**Chapter 19.1 Introduction
**Transfer function

**G(s)**gives the output

**y(t)**when input

**x(t)**is known. The simplest is the

**Proportional unit G(s)=K**transfer function.

**K**is a gain and the relation is simply

**y(t)=K*x(t)**. Other objects aren’t so easy. We have to consider not

**x(t)**and

**y(t)**time functions but their

**Laplace transforms X(s)**and

**Y(s)**. We can write

**Y(s)=G(s)*X(s)**or

**G(s)=Y(s)/X(s)**then.

**Fig. 19-1**

Transfer function

**G(s**) as

**2 Laplace transforms**quotient.

**Chapter 19.2 Transfer function and differential equation when G(s) is a inertial type**

Call Desktop/PID/05_transmitancja/01_inercyjny_skok.zcos

**Fig. 19-2**

Click “Start”

**Fig. 19-3
**It’s typical inertial step response. Don’t consider the output

**y(t)**equation for the time being. The transfer function

**G(s)**is for you “something” from

**Fig. 19-2**which step response is as

**Fig. 19-3**until now. You can foresee other inertal units behaviour –>

**Fig. 19-4**for example.

**Fig. 19-4**

You connote intuitively the

**G(s)**with its

**x(t)**and

**y(t)**that way.

But there is the other, more scientific viewpoint for transfer function

**G(s)**. It’s differential equation counterpart !

Let’s return to the tank with hole

**Chapter**

**17.3**for the present.

There was a differential equation

**x(t) = y'(t) + y(t)**. We will solve it with the aid of

**operational calculus**. We change the differential equation in the operational equation.

**x(t)=y'(t)+y(t)**

**x(t)<=>X(s)**–>

**x(t)=1(t)**step type (

**x(t)**is known!)

**y(t)<=>Y(s)**

**y'(t)<=>sY(s) see**

**Fig.18-6 chapter 18!**The differential equation

**x(t)=y'(t)+y(t)**

is transformed for **operational equation**

**X(s)=s*Y(s)+Y(s) **because **y'(t)–>sY(s)**

We can write the last equation :

**Fig. 19-5
**We know

**Y(s)**now! The clever book with all

**f**

**(t)<—>F(s)**pair is necessary. Look at

**Fig. 18-5 chapter 18**. We have fluke. There is this pair

**f(t)<–>F(s)**here!

**Fig. 19-6**

I remind that

**e=2.71828… is a very important number in mathematics.**The output

**y(t)**was calculated as a differential equation

**solution**. Pure theory.

**Xcos**made this same job.

Let’s check it for

**t=0 sec**,

**t=1 sec**and for

**t=7 sec**.

**Fig. 19-7**

See

**Fig. 19-3**y(1sec)=0.632 and see

**Fig. 19-7**for example y(1sec)=0.632 too! Theory confirms the practice!

**Chapter 19.3 Transfer function and differential equation when G(s) is any.
**The physical phenomenon is described here as a

**3**degree differential equation. Input signal is a

**x(t)**and output signal is a

**y(t)**. More! Both equation sides are differentiated! What’s the transfer function here?

**Fig. 19-8**

The transfer function

**G(s)**is a

**2**polynomians

**quotient**. Nominator

**L(s)**and denominator

**M(s)**are

**polynomians**. The nominator

**L(s)**degree is lower than denominator

**M(s)**

**degree mostly. The**

**L(s)**

**=1**more often than not. The denominator

**M(s)**is a first or second degree polynomians

**product**often. For example

**M(s)=(1+s*T1)*(1+s*T2)**.

We will check the

**G(s)**with the particular

**L(s)**and

**M(s)**parameters:

**M(s)–> a0,a1,a2,a3**

**L(s)–> b0,b1,b2,b3**

Call Desktop/PID/05_transmitancja/02_skomplikowany.zcos

**Fig. 19-9**

Click “Start”

**Fig. 19-10**

Please note that higher degree drivatives are very influential. There are small here! It’s fear to think only when they are bigger! I propose to change a little these parameters.

It’s easy to calculate the

**K**coefficient

**– from the time diagram**

– from the

**G(s)**directly

**Chapter 19.4 Equivalent transfer function -units configuaration**

**Chapter **19.4.1 Introduction

There are **3 **main units connections:

– serial

– parallel

– negative feedback

There are combinations of these connections too.

**Chapter 19.4.2 Serial connected units**

Call Desktop/PID/05_transmitancja/03_polaczenie_szeregowe.zcos

**Fig. 19-11
**The

**G1(s)**output is a

**G2(s)**input.

Click “Start”

**Fig. 19-12**

–

**x(t)**is a input signal

–

**y1(t)**is a intermediate signal

–

**y(t)**is a output signal

It confirms that

**e**

**quivalent transfer function G(s)**of the

**2**serial connected units is:

**Fig. 19-13**

When G1(s), G2(s) are for example amplifiers (proportional units)–>the equivalent gain

**K=K1*K2**.

**Chapter 19.4.3 Parallel connected units**

Call Desktop/PID/05_transmitancja/04_polaczenie_rownolegle.zcos

**Fig. 19-14
**Output

**y1(t)**and

**y2(t)**signals are going to

**sumation node**which output is

**y(t)=y1(t)+y2(t)**. Remember. The

**sumation node**must be used as a output in the parallel connected units!

Click “Start”

**Fig. 19-15**

It’s clearly that

**y(t)=y1(t)+y2(t)**.

**Fig. 19-16**

The

**equivalent transfer function**is a

**sum**of the

**G1(s)**and

**G2(s)**.

**Note**

The output

**y(t)**looks like inertial type type. But it isn’t! Make the common denominator of the sum

**Fig. 19-16**

**.**

**Chapter 19.4.3 Negative feedback linking **

Serial and parallel configurations are easy. It isn’t so nice with the **negative feedback**. What are the cause and effect? We will treat it dispassionately as mathematician so long as.

The **G(s) **is a open loop transfer function-for example **Fig. 19-2**. What’s the closed loop transfer function **Gz(s)**?

**Fig. 19-17**

The people noticed long since that **negative feedback** configuration is very beneficial.

The output signal **y(t) **tries to follow the input signal- set point **x(t)**. The other beneficial is the better dynamics. The output **y(t) **isn’t so lazy now.

**Fig. 19-17****a **shows the signals as time functions **f(t)**. Please notice that **G(s) **input is a difference **e(t)=x(t)-y(t)**. The **G(s)** output **y(t) **is **feed back** with **negative** sign to the **G(s) **and thus is the name **negative feedback**.

**Fig. 19-17b**** **the time functions **f(t) **are transfomed to Laplace **F(s)**.

**Fig. 19-17c **All the transfer functions are quotients **Y(s)/X(s)**–> The closed loop transfer function is **Gz(s)**.

**Fig. 19-17d **Derivation of formula for closed loop transfer function **Gz(s)
Fig. 19-17e **Derivation of formula for closed loop steady gain

**Kz**. The other derivation of formula is

**Kz=Gz(s=0)**.

Are the

**Gz(s)**and

**Kz**formulas true? We will check it for the particular

**G(s)**. You will be convinced, I hope.

**Fig. 19-18
**This particular transfer function

**G(s)**is a double inertial unit with parameters

**K=20, T1=2 sek, T2=5sec**. Will be the

**y1(t)**and

**y(t)**the same?

Call desktop/PID/05_transmitancja/05_sprzezenie_sprawdzenie_wzoru.zcos

**Fig. 19-19**

The block diagrams are the same as

**Fig. 19-18c**.

**Fig. 19-20**

The

**y1(t)**and

**y2(t)**are the same. I am more convinced that the formulas

**Fig 19-17d**and

**Fig 19-17e**are true.

It was a mathematical dispassionately introduction to the

**negative feedback**.

More “passion and intuition” will be given in the next chapter.

**Chapter 19.5 Conlusions
**

**1.**Transfer function is a counterpart of differential equation describing the dynamic object

**G(s)**.

**2.**It’s easy to obtain the steady gain

**K**parameter –>

**Fig**

**. 19-10**

3. Fig. 19-8and

3. Fig. 19-8

**19-9**shows transfer functions as “naked” quotients. The nominator and denominator were the classical polynomians form here. Their forms are a little different often, for example.

**Fig. 19-21**

Why? The answer is simple.

**Fig. 19-21a**and

**19-21b**shows for the first look the

**G(s)**character (double inertial or oscillation type). You don’t see it in the “naked form”

**Fig. 19-21c**.

**4**. Transfer functions of the units connected :

-serial

-parallel

-with the negative feedbac

may be easy replaced by its equivalent tranfer functions.