Measure transmission line Zo – nanoVNA – PVC speaker twin – loss model derivation

The article Measure transmission line Zo – nanoVNA – PVC speaker twin demonstrated measurement of transmission line parameters of a sample of line based on measurement of 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 \(\gamma=\alpha + \jmath \beta\), and from that, MLL.

Measurement with nanoVNA

So, let’s measure a sample of 14×0.14, 0.22mm^2, 0.5mm dia PVC insulated small speaker twin.

Above is the nanoVNA setup for measurement. Note that common mode current on the transmission line is likely to impact the measured Zin significantly at some frequencies, the transformer balun (A 1:1 RF transformer for measurements – based on noelec 1:9 balun assembly) is to minimise the risk of that. Nevertheless, it is wise to critically review the measured |s11| for signs of ‘antenna effect’ due to common mode current.

Above is a plot of the measured |s11| for SC and OC line sections.

Observe that there are no anomalous kinks or the like, but the OC section measurements become a little noise at the lower end.

Traditional matched line loss model

Above is a plot of the calculated MLL (red dots) based on the s11 measurements, and a curve fit to the model \(MLL = k_1\sqrt f+k_2f \text{ dB/m}\).

Allowing for the scatter at the lower frequencies as we are measuring 1m of line with an inexpensive hobby grade VNA, \(MLL=\text{3.26e-5} \sqrt{f}+\text{1.39e-9}f  \text{ dB/m}\) is a pretty good estimator.