Laplace Transforms

The Laplace Transform is a powerful tool used in engineering and physics to solve linear differential equations. Its primary “trick” is to transform a function from the Time Domain ($t$) to the Complex Frequency Domain ($s$), where integration and differentiation become simple algebraic multiplication and division.

The Definition

The Laplace transform of a function $f(t)$ is defined as:

$\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t) dt$

Why Transform?

In the time domain, a system might be described by a messy differential equation: $a \frac{d^2y}{dt^2} + b \frac{dy}{dt} + cy = f(t)$

In the Laplace (s) domain, this becomes a simple equation: $(as^2 + bs + c)Y(s) = F(s)$

You can then solve for $Y(s)$ using basic algebra and “Inverse Transform” back to the time domain.

Transfer Functions and Stability

In control theory, the Transfer Function $H(s) = \frac{Y(s)}{X(s)}$ describes how a system responds to an input.

• If the “poles” (the values of $s$ that make the denominator zero) have negative real parts, the system is Stable (vibrations die down).
• If any pole has a positive real part, the system is Unstable (vibrations grow until the system breaks).

import sympy as sp

# Define symbols
t, s = sp.symbols('t s')
# Define a function to transform: sin(t)
f = sp.sin(t)

# Calculate Laplace Transform
F = sp.laplace_transform(f, t, s)

print(f"Time Domain: f(t) = {f}")
print(f"S-Domain (Laplace): F(s) = {F[0]}")
print("\nNotice how the trigonometric function became a rational algebraic fraction.")

Use Cases

1. Circuit Analysis: Capacitors and Inductors become simple algebraic impedances ($1/sC$ and $sL$).
2. Control Systems: Tuning a thermostat or a drone’s flight controller.
3. Signal Processing: Filtering noise out of audio or sensor data.

Exercises