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 Z0.

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 Z0 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. Continue reading Determination of transmission line characteristic impedance from impedance measurements