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There are five main amplifier designs: Class A, A/B, B, D, and Tube amplifiers. All of these but tube amplifiers are considered "solid-state."
Class A amplifiers are the most sonically accurate. On the other hand, they have some drawbacks that make them not be the most common choice. Class A amplifiers use only one output transistor that is turned "on" all the time, giving out tremendous amounts of heat. Class A amplifiers are very inefficient (~25%). More heat means more heatsink area, so even though most class A amps have built-in cooling fans, they are big. Pure class A amplifiers are usually expensive.
Class B amplifiers are the most common and use two output transistors. One for the positive part of the cycle and one for the negative part of the cycle. Both signals are then "combined". The problem with this design is that at the point when one transistor stops amplifying and the other one kicks in (zero volt line), there is always a small distortion on the signal, called "crossover distortion". Good amplifier designs make this crossover distortion very minimal. Since each transistor is "on" only half of the time, then the amplifier does not get as hot as a class A, yielding to a smaller size and better efficiency (~50%).
Class A/B amplifiers are a combination of the two types described above. At lower volumes, the amplifier works in class A mode. At higher volumes, the amplifier switches to class B operation.
The class D amplifier (known as digital amplifier) is the last of the solid-state types. These amplifiers are not really digital (there is no such thing), but operate similarly in manner to a digital-to-analog converter (DAC). The signal that comes in is sampled a high rates, and then reconstructed at higher power. This type of amplifier produces almost no heat and is very small in size. Efficiency is much higher in class D amplifiers (~80%).
The sound quality of a Class-D amplifier is much lower than that of other solid-state amplifiers, which is why Class-D amplifiers are only used for subwoofers in car audio. This is because the switching speed of the transistors, and lower sound quality are masked by the lower frequencies being reproduced by the subs, since distortion is harder to discern at low frequency.
other variations on a theme:
Class T: Class T (Tripath) is similar to class D with these exceptions: This class does not use analog feed back like its class D cousin. The feedback is digital and is taken ahead of the output filter, avoiding the phase shift of this filter. Because class D or T amplifier distortion arises from timing errors, the class T amplifier feeds back timing information. The other distinction is that this amplifier uses a digital signal processor to convert the analog input to a PWM signal and process the feedback information. The processor looks at the feedback information and makes timing adjustments. Because the feedback loop does not include the output filter, the class T amplifier is inherently more stable and can operate over the full audio band. Most listeners can not hear the difference between class T and good class AB designs. Both class D and T designs share one problem: they consume extra power at idle. Because the high frequency waveform is present at all times, even when there is no audio present, the amplifiers generate some residual heat. Some of these amplifiers actually turn off in the absence of music, and can be annoying if there is too much delay turning back on.
Class G: Class G improves efficiency in another way: an ordinary class AB amplifier is driven by a multi-rail power supply. A 500 watt amplifier might have three positive rails and three negative rails. The rail voltages might be 70 volts, 50 volts, and 25 volts. As the output of the amplifier moves close to 25 volts, the supply is switched the 50 volt rail. As the output moves close to the 50 volt rail, the supply is switched to the 70 volt rail. These designs are sometimes called "Rail Switchers". This design improves efficiency by reducing the "wasted" voltage on the output transistors. This voltage is the difference between the positive (red) supply and the audio output (blue). Class G can be as efficient as class D or T. While a class G design is more complex, it is based on a class AB amplifier and can have the same clean characteristics as well.
Class H: Class H is similar to class G, except the rail voltage is modulated by the input signal. The power supply rail is always just a bit higher than the output signal, keeping the voltage across the transistors small and the output transistors cool. The modulating power supply rail voltage is created by similar circuitry that you would find in a class D amplifier. In terms of complexity, this type of amplifier could be thought of as a class D amplifier driving a class AB amplifier and is therefore fairly complex.
Lastly we have tube amplifiers, which aren't often used in car audio. Tube amplifiers have about 50 to 60% efficiency.
Tube amplifiers are said to sound more musical. The reason is that tube amplifiers produce even ordered harmonics. Musical instruments give off harmonics in even orders. Transistor amplifiers tend to give off harmonics that are odd ordered. These harmonics are not pleasing to the ear as second order harmonics are. Modern solid state amplifiers have very low distortions but their distortions are less tolerated by the ear than even ordered harmonics. This means that when you hear someone say a Tube amp is "warm" sounding, they are actually talking about the second order distortion produced by that tube amplifier, which they find pleasing to the ear. A good example of this is in guitar amplifiers, which often pride themselves on their second order harmonics.
One should note that while most solid state amplifiers have very low distortions (Total Harmonic Distortion) for the left and right channel, other channels are often much higher as these specifications are rarely noted. Subwoofer amplifiers are particularly bad at creating odd ordered harmonics.
I believe that the best tube and solid state amplifiers sound amazingly alike. Bad tube amplifiers sound tubby and slow. Bad transistor amplifiers sound harsh, bright and strident.
Just like you can't judge a good book by its cover, you can learn very little about an amplifier without digging in and seeing what is inside. Generally speaking, the most important component of any amplifier is its power supply. Is it sufficient? Is it accurate? Is it fast? Unfortunately, almost no amplifier company talks about their power supplies or what transformers they use (An example of a good company would be Eclipse, who uses dual toriodal transformers in their amplifier power supplies.)
I think most manufacturers would prefer you not ask.