An operational amplifier (op-amp) integrator circuit is an electronic circuit configuration that utilizes an operational amplifier to perform the mathematical operation of integration on an input signal. Integration is a process that accumulates the integral or area under a signal with respect to time. In this circuit, an op-amp is configured with appropriate feedback components to achieve this integration functionality.
The basic structure of an op-amp integrator circuit consists of an operational amplifier connected in an inverting amplifier configuration with a feedback capacitor (C) in parallel with a resistor (R). The input signal is applied to the inverting input terminal of the op-amp. The output of the op-amp serves as the integrated output of the input signal.
Here's the circuit diagram and the key equations for an op-amp integrator:
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R
Vin --/\/\/\--|___
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\ /
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| \ C
| /
|/
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-| Vout
Key Equations:
Virtual Ground Rule: The inverting input terminal of the op-amp is assumed to be a virtual ground, meaning that the voltage at the inverting input is nearly equal to 0V.
Voltage Across the Capacitor: Since the inverting input is at virtual ground, the voltage across the capacitor is the same as the output voltage: Vc = Vout.
Current through the Feedback Components: The current through the resistor R and capacitor C is the same, which can be described by the equation: I = C * dVc/dt (where dVc/dt is the rate of change of voltage across the capacitor).
Op-Amp Input Configuration: Due to the inverting configuration, the voltage at the inverting input terminal is equal to the input voltage: Vin = Vc.
Op-Amp Ideal Behavior: The op-amp drives its inverting input terminal to match the non-inverting input terminal, which is at virtual ground. This means that the inverting input terminal will always be at 0V.
Combining these equations, you can derive that the output voltage of the integrator circuit is given by:
Vout = - (1 / RC) ∫ Vin dt + V_initial
Where:
Vout is the output voltage.
Vin is the input voltage.
RC is the time constant defined by the resistor (R) and capacitor (C).
∫ represents the integral operation.
V_initial is the initial condition (voltage across the capacitor at t=0).
The op-amp integrator circuit is commonly used in various applications, including analog signal processing, audio processing, and control systems. It provides the mathematical operation of integration, allowing for the accumulation of input signal values over time.