Page 321 - Engineering Electromagnetics, 8th Edition

CHAPTER 10 Transmission Lines 303

L 1 L 2 L 3 L 4

V 0 C 1 C 2 C 3
R

Figure 10.2 Lumped-element model of a transmission line. All inductance
values are equal, as are all capacitance values.

to understanding and quantifying this is to note that the conducting transmission line
will possess capacitance and inductance that are expressed on a per-unit-length basis.
We have already derived expressions for these and evaluated them in Chapters 6 and
8 for certain transmission line geometries. Knowing these line characteristics, we can
construct a model for the transmission line using lumped capacitors and inductors, as
shown in Figure 10.2. The ladder network thus formed is referred to as a pulse-forming
network, for reasons that will soon become clear. 1
Consider now what happens when connecting the same switched voltage source
to the network. Referring to Figure 10.2, on closing the switch at the battery location,
current begins to increase in L 1 , allowing C 1 to charge. As C 1 approaches full charge,
current in L 2 begins to increase, allowing C 2 to charge next. This progressive charging
process continues down the network, until all three capacitors are fully charged. In the
network, a “wavefront” location can be identified as the point between two adjacent
capacitors that exhibit the most difference between their charge levels. As the charging
process continues, the wavefront moves from left to right. Its speed depends on how
fast each inductor can reach its full-current state and, simultaneously, by how fast
each capacitor is able to charge to full voltage. The wave is faster if the values of L i
and C i are lower. We therefore expect the wave velocity to be inversely proportional
to a function involving the product of inductance and capacitance. In the lossless
transmission line, it turns out (as will be shown) that the wave velocity is given by
√
ν = 1/ LC, where L and C are specified per unit length.
Similar behavior is seen in the line and network when either is initially charged.In
this case, the battery remains connected, and a resistor can be connected (by a switch)
across the output end, as shown in Figure 10.2. In the case of the ladder network,
the capacitor nearest the shunted end (C 3 ) will discharge through the resistor first,
followed by the next-nearest capacitor, and so on. When the network is completely
discharged, a voltage pulse has been formed across the resistor, and so we see why this
ladder configuration is called a pulse-forming network. Essentially identical behavior
is seen in a charged transmission line when connecting a resistor between conductors
at the output end. The switched voltage exercises, as used in these discussions, are ex-
amples of transient problems on transmission lines. Transients will be treated in detail
in Section 10.14. In the beginning, line responses to sinusoidal signals are emphasized.

1 Designs and applications of pulse-forming networks are discussed in Reference 1.