## Transmission lines – forward and reflected phasors and the reflection coefficient

Let’s consider the following transmission line scenario:

• Lossless;
• Characteristic Impedance Z0=1+j0Ω; and
• load impedance other than 1+j0Ω, and such that Vf=1∠0 and Vr=0.447∠-63.4° at this point.

The ratio Vr/Vf is known as the reflection coefficient, Γ. (It is also synonymous with S parameters S11, S22… Snn at the respective network ports.)

Above is a  phasor diagram of the forward and reflected voltages at the load. Continue reading Transmission lines – forward and reflected phasors and the reflection coefficient

## PVC speaker twin – loss model applied

One of the many gems of ham lore that I was fed as a beginner almost sixty years ago was that 23/0.076 (0.67mm^2) PVC insulated twin flex was suitable as an RF transmission line at HF, and that it had a Characteristic Impedance Z0 close to 75Ω.

It seems that these claims have been extended to apply to lighter gauge cables often called speaker cable or bell wire.

This article explores two cases of the application of a light grade of speaker twin to a G5RV antenna. The scenarios is a G5RV Inverted V with 7m of speaker twin from dipole to the coax section, and loss is calculated for the speaker twin section at 14.1 and 3.6MHz.

## PVC speaker twin copper / PVC 0.2mm^2 characteristics

The following articles report measurement of a sample of speaker twin, and derivation of a simple loss model:

From those articles, the loss model is copied for reader convenience.

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

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

## Determination of transmission line characteristic impedance from impedance measurements #2

Determination of transmission line characteristic impedance from impedance measurements discussed issues with the short circuit and open circuit terminations used with measurement of Zoc and Zsc for calculation of characteristic impedance of a line section.

Included was a model of the effect of small delay offset in one of the termination parts on an example scenario.

This article gives a Simsmith model that readers might find interesting to explore the effects of line length, offset, line characteristics, and frequency.

I have issues with Simsmith modelling of transmission lines, but nevertheless the model is informing.

The above example is 6m of RG58A/U with 5mm offset in the short circuit termination. Continue reading Determination of transmission line characteristic impedance from impedance measurements #2

## Relationship between radiation efficiency and minimum VSWR for common short helically loaded verticals

For clarity, $${RadiationEfficiency}=\frac{FarFieldPower}{InputPower}$$.

Such antennas are often advertised with a “minimum VSWR” or “VSWR at resonance” figure, but rarely show gain figures. One might wryly make the observation that that is how one might sell dummy loads rather than antennas.

Well, these things do radiate, so they are not very good dummy loads. Lets explore a theoretical example on the 40m band to inform  thinking.

Above is a NEC5.2 model of a vertical on a wagon roof. Continue reading Relationship between radiation efficiency and minimum VSWR for common short helically loaded verticals

## Determination of transmission line characteristic impedance from impedance measurements

Measured impedances looking into a uniform transmission line section with short circuit (SC) and open circuit (OC) terminations can provide the basis for calculation of characteristic impedance Z0.

We rely upon the following relationships:

$$Z_{sc}=Z_0 \tanh (\alpha + \jmath \beta )l\\$$ and

$$Z_{oc}=Z_0 \coth (\alpha + \jmath \beta )l\\$$

Rearranging the formulas and multiplying, we can write:

$$Z_0^2=\frac{Z_{sc}}{\tanh (\alpha + \jmath \beta )l} \frac{Z_{oc}}{\coth (\alpha + \jmath \beta )l}\\$$ $$Z_0^2=\frac{Z_{sc}}{\tanh (\alpha + \jmath \beta )l} Z_{oc}\tanh (\alpha + \jmath \beta )l\\$$

The tanh terms cancel out… provided the arguments are equal. Focus on length l, l for the short circuit measurement might not equal l for the open circuit measurement if the termination parts are not ideal (and they usually are not).

If the tanh terms cancel, we can simplify this to $$Z_0=\sqrt{Z_{sc}Z_{oc}}$$. This is commonly parroted, apparently without understanding or considering the underlying assumption that l is equal for both measurements.

Another big assumption is that it is a uniform transmission line, ie that the propagation constant β is uniform along the line… including any adapters used to termination the line.

The third assumption is that the measured impedance values are without error.

Above is a plot of calculated Z0 for a theoretical case of a line of ~10m length of Belden 8267 (RG213A/U) around the frequency of first resonances. This calculation essentially imitates perfect measurements of perfect DUTs. Continue reading Determination of transmission line characteristic impedance from impedance measurements

## NanoVNA-App v1.1.209-OD15 released

Most of the changes I have made to NanoVNA-App have been to align it with accepted standards and conventions.

This change is to the format of saved Touchstone, .s1p and .s2p, files.

Though the relevant specification is silent on the permitted decimal separator, the only one shown in examples is “.” so it is reasonable to interpret that the required separator is “.” which makes the file format locale independent (as were the first instruments using Touchstone format.

This release of NanoVNA-App writes “.” decimal separator, independent of locale.

The original reading code which was tolerant of either “.” and “,” is maintained, so it will continue to open files which might have been (incorrectly) saved using “,”.

NanoVNA-App-Setup-v1.1.209-OD15

## NanoVNA-H4.3 R44 mod

Whilst following up another matter, I came across the following commit to Hugyen’s NanoVNA-H4 repository.

Remove R44 from NanoVNA-H4 Rev4.3, this resistor may damage U2 and the battery if the NanoVNA-H4 is not used for a long time and the battery is too low.

Above is an extract from the revised schematic committed, the change highlighted by the red arrow. R44 has been changed from 5.1kΩ to not populated. Continue reading NanoVNA-H4.3 R44 mod

## NanoVNA – interpolation – part 5

NanoVNA – interpolation – part 4 and prior articles discussed the possibility of significant error when calibration data is interpolated.

## Test scenario

The test scenario is a NanoVNA-H4 with 5m length of RG58A/U to the reference plane. It has been OSL calibrated at the reference plane using a 1-101MHz 101 point sweep.

## Result without interpolation of the calibration dataset

Above is a zoomed in view of 1-5MHz of a 1-101MHz 101 point sweep, there are measurements at every whole MHz value from 1 to 101. There are only 5 measurement points on this graph. Continue reading NanoVNA – interpolation – part 5

## NanoVNA – interpolation – part 4

NanoVNA – interpolation – part 3 discussed selection of a sweep step size to provide sufficient data points for reasonably accurate interpolation.

## When / where is interpolation used?

The VNA correction process uses measurements of some known conditions to create a calibration dataset, a table if you like of the sweep frequencies and calibration data. Commonly the calibration dataset is a table of the correction factors calculated from measurements of the knowns for each frequency of the calibration sweep. The correction factors are usually calculated for each frequency independently of adjacent frequencies.

When used to sweep a different range, interpolation can be used to interpolate those correction factors to the new measurement frequencies.

A common data flow is that shown above, where the correction terms are calculated for each of the frequencies in the calibration sweeps, and then those terms are interpolated to the frequencies actually used for a DUT measurement sweep. Continue reading NanoVNA – interpolation – part 4