Does the "system gain" (i.e. electrons/ADU) change when one changes the ISO of a digital camera? To put this another way is there an analog amplifier between the sensor and the ADC whose gain is programmed to change when the ISO setting is changed?
is there an analog amplifier between the sensor and the ADC whose gain is programmed to change when the ISO setting is changed
Quite often there is. It can be a multiplier after the ADC, too. Can be that gain does not change (Sigma/Foveon, some digital backs, some other cameras). Can be that it is not for all ISO settings, like "extended low ISO" on Nikons and Canons. Can be a combination of gain and multiplier (intermediate ISO settings on many Canons, Hi ISO on some cameras).
The exact recipe will be a closely guarded trade secret for manufacturers but certain things hold true. Chipworks might give you some background that may help for specific cameras (or it might not).
The actual light captured and charge generated will always be the same at the sensor. There are multiple points where the results can be adjusted.
CMOS sensors have an on-chip gain control / amplifier which makes adjustments during the readout (CCD's don't).
There can a separate amplification stage en-route to the ADC. Necessary for CCD but some CMOS systems have been seen to do this too.
Finally a third layer of digital adjustment may be applied.