Examples of conductor pairs range from parallel conductor transmission lines carrying gigawatts of power to coaxial lines carrying microwatt signals between computers. When these lines become very long, times of interest become very short, or frequencies become very high, electromagnetic wave dynamics play an essential role. The transmission line model developed in this chapter is therefore widely used.
Equally well described by the transmission line model are plane waves, which are often used as representations of radiation fields at radio, microwave, and optical frequencies. For both qualitative and quantitative purposes, there is again a need to develop convenient ways of analyzing the dynamics of such systems. Thus, there are practical reasons for extending the analysis of TEM waves and one-dimensional plane waves given in Chap. 13.
The wave equation is ubiquitous. Although this equation represents most accurately electromagnetic waves, it is also applicable to acoustic waves, whether they be in gases, liquids or solids. The dynamic interaction between excitation amplitudes (E and H fields in the electromagnetic case, pressure and velocity fields in the acoustic case) is displayed very clearly by the solutions to the wave equation. The developments of this chapter are therefore an investment in understanding other more complex dynamic phenomena.
We begin in Sec. 14.1 with the distributed parameter ideal transmission line. This provides an exact representation of plane (one-dimensional) waves. In Sec. 14.2, it is shown that for a wide class of two-conductor systems, uniform in an axial direction, the transmission line equations provide an exact description of the TEM fields. Although such fields are in general three dimensional, their propagation in the axial direction is exactly represented by the one-dimensional wave equation to the extent that the conductors and insulators are perfect. The distributed parameter model is also commonly used in an approximate way to describe systems that do not support fields that are exactly TEM.
Sections 14.3-14.6 deal with the space-time evolution of transmission line voltage and current. Sections 14.3-14.4, which concentrate on the transient response, are especially applicable to the propagation of digital signals. Sections 14.5-14.6 concentrate on the sinusoidal steady state that prevails in power transmission and communication systems.
The effects of electrical losses on electromagnetic waves, propagating through lossy media or on lossy structures, are considered in Secs. 14.7-14.9. The distributed parameter model is generalized to include the electrical losses in Sec. 14.7. A limiting form of this model provides an "exact" representation of TEM waves in lossy media, either propagating in free space or along pairs of perfect conductors embedded in uniform lossy media. This limit is developed in Sec. 14.8. Once the conductors are taken as being "perfect," the model is exact and the model is equivalent to the physical system. However, a second limit of the lossy transmission line model, which is exemplified in Sec. 14.9, is not "exact." In this case, conductor losses give rise to an electric field in the direction of propagation. Thus, the fields are not TEM and this section gives a more realistic view of how quasi-one-dimensional models are often used.