# Automatics

**Chapter 11 Differentiation **

** Chapter 11.1 Introduction**If you know the topic, you can skip this chapter. But if not or “not very much”, then I heartily recommend it. So far, the most commonly used concept was

**Transmittance G(s)**, as

**“something”**that connects the output

**y(t)**with the input

**x(t)**. For systems devoid of dynamics (ideal amplifier, lever, toothed gear ….) it is simply the gain

**K**, i.e. the proportional

**unit**with the transmittance

**K**.

At the first approach to a given phenomenon, technological process, almost everything is a

**proportional unit**. You have an electric heater in your apartment, where the input

**x(t)**is the power knob and the output

**y(t)**is the temperature. You know from experience that if there is a constant temperature outside, e.g. –

**10ºC**, the windows are closed …, then

**x(t)=1 kW**will cause the temperature in the room =

**+20ºC**,

**x(t)=1.1kW–>=+21ºC**,

**x( t)=1.2kW–>=+22ºC…**etc. You treat the entire

**system**with the

**furnace**and the room as a

**proportional unit**.

After some time, however, it will matter that the temperature increase, e.g. by

**ΔT+5ºC**, will last about

**20 minutes**. It is already a

**dynamic object**for you. Quite simply described, but always there. Then you will approximate this system with the

**inertial unit**. Then, more precisely, as

**inertia**with

**delay**, i.e.

**Substitute Transmittance**. Perhaps it will be possible to identify the object as

**four-inertial**, which has

**4 time**constants

**T1, T2, T3, T4**and gain

**K**.

So far, you’ve treated the transfer function

**G(s)**as

**“something”**, which gives the appropriate output signal

**y(t**) to the input

**x(t)**– most often a

**unit step**. You also know what a small lonely letter

**s**in the denominator

**G2(s)**can cause compared to

**G1(s)**, not to mention the letter

**s**in the numerator

**G3(s)**.

**If you don’t know, then quickly return to the relevant chapters.**

Fig. 11-1

Fig. 11-1

After reading

**chapters 11…15**, you will have an initial idea about:

–

**Derivative**

–

**Definite Integral**

–

**Differential equations**, including linear differential equations on which the control theory is based

–

**Operational Calculus**as a tool for solving linear differential equations

–

**Transmittance G(s)**as an equivalent of a differential equation describing a dynamic object.

**Chapter 11.2 The Derivative of a function**I repeated ad nauseam, especially in

**chapter 5**, that the derivative of a function is the

**“speed of the function”**!!! If you don’t feel it, then repeat this chapter. For the derivative term

**G(s)=s*Td**, the output

**y(t)**is proportional to the

**derivative**of the input

**x(t)**, i.e. to

**x'(t)**. And for

**Td=1 sec**, the output

**y(t)**is simply the derivative of the input

**x(t)**.

In

**mathematical analysis**, there are formulas that assign a derivative to

**each function**. Thus, the differentiator in

**Fig. 5-7**in

**Chapter 5**converts the square wave signal

**x(t)**into a

**y(t)=2*t**signal. If you feel like it, go back to this chapter and repeat the experience.

**Fig. 11-2**Every 1st year polytechnic student starts mathematical analysis with these formulas.

**Chapter 11.3. The second derivative, or “derivative of the derivative”**

The **derivative** is a **function**. So you can **recalculate** the **derivative**, i.e. the **second derivative**, from this function.**Fig. 11-3**On a specific example of

**f(t)**as a

**square function**, we will calculate the

**second derivative**of this

**function**.

**Let’s check it by**

Fig. 11-4

Fig. 11-4

**double differentiation**.

**F****ig. 11-5**After the

**first differentiating unit**we have the first derivative

**x'(t)**, and after the

**second**– the

**derivative**from the

**first derivative**, i.e. the

**second derivative**

**x”(t)**. Let’s check whether the theory agrees with practice.

Correct! If you don’t believe me, check with specific values.

If there was also a

**third differentiating unit**, then we would get the

**third derivative**or

**x”‘(t)**as a

**derivative**of the second derivative

**x”(t):**

**x”‘(t)=[(x”(t)]’ = (2)’ = 0**. The derivative of a constant function

**(2)’=0**is always zero because the “speed” or “slope” of a constant function is always

**o.**The third derivative is the

**t**(time)

**axis**of the graph.