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 [Tex]G_1(s)\:\:\:and \:\:\: G_2(s) [/Tex]. 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 [Tex]G_1(s)X(s) [/Tex]

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

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 [Tex]G_1(s)\:\:\:and \:\:\: G_2(s) [/Tex] which are connected in parallel.

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

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

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

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 [Tex]K(s)=X(s)+H(s)Y(s) [/Tex]

The final output will be, [Tex]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)} [/Tex]

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

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 [Tex]Y(s)=G(s)[R(s)+X(s)] [/Tex].

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

Where the output is [Tex]Y(s)=G(s)X(s)+R(s) [/Tex] 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 [Tex]Y(s)=G(s)[R(s)+X(s)] [/Tex] 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 [Tex]Y(s)=X(s)G(s)+R(s) [/Tex]

After shifting of the summation we get the following diagram

The output of the diagram is [Tex]Y(s)=X(s)G(s)+R(s)G(s) [/Tex] 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 [Tex]\frac{1}{G(s)} [/Tex] with the R(s) input.

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

6. Shifting the Take Off Point After The Block

In the above diagram we have [Tex]Y(s)=X(s)G(s),R(s)=X(s) [/Tex]

After we shift the takeoff point after the block we get, [Tex]Y(s)=R(s)=X(s)G(s) [/Tex]. 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

Block diagram

In the above diagram we have [Tex]Y(s)=R(s)=X(s)G(s) [/Tex]

After we shift the takeoff point before the block we get, [Tex]Y(s)=X(s)G(s),R(s)=X(s) [/Tex]. 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.

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.