You can very quickly plot the performance of the transistor under numerous operating conditions much easier than trying to hook multiple meters and power supplies to a transistor on a breadboard. Simulating basic transistor circuits in something like LTCSpice is also very useful for understanding how transistors behave under various conditions. I would add to it that once you have gained a basic understanding of transistor theory you should read transistor datasheets and understand how real transistors perform and how semiconductor manufacturers specify, test and guarantee those devices. This is the only piece of good advice in this post. Pulse it slowly 1/10 on and 9/10 off - this will reduce the energy consumption dramatically.
Then you are probably wasting energy but think about it: you are wasting 10 times as much lighting up the LED. It's a guess that is known to work more often than not and when you're talking about 2 mA, it probably doesn't matter unless you are using really wimpy batteries. Having completed this experiment, you will understand why we commonly guess at 1/10. If this is coming from a uC, use an output pin to drive the base resistor. Is the transistor saturated? Measure Vin so you can see what the driving circuit is doing. Measure V be, I b (measure drop across the base resistor and calculate - measure the resistor first so you have the real number), measure Ic (measure drop across the collector resistor and calculate) and, most important, measure V ce(sat) by probing between collector and emitter. As I said earlier, build the circuit on a breadboard.
Oddly, under different circumstances, every single number on the datasheet is important. But there's a reason that datasheets have a lot of numbers. Understand that every single number I used was a guess. I used 2V for Vf of the LED - it could be as high as 3V at 20 mA according to this typical LED datasheet. The only thing that really matters is how the circuit performs with real numbers. Even if all the parameters stack against the design, the base current will probably be adequate. Since we're only talking about 20 mA of collector current and 2 mA of base current (maybe), the 1/10 thing works pretty well. What if you happened to grab a 2N3904 - no, I haven't looked at the difference, I just assumed a 2N2222A for this discussion. But what you can't do is design a circuit that is dependent on selecting transistors on the basis of gain from a box of parts. Don't like 10% base current? No problem! Pick a different number. Then look at V BE(sat) - I assumed 0.7V but the real value could go as high as 1.2V and that changes the calculation substantially. While you're at it, look at V CE(sat) I assumed 0.2V and the datasheet says a max of 0.3V but they don't specify a 'typical' value. Our circuit still has to work regardless of what the driving device is doing and regardless of which gain 'bin' our transistor came from. Remember, we assumed the driving voltage was 5V but what if the driving device can't quite reach 5V? There are logic families where the output voltage is somewhat less than Vcc. Gain is all over the map and while 1/10 base current is overkill, it is used fairly often. The whole discussion is meaningless without this definition. ETA: I guess I should mention that collector current is equal to h FE * base current. Even at 10 mA (and we're talking about 20 mA), the gain could be as low as 35 (at a very cold -55 degrees C) up to 75. At very light loads it can be as low as 35 and at high loads it could be as high as 300. Look at the datasheet for the 2N2222A and under "On Characteristics" "DC Current Gain" look at the range of h FE (current gain).
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