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It then follows that the base-bias voltage would need to be about 0.601 volts plus about 0.7 volts (internal base-emitter volt-drop). Knowing this allows you to calculate the base-bias resistor values. Here's a simulation that shows what I mean: -

In this example I used a BC547 BJT and the base voltage to ground is 1.269 volts (implying that the base-emitter volt-drop is 0.669 volts). So, it's easy to follow and you should nearly always get predictable results.

Is it a change in supply voltage, or change in transistor beta?

It's both and it's also temperature and it's also circuit repeatability. Here's a side by side comparison: -

It then follows that the base-bias voltage would need to be about 0.601 volts plus about 0.7 volts (internal base-emitter volt-drop). So, around 1.301 volts on the base will ensure that there is approximately 6 volts DC at the collector.

Knowing this allows you to calculate the base-bias resistor values. Here's a simulation where I've tweaked R4 to show what I mean: -

In this example I used a BC547 BJT and the base voltage to ground is 1.269 volts (implying that the base-emitter volt-drop is 0.669 volts). So, it's easy to follow and you should nearly always get predictable results but you'll never get exact results from a single BJT amplifier.

But I'm struggling to understand the exact circumstance where this would be needed. Is it a change in supply voltage, or change in transistor beta?

It's both and, it's also temperature and, it's also circuit repeatability. Here's a side by side comparison: -

To ensure that 2.73 mA consistently flows through Rc means using an emitter resistor. So, if the emitter resistor is 220 Ω (for instance), a current of 2.73 mA flowing from collector into the emitter and then through the resistor would drop about 0.601 volts across it.

It then follows that the base-bias voltage would need to be about 0.601 volts plus about 0.7 volts (internal base-emitter volt-drop). This then allows you to calculate the base-bias resistor values. Here's a simulation that shows what I mean: -

To ensure that a DC current of 2.73 mA flows through Rc consistently requires an emitter resistor. If the emitter resistor is 220 Ω (for example), a DC current of 2.73 mA flows from the collector into the emitter and then through the resistor and, it would drop about 0.601 volts across it.

It then follows that the base-bias voltage would need to be about 0.601 volts plus about 0.7 volts (internal base-emitter volt-drop). Knowing this allows you to calculate the base-bias resistor values. Here's a simulation that shows what I mean: -