Standing waves and Standing wave patterns in transmission lines
Standing waves are created by the superposition of incident and reflected travelling waves in an improperly terminated transmission line. By improper termination, what is meant is, the load impedance has a value different from that of the characteristic impedance of the line.
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If we follow the evolution of the standing wave with time, we will find that its amplitude () oscillates between a maximum value (
) and a minimum value (
), every quarter wavelength apart, the wavelength being that of the voltage wave,
.
is the frequency of the signal being transmitted and
is the phase velocity of the wave, the value of which is slightly less than the free space velocity (
) of an electromagnetic wave. The distance between two successive maxima or minima turns out to be
.
Since the position of points corresponding to
and
remain constant with time, the wave is called a standing wave.
Figure 1. depicts how a standing wave pattern is generated from incident and reflected waves in a lossy () imperfectly terminated (
) line. Since the line is lossy,
and
are not constant but a function of
, decreasing progressively with increasing
. If the line happened to be lossless (
), then however
and
would have been constant throughout the length of the line.
If a line is terminated in an open circuit (), a short circuit (
) or a pure reactive load like
equalling
, the whole of the incident energy at the load is reflected back (
). The standing wave in all such cases is said to be a pure standing wave. Figure 2. depicts such a pure standing wave for a line terminated in an open circuit (
,
) and assumed lossless.
For pure standing waves in lossless lines,
and
. The points corresponding to
, are now called the nodes and those corresponding to
are called the antinodes. The instantaneous standing wave, though oscillating with time, is always zero (
at the nodes and its magnitude is always a maximum at the antinodes, equalling the amplitude (
) of oscillation at the particular instant of time.
Thus we observe, that for standing waves (pure or impure) in any lossless line, and
have the same value throughout the length of the line, and as such, are used to define a parameter called the Voltage Standing Wave Ratio (VSWR) or
for the line as follows:
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So far we have been talking about voltage waves and voltage standing wave patterns. We could also obtain current standing waves if we superpose incident and reflected current waves travelling from the source toward the generator and vice versa respectively as follows:
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Interestingly, if the transmission line is perfectly matched (), there is no reflected wave (
). Such a line is called a flat line, since the standing wave patterns turn out to be a straight lines, as shown in Figure 4. VSWR or
for such a line equals 1.
In conclusion, let us ask the question: What is the practical use of the VSWR?
VSWR is used in slotted line measurements of load impedances. A slotted section of a transmission line at the load end is considered for the purpose. A probe is used to sample the electric field and hence the voltage standing wave at different distances from the load and the position of the first voltage minimum () in the standing wave pattern is found. The reflection coefficient for a lossless line at a distance
from the load end is given by:
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(1) | |
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On the other hand, can be obtained from VSWR measurements, implying we can then obtain
which equals
.
Finally, the load impedance can also be expressed in terms of , as:
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Thus, with the knowledge of standing waves and the VSWR (), one may obtain the load impedance (
).