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Measuring
characteristic impedance
Updated October
4, 2009
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New for October 2009!
On this page we will discuss
an idea we had for measuring characteristic impedance of a transmission
line, based on a question that came our way. It is probably not
much more than a mathematical exercise, but you never know when
it might be useful. If someone gave you a coax cable and didn't
know if it was 75 or 50 ohms, this trick might do the job. Its really
a calculation of Z0 based on a direct measurement
of reflection coefficient.
Here's the premise:
If you plot S11 on a Smith chart,
for a line length of unknown impedance, it traces a circle, with
one point on the circle's circumference at the center of the chart.
The diameter of the circle is mathematically related to the line's
impedance. If the line were a quarter
wave, if you plotted it in frequency up to the quarterwave point
it transcribes a half circle. It is easy to figure out the relationship
of the line impedance for this case.
Below is a 90 ohm line "measured"
between two fifty ohm terminations:

When we plot the reflection
coefficient on the Smith chart, you can see the circle that develops
to the left of the 50 ohm point. When the line reaches a half wavelength,
the circle is complete, then it merely traces over itself as you
go further up in frequency.

At marker m1 the line forms
a classic quarterwave transformer, at this frequency the network
appears like a 162 ohm load:
ZT=SQRT(Zin*Z0)
where ZT is the impedance of
the line (acting as a quarterwave transformer), Zin is the impedance
looking in to it, and Z0 is the system impedance which terminated
both ends of the line.
The impedance of the line is
thus related to the diameter of the circle it's reflection coefficient
traces.
But you don't need a full half-wave
line to trace out the circle. To determine the diameter of a circle
all you need is three points. The circumscribed circle of the triangle
they form is easy to calculate, you can look
up the formula on Wikipedia. Now you can measure the characteristic
impedance of a line from three frequency points of S11 magnitude
and angle!
We made an excel sheet that makes the "diameter" calculation
from each set of three successive frequency points. Then it converts
to impedance versus frequency. The Excel file is called Impedance
Calculator 101.xls, look for it
here.
There are possible errors that need to be minimized for this technique
to produce an accurate assessment of Z0. If there are parasitics
in the measurement (connectors on the ends of a cable of unknown
impedance for example) then the calculation could be off. The S-parameters
need to be fully deembedded for best results. You will see from
the plot that the calculated impedance varies over frequency. Perhaps
the lowest error occurs at the lowest frequency, if someone else
wants to speculate on this we'd like to hear
from you!
There are two solutions to the
math. We didn't figure out how to make the spreadsheet smart enough
to choose the correct one, but it is easy to pick the right one.
If the circle is to the left of center, Z<Z0. To the right, Z>Z0.
Example 1
We were given reflection coefficient
from an electromagnetic simulation of a transmission line. Below
it is plotted (by the spreadsheet) from zero to 10 GHz. The circle
is to the right side of the origin, so the transmission line must
be greater than 50 ohms. Sorry, we were too lazy to plot the data
on an actual Smith chart!
Now here's what the spreadsheet
comes up with for Z0: the characteristic impedance of the line is
somewhere between 69 and 75 ohms.

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