How does base resistance affect a transistor?

I'm not strong with electrical engineering and I have prototyped a circuit which works; however, to my chagrin, I don't understand why.

I'm using an Arduino digital out pin to power a relay. I have an NPN transistor to switch 5 volts through the relay coil that's 125 ohm. At first, I just connected the base to the output pin and it worked fine. Then I noticed people tend to add a base resistor so I stuck a 1K ohm resistor in there and it still worked fine.

I've been trying to learn more about base resistors and I feel like I computed that I should have a larger resistor. Is the purpose of the base resistor just to lower the current to prevent wasting energy? Or to prevent generating heat? I don't fully understand why there's a need for a base resistor, and if there is, why the relay in my circuit worked fine without the resistor and also with a smaller resistor.

This is the datasheet for my transistor: http://pdf1.alldatasheet.com/datasheet-pdf/view/21675/STMICROELECTRONICS/2N2222.html

I don't have a datasheet for the relay (it's part of an old hobby kit . . . you know . . . one of the ones with the breadboard and springs).

Here's my current setup:

simulate this circuit – Schematic created using CircuitLab

Here're my likely incorrect calculations. I computed the collector current as $$I_{c} = \frac{5V}{125\Omega} = 40mA$$

I felt like, based on the spec, a reasonable $h_{FE}$ for 40mA is 75. Thus, $$I_{b (sat)} = \frac{40mA}{75} \approx .5mA$$

Given that, I felt like the best fit for $V_{BE (sat)} = 1.3V$.

Then, I thought I needed to drop 3.7V across the base resistor at .5mA. Thus, $$R_{base} = \frac{5V - 1.3V}{.5mA \div 1000} = 7,400\Omega$$

So, if that's correct, what am I trying to achieve with this resistor? Am I trying to prevent energy loss or heat production? Or, am I trying not to exceed the max base saturation voltage?

Given that 7k4 ohm is not a common resistance, should I drop to a 6k8 ohm resistor or go up to 8k2 ohm resistor?

It seems like 8k2 ohm would keep the base voltage below the saturation max voltage and the 6k8 ohm would keep the base voltage above the max base emitter saturation voltage. Which am I after?

Or, I suppose, alternatively, am I completely off track?

• Suggest you use a more modern transistor than that old hulk- something like a 2N4401 (only 30 yrs old maybe?). Also, shoot for Ic/Ib = 10~20, not 75. Commented Mar 9, 2015 at 1:50
• @SpehroPefhany, what do you mean shoot for an Ic/Ib? Meaning I shouldn't expect hFE of 75? Because I'm at such low collector current, I'm not really going to get much amplification from my transistor? Commented Mar 9, 2015 at 1:58
• You want the transistor to be heavily saturated not just have low Vce, and you want it to be so under all conditions, not only room temperature and such like. Look at where Vce(sat) is specified in the datasheet- it's Ic/Ib = 10 in almost every case. hFE is specified at Vce = 10V and 25°C, not what you want. Commented Mar 9, 2015 at 2:09