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Common Emitter

Continuing from the emitter follower, you learned that the voltage at the emitter is roughly equal to the voltage at the base, independent of the resistor used at the emitter. What does depend on the resistor value used is the current that flows through the emitter resistor.

The current drawn by the resistor is defined by ohm's law
    Ie = Vre/Re
and since the voltage at the emitter resistor is practically the same as the base voltage, then
    Ie = Vb/Re
where Vre is emitter resistor voltage, Re is emitter resistor's resistance and Vb is the base voltage.

You also know that the current comes mostly from the voltage source connected at the collector, since the base doesn't contribute much to the overall emitter current, it can be considered a separate, series circuit.

As a series circuit, you know that the current flowing at any point in the circuit is the same as at any other point in the circuit. You already know the emitter current, so the current through the collector, and any resistor connected to it, will be the same as the emitter current.

The current through the collector resistor causes a voltage drop across it, defined by ohm's law as     Vrc = Ic Rc
where Vrc is the voltage across the collector resistor. Since the collector current is the same as the emitter current, you get
    Vrc = Ie Rc
You also have that the emitter current is defined as
    Ie = Vb/Re

All this data collection is to arrive at an equation for the voltage at the collector
    Vrc = [Vb/Re] Rc
if you rewrite it you get
    Vrc = Vb [Rc/Re].

This final equation gives us an easy definition for the voltage across the collector resistor that is independent of the beta (current gain) of the transistor, characteristic that varies widely among even the same batch of transistor, and that also depends on the temperature of the transistor.

Now, the voltage across the collector resistor is not very useful by itself, but it can be used to obtain the voltage at the collector-resistor connection, in other words, the voltage across the transistor itself.

By Kirchoff's laws, the voltage supplied is the sum of all the voltages induced in the components that form the closed loop. In practice, our loop is the collector resistor, the transistor itself and the emitter resistor. You already know how to calculate the voltage across the resistors, and know that the sum is equal to the supply voltage, so Vcc - Vrc - Vre - Vce = 0, where Vce is the voltage across the transistor's collector and emitter.

Since most of the time, the output of this circuit is connected from the transistor collector to ground, we need to know the voltage at the collector of the transistor with respect to ground, defined as
    Vc = Vcc - Vrc
Or in other terms
    Vc = Vre + Vce
Since it is easier to calculate Vrc than Vce, the first equation is the most widely used.

The design of a common collector amplifier requires that you know all the mayor characteristics of the transistor, like the relationships between collector, emitter and base currents, as well as other properties of circuits like kirchoff's and ohm's laws.

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