Laplace Transform
The Laplace Transform converts a time-domain function into the complex frequency domain (s-domain), enabling differential equations to be solved as algebraic equations. It generalises the Fourier Transform by including a real exponential damping term, making it applicable to a broader class of signals including those that grow with time. It is the primary mathematical tool in control systems engineering, circuit analysis, and linear system theory for analysing stability, transient response, and frequency behaviour.
Key Formula
F(s) = integral from 0 to ∞ of f(t) × e^(-st) dt
LaTeX: F(s) = \mathcal{L}\{f(t)\} = \int_{0}^{\infty} f(t)\, e^{-st}\, dt
| Symbol | Meaning | Unit |
|---|---|---|
| F(s) | Laplace transform (s-domain function) | Complex |
| f(t) | Original time-domain function | Varies |
| s | Complex frequency variable s = σ + jω | rad/s (complex) |
| σ | Real part of s (damping factor) | Np/s |
| ω | Imaginary part of s (angular frequency) | rad/s |
Worked Example
Problem
A first-order RC circuit has the equation: RC·dv/dt + v = V_s where R = 1 kΩ, C = 1 µF, and V_s = 5 V (DC step). Find the output voltage v(t) using Laplace transforms, assuming v(0) = 0.
Solution
Step 1: RC time constant τ = RC = 1000 × 1×10⁻⁶ = 1×10⁻³ s = 1 ms. Step 2: Take Laplace transform of both sides. τ·[sV(s) − v(0)] + V(s) = V_s/s (1×10⁻³)·sV(s) + V(s) = 5/s Step 3: Solve for V(s). V(s)(τs + 1) = 5/s V(s) = 5 / [s(τs + 1)] = 5 / [s(0.001s + 1)] Step 4: Partial fraction decomposition. V(s) = 5/s − 5/(s + 1/τ) = 5/s − 5/(s + 1000) Step 5: Inverse Laplace transform. v(t) = 5 − 5·e^{−1000t} = 5(1 − e^{−t/τ}) V
Answer
v(t) = 5(1 − e^{−1000t}) V, reaching ~63.2% of 5 V (i.e., 3.16 V) at t = 1 ms
Common Laplace Transform Pairs
| Time function f(t) | Laplace Transform F(s) | Region of Convergence |
|---|---|---|
| δ(t) — unit impulse | 1 | All s |
| u(t) — unit step | 1/s | Re(s) > 0 |
| t (ramp) | 1/s² | Re(s) > 0 |
| e^{-at} (exponential) | 1/(s+a) | Re(s) > −a |
| sin(ωt) | ω/(s²+ω²) | Re(s) > 0 |
| t·e^{-at} | 1/(s+a)² | Re(s) > −a |
Interactive Tools
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Related Terms
Fourier Transform
The Fourier Transform decomposes a time-domain signal into its constituent frequency components, expressing it as a superposition of sinusoids of different frequencies, amplitudes, and phases. It provides the frequency-domain representation of a signal and is fundamental to signal processing, communications, image analysis, and solving differential equations. The transform is invertible, meaning the original signal can be perfectly reconstructed from its frequency spectrum.
Transfer Function (control)
A transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input of a linear time-invariant (LTI) system, assuming zero initial conditions. It fully characterises a system's dynamic behaviour in the frequency domain, including its poles, zeros, gain, and stability margins. Transfer functions are the central tool in classical control theory for designing feedback controllers, analysing stability using root locus and Bode plots, and predicting transient and steady-state responses.
Feedback Control
Feedback control is a control strategy in which the output of a system is measured and compared to a desired reference (setpoint), and the difference (error) is used to adjust the system input to reduce that error. Negative feedback — where the output is subtracted from the reference — is the basis of stable automatic control systems in engineering, biology, and economics. Feedback control enables systems to self-correct against disturbances and parameter variations, forming the foundation of servo systems, thermostats, autopilots, and industrial process control.
Named after French mathematician Pierre-Simon, Marquis de Laplace (1749–1827), who developed the transform as a tool for probability theory in his monumental work "Mécanique Céleste." The operational form used in engineering was formalized by Oliver Heaviside in the 1890s through his operational calculus, and rigorously justified mathematically by Thomas Bromwich in 1916.