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Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE) PDF Download

Block Diagram in Control Systems

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.
The Block diagram representation is a combination of these two methods. A block diagram is a representation of a system using blocks.
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.

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Where
R(s) = Input  
C(s) = output  
G(s) = transfer function  
Then, the system can be represented as
C(s) = R(s).G(s)  

Summing Point: When we want to apply a different input signal to the same block then the resultant input signal is the summation of all the inputs. The summation of an input signal is represented by a crossed circle called summing point which is shown in the figure below.

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Take off Point: When there is more than one block, and we want to apply the same input to all the blocks, we use a take-off point. By the use of a take-off point, the same input propagates to all the blocks without affecting its value. Representation of same input to more than one block is shown in the below diagram.

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

How to draw the block Diagram

Consider a simple R-L circuit

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Apply KVL

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Now taking laplace transform of Eq.1 and Eq.2 with initial condition zero

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

From eq. 3 and eq. 4
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)From fig:
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Now taking laplace transform of Eq.5, and Eq.6
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

For the right-hand side of eq.5, we will use a summing point.

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Here the output of summing point is given to the block, and the output of the block is I(s)
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Now the output I(s) is given to another block containing element SL and the output of this block is V0.

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

By combining the above two figures, we get the required block diagram

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Closed loop control system

A system in which a feedback path is there is called a closed-loop control system. In this system, the output is feedback into the error detector and then it is compared with the input signal. The feedback signal can be negative or positive.

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

For positive feedback

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

And for negative feedback
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Block diagram reduction rules

Rule No. 1 Blocks in Cascade
When two or more blocks are connected in series, then the resultant block is the product of the individual blocks.
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Rule No. 2 Blocks in parallel
When two or more blocks are connected in parallel, then the resultant block is the sum of the individual blocks.
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Rule No. 3 Moving a take-off point ahead of a block
When the take-off point is moved ahead of a block (before the block), then the same transfer function is introduced in the take-off point branch.
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Rule No. 4 Moving the take-off point after the block
When the take-off point is moved after the block, then a block with reciprocal of a transfer function is introduced in the take-off point branch.
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Rule No. 5 Moving a summing point beyond the block
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)
Rule No.6 Moving a summing point ahead of a block
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Rule No.7 Interchanging two summing points
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Rule No.8 Moving a take-off point beyond a summing point
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Rule No.9 Moving a take-off point ahead of a summing point
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Rule No.10 Eliminating a forward loop
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Example

Find the transfer function of the following by block reduction technique.

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Step 1: There are two internal closed loops. Firstly, we will remove this loop.
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

Step 2: When the two blocks are in a cascade or series we will use rule no.1.

Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Step 3: Now we will solve this loop.
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)Step 4:
Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE)

The document Block Diagram: Reduction Rules (Detailed Notes) | Control Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Control Systems.
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FAQs on Block Diagram: Reduction Rules (Detailed Notes) - Control Systems - Electrical Engineering (EE)

1. What is a block diagram in control systems?
A block diagram in control systems is a graphical representation of a system using blocks to represent the components of the system and arrows to indicate the flow of signals or inputs and outputs between these components. It provides a visual representation of the system's structure and helps in analyzing and designing control systems.
2. What are reduction rules in block diagrams?
Reduction rules in block diagrams are a set of techniques used to simplify complex block diagrams by rearranging the blocks and eliminating unnecessary components. These rules help in reducing the complexity of the diagram, making it easier to analyze and understand the system.
3. How do reduction rules help in analyzing control systems?
Reduction rules help in analyzing control systems by simplifying complex block diagrams, which in turn makes it easier to understand the system's behavior. By reducing the diagram to a more manageable form, engineers can focus on specific components and their interactions, allowing for better analysis and design of control systems.
4. What are some common reduction rules used in block diagrams?
Some common reduction rules used in block diagrams are: - Series rule: Two or more blocks connected in series can be replaced by a single block representing their combined transfer function. - Parallel rule: Two or more blocks connected in parallel can be replaced by a single block representing their combined transfer function. - Feedback rule: A feedback loop can be simplified by applying the series and parallel rules to the blocks within the loop. - Cascade rule: Two or more blocks connected in cascade can be replaced by a single block representing their combined transfer function. - Branch point rule: A branch point can be simplified by applying the parallel rule to the blocks connected to the branch point.
5. How do reduction rules help in designing control systems?
Reduction rules help in designing control systems by simplifying the block diagrams, which allows engineers to focus on specific components and their interactions. By simplifying the diagram, engineers can identify the critical components and optimize their performance. Reduction rules also help in identifying potential issues or bottlenecks in the system, enabling engineers to make necessary adjustments and improvements in the design.
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