1984. Actually it must be old, since the LM1812 chip that is used here is obsolete since the late nineties. Now what both articles have wrong is the relation between the AL value of a coil and the inductance per winding, which is the number of windings squared.

Now my piezo disc has a capacitance of 6000 pF and a resonance frequency of 512kHz, so if I fill out the formulas, this is what I get:

(Calculations done in SMath Studio, an excellent free program for this)
Secondary inductance should be about 16 uH
I found a coil inductor of 200uH, and counted the number of windings (57). So to get to 16 uH Ill need 4.6 windings. Which seems a very low number, and also nearly impossible to get exactly right. And after I actually tried I can confirm this does not work at all.