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A pictorial representation of the functions performed by each component and of the flow of signals.

**Block diagram**

A pictorial representation of the functions performed by each component and of the flow of signals.

**Basic elements of a block diagram**

o Blocks

o Transfer functions of elements inside the blocks

o Summing points

o Take off points

o Arrow

**Block diagram**

A control system may consist of a number of components. A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals.

The elements of a block diagram are block, branch point and summing point.

**Block**

In a block diagram all system variables are linked to each other through functional blocks. The functional block or simply block is a symbol for the mathematical operation on the input signal to the block that produces the output.

**Summing point**

Although blocks are used to identify many types of mathematical operations, operations of addition and subtraction are represented by a circle, called a summing point. As shown in Figure a summing point may have one or several inputs. Each input has its own appropriate plus or minus sign.

A summing point has only one output and is equal to the algebraic sum of the inputs.

A takeoff point is used to allow a signal to be used by more than one block or summing point. The transfer function is given inside the block

• The input in this case is E(s)

• The output in this case is C(s)

C(s) = G(s) E(s)

**Functional block **–** **each element of the practical system represented by block with its T.F.

**Branches **–** **lines showing the connection between the blocks

**Arrow **–** **associated with each branch to indicate the direction of flow of signal

**Closed loop system**

**Summing point **–** **comparing the different signals

**Take off point **–** **point from which signal is taken for feed back

**Advantages of Block Diagram Representation**

o Very simple to construct block diagram for a complicated system

o Function of individual element can be visualized

o Individual & Overall performance can be studied

o Over all transfer function can be calculated easily.

**Disadvantages of Block Diagram Representation**

o No information about the physical construction

o Source of energy is not shown

**Simple or Canonical form of closed loop system**

R(s) – Laplace of reference input r(t)

C(s) – Laplace of controlled output c(t)

E(s) – Laplace of error signal e(t)

B(s) – Laplace of feed back signal b(t)

G(s) – Forward path transfer function

H(s) – Feed back path transfer function

**Block diagram reduction technique**

Because of their simplicity and versatility, block diagrams are often used by control engineers to describe all types of systems. A block diagram can be used simply to represent the composition and interconnection of a system. Also, it can be used, together with transfer functions, to represent the cause-and-effect relationships throughout the system. Transfer Function is defined as the relationship between an input signal and an output signal to a device.

**Block diagram rules**

Cascaded blocks

**Procedure to solve Block Diagram Reduction Problems**

Step 1: Reduce the blocks connected in series Step

2: Reduce the blocks connected in parallel Step 3: Reduce the minor feedback loops

Step 4: Try to shift take off points towards right and Summing point towards left

Step 5: Repeat steps 1 to 4 till simple form is obtained

Step 6: Obtain the Transfer Function of Overall System

**Problem 1**

**1. Obtain the Transfer function of the given block diagram**

**2. Obtain the transfer function for the system shown in the fig**

**3. Obtain the transfer function C/R for the block diagram shown in the fig**

**Solution**

The take-off point is shifted after the block G2

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