There are many ways to get a good estimate of the characteristic impedance Zo of a transmission line.

One method is to measure the input impedances of a section of line with both a short circuit and open circuit termination. From Zsc and Zoc we can calculate the Zo, and the complex propagation constant \(\gamma=\alpha + \jmath \beta\).

Calculation of Zo is quite straightforward.

The solution for γ involves the log of a complex number \(r \angle \theta\) which is one of the many possible values \(ln(r) + j \left(\theta + 2 \pi k \right)\) for +ve integer k. Conveniently, the real part α is simply \(ln(r) \). The real part of γ is the attenuation in Np/m which can be scaled to dB/m, and the imaginary part is the phase velocity in ** ^{c}**/m. The challenge is finding k.

Let’s take an example from recent measurements of 35m of CCS RG6 coax, and extract the s11 values recorded in saved .s1p files @ 1.87MHz. The saved data in MA format, magnitude and angle (in degrees).

Calculate Zo and gamma is flexible and can accept the MA format data directly.

Above, the results. Zo is 74.73-j1.156Ω, and matched line loss MLL is 0.03281dB/m. This MLL is quite a deal higher than you might find in many line loss calculators, they often fail on CCS cables.

That all seems pretty straight forward, but be warned that the results are very sensitive to small variation in the inputs, so measurement noise can degrade results. In this instance, the magnitude of s11 which is the magnitude of the complex reflection coefficient Γ is almost one, and small changes can have large effects on the results. So this method is not very suited to analysers with 8bit ADCs, but analysers or VNAs with 12bit or better ADC and OSL calibrated should give good results.

It turns out that the kth solution of the log of the complex value mentioned earlier is the correct one, finding that is beyond the scope of this article. We can calculate vf simply from the imaginary part of γ for k=1.

\(vf=\frac{2 \pi}{\beta \lambda}=\frac{2 \pi}{0.04886 \cdot 160.3}=0.802\)

This is a little lower than the nominal vf for this cable, a result of increasing significance of conductor internal impedance at lower frequencies… exacerbated by thin copper cladding of a steel core.

Now if we have an instrument that is not good enough to get accurate resistance component of impedance, we can just work with the reactance values but loose the γ and MLL calculation.

Above is an example where I measured a length of RG58A/U and recorded just the X values of the short circuit and open circuit measurements. If your analyser does not give the sign of X, you must make one of them -ve. Because we have assumed that R=0 in both cases, the solution is for a lossless line and the calculated value of Zo is purely real.