Laplace Transform Table, Formula, Examples & Properties

Laplace transformation is an integral transform that maps a time function f(t), defined for t ≥ 0, into an s-domain function F(s). For a linear differential equation with constant coefficients, differentiation becomes multiplication by s, so the ODE becomes algebra. After you solve for F(s), the inverse Laplace transform returns f(t).

This page covers the unilateral Laplace transform used in circuit and control work: the definition, a transform table, algebraic properties, inversion and worked examples. The same transform is used to write a transfer function. Block-diagram algebra is not developed here.

Many integral transforms exist. Laplace transforms and Fourier transforms are the pair most often used in engineering. The Laplace transform is chosen for causal time functions because the real part of s can make the integral converge when a Fourier integral would not. Algebra in s is still usually easier than integrating the original ODE.

Laplace Transform Table

There is always a table that is available to the engineer that contains information on the Laplace transforms. An example of Laplace transform table has been made below. We will come to know about the Laplace transform of various common functions from the following table .

Laplace Transform Definition

A Laplace table is a lookup sheet. The defining integral is what those table entries come from.

To write the formula, let f(t) be a function of time t for all t ≥ 0.

Then the Laplace transform of f(t), written F(s), is

when that integral converges. The complex variable s = σ + jω may be real or complex, with j = √(-1)

Disadvantages of the Laplace Transformation Method

The method is built for linear time-invariant models with constant coefficients, so differentiation maps to polynomials in s. Nonlinear equations, time-varying coefficients and many PDE problems need another method. Piecewise-linear switching can still be handled with step functions, though the algebra is no longer a single LTI transform pair.

History of Laplace Transforms

The Laplace transform is named after Pierre-Simon Laplace, a French mathematician and astronomer. He used related generating-function integrals in probability, then applied a similar integral to differential equations.

Oliver Heaviside, an English electrical engineer, spread an operational calculus for circuits in the late nineteenth century. Niels Abel, Mathias Lerch and Thomas Bromwich developed inversion and uniqueness results in the same era. Engineering textbooks adopted the named Laplace integral widely around the Second World War, after Heaviside’s operators had already been in use.

Leonhard Euler studied integrals of this type in 1744 and did not push the method far. Joseph Lagrange later adapted Euler’s integrals. Laplace returned to that line in 1782 and, in 1785, used the transform to recast differential equations as operations on F(s). By 1809 he was writing the integral with an infinite upper limit, which is the unilateral form used below.

Method of Laplace Transform

In control engineering the Laplace transform is the usual map from a linear time function to an algebraic function of s. The inverse transform is required to recover the time response. Linearity, shifting, differentiation, integration and convolution are the properties used on those linear models.

Linearity, differentiation, integration, multiplication, frequency shifting, time scaling, time shifting, convolution, conjugation, periodic function. Two limit theorems used with control transfer functions are:

1. Initial value theorem (IVT)
2. Final value theorem (FVT)

Common pairs cover impulse, unit impulse, step, unit step, shifted unit step, ramp, exponential decay, sine, cosine, hyperbolic sine, hyperbolic cosine, natural logarithm and Bessel function. The practical gain is that a higher-order linear ODE with constant coefficients becomes a rational equation in s.

To form F(s) from a given f(t), follow these two steps:

• First multiply f(t) by e^-st, s being a complex number (s = σ + j ω).
• Integrate this product w.r.t time with limits as zero and infinity. This integration results in Laplace transformation of f(t), which is denoted by F(s).

The time function f(t) is recovered from F(s) by the inverse Laplace transform, written £^-1

Laplace Transform Properties

The main properties of Laplace Transform can be summarized as follows:
Linearity: Let C₁, C₂ be constants. f(t), g(t) be the functions of time, t, then

First shifting Theorem:

Change of scale property:

Differentiation:

Integration:

Time Shifting:
If L{f(t) } = F(s), then the Laplace Transform of f(t) after the delay of time, T is equal to the product of Laplace Transform of f(t) and e^-st that is

Where, u(t-T) denotes unit step function.
Product:
If L{f(t) }=F(s), then the product of two functions, f₁ (t) and f₂ (t) is

Final Value Theorem:

This theorem gives the t → ∞ value of a stable time response in feedback design. It does not give the t = 0+ value. The initial-value theorem does that.
Initial Value Theorem:

Let us examine the Laplace transformation methods of a simple function f(t) = e^αt for better understanding the matter.

Comparing the above solution, we can write,

Similarly, by putting α = 0, we get,

Similarly, by putting α = jω, we get,


And thus,

Let us examine another example of Laplace transformation methods for the function



Again the Laplace transformation form of e^t is,

This Laplace form can be rewritten as

Now from the definition of power series we get,

Laplace Transform Examples

Solve the equation using Laplace Transforms,

Using the table above, the equation can be converted into Laplace form:

Using the data that has been given in the question the Laplace form can be simplified.

Dividing by (s² + 3s + 2) gives

This can be solved using partial fractions, which is easier than solving it in its previous form. Firstly, the denominator needs to be factorized.

Cross-multiplying gives:

Next the coefficients A and B need to be found

Substituting in the equation:

Then using the table that was provided above, that equation can be converted back into normal form.
Examples to try yourself
Calculate and write out the inverse Laplace transformation of the following, it is recommended to find a table with the Laplace conversions online:

Solutions:

Worked Laplace transform examples follow:
1) Where, F(s) is the Laplace form of a time domain function f(t). Find the expression of f(t).

Solution

Now, Inverse Laplace Transformation of F(s), is

2) Find Inverse Laplace Transformation function of

Solution

Now,



Hence,

3) Solve the differential equation

Solution
As we know that, Laplace transformation of

4) Solve the differential equation,

Solution
As we know that,

5) For circuit below, calculate the initial charging current of capacitor using Laplace Transform technique.

Solution
The above figure can be redrawn in Laplace form,


Now, initial charging current,

6) Solve the electric circuit by using Laplace transformation for final steady-state current

Solution
The above circuit can be analyzed by using Kirchhoff Voltage Law and then we get

Final value of steady-state current is

7) A system is represented by the relation

Where, R(s) is the Laplace form of unit step function. Find the value of x(t) at t → ∞.
As R(s) is the Laplace form of unit step function, it can be written as

Solution

8) Find f(t), f^‘(t) and f^“(t) for a time domain function f(t). The Laplace Transformation form of the function is given as

By applying initial value theorem, we get,



Applying Initial Value Theorem, we get,

9) The Laplace Transform of f(t) is given by,

Find the final value of the equation using final value theorem as well as the conventional method of finding the final value.
Solution



Hence it is proved that from both of the methods the final value of the function becomes same.

10) Find the Inverse Laplace Transformation of function,

Solution
F(s) can be rewritten as,

11) Find the Inverse Laplace transformation of

Solution
F(s) can be rewritten as,

12) Find the Inverse Laplace transformation of

Solution
F(s) can be rewritten as,

13) Express the differential equation in Laplace transformation form

Solution

14) Express the differential equation in Laplace transformation form

Solution

Where are Laplace Transforms used in Real Life?

Lerch’s theorem (sometimes called Lerch’s cancellation law) states uniqueness of the unilateral Laplace transform. That uniqueness result is not the origin of the method. The Laplace Transform maps a time function to an algebraic function of s. The inverse Laplace transform maps that result back to time.

Control engineers use the transform to write linear models of heating, ventilation and air-conditioning plant, among other loops. Those models appear in many buildings. They are not the only application.

Process control uses the same map when a linear energy or mass balance must be solved for a step or ramp in a manipulated variable. Heat-transfer tests are a common textbook case.

Electrical and mechanical engineers also use Laplace methods on linear circuits, structures and mechanisms. The shared pattern is a constant-coefficient ODE in time.

The control action of an electrical, mechanical, thermal or hydraulic loop can be written as a differential equation from the physical laws of that plant. For a linear model, the Laplace transformation converts that time-domain equation into an algebraic equation in s.

The usual picture is a change of language: write the differential equation in s, solve the algebra, then translate F(s) back to f(t). The inverse step is as necessary as the forward transform.