Electronic – Connecting piezo buzzer with BJT

Ah, the joys of finding a random circuit on the Internet that happens to have a slick presentation. This is actually a very poor design for several reasons.

First of all, the author should have put a resistor between D10 and the capacitor, which would have allowed him to use a much smaller capacitor to get the cutoff frequency he needed. The V_EE pin of an LCD requires only a tiny amount of current. As it is, he's relying on the output impedance of the Arduino pin to limit the current into/out of the capacitor, which is very poor practice.

Secondly, the transistor is being used in a common-emitter mode, not emitter-follower. Using two resistors the way he does doesn't make much sense.

Why isn't a simple capacitor is used in the brightness pin, similarly to the contrast pin?

Two reasons:

1. The Arduino pin by itself can't handle the current required by the backlight. The transistor provides the necessary current gain.

2. In this case, the goal isn't to turn the PWM signal into a DC level, but instead, to use it to turn the backlight LED on and off rapidly to change the apparent brightness.

In the emitter-follower, is the BJT operating in linear or saturation zones?

Like I said, this isn't an emitter-follower. Because of the resistor in the emitter leg, however, it's operating on the cusp between linear and saturated zones.

Why does the author discard the base emitter voltage (0.6V) in his calculations?

I assume that when you say "discard", you mean "ignore". Good question, although it would really be the collector-emitter voltage in this situation. If he was using a circuit configuration in which the transistor would definitely be in saturation, this voltage would be relatively small (about 0.3V), but still significant.

The circuit would be better if the emitter of the transistor were connected directly to ground, and the resistor R1 were placed in the path between Q1's collector and the LED- pin of the display.

Understanding transistors is a bit like peeling an onion- there are many layers. At the simplest large-signal level you can consider the transistor as a current sink that's controlled by the current through the base-emitter junction. The latter behaves like a forward-biased diode. Not much current until you get to some hundreds of mV, and way too much current if you put volts across the junction. As you say, the transistor will conduct excessive current and will be destroyed if you simply connect (say) 5V to the base with emitter grounded. This is in stark contrast to the behavior of a MOSFET.

At a more sophisticated level of understanding (which is required if you want to predict how most amplifiers work) and for small signals the base-emitter junction behaves like a resistor of Vt/Ib where Vt is the thermal voltage, about 26mV at room temperature. So if your base current is 2.5uA (say the beta is 300 and the transistor is biased with 0.75mA collector current), the base-emitter junction looks like about a 10K resistor for small signals. You can consider the transistor as a (somewhat imperfect because of r0) voltage controlled current source with an input resistance of Vt/Ib. This is the hybrid-\$\pi\$ model. Note that the transconductance gm (and thus the voltage gain in a common emitter configuration) is a function of the collector bias current and temperature and beta does not enter into it at all.

I must emphasize that this model is a linearized model about a bias point and is quite invalid if the (change in) input voltage is large (more than some millivolts). In other words we're talking about relatively small changes on top of a fixed base-emitter voltage of perhaps 600 or 700mV.