Effect of changing the reference planes in the S matrix

S parameters relate amplitudes (magnitude and phase) of traveling waves

incident on and reflected from a microwave network, phase reference planes

must be specified for each port of the network

S parameters are transformed when the reference planes are moved from their

original locations

Consider a N-port microwave network, where the original terminal planes are

assumed to be located at zn = 0 for the nth port and where zn is an arbitrary

coordinated measured along the transmission line feeding the nth port

The scattering matrix for the network with this set of terminal planes is denoted

by [S]

Consider a new set of reference planes defined at zn = ln, for the nth port and let

the new scattering matrix be denoted as [S`]

From the theory of traveling waves on lossless transmission lines,

riting the above equation in matrix form,

The phase of Snn is shifted twice the electrical length of the shift in the terminal

plane n, because the wave travels twice over this length upon incidence and

reflection

Cascaded networks

In the cascade connection of two-ports networks as shown in the figure below,