To control beta with a fixed-bias transistor, we add a resistor to the emitter. This provides negative feedback to counter a surge in collector current should a change in Beta happen.
However, for this circuit to be completely independent of beta, either the base resistor needs to be very small or the emitter resistor very large. A larger emitter resistor needs a larger power supply as a result, and a low base resistor needs a independent low voltage power supply.
