Two Port Network

Question

So I'm studying $S$ parameters for two port networks. I've seen some special cases like what happens if port 2 is short circuited or if port 2 is matched with port 1. I was wondering what happens if port 2 is open circuited? I think $V_2$- would be zero for this condition. Am I right?
$$V_{1}^{-} = S_{11}(V_{1}^{+})+S_{12}(V_{2}^{+})$$
$$V_{2}^{-} = S_{22}(V_{2}^{+})+S_{21}(V_{1}^{+})$$

xXx_69_SWAG_69_xXx · Answer

If you leave a port open that is the same thing as having an impedance $infty$ which is the same as having $Gamma = -1$, therefore $V^{-}=-V^{+}$. So in your case the condition imposed on the scattering matrix would be $V^{-}_2=-V^{+}_2$
If you plug that in to the equations above, isolate the $V_2$, and divide the equations you will obtain:
$$V_1^{-}= Big(S_{11}-frac{S_{21}S_{12}}{1+S_{22}} Big) V_1^{+}$$
Where $S_{11}-frac{S_{21}S_{12}}{1+S_{22}} $ is the reflection coefficent seen from port 1.
Notice for a general reflection coefficent at port 2; $Gamma$ the relation would read:
$$V_1^{-}= Big(S_{11}+frac{Gamma S_{21}S_{12}}{1+ Gamma S_{22}} Big) V_1^{+}$$
It is just in this case that $Gamma = -1$

Two Port Network

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