Block Diagram Representation

This lecture we will concentrate on

● Representing system components with block diagrams

● Analyze and design transient response for systems consisting of multiple subsystems

● Reduce a block diagram of multiple systems to a single block representing the transfer function of overall system

The goal of block diagram representation is to obtain a simplified rule between the inputs and the output of the system.

Basics

Transfer function

Adder (summer)

Take of point

Example :

● Armature Controlled DC Motor

Example (cont.)

Rearranging the overall system we have

In a simplified form we have

feedback form

TF of a closed loop system

System having the following form is called closed-loop

: Forward Transfer function

: Feedback Transfer function

Note that

or

Closed-loop Transfer Function

That is

The overall transfer function of the system

Note that

is the characteristic equation

Block Diagram Simplification Rules

1- Cascade

2-Parallel

Block Diagram Simplification Rules

3-Remove a block from the path

Block Diagram Simplification Rules

4- Moving a Summing Junction

Block Diagram Simplification Rules

5. Moving a take of point

How about Multiple Inputs

For Linear systems you can apply Superposition !!!

Meaning when we have multiple inputs, we can tread them independently and sum the outputs

Procedure :

– Step 1 : Set all inputs except one to zero

– Step 2 : Calculate the response for the non-zero input

– Step 3 : Repeat steps 1 and 2 for all input

– Step 4 : Add all responses to obtain the overall

Example

Find for

Solution :

Set U=0 Set R=0

Complicated Examples

Example 1 :

Step 1

Solution :

Step 2

And Finally

The solution

Another Example

Example 2 :

Step 1

Solution :

Step 2

where

And Finally

Combine all

Examples from the Book

Example 3 : find the overall transfer function

All Steps together :)

collapse summingjunctions

form equivalentcascaded systemin the forward pathand equivalentparallel system in thefeedback path;

form equivalentfeedback system andmultiply by cascadedG1(s)

Alternative Approach

Signal Flow Diagrams

– Applied only to linear systems

– Equations must be in algebraic form

– represented by combination of nodes and braches

node : represents variables

branch : dependency of variables

Signal Flow Diagrams

A path is a continous unidirectional successions of braches along which no node is passed more than once

As

or

Signal Flow Diagrams (definitions)

An input node is a node with only outgoing branches

like

An output node is a node with only incomming braches as

A forward path is a path from an input node to an output node

A path gain is the product of the branch gains encountered

A loop gain is the product of the branch gains of the loop

Construction of Signal Flow Diagrams

Simplification of Signal Flow Graphs

Mason's Formulla

where

: the determinant of the graph

or

non- touching2-loops

non- touching3-loops

Simplification of Signal Flow Graphs

Mason's Formulla

where

: the determinant of the graph

or

non- touching2-loops

non- touching3-loops

Path gain of kth forward path

The value of for the part of graph not touching kth forward path

Example ( a warm up)

Simplifiy the following signal flow graph

Solution : Start with finding the forward paths

Example (cont)

Find loops

Calculate and s

re calculate when is removed

The overall transfer function is then

Another Example :

Simplifiy the Block Diagram

Example

Solution:

First form the signal flow graph

Find the forward paths

Example (cont.)

Find loops

Example (still cont.)

Evaluate and s

For recalculate with removed

Similarly for recalculate with removed

Example

Find overall transfer function

which is