Op Amp Integrator Transient Analysis

An operational amplifier (op-amp) integrator circuit is a fundamental electronic circuit used to perform mathematical integration of an input signal and is one of the basic circuits used in analog control systems. Integration in this context refers to the mathematical operation that yields the cumulative area under a signal curve with respect to time. Inverting Op-Amp Configuration: The op-amp integrator circuit is typically configured with the op-amp in an inverting mode. The key component in this circuit is a feedback capacitor (Cf). The capacitor integrates the input voltage signal over time. Input Voltage Integration: When a voltage is applied to the input of the integrator, the capacitor charges or discharges through the resistor (Rf), causing the voltage across the capacitor to be the integral of the input voltage. Mathematical Relationship: The voltage across the capacitor (Vout) can be described by the following equation in the Laplace domain: 𝑉𝑜𝑢𝑡(𝑠) = −1 / (𝑠𝑅𝑓𝐶𝑓) ∙ 𝑉𝑖𝑛(𝑠) where s is the complex frequency variable. Frequency Response: The integrator has a frequency response that rolls off at -20 dB/decade. At low frequencies, the capacitor impedance dominates, causing a high gain. As the frequency increases, the gain decreases, and the output approaches the integration of the input signal.