chapter 2 transfer fuction

7/27/2019 Chapter 2 Transfer Fuction

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TRANSFER FUNCTION & BLOCK DIAGRAM

FKE,UTeM 17 DEKC 3813

CHAPTER 2: TRANSFER FUNCTION & BLOCK DIAGRAM

Contents:

2.1 Introduction

2.2 Transfer function Definition

2.3 Laplace Transform in Transfer Function

2.3.1 Introduction to Laplace Transform

2.3.2 Laplace Transform for Exponential, Step, Ramp, & Sinusoidal Function

2.3.3 Laplace Transform Table

2.4 Control System Block Diagram

2.4.1 Basic Block Diagram

2.4.2 Summing Point

2.4.3 Branch Point

2.5 Block Diagram of Closed Loop System

2.6 Closed Loop Transfer Function

2.7 Open Loop & Feedforward Transfer Function

2.8 Block Diagram Reduction Technique2.9 Signal Flow Graph

Learning Outcomes:

At the end of this chapter, student should be able to:

i. define the term of transfer function

ii. convert function f(t) to function f(s) using Laplace transform table

iii. apply Laplace transform to determine transfer function

iv. recognize the elements of block diagram

v. determine the transfer function of closed loop and open loop system

vi. transform complex block diagram into transfer function

vii. convert block diagrams into signal flow graph

viii. determine transfer function from signal flow graph


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2.1 Introduction

In control system, the analysis of characteristics and performances are simplified

by system representation. The use of Laplace transform is essential to determinea transfer function that describes the relationship between input and output of the

system. Block diagram and signal flow graph are very convenient and natural

tools to represent the components, multiple subsystems and entire control

system. Complex block diagrams of the system can be simplified by using block

diagram reduction technique to obtain the transfer function. The transfer function

also can be obtained through signal flow graph representation.

2.2 Transfer function

The transfer function of a system is defined as the ratio of the Laplace

Transformof the output to the input as show in Figure 2.1:

Figure 2.1

Hence, Transfer function, G(s)=Input

Output=

)(

)(

s

s

R

C

In control system, transfer function of a system is a mathematical model thatrelates the output variable to the input variable of the system. For an example,

the transfer function equation for the system in Figure 2.1 may represent in

numerator and denominator form as shown in Figure 2.2.

Figure 2.2

Transfer Function

G(s)

Input

R(s)

Output

C(s)


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The next section will discuss the overview and application of Laplace Transfrom

in control system transfer function.

Learning Activities

1. Define the term of transfer function

2.3 Laplace Transform in Transfer Function

In control system analysis, Laplace transform plays an important role in order to

determine the transfer function that describes the relationship between input and

output of the system. The review of Laplace transform is essential before transfer

function of control system is determined.

2.3.1 Introduction to Laplace Transform

Laplace transform is use widely in control system analysis. The transfer function

of a control system is defined in sdomain and provides valuable information

about stability and performance of a closed loop or feedback control system.

Laplace transform is a method of operational calculus that takes a function of

time (t-domain) and converts it to a function of complex variable s (s-domain).

The Laplace transform of a function of time, f (t) is defined as

0

)()()( dtetfsFtfL st

where;

f (t) = function in time domain

s = complex variable ( s = + j )

F(s) = Laplace transformation of f(t)

The reverse process of finding the time function f(t) from the Laplace transform

F(s) is called as Inverse Laplace Transform. The inverse Laplace transform of

F(s) is defined as


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)()()(2

1)(1 tutfdsesF

jsFL

j

j

st

where;

u(t) = 1 t > 0

= 0 t < 0

is the unit step function.

The next sub-section will demonstrate some Laplace transform derivation offunction, f (t) to F(s).

2.3.2 Laplace transform for Exponential, Step, Ramp & Sinusoidal Function

Exponential Function

Consider the exponential function;

f(t) = 0 for t < 0

= tAe for t0

where A and are constants. Hence,

0

)(

0

s

AdteAdteAeAeLtssttt

If constant A is unity, then

s

eLt 1

Step Function

Consider the step function;

f (t) = 0 for t0

where A is constant. Hence,


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0 s

AdtAeAL st

For a unit step function,

u(t) = 0 for t0Hence,

stuL

1)]([

Ramp Function

Consider the ramp function;

f(t) = 0 for t0

and A is constant. Hence,

0 20

00

)(s

Adte

s

Adt

s

Ae

s

eAtdtAtetAL st

ststst

Sinusoidal Function

Consider the sinusoidal function,

f(t) = 0 for t0

where A and are constants. By using )(2

1sin jwtjwt ee

jt , hence

02

sin dteeej

AtAL

sttjtj


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= jsj

A

jsj

A

1.

2

1.

2

=22

s

A

2.3.3 Laplace Transform Table

Table 2.1 shows the results for a some representative sample Laplace transform

of time function, f(t) to F(s) that frequently used in control system analysis. By

using Laplace transform table, complex equations derivations are not necessary.

Table 2.1: Laplace Transform Table

f(t) F(s)

(t) 1

u(t)t

1

tu(t)

2

1

s

)(tutn 1

!n

s

n

)(tue t

as

1

)(sin ttu 22

s

)(cos ttu 22

s

s


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Learning Activities

Determine F(s) of the following functions f(t) by using Laplace transform table:

i. f(t) = 5

ii. f(t) = 5t 2e-t

iii. f(t) = sin5t u(t)

2.4 Control System Block Diagram

Control system may consist of a number of components. To simplify the analysisof control system components, a diagram called block diagram is used to

represent the components and system.

2.4.1 Basic Block Diagram

A block diagram of a system is a pictorial representation of the function

performed by each component and the flow of signals.

Figure 2.2 shows elements of the block diagram;

Figure 2.2: Elements of Block Diagram

The arrowhead pointing toward the block indicates the input signal while the

arrowhead leading away from the block represents the output signal. G(s) is

represented the block diagram transfer function.

2.4.2 Summing Point

Refer to Figure 2.3(a) and (b); a circle with a cross is the symbol that indicates a

summing point operation. The plus (+) or minus (-) sign at each arrowheadindicates whether that signal is to be added or subtracted. For system with minus

Transfer Function

G(s)

Input signal Output signal


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(-) sign, it has negative feedback as shown in Figure 2.3 (a) while system with

plus (+) sign has positive feedback as shown in Figure 2.3 (b).

a + a + b

+ b

Figure 2.3(a): Negative Feedback Figure 2.3(b): Positive Feedback

2.4.3 Branch Point

A branch pointis a point, from which the signal from a block goes concurrently to

other blocks or summing point as shown in Figure 2.4;

Figure 2.4: Branch Point

2.5 Block Diagram of Closed Loop System

Figure 2.5 below shows a simple block diagram of closed loop system;

branch point

R(s) C(s)G(s)

+

-

E(s)

a + a - b

- b


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Figure 2.5: Closed Loop System

The output C(s) is fed back to the summing point where it is compared with the

reference input R(s). The output of the block, C(s) is obtained by multiplying thetransfer function, G(s) by the input to the block, E(s) where

C(s) = G(s) x E(s)

When the output is fed back to the summing point for comparison with the input, it

is necessary to convert the form of the output signal to that form of the input

signal using sensor or transducer (example : temperature voltage). Thisconversion is accomplished by the feedback element whose transfer function is

H(s) as shown in Figure 2.6.

Figure 2.6: Closed Loop System with Feedback Element H(s)

The feedback signal that fed back to the summing point is B(s) where

B(s) = H(s) x C(s).

H(s) also called as feedback path transfer function, where itis define as the ratio

of the Laplace transform of B(s) to the C(s), where

R(s) C(s)G(s)

+

-

E(s)

H(s)

B(s)


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)(

)()(

sC

sBsH .

In most cases, the feedback element is a sensor or transducers that measureand convert the output of the system. When the output is compared with the input

signal, the actuating error signal, E(s) is generated where

E(s) = R(s) B(s).

2.6 Closed Loop Transfer Function

From Figure 2.6, the transfer function that relate output, C(s) to the input, R(s) is

called as closed loop transfer function which can be derived as follows;

)()()( sxEsGsC

))().(()()()()( sHsBsRsBsRsE

By replace E(s) into C(s) equation, hence

)()()()()( sCsHsRsGsC )()()()()( sCsHsGsRsG

Re-arrange the equation, hence

)()()()()()( sRsGsCsHsGsC

)()()()(1)( sRsGsHsGsC

)()(1

)()()(

sHsG

sRsGsC

Finally, Closed Loop Transfer Function (CLTF) obtained as


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)()(1

)(

)(

)(

sHsG

sG

sR

sC

When denominator is setting equal to zero, 0)()(1 sHsG , it is known as

characteristic equation that very useful in control system analysis.

Noticed that the above derivation of closed loop transfer function is for negative

feedback loop. For positive feedback loop, the closed loop transfer function is

given as

)()(1

)(

)(

)(

sHsG

sG

sR

sC

.

2.7 Open Loop & Feed Forward Transfer Function

The transfer function of open loop system also can be obtained from closed loop

system block diagram. The ratio of the feedback signal B(s) to the actuating error

signal, E(s) or the product of G(s).H(s) is called as the open loop transfer

function;

Open Loop Transfer Function(OLTF) = )().()(

)(sHsG

sE

sB

The ratio of the output C(s) to the actuating error signal, E(s) is called as

feedforward transfer function orforward path transfer function;

Feedforward / Forward Path Transfer Function = )()(

)(sG

sE

sC

If the feedback element H(s) is unity, then the open loop transfer function and the

feed forward transfer function is equal.


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Learning Activities

Determine the closed loop transfer function (CLTF) for positive feedback loop.

2.8 Block Diagram Reduction Technique

Closed loop control system may contain a large number of blocks and may

involve multiple feedbacks or feed forward paths. It is necessary to reduce the

block diagram to simplified form before an overall transfer function can be

obtained.

The rules of block diagram reduction technique as shown in Table 2.1 can be

used to simplify the complex blocks diagram:

Table 2.1: Rules of Block Diagram Reduction Technique

Rules Original Block Diagram Equivalent Block Diagram

1. Cascadingblock

2. Block inparallel

3. Moving thesummingpointahead ofthe block

4. Moving the

summingpoint


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Rules Original Block Diagram Equivalent Block Diagram

beyond ablock

5. Moving thetake-offpoint ahead of ablock

6. Moving thetake-off

pointbeyondthe block:

7. Eliminatinga feedbackloop:

The following examples will demonstrate the use of block diagram reduction

technique in order to obtain an overall transfer function.

Example 1:

Simplify the blocks diagram in Figure 2.7 and obtain an overall transfer function.

+

R + + C

-G1 G2

G3

G4

Figure 2.7

Solution:


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Step 1: Cascading blocks G1, G2 and parallel blocks G3, G4 (rule 1):

R + C

-

G1G2 G3+G4

Step 2 : eliminate unity negative feedback loop (rule 7);

Step 3 : cascading the two block (rule 1);

R C

211

)43(21

GG

GGGG

Hence, an overall transfer function obtained is211

)43(21

GG

GGGG

R

C

.

Example 2

Simplify the blocks diagram in Figure 2.8 and obtain an overall transfer function.

R CG3+G4

211

21

GG

GG


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-

R + + + C- +

H1

G2 G3

H2

G1

Figure 2.8

Solution:

Step 1 : Moving the summing point 3 outside the positive feedback loop

H1(rule 3)

-

R + + + C

- +G3

H1

G1 G2

1

2

G

H

Step 2 : Cascading blocks G1,G2 (rule 1) and eliminate the positive feedback

loop H1 (similar to rule 7)


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-

R + + C-

G3

1

2

G

H

121121HGG

GG

Step 3 : Cascading block G3 (rule 1)

-

R + + C

-

1

2

G

H

1211

321

HGG

GGG

Step 4 : Eliminate negative feedback loop1

2

G

H(rule 7)

R + C

- 2321211

321

HGGHGG

GGG

Step 5 : Eliminate unity negative feedback loop (rule 7)

R C C

3212321211321

GGGHGGHGGGGG


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Hence, an overall transfer function,3212321211

321

GGGHGGHGG

GGG

R

C

Learning Activities:

Determine the transfer function of block diagrams as follows:

+

R(s) + C(s)

_

G1

G2

G3

2.9 Signal Flow Graph

Signal flow graph is an alternate approach for graphically represented control

system dynamics. It consists of branches which represent systems and nodes

which represent signal. Table 2.2 shows the simple conversion block diagram to

signal flow graph.

Table 2.2

Block Diagram Signal Flow Graph


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Block Diagram Signal Flow Graph

The following example will enhance the conversion technique of block diagram to

signal flow graph.

Example 3

Convert the block diagram in Figure 2.9 to signal flow graph.


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Figure 2.9

Solution:

Step1: Draw the signal nodes

Step 2: Interconnect the nodes.

Step 3: Simplify the signal flow graph by eliminates signal that have single flow in

and out, V2(s), V6(s), V7(s) and V8(s).


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In signal flow graph, there are few terms need to be defined and understood.

With refer to general signal flow graph in Figure 2.10; the following terms can be

determined:

Figure 2.10: General signal flow graph

input node/source node that has only outgoing branches. Node Ris an

example of input node.

R a b c

g

d

i

e

j

S

f h


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output node/sink output node/sink is a node that has only incoming

branches. Node Cis an example of output node.

loop loop is a closed path. Example of loops are bf, dh, j, bcdi& bgd i.

loop gain the loop gain is the product of the branch transmittances of a loop

forward path forward path is the path from an input node to an output node

that does not cross any nodes than once. Example forward path is a,b,c,d,e

and a,b,g,d,e.

Non-touching loop loops are non-touching if they do not possess any

common node. Loop b fnot touching loop dhand loopj

The following example will enhanced the terms definition of signal flow graph.

Example 4

Determine loop gain, forward path gain, nontouching loop and nontouching loopgain.

Figure 2.11

From signal flow graph above, the following terms are identified:


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Loop gain:

G2(s)H1(s)G4(s)H2(s)G4(s)G5(s)H3(s)G4(s)G6(s)H3(s)

Forward path gain:

G1(s)G2(s)G3(s)G4(s)G5(s)G7(s)G1(s)G2(s)G3(s)G4(s)G6(s)G7(s)

Nontouching loop:

Loop G2(s)H1(s) does not touch loops G4(s)H2(s), G4(s)G5(s)H3(s) and

G4(s)G6(s)H3(s)

Nontouching loop gain:

o Nontouching loop gain taken two at a time:

G2(s)H1(s)G4(s)H2(s)

G2(s)H1(s)G4(s)G5(s)H3(s)

G2(s)H1(s)G4(s)G6(s)H3(s)]

o Nontouching loop gain taken three at a time: -nil-

In signal flow graph, transfer function can be obtained by using Masons rule;

Transfer Function,

k

kkT

sR

sC

)(

)(

where

k= number of forward paths

Tk= the kth forward path gain

= 1 - loop gains + nontouching loop gains taken two at a time -

nontouching loop gains taken three at a time + non-touching loopgains taken four at a time -..

k= obtained from by eliminating or removing the loops that touch thekth forward-path


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The following example will demonstrate how Masons rule is applied to determine

the transfer function.

Example 5

Convert block diagram in Figure 2.12 to signal flow graph. Determine the transfer

function using Masons rule.

Figure 2.12

Solution:

From the block diagram, the signal flow graph obtained is

From the signal flow graph,

Number of forward path, k = 1

Forward path gain, T1 = G1G2G3

Loop gain = G1G2H1, -G2G3H2, -G1G2G3


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Hence, = 1 - [G1G2H1 - G2G3H2- G1G2G3] = 1 - G1G2H1 + G2G3H2+ G1G2G3

1 = 1 (forward path gain G1G2G3 touch all loops gain)

Noticed that no nontouching loop in this signal flow graph.

Hence, by using Masons rule,

Transfer Function,321232121

3211

11

1)(

)(

GGGHGGHGG

GGGT

sR

sC

Learning Activities

Convert the following block diagram to signal flow graph. Determine the loop gain

and forward path gain.

+

R + + C

- -

H

G2

G1

G3

Summary

In this chapter, the definition of transfer function has been explained which it is

involved the application of Laplace transform. Some basic functions in time

domain has been derived using Laplace transform. The basic elements of block

diagram have been identified and transfer functions of feedback control have

been determined. The transfer function of complex diagram is obtained by using

method called as block diagram reduction technique. Block diagram of control

also represented as signal flow graph where transfer function is determined using

Masons rule.


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Exercise

1. Define the term of transfer function with aided appropriate diagram

2. Convert the following function f(t) to function F(s) using Laplace transform

table.

i. )()( 3 tuttf

ii. )(5cos)( ttutf

iii. tetf t 5sin)( 2

3. Distinguish the open loop transfer function and forward path transfer function

4. Determine the closed loop transfer function of the following block diagrams

5. Simplify the following blocks diagram and obtain an overall transfer function


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6. Determine the transfer function for following signal flow graph using Masonsrule.

G1

R 1 G2 G3 C

-H

-1

chapter 2 transfer fuction

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