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2 Waves on Transmission Lines – I: (Basic Equations, Multisection Transmission Lines)
Introduction
The two‐wire transmission line is a useful medium for the propagation of the voltage and current waves. It is also useful in the modeling of planar transmission lines. The EM‐wave propagating on multilayered planar transmission lines could also be analyzed with the help of the multisection transmission lines. The primary purpose of this chapter is to review in detail the wave propagation on a transmission line without and with sources. Several other important topics such as the characterization of a line section, nature of wave velocities, dispersion, and reactively loaded lines are further discussed in chapter 3.
Objectives
To formulate Kelvin–Heaviside transmission line equations.
To obtain the solution of the wave equation.
To compute the power flow on a transmission line.
To use Thevenin's theorem on a transmission line section to obtain its transfer function.
To consider the wave propagation on a multi‐section transmission line with the voltage and current sources.
To understand the nature of the wave propagation on a nonuniform transmission line.
2.1 Uniform Transmission Lines
This section presents the basic understanding of wave propagation on a uniform transmission line. The Kelvin–Heaviside transmission line equations are formulated using the lumped circuit elements model of the uniform transmission line.
The voltage and current wave equations are obtained and solved for a terminated line section. The phenomenon of the standing wave is discussed. The Thevenin theorem of the transmission line network is discussed and the transfer function of a line section is obtained.
2.1.1 Wave Motion
The wave motion could be treated as the transfer of oscillation from one location to another location. A harmonic oscillation is described either by a sine or by a cosine function. A periodic oscillation has a fixed period. An oscillation repeats itself after the periodic time (T). In general, an oscillation, i.e. an oscillatory motion can have any shape such as square, triangular, and so on. However, with the help of the Fourier series, such periodic oscillations can be decomposed to the harmonic functions. Likewise, wave motion can also acquire an arbitrary shape. The arbitrary periodic shape of a wave can be decomposed into the harmonic waves.
Figure (2.1) shows the instantaneous amplitude v(t) of the wave generated by an oscillating quantity, such as an oscillating ball (particle) at the location A. It has an angular frequency of oscillation ω radian/sec and maximum amplitude Vmax. The ball is attached to a spring and immersed in a water tank. It generates a wave motion at the surface of the water. The water wave travels with velocity vp and causes another delayed oscillation, at the location B, in a similar ball attached to a spring. Through the mechanism of wave motion, the oscillation is transferred from the location A to the location B separated by distance x. Likewise, the oscillating quantity could be a charge, electric field, magnetic field, or voltage obtained from an oscillator connected to a cable.
The equation of the harmonic oscillation at location A is described by the cosine function,
(2.1.1)
The equation of harmonic oscillation that appears at location B after a delayed time t′ is
(2.1.2)
The wave motion, created by the oscillation at the location A, travels with velocity vp. The time delay in setting up the oscillation at the location B is t′ = x/vp. So the equation of oscillation at the location B is