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Shouldn't it be possible to pass small DC signal or maybe a large AC signal to a diode and observe the effect, if the vice-versa is possible.

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Small signal models are large signal models linearized about an operating point. For example, the dynamic resistance of a diode. That allows simplified analysis, especially in closed form calculations. In the example, we simplify an exponential relationship to a simple fixed resistance- effectively the slope of that exponential curve near a particular point.

Of course if you have a good model that works over a wide range you can use that, however typically the calculations are too difficult to be solved in closed form, so they will have to be done numerically, which is not a problem in many cases with modern computers, but may not yield as much insight into the behavior if the actual operation you are interested in is over a "nearly linear" region.

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