Block Diagram Algebra

In this article, We will discuss about block diagram and its components. We will also discuss about the various rules involved in block diagram algebra along with its equivalent block diagram. In addition to these we will also discuss about the application, advantages and disadvantages.

What is a Block Diagram?

In a control system, there are a number of components. The function of these components are represented by the use of blocks. These blocks are interconnected with each other by using directed lines which indicate the direction of signal flow. Thus, we can say that a block diagram is a representation of a control system with the use of blocks and lines.

Different Elements of a Block Diagram

A block diagram consists of some elements that are used to represent the components of a system in the block diagram. These are:-

• Functional Block:- This symbol represents the transfer function G(s) of a system.
• Summing Point:- This is the point where different output signals from the previous block or different signals of the system are added to form a single signal
• Take Off Point:- It is a tapping point in the system where the desired signal is tapped off to be utilized elsewhere in the diagram.
  

Block Diagram Algebra

Block diagram algebra is a type of algebra which involves the basic elements of block diagram. It is used to find the overall transfer function of system by using block diagram reduction.

Rules for Block Diagram Algebra

First we will look into some connections of block diagram. There some basic connection of blocks in a block diagram. There can be three possible ways of connection between two block. These are :

• Series Connection
• Parallel Connection
• Feedback Connection
• Shifting The Summing Point After The Block
• Shifting The Summing Point Before The Block
• Shifting the Take Off Point After The Block
• Shifting The Take Off Point Before The Block

1. Series Connection

Series connection is one type of connection between two blocks. It is also known as cascade connection. It is similar to the series connection of resistors. Let us take a example to understand this connection.

In the above diagram we have two transfer function G_1(s)\:\:\:and \:\:\: G_2(s) . The input to the system is X(s) and output will be Y(s) .

After passing through G₁ (s) the input for next block would become G_1(s)X(s)

Therefore, the final output of the system will be Y(s)=[G_1(s)G_2(s)]X(s)

From the above equation, we can conclude that the final output of two blocks in series is the product of their transfer function multiplied with the input. From this we can say that two block in series can be replaced by a single block whose transfer function is the product of the transfer function of the two blocks in series.

2. Parallel Connection

It is another type of connection where the blocks are connected in parallel to each other. It is similar to the parallel connection in the resistances. The blocks which are in parallel will have the same input. Let's understand it through the following diagram

The input given is X(s) to the transfer functions G_1(s)\:\:\:and \:\:\: G_2(s) which are connected in parallel.

The output from these transfers function will be Y_1(s)=X(s)G_1(s)\:\:\:and \:\:\: Y_2(s)=X(s)G_2(s)

As we have a summing point before the final output therefore, the final output will be summation of both the outputs.

Y(s)=Y_1(s)+Y_2(s) \newline Y(s)=X(s)[G_1(s)+G_2(s)]

From above we can conclude that the two blocks in parallel can be replaced by a single block whose transfer function is the sum of the transfer functions of the blocks connected in parallel.

3. Feedback Connection

In this type of connection feedback is present in the diagram. When output of the system is fed back to the input to stabilize and reduce error of the system is called as feedback. Feedback can be of positive or negative type. When the feedback loop is added with the input signal it is called as positive feedback and when the feedback is subtracted from the input signal it is called as negative feedback.

The above diagram shows a feedback connection having a positive feedback. We have a input of X(s) and feedback transfer function as H(s) and another transfer function G(s).

As there is positive feedback the feedback is added to the input signal so the summation will give output as K(s)=X(s)+H(s)Y(s)

The final output will be, Y(s)=K(s)G(s) \newline Y(s)=[X(s)+H(s)Y(s)]G(s) Y(s)[1-G(s)H(s)]=G(s)X(s) \newline \frac{Y(s)}{X(s)}=\frac{G(s)}{1-G(s)H(s)}

For negative feedback the transfer function will be, \newline \frac{Y(s)}{X(s)}=\frac{G(s)}{1+G(s)H(s)}

So we can say the feedback can be replaced by the above transfer function as a single block.

4. Shifting The Summing Point After The Block

This involves the shift of the summation point after the block. But after the shift the output result should not change.

Initially the block diagram is in the manner of the given below diagram with input as X(s) and through summation as R(s) and the output is Y(s)=G(s)[R(s)+X(s)] .

When we shift the summation after the block we get the following diagram.

Where the output is Y(s)=G(s)X(s)+R(s) which is not equal to the initial output. To make both the output same we add another block of G(s) with input R(s).

From the above diagram we get the output as Y(s)=G(s)[R(s)+X(s)] which is same as the initial equation.

5. Shifting The Summing Point Before The Block

This involves the shift of the summation point before the block. But after the shift the output result should not change.

Initially the block diagram is in the manner of the given below diagram with input as X(s) and then summation of R(s) after G(s) block.

We will get the output as Y(s)=X(s)G(s)+R(s)

After shifting of the summation we get the following diagram

The output of the diagram is Y(s)=X(s)G(s)+R(s)G(s) which is not equal to case before shifting. To make the equation same we make the following changes to the diagram by adding a block of \frac{1}{G(s)} with the R(s) input.

Now the output will be Y(s)=X(s)G(s)+R(s) which is same as the case before shifting.

6. Shifting the Take Off Point After The Block

In the above diagram we have Y(s)=X(s)G(s),R(s)=X(s)

After we shift the takeoff point after the block we get, Y(s)=R(s)=X(s)G(s) . Here we see that the value of Y(s) remains same but the value of R(s) is changed. To make it same as in original case we add a block of 1/G(s) to R(s). Then the equivalent diagram is shown below.

7. Shifting The Take Off Point Before The Block

In the above diagram we have Y(s)=R(s)=X(s)G(s)

After we shift the takeoff point before the block we get, Y(s)=X(s)G(s),R(s)=X(s) . Here we see that the value of Y(s) remains same but the value of R(s) is changed. To make it same as in original case we add a block of G(s) to R(s). Then the equivalent diagram is shown below.

Application of Block Diagram Algebra

• Block diagrams are used for for simplified representation of control systems.
• Signal processing systems are also represented through block diagram and block diagram algebra is used to analyze these systems.
• It is used for block diagram reduction which is used to find the transmittance of the overall system.

Advantages and Disadvantages of Block Diagram Algebra

Given below are Advantages and Disadvantages of Block Diagram Algebra

Advantages

• Block diagram is used for simplifying complex control systems
• It provides an systematic approach for finding gains of the circuit.

Disadvantages

• It is applicable to linear and time invariant circuits only.
• In case large and complex system it is difficult to design a block diagram and it becomes less effective

Conclusion

In conclusion, Block diagrams are an important part of control system. They are very useful to represent any control system in a simple manner. The block diagram algebra further simplifies the design by reducing the number of block and redesigning as per requirement to calculate the overall gain or transmittance of the system. Although they are very useful in many cases it is limited to linear time invariant system. It is still widely used for analysis of the circuits