It you look closely at the information I posted in post #4, the manufacturer is using a beta of 250 when saturation mode operation is desired. We always use a conservative number to guarantee that the transistor will be saturated. For NPN transistors with betas of 200-300, we still use 10 when using them as switches.

If you're operating in the active region, beta depends on Ic, so the manufacturer will typically give you beta numbers for several currents.

You can see the variation of beta with Ic and temperature.
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Okay. I have a sinking feeling that I have completely misunderstood the base current concept.

Let's say we are using a TIP120 Darlington Pair transistor package.
- Gain: 250 for switching
- Forward Voltage: 1.4V

We have a 12V solenoid that needs to draw 2A in order to operate at full power to pull whatever weight it is rated to pull.

We have a 1N5400 across the solenoid to protect it. Good, check.

Now all we need to do is make sure the transistor opens fully to allow all 2A of current to pass around the solenoid circuit to let it operate at its full 24W.

Key question: Does the base current have to be exactly enough insofar as 250 * I_b = 2A? "I_b" is the base current. OR, is it actually 250 * I_b >= 2A? As in, if I increase the base current from 8mA (=2/250) to 16mA, will the current flowing through the solenoid still be 2A, or will it increase to 4A?

Because if 250 * I_b >= 2A, where I_b = V_R/R and V_R = 5V - 1.4V = 3.6V, then 250 * 3.6/R >= 2.
Simplified: R =< 450 Ohm. So I could actually choose a 450 Ohm resistor or even a 225 Ohm resistor, and the base current would still allow a 2A current to flow to the solenoid regardless, and wouldn't make it go over 2A to 4A?

If this is true, then assuming the gain is even lower, at say 10, we would just need an 18 Ohm resistor at most? What is the point of having a resistor in the first place, then? Is it purely to meet a rated current value into the base and/or out of the controller's output pin? Can I think of this as a sort of "current threshold" analogous to the VGS(TH) of a MOSFET?

 

Key question: Does the base current have to be exactly enough insofar as 250 * I_b = 2A? "I_b" is the base current. OR, is it actually 250 * I_b >= 2A? As in, if I increase the base current from 8mA (=2/250) to 16mA, will the current flowing through the solenoid still be 2A, or will it increase to 4A?

No. Once you drive the transistor (darlington) into saturation, further increases of base current don't result in further increases of collector current. That's an essential part of the definition of saturated mode.

The base current drive is one reason why people prefer MOSFETs. 40 years ago, power MOSFETs weren't generally available, so we used BJT's.

Can I think of this as a sort of "current threshold" analogous to the VGS(TH) of a MOSFET?

Vgs(th) is the voltage at which a MOSFET just starts to conduct. A higher Vgs will decrease Rds(on). When using a MOSFET, you choose one that will have sufficiently low on resistance at the drive voltage.