Block diagrams

Updated

November 5, 2024

1 Simple inteconnection of blocks

We have already seen three basic forms of block diagrams:

Figure 1: Cascade Connection equivalent to \(G_1(s)G_2(s)\).
Figure 2: Parallel Connection equivalent to \(G_1(s) + G_2(s)\).
Figure 3: Feedback Connection equivalent to \(\displaystyle \frac{G(s)}{1 + G(s)H(s)}\).

We can use the basic rules above to simplify block diagrams. We provide a few examples below.

Example 1 Simplify the block diagram shown below.

Block diagram for Example 1

The highlighted blocks are in cascade

The highlighted blocks are in feedback

Thus, the simplified TF is

\[ T(s) = \frac{G_1(s) G_2(s) G_3(s) } { 1 + G_1(s) G_2(s) G_3(s) H_1(s) H_2(s) }. \]

Example 2 Simplify the block diagram shown below.

Block diagram for Example 2
Approach 1: Nested feedback loops

Block diagram for Example 2

The first feedback loop

In the first approach, we can view the system as three nested feedback loops. The first loop is highlighted above, which simplifies to

\[ \dfrac{G_2(s) G_3(s)}{1 + G_2(s) G_3(s) H_1(s)} \]

Simplified block diagram

The second feedback loop

The second loop simplified to

\[ \dfrac{ \dfrac{G_2(s) G_3(s)}{1 + G_2(s) G_3(s) H_1(s)} } { 1 + \dfrac{G_2(s) G_3(s) H_2(s)}{1 + G_2(s) G_3(s) H_1(s)} } = \dfrac{ G_2(s) G_3(s) } { 1 + G_2(s)G_3(s) H_1(s) + G_2(s)G_3(s)H_2(s) } \]

Simplified block diagram

This is a simple feedback loop. Thus, we can simplify it to \[ \dfrac{ \dfrac{ G_1(s) G_2(s) G_3(s) } { 1 + G_2(s)G_3(s) H_1(s) + G_2(s)G_3(s)H_2(s) }} { 1 + \dfrac{ G_1(s) G_2(s) G_3(s) H_3(s) } { 1 + G_2(s)G_3(s) H_1(s) + G_2(s)G_3(s)H_2(s) } } = \dfrac{G_1(s) G_2(s) G_3(s) } {1 + G_2(s) G_3(s) \bigl[ H_1(s) + H_2(s) + H_3(s) \bigr] }. \]

Approach 2: Parallel feedback loops

Block diagram for Example 2

Parallel feedback loop

Simplified diagram

This is a simple feedback circuit. Thus, the simplified transfer function is \[ \dfrac{G_1(s) G_2(s) G_3(s) } {1 + G_2(s) G_3(s) \bigl[ H_1(s) + H_2(s) + H_3(s) \bigr] }. \]

The above two examples show that we can simplify block diagrams using the simple rules for interconnected blocks. However, these simple ideas are unable to simplify more complicated block diagrams shown below.

Example 3 Simplify the block diagram shown below.

Block diagram for Example 3

In order to simplify such diagrams, we need to develop rules for moving pick up points and adders around blocks.

2 Moving pickup points and adders around blocks

There are four basic rules for simplifying block diagrams.

  1. Moving a pickup point after a block to before a block.

    In both cases, \(Y(s) = X(s) G(s)\) and \(Z(s) = X(s)G(s)\).

  2. Moving a pickup point before a block to after a block.

    In both cases, \(Y(s) = X(s) G(s)\) and \(Z(s) = X(s)\).

  3. Moving an adder after a block to before a block.

    In both cases, \(Z(s) = X(s)G(s) + Y(s)\).

  4. Moving an adder before a block to after a block.

    In both cases, \(Z(s) = X(s)G(s) + Y(s)G(s)\).

3 Back to the example

We now see how we can use the rule above to simplify the block diagram of Example 3.

Block diagram for Example 3

Moving a pickup point

We look at the highlighted sub-block and move the pickup point from behind \(G_4(s)\) to after \(G_4(s)\).

Simplified diagram

The simplified diagram is of the form that we can identify the three basic blocks and iteratively simplify it. However, the method is tedious. We can make it much simpler by moving the adders around so that all of them are in the middle.

Simplified diagram

Moving adders

Simplified diagram

The final diagram now has three adders in feedback, similar to Example 2. Therefore, we can simplify the TF as

\[ \dfrac{G_1(s) G_2(s) G_3(s) G_4(s)} {1 + G_2(s)G_3(s)G_4(s) \Bigl[ \dfrac{H_1(s)}{G_2(s)} + \dfrac{H_2(s)}{G_4(s)} + H_3(s)G_1(s) \Bigr]}. \]