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Sketch op-amp integrating circuit with waveform. Explain circuit operator in brief.

Obtained by replacing the feedback resistor $R_f$ in inverting amp configuration by “CF”. Assume that the op.amp used is ideal one.

$V_2→ \text{virtual ground potential.} \\ ∴kCL at V_2, \\ I_1=I_B+I_F--------(1) \\ \text{Due to high i/p impedence,} \\ I_B→∝,∴I_1≈I_f \\ \text{Now current through cap.} \\ I_c=C \frac{dV_c}{dt} \\ \text{But in case of integrater}, I_f=I_cf \\ ∴I_1=Cf \frac{dV_C}{dt} \\ \text{But} I_1= \frac{V_in-V_2}{R_1} \text{and} V_c=(V_2-V_0 ) \\ ∴\frac{V_in-V_2}{R_1} =c_F \frac{d}{dt} (V_2-V_0 ) \\ \text{Using concept of virtual ground,} \\ V_2=0 \\ ∴\frac{V_in}{R_1} =C_f \frac{d}{dt} (-V_0 ) \\ V_c= -\frac{1}{(R_1 C_f )} ∫_0^tV_{in} dt+C→ \text{Constant}$