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.
Let’s check it for t=0 sec, t=1 sec and for t=7 sec.

Fig. 19-7
Is ok!!!

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 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 equivalent 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.

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. 19-17a 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. 19-8
and 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.