tolerance stack-up

Imagine four people each cutting a plank to one metre, each allowed to be off by a centimetre. Line all four planks end to end and the total could be four centimetres short or long — the small errors do not cancel, they pile up. Tolerance stack-up is exactly this in a circuit: every real part is a little off its marked value, and when several imperfect parts work together, their individual errors accumulate into a larger error in the final result. The whole can be sloppier than any single part suggests.

Take a voltage divider meant to put out half the input: two 10 kilohm resistors. If each is 1 percent tolerance, the ratio is not exactly 0.5. In the worst case one resistor sits high and the other low, so the output can be off by roughly the sum of the two tolerances. Add in the resistors drifting with temperature, the reference voltage being a bit off, and the op-amp's own offset, and you have an error budget — a tally of every contributor. There are two honest ways to add them: worst-case adds the magnitudes (pessimistic, all errors aligned against you, simple and safe), while root-sum-square adds them in quadrature, sqrt(e1^2 + e2^2 + ...), which is statistically realistic because they rarely all line up at once.

This matters because it is how precision quietly dies. A designer who picks 1 percent parts and assumes the result is 1 percent accurate is in for a nasty surprise on the bench. The cures are to build an error budget early, to use one precision part where it dominates (a 0.1 percent reference) rather than scattering precision everywhere, to choose a ratiometric topology where errors cancel, or to calibrate the finished unit. Knowing where the error truly comes from tells you exactly which one part is worth paying for — and which dozen parts you can buy cheap.