## 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

## Measurement of various loss quantities with a VNA – a worked example

This article documents a worked example of the matters discussed at Measurement of various loss quantities with a VNA. Above is an air cored solenoid of about 20µH connected between Port 1 and Port 2 of a NanoVNA-H4 which has been calibrated. The whole lot is sitting on an inflated HDPE bag to isolate the DUT from the test bench. Continue reading Measurement of various loss quantities with a VNA – a worked example

## Measurement of various loss quantities with a VNA

Loss, Insertion Loss, and Mismatch Loss terms pre-date VNAs and S parameters, but a VNA can be a very productive way of measuring / calculating these quantities for two port networks.

This article explains the basic S parameters and their use to measure and calculate Loss, Insertion Loss, and Mismatch Loss.

## S parameters

Review of s parameters of a two port network. Above, a two-port network showing incident waves (a1, a2) and reflected waves (b1, b2) used in s-parameter definitions. (“Waves’ means these are voltages, not power.) Continue reading Measurement of various loss quantities with a VNA

## Definitions of important loss terms

Readers of my articles occasionally ask for explanation of the distinction between meanings of:

• Insertion Loss;
• Mismatch Loss;
• Loss (or Transmission Loss).

These terms apply to linear circuits, ie circuits that comply with linear circuit theory, things like that impedances are independent of voltage and current, sources are well represented by Thevenin and Norton equivalent circuits. Continue reading Definitions of important loss terms

## 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

## Thoughts on the ARRL EFHW antenna kit transformer

I have not built and measured the thing, but have done the first step in a feasibility study.

The transformer design is not novel, it is widely copied and this may be one of the copies. The design is usually published without any meaningful performance data or measurements.

The article Select a ferrite core material and sufficient primary turns for a low InsertionVSWR 50Ω broadband RF transformer – comparison of measured and predicted laid out a method for approximating the core loss of a EFHW where the load is adjusted to that input VSWR50=1, ie input Z=50+j0Ω.

That method will be applied here for a good initial estimate of core loss.

I will present calcs for 80m and 40m since there are lots of articles and videos encouraging people to extend the antenna to 80m (with and without a loading coil).

It is quite practical to build an EFHW transformer with less than 0.5dB (11%) core loss.

## Amidon FT240-43 toroid with 2t primary The first point to note is that Amidon’s 43 product of recent years is sourced from National Magnetics Group, and is their H material. It is not a good equivalent to Fair-rite’s 43 mix.

Let’s make a first estimate of core loss at 3.5MHz. We can estimate the complex permeability which is needed for the next calculation. Continue reading Thoughts on the ARRL EFHW antenna kit transformer

## 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