I'm trying to understand the definition of operational transconductance amplifier (OTA). The book on which I'm studying defines it as a class of operational amplifiers which "achieves a high voltage gain with a given transconductance gain, gm, by the use of a very high output resistance". This is the idea behind a lot of basic building blocks, for example the cascode with cascode load of the following image:
The voltage gain is maximized by increasing the gm of the input transistor M1 or by increasing the output resistance (given in this case by the parallel connection of a PMOS cascode and a NMOS cascode).
Question: theoretically, every circuit (for small signals, which ensures linearity) is an OTA. Indeed Razavi, although he never introduces the word OTA in his book, proves the following theorem (which is just a consequence of Norton's theorem): the voltage gain of a linear circuit can be expressed as
where Rout is the output resistance of the circuit and Gm is the shortcircuit transconductance of the circuit
Thus theoretically, just by applying Razavi's theorem, every circuit can be considered an OTA: I find the output resistance and the shortcircuit transconductance, I multiply them with a minus and I have found the voltage gain. But this sounds strange to me: even if this algorithm works for every (linear) circuit from a mathematical point of view (it's a theorem proved by Razavi), why then define the concept of OTA? I suppose that a given circuit must have some features such that it is "meaningful" to consider it as an OTA (i. e. it is "meaningful" finding the voltage gain applying Razavi's theorem instead of doing a standard analysis of the circuit). I also suppose conversely that it would be "meaningless" to consider some circuits, which do not satisfy some features, as OTAs (i. e. it would be "meaningless" finding the voltage gain of these circuits applying Razavi's theorem).