1. Lecture 3 Laplace transform
Physics for informatics
Lecture 3 Laplace transform
Ing. Jaroslav Jíra, CSc.

2. The Laplace transform What is it good for?
Solving of differential equations
System modeling
System response analysis
Process control application

3. The Laplace transform Solving of differential equations procedure
A system analysis can be done by several simple steps
Finding differential equations describing the system
Obtaining the Laplace transform of these equations
Performing simple algebra to solve for output or variable of interest
Applying inverse transform to find solution

4. The Laplace transform The definition
The Laplace transform is an operator that switches a function of real variable f(t) to the function of complex variable F(s).
We are transforming a function of time - real argument t to a function of complex angular frequency s.
The Laplace transform creates an image F(s) of the original function f(t)
where s= σ+ iω

5. The Laplace transform Restrictions
The function f(t) must be at least piecewise continuous for t ≥ 0.
|f(t)| ≤ Meαt where M and α are constants. The function f(t) must be bounded, otherwise the Laplace integral will not converge.
We assume that the function f(t) = 0 for all t < 0

6. The Laplace transform Inverse Laplace transform
Inverse transform requires complex analysis to solve
If there exists a unique function F(s)=L[f(t)], then there is also a unique function f(t)=L-1[F(s)]
Using the previous statement, we can simply create a set of transform pairs and calculate the inverse transform by comparing our image with known results in time scope

7. The Laplace transform Basic properties Linearity Scaling in time
Time shift
Frequency shift

8. The Laplace transform
Another properties
Original
Image

9. The Laplace transform The most commonly used transform pairs Original
Image
Original
Image

10. The Laplace transform Transform pair deduction
u(t) 1
Unit step
The unit step u(t)
is defined by t
The Laplace image
u(t)
Shifted unit step 1 a t
The Laplace image

11. The Laplace transform Transform pair deduction
f(t)
Unit impulse
1/t1
The unit impulse f(t)
is characterized by unit area under its function t1 t
The Laplace image
f(t)
Dirac delta
It is a unit impulse for t1 → 0 t

12. The Laplace transform Transform pair deduction
Exponential function
Linear function
Per partes integration

13. The Laplace transform Transform pair deduction
Square function
Per partes integration
The n-th power function

14. The Laplace transform Transform pair deduction
Cosine function
Sine function

15. The Laplace transform Transform pair deduction
Time shift
f(t)
f(t-a)
The original function f(t) is shifted in time to f(t-a) t a
Frequency shift

16. The Laplace transform Property deduction
Time differentiation
Per partes integration

17. The Laplace transform Property deduction
Per partes integration
Time integration

18. Inverse Laplace transform
The algorithm of inverse Laplace transform
Since the F(s) is mostly fractional function, then the most important step is to perform partial fraction decomposition of it.
Depending on roots in denominator, we are looking for the following functions, where A and B are real numbers:
for a single real root s= a
for a double real root s= a
for a triple real root s= a
for a pair of pure imaginary roots s= ± iω
for a pair of complex conjugated roots s= a ± iω

19. Inverse Laplace transform Basic examples of partial fraction decomposition to find the original f(t)
Two distinct real roots
The equation s2 + 4s + 3= 0 has two distinct real roots s1= -3 and s2= -1
We have to find coefficients
A and B for
Multiplying the equation
by its denominator
Now we can substitute
Decomposed F(s)
so the original function

20. Inverse Laplace transform
One real root
and one real double root
The denominator has a single root s1= -1 and a double root s23= -3
We are looking for coefficients
A, B and C
Multiplying the equation
by its denominator
Now we can substitute to get A,B;
the C coefficient can be obtained
by comparison of s2 factors
Decomposed F(s)
so the original function

21. Inverse Laplace transform
Two pure imaginary roots
Since we know, that
it will be helpful to rearrange
the original formula
Now we can directly
write the result

22. Inverse Laplace transform
One real root
and two pure imaginary roots
We are looking for coefficients
A, B and C
Multiplying the equation
by its denominator
Now we can substitute to get A;
B,C coefficients can be obtained
by comparison of s0,s2 factors
Decomposed F(s)
so the original function

23. Inverse Laplace transform
Two complex conjugated roots
We have to rearrange the denominator in the first step
Decomposed F(s)
Now we have to assemble all necessary relations
so the original function

24. Solving of differential equations by the Laplace transform
Example 1
Find the x(t) on the interval <0,∞)
The image of desired function is
From the former definitions we know, that
Then we can write
The original function

25. Solving of differential equations by the Laplace transform
Example 2
Function on the right side
Necessary relations
Equation in the Laplace form

26. Solving of differential equations by the Laplace transform
Example 2 - continued
A formula for the X(s) after the partial fraction decomposition
after some small arrangements
The original function

27. Solving of differential equations by the Laplace transform
Example 3
Homogeneous second order LDR
Necessary relations
Equation in the Laplace form
The original function

28. Solving of differential equations by the Laplace transform
Example 4
Inhomogeneous second order LDR
Necessary relations
Equation in the Laplace form
knowing that
The original function

29. Solving of differential equations by the Laplace transform
Example 5
Integro-differential equation
Necessary relations
Equation in the Laplace form
The original function