Sind zwei (oder N) Widerstände in Reihe genauer als ein großer Widerstand?

Angenommen, ich habe einen 2 kΩ-Widerstand mit 5% Toleranz. Wenn ich ihn durch zwei 1 kΩ-Widerstände mit einer Toleranz von 5% ersetze, wird die resultierende Toleranz steigen, fallen oder unverändert bleiben?

Ich bin mit Wahrscheinlichkeiten schlecht und ich bin nicht sicher, was genau Toleranz über Widerstand und seine Verteilung aussagt.

Mir ist bewusst, dass es im schlimmsten Fall dasselbe sein wird. Ich bin mehr daran interessiert, was im Durchschnitt passieren wird. Erhöht sich die Chance, einen genaueren Wert zu erhalten, wenn ich eine Reihe von Widerständen verwende (da sich Abweichungen gegenseitig aufheben)?

Auf 'intuitiver Ebene' denke ich, dass dies der Fall sein wird, aber ich habe keine Ahnung, wie ich mit Wahrscheinlichkeiten rechnen und herausfinden soll, ob ich tatsächlich Recht habe.

Der schlimmste Fall wird nicht besser. Das Ergebnis Ihres Beispiels ist immer noch 2 kΩ ± 5%.

Die Wahrscheinlichkeit, dass das Ergebnis näher an der Mitte liegt, wird bei mehreren Widerständen besser, aber nur, wenn jeder Widerstand innerhalb seines Bereichs zufällig ist , was auch einschließt, dass er unabhängig von den anderen ist. Dies ist nicht der Fall, wenn sie innerhalb eines bestimmten Zeitfensters von derselben Rolle oder möglicherweise sogar vom selben Hersteller stammen.

Das Auswahlverfahren des Herstellers kann den Fehler auch nicht zufällig machen. Wenn sie beispielsweise Widerstände mit einer großen Varianz herstellen, dann wählen Sie die Widerstände aus, die unter 1% liegen, und verkaufen Sie sie als 1% -Teile. Dann verkaufen Sie die übrigen als 5% -Teile. Die 5% -Teile haben eine Doppelhöckerverteilung ohne Werte, die innerhalb von 1% liegen.

Da Sie die Fehlerverteilung im Worst-Case-Fehlerfenster nicht kennen und selbst dann der Worst-Case gleich bleibt, ist es für das elektronische Design nicht sinnvoll, das zu tun, was Sie vorschlagen. Wenn Sie 5% -Widerstände angeben, muss das Design mit jedem Widerstand im Bereich von ± 5% ordnungsgemäß funktionieren. Wenn nicht, müssen Sie die Widerstandsanforderung genauer spezifizieren.

Die Antwort hängt stark von der Verteilung der realen Widerstandswerte und Ihrer Frage ab.

I did a simulation, for which I generated a set of 100,000 resistors with 1% tolerance (easier to handle than 5%). From this, I took 1,000,000 times a sample of two and calculated the sum of them.

For the set, I assumed three different distributions:

1. A narrow, perfectly gaussian distribution with $\sigma=2.5$. This means: 63% of all resistors are in the range $1000\pm2.5\Omega$ and 99.999998% are in the range $1000\pm10\Omega$.
   Think of a manufacturer with a reliable production process here. If he wants 1kOhm resistors with 1%, his machine produces them.

2. A uniform distribution where the probability to get any value in the 1% range is equal.
   Think of a manufacturer with a very unreliable production process. The machine produces resistors of any value of a wide range, and he has to pick out the 1%/1kOhm resistors.

3. A wide gaussian distribution ($\sigma=5$), where every resistor outside the 1% range is thrown away and replaced by a "good" one. This is just a blend of the first two cases.
   This is a manufacturer with a better process. Most of the resistors meet the specs, but some have to be sorted out.

Here is the result:

1. When adding two values of the same gaussian distribution, the sum also is a gaussian distribution with a width of $\sigma_{new}=\sqrt{2}\sigma_{old}$.
   The resistors have a tolerance of $\pm 10\Omega$, which converts to a new tolerance of $\pm 14.1\omega$ or $14.1\Omega/2000\Omega=0.7\%$.
   The simulated data shows this, too, as the distribution is slightly wider that 0.5% (vertical green lines)

2. The uniform distribution becomes a triangular distribution. You still get resistor pairs of 1980 or 2020 Ohms (5%), but there are more combinations with lower difference from the nominal value.

3. The result also is a blend of the results of the first two cases...

As said in the beginning, it depends on the distribution. In any case, the probability is higher to get a resistance with less difference from the nominal value, but there's still a probability to get a value which is 1% off.

Further notes:

• Often, a batch contains resistors which all have nearly the same value, which is a bit off the nominal value. E.g. they are all in the range of 995...997Ohm, which is still well in the range of 990...1010Ohm. By combining two resistors, you get a lower spread, but the values are all a little low.

• Resistors show e.g. temperature dependence. The precision is much better than 1% to ensure the resistance stays in the 1% range at different temperatures.

Fun question, Practically, when I was looking at 1% 1/4 W Metal Film R's I found that in a batch, the distribution was far from random. Most of the R's clustered around a value that could be a bit above or a bit below the "target" value. So at least for the R's I looked at it wouldn't make any difference.

There are two important numbers that have to do with your question.

The first is "Worst Case Scenario": In the absolute worst case, one 2k resistor with 5% will be either 2.1k or 1.9k. One resistor of 1k 5% will be 1.05k or 0.95k, added together this comes to either 2.1k or 1.9k. So in the worst case, in series, a bunch of resistors with the same tollerance will always retain their tollerance over the total value and be just as good as one big one.

The other important number is the law of large numbers. If you have 1000 resistors that have an ideal target value and are specified with an absolute maximum error of 5%, of course it's very likely that quite a few of those will be very close to the target value and that the number of resistors with too high a value is about as high as the number with a lower value. The production process for components like resistors falls under a natural statistical process, so it's extremely likely the resulting resistors in a large batch across multiple productions yield what is called a gaussian curve. Such a curve is symetrical around the "desired" value and the manufacturer will try to get that "desired" value to be the value he sells the resistors as, for statistical yield reasons. So you can make an assumption that if you buy 100 resistors, you too get a gaussian distribution. Actually, that may not be the exact case, with resistors a large enough number may have to be 10's of thousands to get a real gaussian distribution. But the assumption is more valid than that all will be off by the worst case in the same direction (all with -5%, or all with +5%)

That's all well and nice, but what does it mean? It means that if you have 10 resistors of 200 Ohms at 5% in series, it's reasonably likely that one will be 201 Ohm, another 199 Ohm, another will be 204 Ohm, yet another will be 191 Ohm, etc etc, and all those "too low" and "too high" values compensate each other and it becomes, suddenly, a big 2k chain with a much better accuracy, through the law of large numbers.

Again, this is only in the specific case of the same value resistors in series. While different values in series are also likely to become more accurate on average, the degree to which this happens or how likely it is, is hard to express correctly without knowing the exact use-case and exact-values.

So, it is, at the least, not at all harmful to place many resistors of same value in series, and usually it gives a much better result. Combine that with the fact that manufacturing a huge amount of boards with just 3 different components is much cheaper than with 30 different components and you often see designs with only 1k and 10k (or maybe 100 Ohm and 100k as well)resistos in cheap, high-volume-production trinkets, where any other value is a combination of the two.

Solid carbon resistors have all but ceased to exist on the market as they catch fire easily and change value with voltage. Now days 'carbon' is normally carbon film.

It is a much more stable resistor, but not as stable as metal-film or ultra-stable like the ceramic resistors made by Caddock. Usually 0.025% is available for about $50 each. A laboratory grade 0.01% or better cost about $150-for now.

Most boards that I work with use 1% metal film smd, which now have a very low cost after being on the market for several decades. Stability with temperature and time is often more important than the absolute value of the resistor.

I sometimes put in a notice in the users guide for my test equipment, to turn it on 15 minutes early so the readings for voltage or current are within 0.1% worst case. If I have to I manually pick series or parallel resistors for absolute value, from a batch that is stable enough over time (10 - 20 years) to be useful in production.

I do not use trim-pots unless mandatory, as their drift is about 200 ppm. If I have to use a trim-pot I use series resistors to keep the trim-pot value low as possible.

For 'surge' resistors I usually had to use 14 awg nickel-chrome wire, 30 strands in parallel to handle 10,000 to 150,000 amp surges of about 20 uS duration each. Exact resistive values were not as important as survivability.

In this sense they were much like wirewound resistors on steroids. The accuracy was seldom better than 10% and they drifted with temperature several percent. They ran too hot to touch, but this was normal, it was about surviving a harsh environment.

We used 6awg wire inductors in series with 0.1 ohm ceramic donut resistors rated for 10,000 amp surges for wave-shaping. Connections were made with buss-bars or 500 mcm locomotive cable. The 'emergency dump' is a water tower resistor made with water and copper sulfate, 3 inch's diamater and about a meter in height. It had a resistance of about 500 ohms but was the only resistor that could dump the charge (30,000 volts) without blowing up.

You can split hairs all you want over deviation, but in the end you build with what works. Sometimes tolerance has to take a backseat to other issues.

I have seen deviation in precision resistors, say reels of 5,000, that seem to drift above or below the ideal value (as measured by a Fluke 87 DVM). It makes finding a series / parallel combination with an exact values nearly impossible. I simply use those that have the closest 'fit' to the value needed.

At ultra-precision levels (<0.025%), controlling temperature drift,board leakage and noise becomes a big issue. Now you have to add parts to keep the 'deviation' over time from becoming an issue.

In terms of measuring with precision equipment (0.01% or better), no combination of series or parallel resistors can be more accurate over time than one resistor that already has a deviation so close to zero as to not be an issue.

Multiple resistors in series or parallel create multiple instances of temperature drift and deviation. To expect them to 'null' out the deviations is absurd, because temperature drift is always an 'additive' function, and deviations tend to drift in one direction on reels of 5,000, yet meet the tolerance spec.

To create a 'perfect' resistor value from multiple values, those with positive deviation would need a negative temperature coefficient, while those in series or parallel that have a negative deviation would need a positive temperature coefficient. Both types of coefficients would have to match to cancel out temperature drift.

From my point of view, during practical normal usage, my answer to @Amomum is NO.

In terms of the maximum/minimum prossible deviation, both cases present the same result.

If you consider the probability of ocurring a 1% deviation to be the same of a 5% deviation, then both cases present the same result.

If you consider the deviation to follow some sort of normal distribution, centered on the resistor design value, still makes no diference. Because even thought the individual deviations will be smaller, the sum will bring them close to the deviations of a bigger resistor. The probability of a 0.5% deviation in a 2kOhm resistor is the same as in a 1kOhm resistor, even though the value of the deviation differs.

The probabilty is

The image of tolerance shows how resitors are sorted during the production process. They are distributes in bins containing specified tolerance, so for example in the bin containing +/-10% you won't find any resistor that has better tolerance than >+/-5%, because those parts are contined in the bin of +/-5%. But if you biuld a series chain a large number of resistors, the mean value will be close to specified

Tolerance means the limit over which the value may get diverge from its actual value. 5% 2k resistor means that the resistance will have value between 1900ohm to 2100ohms. Now for two 1k resistors the value of tolerance will add upand becomes 10%. This a simple rule of Errors. You can read more about this in any Instrumentation and Measurement book. So this means that the value two 1k resistor will vary between between 1800ohm to 2200ohms.