# Cpk - Relation between measurements and resulting tolerances - Trimming resistors

E

#### erik_h

A production machine is trimming resistors to their final, specified value - and within a given tolerance.
The machine is capable of doing this (with some tweaking of the process!) to appr. 0.06% standard deviation (nominal value is 100%).
The machine obviously have an electrical measuring system that does introduce an additional error, but let's ignore that fact for the moment.
In addition some software calulations can be done, and with no error that counts (I hope !).

In some cases 2 resistors of different absolute value should be trimmed to a certain relation.

Lets say:
Ra nominal value should be 1000 ohms +/- 1%
Rb nominal value should be Ra * 4.5 ohms +/- 1% (Which is of course 4500 ohms +/- 1%)
In addition the specification of the final relationship Rb/Ra should be 4.5 +/- 0.5%

The machine setup can be done in several ways.

A:
Trim Ra to 1000 (getting Ra(real))
Trim Rb to 4500 (getting Rb(real))

B:
Trim Ra to 1000 (getting Ra(real))
Trim Rb to Ra(real)*4.5 (getting Rb(real))

C:
Trim Rb to 4500 (getting Rb(real))
Trim Ra to Rb(real) / 4.5 (getting Ra(real))

The question is this:

Could one tell that any of these cases produce a better result when it comes to Cpk - both per resistor and for the relation ?
It seems to me the best way to do this is by methode C. But I don’t have a good rational argumentation for this choice.

Do anyone have comments or good ideas on this ?

#### Tim Folkerts

Trusted Information Resource
Erik,

First, the way you state the specs, it sound like you want Ra = 1000 and Rb/Ra = 4.5 (rather than specifically wanting Rb = 4500. In that case, my first inclination would be to start by getting Ra close to 1000 and then setting Rb = 4.5*Ra.

Whichever two of the three specs you aim for (Ra, or Rb, or Rb/Ra) will have (relatively speaking) a better Cpk, and the third will have (relatively speaking) a worse Cpk.

If you set Ra & Rb, then (assuming the process is pretty well centered) it sounds like you could get a Cpk of about 1%/0.06% = 16 for the individual resistors! So that is no problem. The st dev of the ratio would be about (0.06%^2 + 0.06%^2)^0.5 = 0.085%. Then the Cpk for the ratio would be about 0.5%/0.085 = 6. Still very respectable.

If you set one of the resistors and the ratio, then that resistor will still be Cpk= 16. The ratio will be Cpk = 0.5%/0.06% = 8. The other resistor will have its Cpk lowered because 1) there is more spread, and 2) it will be off-center.

Of course, the error introduced by instrumentation will make these numbers worse and will any degree on off-centeredness, especially in the "third spec". This might be a good time to talk to the customer. Perhaps they can give you some further insight into the relative importance of the three specs (Ra, Rb, & Rb/Ra).

Tim F

P.S You might want to double check the calculations, but I think they are correct.

E

#### erik_h

I think you quite fast grabbed the idea that from the customers site of view the relation is more important than the actual values of the resistors. This is also reflected in his tolerance requirements.
So actually he wants that Ra = 1000 and Rb/Ra = 4.5 !!
However he has to keep the resitors within certain specs. of course.
So - I believe that the conclusion is that we should adjust first one resistor (which we are able to do to a Cpk much better than 10) and then adjust the other one by aiming at the ratio. The last one will then have the worst Cpk, but still well acceptable. The result for the ratio should then be as good as possible.

In real life there is other factors that have to be considered - eg. if the 2 resistors is made out of different materials. Then one material - by its nature - could has far more "noise" in the result than the other. In such case one should take the "noisy" one first - and then adjust the ratio by the easy one - thus acheiving the best ratio Cpk.

#### Tim Folkerts

Trusted Information Resource
Erik,

Your mention of noise in different materials reminds me that temperature will have different effects on different materials. As the materials warm up to operating values, the ratio could change. One more reason to stick with a single material.

Tim F

E

#### erik_h

Well - differences in material is a well known issue in this production.
Actually I should say that the materials is of a different "mix" to cope with the great difference that exists between a resistor of some ohms and another of some megohms.

Long term stability is another concern that also varies - but all this is well know within this industy.

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