A transmission line is usually described by its ABCD-matrix. (Note that in ABCD-matrices, i.e. the chain matrix representation, the current is defined to flow out of the output port.)
These can easily be recalculated into impedance parameters.
Or in admittance parameter representation it yields
whence denotes the propagation constant and is the length of the transmission line. represents the characteristic impedance of the transmission line. The Y-parameters as defined by eq. (9.199) can be used for the microstrip line. For an ideal, i.e. lossless, transmission lines they write accordingly.
The scattering matrix of an ideal, lossless transmission line with impedance and electrical length writes as follows.
With = 299 792 458 m/s being the vacuum light velocity. Adding attenuation to the transmission line, the quantity extends to:
Another equivalent equation set for the calculation of the scattering parameters is the following: With the physical length of the component, its impedance and propagation constant , the complex propagation constant is given by
where is the attenuation factor and is the (real) propagation constant given by
where is the effective dielectric constant and is the TEM propagation constant for the equivalent transmission line with an air dielectric.
The expressions used to calculate the scattering parameters are given by
with being the normalized impedance and is the normalized admittance.