I have heard that Op-amps are like special transistors with none of the drawbacks and all of the advantages. What properties make an operational amplifier a "better" amplifier then a biased transistor, and how are transistors constructed to make an op-amp?
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You could successfully build a audio amp from many different types of BJTs. It will be the circuit, not the transistor, that makes the amp work well. I'd pick jellybean parts like the 2N4401 (NPN) and 2N4403 (PNP) and stick with them for everything except for the final power output transistors. Lots of parts could fill that role. If you have your own favorite jellybean small signal transistors, use them if you prefer. The ones I mentioned have reasonable gain and can handle up to 40 V, which should be plenty good enough to allow for a amp to impress your profesor with.
There are lots of possible power transistors to use as the final output. If you are aiming for a few Watts, I'd probably go with basic parts like the TIP41 (NPN) and TIP42 (PNP).
Again though, it's not the choice of transistor that will make or break this project. You can certainly create a impressive audio amp with the transistors I mention, but you can also make a mess. It's really up to the design. In audio, overall noise and harmonic distortion are high priorities. Those come from careful circuit design and attention to these parameters at every step along the way.
You can also use other types of transistors, like JFETs or MOSFETs. Those would require a different circuit topology to utilize properly, but can be used to make a good amp too. Since you will be going over BJT details more thoroughly, I'd stick to them for now. This will be a great learning exercise. Designing a amp with very low noise and very low distortion is not trivial.
What intuitions or rules of thumb would faithfully guide me on when to consider an OTA instead of a "regular" op amp; perhaps illustrated by any "classic" applications where an OTA would be preferred (and why)?
You can't really compare an OTA with a regular OpAmp. OpAmps are simple building blocks that you'll usually "configure" to do one fixed operation by adding components around it.
OTAs are similar but have the added benefit that once you've "configured" them you can still control certain aspects of the operation (lets say amplification) by applying a control current.
The key difference is, that an OTA has three inputs while your OpAmp has only two. Besides the two differential input terminals that an OTA and an OpAmp share, the OTA has a third input that lets you set the gain of the amplifier by applying a current.
This third input enables you to do things that you just can't implement with a simple OpAmp: The OTA is able to multiply two time varying signals!
The OpAmp on the other hand is able to multiply (or amplify) as well, but only one signal is time varying (the one at the differential input). The other factor that goes into the multiplication is constant and defined by the feedback resistors.
Typical use-cases of OTAs are "Voltage Controlled Amplifiers".
Lets say you want to control the volume of an audio signal. For a stereo signal you can use a stereo potentiometer, attenuate the signal and then buffer it using an OpAmp. Fine, but how would you accomplish the same thing if you are dealing with more than two channels? A 5.1 sound-system for example? You'll probably won't find potentiometers with more than two channels.
Here OTAs come to the rescue: You can use a single potentiometer to generate a control voltage and feed it to any number of voltage controlled amplifiers. With the turn of a single knob you can now control the volume of any amount of audio channels as you like.
Another common uses are automatic gain controls. Here a signal gets amplified based on it's amplitude. A signal with low amplitude gets amplified a lot while a signal with high amplitude will just get buffered. The goal here is to generate a signal with less dynamic range at the output. This may avoid clipping the signal and prevent low amplitude parts to be buried in noise. 20 years ago you found these kind of circuits in dictating machines, telephones, tape recorders etc. Nowadays the job is cheaper done in software.
Another big field where OTAs are used are "voltage controlled filters". Here you don't control the amplification of a signal but the cut-off frequency of a filter. Around the half of all analog synthesizer filters from the eighties are based on OTAs.
From the circuit design point of view OpAmps and OTAs are also used differently:
OpAmps are almost always used in closed-loop configuration. E.g. You'll almost always find a resistor or other component that goes from the output to the negative input. As you probably know this used to bring down the very high open-loop gain of an OpAmp down to some useful level.
OTAs on the other hand are very rarely used in closed loop configuration, e.g. you won't find the typical resistor from output to negative input. This is because they don't have the high open-loop gain to begin with. The gain of the OTAs are defined by the current going into the gain control input after all.
This has several consequences: Think about a voltage follower built around an OpAmp. The output of the OpAmp directly connects to the negative input. If you apply a voltage to the positive input the negative feedback makes sure that the voltage difference between the differential inputs is almost zero.
Since there is rarely negative feedback in OTA circuits there is also no mechanism to keep the differential inputs at the same voltage. Instead you'll find a huge voltage divider before the inputs that keep the maximum voltage difference of the input terminals at 10mV to 30mV (rule of thumb). If you go above this the OTA will become more and more nonlinear and will output a highly distorted signal.
Regarding your voltage regulator: This is really a bad use-case for an OTA because you don't need the gain program-ability feature. You could build one using an OTA, but the cool feature of the OTA would not be of any use.
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A transistor is a single electronic element. By itself, it can do nothing. You use the term "biased transistor" which implies a transistor with other circuit components (usually resistors) to make a more useful entity, usually referred to as a circuit.
A transistor can be made to perform many functions such as amplification, rectification, filtering, etc. when it is combined with other circuit elements (resistors, capacitors, and even other transistors).
An operational amplifier is a specific combination of these circuit elements that forms a building block for the same functions plus many more. It is more capable than a single transistor because it contains many more circuit elements that allow for very flexible circuit configurations. It is so generally useful that it is made as an integrated circuit which is easily applied as a single circuit element and is much cheaper and smaller than it would be if built from discrete components.
An operational amplifier is the equivalent of many transistors and is thus able to perform much better than a single transistor (e.g. higher input impedance, lower output impedance, higher gain, differential inputs and/or differential outputs, etc.).
There are instances, however, where a discrete transistor can outperform an operational amplifier. One is noise performance. Special discrete transistors are available which have lower voltage noise than the best operational amplifiers and are used when the very lowest noise levels are needed. One such application is in sonar hydrophone amplifiers which have to work with signals at sub-microvolt levels.
In general, though, IC operational amplifiers out-perform single transistors and discrete operational amplifiers.