Page 322 - Engineering Electromagnetics, 8th Edition

304 ENGINEERING ELECTROMAGNETICS

Finally, we surmise that the existence of voltage and current across and within the
transmission line conductors implies the existence of electric and magnetic fields in
the space around the conductors. Consequently, we have two possible approaches to
the analysis of transmission lines: (1) We can solve Maxwell’s equations subject to the
line configuration to obtain the fields, and with these find general expressions for the
wave power, velocity, and other parameters of interest. (2) Or we can (for now) avoid
the fields and solve for the voltage and current using an appropriate circuit model. It is
the latter approach that we use in this chapter; the contribution of field theory is solely
in the prior (and assumed) evaluation of the inductance and capacitance parameters.
We will find, however, that circuit models become inconvenient or useless when
losses in transmission lines are to be fully characterized, or when analyzing more
complicated wave behavior (i.e., moding) which may occur as frequencies get high.
ThelossissueswillbetakenupinSection10.5.Modingphenomenawillbeconsidered
in Chapter 13.

10.2 THE TRANSMISSION LINE EQUATIONS
Our first goal is to obtain the differential equations, known as the wave equations,
which the voltage or current must satisfy on a uniform transmission line. To do this,
we construct a circuit model for an incremental length of line, write two circuit
equations, and use these to obtain the wave equations.
Our circuit model contains the primary constants of the transmission line. These
include the inductance, L, and capacitance, C,as well as the shunt conductance, G,
and series resistance, R—all of which have values that are specified per unit length.
The shunt conductance is used to model leakage current through the dielectric that
may occur throughout the line length; the assumption is that the dielectric may possess
conductivity, σ d ,in addition to a dielectric constant, r , where the latter affects the
capacitance. The series resistance is associated with any finite conductivity, σ c ,in
the conductors. Either one of the latter parameters, R and G, will be responsible for
power loss in transmission. In general, both are functions of frequency. Knowing the
frequency and the dimensions, we can determine the values of R, G, L, and C by
using formulas developed in earlier chapters.
We assume propagation in the a z direction. Our model consists of a line section
of length z containing resistance R z, inductance L z, conductance G z, and
capacitance C z,as indicated in Figure 10.3. Because the section of the line looks
the same from either end, we divide the series elements in half to produce a symmet-
rical network. We could equally well have placed half the conductance and half the
capacitance at each end.
Our objective is to determine the manner and extent to which the output voltage
and current are changed from their input values in the limit as the length approaches
avery small value. We will consequently obtain a pair of differential equations that
describe the rates of change of voltage and current with respect to z.In Figure 10.3,
theinputandoutputvoltagesandcurrentsdifferrespectivelybyquantities V and I,
which are to be determined. The two equations are obtained by successive applications
of Kirchoff’s voltage law (KVL) and Kirchoff’s current law (KCL).