Measuring characteristic impedance

Updated October 4, 2009

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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.