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In the fixed.bias biasing several key parameters of the amplifier depend on the Beta of the transistor. This parameter has a wide variation (often 5 to 1 or more) for a particular transistor type. Thus, if you need to make a production run of a number of identical amplifiers, this variation in the Beta is unacceptable.

In the voltage-divider stabilized method the performance of the amplifier is independent of Beta (or almost). Thus, this design is suitable to be used in the production of a large number of identical devices.

In terms of biasing, a voltage divider circuit helps offset the effect of changes in the beta value of the transistor. So it's said to be more 'stable' because heating of the transistor, or swapping transistors entirely (both of which affect beta) will not greatly affect the output of the circuit.

As far as the actual output is concerned, I believe both circuits have similar input/output impedance ranges and similar voltage gains. But the fixed biased circuit typically has more current gain.

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