## Measuring velocity factor

This article discusses measuring velocity factor using the NanoVNA. The DUT is coax with N type connectors as it provides a better example to demonstrate and learn from. Having acquired competency, extension to two wire lines is just a matter of attending to the matters of a suitable transformer, and appropriate SOL calibration parts.

## N type connectors

The ‘standard' reference plane on N connectors is shown in the diagram above. For the purpose of this article, length measurements were made between the reference planes at both ends of the cable.

The N connector maintains Zo=50Ω through the connector, though a 9.3mm section of the path is usually air dielectric which needs to be taken into account for mm accuracy as needed for some applications (eg tuned line lengths used in repeater filters).

For the measurement of resonances here, a short circuit (SC) line termination is used as it provides the best accuracy of Zo in the region of that termination, albeit the last 9mm is at VF=1.

If you do not have a good SC termination, use a good quality female to female adapter, the same adapter for all measurements, with or without a N male open circuit (OC) termination. Measure the cable length between the reference planes on the male connectors, the extra length of the F-F adapter will be deducted in the calculator used below.

## Lengths

We must differentiate between electrical length and physical length of the line, the velocity of the wave (phase velocity vp) is reduced by the dielectric, we define velocity factor \(vf=\frac{v_p}{c_0}=\frac{1}{\sqrt \epsilon_r}\).

\(ElectricalLength=\frac{ReferencePlaneLength-0.0186}{vf}+0.0186 \text{ m}\\\)Rearranging to make ReferencePlaneLength the subject:

\(ReferencePlaneLength=(ElectricalLength-0.0186)vf+\\0.0186 \text{ m}\\\)The technique is to measure the connectors from reference plane to reference plane and calculate the velocity factor, the connectors are subtracted in the calculator to find the vf of the cable itself.

## Cable and connectors

The measurements made here are of sections of RG213 (with solid polyethylene dielectric) with exactly the same crimp male N connectors used on all ends.

## Measure physical length and first resonance of two different lengths of the same cable and connectors

Above, measurement of the first resonance of the shorter line section with SC termination.

Above, measurement of the first resonance of the longer line section with SC termination.

Now use Velocity factor solver to find the velocity factor of the cable itself.

Plugging the measured resonant frequencies and line lengths into the calculator solves for velocity factor, and also gives us the fixture offset in ps, and e-delay used by some VNAs to approximately correct the reference plane.

## Reconciliation check

We will measure the resonant frequency having set e-delay to remove the fixture delay from the measurement, and from that the electrical length.

Above, measurement of the first resonance of the shorter line section with SC termination and VNA corrected for the fixture offset (e-delay=264ps). The VNA reference plane is now the reference plane of the N near connector.

\(ElectricalLength = \frac{c_0}{4 f_r} = \frac{299792458}{4 \cdot 7108080} =10.54 \text{ m}\\\)Now calculating the electrical length from the ReferencePlainLength with correction for the air dielectric within the connector.

\(ElectricalLength= \frac{ReferencePlaneLength-0.0186}{vf}+0.0186=\\ \frac{6.9738-0.0186}{vf}+0.0186= 10.54\text{ m}\)The two rounded to the fourth significant digit reconcile.

The same techniques can be used with two wire line and a suitable balun, the instrument + balun being OSL calibrated at the DUT side of the balun.