block diagrams david w. graham ee 327. 2 block diagrams symbolic representation of complex signals...
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Block Diagrams
David W. Graham
EE 327
2
Block Diagrams
• Symbolic representation of complex signals– Easier to understand the relationships
between subsystems using block digrams than by looking at the equations representing them, or even schematics
– Often easier to obtain a transfer function for the overall system by first drawing block diagrams
3
Basic Elements
1. Block H(s)X(s) Y(s) = H(s)X(s)
2. Adder X1(s) +
X2(s)
Y(s) = X1(s) + X2(s)
3. Node X(s)
X (s)
X(s)
4
Basic Connections
• Series / Cascade
• Parallel
• Feedback
5
Series / Cascade Connections
H2(s)W(s) = H1(s)X(s)
H1(s)X(s) Y(s) = H2(s)W(s) = H1(s) H2(s)X(s)
Equivalent to a single block of
H1(s)H2(s)X(s) Y(s)
6
Parallel Connections
+
H1(s)
H2(s)
X(s) Y(s)
Y1(s)
Y2(s)
Y1(s) = H1(s)X(s)Y2(s) = H2(s)X(s)
Y(s) = Y1(s) + Y2(s)= [H1(s) + H2(s)]X(s)
Equivalent to a single block of
H1(s) + H2(s)X(s) Y(s)
7
Feedback Connection(Negative Feedback)
+
H2(s)
X(s) Y(s)E(s)
-H1(s)
Y(s) = H1(s)E(s)E(s) = X(s) – H2(s)Y(s)
Y(s) = H1(s)[X(s) – H2(s)Y(s)]= H1(s)X(s) – H1(s)H2(s)Y(s)
Y(s)[1 + H1(s)H2(s)] = H1(s)X(s)
Y(s) = H1(s)X(s) 1 + H1(s)H2(s)
(Equivalent Block)
8
Complex System Modeling1. Obtain a transfer function for each subsystem2. Determine equations that show the interactions of each
subsystem3. Draw a block diagram (Hint. Start from input and work
to output or vice versa)4. Perform block-diagram reduction
H1(s)H2(s)X(s) Y(s)H2(s)H1(s)X(s) Y(s)A.
+
H1(s)
H2(s)
X(s) Y(s)B. H1(s) + H2(s)X(s) Y(s)
+
H2(s)
X(s) Y(s)-
H1(s)C. H1(s)1 + H1(s)H2(s)
X(s) Y(s)
+ H(s)
X2(s)
Y(s) = H(s)[X1(s) + X2(s)]X1(s) X1(s) +H(s)
X2(s)
Y(s) = H(s)[X1(s) + X2(s)]
H(s)D.
9
Block-Diagram Reduction Example 1)(1 sH )(2 sH
)(3 sH
)(sX+
-)(sY
)()( 21 sHsH
)(3 sH
)(sX+
-)(sY
)(sX )(sY)()()(1
)()(
321
21
sHsHsH
sHsH
10
1
2
110 321
s
ssH
ssHsH , , If
3022
1010
102110
1
210
)()()(1
)()(
2
321
21
ss
s
sss
ssHsHsH
sHsHsH
10
Block-Diagram Reduction Example 2
s
1
2
2
s
)(sX+
-)(sY
1
1
s
3
+-
Divide by 1/s block
++
s
1
2
2
s
s
)(sX+
-)(sY
1
1
s
3
+-
++
11
s
1
2
2
s
s
)(sX+
-)(sY
1
1
s
3
+-
3
1
31
1
1
s
s
s Block Feedback
++
3
1
s
2
2
s
s
)(sX+
-)(sY
1
1
s++
12
3
1
s
2
2
s
s
)(sX+
-)(sY
1
1
s
3
1
s
++
3
1
s
2
2
s
s
)(sX+
-)(sY
1
1
s++
Feedback
Parallel
3
11
s)(sX )(sY
322
11
1
11
sss
s
s)(sX )(sY
ssss
ss
2132
42
13
Large System ExampleSatellite Tracking System
• Trying to track a satellite with an antenna• Input to the system is a control angle (i.e. the angle we want to point the antenna)• Output is the actual angle of the antenna• The overall system requires several components to complete its task
+-
N
1
TrackingReceiver
θm
θc
θL
TachometerMotor
AmplifierV1
V2
Va
Gear Box
-
14
Subsystem Equations
1. Tracking ReceiverInput θc, θL
Output V1
+-
N
1
TrackingReceiver
θm
θc
θL
TachometerMotor
AmplifierV1
V2
Va
Gear Box
-
LcsHV 11
H1(s) is a complicated transfer function
2. AmplifierInput V1 – V2
Output Va
21 VVKV aa
aKsH 2
3. MotorInput Va
Output θm
aTmbTm VKKKJ
bT
T
aTmbTm
KsKJs
KsH
VKKsKJs
23
2 is TF
System Input θc,
System Output θL
System Signalsθc, θm, θL, V1, V2, Va
System ConstantsJ, Ka, Kb, Kt, KT
4. TachometerInput θm
Output V2
mtKV 2
tsKsH 4
5. Gear BoxInput θm
Output θL
mL N 1
N
sH1
5
15
Create a System Block Diagram(Let us work from input to output)
)(1 sHc+
-L
1V
+-
2V
aVaK
bT
T
KsKJs
K
2
m
tsK
N
1L
Subsystem Equations
1. Tracking ReceiverInput θc, θL
Output V1
LcsHV 11
H1(s) is a complicated transfer function
2. AmplifierInput V1 – V2
Output Va
aKsH 2
3. AmplifierInput Va
Output θm
bT
T
KsKJs
KsH
23
4. TachometerInput θm
Output V2
tsKsH 4
5. Gear BoxInput θm
Output θL
N
sH1
5
Connect like nodes
16
Perform Block-Diagram Reduction
)(1 sHc+
-L
1V
+-
2V
aVaK
bT
T
KsKJs
K
2
m
tsK
N
1L
Series Connection
)(1 sHc+
-L
1V
+-
2VbT
Ta
KsKJs
KK
2
m
tsK
N
1L
Feedback Connection
17
)(1 sHc+
-L bT
Tta
bT
Ta
KsKJsKKsKKsKJs
KK
2
2
1 N
1L
c+
-L
btaT
Ta
KKKsKJs
KK
N
sH
21 L
c L TabtaT
Ta
KKsHKKKsNKNJs
KKsH
12
1