Hello,
In Ron's circuit it also helps to know that a constant voltage on the base keeps a pseudo constant voltage across each emitter (sense) resistor. A constant voltage across a resistor results in a constant current through that resistor and that's the goal.
Since this current can be varied with either the emitter resistor value or the base voltage, we elect to change the base voltage so that we can control all N stages.
The current through the collector is the current through the emitter minus the current in the base emitter, and the current in the base emitter comes from the voltage setting and the emitter resistor. The Beta of the transistor used must be high enough to ensure the current in the collector can always be produced by the small base current. Roughly a gain of 40 should do it which i think is what most small transistors can do these days. It can vary up or down quite a bit too as the following should illustrate.
The collector current (our goal) can be shown to be approximately equal to:
Ic=((Vb-Vd)*B)/(Re*(B+1))
where Vb is the base voltage to ground, Vd is the base emitter voltage drop, Re is the emitter resistor, and B is the DC Beta.
For small variations Vd does not change much, and Re is a constant resistance, and holding Vb constant (our set point) we can reduce this to:
Ic=K*B/(B+1)
where K is a constant, so we can see that Ic is proportional to B/(B+1).
Computing this for B of several values, we get:
B=40, B/(B+1)=40/41=0.9756 (nominal value)
B=20, B/(B+1)=20/21=0.9524 (half value)
B=80, B/(B+1)=80/81=0.9877 (double value)
Now comparing the two extremes for half and double the Betas, we see that the collector current will only change by about 4 percent for these fairly wide extremes. So we end up with a relatively good current regulator.