S-Parameters of lossy two-port network

Question

I have a lossy passive two-port network with some input mismatch, so, $$ S_{11} \neq 0 $$ By simulation, I find the S11 and S21 parameters. What I would like to know is the losses within the network, when a wave is incident at port 1. I first thought it should be $$ 1 - |S_{11}|^2 - |S_{21}|^2 $$ because what is not reflected and not transmitted must be lost within the network. However, a discussion with a colleague gave rise to the idea that $$S_{21}$$ is somehow contained within $$S_{11}$$... So, is it possible to find the losses from S-Parameters?

Answer 1

How is \$S_{21}\$ contained in \$S_{11}\$? By the definition of S-parameters, \$S_{11}\$ is the amount of wave leaving the two-port from port 1 when all other ports are terminated in a matched load, and a wave is incident on port 1. \$S_{21}\$ is the wave energy leaving port 2 due to an incident wave on port 1. Therefore, your original statement is correct, in that the power lost (assuming all energy absorbed in the two port is loss) is:

$$ 1 - |S_{11}|^2 - |S_{21}|^2 $$

The amount of incident wave power, minus the reflected power, minus the power that makes it through the system. In fact, this equation is often used to define a lossless, passive two-port (although it's usually written in a different form, and it has to apply for both directions). One could write this as:

A passive two port is lossless when:

$$ |S_{11}|^2 + |S_{21}|^2 = 1$$ $$ |S_{22}|^2 + |S_{12}|^2 = 1$$

*Usually this is written in matrix equations, where they have the added benefit that they can be generalized to N-port networks.