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Looking at this data sheet: http://www.datasheetcatalog.org/datasheet/stmicroelectronics/4136.pdf

How much voltage has to be applied across the base and emitter to switch on a "TIP 3055" transistor? The V-be row is the only thing that looks like it is telling about that, but I don't think that's what I'm looking for (mabey it is?).

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Bipolar transistors are controlled by current, not voltage. The important factor here is HFE - The large signal collector current gain. Apply a 100mA current through the B-E junction and you'll get from 20 to 70 times that flowing C-E as the range for HFE is 20 to 70.
If you're using this for switching, work out what your load would draw if you connected it straight to 0V. Then divide this by 20 to find the base current you need, and add 20% or so to make sure it's turned on hard (saturated). The absolute maximum base current for that (IBmax) is 7A(!). The maximum collector current is 15A.
If you're switching at high speed, things get more complicated.
The Vbe you mentioned is the voltage you would see between those pins at a collector current of 4 A, assuming you were applying a base current to do that. Vce(sat) is the Collector to emitter voltage when turned hard on at the currents specified.

Thank you, I thought it was voltage controlled but it is very helpful to know it's current controlled. I am planning to use a low power oscillator circuit with a JFET to pulse a transistor or an IGBT on and off to drive a transformer for a capacitor charger. But if the transistor being pulsed is current controlled I should use a few JFETs in parallel in the oscillator circuit because I don't think they can handle much current.

I was unable to reply to your last post once you'd marked best answer, but I wanted to note the following distinction to aid you in future efforts:

FET transistors differ from BiPolar transistors in that they ARE voltage controlled, unlike their cousins the bipolars.

Just in case you may have interpreted AndyGadget's good answer regarding the operation of Bipolar transistors

Good point -  Thanks Sean.

meh. np. I noticed that he said in his followup that he's considering using some JFETs and figured it might be sensible to clarify, before he gets magic smoke or no effect..

(the word in my ref to you was supposed to be "misinterpretted", not "interpretted" )

Form the data sheet as you say:

For a base Emitter voltage of 1.8 volts ad a emitter collector voltage of 4 volts you should get a collector current of 4 amps.