Preliminary Automatics Course
Chapter 18. Operational calculus
Chapter 18.1 Introduction
The appropriate Laplace transform F(s) is assigned to every time function f(t).
F(s) is a quotient of 2 polynomians.
f(t) ==>F(s) (Fig. 18-1a)
And vice versa.
The appropriate time function f(t) is assigned to every Laplace transform F(s).
F(s) ==>f(t) (Fig. 18-1b)
f(t)<==>F(s) (Rys. 18-1c)
Symbol s is a complex number. Don’t you know these numbers? Don’ bother yourself.
The particular example of the Laplace transform F(s)
F(s) for f(t)=cos(t)
Remember that f(t)=cos(t) for t>=0. For t<0 f(t) =0!!!! This rule involves all the time functions f(t).
The expression f(t)=F(s) is a absolute nonsense.
Chapter. 18.2 F(s) formula
It’s possible to calculate F(s) for easy functions f(t). For more complicated-no. I wondered to set this formula out even. Never mind If you aren’t math specialist. But you have to realise only, that there is a very clever book with all pairs f(t)<==>F(s). The pair from Fig. 18-2c belongs to this book.
The Fig.18-4 represents some of the most popular time functions f(t) used in the automatics. The Fig.18-5 are the pairs f(t)<==>F(s). This is a small part of this clever book.
Note that all these functions are null for t<0. All begins in time t=0 in automatics usually.
Number e=2,7182… used in 3 formulas is the most known in the mathematics as 0, 1 ,Π…
There is pair δ(t)<==>1 with Dirac pulse too.
Chapter 18.3 Laplace transform and derivation
The most important operational calculus formula.
This formula may be generalise for higher derivatives.
These formulas enable easy linear differrentiationd equations solution. You will be convinced in the next chapter.