An operational amplifier (op-amp) integrator is a specific configuration of an op-amp circuit that performs analog integration of input signals. Integration, in this context, means calculating the accumulated sum of the input signal over time, which is equivalent to finding the area under the input signal curve.
The basic op-amp integrator circuit consists of an op-amp connected with a feedback capacitor (C) in the feedback path. The input signal (Vin) is applied to the inverting (-) input terminal of the op-amp, and the output (Vout) is taken from the output terminal. The non-inverting (+) input terminal is usually grounded.
The integrator circuit takes advantage of the fact that the voltage at the inverting input terminal of the op-amp is virtually equal to the voltage at the non-inverting input terminal due to the high open-loop gain of the op-amp (ideally infinite). This results in a virtual ground at the inverting input, meaning the voltage at this terminal is effectively zero.
Here's the basic op-amp integrator circuit:
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+--------------+
Vin --| |
| |
| Op-Amp | Vout
| +--|___
| | | |
+--|| C | |___|
R |
| |
+--------------+
The operation of the op-amp integrator is based on the capacitor charging and discharging through the feedback resistor (R). When an input voltage is applied, the capacitor starts to charge or discharge at a rate determined by the RC time constant (R Γ C). As the input voltage varies, the capacitor voltage continuously integrates the input signal over time.
The output voltage (Vout) can be mathematically described as follows:
Vout = - (1 / RC) * β«Vin dt + Vinitial
Where:
Vout is the output voltage.
Vin is the input voltage.
RC is the time constant of the feedback network (R Γ C).
β« denotes the integral operation.
Vinitial is the initial voltage across the capacitor (at t = 0).
It's important to note that the ideal op-amp integrator has an infinite gain at DC (low-frequency), meaning it integrates the input signal perfectly. However, in practice, real op-amps have limitations due to their finite gain and bandwidth, leading to errors and distortion in the integrated output for high-frequency signals. To improve the performance of real op-amp integrators, additional compensation techniques or active filtering may be employed.