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A small-signal amplifier configuration such as a common emitter amplifier may step into nonlinearity if the small-signal voltage input into the circuit is not of a small amplitude. Why is the indicated phenomenon possible? How does one deal with the problem through design details? Is having an unbypassed emitter resistor on the order of 100 Ω related to the concept pointed out?

The amplification could become non-linear if the transistor enters saturation or if the transistor exits from the active mode.

For this configuration (common emitter), the DC bias of the transistor gives Vc when the signal is zero.

That implies the output (the voltage at the collector, Vc), cannot numerically decreases less than the voltage initially blocked by the transistor, Vc and so, the input signal canot be less than Vc/beta.

Furthermore, the output cannot exceed the voltage supplied by the source either. So the input cannot exceed (Vcc - Vc)/ beta.

We must adjust for the voltage drop (diode like), Vbe_on, when the transistor conducts, mainly if the voltage of the source is, itself, small in comparison.

The resistor Re is, generally, used to alleviate the variation of beta (manufacturing, temperature). Its presence has a tendency to decrease the amplification, though, unless a capacitor is used in parallel. With that capacitor, a first analysis of the signal may ignore Re for the signal (but consider it for the DC bias) with a small inconvenient to add a light phase-out between the input and the expected output. Indeed, if the frequency of the input is high enough, that capacitor acts like a plain wire in parallel to Re and thus, as for the signal is implied, Re does not exist.