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. Continue reading Determination of transmission line characteristic impedance from impedance measurements
Last update: 13th May, 2023, 10:15 AM
