Quick DAC Question

Hello, I was working on some problems that my prof had assigned but his solutions were pretty unclear and did not reason his answers very justly. I have finished most of the question but I'm wondering if someone can help me out.

Here is the question:

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My solutions:

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I finished the first part and received the correct solutions, but I have no idea how to to the second part, where they ask to find a value for resistor R that will cause the gain error to be zero, noting that all the switches have an on-resistance of 200 ohms in this case.

I am not sure what they are relating the gain error to, as the gain is not constant in the previous example. Am I to pick an R that meets specification with an ideal case?

The solution that Jaeger and Blalock (Q12.79 - 4th Edition) posted was R = 1.0742 kohms. No idea how they came to that though.

Can anyone lend a hand?

Thanks!

This isn't clear at all. Does he intend "zero gain error" to mean a gain of -1? That circuit can't swing positive. And if so, are we to assume the intended output range is -5V to 0V, because Vref is +5V?

If these assumptions are correct, that 1074.2Ω R won't do it - that gives a span of -3.63V to 0V, yielding a gain of -.726, which hardly seems like a "zero gain error" to me.

A 1477Ω R would get your -5V to 0V span... if THAT's what he means.