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Several times here on the EE Stack Exchange, I've seen people say not to connect a load between the emitter of an NPN transistor and ground, but instead connect it between the power source and the collector. In that case the transistor is a current sink, not a current source.
Why shouldn't you draw current from the emitter, like this below?
simulate this circuit – Schematic created using CircuitLab
Your question says "current source" but I think you're really asking about switching voltage to a load.
With a emitter-follower you're stuck with around 600mV of voltage drop (plus whatever the driving circuit drops when supplying the base current) unless you have a higher voltage source available. So if you are driving an LED and have a 3.3V power supply and a GPIO that has 2.9V on it when driving base current, you're going to get about 2.3V to your LED, which will be rather limiting, plus the transistor dissipates power because of the current and voltage lost.
If you use a saturated transistor (PNP or NPN) then your voltage drop across the transistor might go down to a few tens of millivolts, and almost the full supply voltage is available for the load, and the transistor stays cool.
On the other hand, there are circumstances favorable to the use of an emitter follower, for example if you wish to draw current from a higher voltage (perhaps unregulated) supply and provide a relatively constant voltage (when 'on') to a load from that.
Most of the simple questions here don't require that kind of circuit, but I've certainly used it to benefit in commercial products where saving a bit of money can make a big difference.
A current source should have high impedance.
Looking in to the emitter of a BJT in forward active mode you see a low impedance.
This means the base acts more like a constant voltage source (with a value of Vbase - 0.7 V) than like a constant current source.
This circuit is commonly used when a voltage source is wanted. It is called an emitter follower. The controlling signal is applied to the base, not the collector.
Another similar configuration is the common base amplfier. For common base, the controlling signal is applied to the emitter, and an output current is produced at the collector. This acts as a near-unity gain current buffer, much like the emitter follower is a unity-gain voltage buffer.
The main reason is because bjt current through the the emitter is proportional to the Base-emitter voltage difference in an exponential relation.
With the base set at a certain voltage, and ground not moving because it's ground you will always get the same amount of current flow through the BJT regardless of what the load resistance is.
In your case though, with a set base voltage, your emitter voltage is no longer pinned. This results in exponentially more current if your load resistance drops. There's no current control at all at that point because you have virtually no control over the emitter voltage.
From Ebers-Moll model on wikipedia:
$$I_E=I_{ES}\Big(e^{V_{BE}/V_T}-1\Big)$$
Although you mention that people often said that, the very nature of the different ways (call it "modes") of using a transistor requires that one consider the circumstances where those statements could have been done. Generally speaking, the configuration you mention, aka "emitter follower"/"common collector" is as good as any other configuration (say "common emitter" or "common base"). The fact that 3 different modes exist and are used everywhere is simply because each one has its own pros and cons so, it's not correct to generally state that this configuration is not advised "period". To drive a lamp, as in your example, both "common collector" (your example) or common emitter can be used effectively. Which one to choose, depends on what kind of control you want for the lamp: 1. If you want just ON/OFF control, and if you can select a lamp voltage that matches the power supply or if your power supply matches the lamp voltage, using the common emitter configuration gives you the advantage that your control voltage doesn't need to be the same voltage of the lamp. As long as your control voltage is enough to bias the transistor (0.7 V) and can supply enough current to the transistor's base to saturate it (roughly anything higher than the lamps current divided by the transistor's current gain), you're good to go. The transistor will run cool (because it's saturated and therefore has very little voltage drop on it) and you can - say - switch on/off a 110 V lamp with a control signal's voltage less than an AA battery (caution, TRANSISTOR must be able to stand the power supply's voltage).
In the explanation above I'm tring to keep it to the "basics". Of course that a lot more can be said, and it's all well covered in the transistor's mathematical models and how they translate depending on how you connect the transistor (common emitter, common collector or even "common base" which is not mentioned in this explanation because it doesn't suit well this particular need of driving a lamp, but has many advantages for other needs).
