combining blocks in series: changing the direction of a block: block diagram manipulation
TRANSCRIPT
Combining blocks in series:
Changing the direction of a block:
Block Diagram Manipulation
Move a block to before a summing junction:
Move a block to after a summing junction:
Block Diagram Manipulation
Source Nise 2004
Move a block to before a take-off point:
Move a block to after a take-off point :
Block Diagram Manipulation
Can be expressed as functions in respect to the changing variable:
We can form three equations:
1)
2)
3)
Negative in the Feedback Loop
sX
sH
sG sC sR sE+_
sXsRsE
sEsGsC
sCsHsX
Combine 1 & 3
2 & 4
Multiply out bracket
Collect output terms
Rearrange
We need to form a relationship between input & output by removing the intermediate variables:
4) sCsHsRsE
sCsHsRsGsC
sCsHsGsRsGsC
sRsGsCsHsGsC
sRsGsHsGsC 1
sX
sH
sG sC sR sE+_
sX
sH
sG sC sR sE+_
sHsG
sG
sR
sC
1Forward
1 - Loop=
Common in useful systems:
Can be expressed as functions in respect to the changing variable:
Negative in the Feedback Loop
sHsC sH
sG sC sR sE+_
+_Input Error
PlantOutput
Feedback
Useful parts:
Create transfer function from the
variables (input and output) and
constants (bits inside the boxes).
Output = Forward
Input 1 - Loop
Negative in the Feedback Loop
+_Input
PlantOutput
Feedback
Forward
Loop
As an introduction, consider dynamic systems:
these change with time
As an example consider water system with two tanks
Water will flow from first tank to second
Dynamic Systems
IF
L
Stops when levels equal.
Plot time response of system….?
Water flows because of pressure difference (Ignore atmospheric pressure - approx equal at both ends of pipe)
If have water at one end - what is its pressure? (Tanks with constant cross sectional area A)
Pressure is force per unit area, = Force / AForce is mass of water * gMass of water is volume of water * density
M = V * Volume is height of water, h, times its area A:
V = h * ACombining:
pressure is
Dynamic Systems
g**hA
g**A*h
FPressure
Pressure
For first tank, pressure is I * * gFor second tank, pressure is L * * g
Thus flow depends on (I-L) * * g as well as on the pipe (its restrictance, R)
Flow =
Flow changes volume of tanks: Volume change = A * change in height (L) = Flow
Thus change in L with time =
A tank has a capacitance, C =
Thus change in L with time is
Flow stops, and there is no change in height, when I = L
Dynamic Systems
g*A
dt
dL
To justify the block diagram:
Flow F = (I - L) * 1/R
Transfer function of Pipe = 1/R
L = 1/C * Integral of F
Transfer function of tank = 1/C * 1/s = 1/sC
Water system
FI
L
LI F1/sC1/R
I-L
FI
L
LI F1/sC1/R
I-L
sCR1
1
sCR11sCR
1
sC1*R
1--1sC
1*R1
Loop-1
Forward
I
L
Dynamic change in height :
Rearrange
The transfer function:
Multiplying both sides by I * (1+sCR) gives:
or
that is
Slightly rearranged, this gives
Water system
FI/R
tT
0.37I/R0.63I
T
L
t
I
CR
L-I
dt
dL
sCR1
1
I
L
IsCR)L(1
Idt
dLCR L
CR
L-I
dt
dL
L + sCR L = I
We have dynamic equation:
Put into ‘s’ domain and rearrange:
Integrate a step input with respect to time:
Variation of L is
As t gets larger, exponential term disappears, L tends to I.
At t = T, L is 63% of final value, IAt t = 5T, L is within 1% of final value.
Time Response
Tt
-e*I-IL
FI/R
tT
0.37I/R0.63I
T
L
t
I
R*C
LI
dt
dL
R*C
LIsL
IL R*C*sL s R*C1
1
I
L
s T1I
O
K
Any system of the form:
Has a time response (depending on input):
Time Response of System
sT
K
I
O
11
Time
Output
Exponential
Has a time response to a unit step input:
Unit Step Response of System
Time
Output Output when K = 1
Input
Output Output when K = 1.6
Input
Has a time response to a unit step input:
Unit Step Response of System
Time
Output Output when K = 1, T= 0.01
Input
Output Output when K = 1, T= 0.02
Input
Exercises
Two tanks are connected by pipe.
Heights of liquid are 0.1m and 0.02m
Liquid density, = 2 kg/m3, g = 9.8 ms-2
What is the pressure difference?
If restrictance is 0.1 N s / m5, what is flow?
Write down differential equation for system if the area of the second tank is 0.5 m2
Write down equations showing variation of F and L.
(In each case, use equations given in notes, and put in appropriate component values)
Exercises