The output of the common emitter amplifier is inverted because increasing the base-emitter current causes a proportional increase in collector-emitter current. That increase in collector-emitter current pulls the collector towards the emitter, so the voltage on the collector will go down when the biased base voltage goes up, and vice versa. This is the characteristic of the Class A Common Emitter amplifier.

Responding to a request for more details...

This is for the NPN transistor. It applies to the PNP transistor as well, but directionality of voltage and current increases is reversed in that case.

Start with the base-emitter circuit. You have some kind of bias network holding the base at a certain voltage. That voltage represents a certain current, which goes through the base-emitter junction and emitter resistor, if there is one. Typically, you consider that the emitter voltage is less than the base voltage by the amount of one diode junction, or about 0.7 volts. If you were to increase the input voltage, you would cause a corresponding increase in base-emitter current.

The transistor has gain, beta-dc or hFe, which is basically the ratio of collector-emitter current over base-emitter current, so the base-emitter current is controlling a larger collector-emitter current.

Now, focus on the collector-emitter circuit. You have some kind of resistor in the collector, and you might have some kind of resistor in the emitter. (More on the emitter resistor later.) Think of this circuit as three resistors in series, the collector resistor, the equivalent resistance of the collector-emitter junctions, and the emitter resistor. This also represents a current, one that is being controlled by the base-emitter resistance.

Note that the base-emitter current is being added to the collector-emitter current, so the emitter current, by Kirchoff's current law, is the sum of the base and collector currents. Since the gain is relatively high, however, the contribution from the base is generally negligible. (In high power transistor amplifiers, gain is usually low, so base current is not negligible, so we do take it into account.)

The crucial factor here is that the collector current is proportional to the base current, in the ratio of beta-dc, or hFe. If, for instance, base current were increased by 1 ma, with an hFe of 200, then the collector current would increase by 200 ma.

Well, sort of....

You have to consider the transistor's limits, and you have to consider whether or not you are opereating in linear mode. Limits are easy, just check the specs. Lets look at linear mode...

If you attempt to pull more collector current than the collector-emitter circuit would allow, i.e. to make the equivalant collector-emitter resistance go to zero, then the transistor starts operating in saturated mode. In saturated mode, the transistor is acting as a switch, and it is distinctly non-linear. Even if not saturated, the transistor can be poorly linear when operating at the ends of the linear range.

This is why any good design includes consideration of linear mode range. You want to operate in the center of the linear range, which simply means that we bias the base to cause the collector to be in the middle of its optimal range, giving maximum linearity.

Summarizing so far, we have a transistor that is multiplying its base current by some factor, beta-dc or hFe, causing a proportional collector current. With this viewpoint, the amplifier is non-inverting because increasing base current causes collector current to increase.

We call the circuit inverting, however, because we want to think of the collector voltage rather than the collector current.

Remember that the transistor has an equivalent resistance. In particular, the collector-emitter resistance changes in response to stimuli on the base. In order for the collector current to increase, the equivalent resistance must decrease.

Looking at the collector-emitter circuit, you have a voltage divider, collector resistance at the top, and the sum of equivalent collector-emitter resistance and the emitter resistance at the bottom. It is easy to see that, if the collector current increases, the collector voltage must decrease. That is why we call this an inverting amplifer.

Back to the emitter resistor...

When we say "common emitter", we mean that the emitter is common and we analyze everything else. You can design and operate this amplifer with no emitter resistor, and that would be a true common (or grounded) emitter configuration. Problem is the circuit will not be stable...

First, gain varies amongst transistors, even amongst transistors of identical design. It is common to state that hFe ranges from 80 to 400, as an example. The circuit design must consider this variability. If you want predictable and stable gain, you must compensate for gain variation. You design the circuit for minimum hFe, but you look at what happens with maximum hFe.

To make matters worse, the junction voltage at any particular current varies with temperature, sometimes substantially. This means that your beautifully designed circuit is unpredictable when it gets warm, and all circuits that manipulate power, even small amounts of power, get warm. Its all a matter of degree.

There are many ways to compensate for gain variations. One of them is to use an emitter resistor. This effectively places a limit on gain by moving the primary factor for gain from the transistor to the circuit.

The gain of a common emitter amplifier is hFe. When there is an emitter resistor, however, the gain is collector resistance divided by emitter resistance. If that ratio is less than hFe, then hFe variability will not affect gain.

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