Practical_Antenna_Handbook_0071639586

C h a p t e r 4 : T r a n s m i s s i o n L i n e s a n d I m p e d a n c e M a t c h i n g 121
2 f
β = π = 2π fZ0C' (4.16)
cv
Thus we can predict the variation in the amplitude of a wave of frequency f simply
from knowing the attenuation (usually expressed in dB/100 ft) and the velocity factor,
v F , of the transmission line.
F
Transmission Line Responses
When we employ a transmission line to deliver RF energy from a transmitter to an antenna
we are normally interested in its performance at one frequency (as when we are
on-off CW keying the transmitter) or for a very small range of frequencies (as in voice
transmissions). For much of our analysis of the antenna and feedline system we need
consider only the cable’s steady-state ac response, especially when the load (antenna) impedance
is perfectly matched to the characteristic impedance of the transmission line.
Often, however, the antenna and transmission line impedances are not exactly the
same and we need to analyze what happens in such cases. Then it becomes extremely
helpful to our understanding of how transmission lines function if we consider the
line’s step-function response. Step-function analysis involves the application of a single
voltage transition at the input of the line: The input voltage snaps from zero (or an initial
steady value) to a nonzero (or final steady) value “instantaneously”, and is held
there until all action along the entire transmission line has died out.
Through a mathematical technique known as Fourier analysis, a voltage step can be
shown to consist of many individual sine waves of differing amplitudes and phases
and extending over an extremely wide range of frequencies. Thus, by exciting the transmitter
end of the transmission line with a single voltage step and examining the resulting
waveforms at other points along the line and at the junction with the load, we can
get an immediate feeling for the behavior of the line over a very wide range of frequencies,
as well as a detailed understanding of what happens when there is a mismatch
between the line and the load.
Step-Function Response of a Transmission Line
Before we get into the details of transmission line step response, let’s look at a mechanical
analogy that should be familiar to all of us. A taut rope (Fig. 4.3A) is tied to a rigid
wall that does not absorb any of the energy in the pulse propagated down the rope.
When the free end of the rope is given a vertical displacement (Fig. 4.3B) by yanking up
on it once, a wave is propagated along the rope, toward the wall, at velocity v (Fig.
4.3C). When the pulse hits the wall (Fig. 4.3D), it is reflected (Fig. 4.3E) and propagates
back along the rope toward the free end (Fig. 4.3F).
If a second pulse is propagated down the line before the first pulse dies out, then
there will be two pulses on the line at the same time (Fig. 4.4). When the two pulses are
both present on the rope, the resultant deflection of the rope will be the algebraic sum
of the two pulses. If a continuous train of pulses is applied to the line at certain resonant
repetition rates, the net deflection resulting from the combined incident and reflected
pulse amplitudes at each point along the rope will create standing waves—one example
of which is shown in Fig. 4.5—that do not propagate in either direction on the rope but,
instead, go straight up and down at each point on the rope.