Determination of transmission line characteristic impedance from impedance measurements

Measured impedances looking into a uniform transmission line section with short circuit (SC) and open circuit (OC) terminations can provide the basis for calculation of characteristic impedance Zo.

We rely upon the following relationships:

\(Z_{sc}=Z_0 \tanh (\alpha + \jmath \beta )l\\\) and

\(Z_{oc}=Z_0 \coth (\alpha + \jmath \beta )l\\\)

Rearranging the formulas and multiplying, we can write:

\(Z_0^2=\frac{Z_{sc}}{\tanh (\alpha + \jmath \beta )l} \frac{Z_{oc}}{\coth (\alpha + \jmath \beta )l}\\\) \(Z_0^2=\frac{Z_{sc}}{\tanh (\alpha + \jmath \beta )l} Z_{oc}\tanh (\alpha + \jmath \beta )l\\\)

The tanh terms cancel out… provided the arguments are equal. Focus on length l, l for the short circuit measurement might not equal l for the open circuit measurement if the termination parts are not ideal (and they usually are not).

If the tanh terms cancel, we can simplify this to \(Z_0=\sqrt{Z_{sc}Z_{oc}}\). This is commonly parroted, apparently without understanding or considering the underlying assumption that l is equal for both measurements.

Another big assumption is that it is a uniform transmission line, ie that the propagation constant β is uniform along the line… including any adapters used to termination the line.

The third assumption is that the measured impedance values are without error.

Above is a plot of calculated Zo for a theoretical case of a line of ~10m length of Belden 8267 (RG213A/U) around the frequency of first resonances. This calculation essentially imitates perfect measurements of perfect DUTs.

Lets introduce a small length change, short circuit termination results in an equivalent 9mm increase in length in the DUT (approximately the offset of a common SC termination wrt a common OC termination).

Above is the theoretically calculated Zo based on the offset caused by a pair of common pair of N type terminations.

Above is a measurement over a wider frequency range of a 10m length of Belden 8267 (RG213A/U) ,  the first resonances being around 5MHz due to the length of the DUT. A similar glitch occurs, and it is explained by the termination offset which violates the underlying assumptions used above in deriving the formula for Zo.

It turns out that this calculation of Zo is particularly sensitive to non-ideal short circuit and open circuit terminations, eg small differences in l between the two DUTs in the regions near quarter wave and higher resonances.

As mentioned, adapters can also contribute to errors.

Measured impedances are subject to instrument error, and in the case of VNAs, calibration and correction based on incorrect characterisation of the calibration parts and fixture contribute to error which is often most apparent when measuring extremely low and extremely high impedances (ie around the DUT resonances) and that can further exacerbate glitches caused by ignoring the offset (and other imperfections) of terminations used for DUT measurement.

These problems do not make the technique worthless, the measurement technician must work to minimise them, and then understanding the cause of some anomalous values, exclude them from the data so as to expose the underlying transmission line behavior.

There are automated tools that attempt to report Zo for users who desire turn key simple click interfaces without understanding… I don’t trust them blindly.