Transmission Lines

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 37.) 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.),

(See Exercise 7.) These two equations can be combined to give

(414) |

together with an analogous equation for . In other words, and both obey a wave equation of the form (383) in which the associated phase velocity is

(415) |

Multiplying (412) by , (413) by , and then adding the two resulting expressions, we obtain the energy conservation equation

(416) |

where

(417) |

is the electromagnetic energy density (i.e., energy per unit length) of the line, and

(418) |

is the electromagnetic energy flux along the line (i.e., the energy per unit time that passes a given point) in the positive -direction (ibid.). Consider a signal propagating along the line, in the positive -direction, whose associated current takes the form

(419) |

where and are related according to the dispersion relation

(420) |

It can be demonstrated, from Equation (412), that the corresponding voltage is

(421) |

where

(422) |

Here,

(423) |

is the characteristic impedance of the line, and has units of ohms. It follows that the mean energy flux associated with the signal is written

(424) |

Likewise, for a signal propagating along the line in the negative -direction,

(425) | ||

(426) |

and the mean energy flux is

(427) |

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

(428) |

where is the

(429) |

where is the

(430) |

which, of course, is the

(431) |

Hence, signals always propagate down such lines at the velocity of light in vacuum. The impedance of a parallel strip transmission line is

(432) |

where the quantity

is known as the

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 2012)

(434) |

Thus, the impedance of a parallel wire transmission line becomes

(435) |

For the case of a co-axial cable in which the radii of the inner and outer conductors are and , respectively, the capacitance per unit length is (Fitzpatrick 2008)

(436) |

Thus, the impedance of a co-axial transmission line becomes

It follows that a parallel wire transmission line of given impedance can be fabricated by choosing the appropriate ratio of the wire spacing to the wire radius. Likewise, a co-axial transmission line of given impedance can be fabricated by choosing the appropriate ratio of the radii of the outer and inner conductors.