Development of Block Diagram

1. Lecture: 33
DEVELOPMENT OF
BLOCK DIAGRAM

2. SYSTEM
Figure 1: Control system for a stirred-tank heater

3. BLOCK DIAGRAM
Figure 2: Block diagram of a simple control system.

4. DEVELOPMENT OF
BLCOK DIAGRAM
 Each block in Fig.2 represents the functional
relationship existing between the input and output of
a particular component.
 In the previous lectures, such input-output relations
were developed in the form of transfer functions.
 In block-diagram
 representations of control systems,
 the variables selected are deviation variables,
 and inside each block is placed the transfer function
relating the input-output pair of variables.
 Finally, the blocks are combined to give the overall
block diagram.
 This is the procedure to be followed in developing
Fig. 2.

5. PROCESS
 Consider first the block for the process.
 This block will be seen to differ somewhat from those
presented in previous lectures in that two input
variables are present;
 However, the procedure for developing the transfer
function remains the same.
 An unsteady-state energy balance around the tank
gives:
 Where To is the reference temperature.
 At steady state, dT/dt=0, and Eq.(1) becomes:
dt
dT
CV
T
T
wc
T
T
wC
q o
o
i 




 )
(
)
( …….……….(1)
0
)
(
)
( 



 o
s
o
is
s T
T
wc
T
T
wC
q …….……….(2)

6. PROCESS
 Subtracting Eq.(2) from Eq.(1) gives:
 Notice that the reference temp. To cancels in the
subtraction.
 If we introduce the deviation variables:
 Eq.(3) becomes
 Taking the Laplace transform of Eq.(7) gives:
………….(3)
is
i
i T
T
T 

'
dt
T
T
d
CV
T
T
T
T
wC
q
q s
s
is
i
s
)
(
)
(
)
[(






 
s
q
q
Q 

s
T
T
T 

'
…………………………………..(4)
…………………………………..(5)
…………………………………..(6)
dt
dT
CV
T
T
wC
Q i
'
)
'
'
( 


 ……………………..(7)
)
(
'
)]
(
'
)
(
'
[(
)
( s
CVsT
s
T
s
T
wC
s
Q i 


 ………………..(8)

7. PROCESS
 Rearranging Eq.(8) gives:
 This last expression can be written as:
 Where τ =ρV/w
 If there is a change in Q(t) only, then Ti’(t)=0 and the
transfer function relating T’ to Q is:
 If there is change in Ti’(t) only, then Q(t)=0 and the
transfer function relating T’ to Ti’ is:
)
(
'
)
(
1
)
(
' s
T
wC
s
Q
s
w
V
s
T i









 ………………..(9)
)
(
'
1
1
)
(
1
/
1
)
(
' s
T
s
s
Q
s
wC
s
T i






………………..(10)
1
/
1
)
(
)
(
'


s
wC
s
Q
s
T

………………..(11)
1
1
)
(
'
)
(
'


s
s
T
s
T
i 
………………..(12)

8. PROCESS
 Eq.(10) is represented by the block diagram shown in
Figure-3.
 This diagram is simply an alternate way to express
Eq.(10) in terms of the transfer functions of Eqs.(11)
and (12).
)
(
'
1
1
)
(
1
/
1
)
(
' s
T
s
s
Q
s
wC
s
T i






………………..(10)
 Notice that a symbol called “SUMMING
JUNCTION” in Fig-3.
 Subtraction can also be indicated with this
symbol by placing a minus sign at the
appropriate input.
 This symbol was used previously as the
symbol of comparator of the controller.
 This symbol may have several inputs but
only one Output.
Figure 3: Block diagram
of Process