Transmission lines

An idealized transmission line consists of two parallel conductors of uniform cross-sectional area. The conductors possess a capacitance per unit length, , and an inductance per unit length, . Suppose that measures the position along the line.

Consider the voltage difference between two neighbouring points
on the line, located at positions and , respectively.
The self-inductance of the portion of the line lying between these two
points is . This small section of the line can be thought of as
a conventional inductor, and, therefore, obeys the well-known equation

(990) |

Consider the difference in current between two neighbouring points on the
line, located at positions and , respectively. The capacitance
of the portion of the line lying between these two points is
. This small section of the line can be thought of
as a conventional capacitor, and, therefore, obeys the well-known equation

(992) |

Equations (991) and (993) are generally known as the

Differentiating Eq. (991) with respect to , we obtain

(994) |

(995) |

This is clearly a wave equation, with wave velocity . An analogous equation can be written for the current, .

Consider a transmission line which is connected to a generator at one end
(), and
a resistor, , at the other (). Suppose that the generator outputs
a voltage
. If follows that

(997) |

where . This clearly corresponds to a wave which propagates from the generator towards the resistor. Equations (991) and (998) yield

(999) |

(1000) |

(1001) |

The most commonly occurring type of
transmission line is a co-axial cable, which consists of
two co-axial cylindrical conductors of radii and (with ). We
have already shown that the capacitance per unit length of such a cable is
(see Sect. 5.6)

(1002) |

(1003) |

(1004) |

(1005) |

(1006) |

(1007) |

If we fill the region between the two cylindrical conductors with a
dielectric of dielectric constant , then, according to the
discussion in Sect. 6.2, the capacitance per unit length
of the transmission line goes up by a factor . However,
the dielectric has no effect on magnetic fields, so the inductance
per unit length of the line remains unchanged. It follows that the
propagation speed of signals down a dielectric filled co-axial cable
is

(1008) |

We have seen that if a transmission line is terminated by a resistor whose
resistance matches the impedance
of the line then all of the power sent down the
line is absorbed by the resistor. What happens if ? The answer is
that
some of the power is reflected back down the line. Suppose that the
beginning of the line lies at , and the end of the line is at .
Let us consider a solution

(1010) |

(1011) |

(1012) |

(1013) |

(1014) |

Clearly, if the resistor at the end of the line is properly matched, so that
, then there is no reflection (*i.e.*, ), and the input impedance of
the line is . If the line is short-circuited, so that , then there is total
reflection at the end of the line (*i.e.*, ), and the input impedance becomes

(1015) |

(1016) |

Thus, a quarter-wave line looks like a pure resistor in the generator circuit. Finally, if the length of the line is much less than the wave-length (

Suppose that we wish to build a radio transmitter. We can use a standard half-wave antenna (*i.e.*, an antenna whose length is half the wave-length
of the emitted radiation)
to emit the radiation. In electrical circuits, such an antenna acts like a resistor of resistance
73 ohms (it is more usual to say that the antenna has an impedance of 73 ohms--see Sect. 9.2).
Suppose that we buy a 500kW
generator to supply the power to the antenna. How do we transmit
the power from the generator to the antenna? We use a transmission line, of course.
(It is clear that if the distance between the generator and the antenna is of
order the dimensions of the antenna (*i.e.*, ) then the constant-phase
approximation breaks down, and so we have to use a transmission line.)
Since the impedance of the antenna is fixed at 73 ohms, we need to use a
73 ohm transmission line (*i.e.*, ohms) to connect the generator to
the antenna, otherwise some of the power we send down the line is reflected
(*i.e.*, not all of the power output of the generator is converted into
radio waves). If we wish to use a co-axial cable to connect the generator to
the antenna, then it is clear from Eq. (1009) that the radii of the
inner and outer conductors need to be such that
.

Suppose, finally, that we upgrade our transmitter to use a full-wave antenna
(*i.e.*, an antenna whose length equals the wave-length of the emitted radiation).
A full-wave antenna has a different impedance than a half-wave antenna. Does
this mean that we have to rip out our original co-axial cable, and replace it
by one whose impedance matches that of the new antenna? Not necessarily.
Let be the impedance of the co-axial cable, and the impedance of
the antenna. Suppose that we place a quarter-wave transmission line (*i.e.*, one whose
length is one quarter of a wave-length) of characteristic
impedance
between the end of the cable and the
antenna. According to Eq. (1017) (with
and
), the input impedance of the quarter-wave line
is
, which matches that of the cable. The output impedance
matches that of the antenna. Consequently, there is no reflection of the power
sent down the cable to the antenna. A quarter-wave line of the appropriate impedance
can easily
be fabricated from a short length of co-axial cable of the appropriate .