Introduction. Transmission line theory

Electromagnetic energy, once generated in one place, has a natural tendency to spread in the whole space at a speed close to 300.000 Km/s. In telecommunications this behavior can be useful when the user position is not known in advance, as in a broadcasting system or in a cell phone network. In other applications, instead, electromagnetic energy must be transferred from one place to the other along a well deﬁned path without any spreading at all: an example is the cabling of a building.

In the most general terms, a transmission line is a system of metal conductors and/or dielectric insulating media that is capable of “guiding” the energy transfer between a generator and a load, irrespective of the bends that the line undergoes because of installation needs.

There are many types of transmission lines, some examples of which are shown in Fig. 1.1. The various line types are used for diﬀerent applications in speciﬁc frequency ranges. Striplines and microstrips are used only inside devices, such as ampliﬁers or ﬁlters, and their lengths never exceeds some centimeters. Twisted pairs and coaxial cables are used for cabling a building but coaxial cables can also be used for intercontinental communications. Waveguides are used to deliver large amounts of microwave power over short to moderate distance. Waveguides can also be made of dielectric materials only, as in the case of optical ﬁbers.

a b c

d e

Fig. 1.1. Examples of transmission lines: (a) coaxial cable, (b) two wire line, (c) optical fiber, (d) microstrip, (e) stripline.

Long line – regular transmission line of length greater than of wave in line λ.

Two round conductors (wires) will be assumed to be perfect (i. e. have infinite conductivity). Peculiarity of long line is interference of incident from a generator and reflected from a load EMW. These waves have opposite directions.

Line has next parameters (distributed):

[Ω/m] – resistance; [1/Ωm] – conductivity; [H/m] – inductance; [E/m] – capacity.

Equivalent circuit of line element dz is shown in Fig. 1.2.

Fig. 1.2

and values depend on wire material and dielectric quality. and are defined by form and dimensions of conductor’s cross-section and distance between them.

Equivalent circuit is fair for symmetric and asymmetric transmission lines.

Parameters of length element dz

(1.1)

where , , , – line parameters (along 1 meter).

Current and voltage increments

or using (1.1)

where – line impedance, – line admittance.

Whence there are telegrapher’s equations of long line

(1.2)

Equation show the connection between voltage U and current I in arbitrary section of line.

In order to solve (1.2) let’s find differential by z

(1.3)

In regular line the line parameters are constant

(1.4)

Substituting U and I from (1.2) to (1.3) and after transformation will obtain homogeneous equations of long line

(1.5)

where – propagation factor of EMW in line.

Solution of (1.5)

where – complex amplitudes of voltage U and current I for incident and reflected waves ( , ). If there is “+” in power the wave propagates in negative direction of axis z, “-” – in positive direction.