# Operational Calculus

**Chapter 14 Operational Calculus**

**Chapter 14.1 Introduction**Each time function

**f(t)**is assigned its

**Laplace transform F(s)**

which is most often the quotient of

**2**polynomials of the complex variable

**s**.

*****

**f(t) ==>F(s)**(Fig. 14-1a)

And vice versa Each Laplace transform

**F(s)**is assigned a time function

**f(t)**

**F(s) ==>f(t)**(Fig. 14-1b)

Ultimately, “it works both ways”

**f(t)<==>F(s)**(Fig. 14-1c)

*If you don’t know complex numbers, don’t worry. Treat them like regular

**s**variables. Below is an example where

**f(t)**is a function of time and

**F(s)**is a fraction. In it, the numerator

**L(s)**is a polynomial of degree

**2**and the denominator of degree

**3**:

**Relations between**

Fig. 14-1

Fig. 14-1

**f(t)**and

**F(s)**

An even more concrete example

**Example for the function**

Fig. 14-2

Fig. 14-2

**f(t)=cos(t)**

**Note 1**

In fact,

**f(t)=cos(t)**only for

**t>=0**. For

**t<0 f(t) =0**!!! The rule applies to all functions

**f(t)**listed below. This is due to the fact that in automatics everything starts at a specific time, most often

**for t=0**. For example, the function unit step

**x(t)=1**for

**negative**

**t**is

**zero**.

**Note 2:**

The expression

**f(t)=F(s)**would be absolutely pointless!

**Chap. 14.2 Relationship between F(s) and f(t)**

The time function f(t) and its transform F(s) are related by the equation**Fig. 14-3**Formula for

**F(s)**for function

**f(t)**

For simple functions

**f(t)**it can still be calculated. For more complicated ones, too, but only once to pass the

**student test**. I even thought about giving this formula. Especially since, apart from the concept of

**integral**, the number

**s**is an obstacle. There is the so-called a

**complex number**, and to make it even more fun in the

**exponential function**.

If it bothers you, then you can let it go. Then treat the

**Laplace transform**as an ordinary assignment

**f(t)<=>F(s)**. As if there was a very

**clever book**somewhere, and there were pairs

**f(t)<=>F(s)**. Even an automatics theorist sometimes uses such a book. Its piece, i.e. only one pair for

**f(t)=cos(t)**is

**Fig. 14-2c**. More pairs are

**Fig. 14-5**. As for the complex number

**s**, treat it like a good real number. Just be aware that it isn’t.

**Functions**

Fig. 14-4

Fig. 14-4

**f(t)**for which appropriate transforms

**F(s)**are assigned in

**Fig. 14-5**.

The time charts are there to emphasize that all

**f(t)**functions have a

**zero**value for

**t<0**. It is generally the case in automation that something starts at

**t=0**, e.g. unit step

**f(t)=1(t)**.

**The figure also shows the pair**

Fig. 14-5

Fig. 14-5

**δ(t)<==>1**, i.e. the

**Dirac impulse**and its transform

**F(s)=1**.

The number

**e=2.7182…**in

**3**formulas is one of the “most famous” in mathematics next to

**0, 1, PI…**

**Chapter 14.3 How does differentiating f(t) affect its transform F(s)?****Fig. 14-6**This is the most important theorem of the

**Operatotional Calculus**. Without it, there would be no

**Operatotional Calculus**!

**Note**

In fact, the formula for the

**derivative transform**is a little different, but it can be used in most cases. And so be it. After all, this course is just an

**introduction**to automatics.

The formula can be easily generalized to further derivatives

**Calculating the**

Fig. 14-7

Fig. 14-7

**derivative**of a

**function**requires some effort. However, calculating the

**transform**of the nth

**derivative**is a piece of cake. Just multiply

**F(s)**by the

**nth**power of

**s**. This rule makes it easy to solve l

**inear differential equations**. You will find out in the next chapter.