Block Diagrams David W. Graham EE 327. 2 Block Diagrams Symbolic representation of complex signals –Easier to understand the relationships between subsystems.

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