Author Topic: Using a fast pulse and DSO for vector network analysis  (Read 3137 times)

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Offline rhb

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Using a fast pulse and DSO for vector network analysis
« on: March 07, 2019, 01:46:14 am »
I received one of Leo Bodnar's fantastic 100 pS pulse generators yesterday and spent most of today playing with it hooked up to my LeCroy DDA-125 which is a 1.5 GHz, 4 GSa/S 8 bit DSO.

As a consequence of some ill advised online purchases, I wound up with a considerable number of RF adaptors and 50 ohm terminators of dubious quality.

As an experiment I connected the pulser to a tee connected to the DSO BNC input set for 50 ohm DC coupling.  I attached a ~ 2 ft 50 ohm cable with good quality connectors and tested the reflections from the 50 ohm terminators using a known good F-F BNC coupler.  This resulted in 4 BNC terminators going into the reject bin.

By setting amplitude cursors at the peaks of the reflection from a good device I was able to screen a large number of adaptors as quickly as I could change them.  I culled 11 more devices out of 60 or 70 that I tested.

VNAs commonly have software available to do TDR measurements by Fourier transform from frequency to time.  In reflection seismology the recording system is typically evaluated by recording an impulse response. Before digital recording this was simply done as a system check.  After the advent of digital recording it became common practice to zero phase the data using the impulse response of the instrument and the source to make interpretation of the data easier.  This is usually accompanied  by a Wiener filter to flatten the amplitude spectrum.

I've gotten rather interested in using a DSO and a fast pulse to do vector network analysis.  Naturally this is limited to the BW of the DSO, but 100 MHz DSO's with deep memory are cheap.  There is the issue of dynamic range, however, adroit DSP work can overcome that.  And an Owon XDS2102A offers 12 bit ADC for  $400.

I just bought an 8753B and 85046A in very nice shape.  I also have a VNWA 3E and an xaVNA though I have not yet attempted to use any of them.  So I have excellent instruments for verifying the results once I collect the various bits I still need to get.

My interest in using a fast pulse and DSO  for vector network analysis is for the benefit  of those who cannot afford such toys, but want to build HF gear.  One of Leo's 100 pS pulsers is under $80 delivered to the US.  The required software is fairly trivial, at least relative to what I'm used to dealing with.  It requires retrieving data from the DSO, performing sums and FFTs and plotting.

I'm very familiar with the reflection seismology literature relevant to this, but not the EE literature. Does anyone know of any published work on using TDR for VNA? Historically fast pulses and sampling were very expensive.  And even today, using TDR for VNA is not practical at UHF and above.  My interest in prior work and literature is because when I write software I like to include the relevant citations in the comments.  And I rather doubt that a ham building a QRP station would appreciate being dragged through reflection seismology.

 

Offline dcarr

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Offline dcarr

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #2 on: March 07, 2019, 01:54:44 am »
I've tried this at 10ghz using a SD-24 on a mainframe Tek sampling scope and the main issue I have is pulse-to-pulse jitter (~3ps) which distorts the phase data.  I think it would be possible to back this jitter out, but I haven't but put a lot of effort into it.  That said, I find this super useful for scalar data up to ~20GHz.
 

Offline rhb

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #3 on: March 07, 2019, 03:35:17 am »
Oh, thank you!!!

I was dreading idiotic "it can't be done" responses.  Your first link will keep my busy for a week.

As you mentioned an SD24 I'd like to solicit your advice.  I'm considering buying a CSA803 and SD24 and SD30 or SD32 sampling heads.  I posted to the tekscopes list asking for advice,  but not  have gotten a useful response. What do I need to watch out for?  Looking on eBay, it appears I should be able to get the mainframe and two heads for under $3k if I am patient.

I'm a 65 year old retired PhD level geoscientist.  I spent my career in seismology because there was no work in economic (mining) geology when I graduated with my MS.  I quickly got hooked on DSP and went  back to school at UT Austin.

Because I have been extremely frugal (one new vehicle at age 40, a base model Toyota pickup for $7800) for my entire life I now find myself able to buy any toy I want.  I've now spent more in the last 2 years on T&M kit than I have on motor vehicles in my whole life.  Which is still only about $15K.  Other people who made less drove 911s.  And the average price of a new car is less than what I've spent on cars  in my entire life.

I vividly remember the pain of trying to build simple radios without adequate T&M gear.  I failed and I knew why I failed, but I could not justify buying what I needed to solve the problem.  It was just a hobby.

I want to do something that addresses that.  I fully realize that less than 1% of young people today are interested.  I don't care.  If my work helps one kid succeed I'm satisfied.

The way to remove the jitter is to collect data over the display  gate, Fourier transform, make a linear fit to the phase and adjust the phase so all the transforms have the same phase.  then sum.  It eliminates the jitter and every time you double the number of samples you add 6 dB of dynamic  range.

A $400 Owon XDS2102A will collect a single 12 bit  channel at 500 MSa/S.  With a 20 Mpt  memory depth that will provide ~100 dB of dynamic range.  The single channel reflection response can be solved for the transmitted response.  I don't recall who published it, but it was done 40 years ago by one of two people I have had the pleasure of chatting with several times.  I don't think anyone has solved reversing the reflected and transmitted responses as done by an S parameter test set, but I'm fairly certain that it can be done.  If both the reflected and transmitted response contain all the information in one direction, they must contain all the information in the opposite direction.

Thank you *very* much for your reply.
Reg
 

Offline dcarr

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #4 on: March 07, 2019, 04:16:51 am »
Reg,

I bought my 11801B with 1 SD-24 TDR head (two channels) for about $800.  My instrument has a lot of hours on it (30,000!!!) but it passes all self tests and generally seems to work as expected.  As you probably know the SD-24 has two channels, both with ~30ps rise time step generators and 20GHz bandwidth samplers.  Mine seems to work "nominally" with the caveat that one of my step generators has a slight ringing in it that I cannot completely tune out.  The other thing that you should note is that there is a stupid bug in some 11801B firmware that causes the trace to jitter back and forth a few ps as a front panel LED is flashed on and off.  There was a firmware fix for this (replacement EEPROMs), but I don't have it.  Instead I've found that loading a "memory" trace on screen and selecting it during acquisition disables the LED and thus eliminates this issue.  It's a little annoying but workable.

To get a clean looking trace, I need to enable 8+ waveform averaging when using TDR.  An issue arises that any jitter between subsequent waveforms when averaging is enabled then gets "blended" into the measurement.  One could probably single shot every waveform and do the "de-jittering" and averaging in post processing, but as I mentioned, I haven't attempted this in part because I use the slow RS232 interface to acquire waveforms instead of the GPIB bus which may be faster.

I didn't completely follow your question about separating the transmitted and reflected responses, but because the SD-24 has two samplers and two pulsers it should be possible to take both through and reflected measurements on both ports (of a two port device) and in both directions without moving cables.  Maybe that helps.

If you get serious about this and buy one of these scopes, post here again.  I have written python/octave scripts that talk to the instrument and implement "VNA" algorithms listed on the website mentioned previously.  One thing that strikes me as odd in those is the hybrid triangle wave/square wave method that is used to compute the Fourier transform of the step response.  I'm not knowledgable enough to know for sure, but it's not clear to me if this approach was a computational convenience from when computing was more expensive or really the best way to compute the transform of a step.

Hope that helps.
David
« Last Edit: March 07, 2019, 04:21:09 am by dcarr »
 

Offline dcarr

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #5 on: March 07, 2019, 04:28:39 am »
Here are |S11| and |S21| measurements of a 9.4GHz bandpass filter taken using this setup:



 

Online _Wim_

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #6 on: March 07, 2019, 06:54:55 am »
Another nice affordable toy (+-100$) to buy is a Tektronix 1502B or C (they appear quite often on Ebay, with most of them the LCD backlight is gone, but they are perfectly usable without). With one of these I discoverd some of my cheap BNC-BNC cables with printed RG-58 numbering were actually RG-59 of another 75 ohm coax... Lost cost has indeed some traps. OTOH, some of the RG316 BNC-BNC low cost ones were actually quite good
 

Offline DaJMasta

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #7 on: March 07, 2019, 07:11:29 am »
Definitely possible, but I think very difficult to get SNR comparable to more standard VNA architectures.

I'd take a look at the datasheets/info for the LeCroy SPARQ series VNAs.  They use TDR to derive all their data, and have very wide bandwidth, but above pretty low frequencies, it's dynamic range is pretty limited - I think this is because of the high speed converters necessary to make the time domain measurements at full frequency vs. the downconverted measurement that VNAs use.


Also definitely can be done with the aforementioned Owon scope, but your pulser risetime demands are much, much lower due to the input bandwidth of the scope (and I guess the 250MHz nyquist of the sample rate).  At this point with the lower bandwidth, though, you could just use a sig gen and a directional coupler to do the whole thing with CW tones instead of TDR.
 

Offline rhb

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #8 on: March 07, 2019, 04:03:01 pm »
Excellent responses.  Thanks to all.

I was using the 2 way delay of the reflection to separate the incident and reflected waves in time and triggering on the incident pulse.

The Owon in 12 bit mode will give me a 20 Mpt record. So by segmenting the record, doing an FFT and phase aligning before summing I should get around 100 dB of dynamic range.  The phase alignment is needed to eliminate jitter.

You can determine a fair bit about the phase and frequency from the appearance of a reflection, at least if you're used to looking at time domain responses.  But a proper analysis requires a transform to magnitude and phase in the frequency domain.  What I was doing is only suitable for quick screening.

The step response will do the same thing, but the mathematics are more involved.  The nice thing about a spike is it is constant amplitude in the frequency domain.  A triangular pulse is a sinc(f)**2 in frequency, so you have to normalize by that.  Of course, Leo's pulser is not putting out a true spike, so it will also need to be normalized if the method is used above 100-200 MHz.  But it's pretty close to a minimum phase Gaussian.  The TDR step is actually the leading edge of a square wave.  That's where the complications come in as sinc(f) rings forever.

I've ordered Joel Dunsmore's book and downloaded his dissertation which looks at the problem.  The great thing for me is that I derived the full scattering response for the elastic wave equivalent in grad school using the Z transform.  That lets me think about the problem without the clutter of new jargon for everything.  As I'm also getting started playing with FPGAs I've just about reached a toxic dose of acronyms.

https://www.amazon.com/Handbook-Microwave-Component-Measurements-Techniques/dp/B00CB1YS04

There's a lot of good stuff here:

https://www.kirkbymicrowave.co.uk/Support/Links/#books

My principle goal in this work is to provide a way to do network analysis on a tight budget.
 
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Offline David Hess

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #9 on: March 07, 2019, 08:08:23 pm »
Tektronix and LeCroy have published application notes about using this method for network analysis.  I am sure HP/Agilent has as well.

The dynamic range is low but the potential real time operation makes it very useful in development applications involving feedback control loops.
 

Offline artag

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #10 on: March 07, 2019, 09:09:07 pm »
Couldn't the dynamic range be improved by aggregating multiple measurements ? You don't need to do single shot capture and, by the same token, you don't need real-time capture. So by taking longer over the measurements (and a traditional VNA would have to do a sweep) you can improve the accuracy.
 

Offline TAMHAN

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #11 on: March 07, 2019, 09:45:58 pm »
rhb, well, I am semiretired too now that I have taken possession of my lab - so no excuse not to get up and type a response to this. Attention: I am not an RF engineer. I never used a VNA in anger even though I own one.

I was hanging out at EmbeddedWorld last week or so, and at the PicoTest booth there was a Steven M Sandler (the author of the Power Integrity book) and he showed off something similar. It basically was an USB stick like trinket (cost like 3k2 USD) which farted out a very steep pulse. This pulse went into a splitter and into two channels of a Rohde and Schwarz analytic scope (one of the heavy ones). He then ran some kind of formula on the DSO, and had a TDR like trace on the screen.

I see two issues for our practical application:
  • The Rohde scope he used had like 10GhZ (ten!!!) of bandwidth
  • I use a LeCroy 9354AM which is similar to yours. AFAIK, these scopes can not crunch an arbitrary number

If you want me to, I can look up the pics of the trinket and the structure. But email me at tamhan aht tamoggemon roundthing com so that I can send them easily. Sadly, all else I can offer is to put you in touch with Sir Sandler, other than that I do not think I am of help (unless you want some old GPIB code for Visual C#)
Feel like some additional tamile wisdom? Visit my YouTube channel -> https://www.youtube.com/user/MrTamhan for 10min tid-bits!
 

Offline rhb

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #12 on: March 07, 2019, 10:36:57 pm »
Leo's pulser is powered via USB and also can be adjusted using a windows app.  I just use it as is, but you can change the voltage if you want to test something fragile.  I have both the square wave pulser and the spike pulser.  They were under $80 each delivered to the US.  A PC is only needed if you want to change the settings.

https://www.eevblog.com/forum/projects/yet-another-fast-edge-pulse-generator/

@artag did you not see this?

Quote
The Owon in 12 bit mode will give me a 20 Mpt record. So by segmenting the record, doing an FFT and phase aligning before summing I should get around 100 dB of dynamic range.  The phase alignment is needed to eliminate jitter.

You gain 6 dB of dynamic range every time you double the number of samples which are summed after doing the phase alignment.

I'm getting a clean 10 nS pulse from my F***Tech FY6600  on my LeCroy DDA-125 which goes to 1.5 GHz.  The LeCroy shows a ~700 pS pulse trace from Leo's 100 pS pulser, so that's an accurate measure of the F***Tech set to 10 nS output at the 2 V setting..  Actual output is 1 V into 50 ohms.

So I'm going to write software that will perform vector network analysis up to the 6 m band using a F***Tech and a 100 MHz DSO.  I'm likely to just do the conversion to magnitude and phase vs frequency and leave the Smith chart to someone else.

I'll post more details after I read Dunsmore's book and dissertation.  While I'm at it I'll also write a decent FFT  for spectrum analysis on a PC using data from a DSO.

 

Offline DaJMasta

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #13 on: March 08, 2019, 03:11:38 am »
I am by no means an expert, but I mentioned before I thought it could be done with the Owon scope you mentioned, and now I'm having my doubts.

The issue is that I don't think accurate TDR measurements can be made without a high bandwidth digitizer and a fast pulse.  Basically, if you use a fast pulser with a rise time dramatically faster than the scope you're measuring the reflection with, won't the reflection be entirely covered up by rise time of the scope?  Won't all the data on the details of the reflection be between two sample points at 500MS/s whereas you'd catch them at 10s of GS/s?

Similarly, if you slow down the pulser to match better with the scope's risetime, won't you only be able to see reflections at a very long distance from the scope?  So a 1m BNC that you're measuring the termination on the opposite end of would have the reflection arrive too quickly back to the source to be able to see in detail on the scope (similar to the last problem).


Not sure if these concerns are 100%, like I said I'm not terribly familiar with the intricacies of TDR, but I've been unable to find any discussion of TDR at 100MHz frequencies or lower... it always seems to be with a source/scope in the 100s of ps or less performance, and I get the impression you will need that to see enough detail to make usable VNA measurements.  Worth confirming before you buy any hardware, at least.  Maybe putting in the LeCroy's bandwidth filter and doing some low frequency measurements could tell you if it was a potential issue.
 

Offline rhb

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #14 on: March 08, 2019, 03:49:10 am »
The key is to use a delay line between the test fixture and the DSO.  With the source at the DSO, the incident wave will trigger the scope, then the reflection will appear a little later.  I'll set up  different DSO and get a screen dump.  The LeCroy will only print and I'm out of thermal paper.

It will be obvious once you see a picture.  The problem is my description which assumes you're familiar with the 1D reflection problem.  My failure, not yours.
 

Offline OwO

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #15 on: March 08, 2019, 04:25:07 am »
I think it's probably inherently more difficult to get good measurements with pulses in the time domain because all the reflection information is contained within a very short span of time, whereas with sweeped sinusoid you can coherently detect and average over the entire duration of the sweep, and keep the amplitude closer to the maximum of the ADC to fully exploit its dynamic range. However that does point to other possibilities, for example instead of using a pulse (where all frequency components are equiphase) you could use a pre-generated "noise" signal that is flat in the frequency domain but with random phases. In the receiver you would "rotate" the frequency bins to cancel out the phase error since the "noise" signal spectrum is known.
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Offline loxodes

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #16 on: March 08, 2019, 06:01:24 am »
It is possible to throw together a sampling oscilloscope/TDR with a digital delay chip and a few high speed comparators.
This might provide bandwidth an order of magnitude or two cheaper than going for real time oscilloscopes.

I'm currently testing a 90 GS/s (equivalent time sampling) TDR with a bandwidth of maybe 8 GHz using a topology pulled out of the AD9500 datasheet.
(https://github.com/loxodes/tdr)
« Last Edit: March 08, 2019, 06:03:01 am by loxodes »
 

Offline rhb

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #17 on: March 08, 2019, 01:53:22 pm »
I think many don't understand what this is about.

1 - I've learned that the use of TDR for VNA work has a long history.  I was not aware of that which is why I started the thread.

2 - The purpose is to provide VNA capability for the HF bands using a hobby grade  DSO.

3 -  The actual DSP will be done on a PC using a maximum buffer depth DSO trace.

I'm getting a clean,  sinc(t)**3 15 nS pulse from a F***Tech FY6600 on an Owon XDS2102A.  There is a little bit of ringing in the AFE of the Owon. I know most of it is caused by the AFE because it's not the same as on the 1.5 GHz LeCroy I was using yesterday.  The ringing is easily corrected in the DSP.  This implies good frequency response to over 100 MHz and is entirely adequate for use as a VNA for the HF bands.

A  sub 15 nS pulser can be produced using a single 74HC00 wired up to produce a single shot pulse and retriggered by a 555.  That is *very* cheap to build.  The connectors and cables for the test fixture will cost more.   I'm hoping that I'll have a bit of luck today and find the 7400 based single shot I built 30+ years ago.

The Owon provides 12 bits at 500 MSa/S with a buffer depth of 20 MPts.   That will provide a dynamic range of 100 dB which is in the same ballpark as an HP 8753B I'll be comparing the results with once I get adaptors to fit the 85046A S parameter test set and learn how to use it.

The same results can be obtained with an 8 bit DSO with a shallower buffer by using multiple traces.  Every time you double the number of sweeps in the time window of interest you get another 6 dB of dynamic range (i.e. one more bit).

If you're not interested in radio this is probably of little or no interest.  But if you want to build ham gear or need a notch filter to suppress a nearby transmitter which is saturating the front end of your SDR, it is extremely valuable.  The dynamic range won't be as good in the FM broadcast band, but it will still be usable.

I spent 12 years in university, so I was very poor for a long time and *very* frustrated by T&M limitations.  All I had was a 5 MHz, single trace, recurrent sweep Heathkit IO-18, a VOM, a DMM,  a 5 V PSU I built and a 12 V PSU I repaired.  I failed miserably in my attempt to build a 40 m DC receiver because I could not measure the response of the filters I built.  I now have the benefit of 30 years of professional DSP experience and a budget that lets me buy anything I need.  My strongest interest is in low cost T&M kit. Yhings a person on a tight budget can afford.  There is a lot of cheap Chinese T&M gear and eval board modules available.  I'm looking for opportunities to improve the benches of those who cannot afford to spend much money.  If you have a F***Tech or similar you can sweep an HF filter.  But if you just bought a DSO and don't have the money for an F***Tech, a 74HC00 and a 555 will do until you can afford it.
 

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #18 on: March 08, 2019, 02:26:47 pm »
If this is just about building the cheapest possible VNA I'm on it already, prepare for a sub-$30 VNA to hit the market soon ;)
If you want even cheaper then bypassing my profits and just building one yourself from a si5351, stm32, mixer, and some passives isn't very hard. I'll be publishing design files.
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Offline rhb

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #19 on: March 08, 2019, 04:12:59 pm »
I'm looking forward to seeing what you've done.
 

Online _Wim_

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #20 on: March 16, 2019, 08:34:50 am »
It is possible to throw together a sampling oscilloscope/TDR with a digital delay chip and a few high speed comparators.
This might provide bandwidth an order of magnitude or two cheaper than going for real time oscilloscopes.

I'm currently testing a 90 GS/s (equivalent time sampling) TDR with a bandwidth of maybe 8 GHz using a topology pulled out of the AD9500 datasheet.
(https://github.com/loxodes/tdr)

 :-+  Impressed
 

Online _Wim_

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #21 on: March 16, 2019, 08:36:30 am »
If this is just about building the cheapest possible VNA I'm on it already, prepare for a sub-$30 VNA to hit the market soon ;)
If you want even cheaper then bypassing my profits and just building one yourself from a si5351, stm32, mixer, and some passives isn't very hard. I'll be publishing design files.

Can we pre-order?  ;D

What bandwidth? What specifications are already know or what is your target?
 

Online _Wim_

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #22 on: March 16, 2019, 08:48:48 am »
@rhb

How will you configure the correction for the Owon? Would it be possible to "import" corrections for other 12-bit scopes with high memory depth. This would be a great application for my 12-bit picoscope also.  ::)

Could the correction be something like recording the step response directly on the scope (without reflections) where your software uses some kind of convolution to correct for the scope ringing, phase and bandwidth limitations? Or is this a bit optimistic?
 

Online tggzzz

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #23 on: March 16, 2019, 09:29:16 am »
Another nice affordable toy (+-100$) to buy is a Tektronix 1502B or C (they appear quite often on Ebay, with most of them the LCD backlight is gone, but they are perfectly usable without). With one of these I discoverd some of my cheap BNC-BNC cables with printed RG-58 numbering were actually RG-59 of another 75 ohm coax... Lost cost has indeed some traps. OTOH, some of the RG316 BNC-BNC low cost ones were actually quite good

Go for a Tek 1502 without the suffix. Higher bandwidth, shorter range, static-sensitive tunnel diode step generator. They will not turn on unless there is a functioning battery in them.

50ps risetime, system roundtrip bandwidth 140ps, so you can work out the resolution.

I've sold some by having a 300mm piece of microstrip, and letting people watch the trace wiggle as I lightly touch the microstrip with first and index fingers, and move them apart and together. That viscerally convinces them of the resolution and sensitivity. Plus the lovely old-world sampling display :)
« Last Edit: March 16, 2019, 09:30:48 am by tggzzz »
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Online tggzzz

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Re: Using a fast pulse and DSO for vector network analysis
« Reply #24 on: March 16, 2019, 09:41:41 am »
A  sub 15 nS pulser can be produced using a single 74HC00 wired up to produce a single shot pulse and retriggered by a 555.  That is *very* cheap to build.  The connectors and cables for the test fixture will cost more.   I'm hoping that I'll have a bit of luck today and find the 7400 based single shot I built 30+ years ago.

I now have the benefit of 30 years of professional DSP experience and a budget that lets me buy anything I need.  My strongest interest is in low cost T&M kit. Yhings a person on a tight budget can afford.  There is a lot of cheap Chinese T&M gear and eval board modules available.  I'm looking for opportunities to improve the benches of those who cannot afford to spend much money.  If you have a F***Tech or similar you can sweep an HF filter.  But if you just bought a DSO and don't have the money for an F***Tech, a 74HC00 and a 555 will do until you can afford it.

https://www.eevblog.com/forum/testgear/show-us-your-square-wave/msg1902941/#msg1902941

I too have an "unlimited" budget for the same reasons as you, but, for the reason in my .sig, I too enjoy doing more with less.
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
Having fun doing more, with less
 


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