So I'm dicking around with excel and calculating how to replace some
precision¹ resistors in a few pieces of vintage test equip I have and the thought has occurred to me:
I can build up series strings to hit my target value, or I can build up parallel strings to do the same. I could even go further and do
both² in the same string if I really wanted to.
My current parameter is no more than 4 resistors total to achieve target value, strongly preferring combinations of 2 from the E96 set, but if they're E24 or E12 values I'll go up to 3 units (cost is less, usually.) and if they're some happy combination of E12 values I'll give 4 a try.
Actual example: I need a 7.00 MOhms resistor, the current one is a 1% spec unit and it is measuring 7.189 MOhms, so it is currently 1.7% out of tolerance(2.7% total deviation from spec.) and needs replaced. I can't get a 7 Meg resistor in 1% or better for any
sane price, but I could get a 5 and a 2 and series them for all of $1.85, both are 1% resistors. I could also get a 33 and an 8.87 for $1.22 and parallel them for 6.991 MOhms which is 0.1% from target.
So my question is, if I series them, the tolerance error of the string is the same as the units themselves. (So 1% if I use all 1% units.) If I'm paralleling 1% units, is the additional 0.1% margin from not hitting the target value precisely more or less helpful than the series option? Its one order of magnitude less than my target tolerance.
Now if I did something really
silly³ and sourced 0.5% or even 0.25% units, even with the additive 0.1% error I'm going to net to something less than 1%, right? 0.6% and 0.35% respectively.
Returning things to the less
side of the line, in the parallel example I *know* I'm going to be 9.076 KOhms short of ideal, so lets fix that. Now I'm going with a 33 and 8.87 in parallell, and then put a
9.09 KOhms in series, I get 7,000,090 Ohms, which is 0.0013%
over target value.
† Total cost on this is: $1.37, a savings of $0.48 over the straight 1% tolerance series combination.
I'm basically asking someone to confirm my math here, it is entirely possible I haven't paid enough attention to how tolerances work when making strings. I'm operating under the supposition that if I string together two 1% tolerance units I'm actually statistically more likely to get a value that's off by less than 1%. (Say the first unit is over by 0.39%, and the second one is under by 0.24%, if they were both the same marked value then in series they'd be 0.15% over marked x 2.) I'm not entirely sure how that kind of thing works with parallel. I am sure it would be a very serious spreadsheet indeed, because the % deviation of the higher value affects the equation differently than an equivalent % deviation of the smaller value would.
Additionally, I'm actually restoring three electrically identical units of one device, and two of another. So in both cases, I could measure the resistors I receive and possibly pair them up to hit even closer to target when the same resistor is out of spec in more than just one unit.
I'm interested to hear what you have to say on the subject. I do
value‡ your input.
¹ For the purposes of our discussion this will be the original stated design tolerance of 1%. I'm aware of the fact that what I'm doing here is seen by some as stupid. Sure, I do own very nice modern test equipment that is also quite accurate. I can see no reason why I can't improve the accuracy of things I work on beyond their original design spec by replacing out of spec things with original spec value but to the tighter tolerances that are easily affordable in modern times... Yes, I am taking the temp-co of the components purchased into consideration, as that affects the actual accuracy more so than stated tolerance and value at 19C.
² I will agree that this is slightly over the "obsessive" line, for certain. However, sometimes you can get three more common value resistors for much less than two not-so-common but not-quite-unavailable values.
³ Purely a what-if scenario. The cost to do this is
.
† Because, you guessed it! 9.00 KOhms isn't a sane value. Sure, I could get one, for a whopping $2.11, but
. Plus its quite large, physically...
‡ What unit do we use to express that? dB? Possibly dB/fg? (Decibels referenced to a single "fuck given?"
)