Transmission Lines

For the case of a standard “mains” circuit, which oscillates at 60 Hz (in the US), we find that
.
Hence, we deduce that the single-phase assumption is reasonable for such circuits. For the
case of a telephone circuit, which typically oscillates at 10 kHz (i.e., the typical frequency of human speech), we find that
. Thus, the single-phase assumption definitely breaks down for long-distance
telephone lines. For the case of internet cables, which typically oscillate at 10 MHz, we find that
.
Hence, the single-phase assumption is not valid in most internet networks. Finally, for the case of TV circuits,
which typically oscillate at 10 GHz, we find that
. Thus, the single-phase
approximation breaks down completely in TV circuits. Roughly speaking, the single-phase approximation is
unlikely to hold in the type of electrical circuits involved in *communication*, because these invariably
require high-frequency signals to be transmitted over large distances.

A so-called *transmission line* is typically used to carry high-frequency electromagnetic
signals over long distances; that is, distances sufficiently large that the
phase of the signal varies significantly along the line (which implies that the line is much longer than the
free-space wavelength of the signal).
In its simplest form, a transmission line consists of *two* parallel conductors
that carry *equal and opposite electrical currents*, , where measures
distance along the line. See Figure 6.1. (This combination of two conductors carrying equal and
opposite currents is necessary to prevent intolerable losses due to electromagnetic radiation.) Let be the instantaneous voltage difference between the two
conductors at position . Consider a small section of the line lying
between and
.
If is the electric charge on one of the conducting sections, and the charge on the other, then charge
conservation implies that
. However, according to
standard electrical circuit theory (Fitzpatrick 2008),
, where is the *capacitance per unit length*
of the line. Standard circuit theory also yields
(ibid.), where
is the *inductance per unit length* of the line.
Taking the limit
, we obtain
the so-called *Telegrapher's equations* (ibid.),

(6.55) |

(6.56) |

(6.57) |

(6.58) |

(6.59) |

(6.60) |

(6.61) |

(6.62) |

(6.63) |

(6.64) |

(6.65) |

(6.66) | ||

(6.67) |

(6.68) |

As a specific example, consider a transmission line consisting of two uniform parallel conducting strips of width and perpendicular distance apart , where . It can be demonstrated, using standard electrostatic theory (Grant and Philips 1975), that the capacitance per unit length of the line is

(6.69) |

(6.70) |

(6.71) |

(6.72) |

(6.73) |

Practical transmission lines generally consist of two parallel wires twisted about one another (for example, twisted-pair ethernet cables), or two concentric cylindrical conductors (for example, co-axial TV cables). For the case of two parallel wires of radius and distance apart (where ), the capacitance per unit length is (Wikipedia contributors 2018)

(6.75) |

(6.76) |

(6.77) |