Let’s demonstrate the measurement of Rin of an o/c resonant section around the 160m band, which we will then use to calculate MLL.
Above, the AA-600 connected to the cable using a F(F)-N(M) adapter, the cable is 305m in length and the far end is open circuit.
So lets scan around 1800kHz looking for a convenient resonance.
Ok, there is a resonance just a bit above 1800kHz. Let’s move the cursor using the freq+ arrow key, press the range- key a few times to narrow the scan and press to rescan.
Ok, closer now, but overshot a bit. Let’s move the cursor again and switch to All mode.
Above is All mode, X is just greater than zero, so we will shift freq down using the freq- arrow key watching X for the zero crossing.
Above, this is close enough. We can read Rin as 19.0Ω.
Above, calculating MLL from the input data we have 0.0074dB/m or 0.74dB/100m.
This whole procedure takes less than a minute using the stand alone instrument… in part due to the very effective human interface on the AA-600, you might think the designer knew how the instruments would get used, something you cannot say for a lot of analysers and VNAs (which are now the fashion).
The technique is particularly useful if the far end of the cable is not accessible, eg it is sometimes hidden in the interior of the drum.
(Ikin 2016) proposes a different method of measuring noise figure NF.
Therefore, the LNA noise figure can be derived by measuring the noise with the LNA input terminated with a resistor equal to its input impedance. Then with the measurement repeated with the resistor removed, so that the LNA input is terminated by its own Dynamic Impedance. The difference in the noise ref. the above measurements will give a figure in dB which is equal to the noise reduction of the LNA verses thermal noise at 290K. Converting the dB difference into an attenuation power ratio then multiplying this by 290K gives the LNA Noise Temperature. Then using the Noise Temperature to dB conversion table yields the LNA Noise Figure. See Table 1.
The explanation is not very clear to me, and there is no mathematical proof of the technique offered… so a bit unsatisfying… but it is oft cited in ham online discussions.
I have taken the liberty to extend Ikin’s Table 1 to include some more values of column 1 for comparison with a more conventional Y factor test of a receiver’s noise figure.
Above is the extended table. The formulas in all cells of a column are the same, the highlighted row is for later reference.
A test setup was arranged to measure the noise output power of an IC-7300 receiver which has a sensitivity specification that hints should have a NF≅5.4dB. The relative noise output power for four conditions was recorded in the table below.
Ikin’s method calls for calculating the third minus second rows, -0.17dB, and looking it up in his table. In my extended table LnaNoiseDifference=-0.17dB corresponds to NF=3.10dB.
We can find the NF using the conventional Y factor method from the values in the third and fourth rows.
The result is NF=5.14dB (quite close to the expected value based on sensistivity specification).
Ikin’s so called dynamic impedance method gave quite a different result in this case, 3.10 vs 5.14dB, quite a large discrepancy.
The chart above shows the relative level of the four measurements. The value of the last two is that they can be used to determine the NF using the well established theory explained at AN 57-1.
The values in the first columns are dependent on the internal implementation of the amplifier, and cannot reliable infer NF.
Let’s review of the concepts of noise figure, equivalent noise temperature and measurement.
Firstly let’s consider the nature of noise. The noise we are discussing is dominated by thermal noise, the noise due to random thermal agitation of charge carriers in conductors. Johnson noise (as it is known) has a uniform spectral power density, ie a uniform power/bandwidth. The maximum thermal noise power density available from a resistor at temperature T is given by \(NPD=k_B T\) where Boltzman’s constant k_{B}=1.38064852e-23 (and of course the load must be matched to obtain that maximum noise power density). Temperature is absolute temperature, it is measured in Kelvins and 0°C≅273K.
Noise Figure NF by definition is the reduction in S/N ratio (in dB) across a system component. So, we can write \(NF=10 log \frac{S_{in}}{N_{in}}- 10 log \frac{S_{out}}{N_{out}}\).
One of the many methods of characterising the internal noise contribution of an amplifier is to treat it as noiseless and derive an equivalent temperature of a matched input resistor that delivers equivalent noise, this temperature is known as the equivalent noise temperature Te of the amplifier.
So for example, if we were to place a 50Ω resistor on the input of a nominally 50Ω input amplifier, and raised its temperature from 0K to the point T where the noise output power of the amplifier doubled, would could infer that the internal noise of the amplifier could be represented by an input resistor at temperature T. Fine in concept, but not very practical.
Applying a little maths, we do have a practical measurement method which is known as the Y factor method. It involves measuring the ratio of noise power output (Y) for two different source resistor temperatures, Tc and Th. We can say that \(NF=10 log \frac{(\frac{T_h}{290}-1)-Y(\frac{T_c}{290}-1)}{Y-1}\).
AN 57-1 contains a detailed mathematical explanation / proof of the Y factor method.
We can buy a noise source off the shelf, they come in a range of hot and cold temperatures. For example, one with specified Excess Noise Ratio (a common method of specifying them) has Th=9461K and Tc=290K. If we measured a DUT and observed that Y=3 (4.77dB) we could calculate that NF=12dB.
This method of noise figure measurement is practical and used widely. Note that the DUT always has its nominal terminations applied to the input and output, the system gain is maintained, just the input equivalent noise temperature is varied.
Some amplifiers are not intended to be impedance matched at the input (ie optimised for maximum gain), but are optimised for noise figure by controlling the source impedance seen at the active device. Notwithstanding that the input is not impedance matched, noise figure measurements are made in the same way as for a matched system as they figures are applicable to the application where for example the source might be a nominal 50Ω antenna system.
So, NF is characterised for an amplifier with its intended / nominal source and load impedances.
Nothing about the NF implies the equivalent internal noise with a short circuit SC or open circuit OC input. The behaviour of an amplifier under those conditions is internal implementation dependent (ie variable from one amplifier design to another) and since it is not related to the amplifier’s NF, it is quite wrong to make inferences based on noise measured with SC or OC input.
So this raises the question of NF measurements made with a 50Ω source on an amplifier normally used with a different source impedance, and possibly a frequency dependent source impedance. An example of this might be an active loop amplifier where the source impedance looks more like a simple inductor.
Well clearly the measurement based on a 50Ω source does not apply exactly as amplifier internal noise is often sensitive to the source impedance, but for smallish departures, the error might be smallish.
A better approach might be to measure the amplifier with its intended source impedance. In the case of the example active loop antenna, the amplifier could be connected to a dummy equivalent inductor, all housed in a shielded enclosure and the output noise power measured with a spectrum analyser to give an equivalent noise power density at the output terminals. Knowing the AntennaFactor of the combination, that output power density could be referred to the air interface. This is often done and the active antenna internal noise expressed as an equivalent field strength in 1Hz, eg 0.02µV/m in 1Hz. For example the AAA-1C loop and amplifier specifies Antenna Factor Ka 2 dB meters-1 @ 10 MHz
and MDS @ 10MHz 0.7 uV/m , Noise bandwidth =1KHz and
to mean equivalent internal noise 0.022µV/m in 1Hz @ 10MHz at the air interface. 0.022µV/m in 1Hz infers Te=6.655e6K and NF=43.608dB again, at the air interface. These figures can be used with the ambient noise figure to calculate the S/N degradation (SND).
A spectrum analyser or the like can be used to measure the total noise power density at the output of the loop amplifier with the input connected to a dummy antenna network (all of it shielded) and to calculate the equivalent noise temperature and noise figure at that point. For example, if we measured -116dBm in 1kHz bandwidth, Te=1.793e+5K and NF=27.9dB. Knowledge of the gain from air interface to that reference point is needed to compare ambient noise to the internal noise and to calculate SND, that knowledge might come from published specifications or a mix of measurements and modelling of the loaded antenna.
The mention of a spectrum analyser invites the question about the suitability of an SDR receiver. If the receiver is known to be calibrated, there is no non-linear process like noise cancellation active, and the ENB of the filter is known accurately, it may be a suitable instrument.
In both cases, the instruments are usually calculated for total input power, ie external signal and noise plus internal noise, so to find external noise (ie from the preamp) allowance must be made for the instrument NF (ie it needs to be known if the measured power is anywhere near the instrument noise floor).
Field strength / receive power converter may assist in some of the calculations.
The foregoing discussion assumes a linear receiver, and does not include the effects of intermodulation distortion IMD that can be hugely significant, especially in poor designs.
Part of the problem of IMD is that the effects depend on the individual deployment context, one user may have quite a different experience to another.
There are a huge number of published active loop antenna designs and variant, and a smaller number of commercial products. Most are without useful specifications which is understandable since most of the market are swayed more by anecdotal user experiences and theory based metrics and measurement.
Online posters are excited that it supports some versions of nanoVNA, and one thread attempts to answer the questions:
The SWR image shows that the SWR minimum is at the center phase angle as you would expect. My question is:
what are the other points that look like resonance,
and should I trim my antenna based on phase?
If so which one?
They are interesting questions which hint the ham obsession with resonance as an optimisation tartget.
Properly interpreting VNA or analyser measurements starts with truly understanding the statistic being interpreted.
In this case, the statistics being discussed are Phase and VSWR, and their relationship.
What is the Phase being discussed?
Above is an Antscope2 phase plot for an archived antenna measurement. The measurements are of a 146MHz quarter wave mobile antenna looking into about 4m of RG58C/U cable. We will come back to this.
Above is the SWR (VSWR) plot.
Let’s analyse both of these.
What is VSWR?
Well you do need to understand the meaning of VSWR, the context, and expected values for the type of antenna being measured.
Minimum VSWR occurs at about 147.625MHz in this case, and at 1.3, it is quite within expectations for this type of antenna.
What is phase?
Let’s assume the reported complex Z at the cursor in the phase plot is correct, and calculate some interesting values from it.
Above, the phase of Γ is -160.6° which reconciles with the plotted phase… so it would seem that the plot of phase is actually a plot of the phase of the complex reflection coefficient Γ.
Is that useful in the context of the questions posed above about optimising VSWR and resonance?
No.
Above is the phase plot from Antscope (1) for the same file.
Phase is reported at 147.625 as -11.6°.
Can Antscope(1) and Antscope2 be plotting the same thing on their respective phase plots?
No. Antscope(1) plots the argument of Z and Antscope2 plots the argument of Γ.
Are either of them directly useful to determine if the antenna itself is resonant?
No. We are looking into a feed line, and without de-embedding the feed line from the measurements, neither phase directly indicates resonance of the antenna itself.
In fact, for a quarter wave ground plane antenna, minimum VSWR occurs very close to resonance, and feed line losses are least at minimum VSWR so it should be the optimisation target.
]]>Warned of a potential quality issue, I measured my own AA-600.
Above, the test of the inner pin forward surface distance from the reference plane on the N jack on the AA-600. The acceptable range on this gauge for the female connector is the red area, and it is comfortably within the red range.
Above is a table of critical dimensions for ‘ordinary’ (ie not precision) N type connectors from Amphenol.
This dimension is important, as if the centre pin protrudes too much, it may damage the mating connector.
Pleased to say mine is ok, FP at 0.192″.
I used a purpose made gauge to check this, but it can be done with care with a digital caliper (or dial caliper or vernier caliper), that is what I did for decades before acquiring the dial gauge above.
]]>quite normal, this article suggests there is an obvious possible explanation and that to treat it as quite normal may be to ignore the information presented.
Above is a partial simulation of a scenario using Rigexpert’s Antscope. It starts with an actual measurement of a Diamond X-50N around 146MHz with the actual feed line de-embedded. Then a 100m lossless feed line of VF=0.66 is simulated to produce the plot that contains a ripple apparently superimposed on an expected V shaped VSWR curve.
This is the type of ripple that the expert’s opine is quite normal.
Let’s apply some thinking to this.
The ripple is periodic with frequency, the pattern repeats about every 1MHz around 146MHz, or about 0.68% of frequency, or 1/146 of frequency.
The question that should be asked is, is this periodicity related to the electrical length of the feed line? Recall that VSWR patterns repeat every half wavelength along a transmission line, so we are interested in the feed line length in half wavelengths.
The feed line is 73.8λ at 146MHz, or 147.6 half wavelengths. This means that with a frequency change of 1/147.6, we will increase the electrical length by one half wavelength.
In this case, the frequency periodicity of the plotted VSWR curve is approximately the same as the frequency periodicity of the electrical length of the line, and is a likely explanation of the VSWR curve if the reference impedance for the VSWR plot is significantly different to the characteristic impedance of the transmission line section.
Since Zref of the instrument is 50Ω we might properly suspect that the VSWR plot ripple is caused by depart of the transmission line from 50Ω and using Antscope, explore other Zrefs to minimise the ripple.
Above is a plot with Zref=45Ω, it is more like what might be expected, and it suggests that the transmission line has Zo=45Ω and that is the cause of the ripple.
If the transmission line was specified to be 50Ω, analysis of the VSWR ripple suggests the transmission line does not meet specification and warrants a more conclusive test by placing a known frequency independent load at the far end and measuring VSWR, observing ripple and finding Zref that minimises the ripple.
Of course, the VSWR ripple may be caused because Zref of the instrument is not correct, so using an instrument that is incorrectly calibrated might produce similar ripple.
Above is a simulation where the 100m transmission line Zo=50Ω and Zref=45Ω. Check that instrument Zref properly suits the transmission line before making measurement.
]]>This has been commented on by online experts stating that Hugen, the designer of this board, posted notes about his efforts to keep the grounds for tx and rx port circuits isolated to some extent.
Opinion by some is that the modification I performed above which electrically bonds the two connectors through a brass bar of about 60mm length is likely to significantly degrade performance.
Let us look at currents that might be induced in the outside surface of the shield in a typical test setup. Unavoidable ambient RF fields will induce a voltage in the loop circuit formed by the outer surface of the shield.
In the case of the unmodified nanoVNA-H, the extraneous shield currents flow approximately around the green path, traveling though the ground planes of tx and rx port circuitry before being joined approximately in the middle of the instrument.
In the case of the modified nanoVNA-H, the extraneous shield currents flow approximately around the magenta path, with far less of it flowing in through the ground planes of tx and rx port circuitry and creating ground conductor noise voltage.
It is difficult to understand the claim that the green current path should be better, and that the magenta current path degrades performance. Rather one might question the wisdom of isolating the tx and rx port grounds to drive these currents further into the box.
Of course I defer to credible measurement that shows that the modification does degrade performance.
]]>Above is a demonstration circuit in Simsmith, a linear source with Thevenin equivalent impedance of 50-j5Ω. The equivalent voltage is specified by useZo, which like much of Simsmith is counter intuitive (as you are not actually directly specifying generator impedance):
Vthev and Zthev are chosen so that ‘useZo’ will deliver 1 watt to a circuit impedance that equals the G.Zo. Zthev will be Zo*.
So, in a lossless conjugate matched circuit we would expect the load power to be 0dBW.
In the circuit above, the load is not the conjugate of the source impedance so there will be some mismatch loss, to mean the ratio of power in a matched load to the power in the mismatched load.
In this scenario, the calculated power in the mismatched load is -0.307dBW, so mismatch loss is 0 – -0.307=0.307dB.
This mismatch load is often calculated as \( ML=10\cdot log_{10}(\frac1{1-|Γ|^2}) \) where Γ is calculated wrt 50+j0Ω, and the value is shown above at ML as 0.3856dB, quite different to the actual mismatch loss.
KML is mismatch loss calculated using Kurokawa’s power reflection coefficient, and at 0.3069dB (rounds to 0.307dB) it reconciles with the 0.307dB calculated from the displayed power levels.
The expressions used for the displayed G quantities are show above.
What if we could introduce a circuit element that hid the true nature of the source and made it look like an ideal 50+j0Ω source?
The scenario above inserts an ideal isolator between source and load (the N block as it is known in Simsmith).
Note that this is not your Hammy Sammy isolator. Isolator is one of the many terms with a well understood meaning in industry, appropriated by hams and give new / incompatible meaning just to confuse.
An isolator, this isolator, is a two port device that allows RF to flow in one direction and not the other for a specified Zo. So in this instance, Zo is specified as 50+j0Ω, and power can flow from source to load (with no loss), but no power can flow from load to source.
An isolator is characterised by \(S=\begin{bmatrix}0 & 0\\1 & 0\\\end{bmatrix}\) using conventionial notation, but Simsmith does not follow convention, you must transpose P1 and P2.
Above is Simsmith’s “backwards circuit” display option, the departure from convention in port labelling is more obvious, so to understand things in the Simsmith world, \(S_{simsmith}=\begin{bmatrix}0 & 0\\0 & 1\\\end{bmatrix}\) but to program it, you can undo the port transposition in the wiring and use the conventional S matrix, see below.
//Isolator P1 w2 gnd; P2 w1 gnd; sprm1 {w1 gnd w2 gnd} {{0,0},{1,0}} {50};
A consequence of the isolator is that the output of the isolator (P1… yes, more backwards confusion from Simsmith) appears to have a source impedance of 50+j0Ω. The displayed quantity L_sZ is the calculated impedance looking from load L back towards the source, it is 50+j0.0001247Ω… slightly off, probably due to rounding errors in a fairly complicated calculation.
Note that the effect of the isolator is that there is no reflected energy (wrt isolator Zo), so the source sees a load of 50+j0Ω.
Power into the isolator is -0.0108dBW (mismatch loss at this node is 0.0108dB) and power into the load is 0.396dBW, mismatch loss at the load input is 0.385dB. Both calculated (traditional) ML and KML reconcile with the calculated power levels.
Is an isolator the magic component that can deliver a transmitter with Thevenin source impedance equal to 50+j0Ω? Well, low loss isolators are practical at microwaves, and devices can be made to cover HF, the do it with considerable loss. For an ordinary HF SSB telephony transmitter there is little benefit and severe cost and efficiency issues.
… or where a little knowledge is dangerous.
A common newby online question is “my 50Ω VSWR is 3, surely that is really bad?”
The expert answers tend to go along the line “Don’t worry, reflected power is 25%, which means your transmitted power is 75%, that is just 1.3dB lower, a quarter of an S-point and no one will notice it.” Some may provide a link to a handy dandy table of these magic values.
The experts have assumed the transmitter is well represented by a Thevenin equivalent circuit, and that Zs is 50+j0Ω, an unwarranted assumption. Not only is the transmitter unlikely to be sufficiently linear to apply linear circuit theory in that way, practical transmitters often include protection systems that may reduce power output to limit ‘reflected power’ to 10% of rated power or less.
The fact that in many transmitters, protection circuits would have kicked in is a salutary warning. Operation well outside of specified load range might well result in degraded distortion products.
Calculation of mismatch loss requires an understanding of the characteristics of the source.
It is unsafe to assume that a transmitter that is designed to work into a 50Ω load is well represented as a Thevenin source with Zs=50Ω.
Traditional mismatch formula fails under some load and source impedance combinations.
Kurokawa’s power reflection coefficient may be a usable metric for linear systems, provided that actual source and load impedance are known.
The answer is on the face of it quite simple.
Jacobi’s Maximum Power Transfer Theorem extended to alternating current tells us that in a linear circuit, maximum power is transferred from a source to its load when the load impedance is the complex conjugate of the Thevenin equivalent source impedance.
This would seem to have obvious direct application in getting maximum power radiated from a transmitter.
But in most practical cases there are two important points that invalidate this thinking:
In the days before common use of VSWR meters, it was common practice to use an antenna current meter or a remote field strength meter to indicate maximum radiation power than transmitter adjustments were made to maximise that (within limits such asdesign or specification limits on PA device current).
Of course maximising power radiated regardless of any and everything else is a fairly inadequate technique, but with its roots in A1 Morse code transmitters and a user base who were not progressive, it survived until the ‘magic’ of VSWR meters penetrated the user base.
Modern SSB telephony transmitters will usually be solid state and designed to work into some fixed nominal load impedance. Some incorporate a wider range matching network (often known as an Antenna Tuning Unit) for greater flexibility.
For the purpose of this discussion, the term ‘transmitter’ is taken to mean the power amplifier and any necessary filters, but does not include an internal Antenna Tuning Unit.
Transmitter design includes a very wide range of parameters, and whilst it is possible to control source impedance at the output, it has almost no advantage and a lot of disadvantage to other aspects of the implementation… so it is not usually done for this type of transmitter.
Transmitters are usually designed to suit a given range of load impedance, often specified as a nominal value with some notional VSWR tolerance, eg a nominal 50Ω antenna with maximum VSWR of 1.5. It should be safe to operate the transmitter into a load that satisfies that criteria, and the buyer might hope that it should deliver its rated performance including distortion performance. That said, it is not common practice to test at these limits, and in practice specified power, distortion etc might only occur at very close to the nominal load impedance. None of this is to imply that maximum output power occurs in the nominal load (eg 50+j0Ω).
So, to obtain the rated performance, the objective is to deliver a load that is within it rated range, and preferably as close to the nominal value as practicable.
Given the power delivered to that transmitter load, the matter of how much power is radiated falls to the other system elements.
That being the case, the first optimisation objective should be to deliver the transmitter its rated load. Next, address performance of the rest of the system in terms of maximising radiated power.
Is there room for mathematics and theory here?
The greatest problem in applying conventional linear circuit theory to the problem is that in most cases, ordinary ham SSB telephony transmitters are not well represented by a Thevenin equivalent circuit and analysis based on that is simply invalid.
There is a whole vocabulary that flags woolly thinking, conjugate matching, flywheel effect, mismatch loss, re-reflection, an alternate expression for Γ (the complex reflection coefficient), the new SWR* (conjugate SWR, yes complex conjugate of a scalar) to name a few.
There is discussion in pseudo maths, or junk maths pretending to be science. All grist for the mill for a hobby that is less and less science based with the passage of time.
The proceeding discussion did not need to talk about “happy” transmitters, antennas or other system components. Antropomorphism is a technique often used to ‘communicate’ with hams who do not have a clue… and often by those who also do not understand the problem and often want to preserve their status by having others satisfy their thirst for knowledge with their own inadequate understanding.
Specs for SMA connectors range from minimum of 0.2Nm torque to maximum of 1.7Nm, but 0.6Nm and 1.0Nm are common commercial practice.
Some nanoVNA sellers state:
As the SMA ports are made of cast copper, please not connect hard 50-7 / RG213 and other cables directly to the SMA ports through M-to-SMA connector to avoid damaging the SMA ports. You can use the included SMA cable to connect to the SMA port as shown in the picture below, and then use M to SMA connector.
Clearly Chinese Cheats, they will say anything to make a sale and anything to avoid commitment to quality. These connectors are very unlikely to be copper, but are likely to be a copper alloy: brass. What they also avoid in the above statement is claim for PCB damage due to flexure of the SMA connectors torqued to accepted industry torque for reliable connections and measurement.
Above is a pic of a modification to reinforce the connectors. This article sets out the analysis of just the solder joint within the cross section of the brass pieces.
A side effect is that this modification bonds the ground planes for the input and output parts of the nanoVNA via the brass bar where they have been kept isolated to some extent.
I should note that there has been much discussion online as to whether the noise floor of the nanoVNA is degraded by the shields fitted to the board, and various modifications to the shields. Some of this discussion proposes that the issue is mostly around the mixers and noise loops, and I note that in -H designs prior to v3.3, the mixer power supply was not adequately decoupled. It is possible that electrical connection of the SMA connectors in this way degrades noise performance at some frequencies. No significant change was observed in the noise floor of s11 or s21 channels from 1 to 300MHz (I don’t regard instrument performance to be good above 300MHz). I have not seen credible evidence of degradation of the nanoVNA-H v3.3 build.
If indeed bonding the two SMA connectors close to the instrument increases the noise floor or has other performance impacts as suggested, it questions whether the currents on the exterior of the coax influence measurement (which it should not) and it questions whether two port measurement fixtures or adapters should be attached close to the nanovna.
(See also Reinforcement of nanoVNA-H connectors – performance discussion.)
At first, the strength of the butt soldered joint might seem a simple case of beam analysis where the beam is of cast solder of the same cross section l x w as the soldered joint.
I think that is not a good model, because the very thin solder layer probably invalidates the assumption of a neutral axis in the middle of the section. My initial thinking is that the strength can be worked up assuming the the solder joint ‘hinges’ on the compression side of bending and the bending moment limit is that value that stretches the extreme fibres on the far side to the tensile strength of cast 60/40 solder.
On that basis, we can assume that the pressure within the solder rises linearly from zero at one side to maximum at the other, ie that \(p(x)={\sigma}\frac xl\).
So the total torque when loaded to σ is the sum of the pressure times area times distance for the incremental areas across the section.
\(T=\int^l_0{\sigma}\frac xlwxdx\)
\(T=\int^l_0{\sigma}\frac{wx^2}ldx\)
\(T=[\frac{{\sigma}wx^3}{3l}]_0^l\)
\(T=\frac{{\sigma}wl^2}{3}-0\)
Taking \({\sigma}=50MPa,w=0.0016,l=0.00635\):
\(T=\frac{5e7\cdot0.0016\cdot0.00635^2}{3}-0\)
\(T=1.075 Nm\)
So, it looks promising.
A test piece was constructed and tested to destruction at 1.8Nm.
It is a little stronger than expected, possibly due to the solder used, possibly contribution of the small excess outside the brass cross section, and it is only a single test sample. The experiment suggests that the analysis approach is better than treatment as a long slender beam (which would result in a lower predicted strength).
The final test is that the instrument has been in use for weeks with the reinforcement and connectors torqued to 1Nm with no signs of cracking or sponginess. Beyond this test period, it is my intention to tighten the connectors to just 0.6Nm (common commercial practice).
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