Theory
In any transmission
system, a source sends energy to a load, such as an antenna. Ideally,
we design the transmission network such that the characteristic impedances
of the source, the transmission line and the load are all identical. Unfortunately,
many real-world situations prevent the match from being perfect.
For example, we might
want an antenna (the load) to be useful over a broad range of frequencies.
But the characteristic impedance of an antenna is unlikely to stay constant
with frequency, especially if the frequency span is great.
When the transmission
line impedance does not match that of the load, part of the transmitted
waveform is reflected back towards the source. The reflected wave, which
varies in phase and magnitude, adds to the incident (transmitted) wave
and the sum is called a Standing Wave.
The reflected wave
causes the amplitude to vary as a function of position along the transmission
line. The Standing Wave Ratio (SWR), which is the ratio between the maximum
and minimum amplitudes of the total waveform, will in this case be greater
than one.
If there is no reflected
wave, i.e., if the impedance match is perfect, the amplitude of the total
waveform (incident plus reflected wave) will be the constant, regardless
of where we measure it along the transmission line. The result is a SWR
of 1. SWR = 1 indicates maximum power transfer to the load.
SWR can be inferred
by measuring the reflection coefficient of the circuit. The network analyzer
is a tool that enables us to do just that.
If we know the reflection
coefficient, we can determine the characteristic impedance of the load
by using a Smith Chart. The Smith Chart has circles of constant resistance
and arcs of constant reactance. The relationship between reflection coefficient
and characteristic impedance is shown in the diagram. At first glance,
the Smith Chart appears complicated, but its elegance soon becomes obvious.
The Smith Chart can
help us translate the reflection coefficient into impedance. First, measure
the reflection coefficient with a network analyzer (or invent one of your
own choosing). Place the reflection coefficient, by using either the mouse
or the drop-down input boxes, at the desired value (real + imaginary)
on the Smith Chart. Hit the Play button (triangle), and the program will
display a circle with a radius equal to the reflection coefficient magnitude
(constant VSWR circle). Notice that if you move the reflection coefficient
anywhere on this circle, you can see from the waveform at the left that
the SWR is the same, only its phase changes. (Phase values are not shown
around the chart in this program; however, the phase is calculated and
shown at the left side of the screen.)
In general, only
the horizontal line (diameter) is labeled with (normalized) resistance
values and only the unit (outer) circle is labeled with (normalized) reactance
values. To read the desired values, it is necessary to follow the appropriate
circle of constant resistance to the diameter line, and to follow the
appropriate arc of constant reactance to the unit circle. Hit Play again,
and the program will display the constant-resistance circle and the constant-reactance
arc for you. (The actual values are calculated and shown at the left side
of the screen.)
Experiment with the
simulator and see if you can predict the shape of the standing wave before
you move the reflection coefficient.
What is the range
of the SWR?
What is the significance of the center of the chart?
What assumptions are made about the transmission line?
What assumptions are made about the source?
|