Ahh, this should help -- when we measure incremental resistance, we're using an external signal to perform the measurement. Not the (in this case) op-amp's own movement. Though that can be useful, too, or even equivalent.
Suppose the op-amp's output can be modeled as a Thevenin equivalent source. It produces some ideal voltage V1, and has some source resistance Rs1. If V1 = 0, we can't draw any current from the source by itself; but this doesn't prevent it from having a source resistance. We just have to be cleverer to measure it.
If we connect it to another source, of voltage V2 != 0 and known resistance Rs2, then we simply get a voltage divider, from V2 through Rs2 and Rs1, to V1 = 0V. Then we can measure the voltage in the middle, and solve for Rs1. Indeed, we can do this even if V1 != 0, by measuring V1 when I = 0 (so that V(Rs1) = 0), and subtracting it out: Vmid = V1 + (V2 - V1) Rs1 / (Rs1 + Rs2). (I think I wrote that right; exercise: write it out and double check.)
If we don't generally know the DC value of V1, we can do the same thing at AC. Use a series coupling capacitor, Xc << (Rs1 + Rs2). The same relation applies, but we can replace the DC voltages with the small-signal AC voltages; and now the signal voltages can be small, perhaps mV in practice (or in theory, the amplitude doesn't matter, but theory is assuming linearity, which our real circuits only provide over a limited range).
Also, at AC, we might be concerned with the phase shift and frequency dependence of these results; in short, we're not measuring a resistance, but some combination of resistance and reactance (including one or the other being zero). If we use sinusoidal voltages, we can express phase as sine and cosine components, and use complex numbers to represent impedances (the "complexity" only meaning that we use one number a+jb to represent pairs of those components).
Still another way to see this, is to apply superposition. In the connected-Thevenin-sources test, consider all but one source inactive (set to zero), for each source in the system. Calculate the current flow, then sum the results over all conditions.
And we've already considered the one-source-zero case, and there are no other cases (just exchange V1/V2, Rs1/Rs2), so we're complete!
If the source has a nonzero voltage, we can measure its output unloaded (I = 0), then again with some load resistance. We know the output voltage and load resistances in both cases, so we can again calculate Rs1. (Notice a load resistor by itself, is also a Thevenin source of V = 0 -- this really isn't stating anything different from the above!)
Practical downsides are, if the unloaded voltage is say 1.337V, and the loaded voltage is 1.336V, well, that's really not saying much when our resolution is +/-0.001V. Whereas if we read 0.00 vs 1.02mV, we have very good confidence in the result. So while circuit theory by itself doesn't care, we may still prefer one approach over the other. This isn't hard to anticipate, it's just incorporating other theory: namely, the statistics of random variables under addition and multiplication.
In any case, this spans roughly the first year of EE theory and labs, so, if you haven't studied it much yet, you'll have to look up these terms... But these are among the subjects taught, and with study you will become comfortable with them.
The most important takeaway is that circuit theory is like any other algebra: there is a set of operations, which have no effect on the correctness of the system (the states before and after, are equivalent), that allow you to expand or reduce the system in certain ways, hopefully allowing you to solve it. Even better(?), it's done with a graphical language, not just flat equations.
Anyway, at no point is there an actual 10k resistor: we are imagining or testing load and source conditions with respect to that as a reference value. The op-amp might have a few ohms output resistance, which is much less than 10k, safe to use. The ADC being insensitive to sources up to 10k, and having a resolution of about 1/4000th of full scale, implies 10k * 4000 or 40M for its effective load resistance. (I said over 10M (10 bits equivalent), as the 12 bits can be optimistic and depends on reasonably good design to actually meet.)
For the LEDs, I would use some filtering, yes. Enough to "take the edge off", maybe a time constant of fractional ms (e.g., 470 ohms in series, 1nF to GND). Might be a good idea to put this in front of the current sense amp; usually the datasheet/appnote provides hints on this. I would also consider a window comparator, to do the check directly: no need to ask an ADC to measure things.
Tim
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