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)
Conclusion
f(t)<==>F(s) (Rys. 18-1c)
Symbol s is a complex number. Don’t you know these numbers? Don’ bother yourself.
Fig. 18-1
The particular example of the Laplace transform F(s)
Fig. 18-2
F(s) for f(t)=cos(t)
Note 1
Remember that f(t)=cos(t) for t>=0. For t<0 f(t) =0!!!! This rule involves all the time functions f(t).
Note 2:
The expression f(t)=F(s) is a absolute nonsense.
Chapter. 18.2 F(s) formula
Fig. 18-3
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.
Fig. 18-4
Note that all these functions are null for t<0. All begins in time t=0 in automatics usually.
Fig. 18-5
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
Fig. 18-6
The most important operational calculus formula.
This formula may be generalise for higher derivatives.
Fig. 18-7
These formulas enable easy linear differrentiationd equations solution. You will be convinced in the next chapter.