Electronic – Problems with Transistor TIP122

TIP122 datasheet is here

Fig 2 of the data sheet shows that Vbe is typically 1.75V "saturated" when Ic = 3A. Using Vbe = 2V is safer. Using 2.5V does no harm.

Fig 2 shows that Vce = 1V typical at 3A. So the transistor will dissipate Power = V x I = 1V x 3A = 3 Watt. If you use eg a TO220 package you will need a modest heatsink. If you use a surface mount package you may need to check heat sinking issues.

Your circuit should look like the diagram below. Note that the motor connects from 12V to collector.

Fig 1 in datasheet shows that gain improves wuth IC upto about 3 and that a gain of 4000 may be expected. Do not trust it :-).
Designing with a gain of 1000 will work well.

For Ic = 3a you need 3 mA drive at gain = 1000 as you noted. If we assume a 5V supply then R1 is such as to allow 3 mA to flow when Vbe = 2V (see above)
R = V/I = (5-2) /0.003 A = 1000 ohms.

If the process ior was running on 3V3 say then
R1 = V/I = (3.3-2) / 0.003 A = 430 ohms =say 390 ohms

Worst case if you allowed Vbe = 2.5V and drove it with a 3V3 processor pin then
R = V/I = (3.3-2.5)/0.003 = 266 ohm = say 270 ohms.

So R1 may have a 4:1 variation depending on what Vbe you decide to choose* and what processor supply voltage is. In fact 1000 ohms would probably work O in all cases and 470 ohms is probably a good all round compromise. (* I said "what Vbe you decide to choose" but in fact the transistor does the choosing - actual Vbe will depend on the transistor in the given circumstances and we choose a Vbe for design purposes based on what the datasheets tell us. ).

A motor is inductive. When you turn it off the motor current cannot styop instantaneously and MUST go somewhere. Diode D1 gives it somewhere to go.
Without D1 you will get a LARGE inductive driven voltage spike. The voltage will rise until the motor current finds somewhere to go ! :-). This can be fatally bad for the transistor and for other electronics, ALWAYS include a D1 equivalent on such cases.

To be safe D1 should be rated at motor current. In practice this may not be needed depending on how you are driving the motor. If you are using fast PWM then D1 should be a fast diode BUR in almost all cases a standard power diode will do OK. 1 x 1N400x will probably do. 2 o3 3 in parallel will be better. A single power diode rated at3A or more is better still - but 1N400x are cheap and more usually available.

No no no !!!; You mentioned driving the transistor base directly with no resistor. This is very bad practice as it very probably violates the processor's datasheet specifications. Once you do that anything can happen and anything might. In some cases (not in this one) you may also destroy a driven device doing that. Circuit components should always sbe designed !.

The TIP120 does not need 120mA at the base for normal operation, that's the absolute maximum rating, above which you don't want to go.
The spec you are mostly interested in is the hFE (current gain), which for a darlington is very high, since it's two transistors connected in a way so the current gains multiply. For the TIP120 it's given as minimum 1,000 (compare with a typical 200 for a single bipolar transistor)
Also important are the max collector current (5A) and the collector emitter voltage (60V)

The main disadvantages are that the base-emitter voltage is doubled compared to a single transistor (~1.4V), and the saturation voltage is higher (typically ~0.8V compared to ~0.2V at low currents)
These points are rarely a problem for a simple switch driven from a micro pin. At higher collector-emitter currents though, the Vsat rises and can interfere with desired operation and cause problems with dissipation.
For example, in the TIP120 datasheet note that at 3A Ice, Vsat is given as 2V, but at 5A it has risen to 4V. That's 20W of dissipation, a lot to heat to try and get rid of to keep the temperature down. So when switching a large current you need to take these factors into account, and maybe decide to look at a more suitable part (e.g. logic level, low Rsdon power MOSFET)

Since we have a gain of 1000, we hardly have to draw anything from the micro pin. Let's say we want to switch 1 Amp:

1A / 1000 = 1mA into the base needed.
If we have a drive voltage of 5V, then we subtract the Vbe from the drive voltage and divide by the current:

(5V - 1.4V) / 1mA = 3.6k resistor. To give it a bit of leeway select something a bit smaller like 2.2k. This still only draws ~1.6mA.

I wouldn't read too much into the different prices - the price of components is often dictated by how popular they are, the more they sell the less they cost. If you see better specs at a cheaper price, go for it ;-)
You can some pretty odd prices when the component is scarce/new/obsolete - I saw a 10uF ceramic capacitor priced at £7.50 (qty 1) on Farnell the other week...