integratorer

Integratorer is the plural form used in some languages for devices or systems that output the time integral of their input. In continuous-time engineering terms, an ideal integrator produces y(t) = ∫ u(τ) dτ + y(0). In the Laplace domain its transfer function is H(s) = 1/s, which implies a 90-degree phase shift and a magnitude that grows with frequency. In discrete time, a simple accumulator implements y[k] = y[k−1] + T·u[k], where T is the sampling interval.

Real-world integrators are not perfect. Practical implementations exhibit finite DC gain, drift, and saturation. A leaky integrator adds a small decay term, modeling ẏ = u − αy, to prevent unbounded growth. In control loops, integrators can experience windup when actuators saturate; anti-windup methods are used to limit or reset the integrated signal.

Implementation approaches vary. Analog integrators often use an op-amp with a capacitor in the feedback path, producing continuous integration but being sensitive to bias currents and component tolerances. Digital integrators accumulate discrete samples, with considerations for numerical precision, sampling rate, and potential overflow. Variants include resettable integrators and programmable or fractional-order integrators in more advanced systems.

Applications are broad. In control systems, the integral term of a PID controller reduces steady-state error. In signal processing and data analysis, integrators are used for smoothing, feature extraction, or numerical analysis. In mathematics and physics, the integral operator underpins area calculations and cumulative effects over time, linking instantaneous input to accumulated output.