Electronics > Metrology

PCR versus TCR

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Zeranin:

--- Quote from: DiligentMinds.com on April 21, 2016, 05:53:07 pm ---And YES I do consider the mystery solved.  When the resistor self-heats, there is a temperature difference, and that explains the PCR.  What is actually happening is probably a very complex interaction between the TCE of the various materials, and the TCR of the Zeranin-30 element, combined with less than 100% efficiency in transferring  the heat from the element ...

--- End quote ---

And I maintain that you are 'hand-waving', without understanding what is actually going on.  :)

It's time for me to stick my neck out. I have always claimed to know what was going on, and that I would get around to explaining my ideas. What I have been doing so far is shamelessly using the excellent brains here to see what ideas others would come up with, which helps crystallize my own thoughts. I now feel sufficiently confident to present what I believe is happening, in full detail. Any detailed explanation will need to correctly predict the observed direction of resistance shift, as well as get the magnitude approximately right. I welcome critical scrutiny, that is one of the great benefits of presenting ideas on a forum.

 

Zeranin:

--- Quote from: d-smes on April 21, 2016, 08:19:54 pm ---Zeranin - Thanks for bringing your problem to the forum and for your detailed descriptions and responses.  Everyone loves a mystery!

I would like to question your assumption that the Zeranin is constrained in the X and Y axis.  As-built, the Aluminum X & Y are locked to the Zeranin X & Y through the heat-bonding, electrically insulating film.  But that film has mechanical compliance and its own TCE (which is safe to ignore).  The mechanical compliance can be thought of as an array of stiff springs connecting each Al X-Y coordinate to the same Zeranin X-Y coordinate at room temperature (as-built).  Now when heated uniformly in an oven (no current), the Al and Zeranin want to grow to two different sizes because of the differing TCEs but they can't because they are constrained by all these stiff springs.  So, does the Al get compressed to match the Zeranin hot dimensions or does the Zeranin get stretched to match the Al hot dimensions?  My guess is that the Al, even though it's softer, is thicker and ultimately has the higher stiffness.  So the Zeranin gets stretched when the assembly is heated.  But the springs of the insulating film also get stretched such that the hot X-Y coordinates no longer match; the Zeranin is stretched to slightly smaller X-Y dimensions than the Al.  In this regard, the rubbery thermal interface may have and advantage because it has more "give" (weaker springs).

Also recognize that in the dimension of the zig-zags of the Zeranin, there are gaps which will take up a lot of the differential TCE in that dimension.  However, these put additional strain on the corners where the zig bends to become a zag.  But I agree, that within each 25mm zig and zag the Zeranin is still getting stretched in both dimensions.

Now apply current.  The Zeranin gets 0.25K warmer and would have slightly larger dimensions, if it weren't constrained, by an amount I'll call dx and dy.  In this case, the Al X-Y coordinates stay the same (assumed constant temperature) but what will the Zeranin's X&Y's become?  I assume the spring constant of the insulating film remains constant and stretches the Zeranin by the same amount.  It follows then that the differential heated dimensional difference of dx and dy shows up as Zeranin's X&Y coordinates growing by dx and dy (in reality, slightly less than dx & dy due to slightly less spring force).

I believe there is also a slight temperature difference between the cooled side/face of the Zeranin and the top clamp side of the Zeranin.  This means the clamped face area expands slightly more than the cooled face causing the Zeranin to want to curl or cup (convex on the warm side, concave on the cool side).  Given the thinness and good thermal conductivity of the Zeranin, this is probably a second-order effect at best.  But it also results in different stress/strain and dimensional changes that may explain the powered vs un-powered difference in R-T.

--- End quote ---

I also love a technical/engineering mystery, and glad others do too. Personally I need to understand everything I observe and measure, otherwise I would go mad, even if it means considerable head-bashing and lost sleep until I see the light.

Thank you for thinking in detail about the stresses and strains that we would expect. I take your point that the thin bonding film will be very slightly compliant, so that the zeranin does not exactly follow the movement of the aluminium substrate. It remains my belief though, that for all practical purposes, the two are completely bonded together. The full explanation of what I believe is happening is rather long and subtle, and so far I have been flat out reading and responding to the many replies on this topic. Time has come for me to give my explanation, which does completely explain all observations, and I hope you will be convinced by it. Cheers.

 

Zeranin:
I have started a sister thread entitled Resistivity vs Temperature – flatter is better?, as a precursor to explaining what I believe is going on to produce the 5ppm drift that I observe immediately after the current through my Zeranin 30 shunt is switched from zero to 16A, and why the R-T curve does not apply.

If and when we have agreement on the sister thread, I'll resume my discussion here.

Zeranin:

--- Quote from: DiligentMinds.com on April 21, 2016, 05:53:07 pm ---You stated earlier that it is time to deliver to your client, and so there is nothing drastic that can be done, so you have to work with what you have [which, is pretty good-- so what's all the complaining about?]  That is why I suggested compensation at this stage of the design cycle.  Since the laws of physics are not going to change for your project, you will *never* reduce the natural PCR to zero, and if you need it to be zero, then compensation is the only way to make that happen.  5ppm is not that awful to compensate for as long as the PCR is predictable and reproducible.

If I were designing such a thing, I think I would start with Evanohm-R instead of Zeranin-30.  Very thick Evanohm bar [or "strap" or "sheet" or whatever].  This would be TIG welded to a heavy copper bar on each end.  The whole thing would be bent into the familiar "potato masher" shape, and then gold plated.  Then, a heat treatment process would begin by measuring the TCR, and heat treat again [iterate until the TCR becomes immeasurable].  Then, place the whole thing in an oil bath that is controlled to +/-0.01C ...  End of problem, but considering the labor involved, quite expensive.  This kind of shunt would be extremely stable [less than 0.1ppm/a drift].  If you were to contract with a supplier for a few of these, I would guess that it would end up costing you between US$5K and US$10K per shunt-- [most of which is labor].

*** EDIT ***
Almost all high-end DMMs compensate for internal shunt PCR [and some even compensate for the TCR]; and they do this in software.

--- End quote ---

I agree with all of that. I am not in a position to make major changes at this stage, though it would make sense for me to make an effort to reduce thermal resistance from zeranin to constant-temperature heatsink. I still have time and budget to at least have a go at that, and conceivably could gain a factor of x2 improvemnt. I hear what you are saying re compensation. Can't do it in software, as there is no software in this current driver. As the effect is non-linear because the heating goes as I^2, hardware compensation would be ugly and messy, and I have no intention of re-designig and re-building the current driver PCB.

Your suggested shunt design would perform superbly. To properly eliminate PCR effects, we agree that you really need to go naked.   

Zeranin:
Here is what I believe is going on.

When the current is switched from zero to 16A, the temperature of the zeranin sheet rises above the constant-temperature substrate, changing the resistance for reasons as yet unexplained. This small 0.25K rise in zeranin temperature won’t produce any measurable change in resistance by way of the measured R-T curve, because I’m operating on the flat part of the curve where dR/dT=0. As it is only the temperature difference between zeranin and substrate that produces a resistance change, I can equally well consider a drop in temperature of the substrate of 0.25K, an approach that simplifies the analysis. Therefore I will analyse what happens to the zeranin resistance when the substrate temperature falls by 0.25K, realizing that the result will be the same as if the zeranin temperature was to rise by 0.25K.

The aluminium substrate is much thicker than the zeranin, and they are bonded together, so any dimensional change in the substrate in the X-Y direction will be forced to also occur to the Zeranin. The expansion coefficient of aluminium is +22 ppm/K. If the substrate temperature falls, then it will contract by 22 ppm/K equally in all directions. The substrate contraction in the Z-direction will do nothing, because nothing is constrained in the Z-direction, as discussed in previous postings. However, the zeranin will be forced to follow the substrate in X and Y, with the result that the zeranin dimensions will contract by 22ppm/K in X and Y, but this will result in no change in resistance, because these two dimensional changes cancel, one being a Length term, and the other a Width term.

I am confident that the explanation so far is correct, predicting that an increase in the zeranin temperature above the substrate (or fall in substrate temperature relative to zeranin, same thing) will produce no change in resistance, darn it. That is not the answer we wanted, because we observe that the resistance damned well DOES change, that’s the whole problem.

Clearly there is some other dimensional effect going on in addition, that has not yet been considered, that causes the observed decrease in zeranin resistance, when the zeranin temperature rises relative to the substrate.

The explanation is found in the ‘Poisson effect’. Imagine elastically stretching a length of wire. Of course, the resistance increases because the length is increased. However, what also happens is that the diameter elastically decreases, thus increasing the resistance even further. This is called the Poisson effect, and Poisson’s Ratio is the ratio between the ppm change in length, and the ppm change in diameter, with a value of 0.5 corresponding to an overall conservation of volume. Now apply this to our case where the zeranin has been compressed by 22 ppm/K in X and Y, resulting in an increase in thickness in the Z-direction. In effect, when the zeranin is compressed in X and Y, it responds by ‘popping out’ in the Z-direction, in an attempt to maintain the original volume.
 
The maximum possible extent of this effect would be an increase in zeranin thickness of 44 ppm/K, being a 22 ppm/K contribution from X and Y. However, for typical values of Poisson’s ratio, the actual increase in thickness will be less than that, say around half, leading to an increase in zeranin thickness of 22 ppm/K. The observed rise in temperature is 0.25K, so this would lead to a decrease in resistance of 0.25 x 22 = 5.5 ppm, which is remarkably close to the ~5ppm decrease that is observed.

This analysis explains why the zeranin resistance does not change when the zeranin and substrate are heated in unison ( dR/dT=0 on R-T curve), but the resistance does change when the zeranin self-heats above the substrate, and both the direction and magnitude of the resistance change are correctly predicted. I feel confident that this explanation is correct.

If we translate the above analysis into a formula, we get :-

dR  = P x Rth x B x EC  (equation 1)
     
where
dR is the decrease in resistance, in ppm, as a result of self-heating
P is the power dissipation in Watts, self-heating the zeranin
Rth is the thermal resistance from zeranin to substrate, in K/W
B is a constant related to Poisson’s Ratio, ~1.0, but <2.0
EC is the thermal expansion coefficient of the substrate, in ppm/K

For my zeranin shunt example :-
dR = 25 x 0.01 x 1.0 x 22 =  5.5ppm

One can also rearrange equation 1, to give an expression for the Power Coeffcient of Resistance (PCR), in units of (ppm/K) per watt of dissipation.

dR/P = PCR = Rth x B x EC   (equation 2)


Knowledge is power. Now we can clearly see exactly what will help in reducing the self-heating-induced resistance drift, and what will not.

From equation1, we can reduce the resistance drift, dR, by reducing the dissipated power, P, or the thermal resistance from foil to substrate, Rth. No surprises there, we knew that already.

Some people suspected that the resistance drift was caused by difference in expansion coefficient, EC, of the zeranin and substrate, but not so. The EC of the zeranin foil doesn’t show up in the analysis or equations at all, and therefore there is nothing to be gained from choosing a substrate material that matches the EC of the resistive foil, at least as far as minimizing PCR is concerned. The formula clearly shows the only thing that matters for PCR is the EC of the substrate.

The self-heating-induced resistance drift scales directly with the EC of the substrate. Thus, an aluminium substrate (22ppm/K ) is a poor choice. Copper would be better, and steel significantly better, though the thermal conductivity of steel is less than ideal. Invar would be best, except that the thermal conductivity is so low as to be useless. The resistor manufacturer has further options with ceramics.

Note that whatever substrate material is chosen, it will always be necessary to arrange for dR/dT to be zero or small at the foil operating temperature. If the ECs are not matched, this will modify the ‘naked’ R-T curve which in my case I can account for (within reason) by operating at whatever temperature the sweet spot (dR/dT=0) happens to be at. The resistor manufacturer can account for this by tweaking the resistive alloy.
 
Please tear this explanation apart, and/or offer an alternative explanation that fits the measurements. Comments, please. If the explanation withstands scrutiny, then the mystery is solved and understood.

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