## Measuring Henrik's RF bridge

19.11.2020 17:03

I've recently made a copy of a bridge-based directional coupler that was published by Henrik Forstén on his blog. After some difficulties with soldering the components I now have one fully assembled device. The question is of course whether it also functions correctly. Along with his designs Henrik also published the results of some measurements of his couplers with a professional VNA. Unfortunately I don't have access to an instrument like that. Nonetheless I performed some similar measurements with the equipment I have at hand and tried to compare the performance of my copy with the original.

The first test I did was to measure the S-parameters. I replicated Henrik's measurements with a NanoVNA-H. I measured the coupler as a two port device. For each pair of ports I connected one port to CH0 and the other to CH1 while terminating the third port on the coupler with the calibration load that came with the NanoVNA.

Ports are numbered as following: 1 - DUT, 2 - IN and 3 - DET.

S12 is the through path from signal generator to the device-under-test. Ideally it should be 0 dB (I think a more correct value to show here would be S21, but Henrik shows S12 - in any case, the difference is minimal). S13 is the coupled path for the reflected wave to the detector. The design of the bridge sets this at -16 dB. S23 is unwanted coupling of the forward wave to the detector. Ideally it should be at minus infinity dB. My measurements are the thicker lines that go up to 1500 MHz (the limit of my NanoVNA). Henrik's measurements with a commercial VNA are the thinner lines. They extend up to 10 GHz so only a small part is shown.

As you can see, the gain measurements match almost perfectly for the small frequency range that I could cover. Directivity (S13 - S23) is better than 25 dB.

The reflection part of S-parameters doesn't match that well. Again, the bold traces that go to 1500 MHz are my measurements while the faded out lines are taken from Henrik's graphs. I'm seeing around 10 to 20 dB worse return loss.

I don't know whether this due to a problem with my measurement or a problem with my bridge construction. On one hand, I suspect that NanoVNA can't compensate for imperfect matching of CH1 in two-port measurements. It implements only one-half of a full two-port VNA and lacks the CH1 reflection measurement it needs to compensate for that. On the other hand, later measurements also show that impedance matching seem to be a persistent problem in my setups.

Next test I did was the same scalar directivity measurement that I did with the "Transverters Store" bridge. I used ERASynth Micro as a signal source connected to IN and a rtl-sdr DVB-T dongle as a power detector connected to DET. I measured the power when the DUT port was terminated with a short and when it was terminated with the calibration load from the NanoVNA set. Estimated directivity was the ratio of these two measurements.

Comparing the two bridges it appears that the Transverters bridge has a higher directivity at low frequencies. However above 1 GHz Henrik's bridge works better. Based on the construction of the two devices that kind of makes sense. I'm not sure what's with the directivity spiking up like that at 1500 MHz. Anyway, I don't plan to use this bridge for scalar measurements since I now have a vector setup. I was just curious how it compares to the old bridge.

Finally I performed a 1-port short-open-load calibration of my vector measurement setup using the new bridge and the NanoVNA calibration kit. The graph below shows the error network terms that I calculated from the calibration measurements using scikit-rf. Again, Henrik's results are shown as well for comparison.

This isn't a comparison of bridge performance but rather of the whole measurement system, since these error terms include the effects of other components as well. Obviously the two systems are completely different in these two cases, but still it gives some idea of how well my setup performs compared to Henrik's home-made VNA.

Note that the directivity in my previous scalar measurement is defined slightly differently than here. It is roughly equivalent to reflection tracking divided by directivity from the error terms graph. Slide 48 in the Agilent Network Analyzer Basics presentation has a good illustration of the meaning of the error terms.

A difference in reflection tracking is expected since my and Henrik's VNA probably have different overall gains in the system. I'm happy that it's reasonably flat, meaning that the losses in the bridge don't increase much towards higher frequencies.

My directivity seems slightly better and this result also shows a peak at around 1500 MHz that I saw in the scalar measurement. Source match is roughly 20 dB worse. This agrees with the NanoVNA measurements that have shown quite a bad reflection loss on all bridge ports and also my previous results showing that overall I have quite bad matching in the whole system. I might experiment a bit with this in the future and try to correct it.

For another comparison, here are the error terms calculated using the same method, but using the "Transverters Store" bridge. Here the bridge starts attenuating the signal towards higher frequencies, and hence losing the effective directivity. The source match seems to be better at low frequencies though.

In conclusion, my copy of the bridge design seems to work just fine. Its effective directivity is around 25 dB in the range I was able to measure. The multiplex board doesn't seem to affect it much and the only problem seems to be relatively poor 50 Ω matching, but that might be because of my other equipment. For low frequency measurements below 1 GHz using the Transverters Store bridge should yield better results. A true test of the new bridge however would be at higher frequencies, beyond the 2 GHz that my current setup is able to handle. The only limiting factor is the rtl-sdr receiver however. I plan to get a better one though in the future and that should enable me to make measurements up to 6 GHz. Judging by Henrik's results the new bridge should handle that as well.

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## Assembling Henrik's RF bridge

15.11.2020 10:54

Last week I wrote some thoughts on the design of a bridge-based directional coupler that was published by Henrik Forstén. I re-drew his Gerber files with some minor modifications so that I could order a copy of the PCB and make a coupler to use with my measurement setup. The PCBs and the rest of the components arrived as planned and I spent a morning assembling everything together. As I suspected, it was quite a challenging soldering job. Here are some notes on how it went.

I ordered the PCBs at AISLER. I was surprised to see that they arrived in a little extra breakaway frame you can see above. The width of the PCB is right at the 15 mm limit of what they accept for manufacture, so the frame might be related to that. Or it might be to protect the thin and fragile-looking board from damage during shipping? In any case, all the PCBs arrived in good condition. Unfortunately, the breakaway tabs were again right at the spots where edge-mounted SMA connectors go, so some sanding was necessary after removing the frame.

Soldering the components on the PCB was really tricky - it's a hellish combination of tiny 0402 components, large thermal mass of the SMA connectors and a coax cable with a heat-sensitive insulation. I did it all with my soldering iron and rosin-core Sn-Pn solder. Hot air seemed like a bad choice because it would be hard to avoid heating up the coax too much.

I soldered components in the following order:

1. Tack each SMA connectors in one spot just to hold it in place. This is because if any solder gets (accidentally) deposited onto their footprints it's hard to fit them on the edge of the board.
2. Solder all the 0402 resistors.
3. Solder the coax cable on three spots, leaving only the ground/shield connection at the IN port unsoldered and unconnected. After the shield makes contact to the ground at that point it's impossible to measure with a multimeter if the R3, R4, R5 and R6 are correctly soldered.
4. Finish up soldering of the SMA connectors.
5. Verify that all legs of the bridge have their expected resistances, fix anything that's wrong.
6. Solder the shield of the coax to the ground plane, verify that there is no short between coax core and shield.

Soldering 0402 parts requires a very fine soldering iron tip, however that will not have enough power to heat up the connector and the ground plane. Hence I ended up exchanging tips a few times during assembly. I also adjusted the temperature of the iron as low as I could get it while still melting the solder each time I was doing anything in proximity to the coax cable.

I still messed up the coax once and had to start over. Henrik lists RG-405 coax in his BOM, but I couldn't get this type for a reasonable price and used a type that looked similar. This one has FEP insulation and isn't as heat sensitive as some other cables I've worked with, but it will still soften and melt through if you're not careful.

The spot that gave me the most trouble is the gap where R3 and R4 are placed. Here you have two tiny chips between two huge blobs of solder - the coax shield on one end and the SMA connector shield on the other. I was constantly getting a short with a solder bridge over the gap and the resistors simply floating away on the melted solder. Any solder bridge at that point is hard to remove since it's hard to melt both ends at the same time and not ruin the coax in the process. It took a lot of tries to get to something that wasn't absolutely terrible.

If I would be redoing the PCB I would leave a bit more space for R3 and R4. I had no such problems with R5 and R6 which only have one end embedded in the blob that's holding the coax shield. Also, there's really not enough space to properly land the coax core onto the PCB trace. On the input side the best I could do was to solder the core directly onto the connector pad. On the other side it landed onto the R1. So it would probably be better to leave out the small stubs altogether.

I also suggest changing the connectors to a gold-plated variant. I'm not sure if mine were dirty, but I had problems getting the solder to properly wet their pins. I had no such problems with similar, gold-plated ones on the multiplex board.

A minor thing that bothers me is that ferrite cores are just rattling around. By the ludicrous amount of packaging Mouser put them in for shipping I suspect they get chipped easily. Replacing them would be a nightmare. I'll probably make a small enclosure that will fit snugly over the cut-out to hold the rings and protect them.

In conclusion, assembling these bridges isn't simple. As promised, I've published my modified designs if you want to make your own. After some initial experiments I'm very happy with my copies. I'll publish some measurement results later, but so far they seem to behave very similarly to what Henrik's measurements show.

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## NanoVNA-H output signal level

11.11.2020 19:13

When searching the web for the stimulus signal level that is present on port CH0 of NanoVNA, most results seem to refer to the RF output figure given in the User Guide: between -13 and -9 dBm. I've seen posts also specifically claim that NanoVNA is suitable for measuring active components because of the low signal level at which it performs measurements:

However there seem to be a lot of different NanoVNA variants floating around, differing in both hardware and software. It's always good to double check these things. This is how the output signal of a NanoVNA-H I have (firmware version 0.4.5-1-gfbbceca, PCB revision v3.4) looks on an oscilloscope. CH0 on NanoVNA was connected to a 50 Ω terminated input on the scope. NanoVNA was set to CW stimulus at 1 MHz:

The measured RMS voltage is 326 mV, which is around 3.3 dBm at 50 Ω. Much higher than what is stated in the manual. It falls slightly with frequency. At 10 MHz I've measured 2.8 dBm. This is still uncomfortably high. For example, it will likely damage a HackRF if you tried to measure the return loss of its antenna input. The E4000 tuner in an rtl-sdr has a absolute maximum RF input level of 10 dBm, so strictly speaking it will probably survive. Still, any kind of S11 measurement will show much worse results than in real life for low-level signals since it will overdrive the RF front end of the receiver.

According to this post, some versions allow you to set the signal level from the user interface or by using serial commands. As far as I can see my version only supports the serial command method.

These are the output levels I measured at 1 MHz using different power settings. For some reason the power serial command doesn't work if the NanoVNA is in the CW stimulus mode - the command only had an effect if I sent it when full sweep was enabled on the device. However after the command was accepted, the output power in CW mode was also affected:

serial command RMS voltage [mV] power @ 50 Ω [dBm]
power 0 88.7 -8.0
power 1 173 -2.2
power 2 253 1.1
power 3 324 3.2

From this it seems that power 3 setting is the default at power on. As you can see, there's really not much difference between the settings in the logarithmic scale. Even the lowest setting is still higher than what is stated in the manual. I also wouldn't trust these results too much. From what I read the output power is adjusted relative to output frequency and I can only measure it at the very low end. Hence I don't plan on using NanoVNA for measuring any kind of sensitive components.

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## S21 curves for SMD resistors

09.11.2020 20:19

Some time ago I've stumbled upon the following figure that appears in the Building VNA Calibration Loads video by W0QE. The discussion about it starts around the 13:42 mark. It supposedly comes from some work done at CERN. The "Fig 2" in the caption suggests it appears in a paper, but after searching for it I've failed to find a public source.

Image by W0QE

What this measurement shows is how the impedance of surface mount resistors of different values change with frequency. If the resistors would be ideal all values would have perfectly parallel horizontal lines. However at high frequency the plots for high value resistors curve upwards. The line that they approach asymptotically is one of a 50 fF parasitic capacitor that appears across the resistor. On the other hand, the line for a 0 Ω resistor curves downwards due to parasitic inductance. W0QE discusses the resistor model in more detail, so watch the video for the full explanation.

According to this result the 100 Ω resistor is best at maintaining constant S21 towards high frequencies. This is the base of the argument that making a 50 Ω termination with two parallel 100 Ω resistors is best, and more accurate than using a single 50 Ω resistor or four 200 Ω resistors.

Some thoughts on this: the caption says that S21 curves shown are for 1206 size resistors. Very likely they look differently for other sizes since the parasitics will change with physical dimensions. Hence the optimal choice of a termination resistor might be different if using other sizes.

Another thing that W0QE doesn't mention is that the lines also curve towards the low frequency end of the scale. I suspect this is some kind of a measurement error. I see no reason why a resistor would not have the correct value at DC. This feature also does not appear in W0QE's simulation that is also shown in the video alongside this experimental result.

Anyway, the whole video is well worth a watch. I just wanted to give some more visibility to this particular figure and also have it here for my reference. Since it only appears in a video it's kind of hard to search for it. I already forgot where I saw it once and it's frustrating not to be able to find a thing I know I've seen somewhere when I want to refer to it.

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## Notes on Henrik Forstén's RF bridge

06.11.2020 19:03

For my tinkering with vector signal measurements I've needed a directional coupler. So far I've been using the "Transverters Store" RF bridge for this purpose. Unfortunately I found it to be less than ideal above 1 GHz. Its design also requires the use of a separate, reference 50 Ω load on a SMA connector, which over time proved to be unreliable and an unnecessary source of problems. I've been wanting to replace it with something better. Mini-Circuits ZHDC-10-63-S seems to be a popular directional coupler with good performance, but it's either perpetually out of stock or not shipping to my part of the world.

I've recently stumbled upon Henrik Forstén's blog. He made his own home brew VNA and generously shared all the design documents, including those for his directional couplers. The directivity measurements he posted show that his couplers are significantly better at high frequencies than what I currently have. He saw better than 25 dB directivity at up to 5.5 GHz. Compare this to the Transverters bridge where directivity falls below 25 dB at around 1 GHz. Making my own copies of his couplers seemed like a straightforward way of improving my measurement setup.

Henrik's coupler design is based on a 2015 IEEE article by Drobotun Nikolay and Mikheev Philipp (at the moment, the paper is freely accessible here). Similar to the Transverters board, the principle of operation is a based on a Wheatstone bridge with a balun made from a coaxial cable and ferrite beads. However the bridge topology as well as the balun design are quite different. This balun uses only one coaxial cable instead of two, and interestingly uses 3 different types of ferrites. The 50 Ω reference is integrated onto the bridge itself. The PCB design uses 4 layers. The coupling factor is 16 dB.

At first I thought getting the PCB made would be as simple as zipping up Henrik's Gerber files and sending them off to AISLER. Unfortunately that didn't work out. As much as I fiddled with the Gerber files I couldn't get the AISLER's on-line ordering system to accept them. I tried deleting the "CUTOUT" letters on the board outline layer and it didn't help. In the end it might have been the fact that the board is slightly narrower than the 15 mm minimum.

I ended up redrawing the whole thing in a PCB design program that-shall-not-be-named and making a fresh new set of Gerbers. In the end I think that turned out for the best, because it made me look a bit deeper into this design and do a few modifications.

Henrik had his PCBs made using OSH Park 4 layer service. This process uses a high-frequency, low permeability substrate FR408. My AISLER boards will be made with the 4-Layer HD stackup which uses just plain old FR-4. So when redrawing the designs it occurred to me to also adjust trace widths so that characteristic impedances would stay the same.

One thing I could not figure out were these narrowed sections of the traces around the bridge section. The wider part of the trace is approximately 50 Ω based on OSH Park's process parameters. The narrower part is closer to 75 Ω. I'm not sure why the narrow part is necessary. I can't see it in the figures in the original IEEE article either, although authors don't provide a good picture of the final design so it's hard to say.

The only reason I can think of is thermal relief, to avoid the SMA connectors from sinking too much heat when soldering the resistors and the balun. Since the whole divider part is only around 4 mm across I doubt trace impedances play much role in signal integrity. 1/10 wavelength rule gives a maximum frequency of around 4 GHz for traces without a controlled impedance.

Image by Drobotun Nikolay and Mikheev Philipp (modified)

Where the traces meet the SMA connectors they need to be widened so that the connector pin lands properly. The IEEE article mentions using microstrip tapers for better matching in this part. Henrik's design doesn't use these, however it keeps the connector center pin footprint at about 50 Ω by dropping the ground plane by one layer beneath the footprint.

Another difference I found between the paper and Henrik's design were the resistor values. The paper mentions that the resistor next to the balun needs to be about 10% larger (10.3 Ω) than the theoretical value (9.3 Ω) due to some unspecified effect of the balun. Henrik lists the uncorrected value in his BOM.

Another thing to note about the resistors is that the pads on the PCB fit tiny 0402 size packages. I suspect these will be quite a challenge to solder manually, especially since they're very close to where the coaxial cable lands. Actually, the whole coupler is just 40 x 15 mm - much smaller than it appears on the photos.

In my modified design I've re-calculated all the trace widths to 50 Ω at AISLER's process parameters and left out the narrowed sections. I also slightly enlarged to board so that it's now exactly 15 mm wide and made sure that it complies with the 0.3 mm edge clearance design rule. I was tempted to switch to larger 0603 resistors, but I'm guessing these come with worse high-frequency response as well, so I left them at 0402. In the end I also added some helpful labels on the silkscreen print layer.

Now I'm waiting for the PCBs to be delivered. Thankfully I didn't have any problems ordering the exact ferrite types. Again I'm really grateful that Henrik went through the trouble of documenting his instrument and publishing the designs. We'll see how well my copies perform. After I assemble one and check that it works reasonably well I plan to also publish my modified Gerbers and BOM for anyone else that wants to make a copy of these devices using a FR-4 process.

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## Vector measurements with the rtl-sdr, 7

16.10.2020 18:44

Looking back at my last few posts I feel that I was going into increasingly theoretic guesswork about the impedance mismatches in my rtl-sdr measurement setup. Even though I still think it could perform better, I would like to share some actual antenna measurements I made with it. I've also now got my hands on a NanoVNA-H, one of the many variants of the popular low-cost vector network analyzer. It's interesting to compare the measurement results between the two instruments.

Yesterday I've measured the VSWR of a few compact LTE antennas. These antennas have an adhesive backing and are meant to be mounted into mobile devices that use the cellular network for connectivity. All of these antennas come with a coax cable tail that terminates with a female U.FL connector. This was a bit problematic since both my instruments terminate in a SMA connector. So to make a connection I've used a U.FL-to-U.FL "thru" on a RF demo kit PCB I got with my NanoVNA and another short U.FL-to-SMA patch cable.

I've calibrated both my rtl-sdr setup (vect-meas-mux in the following) and the NanoVNA with the short-open-load one port method. I've used the U.FL calibration standards on the same RF demo kit. This resulted in the calibrated measurement plane on the U.FL end of the U.FL-to-SMA patch cable. Hence the measurements include effects of the "thru" and the coax cable tail that was attached to each antenna.

The vect-meas-mux setup consisted of the ERASynth Micro, rtl-sdr, my custom multiplex board and the Transverters RF bridge as the directional element. It allows measurements up to 2000 MHz, with a gap around 1200 MHz, defined by the rtl-sdr's tuning range. It's the same setup as I described in the past blog posts in this series. The NanoVNA-H was bought from Nooelec and the particular version I have is no longer listed on their website. It shows firmware version 0.4.5-1 and allows measurements up to 1500 MHz. I transferred the S11 measurements from the NanoVNA to the computer using code based on this example notebook and then calculated VSWR from S11.

In the graphs below, the VSWR measured with vect-meas-mux (orange) and NanoVNA (blue) is compared to the VSWR given in the datasheet (black) of the respective antenna. I was mostly interested in how both instruments agree with each other and how well the results were repeatable. The datasheet curve is there as a reference. I wasn't expecting to get a good match. I took care to have a reasonably clean measurement environment: as much open space in all directions around the antenna as possible, antennas mounted on a non-conductive plastic holder, etc. However obviously that can't compare with a measurement in an anechoic chamber and random objects in the vicinity were affecting the results. I also didn't use a choke on the coax tail to prevent it from radiating.

I'm pretty happy with how these measurements turned out. The results from my own instrument match those from the NanoVNA quite well. The agreement with the datasheet is often reasonably good as well although, as I said before, that's probably due to my poorly improvised environment around the antenna.

After doing many measurements the main problem with the rtl-sdr method turned out to be its slowness. NanoVNA displays the result nearly instantaneously. On the other hand a 200-point sweep from 55 to 2000 MHz with vect-meas-mux takes around 7 minutes and 30 seconds. Most of this is due to hugely inefficient way I collect signal samples from the rtl-sdr: I run the rtl_sdr binary once for each of the 200 frequency points. This results in the setup and tear down time of the rtl_sdr binary dominating the total run time. I only need to dwell on each point for about 100 ms, which would mean a sweep time of about 20 seconds. I'll probably rewrite my code to use the rtl_tcp server to fetch the I/Q samples, which should reduce the overhead significantly.

In the future I would also like to be able to estimate the error intervals of the measurement, like I did with my scalar experiments. Because this was a vector measurement and I used calibration, the relatively poor directivity of my RF bridge affected the results much less than with scalar measurements. A bigger factor was probably the limited dynamic range of the rtl-sdr. scikit-rf has some support for estimating error intervals, but I still need to figure out how to integrate that with my code.

I'm also looking for extending the frequency range of my measurements, but that will require switching to a more expensive SDR. Perhaps something like the HackRF or an USRP.

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## Vector measurements with the rtl-sdr, 6

06.10.2020 19:58

This is another update on my project where I'm developing a system for vector signal measurements with the rtl-sdr. Back in August I was testing the performance of the time multiplex board I made. One of the more puzzling things I discovered then was that apparently the signal losses in the multiplexer varied a lot with frequency. I measured a dip in signal level of almost 10 dB at 1700 MHz. I expected a mostly frequency-flat characteristic due to attenuation on the PCB and in the MMIC switches.

My first suspicion was that I messed up the design of the grounded co-planar waveguides (GCPW) on the board. After some research however I found conflicting information on that. Some sources say that my design is correct and others say it's not. I also got the impression that even if the trace dimensions I used were slightly wrong, they shouldn't result in the signal losses I was seeing due to relatively short trace lengths.

I currently don't have any professional RF equipment at hand. My gain (i.e. S12) measurements were done by using ERASynth Micro as a signal source and rtl-sdr as the detector. I measured the losses in the circuit board by routing the signal through the board, sweeping the signal in frequency at a constant output power, and recording the received power at rtl-sdr using rtl-power-fftw. I used a measurement of a male-to-male SMA through as a reference in an attempt to subtract any variability of ERASynth's output power and rtl-sdr's sensitivity.

Since I suspected that the variability in the gain I measured might also be due to impedance mismatches I did some further measurements with ERASynth and rtl-sdr to better understand what is going on. These measurements were performed in the same way as my previous one. ERASynth Micro was set to -40 dBm output level and was swept in steps from 55 MHz to 2 GHz. On each step, the rtl-sdr was tuned to the same frequency and the baseband signal power was measured with rlt-power-fftw (RF gain was set to 15 dB).

The picture below shows the six signal paths I measured:

"thru" measurement was a single rigid male-to-male SMA adapter. "thru3" was a combination of 3 adapters (male-to-male, female-to-female, male-to-male). "gcpw" was a measurement of a single leg of the PCB trace on a spare board where I soldered a SMA connector in place of the switch IC.

During the measurements involving the assembled multiplex board the switches were powered up and configured to allow the signal to pass in the measured direction. Also worth mentioning is that for measurements involving both PCBs, two male-to-male adapters were also in the signal path (one on the ERASynth side, the other on the rtl-sdr side).

Following graph shows the results. In contrast to the graph above from August, this graph is not normalized in any way and shows raw values returned by rtl-power-fftw. The gap between 1100 MHz and 1250 MHz is the band to which rtl-sdr can't tune to. The results for the "ref in - det out" path had +10 dB added on this graph to make it comparable with other traces since this path goes through a 10 dB attenuator on the board.

These results show several interesting things. The first good news is that the two identical paths through both switches show nearly identical results ("ref in - dut out" and "dut in - det out" traces). This is reassuring. It's unlikely that my board isn't soldered well or that I damaged one of the switches.

The other interesting part is that all other traces differ a lot from each other. Even just adding two adapters changes things significantly ("thru" and "thru3" traces). This suggests that the variability I was seeing is due to some mismatch between the ERASynth and rtl-sdr and not the fault in my board.

It's also telling that the only path that involves an attenuator ("ref in - det out") shows the least variability and also seems to be lower or equal to all other traces. The attenuator provides isolation between the source and the load and reduces the impact of their impedance mismatches. It changes the situation from the mismatched load and source (where gain varies with frequency) to a mismatched load and mismatched source separately (where gain does not vary with frequency). Also, in contrast to the GCPW traces, I'm pretty sure I got the 50 Ω impedance of the attenuator correct.

k = \frac{u_{max}}{u_{min}} = 2.5 \qquad [8 \mathrm{dB}]
|\Gamma_g\Gamma_l| = \frac{k-1}{k+1} = 0.43 \qquad [-7.3 \mathrm{dB}]

The measured power in the un-attenuated paths at frequencies above 1 GHz varies for about 8 dB. This translates to a product of reflection coefficients of ERASynth and rtl-sdr of about 0.4 (-7 dB). I don't have any data for ERASynth. The datasheet for E4000 tuner in the rtl-sdr states a typical return loss of 15 dB for a 50 Ω system, but my rtl-sdr has some components in front of the tuner which might make it worse. I measured the return loss of my rtl-sdr at about 11 dB at 70 MHz, so 7.3 dB at 1 GHz combined with ERASynth doesn't sound too much off.

In summary, this all points to the rtl-sdr's poor 50 Ω matching to be the culprit. There are some ways I could make the frequency response flatter. The simplest solution would be adding another attenuator to the board in front of the DET OUT connector. I'm using the lowest RF gain setting on the rtl-sdr so there should be enough reserve in terms of signal level. I don't want to do anything more elaborate like a rtl-sdr specific matching network since I want this board to be usable with other small SDRs as well.

Anyway, I think I'll leave the multiplex hardware as it is for now. I still have some planned improvements on the clock recovery code. I've also found Henrik's blog and his wonderful posts about the homemade VNA. His project is orders of magnitude more advanced than mine, but some of the details he writes about are still very useful for me. Thanks to him I found out about the scikit-rf Python library which I will gladly use instead of my home brew calibration code. I also found his directional coupler design very interesting and I might eventually make a copy of it to replace the RF bridge I'm currently using.

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## Transmission line mismatched on both ends

14.09.2020 20:30

In one of my previous posts I mentioned a result of a RF gain-versus-frequency measurement that looked like a sine wave. I said the sine wave probably means that there is some impedance mismatch in the measured signal path. I wasn't completely sure back then so I did a quick refresh on transmission line theory. Indeed, if you have a transmission line that is driven by a mismatched source and also has a mismatched load on the other end at the same time, the amplitude on the load will vary periodically in relation to the signal frequency.

Consider the following circuit. You have a sine wave source ug with a source impedance that has a reflection coefficient Γg. The source is connected through a transmission line to a load with its own reflection coefficient Γl. The transmission line has a characteristic impedance Z0, length d and propagation speed c. I've also marked the forward voltage wave in the transmission line uf and the reverse voltage wave ur.

We keep the amplitude of the signal source ug constant and vary the frequency. If we measure the amplitude of the signal on the load ul the amplitude will vary with applied frequency in a sinusoid like on the following graph. Note that the horizontal axis is frequency, not time. The vertical axis shows amplitude, not instantaneous voltage:

The amplitude on the load ul will show peaks every Δf. At the peaks, the signal will have the amplitude umax and in the valleys it will have the amplitude umin.

It turns out that Δf has the following relation to transmission line length and propagation speed:

\Delta f = \frac{c}{2 d}

This equation can be useful in debugging since it can point to the part of the signal path where the mismatch is happening. For example, if you know the propagation speed it allows you to calculate the length of the mismatched segment.

The ratio of the maximum and minimum amplitude (in linear scale) on the load depends on the reflection coefficients at the source and the load ends of the transmission line:

\frac{u_{max}}{u_{min}} = \frac{1 + |\Gamma_g\Gamma_l|}{1 - |\Gamma_g\Gamma_l|}

This last equation is interesting. At the first glance it makes sense. If either of the ends is perfectly matched (Γg = 0 or Γl = 0), then the ratio is 1 and there is no dependency on frequency. This is the expected result. A transmission line that is mismatched on only one end still has a non-optimal power transfer but the amplitude on the load is constant and doesn't depend on the signal frequency.

A signal flow diagram, similar to the one mentioned in this article, can also be used to verify its correctness.

The equation for the ratio of amplitudes on the load is very similar to the equation for the voltage standing wave ratio:

\mathrm{VSWR} = \frac{1 + |\Gamma_l|}{1 - |\Gamma_l|}

In a VSWR calculation you only have a mismatched load. In the case where you have two mismatched ends, it then kind of makes sense that you multiply both reflection coefficients together.

However if you think about it, it's quite weird that it turns out this way. In the VSWR case you are calculating the ratio of the sum and difference of the forward wave (uf = 1) and the reflected wave (reflected once off the load, hence ur = Γl). Since the source is perfectly matched in that case, the reflected wave doesn't reflect back the second time. It's obvious that the formula for the ratio is this simple.

On the other hand in the case where both the source and load are mismatched, you get an infinite number of reflections. When the source first gets switched on, the first forward wave driven by the source reflects off the load with Γl. When the reflected wave gets to the source it reflects off the mismatched source impedance with Γg and travels back to the load superimposed on the original forward wave. This combination of the signal driven by the source and the first reflection again reflects off the load and so on to infinity. The steady-state amplitude of the signal on the load is the sum of all those infinite reflections.

If you think about it this way, it's surprising that the ratio ends up being this simple equation that is the same as if the signal only reflected once from the load and once from the source.

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## P57 feed-through terminator

26.08.2020 17:57

A quick note about this thing. It's a BNC feed-through terminator I've bought for cheap off AliExpress when I was on a kind of an RF accessory buying binge. After months of shipping delays it recently dropped into my mailbox. This one is marked "P57 load resistor 50 Ω". I've seen very similar looking devices being sold under various other names and brands. I've bought it because my TDS 2002B scope does not have a low-impedance input option.

The label says that the terminator is rated from DC to 1 GHz. The analog bandwidth of my scope is only 60 MHz, so that shouldn't be an issue. The DC resistance between signal pin and ground measures exactly 50.0 Ω on my Keysight U1241C. That's a good sign that it doesn't just have a standard carbon 51 Ω resistor inside.

The build quality looks fine at the first glance, although with the plastic body I wouldn't use this where any kind of significant power would be dissipated on the load.

This is the result of a quick test I did. I connected the ERASynth Micro to the oscilloscope CH 1 over a coax cable. The red plot shows the signal amplitude measured at various frequencies without the terminator (so terminated with 1 MΩ at the scope's probe input). The blue plot shows the amplitude with the cable terminated with P57 on the scope end. The amplitudes were measured with the FFT function and hence only take into account the base frequency, without any harmonics.

The ERASynth Micro was always set to 0 dBm output level. If everything would be perfect, the blue plot would be at -13 dBV and the red plot would be 6 dB higher (twice the amplitude). Falling amplitude beyond 60 MHz is expected because of the limited analog bandwidth of the scope's front end.

I've measured between 8 and 5 dB difference between the terminated and unterminated amplitudes, which seems fine. Or at least not excessively wrong. There's a lot of unknown errors in this measurement. Cable and adapter loss, ERASynth Micro output matching and level accuracy and so on.

In conclusion, it does what it says in the description and seems good enough for my purpose.

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## Vector measurements with the rtl-sdr, 5

18.08.2020 18:01

Here is a quick update on my project to measure the phase of a reflected RF signal using rtl-sdr and what is basically a simple home-brew vector network analyzer. I'm using a cheap RF bridge from eBay as the directional element in the measurement and a board I've developed to multiplex two signals into the rtl-sdr's single ADC. In my last post I've assembled the time multiplex board. I've also shown some basic tests of its performance using the ERASynth Micro microwave synthesizer, such as signal attenuation on the PCB traces and cross talk through the switches. Now I finally have tests to show of the complete vector measurement system, although only in pass-through (i.e. S12) mode without using the reflection bridge.

To test out if I can correctly measure the phase of the signal using my system, I performed several tests where I connected the DUT IN to OUT with different lengths of a coaxial cable. In theory, different lengths of the cable should introduce different delays into the signal. I should be able to measure the delay as a difference in the phase of the signal arriving into the DUT IN input.

I've swept the frequency of the signal generator from 100 to 1000 MHz, which covers most of the useful region of the rtl-sdr. For each frequency, I performed the vector measurement using the delay-and-divide method I outlined earlier. I ran the test with three lengths of SMA patch cables I had at hand: 15, 30 and 45 cm.

The measured amplitude in all cases should be around +10 dB. The reference signal, against which the input signal is compared, is attenuated by 10 dB in the multiplex board. Because of this, an unattenuated 0 dB signal passing between DUT OUT and IN compared to a -10 dB reference is seen as +10 dB by my device. I did this to optimize the receiver dynamic range, since many directional couplers and RF bridges have around 10 dB of coupling.

Anyway, this measurement is about what I would expect. For all lengths of cable the mean is around 10 dB. Cable loss is too small to be visible. The variations in amplitude of around ±2 dB depending on the frequency are more than I would like. They are probably because of multiple bad impedance matches somewhere in the signal path. I've also seen these in my previous measurements.

This is the phase component of the same measurement. It shows that the input signal lags versus the reference and that the phase difference increases with frequency. It's also clearly visible that the slope depends on the length of the cable. Such a linear phase characteristic is exactly the expected result for a constant signal delay. The phase plot has been unwrapped here using the numpy.unwrap method.

By dividing the measured phase with the angular frequency, the measurement can be shown directly in terms of time delay:

This is a very nice result. Each lengthening of the cable by 15 cm increased the average time delay by about 0.8 ns. This gives a relative velocity of propagation through the cable of around 0.63 c, which is a reasonable number. I don't have the exact data for my no-name patch cables. A cable with a solid polyethylene dielectric, typical for low-end cables, has the velocity factor of 0.66. This is close enough to my result and the electrical length of my cables isn't very well defined anyway.

Another interesting thing I can get from these results is the delay for a theoretical cable of length 0 cm. This is the intrinsic signal delay in my system and is about 0.15 ns based on these measurements. This should correspond to the difference in electrical lengths on my multiplex board between the reference signal path and the DUT connector path. If I estimate the velocity factor of the coplanar waveguides to be:

VF = \frac{1}{\sqrt{\varepsilon_r}} = 0.48

This gives a length difference of about 22 mm. This is again a reasonable result. By measuring PCB trace and connector lengths I roughly estimate the real difference to be about 30 mm.

In conclusion, I'm quite happy with these results. They show that my basic time multiplex idea works correctly for vector measurements. As far as the whole system is concerned there are still several rough spots however. Clock recovery code still needs some more polishing since running it on real data revealed some corner cases that didn't show up in simulations. Thankfully I see nothing fundamentally wrong with it as far as I can see and fixing it should be just a matter of writing some better software.

The bigger problem is the bad overall RF performance of the multiplex board. I've discussed before that there seems to be something very wrong with impedance matching which causes large deviations in measured signal amplitude. I still haven't completely figured out what's wrong there. One mistake I did found on the PCB design was that my coplanar waveguides don't have the width much larger than height over the ground plane (see e.g. slide 41 in RF/Microwave PC Board Design and Layout). This is one of the assumptions of the equations for their Z0 I used. This error might be what's causing some of my problems, but fixing it unfortunately means making a new PCB.

Another, even more puzzling problem, is that the RF bridge seemingly loses its directivity when used in this system, even at low frequencies. Since I made this vector measurement system specifically for reflection measurements (S11) this is kind of disappointing. I don't really understand yet why that happens. It's not because of switch cross-talk. I've measured that and it should be negligible. The mismatches on the board also shouldn't be affecting the bridge in this way, especially at low frequencies where they don't have much effect. Scalar measurements work just fine with rtl-sdr and ERASynth Micro. I've also measured that my bridge has reasonable directivity below 1 GHz when used without the multiplex.

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## Vector measurements with the rtl-sdr, 4

31.07.2020 15:29

I'm developing a method for performing vector reflection measurements using a cheap, single channel software-defined radio receiver. To measure the phase of the reflected signal I need to receive two RF signals coherently, a reference and the actual measured signal. I'm using time multiplexing to do that since I'm restricted to only a single analog-to-digital converter. In my last post I've described a circuit I developed that performs the multiplexing in the analog domain. The demultiplexer as well as all other signal processing is done by software in the digital domain.

I ordered the bare printed circuit boards from AISLER using their 2-layer/HAL surface finish process. This is the first time I've used their service. I was looking to have the boards made in Europe since I'm seeing large shipping delays recently due to COVID-19 (I'm still waiting for some components I ordered from overseas back in April). After looking at a few local PCB prototyping services, AISLER looked like by far the best bet. I got the boards in my mailbox in less than a week for a price that was comparable to ordering them from China.

Between the HAL and ENIG surface finishes offered I decided on HAL since it should have better performance at high frequencies. I also removed the soldermask from the top of the 50Ω coplanar waveguides in an attempt to further reduce the losses. I was pleasantly surprised that AISLER offers precise stackup specifications, including εr. That's something that I've often missed in similar services and it probably made my trace impedance calculations somewhat more accurate. However this wasn't manufactured as a controlled impedance board.

I assembled the boards manually using a hot air station. The boards had some leftovers from panelization breakaways right at the locations of the edge-mounted SMA connectors. I had to smooth the edges with some sandpaper to get the connectors to fit nicely. The QFN packages of the MMIC switches were also a bit tricky to solder, but I recently got a lot of experience soldering tiny AVRs at work so that wasn't too troublesome. The only thing I'm never completely sure with QFNs is how well the ground pad gets connected. The datasheet says that is important for good RF performance of the switch.

The big question of course is whether it works or not. For testing I've connected the board to my rtl-sdr receiver and the ERASynth Micro microwave synthesizer. I've used short, rigid male-to-male SMA adapters between the instruments to keep down the losses. I've also shorted the device-under-test in/out (DUT) ports with a short length of a coax cable. In this setup the expected 100 Hz 3-state multiplex pattern is visible on the rtl-sdr baseband. The board switches between an off state (no signal), the reference signal which is the input attenuated by 10 dB, and the signal that passes through the DUT (which in this case is the full input signal, minus any attenuation in the coax cable):

So the basic multiplex functionality appears to be working, but is it any good? What are the losses and crosstalk in the switches and the PCB? Estimating the RF performance is where it gets a bit tricky. I've attempted to do that using the setup pictured above. To estimate the attenuation of the signal I've swept the input frequency and measured the received signal power at the rtl-sdr using rtl-power-fftw. I've performed the same measurement with the multiplex board locked into dut state and into ref state.

Since the rtl-sdr isn't a calibrated power meter I've also performed a third measurement where I directly connected the rtl-sdr to the signal generator using one of the rigid SMA adapters. This was the reference against which I compared the other two measurements. It's shown as 0 dB and the dotted black plot in the figure above. This way I subtracted any variability in the rtl-sdr sensitivity versus frequency. In all cases I manually locked the gain of the rtl-sdr to 14 dB.

Ideally, I would expect the blue dut plot to be near 0 dB. It should show only the loss in the PCB, switches (around 0.7 dB according to the datasheet) and the coax cable. Similarly, the ideal orange ref plot should be near constant -10 dB, because the board includes a 10 dB attenuator in that signal path.

As you can see, the reality is not nearly as perfect as the theory. The gain of both signal paths varies quite a lot with frequency. There seem to be one slowly varying component that's common to both paths (with minimums at around 600 MHz and 1600 MHz) and one that's only on the dut path (with periodic minimums every 250 MHz or so). This seems to suggest that there is more than one bad impedance match somewhere. It's not necessarily in the design of the board itself. I don't know how well ERASynth Micro is matched to 50Ω and I also don't have any specs for the rtl-sdr (I've been speculating on it's return loss before). I also don't have much confidence in the quality of SMA adapters and the coax cable I was using.

Finally, this is the result of the crosstalk test I did. Here I'm comparing the signal power with the board in the off state (blue plot) with the receiver noise floor (dotted black plot). The measurement in the off state was done with the same setup as above. I've measured the receiver noise floor by terminating the rtl-sdr input with a 50Ω terminator. As you can see, both plots are basically identical, which shows that there's no significant leakage of the signal when the switches are turned off.

A better way to show this result is by subtracting the noise floor from the off measurement in linear scale. This new plot now gives directly the isolation of the signal when the board is in the off state. 0 dB on the graph is again the input signal level, same as on other graphs.

Ideally, this plot would be at minus infinity. It is not because the two switches are not perfect (around 60 - 70 dB isolation per switch according to the datasheet) and some signal also probably leaks through the PCB and the space around it. When I was measuring this I also realized that the layout where I have the signal generator and the detector right next to each other might not have been the best choice to keep isolation high. Neither ERASynth Micro nor the rtl-sdr have metallized enclosures, so some signal might also leak out that way.

In any case, I think 50 dB isolation is more than good enough for my purpose. It goes down to 40 dB near 2 GHz, but that might be because rtl-sdr sensitivity is getting really bad in that range. This measurement is not very precise, since the crosstalk signal is below the noise floor of the receiver. It's unlikely that I will be using this board with bridges or directional couplers that would have directivity better than 40 dB. Hence isolation in the multiplex board shouldn't be the limiting factor in the accuracy of my measurements.

I would like to better understand where the unexpected variations in the signal path attenuation come from. Did I make an error in the board design or is it caused by cables and other equipment I'm using? I have some more experiments planned related to this. I also still need to take my signal processing code out of the simulation I wrote and use it on the data from the real hardware. In the end the only thing that matters is the quality of the final measurement result.

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## Vector measurements with the rtl-sdr, 3

12.07.2020 18:48

After doing some scalar reflection measurements using an rtl-sdr and an RF bridge I recently started exploring the possibility of capturing phase information as well using a similar setup. I came up with a relatively simple method that does most of the signal processing in software. I already validated that the method works well enough in simulation. All I need now to try it out in practice is a simple multiplex circuit board. I need that to coherently record two RF signals with rtl-sdr's single channel analog-to-digital converter.

At the time of my last post I already had most of the circuit sketched out on paper and the basic calculations done. I've spent the last couple of days finishing up the design and transferring it from paper into the computer. This is a 3D render of the current draft of the circuit board:

It's a two-layer design on a 1.6 mm FR-4 substrate. All RF and almost all the other tracks are routed on the top layer, leaving the bottom for a mostly uninterrupted ground plane. I'm not sure yet about the surface finish and solder mask. Signal connections are using edge-mounted SMA connectors.

I had to revise the schematic a few times when I was making the PCB layout. The 50 Ω coplanar waveguides can't overlap or change layers and I wanted to have the RF part of the circuit laid out as cleanly as possible. Fortunately, the input and output ports on the MMIC switches I'm using are interchangeable. This gave me enough flexibility to come up with a combination of ports where such a layout was possible. By a lucky coincidence the exact combination I ended up using also inverted the sense of one switch. This had an added benefit that I could omit an extra quad-NOR gate from the driver circuit.

The rest of the circuit is quite straightforward. I'm using a HEF4017 Johnson counter to drive the switches in the correct sequence. I ended up going with a three-state "off-ref-dut" switching cycle to aid the clock recovery like I mentioned in one of my earlier posts. There's a selector switch that allows the sequence to be driven by an internal oscillator, a manual button or an external clock coming into the fifth SMA connector on the bottom right of the board.

I've added the manual button to aid in testing. It will allow me to lock the RF circuit in a specific configuration and perform measurements on that specific signal path. Two LEDs show the currently selected path.

The internal oscillator is a simple 100 Hz astable multivibrator using a low-voltage variant of the classic 555 timer. Its frequency will be very inaccurate, but that shouldn't be a problem. The software needs to do full clock recovery anyway over several full switching cycles and the clock only needs to be roughly in the vicinity of 100 Hz and stable for around 10 cycles. If it later turns out that I've missed something and do need a better clock source I can still bring it in from an external source.

I probably need to go over the design once more time in case I missed something, but otherwise the PCB layout seems ready to be sent out to a prototyping fab. I getting curious. For the sub-2 GHz frequency range that the rtl-sdr and my current RF bridge can handle I think the hardware should work well enough. For the full 8 GHz rating of the switches I have some more doubts, mainly due to the 10 dB attenuator made from 0603 components and loses in the FR-4 substrate.

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## Vector measurements with the rtl-sdr, 2

05.07.2020 10:34

In my last post I've talked about a setup for performing vector reflection measurements with the rtl-sdr. I've come up with an idea for a simple time multiplex hardware so that I could receive both the reference and the measured signal with rtl-sdr's single channel ADC. I did some simulations of the setup and I mentioned that I saw some ±5 degree phase errors. I didn't investigate the source of that error at the time.

After spending some more time thinking about it it turned out that the phase errors I've seen in simulations are due to switch cross-talk. It's quite obvious in retrospect. The measured and reference signals get mixed in the switches and this changes their phase a little. It's best to show this with a phasor diagram:

These are the ideal signals. The reference Uref and the measured signal Udut that passed through the device under test. Udut has a different phase and amplitude compared to the reference and I want to measure that difference.

Due to switch cross-talk, what I'm actually measuring is U'ref and U'dut. U'ref is a sum of both the ideal reference and ideal measured signals, but the measured signal has been attenuated by k, which is the switch cross-talk in linear scale. Vice-versa for Udut. εref and εdut are the phase errors due to this addition.

\varepsilon = \varepsilon_{ref} + \varepsilon_{dut}

The combined phase error depends on the phase shift α in the signal caused by the device under test and the attenuation of the measured signal. The error is largest when α = 90° and the amplitude of the measured signal is smallest. Some geometry also shows that this maximum phase error ε in radians is, for small errors, roughly the same as the switch cross-talk (CT) minus the attenuation of the DUT (A, ratio between Uref and Udut) in linear scale:

I expect this error to be smaller in practice than what I had in this simulation. First reason is that I made an error and only accounted for one switch in the simulation. I reality there will be two switches and hence, at least in theory, double the attenuation on the unused signal path. The second is that I now plan to use Renesas F2933 switches, which have a much better rated isolation than the F2976 I've considered in my simulation.

Given the limited dynamic range of the rtl-sdr, -80 dB cross-talk or less should probably suffice for a reasonable accuracy over the entire measurable range. I also expect this is the kind of error that I can compensate for, at least to some degree, in software with short-open-load-through (SOLT) calibration. I have to lookup some of my old notes on the math behind that.

Talking about practice, I have the circuit schematic I want to make roughly drawn up on paper. I've decided on all the components I will use. The digital part for driving the switches will be low-voltage 3.3V CMOS logic, since that's compatible with F2933 inputs. For testing purposes I want to also be able to drive the switches from an external signal source and select the signal path manually. Next step is to draw the circuit in some EDA software and design the PCB layout.

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## Vector measurements with the rtl-sdr

21.06.2020 11:41

Previously I was writing about some experiments with reflection measurements using an rtl-sdr receiver. I used the rtl-sdr as a simple power meter together with an RF bridge to measure VSWR. This was a scalar measurement. All the phase information from the signal was lost and with it also the angle information about the complex impedance of the load I was measuring. Since I was happy with how the method performed in that experiment I was wondering if I could adapt the setup to measure the phase information as well.

With a vector measurement I need a reference signal to compare the phase of the measured signal to. This is a problem, since the rtl-sdr only has one input and can only sample a single signal at a time. My idea was that perhaps I could multiplex the reference and the measured signal onto the single input. Both time and frequency multiplex seemed doable, but time multiplex seemed by far simpler to implement in hardware. Integrated microwave switches with usable characteristics, such as Renesas F2976, are reasonably cheap these days.

Previously I recorded a 2 second array of digital samples of the measured signal for each scalar measurement point. With the time multiplex setup, I could record similar 2 seconds of input, but that input would now contain both the reference and the measured signal. I could then de-multiplex and process it in software. The new setup would look something like the following. The device under test could again be a bridge, or something else:

The time multiplex board contains two SPDT switches. In one position, the switches direct the reference signal to the rtl-sdr. In the other position the signal passes through the device under test and then back to the rtl-sdr. The switch frequency I'm thinking about is somewhere around 100 Hz. A complex baseband signal recorded by the rtl-sdr would then look something like this:

Most of the complexity in this setup would be in software. The software would need to find out which parts of the recording is the reference and which part is the measured signal. This is similar to clock recovery in a receiver. It would then compare the two signals and do some filtering. This is a rough block diagram of the processing pipeline:

The reality is a bit more complicated though. Especially clock recovery seems tricky. My original intention was to use auto-correlation of the signal but it turned out much too slow. Right now I'm just using simple amplitude thresholding, which works as long as DUT is attenuating the signal enough compared to the reference. There's also some additional processing required to account for the fact that my delay is an integer number of samples, which introduces an additional random phase shift that needs to be accounted for.

So far I've only performed some proof-of-concept simulations of this setup using the performance of the rtl-sdr I've seen in my scalar measurements and the properties of the switches from the datasheet. It does seem feasible. Here are simulated vector measurements of three points on the complex plane. For example, these might be complex reflection coefficient measurements on the Smith chart. The gray dots show the true value and the blue dots show the simulated measurements:

There is some angle and amplitude error, but otherwise the principle seems to work fine. These are the histograms of the errors over a large number of simulated measurements of random points on the complex plane, where the measured signal was well above the receiver noise floor:

I'm not sure yet what part contributes the most to these errors. I'm simulating several hardware imperfections, such as switch cross-talk, frequency and phase inaccuracies and receiver noise. The most complicated part here is the clock recovery and I suspect that has the largest effect on the accuracy of the output. The problems with clock recovery actually made me think that having a four-state "off-dut-off-ref" cycle instead of a two-state "dut-ref" for the switches would be better since that would gave a much stronger pattern to match against. Another idea would be to lock the multiplex clock to the actual signal. ERASynth Micro does provide a 10 MHz reference clock out, but dividing it down to 100 Hz would need a huge divider.

Anyway, the simulations so far seem encouraging and I probably can't get much further on paper. I'm sure other factors I haven't thought of will become evident in practice. I plan to make the time multiplexing board in the future and try to do some actual experiments with such a setup.

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## Resistor tolerance and bridge directivity

13.06.2020 14:20

Another short note related to the RF bridge I was writing about previously. The PCB has four resistors soldered on. Two pairs of 100 Ω in parallel. Each pair forms one of the two fixed impedances in the two branches of the bridge circuit. The two variable impedances in the bridge are the device under test (left) and the termination on the REF terminal (right). The black component on the bottom is the RF transformer of the balun circuit.

My father pointed out the fact that these resistors don't look particularly high precision. The "101" marking (10 times 101 ohms) is typical of 5% tolerance resistors. 1% parts more often have "1001" or the EIA-96 character code. Unfortunately I can't simply measure them in circuit with a multimeter, because the balun forms a DC short circuit across them. I don't want to desolder them. Still, I was wondering how much variances in these resistors would affect the bridge directivity.

Following is the result of a Monte Carlo simulation showing three histograms for bridge directivity. Each was calculated for one possible tolerance class of the 4 resistors used. The assumption was that individual resistor values are uniformly distributed between their maximum tolerances. The effect of two parallel resistors on the final distribution was included. The peak on each histogram shows the value for directivity that is most likely for a bridge constructed out of such resistors.

Each tolerance class defines the lowest possible directivity (where the two resistors are most mismatched). On the high end the histogram isn't limited. In any tolerance class there exist some small possibility that the resistors end up being perfectly matched, however the more you move away from the average directivity the less likely that is, as the probability asymptotically approaches zero.

This is the same data shown as an estimate of the cumulative distribution function. The annotations on the graphs show the 90% point. For example, for 5% resistors, 90% of the bridges would have higher than 32.1 dB directivity. You gain approximately 20 dB in directivity each time you reduce the resistor tolerance by a factor of 10.

It's important to note that this was calculated using a low-frequency bridge model. In other words, none of the high-frequency effects that cause the real-life directivity to fall as you go towards higher frequencies are counted. Any effects of the balun circuit and the quality of the REF termination were ignored as well. So the directivity numbers here should be taken as the best possible low-frequency case.

Anyway, I thought this was interesting. Similar results apply to other devices that use a resistor bridge circuit as a directional coupler, such as the NanoVNA and its various variants. Also somewhat related and worth pointing out is this video by W0QE where he talks about resistor matching for calibration loads and how different SMT resistors behave at high frequencies.

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