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Amplifier is the general name for devices that can amplify signals (power, voltage and/or current). They can be built with transistors, tubes (valves), MOSFETS.
Operational amplifiers are high performance integrated differential amplifiers(high gain, high input impedance, low output impedance, accurate and high bandwidth response) . They usually are small signal amplifiers and have two inputs (inverter and non inverter) and one output. Their name derives from the past analog computers parts, as they were design to perform math operations (so the name "operational") using analog voltages. Operational Amplifiers can be used inside an amplifier, as oscillators, in instrumentation, and many other uses.
Main task of amplier is to amplify an incoming signal according to their gain. Opamp Amplifier is actually an amplifer but they are used more in dc based application like in feedback system to perform the math operation like comparing, adding dc and so on....
An amplifier designed to deliver maximum power output to a load. Example: In an audio system, it is the power amplifier that drives the loudspeaker.
Op-Amps are high gain DC amplifier that has a high input impedance and a low output impedance. Op-amps are the most basic type of linear integrated circuits.
Op amps can do mathematical operations in a circuit, it can perform the following depending on the configuration: Addition (voltage summer), Subtraction (Subtracter), Differentiation/integration (integrator/differentiator) thus the name implied operational.
So, we can now conclude that Op-Amp is just another form of an amplifier.
Very high input impedance, low output impedance, high bandwith capacity
and very little low offset voltage (supposedly)
OP-AMP is not a DC amplifier. It has a BandWith (BW) of useful frequency range. So an OP-AMP also be used for AC signals. At DC it has usually very high gain but for a given frequency gain drops (higher the frequency lower the gain, OP-AMP may have gain until its Unity Gain Frequency reached where gain reaches to zero).
The term "Operational Amplifier" was coined in the early days of computation when the bulk of simulation work was done using analog computers. The operational amplifier was used, in conjunction with a few external components, to perform a mathematical "operation", such as summation, integration, differentiation. The idea was to provide an amplifier whose characteristics (input imedance, gain, offsets, etc) were good enough such that the closed loop characteristics of a circuit were dependent only on the external components, and not on the characteristics of the amplifier itself.
Amplifier is a generic term that amplifies the given input signal to the desired output level .
Operational amplifier is again an amplifier that which has some advantages over other available ampliers like common mode rejection , very high input impedence and very low output impedence etc.