transfer function [control engg]

8/14/2019 Transfer Function [Control Engg]

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Dr. D.V. GIRISH

Professor & Head

Department of Mechanical Engineering

Malnad College of Engineering

Hassan - 573201

ME 55 Control Engineering


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Dr.D.V.Girish, Professor & Head (Mechanical) MCE, Hassan

Transfer Functions:It is defined as the ratio of the laplace transform ofoutput (response) to the laplace transform of input

(excitation) assuming all the initial conditions to

be zero.

g (t)r(t) C(t)

Fig

G (S)R(S) C(S)

(a) System in

time domain

Fig (b) System in

Laplace domain

Fig: Transfer Functions of a system


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If G(S) be the transfer function of the system, we can

write mathematically

L.T. of output

L.T. of InputG(S) = all initial

conditions are Zero

C(S)R(S)

=

This Transfer function is a property of the system itself,

independent of the input or driving function. The T.F includes

the units necessary to relate the input to the output, However, itdoes not provide any information concerning the physical

structure of the system. i.e., the T.F of many physically

different systems can be identical


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Consider a spring-mass-damper (k-m-c) system

on which the force F acts and displacement xof the mass is the output.

x

F

m

0

x

Fm

0

kx cx

Draw the free body diagram as shown Equation of Motion

= F cx kx = mx...

mx + cx + kx = F.

=..

k

c

.


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Taking laplace transform of each term of this

equation (assuming Zero initial condition), we can

write,

F(s) = ms2 X(s) + cs X(s) + kX(s)

Now, taking the ratio of X(s) to F(S) we can write,

the transfer function of the system

X (S)

F (S)= = G(S) =

1

ms2+cs+k


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Here the highest power of the complex variable

S, in the denominator of the transfer functiondetermines the order of the system. Thus the

k-m-c system under consideration is a second

order system.

Similarly we can write for k-m system

i.e., c = 0

X (S)

F (S)=

1

ms2+K

and if m=0, we can write

X (S)

F (S)=

1

CS+K


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BLOCK DIAGRAMWe know that,Input-Output behavior of a Linear System or

Element of a Linear System is given by Transfer

Function,

G(s)= C(s)/R(s)

Where, R(s) = Laplace transformation of the

input Variable

C(s) = Laplace transformation of the

output Variable




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A Convenient graphical representation of thisbehaviour, i.e., short hand pictorial representation of

the cause and effect relationship between the

input and out put of a physical system is known as

BLOCK DIAGRAM

G (S)R (S) C (S)

(Input) (Output)

This is shown in Fig (i)



Fig (i)


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In addition to this summing or differencing of

signals is indicated by the symbols shown in the

fig (ii) (a) (b) & (c), while take off point of a

signal is represented by fig(iii)

i.e., in summing point two or more signalscan be added or subtracted.



A

B

A+B

Fig (ii) (a)

A

B

A-B

Fig (ii) (b)

A +A-B+C

C

+

-B

Fig (ii) (c)


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The points at which the output signal of any

block can be applied to two or more points is

known as Take off Point

(This output is analogous to voltage but not tocurrent)

Fig (iii)


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Output

C (S)G1(S)

R (S)

Input

G2(S) G3(S)R(S) G1(S) R (S) G1(S) G2(S)

= R (S) G1(S) G2(S) G3(S)

Output

Input =C(S)

R(S)G1(S) G2(S) G3(S)=

The direction of flow of signal from Input to output is

Known as Forward Path.

Forward Path:


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Feedback Path:The direction of flow of signal from output to

Input is Known as Feedback Path.



Feed Forward

R(s)G (S)

H (S)

C(s)

Output

Input =C(S)

R(S)= ?

Feedback

_

+


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The following Equations refer to a Tension Regulating

Apparatus such as used in the Paper Industry.

(a) Main Input x =Fr A.,

(b) Lever Measurement. e = (x-y)/2

(c) The change in torque provided

by motor tm = [Km/(1+ p) e

(d) Roll tension Fc = tm/R

(e) Tension of Control Spring y = 2 Fc/K

Draw individual Block Diagram and

Determine overall Transfer function




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Solution : Individual Block Diagrams:

(a) Main Input x =Fr A.,

(b) Lever Measurement. e = (x-y)/2

AFr x

1/2x-y

y

x e

Km/(1+ p)tme(c) The change in torque

provided by motortm= (Km/(1+

p))e


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(d) Roll Tension, Fc = tm/R 1/RFctm

2/KyFc



(e) Tension of ControlSpring,

y = 2 Fc/K


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Dr D V Girish Professor & Head (Mechanical) MCE Hassan

Complete Block diagram andoverall transfer function :

Km

(1+ p)

tm1/2A

Fr e

1/R

2/K

Fcx-y

Fc

Fr

= ?Overall

Transfer Function=

transfer function [control engg]

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