SDomäne

Sdomæne, or the s-domain, is the complex frequency domain used in the Laplace transform to analyze linear time-invariant systems. In this representation, signals originally defined in time, x(t), are mapped to X(s) in the complex plane. The complex variable s = σ + jω combines exponential growth/decay (σ) with oscillatory behavior (ω). The Laplace transform can be taken as a unilateral transform, X(s) = ∫_0^∞ x(t) e^{-st} dt, or as a two-sided transform, X(s) = ∫_{-∞}^{∞} x(t) e^{-st} dt.

In the s-domain, many time-domain operations become algebraic. Differentiation in time corresponds to multiplication by s, while integration corresponds to division by s. Convolution in time becomes multiplication in s. A linear time-invariant system is often described by its transfer function H(s) = Y(s)/U(s), representing the system’s output Y(s) to a given input U(s).

Poles, zeros, and the region of convergence (ROC) are key features. The ROC is the set of s values for which the Laplace transform converges. For causal systems, the ROC lies to the right of the rightmost pole. Stability (BIBO stability) requires all poles to have negative real parts (left half-plane), which guarantees the ROC includes the imaginary axis and a bounded output for bounded input.

Practical uses include filter design, control theory, and system analysis, where the s-domain facilitates understanding dynamics, stability, and frequency response. The s-domain is related to the Z-domain for discrete-time systems, via substitutions like s ≈ (1/T) ln Z and frequency mappings. Simple examples include the transform of a unit step, 1/s for Re(s) > 0, illustrating how time-domain behaviors are represented in s-domain terms.