Electronic – Estimate capacitor parameters

When calculating the S-parameters, you should terminate all of the ports that don't have stimulus applied. So, in your situation, to calculate \$S_{11}\$ and \$S_{21}\$, you'd be working with this circuit:

^{simulate this circuit – Schematic created using CircuitLab}

Notice that the current passing through the capacitor from port 1 to port 2 is \$i_1\$ and \$-i_2\$ at the same time.

Now you can easily calculate \$i_1\$ and \$i_2\$ using the series impedance rule:

$$i_1 = -i_2 = \frac{v_s}{2Z_0+1/j\omega{}C}$$

And you can also calculate \$v_1\$ and \$v_2\$ using the voltage divider rule,

$$v_1 = \frac{Z_0+1/j\omega{}C}{2Z_0+1/j\omega{}C}v_s$$

$$v_2 = \frac{Z_0}{2Z_0+1/j\omega{}C}v_s$$

Then you can convert these voltages to the incident and reflected wave variables at each port by

$$a_n=\frac{1}{2}\frac{v_n+Z_0i_n}{\sqrt{\Re\left({Z_0}\right)}}$$

$$b_n=\frac{1}{2}\frac{v_n-Z_0^\star{}i_n}{\sqrt{\Re\left({Z_0}\right)}}$$

(which get a lot less hairy when you start plugging in numbers, since your \$Z_0\$ is purely real)

And then you have

$$S_{11}=\frac{b_1}{a_1}$$

and

$$S_{21}=\frac{b_2}{a_1}$$

I'm pretty sure this will show you that your equations for the off-diagonal parameters aren't quite right. You might be thinking that \$S_{11} + S_{21} = 1\$ because of conservation of energy, however this isn't correct because the travelling wave parameters are not proportional to the signal power but to its square root. To get the correct conversions from Z-parameters to S-parameters, see the Wikpedia page on Z-parameters.

Assuming the plates are connected in an interleaved fashion as normal, there are n-1 gaps with n electrodes. Each one acts as a capacitor in parallel with the others.

Here is an image from this website.

You can see that there are 11 plates and 10 gaps resulting in effectively 10 parallel capacitors of area equal (approximately) to the area of the dielectric slabs.