# Determination of transmission line characteristic impedance from impedance measurements – eighth wave method

For a lossless line, the reactance looking into short section and open circuit terminated line sections is $$X_{sc}=Z_0 \tan \beta l$$ and $$X_{oc}=Z_0 \frac1{\tan \beta l}$$.

Noting that when $$\beta l= \frac{\pi}{4}, \tan \beta l=1$$ so when the line section is π/4ᶜ or 45° or λ/8, then $$|X_{sc}|=|X_{oc}|=Z_0$$.

We can use this property to estimate Z0 of an unknown practical low loss transmission line by finding the frequency where $$|X_{sc}|=|X_{oc}|$$ and inferring that $$Z_0 \approx |X|$$. Above is a chart created using Simsmith's transmission line modelling of the reactance looking into short section and open circuit terminated 10m sections of RF174. The blue and magenta lines intersect at X=51.16Ω whereas red R0=51.85Ω, about -1.3% error. The error depends on line loss, line length, frequency and the characteristics of the terminations.

With care, the method is quite useful, it is an approximation, but a good one for reasonably low loss line sections.

The eighth wave property that is used also occurs at odd multiples of an eight wavelength, but becomes less accurate as line loss disturbs results.