Transmission Line

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.

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