Table of contents  
What is a Block Diagram?  
Elements of Block Diagram  
Rules for Block Diagram Reduction  
Example of Block Diagram Reduction 
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Any system can be described by a set of differential equations, or it can be represented by the schematic diagram that contains all the components and their connections. However, these methods do not work for complicated systems. Block diagram comes in role for such complicated systems.
Block diagrams consist of a single block or a combination of blocks. These are used to represent the control systems in pictorial form.
For representing any system using block diagram, it is necessary to find the transfer function of the system which is the ratio of Laplace of output to Laplace of input.
Where,
R(s) = Input
C(s) = output
G(s) = transfer function
Then, the system can be represented as:
C(s) = R(s).G(s)
Note: Sometimes, block diagrams become complex for some of the control systems and hence evaluation of their performance become difficult. For that, we require simplification (or reduction) of the block diagram which is carried out by block diagram rearrangement.
The various points for analyzing and reduction of these units are explained below:
A block diagram
The transfer function of the block shows the mathematical function of the element present as a block.
The elements used for the connectivity in a block diagram are given as follows.
It refers to the summing operation on the incoming signals. The point in the block diagram representation where the multiple signals are compared is known as the summing point.
(i) Summing Point Representation in a Block Diagram:
It should be noted that the dimensions of the incoming signals on which the summing operation is performed, must be same.
In a closedloop system, part of the output is fed back to the input that acts as feedback for the system. So, the takeoff point is that point from where a portion of the signal is taken as feedback for the system.
(i) Takeoff Point Representation in a Block Diagram:
When a number of blocks are connected in block diagram then overall transfer function can be obtained by block diagram reduction technique.
Suppose we have two blocks in cascade connection as shown below:
We can calculate:
Suppose two blocks are connected parallely as given below:
Suppose we have a combination of takeoff point and block as shown below:
Here (1) = R(s)
Suppose there is a takeoff point ahead of block as given below:
Let's Suppose we have a configuration of summing point and block as given below:
Suppose we have a combination where we have a summing point present after the block as shown below:
Consider a combination of two summing points directly connected with each other as shown below:
We can use associative property and can interchange these directly connected summing points without altering the output.
The figure below shows the abovediscussed configuration:
We have already derived in our previous article that the gain of a closedloop system with positive feedback is defined as:
So to remove the feedback loop, feedback gain must be used.
Therefore, we can have:
Till now we have seen the important rules to be kept in mind while reducing the block diagram. Let us now see an example to have a better understanding of the same.
First, see the procedural steps to be followed for solving block diagram reduction problems:
Q. Consider a closedloop system shown here and find the transfer function of the system:
Solution:
Reducing the 3 directly connected blocks in series into a single block, we will have:
Further, we can see 3 blocks are present that are connected parallely. Thus on reducing blocks in parallel, we will have:
Further on simplifying the internal closedloop system, the overall internal gain will be:
So, we will have:
Now reducing the two blocks in series:
So, this is the reduced canonical form of a closedloop system.
We know gain of the closedloop system is given as:
Therefore,
On simplifying the equation
This is the overall transfer function of the given control system.
54 docs45 tests

54 docs45 tests
