# Calculate the scattering matrix of a 3-port lumped element network [closed]

I can get S11 and S12 by terminating port 2 and 3 with Z0 and Zr respectively. But when I try to calculate S13 it seems less straight forward. The following is the network, taken from Gao, J. The Physics of Superconducting Microwave Resonators. It is also assumed that the coupling capacitance Cc is small.

The S-matrix given by the original reference is:

But the way I calculated it is

Just to re-type the calculation for easier view:

define $$\delta_0=\omega C Z_0$$ and $$\delta_r=\omega C Z_r$$. Assume $$C$$ is small so that $$\delta_r, \delta_0 \ll 1$$.

We have $$S_11$$ by the usual formula of reflection:

$$S_{11}=\frac{Z_{in}-Z_0}{Z_{in}+Z_0}\approx -j\delta_0/2$$

where $$1/Z_{in}=\frac{1}{Z_0}+\frac{1}{Z_r+\frac{1}{j \omega C}}$$

Then I have $$S_{31}$$ as

$$S_{31}=\frac{V_3^-}{V_1^+}|_{V_2^+=V_3^+=0}\\ =V_3/(V_1^+)\\ =(1+S_{11})\frac{Z_r}{Z_r+\frac{1}{j \omega C}}\\ \approx j\delta_r$$

## closed as off-topic by John Rennie, ZeroTheHero, Buzz, Kyle Kanos, FGSUZJan 25 at 22:28

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• Can you show your calculations when trying to calculate S13? – The Photon Jan 23 at 21:49
• Hi, i added my calculation, as well as the result given in the reference. Thanks. – xi dai Jan 23 at 23:53
• Note that we have an equation editor built into the site which helps for readability of your post. – Kyle Kanos Jan 24 at 11:09
• i don't understand why this is marked as off-topic? – xi dai Jan 26 at 17:23
• @KyleKanos. any chance we can reopen this question? – xi dai Jan 27 at 19:20