Measure transmission line Zo – nanoVNA – PVC speaker twin

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.

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.

The transformer seen above is a 1:1 transformer as described at A 1:1 RF transformer for measurements – based on noelec 1:9 balun assembly. The system was OSL calibrated at the pin socket underneath the grey connector.

Scans were made from 1-60MHz of the 1m section of line with short circuit and open circuit termination.

Above is the short circuit scan, the local dip around 30MHz would appear to be an artifact of the test environment, so that area will be avoided.

Above is the open circuit scan which looks generally smooth.

Both scans were saved as .s1p files.

Calculations will be offered at 40 and 14MHz.

40MHz

Above, the results. Zo is 142-j3.5Ω, and matched line loss MLL is 0.3dB/m. This MLL is quite a deal higher than you might expect, even of PVC insulated cables.

Because the line is well less than \(\frac{vf \: \lambda}{4}\) (guessing at the likely value of vf), we can calculate vf simply from the imaginary part of γ for k=0.

TWLLC model

PVC / copper speaker twin 0.22mm^2 @ 40.53MHz

Parameters
Conductivity 5.800e+7 S/m
Rel permeability 1.000
Diameter 0.000500 m
Spacing 0.001700 m
Velocity factor 0.660
Loss tangent 5.000e-2
Frequency 40.530 MHz
Twist rate 0 t/m
Length 1.000 m
Results
Zo 150.75+j2.90 Ω
Velocity Factor 0.6600
Length 73.742 °, 0.204839 λ, 1.000000 m, 5.054e+3 ps
Line Loss (matched) 0.346 dB

Above, a TWLLC model was constructed to have about the same MLL. The model would suggest that dielectric loss is quite high.  In my experience, PVC LossFactor ranges widely and would seem to depend on pigments, plasticisers and fillers used in manufacture.

14MHz

Above is a calc from the same sweep, but at 14MHz. MLL is pretty appalling for this sample, yet I see people recommending this stuff and stuff like it for feed line for a dipole, particularly for portable work. Larger size PVC twin lines are better, but never good. The figure-8 cable / zip cord dipole of ham lore is one of those things that doesn’t pass analysis.