Introduction to Feedback Systems / © 1999 Önder YÜKSEL

AN EXAMPLEAN EXAMPLE

04/19/23 1.7 An example 2

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

VR

motor

VL

generator

tank1 tank2turbine

valve

04/19/23 1.7 An example 3

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Comparator

VR

VL

K

VM

vM=K(vR-vL)

K+

-

VR

VL

VM

04/19/23 1.7 An example 4

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

vM

Motor(armature controlled)

Motor data:

Armature winding resistance: Ra

Armature winding inductance: La

Motor constant: KM

Back e.m.f. constant: Kb

TMTM

MM

Rotor inertia: JM

Shaft friction: BM

04/19/23 1.7 An example 5

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Armature controlled D.C. motor - revisited

amiKT

ωKe bb

i f=cnt.

+

eb

-

ia

T

La Ra

Jm Bm

04/19/23 1.7 An example 6

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Model: A block diagram

sJB

1

mm

sLR

(s)e(s)e(s)i

aa

baa

amiKT

TT

ωBm

ωJm

s)ωJ(BT(s) mm

ωKe bb

++

--

ea

eb

sLR

1

aa

ia TTKm

Kb

04/19/23 1.7 An example 7

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

For the problem at hand

eb

T

vM

TM

M

BUT

Mechanical load on the shaft should be the total load

Including that due to the valve also

04/19/23 1.7 An example 8

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Block diagram for the motor

sJB

1*

m*

m

M

Where BM* & JM

* show the total including the load

VM +

-eb

sLR

1

aa

ia TMKm

Kb

04/19/23 1.7 An example 9

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Valve

TMTM

MM

q0q0

Data

q0=KvM

Shaft friction: Bv

Shaft inertia negligible

04/19/23 1.7 An example 10

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Valve model

s1

M

Kv

M q0

and

BM*=BM+Bv , JM

*=JM

q0=KvM sM=M

04/19/23 1.7 An example 11

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Double tank:Definitions

h1

A1A1 A2

A2

h2

q0q0

q1q1 q2

q2

p1 p2

R1R1 R2

R2

p3

p1=h1

p2=h2

04/19/23 1.7 An example 12

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Analysis

h1

q0

q1

p1

Rate of change of heightXcross sectional area

=added volume per unit time

1011 qqhA

1A1

s1+

-

q0

q1

h1

04/19/23 1.7 An example 13

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

q1

p1 p2R1

1

21

1

211 R

hhR

ppq

1R1+

-

h1

h2

q1

04/19/23 1.7 An example 14

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

h2

q1q1 q2

q2

p2

2122 qqhA

2A1

s1++

--

q1q1

q2q2

h2h2

04/19/23 1.7 An example 15

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

2

32

2

322 R

ph

R

ppq

2R1++

--

h2h2

p3p3

q2q2

q2q2

p2 p3R2

R2

04/19/23 1.7 An example 16

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Block diagram for the double tank system

1A1

s1+

-

q0

q1

h1-

h2

1R1+ q1

2A1

s1+

-

q2

h2

04/19/23 1.7 An example 17

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Turbine

q2

G

TG

p3

0

Turbine inertia & friction negligible

GG32 Tωpq

Np

ω

T

q

3

G

G

2

A lossless (transformer-like) element

04/19/23 1.7 An example 18

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Generator

G

TG

vL

iF

Field controlled type

Rotor inertia: JG

Bearing friction: BG

Field inductance: LF

Field resistance: RF

04/19/23 1.7 An example 19

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Field controlled D.C. motor

ia=Cnt.if

T

Fix armature current!LfRf

+ef

+ef

Jm Bm

Tm=Kmif

04/19/23 1.7 An example 20

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Block diagram model

sJB

1

mm ef

ef

(s)esLR

1(s)i f

fff

fmm iK(s)T

T(s)sJB

1(s)ω

mmm

sLR

1

ff

if TTKm

04/19/23 1.7 An example 21

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

ia=Cnt.if

G

TG

Lf Rf

+vL

+vL

JmBm

ILIL

Jm JG

Bm BG

04/19/23 1.7 An example 22

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

iF=KGTG

fFFL )isLRv (

GGG

G TsJB

1ω

Nωph G

22

NqT 2

G

KG iFFF sLR

vLvL

h2

GN

TGBG+sJG

Nq2

04/19/23 1.7 An example 23

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

K+

-

VR

VL

VM

s

1Kv

Mq0

comparator motor valve double tank turbine-generator

+s

1

-

h1 -1R

1+q1

2A1

s1+

- h2

1A1

q2

N BG+sJG KG RF+sLF

N

ωG TGif VL

bKbe

+

-sLR

1

aa mKsJB

1*

m*

m

Mia

04/19/23 1.7 An example 24

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

End of Chapter

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