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Unfortunately, there are problems with this simple design. Leaving out details of electrical engineering, we need N-1 resistors to get N output. If we want 12 bits of resolution, 212 = 4096, and that's a lot of resistors. Isn't there a way to use fewer resistors? It turns out that there's a method that uses only 2N resistors. It's called an R/2R ladder.

R/2R network
A 3-level R/2R Network

Start at the far right. How does V3 compare to V2? By analogy to the previous figure, V3 = V2/2. At first, it looks like computing how V2 and V1 relate could be ugly. However, there are 2 routes to ground from V2. Both of them have a total resistance of 2R. The effective resistance of two resistors in parallel relate as 1/Reffective = 1/Rpath 1 + 1/Rpath 2. The effective resistance from V2 to ground via the two parallel paths is R, the same as the resistance of the individual resistors! This means that each numbered voltage is 1/2 the potential of the next lower numbered voltage. Suddenly, we have a way to get 1/2n fractions of a reference potential for any n, just by using a large enough number of identical resistors. Actually, there's a limit; the precision of the resistors has to be such that the errors in each stage are smaller than the smallest fractional voltage. That means that, if we have a 12 bit ladder, all resistances must be precise to 1 part in 212 = 1/4096. That's about 0.025%. That sounds difficult to achieve, but resistors can be trimmed to 0.1% fairly easily, 0.01% with effort, and even 0.001% if temperature is carefully controlled and adequate standards are available.

We thus have a way to look at a reference potential and 1/2, 1/4, 1/8, ... of that reference. How do we combine those potentials so we can get potentials that vary in small, equal steps from 0 to Vin? We need to combine the current in the R/2R network with one amplifier. Let's take a moment to examine the Inverting Operational Amplifier With Gain, one of the most common analog electronic circuits.



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