Influence of switch resistance in Hamon Dividers

[Lesolee]

Thanks!

Just for the clarification: You're describing the series resistance from the switch to the input of the divider.

Yes, just as Kleinstein noted earlier. They didn't show the switch(es) in their simplified circuit.

[Lesolee]

Although not requested, I saw a note in the manual (3.21) claiming that by adjusting the (ideal) parallel value you make the (ideal) series value exact.

I found this a bit dubious so I tried it, and wrote it down in case anyone else is interested.

[Dr. Frank]

Lesolee,

if you evaluate your calculation for 2nd order terms, you'll derive that this 0.01% = 10^-4 mismatch transforms into a quadratic error of the ratio, i.e. to 10^-8
That's why the SR-1010 ratio standards can give <1ppm ratio for 100:1 transfers, with only 20ppm precise, fixed resistors.

Frank

[e61_phil]

Please don't mix up the series to parallel configuarion and the switch resistance.

For the ratio the combination of the resistors isn't that important. But the switch resistance has exactly the influence I described in the first post. The is no magic hamon stuff here. It is a very simple linear network.

[e61_phil]

This Hamon adjustment condition you always miss in your calculations!

I don't miss it. It was shown as an example in the Spice simulation.


Frank: If you think there is a wrong assumption or something else, please do a detailed calculation from the first picture of my first post. Without some magic hamon formulars.

Or: Do you disagree with the LTSpice networks? What do you think, needs to be changed to be correct?


Edit:
Just for demonstration I adjusted every of the four possible resistors in the textbook circuit. (R1-R3 series/parallel resistors; Rout lower 40k resistor).
I printed all values, so it should be easy for you to verify that.

R1 adjusted:


R2 adjusted:


R3 adjusted:


Rout adjusted:

[Dr. Frank]

Here's my manual calculation, first with the 3 legs directly in parallel, which effectively gives 2 times the switch resistance of 10mOhm, or 0.05ppm error as expected, and 2nd with star / triangle calculation which yields 3 times R_sw, or  30mOhm (not intuitively) or 0.075ppm.
All w/o any magic, only the Hamon adjustment criterion used.

Sorry for the bad quality, the camera has got mumps, obviously.

Your SPICE calculus yields even 8 times the switch resistance, as the adjusted 120k leg is  80mOhm off.
There's something fishy, very obviously, but I can't figure out, where this happens.
Maybe the triangle / star ansatz is already wrong, as the 3 times error that I have calculated, is also counter-intuitive.. I would expect some reduction /compensation for this switch error, instead.

Frank

[e61_phil]

Hi Frank,

in the very first line you add dR, but I cannot see why.
Maybe, we should try to solve it with LTSpice only to not run into some calculation error.
Therefore, my first question is: How should we trim the bridge? Is one of the resistors in the upper leg ok?

Edit:
I attached a Spice simulation and I used the value I calculated for a trimming of R2 (see above). As you can see the bridge is perfectly nulled and after switching into the 1:10 mode you will get an error of 0.2ppm.

[Dr. Frank]

That dR are simply the switch resistances of 10mOhm each. The middle leg is trimmed so that that X is the correction value which later turns out to be these - 2 dR.
Very obvious.

I prefer old school  paper math, as just simply putting it into a simulation tool often swallows / covers errors.

Your Spice should yield the same result, as my paper stuff, so there's obviously something wrong. 8 times deviation of the middle resistor leg makes logically absolute no sense.. where should that come from?

Frank

[iMo]

Be careful with LTSpice results. With the default settings you will not see expected results when messing with high-res inputs/outputs.
See below examples how to get high-res results.

[e61_phil]

I prefer old school  paper math, as just simply putting it into a simulation tool often swallows / covers errors.

My paperwork agreed from the beginning. Spice was in the first post only used to double check the results.

I attached the same calculation again. This time everything with paper and calculator (HP48). The result is still exactly the same. I wrote down many steps in between to make it easier to verify.

[SilverSolder]

[...]
I prefer old school  paper math, as just simply putting it into a simulation tool often swallows / covers errors.
[...]

Or even worse, introduces new errors specific to the tool...

[e61_phil]

For additional verfication I made a quick and very dirty experiment. I took three 100k resistors (100ppm/K ones only) from the drawer. Soldered them together and meaures every one with a 34401A. I also used two 1.2R resistor (intentionally higher value to make the effect more visible). The "shorting" resistors are applied with cables and clips. I measured the shorting resistors togehter with cables and clips of course.
After applying the shorts I measured the resistance of the parallel configuration and the measurement agrees within 20ppm with the calculated values. The Spice simulations again agreed perfectly with the calculation, which makes numerical problems very unlikely.


PS: If you calculate the deviation between the simple parallel circuit of R1, R2 and R3 and compare that to the measured one, you end up with ~33ppm. That is roughly 100x more than 0.2ppm which was calculated with 100x smaller shorting resistors.
On the other side it is still only 33ppm. Perhaps, I should repeat this with even higher shorting resistors.

[e61_phil]

I couldn't resist

This time I used 100R resistors for the shorting. That gave an error of ~2700ppm versus ideal shorts. The measurement and the calculation agreed again within 18ppm. The Spice simulations also agreed again perfectly.

And again: 2700ppm / 100R * 10mR ~ 0.2ppm

[Lesolee]

Please excuse me for repeating what has already been done. I just wanted to check it independently.

I have used SIMetrix 8.20. This is SPICE-like but has its own custom computational engine.

Part A: "First test your test-equipment"



The upper chain is 0.01 ppm high. The result is 0.005 ppm low at the divided point. This agrees with the Windows 10 scientific calculator (and is obvious anyway).

In standard convergence mode the divider read +2mV on the x1,000,000 output. (0.002 ppm high). NOT GOOD ENOUGH for our use. Therefore it was put into Extended Precision mode to get the correct result.  Test method validated. 

Part B: Calibrate the parallel configuration

I tuned the top resistor. The Fluke manual doesn't say which exact resistor is to be trimmed. Yes, I did go a bit over-the-top getting it spot on!



Part C: Division!



I am now very happy that whatever Fluke have actually done is not shown in their description, because the actual error is 5x bigger than they claim.

Yes I know that has already been clearly stated (several times), but there were issues about the validation of the result.

[iMo]

With LTspice you may get the same results..

[e61_phil]

With LTspice you may get the same results..

And with Python, Excel, HP calculator...

[Lesolee]

I had to try it. I adjusted the middle resistor to see if it made much difference. The value was reduced by 26 nano-ohms to get the new balance. The series mode was barely  different.

[e61_phil]

I had to try it. I adjusted the middle resistor to see if it made much difference. The value was reduced by 26 nano-ohms to get the new balance. The series mode was barely  different.

In reply #29 I solved it for every resistor. You can verify the results within your simulation.

[Kleinstein]

One can calculate the error to the parallel connection approximately to first order relatively simple:
The current through the switches is very close to  2/3 the total current. So the voltage drop for every switch is 2/3 * I* R_switch.
The upper and lower resistors sees one such voltage drop, while the center resistor sees 2 of the switches.  So the average extra voltage is 4/3  * 2/3 * I * R_switch. So the extra resistance for the parallel connection is very close to 8/9  times the switch resistance.  No real need to calculate more accurate as the three 120 K resistor may very likely differ more than R_switch from each other.  This result is very close to the initial number - withing the rounding used.

The Flue formula  (3-1) was for a different error. So for the parallel connection Fluke either used better switches or  some form of the compensation circuit. The resistors for the compensation only need to be larger than the switch resistors by something like a factor of 10 to get a reasonable good compensation - this may be just some cable / PCB trace parts. They don't give the details, but vague indication:

[e61_phil]

So the extra resistance for the parallel connection is very close to 8/9  times the switch resistance. 

Very nice analysis!

You couldn't be closer with the added resistance

[Lesolee]

The extra resistance needed in the series configuration is just under 8x the switch resistance, so that's 80 milliohms. We then have to suppose that the switch resistance is absolutely stable.

I can’t think that adding 10 series contacts in the series path is reasonable. On the other hand adding 10 parallel contacts in the parallel circuit (and one in the series circuit) would make a more robust system.

[e61_phil]

And now that we've verified that the equations here match, the influence of the leakage resistance.

10mR during calibration and the sweep during dividing.

[Lesolee]

And now that we've verified that the equations here match, the influence of the leakage resistance.

10mR during calibration and the sweep during dividing.

Confirmed at 100G and 1T. Your curves are very useful  .

[Dr. Frank]

I finally got time to check my calculation again, and found the error, third line on the 2nd page. It's 63*R*X on the left side, instead od 33*R*X, so that really gives a mis-alignment of -8*R_s, what I already found for the other configuration also, and what e61_phil found by means of his SPICE calculus.

Next step was to measure this R_s. On my 15 year old precision grade ELMA switch, all measured contacts (still) showed 5mOhm only, with a slight variation by switching actuation of about +/- 1mOhm.
These 5mOhm therefore give about -40mOhm adjustment error or ~ 0.16ppm. (with 100 * 25kOhm resistors)
That's still good enough, but obviously can be compensated, by the remark in the Fluke manual "... adjustment of the interconnection resistances.."
This switch resistance variation gives +/- 8 mOhm, or an estimate of +/- 0.032ppm, which might be the root cause for these 0.042ppm worst case, Fluke mentioned.
Maybe they also include in this parameter a permanent increase/ageing of the switch contacts.

All three parameters, the low nominal switch contact, its variation over actuation, and its deterioration, I would expect inside the specification of their custom made switches.

Third step was to compare my findings on my box with the 752A schematics, and to identify this compensation / adjustment of the -8*R_s, and of further cable / switch resistances.

I have to admit, that I didn't have the schematics available 15 years ago, and I did not check this paragraph 3-30 carefully enough.

Long story short, this compensation is done very obviously by a strange long cable, denoted as: '36.75 inches long AWG 22 solid Cu'.
That exact specification gives 49.4 mOhm, or 0.12 ppm compensation.
This cable-resistor is connected in series with the upper 360 kOhm leg of the divider, but only in the unfolded, divider mode.

So that's the way, how to compensate these said effects.
The value of this resistance also has the correct order of magnitude, so one can conclude, that the switches of the 752A also have about 5 mOhm contact resistance.
Fluke obviously made a precise analysis and summation of all involved cable lengths and number of switch contacts and then determined this specific length of this ominous cable.
It should be located between both switches, coiled up, although I think there is another failure in the schematic, as this compensation is not effective in the 100:1 divider.

Anyhow, I think that I now have fully understood this aspect. Sort of black magic inside the 752A, which is engineered very well, over all.
I also think, that it's possible to replicate its high accuracy using volt-nuts grade facilities.

Frank

 

[e61_phil]

and what e61_phil found by means of his SPICE calculus.

Not only spice. Spice was only to verify the calculations done in python.

Did you measure the 5mR including the cables?
What do you expect for the leakage resistance? With 1e12 Ohms you would end up with roughly 0.35ppm in my calculation with your configuration (75k instead of 120k). At 1e13 and higher the error is dominated by the 5mR contact resistance.