Differentiatorer

Differentiatorer, or differentiators, are devices, circuits, or mathematical operators whose output corresponds to the derivative with respect to time or another variable of their input. In Scandinavian languages such as Danish and Norwegian, the term differentiatorer is used as the plural form. In mathematics, the differentiation operator is usually denoted D or d/dx, and maps a function f(x) to its derivative f'(x).

In continuous-time signal processing, a differentiator produces an output proportional to the rate of change of the input. The ideal differentiator has a transfer function H(s) = k s, implying increasing gain with frequency and making it sensitive to high-frequency noise. In discrete time, a common approximation uses finite differences, such as y[n] ≈ (x[n] - x[n-1]) / T, where T is the sampling period.

Practical differentiators are designed to reduce noise amplification and improve stability. In electronics, an ideal differentiator is typically implemented as an operational amplifier with a capacitor in the input path and a resistor in the feedback path, complemented by additional elements to bound gain at high frequencies. Band-limited or compensated differentiators can also feature a low-pass element to moderate behavior at high and very low frequencies.

Practical differentiators are used in control systems, edge detection, and instrumentation to measure rate of change. They face limitations from noise and drift, require careful tuning, and are often paired with filters or integrators to form robust compensators. See also differentiation, Laplace transform, transfer function.