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Chapter: Control Systems - Systems and their Representation

Block diagram reduction Techniques - Transfer Function

A pictorial representation of the functions performed by each component and of the flow of signals.

Block diagram

 

A pictorial representation of the functions performed by each component and of the flow of signals.

 

Basic elements of a block diagram

 

o   Blocks

 

o   Transfer functions of elements inside the blocks

 

o   Summing points

 

o   Take off points

 

o   Arrow

 

Block diagram

 

A control system may consist of a number of components. A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals.

 

The elements of a block diagram are block, branch point and summing point.

 

 

 

 

Block

 

In a block diagram all system variables are linked to each other through functional blocks. The functional block or simply block is a symbol for the mathematical operation on the input signal to the block that produces the output.



Summing point

 

Although blocks are used to identify many types of mathematical operations, operations of addition and subtraction are represented by a circle, called a summing point. As shown in Figure a summing point may have one or several inputs. Each input has its own appropriate plus or minus sign.

 

A summing point has only one output and is equal to the algebraic sum of the inputs.


A takeoff point is used to allow a signal to be used by more than one block or summing point. The transfer function is given inside the block

 

 The input in this case is E(s)

 The output in this case is C(s)

 

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


Functional block  each element of the practical system represented by block with its T.F.

 

Branches  lines showing the connection between the blocks

 

Arrow  associated with each branch to indicate the direction of flow of signal

 

Closed loop system

 

Summing point  comparing the different signals

 

Take off point  point from which signal is taken for feed back

 

 

Advantages of Block Diagram Representation

 

o   Very simple to construct block diagram for a complicated system

 

o   Function of individual element can be visualized

 

o   Individual & Overall performance can be studied

 

o   Over all transfer function can be calculated easily.

 

 

Disadvantages of Block Diagram Representation

o   No information about the physical construction

 

o   Source of energy is not shown

 

Simple or Canonical form of closed loop system


R(s) – Laplace of reference input r(t)

C(s) – Laplace of controlled output c(t)

E(s) – Laplace of error signal e(t)

 

B(s) – Laplace of feed back signal b(t)

G(s) – Forward path transfer function

H(s) – Feed back path transfer function

 

Block diagram reduction technique

 

Because of their simplicity and versatility, block diagrams are often used by control engineers to describe all types of systems. A block diagram can be used simply to represent the composition and interconnection of a system. Also, it can be used, together with transfer functions, to represent the cause-and-effect relationships throughout the system. Transfer Function is defined as the relationship between an input signal and an output signal to a device.

Block diagram rules

Cascaded blocks

 






 



Procedure to solve Block Diagram Reduction Problems

 

Step 1: Reduce the blocks connected in series Step

 

2: Reduce the blocks connected in parallel Step 3: Reduce the minor feedback loops

 

Step 4: Try to shift take off points towards right and Summing point towards left

 

Step 5: Repeat steps 1 to 4 till simple form is obtained

 

Step 6: Obtain the Transfer Function of Overall System

 

 

Problem 1

 

1. Obtain the Transfer function of the given block diagram

 



 

2. Obtain the transfer function for the system shown in the fig


 

3. Obtain the transfer function C/R for the block diagram shown in the fig


Solution

The take-off point is shifted after the block G2

 



 

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