You can also mention in your proof that the voltages between the two parallel resistors are identical at each point and thus there is no current between them. Otherwise, it would affect the analysis.
Also, I presume that the resistance/square depends also on the thickness. Is the resistance per cube (face to opposite face) also fix? If so it may be an even more canonical characteristic.
You can use the same idea for cubes: 2mm^3 can be divided in four 1x2m^3 resistors, so if a 1mm^3 resistor has n ohm, then a 2mm^3 resistor of the same material has n/2 ohm (and a 3mm^3 cube has n/3 ohm).
Thanks for the Video, Dave!
Actually, I need to jump in here: The example you show here is a somewhat "low-end" example of recent designs by Fluke / Danaher. This technique of laser trimming through glass is not new (been around since at least late 60's / early 70's) and one of the main problems with what you see here is the fly-ash contamination that results from the laser trim process. That ash deposited on the inside glass is not completely inert, and eventually finds its way back onto the network circuit (light, vibration, static fields etc all promote migration). Let alone the mechanical stress on the substrate when you burn into it with the laser. All of this can show up years down the road as a bit of instability in the instrument if its not calibrated often. This is no longer considered "best practices" in precision measurement circles. A stable circuit is a clean circuit!
Very stable resistor networks are done on a substrate using transparent sapphire, not the white (probably) alumina as shown here. Both materials are in the same "ceramic" family, but sapphire can offer a superior end product with better thermal properties. (Its very expensive to work with though and a pain to cut it apart). If you want the resistor network to remain stable, you -don't- touch it with laser trim, no matter what the substrate. The resistor assembly would be cleaned of -all- contaminates, welded into a shielded light-tight / humidity-proof and vibration proof hermetic metal package, and mounted onto a PC board with something besides that wobbly, cheap SIP technique. That's how we do it for military & aerospace applications anyway.
Watch out for thermals around the pins of that SIP package the in the example shown here. This might be only for high voltage input, but if you were trying to attenuate lower level voltages with any sort of precision, the resistor network would not be mounted on the PC board like that. In general, those pins all need to be at the same temperature for low level signals. It could be they weren't after that, since that ~9M ohm resistor is going to make all sorts of thermal noise on its own.
As another poster pointed out, if you have to trim a resistor network to this level, you're not doing the software right somewhere else, maybe. For the last decade or so we have cheap 32 and 64 bit MCU's to handle calibration duties much easier than this, and it results in a much more stable, reliable product. You want a STABLE resistor divider; trying to hit some cardinal divide ratio is usually not necessary at all if you calibrate in software. Just build the resistor divider very close to what you need, then take care of as much as possible for trimouts during software calibration. We haven't needed to use a precision-trimmed resistor divider in years...kind of obsolete like CD's and cassette tapes. And we build precision test / measurement devices all the time.
Of course if low noise is what you need, then good precision Wire Wound resistors, coupled with a thermocouple for temperature compensation at the controller end, can easily outperform printed or diffused resistors used on front-end attenuators for freqs < 250kHz or so.
Thin / thick film network circuits were used extensively by Beckman Instruments, General Instruments, Bourns and others even back to the mid 50's that I know of. The substrates were on glass, sapphire, alumina and any other hard, temperature stable material. In pre-software days the resistors would be trimmed with abrasive techniques (sand blast, bead blast), water jets, diamond saws, microwave / maser, ultrasonic probe, radiation sources, etc. Generally these were for aerospace & military equipment, and in use for many years before Fluke started selling calibration equipment.
Resistance per cube is called bulk resistivity (e.g., copper is around 17 nOhm m).
It's not "per cube" because it *does* vary with a dimensional unit.
Interesting fact about quantities: when you see a bulk parameter that's got a single order unit (m or m^-1), it's usually a cubic dimension, not simply a linear dimension. That is, you factor the length of the region in the same proportion as the given unit, and its cross-sectional area inverse to that. So, bulk resistivity has meters on top, so multiply by length (= resistance is proportional to length) and divide by area to get ohms. Conversely, conductivity is S/m, so divide by length and multiply by area.
Tim
Resistance per cube is called bulk resistivity (e.g., copper is around 17 nOhm m). It's not "per cube" because it *does* vary with a dimensional unit.
It was actually resistance per square.
It's quite simple, really.
R = rho * L / A
The thickness of the copper layer that gets deposited on the ceramic substrate is fairly accurately controlled, so you can divide resistivity by that thickness and you get what is commonly called "sheet resistance".
So:
R = Rsh * L / W
If you look at a square of copper, L / W = 1 / 1, so the resistance of the opposite sides of that square are exactly Rsh.
But here's the trick:
opposite sides. You can't simply add squares like Dave added them in the video. The resistance of a 90° corner square would be 0.56*Rsh, for example. The resistance of a square, where the current enters and leaves on the same side of the square (the top square in Dave's example), would be something different entirely.
Hm, LTFLU voltage reference in DM7510? That's surprising Expected to see LTZ there.
I thought it was indentical to the LTZ? Just labeled for Fluke, and perhaps binned better?
It's quite simple, really.
R = rho * L / A
The thickness of the copper layer that gets deposited on the ceramic substrate is fairly accurately controlled, so you can divide resistivity by that thickness and you get what is commonly called "sheet resistance".
Yeah, when you have one dimension of the cross sectional area predefined, it becomes per square (dimensionless, so depends only on the geometric ratios, not the scale). For example, copper at 17 nOhm m / 35um as in PC board = 0.48 mohm/sq.
Tim
I would like to know what the VPG hermetic foil is for-- [VHP103?] Maybe for an "auto-cal" feature for ohms?
That was my guess, yes, just like in the new Keysight.
Looking at the image of those networks, I see loads of parasitic capacitors and inductors. The AC response would be dramatically complex. Could you elaborate a bit on that Dave? I can imagine an RMS to DC converter is placed before the resistor network, but then that converter would need similar accuracy (or at least stability so an error can effectively be calibrated out) as the resistor network.
There is a current density and potential distribution for one of the resistors in the video.
I did only a single layer resistance - constant value.
Looking at the image of those networks, I see loads of parasitic capacitors and inductors. The AC response would be dramatically complex.
The DMM7510's AC performance specification only goes up to 300kHz, at which its accuracy is only within 4% reading + 0.5% range.
I doubt the 9.9M resistor has more than 1nF and 1nH worth of overall capacitance and inductance. That would affect accuracy on a 300kHz input signal by less than 0.001%. Even if my guestimate was off by two orders of magnitude, it would still be well within tolerances.
Looking at the image of those networks, I see loads of parasitic capacitors and inductors. The AC response would be dramatically complex.
The DMM7510's AC performance specification only goes up to 300kHz, at which its accuracy is only within 4% reading + 0.5% range.
I doubt the 9.9M resistor has more than 1nF and 1nH worth of overall capacitance and inductance. That would affect accuracy on a 300kHz input signal by less than 0.001%. Even if my guestimate was off by two orders of magnitude, it would still be well within tolerances.
The layout does it's best to make power dissipation per unit area as close to equal as possible, and the long and convoluted tracks are going to have low self inductance, and reasonably low capacitance between tracks with large voltage potential, and as they all are in series with each other it will be quite non inductive and reasonably low capacitance. Designed to be as easy to compensate as possible with only a single low value high voltage capacitor or to have a easy to compensate frequency response.
Designed to be as easy to compensate as possible with only a single low value high voltage capacitor or to have a easy to compensate frequency response.
I doubt there is any frequency compensation to be done here: if each loop is 20pH and there are 200 of them, that's 4nH and at 300kHz, that's 7.5mOhm vs 10Meg or 1ppb of error. That's five orders of magnitude beyond the meter's 900ppm accuracy on its best AC ranges, never mind the 45000ppm accuracy spec at 300kHz. The unnecessary AC compensation components would likely have a much worse detrimental effect on DCV accuracy and drift specs than any AC accuracy gain it could provide.
I doubt they would risk losing 1ppm on DC accuracy to improve AC accuracy by an effectively non-detectable amount.
The complex LC response of such a network will be much worse than you'd imagine, and far more involved than a simple lumped capacitance and inductance figure.
Hairpin loops do have low inductance, but it's not zero. I'd have to guess more like 100nH. But this will be utterly negligible, because with R so large, the L/R time constant is minuscule (e.g., 0.1u / 10M = 10 fs!). Likewise, stray capacitance will be more like 10pF, but it's distributed over the whole thing, so it's more like a chain of RC filters, half to ground and half bridging across various points in the path itself. It will have extreme "hook" -- there's a Tektronix appnote on this subject, as it applies to oscilloscope attenuators and probes.
They may well simply add a dominant pole-zero (i.e., swamping the stray capacitance with external capacitors), giving a significantly reduced input impedance at AC, in exchange for an at least reasonably predictable frequency response.
Tektronix' preference for this sort of thing is a chain of small resistors with capacity hats, keeping everything as small and simple as possible, while still maintaining sufficient voltage standoff and power rating to handle the high voltage. The, P6015 probe is it? has a dielectric fluid (freon??) backfill, to maintain the voltage and power rating of its components while keeping the frequency response right.
Tim
The complex LC response of such a network will be much worse than you'd imagine, and far more involved than a simple lumped capacitance and inductance figure.
In any case, my point to the original post I responded to was: the DMM7510 has nowhere near the AC precision and bandwidth necessary for the resistor network's parasitics to have any meaningful effect on AC measurement accuracy even if my guesstimates (or yours) were a few orders of magnitude off.
While I have no doubt there are some very fancy resistor networks and construction techniques in high-frequency, high-voltage probes and oscilloscope front-ends, the DMM7510 is a multimeter with only 4.5% AC accuracy at 300kHz. Its AC accuracy is already becoming miserable decades before the network's parasitics are likely to become remotely significant.
There is a current density and potential distribution for one of the resistors in the video.
I did only a single layer resistance - constant value.
What did you use to simulate that? looks pretty neat.
So how much does this kind of network gain the instrument? That resistor network likely costs more than all the components in the 2002 put together.
Not really, 2002 have hermetic foil (which are the most expensive) and ceramic networks as well.
Main cost is in aging and training for these custom parts to ensure their stability.
Not really, 2002 have hermetic foil (which are the most expensive)
3 individual ones for the resistance and current measurement ... but they are not bespoke and soldering stuff into metal cans is only artificially expensive. I seriously doubt they get anywhere near the cost of this monster.
