Transforms

Laplace & Z-Transform Toolkit

Controls

Browse the complete Laplace transform reference table on the right.

Search f(t) or F(s)

ay'' + by' + cy = g(t)

a (y'') b (y') c (y) y(0) y'(0)

g(t) forcing function

Browse the Z-Transform reference table on the right.

f(t) (time domain)

F(s) = ℒ{f(t)}

Conditions

Key Theorems

Linearity

ℒ{af(t)+bg(t)} = aF(s)+bG(s)

First Derivative

ℒ{f'(t)} = sF(s) - f(0)

Second Derivative

ℒ{f''(t)} = s²F(s) - sf(0) - f'(0)

s-shifting

ℒ{eᵃᵗf(t)} = F(s-a)

t-shifting

ℒ{f(t-a)u(t-a)} = e⁻ᵃˢF(s)

Convolution

ℒ{(f*g)(t)} = F(s)·G(s)

Initial Value Thm

f(0⁺) = lim[s→∞] s·F(s)

Final Value Thm

f(∞) = lim[s→0] s·F(s)

x[n]

X(z)

ROC

Laplace Transforms & System Response Handbook

Mathematical Principles & Theorems

The unilateral Laplace Transform \(\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty f(t) e^{-st} dt\) converts differential equations into algebraic equations in complex frequency \(s = \sigma + j\omega\). Key properties: Linearity, Differentiation \(\mathcal{L}\{f'(t)\} = sF(s) - f(0)\), Integration \(\mathcal{L}\{\int_0^t f(\tau)d\tau\} = \frac{F(s)}{s}\), and Convolution \(\mathcal{L}\{f * g\} = F(s)G(s)\). Transfer function \(H(s) = \frac{Y(s)}{X(s)}\) poles determine system stability (stable if \(\text{Re}(p_i) < 0\)).

Operating Instructions

Select test function \(f(t)\) (Step, Impulse, Exponential, Sinusoid, Polynomial) or enter transfer function \(H(s)\).
Configure system parameters and initial conditions using numeric inputs.
Click Transform to view analytic Laplace domain representation \(F(s)\).
Inspect the impulse response \(h(t)\), step response curves, and pole-zero s-plane constellation.