3.6:
Integrator and Differentiator
Consider a car with cruise control using an op amp circuit to maintain a constant speed. This circuit employs the function of an op amp as an integrator and differentiator.
An ideal integrator and differentiator can be created from an inverting operational amplifier circuit by replacing the feedback and input resistors with capacitors, respectively.
An integrator is an op amp circuit that generates an output in proportion to the integral of the input signal.
Conversely, a differentiator produces an output proportional to the rate of change of the input signal.
By applying Kirchhoff's current law at the inverting node of both circuits and utilizing the current-voltage expressions for the resistor and capacitor, an equation relating the output and input voltages can be derived.
Upon integrating the equation for the integrator circuit on both sides and assuming the capacitor discharges before signal application, the final expression demonstrates that the output voltage is proportional to the input's integral.
In contrast, for the differentiator circuit, the output voltage is proportional to the derivative of the input signal.
3.6:
Integrator and Differentiator
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.