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.
(9.194) | ||
(9.195) |
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.
(9.197) | ||
(9.198) | ||
(9.199) | ||
(9.200) |
The scattering matrix of an ideal, lossless transmission line with impedance and electrical length writes as follows.
(9.201) |
(9.202) |
(9.203) |
With = 299 792 458 m/s being the vacuum light velocity. Adding attenuation to the transmission line, the quantity extends to:
(9.204) |
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
(9.205) |
where is the attenuation factor and is the (real) propagation constant given by
(9.206) |
where is the effective dielectric constant and is the TEM propagation constant for the equivalent transmission line with an air dielectric.
(9.207) |
The expressions used to calculate the scattering parameters are given by
(9.208) | ||
(9.209) |
with being the normalized impedance and is the normalized admittance.