Op-amp circuits have significant applications in various fields, including automotive engineering. One such application is cruise control systems in cars, where op-amp circuits are integral for maintaining a constant speed. In these systems, op-amps function as both integrators and differentiators.

An integrator within an op-amp circuit produces an output directly proportional to the integral of the input signal. This is achieved by replacing the feedback resistor in a typical inverting amplifier circuit with a capacitor, resulting in an ideal integrator. An equation relating output and input voltages is derived by applying Kirchhoff's current law and utilizing current-voltage relationships for resistors and capacitors. When integrated, this equation demonstrates that the output voltage corresponds to the integral of the input signal.

Conversely, a differentiator within an op-amp circuit yields an output proportional to the input signal's rate of change. Achieving this involves replacing the input resistor with a capacitor in a standard inverting amplifier, creating a differentiator circuit. An equation linking output and input voltages is established by applying Kirchhoff's current law and employing current-voltage relations. In this case, the equation indicates that the output voltage is proportional to the derivative of the input signal.

It is worth noting that these op-amp circuits are valuable in energy storage applications and are often designed using resistors and capacitors due to their compactness and cost-effectiveness. While integrators are widely employed in analog computers and various applications, differentiators are less common in practice due to their tendency to amplify electrical noise, making them electronically unstable.