Circuit Theory-2 #13 | The Laplace Transform - Definition, Step, Impulse

Instructor: Dr. Süleyman Burak ÇELİK

In this Circuit Theory-2 lesson we close the AC steady-state chapter and open the s-domain. The Laplace transform converts differential equations into algebra and unifies transient and steady-state analysis under one tool. You will learn: - Why we need the Laplace transform: ODE → algebra - The one-sided Laplace integral F(s) = ∫₀^∞ f(t) e^{-st} dt - s as a complex frequency: σ + jω, decay + oscillation - Pair 1: u(t) ↔ 1/s, ROC Re(s) is greater than 0 - Pair 2: δ(t) ↔ 1, via the sifting property - Pair 3: e^{-at} u(t) ↔ 1/(s+a), ROC Re(s) is greater than -a - Pairs 4-5: cos(ωt) and sin(ωt) via Euler - 3D visualization: f(t) e^{-st} as a damped spiral in space Timestamps: 00:00 The Laplace Transform 00:03 From AC steady state to the s-domain 00:28 From calculus to algebra 01:06 The (one-sided) Laplace transform 01:42 s = σ + jω: decay + oscillation 02:27 Pair 1: u(t) ↔ 1/s 03:15 Pair 2: δ(t) ↔ 1 03:51 Pair 3: e^{-at} u(t) ↔ 1/(s+a) 04:36 cos and sin via Euler 05:15 Six fundamental pairs 05:51 3D Laplace integrand spiral 06:45 Wrap Next lesson: shift, differentiation, and integration in the s-domain. Part of the Circuit Theory-2 playlist by AcEdumy.