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Laplace Transforms

Laplace transforms convert a function f(t) in the time domain into function in the Laplace domain F(s).

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

As an example of the Laplace transform, consider a constant c. The function f(t) = c and the following expression is integrated.

$$\mathcal{L}(c)=\int_0^\infty c \, e^{-s\,t} dt = -\frac{c}{s}e^{-s\,t} \biggr\rvert_0^\infty = 0 - \left(-\frac{c}{s} \right) = \frac{c}{s}$$

Mathematicians have developed tables of commonly used Laplace transforms. Below is a summary table with a few of the entries that will be most common for analysis of linear differential equations in this course. Notice that the derived value for a constant c is the unit step function with c=1 where a signal output changes from 0 to 1 at time=0.

Laplace Transform Table

f(t) in Time Domain F(s) in Laplace Domain
$$\delta(t)\quad \mathrm{unit \; impulse}$$$$1$$
$$S(t) \quad \mathrm{unit \; step}$$$$\frac{1}{s}$$
$$t \quad \mathrm{ramp \; with \; slope = 1}$$$$\frac{1}{s^2}$$
$$t^{n-1}$$$$\frac{(n-1)!}{s^n}$$
$$e^{-b\,t}$$$$\frac{1}{s+b}$$
$$1-e^{-t/\tau}$$$$\frac{1}{s(\tau s + 1)}$$
$$\sin(\omega t)$$$$\frac{\omega}{s^2+\omega^2}$$
$$\cos(\omega t)$$$$\frac{s}{s^2+\omega^2}$$
$$\frac{1}{\tau_1-\tau_2}\left(\exp{\left(-t/\tau_1\right)} - \exp{\left(-t/\tau_2 \right)} \right)$$$$\frac{1}{\left(\tau_1s+1\right)\left(\tau_2s+1\right)}$$
$$\frac{1}{\tau^n \left(n-1\right)!}t^{n-1}\exp{\left(-\frac{t}{\tau}\right)}$$$$\frac{1}{\left(\tau s+1\right)^n}$$
$$\frac{1}{\tau \sqrt{1-\zeta^2}} \exp{\left(-\frac{\zeta \, t}{\tau}\right)} \sin{ \left( \sqrt{1-\zeta^2} \frac{t}{\tau} \right) }$$$$\frac{1}{\tau^2 s^2 + 2 \zeta \tau s + 1}$$
See 2nd Order Systems$$\frac{1}{s\left(\tau^2 s^2 + 2 \zeta \tau s + 1\right)}$$
$$\frac{df}{dt}$$$$sF(s)-f(0)$$
$$\frac{d^nf}{dt^n}$$$$s^n F(s) - s^{n-1} f(0) - s^{n-2}f^{(1)}(0) - \ldots \\ - sf^{(n-2)}(0) - f^{(n-1)}(0)$$
$$\int f(t)$$$$\frac{F(s)}{s}$$
$$f\left(t-t_0\right)S\left(t-t_0\right)$$$$e^{-t_0s}F(s)$$

Note that the functions f(t) and F(s) are defined for time greater than or equal to zero. The next step of transforming a linear differential equation into a transfer function is to reposition the variables to create an input to output representation of a differential equation.

Laplace Transforms with Python

Python Sympy is a package that has symbolic math functions. A few of the notable ones that are useful for this material are the Laplace transform (laplace_transform), inverse Laplace transform (inverse_laplace_transform), partial fraction expansion (apart), polynomial expansion (expand), and polynomial roots (roots).

import sympy as sym
from sympy.abc import s,t,x,y,z
from sympy.integrals import laplace_transform
from sympy.integrals import inverse_laplace_transform

# Laplace transform (t->s)
U = laplace_transform(5*t, t, s)
print('U')
print(U[0])
# Result: 5/s**2

# Inverse Laplace transform (s->t)
X = inverse_laplace_transform(U[0],s,t)
print('X')
print(X)
# Result: 5*t*Heaviside(t)

# Function
F = 5*(s+1)/(s+3)**2
print('F')
print(F)
# Result: (5*s + 5)/(s + 3)**2

# Partial fraction decomposition
G = sym.apart(F)
print('G')
print(G)
# Result: 5/(s + 3) - 10/(s + 3)**2

# denominator of transfer function
d1 = (s+1)*(s+3)*(s**2+3*s+1)

# expand polynomial
d2 = sym.expand(d1)
print('d2')
print(d2)
# Result: s**4 + 7*s**3 + 16*s**2 + 13*s + 3

# find roots
print(sym.roots(d2))
# Result: {-1: 1, -3: 1, -3/2 - sqrt(5)/2: 1, -3/2 + sqrt(5)/2: 1}

Assignment

See Laplace Applications