## Relationship between angle of reflection coefficient and angle of impedance

It was stated above that the angle (or phase) of s11 or Γ is not the same as the angle (or phase) of Z.

Given Zo and Γ, we can find θ, the angle of Z.

\(

Z=Z_0\frac{1+\Gamma}{1-\Gamma}\)Zo and Γ are complex values, so we will separate them into the modulus and angle.

\(

\left | Z \right | \angle \theta =\left | Z_0 \right | \angle \psi \frac{1+\left| \Gamma \right | \angle \phi}{1-\left| \Gamma \right | \angle \phi} \\

\theta =arg \left ( \left | Z_0 \right | \angle \psi \frac{1+\left| \Gamma \right | \angle \phi}{1-\left| \Gamma \right | \angle \phi} \right )\)We can see that the θ, the angle of Z, is not simply equal to φ, the angle of Γ, but is a function of four variables: \(\left | Z_0 \right |, \psi , \left| \Gamma \right |, \& \: \phi\) .

It is true that if ψ=0 and φ=0 that θ=0, but that does not imply a wider simple equality. This particular combination is sometimes convenient, particularly when ψ=0 as if often the case with a VNA.

This article offers a simulation of a load similar to a 7MHz half wave dipole.

The load comprises L, L1, and C1 and the phase of s11 (or Γ) and phase of Z (seen at the source G) are plotted, along with VSWR.

Firstly, note that the two phase plots are very different, but in this case they cross over at phase=0 at 7.077MHz.

Secondly, note that even thought both phases are zero at 7.077MHz, the VSWR is 1.20. Neither phase demonstrates the best conditions for least feed line loss, minimum VSWR is slightly lower at 7.099MHz.

Maximum power in the load coincides with minimum VSWR at 7.099MHz.

Beware of claims that phase (of something) is the optimisation target, the author probably doesn’t really understand this stuff.

]]>(Duffy 2005) showed that the hybrid feed is susceptible to high losses in the low Zo line as it is often longish, is relatively high loss line and operates with standing waves.

Lets look at measurement of a real antenna, broadly typical of the G5RV. The antenna measured is a G5RV rigged in Inverted V form, 11m height at the apex and around 8m at the ends. The feed line is 2mm diameter copper spaced 50mm with occasional plastic insulators.

To some extent, the measurements are dependent on the environment, and whilst there will be variation from one implementation to another, the measurements provide a basis for exposing the configuration challenge.

Above is a plot of VSWR(50) essentially at the lower end of the matching section and low Zo line. The measurement is made looking into 0.5m of RG142 and a Guanella current balun that uses about 1m of 110Ω pair, it is essentially the load end VSWR of a hybrid feed were it used.

The plot shows that VSWR dips close to four pre-WARC HF amateur bands.

The optimisation challenge is to steer these minima into the desired bands, especially the lower ones where the dips are narrowest, the low Zo line can easily account for most of the loss.

Fractional G5RV antennas seem very popular in the US market, and they appeal to hams wanting multi band performance in small space.

One of the offerings is the quarter size G5RV, commonly marketed as the G5RV Mini.

The original concept set out by G5RV was a combination of a centre fed dipole and open wire transformation section to successfully deliver a lowish VSWR(50) on several of the pre-WARC bands. This enabled arbitrary length low Z feed extension to the transmitter, and allowed direct attachment to transmitters of the common design of the day (1950s).

The quarter size G5RV is as the name suggests one quarter length of both the dipole and the transformation section. Radio Oasis shows dimensions of a 25.5′ dipole and 8′ transformation section.

An NEC model was constructed assuming Wireman 551 nominal 450Ω tranformation section, though ignoring loss. The dipole is made of 2mm diameter copper and is 10m above ‘average’ ground (σ=0.005, εr=13).

Above is a plot of impedance Z looking into the transformation section, swept from 10-35MHz. As expected, there is a high impedance resonance around 21MHz, and low Z resonances around 13.5 and 31MHz. It is the latter that offers the best prospect of low VSWR(50).

Above is the VSWR(50) plot with minima at 13.9 (VSWR=3) and 31.5MHz (VSWR=1.04).

The combination could be shortened a little to move the 13.9MHz VSWR minimum up to say 14.2MHz, and of course that will push the 31.5MHz resonance higher.

Above, the dipole shortened a little moves the minimum VSWR up to 14.2MHz, and the next minimum even further out of the 10m band. VSWR at 28.5MHz is 8.4, and higher in all the bands between that and 14MHz.

This configuration does not deliver G5RV’s designed lowish VSWR on multiple bands, and the VSWR=3.3 on 20m is achieved by slight tuning and is barely lowish. It perhaps could be described as a quite poor monoband antenna.

- Duffy, O. 2001. RF Transmission Line Loss Calculator (TLLC). VK1OD.net (offline).
- ———. Oct 2005. Feeding a G5RV.
- ———. May 2006. Optimising a G5RV.
- ———. 2008. A model of a practical Guanella 1:1 balun. https://owenduffy.net/files/GuanellaBalun01.pdf.
- ———. 2010. Additional loss due to VSWR.
- ———. 2011. Estimating parameters of two wire transmission lines
- Varney, Louis. July 1958. An effective multi-band aerial of simple construction In RSGB Bulletin July 1958.

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Having skimmed a presentation published on the net, an interesting example is presented of an 80m half wave centre dipole with feed line and various plots from the nanoVNA used to illustrate the author’s take on things.

The author is obsessed with resonance and obsessed with phase, guiding the audience to phase as ‘the’ optimisation target. Phase of what you might ask… all the plots the author used to illustrate his point are phase of s11.

I have constructed an NEC-4.2 model of a somewhat similar antenna to illustrate sound concepts. Since NEC-4.2 does not model lossy transmission lines (TL elements), we will import the feed point data into Simsmith to include transmission line loss in the model.

Above is the Simsmith model.

The dipole is fed with around 20m (66′) of RG58A/U 50Ω line with vf=0.67.

Above is the Smith chart looking into the feed line (click for a larger image).

There are three markers, from right to left:

- 3.738MHz: phase of s11 is very low (0.006°, approximately zero), Z=129.9+j0.0153Ω, VSWR=2.60;
- 3.636MHz: phase of s11 is very low (0.17°, approximately zero), Z=74.18+j0.0902Ω, VSWR=1.48; and
- 3.600MHz: phase of s11 is nowhere near zero (30.8°), Z=63.22+j9.443Ω, VSWR=1.33

Lets look at the power delivered to the antenna.

Above, we see that the power is maximum at approximately the same frequency as where VSWR is minimum. That is no coincidence, standing waves result in higher line loss in this scenario.

Again the same three markers, and the two frequencies where phase of s11 is approximately zero have higher loss, less power reaching the antenna, than the one at minimum VSWR where phase of s11 is 30°.

It turns out that with practical feed lines and antenna conductors with this type of antenna, the loss in the antenna conductors is small compared to the feed line:

- dipole radiation efficiency is not very sensitive to frequency, small departure from resonance does not change the radiation efficiency of the dipole itself much; and
- feed line loss will be lowest where VSWR is minimum.

So, this antenna system will have best radiation efficiency at 3.6MHz because feed line losses are least.

Where the reference impedance for calculation of the complex reflection coefficient Γ or s11 is real, then when the angle of gamma is zero at some point, the ratio or V/I or impedance at that point is purely real (ie, zero reactance).

That says nothing for the value of the resistance, nor of the magnitude of Γ or s11 or of the VSWR.

Further, when the magnitude of s11 is very small (and VSWR is close to unity), the angle of Γ or s11 as measured is dominated by instrument noise and at the limit is essentially a random number and is meaningless.

So any optimisation based simply on angle of Γ or s11 is naive in the extreme.

Optimisation for phase of load impedance at the source equal to zero means reactance X is equal to zero, but says nothing about the resistance component, and therefore about VSWR.

In the example above, at the frequency of minimum VSWR (least line loss, maximum power delivered to the antenna) the impedance looking into the line has a phase angle of -8.5°.

So any optimisation based simply on angle of impedance seen by the source is naive in the extreme.

It was stated above that the angle (or phase) of s11 or Γ is not the same as the angle (or phase) of Z.

Given Zo and Γ, we can find θ, the angle of Z.

\(

Z=Z_0\frac{1+\Gamma}{1-\Gamma}\)

Zo and Γ are complex values, so we will separate them into the modulus and angle.

\(

\left | Z \right | \angle \theta =\left | Z_0 \right | \angle \psi \frac{1+\left| \Gamma \right | \angle \phi}{1-\left| \Gamma \right | \angle \phi} \\

\theta =arg \left ( \left | Z_0 \right | \angle \psi \frac{1+\left| \Gamma \right | \angle \phi}{1-\left| \Gamma \right | \angle \phi} \right )\)

We can see that the θ, the angle of Z, is not simply equal to φ, the angle of Γ, but is a function of four variables: \(\left | Z_0 \right |, \psi , \left| \Gamma \right |, \& \: \phi\) .

It is true that if ψ=0 and φ=0 that θ=0, but that does not imply a wider simple equality. This particular combination is sometimes convenient, particularly when ψ=0 as if often the case with a VNA.

So, pursuit of:

- feed point X equal to zero;
- feed point phase of Z equal to zero; and
- some notion of ‘resonance’ at the feed point

for this type of antenna system are all misguided.

Claims that phase alone is some magic quantity that drives optimisation is misguided.

This type of antenna system can be optimised with the nanoVNA, and the optimisation target is VSWR.

Read widely, don’t accept plausible looking explanations that appeal simply because they appeal, without understanding them, specious works are widespread and swallowed by the gullible.

]]>Above, the core is 35x21x13mm, a mid sized core, two used in my redesign of a commercial balun and implemented by VK4MQ . The mid size limits dissipation, but compactness can be an advantage. The cores sell for less than $4.00 per core and are readily available in Australia.

The core is almost certainly made in China, and Jaycar does not publish complex permeability curves for the material, but above are my measured characteristics over HF. The Chinese factor does raise questions about continued supply of consistent quality product.

Above is a plot of Fair-rite’s complex permeability for #31 ferrite material.

If you compare the two, they are not identical, but are very similar and you could conclude that applications where #31 is a good material selection would be well served by L15. Notably #31 is a MnZn ferrite and the L15 appears to be a NiZn ferrite based on its very high resistivity.

It is interesting to observe the fashions in online discussions of the best balun material

that the current fashion amongst online experts is #31. #31 is certainly a good candidate for applications with emphasis on lower HF, but its suitability for a specific applications needs also to consider other factors like its loss.

For the same reason, Jaycar’s LO1238 using L15 material may be quite suitable to those type of applications.

]]>Is this Segal’s law at play?

There are several common contributors including:

- faulty, dirty, or not properly mated connectors and cables;
- VSWR meters that are not accurate at low power levels; and
- influence of the common mode current path on VSWR.

The first and obvious question is, are all the cables and connectors sound and properly tightened? Some types of connector (eg UHF series, SMA) depend on tightness of the screw ring / nut for proper electrical contact of the outer conductor. An aerosol can of Isopropyl Alcohol (IPA) and some cleaning swabs / brushes are very handy to ensure connectors are clean.

Should you trust your VSWR meter – detector linearity discusses the second issue.

To the third issue mentioned, if you truly want to compare the two instruments, you must measure EXACTLY the same thing… and that means that when you disconnect the coax from the back of the radio and connect it to the analyser, you must RESTORE the common mode current path. I usually do this by holding the analyser coax connector (with antenna cable attached) firmly against the transceiver connector outer threads, and preferably isolating my fingers from the metal using an insulating sheet.

Above, an example of the nanoVNA with minimal adapters to UHF series, antenna patch lead attached and the shell of the connector held in good contact with the transceiver connector at far left. The white sheet is a silicone sheet to insulate my hand from the other stuff so that it is as close to the operating configuration as reasonably possible.

In this case, the minimum VSWR reconciles very well with the IC-7300 VSWR meter, provided the tx power is more than 60W.

]]>Let’s work an example using Simsmith to do some of the calculations.

Scenario:

- 20m ground mounted vertical base fed against a 2.4m driven earth electrode @ 0.5MHz;
- 10m RG58A/U coax; and
- Receiver with 500+j0Ω ohms input impedance and Noise Figure 20dB.

An NEC-4.2 model of the antenna gives a feed point impedance of 146-j4714Ω and radiation efficiency of 0.043%, so radiation resistance \(Rr=146 \cdot 0.00043=0.0063\).

Above, the NEC antenna model summary.

Above is a Simsmith model of the system scenario.

R1 and G model the antenna, G uses Rr for Zo, and R1 contains the balance of the feed point impedance.

With the useZo source type, the source would deliver 1W or 0dBW to a conjugate matched load.

The next important figure is the power into the 500Ω load L. it is -58.3dBW. Simsmith has calculated the solution to the antenna loss elements, mismatches and coax loss under standing waves. Effectively, the average gain of the antenna system (everything to the right of L) is -58.3dB. Such an antenna is likely to have a Directivity of around 4dB, in fact the NEC model calculates 4.8dB. So the maximum gain is -58+4.8=-53.2dB.

The burning question is whether it is sufficiently good to hear most signals. Well, a better question is how much does it degrade off-air signal to noise ratio (S/N). All receivers degrade S/N, but how much degradation occurs in this scenario.

We need to think about the ambient noise. Lets use ITU-R P.372 for guidance on the expected median noise in a rural precint.

Above, ambient noise figure @ 0.5MHz is 75.54dB.

Now lets calculate the Signal to Noise Degradation (SND).

At 4.58 dB it is not wonderful, the weakest signals (ie those with low S/N) we be degraded significantly, stronger signals (those with high S/N) will be degraded by the SAME amount, but for instance reducing S/N from 20 to 15dB is not so significant.

Applying this to your own scenario

The information fed into the calculations included:

- Rr;
- feed point impedance;
- transmission line details;
- Rx input impedance and NF; and
- Ambient noise expectation.

To calculate your own scenario, you need to find these quantities with some accuracy.

Tools:

]]>Should we have expected this outcome?

Let us solve a very similar problem analytically where measurement errors do not contribute to the outcome.

Taking the load impedance to be the same 10.1+j0.2Ω, and calculating for a T match similar to the MFJ-949E (assuming L=26µH, QL=200, and ideal capacitors) with Simsmith we can find a near perfect match.

The capacitors are 177.2 and 92.9pF for the match.

Also calculated is the impedance looking back from the load to the source shown here as L_revZ. The impedance looking back towards the 50Ω load is 17.28-j0.6216Ω, which is quite close to the value obtained by measurement, 18.0-j0.8Ω (which is dependent on the actual Q of the ATU elements).

Is there some smoke and mirrors in calculation of L_revZ? Lets turn the network around.

Now turning the network around by swapping the capacitors and changing the load to 50+j0Ω.

Above, the impedance looking back towards the 50Ω load is 17.28-j0.62Ω, which consistent with the L_revZ calculation and is quite close to the value obtained by measurement, 18.0-j0.8Ω (which is dependent on the actual Q of the ATU elements).

So, in answer to the question Should we have expected this outcome?

, the answer is yes, it is not surprising and quite similar to what we might expect from a network of this type.

Walt Maxwell’s Conjugate Mirror (Maxwell 2001 24.5) which imbues a magic system wide conjugate match with certain benefits, a utopia, which does not apply to systems that include any loss, it does not apply to real world systems. Maxwell does not state that limitation of his proposition.

Is a ham transmitter conjugate matched to its load? Watch for a follow up post.

- KL7AJ on the Conjugate Match Theorem
- Maxwell, Walter M. 2001. Reflections II. Sacramento: Worldradio books.
- Kl7ajConjugateMatch Simsmith models

]]>

- an inverted V dipole;
- Guanella 1:1 balun; and
- a ‘tuned’ length of RG6/U CCS coax.

The antenna system will be centred on 7.080MHz to suit my own operating preferences.

The coax is that featured at nanoVNA – RG6/U with CCS centre conductor MLL measurement and the matched line loss is taken from measurement as 4.1dB/100m @ 7.1MHz (all conductor loss). The feed line cost $50 for 100m incl delivery, so this project uses $12 worth of cable.

The broad concept is that the dipole is tuned a little shorter than a half wavelength to excite a standing wave on the coax. The VSWR desired is a little over 1.5, and the length of the coax is tuned so that the impedance looking into the coax is close to 50+j0Ω. “A little over” is so that the VSWR at the source end is very close to 1.5.

Above, the topology of the Inverted V Dipole with modelled current distribution in green. The apex of the dipole is at 11m and it is over ‘average ground’ (σ=0.005 εr=13).

A guanella 1:1 balun is incorporated in the Simsmith model by way of 1m of lossless 110Ω transmission line with VF=0.9. This is to model a balun wound with twisted pair, eg the balun shown above which is wound with XLPE insulated wire which is lower loss and better voltage withstand than PVC.

Let’s look at the form of the solution in Simsmith, well it is the actual solution but we will talk just about the form of the solution first.

Above, the specified load is the impedance looking into the dipole centre from the NEC model. T1 is a length of nominally 75Ω transmission with some loss (4.1dB/100m @ 7MHz as per previous measurement), note the spiral due to line loss. T2 represents the balun by way of 1m of 110Ω transmission line with VF=0.9. So this is the form of the solution, a shortened dipole with some significant -ve reactance such as to launch a VSWR(75)≈1.5 standing wave on the 75Ω line, and the line is cut where the impedance looking into the line is 50+j0Ω. From the chart, feed line loss under the specific mismatch conditions is 0.958dB, or about 20%.

Now to the NEC model to design the antenna.

Iteratively it was found that a leg length of 9.985m gave the desired input impedance for the desired input end VSWR on the 75Ω feed line.

Above, a summary of the model.

Above is the gain pattern from the model, but note that there is a small loss in the feed line not included in the model. The pattern is typical of a low Inverted V Dipole, maximum gain at the zenith.

In a world where the only thing of merit is DX, then a flat top dipole has a slightly better gain at low elevation… but if you don’t want to hear the DX contesters then the ‘cloud warmer’ pattern has relative advantage of ‘local’ contacts.

Let’s look again at the Simsmith model swept over a frequency range.

The load impedance is exactly that drawn from the NEC model (in fact it is imported from the NEC output file), and the feed line configured to match the RG6/U CCS. The feed line length was adjusted to minimise VSWR(50) looking into the line.

To find the SYSTEM radiation efficiency, we must multiply the NEC model radiation efficiency (70.05%) by the transmission line efficiency (80.2%) to obtain 56.2%. That is a moderate system radiation efficiency for an antenna of this type, and there is 20% of power lost in the 20m of quite low cost feed line.

Above is the VSWR curve from the Simsmith model, VSWR 1.5 bandwidth is well over 200kHz, so it is quite a practical matching scheme.

The antenna can be used without an ATU, operationally an advantage and avoiding the need for a costly ATU if high power is involved. Whilst QRP ATUs are less expensive, they also tend to be less efficient… so you achieve QRP^2 when using them.

Note that this is designed as a single band antenna, the matching scheme uses a ‘tuned’ length of coax.

Comparing this antenna with Applying the RG11A/U to a 40m Inverted V Dipole antenna, system radiation efficiency is 56% vs 66%, about 0.7dB poorer. With almost identical gain pattern, gain is likewise 0.7dB poorer.

]]>- an inverted V dipole;
- Guanella 1:1 balun; and
- a ‘tuned’ length of RG11A/U CCS coax.

The antenna system will be centred on 7.080MHz to suit my own operating preferences.

The coax is that featured at Checkout of a roll of Commscope 4510404 CCS RG11A/U – Zoc, Zsc based MLL calculation and the matched line loss is taken from measurement as 1.2dB/100m @ 7MHz (all conductor loss). The feed line cost $99 for 305m incl delivery, so this project uses $6.50 worth of cable. The feed line is not good because it is cheap, it is good because it suits the application very well, and as a bonus, it is inexpensive.

The broad concept is that the dipole is tuned a little shorter than a half wavelength to excite a standing wave on the coax. The VSWR desired is a little over 1.5, and the length of the coax is tuned so that the impedance looking into the coax is close to 50+j0Ω. “A little over” is so that the VSWR at the source end is very close to 1.5.

Above, the topology of the Inverted V Dipole with modelled current distribution in green. The apex of the dipole is at 11m and it is over ‘average ground’ (σ=0.005 εr=13).

A guanella 1:1 balun is incorporated in the Simsmith model by way of 1m of lossless 110Ω transmission line with VF=0.9. This is to model a balun wound with twisted pair, eg the balun shown above which is wound with XLPE insulated wire which is lower loss and better voltage withstand than PVC.

Let’s look at the form of the solution in Simsmith, well it is the actual solution but we will talk just about the form of the solution first.

Above, the specified load is the impedance looking into the dipole centre from the NEC model. T1 is a length of nominally 75Ω transmission with some loss (1.2dB/100m @ 7MHz as per previous measurement), note the spiral due to line loss. T2 represents the balun by way of 1m of 110Ω transmission line with VF=0.9. So this is the form of the solution, a shortened dipole with some significant -ve reactance such as to launch a VSWR(75)≈1.5 standing wave on the 75Ω line, and the line is cut where the impedance looking into the line is 50+j0Ω. From the chart, feed line loss under the specific mismatch conditions is 0.276dB, or about 6%.

Now to the NEC model to design the antenna.

Iteratively it was found that a leg length of 10.03m gave the desired input impedance for the desired input end VSWR on the 75Ω feed line.

Above, a summary of the model.

Above is the gain pattern from the model, but note that there is a small loss in the feed line not included in the model. The pattern is typical of a low Inverted V Dipole, maximum gain at the zenith.

In a world where the only thing of merit is DX, then a flat top dipole has a slightly better gain at low elevation… but if you don’t want to hear the DX contesters then the ‘cloud warmer’ pattern has relative advantage of ‘local’ contacts.

Let’s look again at the Simsmith model swept over a frequency range.

The load impedance is exactly that drawn from the NEC model (in fact it is imported from the NEC output file), and the feed line configured to match the Commscope 4510404 CCS RG11A/U. The feed line length was adjusted to minimise VSWR(50) looking into the line.

To find the SYSTEM radiation efficiency, we must multiply the NEC model radiation efficiency (70.04%) by the transmission line efficiency (93.8%) to obtain 65.7%. That is a pretty good system radiation efficiency for an antenna of this type, and there is only 6% of power lost in the 20m of quite low cost feed line.

Above is the VSWR curve from the Simsmith model, VSWR 1.5 bandwidth is over 200kHz, so it is quite a practical matching scheme.

The antenna can be used without an ATU, operationally an advantage and avoiding the need for a costly ATU if high power is involved. Whilst QRP ATUs are less expensive, they also tend to be less efficient… so you achieve QRP^2 when using them.

Note that this is designed as a single band antenna, the matching scheme uses a ‘tuned’ length of coax.

]]>The screenshot above is of SpectrumLab waterfall display of the receiver output and a view of the position of relevant aircraft.

All three signals show Doppler shift

Two of the aircraft are behind the path, ie a backscatter path.

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