# Scattering matrix of matched a device with matched ports

The scattering matrix of a device with matched ports has zeros at the position of the corresponding diagonal element(I am referring to equation 7.2 or 7.4 in Microwave Engineering by David M Pozar, 2nd Edition). I've read and understood this from various sources on and off the internet.

However, what eludes me is the following: The S-matrix is meant to relate the reflected voltage at a port to the voltage incident at every port for a given microwave network. The S-matrix is itself independent of the voltage that appears across its ports or the kind of load that is attached to a port and that's why it's helpful. Hence, if we say that the port which is terminated by a matched load has a corresponding zero entry in the matrix, doesn't it violate the fact that the S-matrix is a characterization of the system's reflected-incident voltages independent of the conditions a network is made to operate at?

• no because ideal=Zo-Zin(f)=0 while reflected voltage = I(load)* delta Zo/sumZ as a percentage of any voltage or -dB or+ return Loss. this "per unit" method is normalized to Zo=TBD e.g. 50,75 , 120 etc Apr 7 '18 at 14:27

The S matrix is only defined properly when all its ports are terminated by a matched load. The definition in Pozar is $$S_{ij}=\frac{V_{i}^{-}}{V_{j}^{+}}\Biggr|_{V_{k}^{+}=0\,\text{for}\,k\neq j}$$ $V^+$ for incoming waves and $V^-$ for outgoing (reflected) waves
To summarize, the actual S parameters are linear, thus they are independent of the voltage across them (no saturation). But the assumption that the scattering parameters of a device will be the same no matter to what network it is connected is not correct, to quote the book. "An important point to understand about scattering parameters is that the reflection coefficient looking into port n is not equal to $S_{nn}$ unless all other ports are matched"