InstrumentTypes

andPerfo

rmance

Characteristics

17Mar

15

Que

stions

Howistheaccuracy

ofan

instrumen

tusuallydefin

ed?

Whatisthe

diffe

rencebe

tweenaccuracy

andprecision

?To

illustratethedynamiccharacteristicso

fthe

following:

(a)zeroorde

rinstrum

ent

(b)firstorde

rinstrum

ent

(c)secon

dorde

rinstrum

ent

Atungsten

resistancethermom

eter

with

arangeof

–270

Cto

+1100Chasa

quoted

inaccuracy

of1.5%

offullscale

reading.Whatisthe

likelymeasuremen

terror

whe

nitis

readingatempe

rature

of950C?

3

InstrumentT

ypes

PassiveandActiveInstruments

instrumento

utpu

tisp

rodu

ced

entirelyby

thequ

antitybe

ing

measured

thequ

antitybe

ingmeasured

simplymod

ulates

themagnitude

ofsomeexternalpo

wer

source.

In a

ctiv

e in

stru

men

ts, t

he

exte

rnal

pow

er s

ourc

e is

us

ually

in e

lect

rica

l for

m,

but

in s

ome

case

s, it

can

be o

ther

form

s of

ene

rgy,

such

as

a pn

eum

atic

or

hydr

aulic

one

.5

NullTypeandDe

flectionType

Instruments

defle

ctiontype

instrument

thevalueof

thequ

antitybe

ingmeasuredis

displayedinterm

softhe

amou

ntof

movem

ento

fapo

inter.

nulltype

instrument

pressure

measurementism

adein

term

softhe

valueof

theweights

need

edto

reachthisnu

llpo

sition

Ingene

ral,nu

lltype

instrumentsaremoreaccurate

than

defle

ctiontype

s,bu

tdeflectiontype

instrument

isclearly

moreconven

ient

6

Analogue

andDigitalInstrum

ents

Analogue

Instrumen

tsou

tput

that

varie

scon

tinuo

uslyas

thequ

antitybe

ingmeasured

changes

DigitalInstrum

ents

output

that

varie

sindiscrete

step

sandso

canon

lyhave

afin

itenu

mbe

rofvalue

s.

7

Indicatin

gInstrumentsandInstrumentswith

aSignal

Outpu

t

Indicatin

gInstruments

Instrumentswith

aSignalOutpu

t

Instrumen

tsthat

have

asig

naltypeou

tput

areused

common

lyas

partof

automaticcontrolsystems.

8

StaticCh

aracteristicso

fInstrum

ents

Accuracy

ofmeasurementisthu

sone

considerationinthe

choice

ofinstrumentfor

aparticular

application.

Other

parameters,such

assensitivity,linearity,andthe

reactio

nto

ambienttem

perature

changes,arefurthe

rconsiderations.

Values

quoted

forinstrum

entcharacteristicsinsuch

adata

sheeto

nlyapplywhe

ntheinstrumentisu

sedun

derspe

cifie

dstandard

calibratio

ncond

ition

s.

9

Data

Sheet(Example)

10

Accuracy

Anindicatoro

fhow

closetheou

tput

readingof

the

instrumen

tistothecorrectvalue

.measuremen

tuncertainty,inaccuracy

Percentage

ofthefullscale(f.s.)reading

ofan

instrument

ifwearemeasurin

gpressuresw

ithexpe

cted

values

betw

een0and

1bar,wewou

ldno

tuse

onewith

ameasuremen

trange

of0–10

bar.

Exam

ple

11

Precision

/Rep

eatability/Re

prod

ucibility

spread

ofreadings

ofthesamequ

antity.

Repe

atability:closene

ssof

output

readings

with

constant

measuremen

tcon

ditio

ns.

Reprod

ucibility:closene

ssof

output

readings

with

varying

measuremen

tcon

ditio

ns.

12

Tolerance

themaxim

umerrorthatistobe

expe

cted

insomevalue.

Whe

nused

correctly,tolerance

describ

esthemaxim

umde

viation

ofamanufacturedcompo

nent

from

somespecified

value.

Exam

ple

13

Range(Span)

andLine

arity

Range

theminim

umandmaxim

umvalues

ofaqu

antitythat

theinstrumentis

desig

nedto

measure

Line

arity

Non

linearityisde

fined

asthemaxim

umde

viationof

anyof

theou

tput

readings

markedofrom

thisstraight

lineandisusually

expressedas

ape

rcen

tage

offullscalereading.

14

Sensitivity

ameasure

ofthechange

ininstrumento

utpu

tthato

ccurs

whe

nthequ

antitybe

ingmeasuredchangesb

yagivenam

ount.

theslo

peof

thestraight

lineof

bestfit

Exam

ple

15

Threshold,

Dead

SpaceandHy

steresisEffect

Threshold

Theminim

umlevelofinp

utleadstoachange

intheinstrumen

toutpu

t

Dead

space

therangeof

diffe

rent

inpu

tvalues

over

which

there

isno

change

inou

tput

value.

Hysteresiseffects

Diffe

rent

characteristicsfor

increasin

gandde

creasin

gcaseso

fmeasurin

g.magne

tichysteresis:

instrumen

tsthat

containelectricalwindingsformed

roun

dan

ironcore.

16

Summaryof

nonlinearties

17

Resolutio

nandEffectso

fDisturbance

Resolutio

nalower

limiton

themagnitude

ofthechange

intheinpu

tmeasured

quantitythat

prod

uces

anob

servablechange

intheinstrumento

utpu

t.

Effectso

fDisturbance

Allcalibratio

nsandspecificatio

nsof

aninstrumen

tare

onlyvalid

unde

rcontrolledcond

ition

softem

perature,pressure,andso

on.The

sestandard

ambien

tcon

ditio

nsareusually

defin

edintheinstrumen

tspe

cification.

zero

drift

(bias)

sensitivity

drift

18

Effectso

fDisturbance

19

Exam

ple:Zero

Drift

20

Exam

ple:Zero

Drift

+Sensitivity

Drift

21

Characteristicso

fInstrum

ents

22

Thestaticcharacteristicso

fmeasurin

ginstrumentsare

concerne

don

lywith

thesteady

statereadingthat

the

instrumentsettle

sdow

nto,suchas

accuracy

ofthereading.

Thedynamiccharacteristicso

fameasurin

ginstrument

describ

eits

behavior

inthetim

eintervalbefore

the

instrumentreaches

asteady

valueinrespon

se.

StaticRe

spon

seCh

aracterization

MeasurementInstrum

ents

are

notp

erfect:steadystate(static)

andtransie

nt(dynam

ic).

Calibratio

n:to

obtainthe

relatio

nbe

tweenthemeasured

varia

bleandinstrumento

utpu

tvaria

ble.

Type

sofcalibratio

ns:static

and

dynamic

Sensitivity:thesm

allestchange

inaqu

antitythat

aninstrument

cande

tect

Stat

ic c

alib

ratio

n cu

rve

23

Calibratio

nAccuracy

Absolute

error:

Relativeerror:

Calibratio

naccuracy:

whe

nthecharacteristicso

fthe

instrumentw

illhave

drifted

from

thestandard

specificatio

nby

anun

acceptableam

ount.

Whe

nthissituatio

nisreache

d,itisne

cessaryto

recalibrate

theinstrumentb

ackto

standard

specificatio

ns.

24

Exam

ple

25

Solutio

n

U

E2

26Sign

ifica

nt fi

gure

s =

3

Inpractice,thereisatim

ede

layandam

plitu

dediffe

rencebe

tween

themeasuredvalueandtrue

valuefortim

evaryinginpu

tvariableat

agiventim

e.Thisdiffe

renceisreferred

toas

thedynamicerror.

Ingene

ral,thedynamicrespon

seof

ameasuremen

tinstrum

entcan

bede

scrib

edby

azero,1

stor

2ndorde

rsystem.

DynamicRe

spon

seCh

aracterization

27

30 s

< 1

s

3

min

utes

Descrip

tionof

DynamicBe

haviou

r

28

Timeinvaria

ntmeasurin

gsystem

,

Zero

orde

r:

Firsto

rder:

Second

orde

r:

Inpu

t

0

1

2

Introd

uctio

nto

MatlabandSimulink

29

Hel

p ic

onW

indo

w fo

r in

tera

ctiv

e co

mm

ands

Clic

king

the

Sim

ulin

kic

onTo

sta

rt

New

file

Simulinkforclassical

controlsystem

analysis

Cre

atin

g a

New

Mod

el

Sele

ct t

he o

bjec

tive

blo

ck in

the

Si

mul

ink

Libr

ary

Bro

wse

r, th

en

drag

it t

o th

e m

odel

win

dow

.

To a

dd b

lock

s to

you

r m

odel

Con

nect

ing

Blo

cks

in t

he M

odel

Win

dow

1.Po

sitio

n th

e m

ouse

poi

nter

ove

r an

inpu

t/ o

utpu

t po

rt.

2.D

rag

a lin

e fr

om t

he in

put/

outp

ut p

ort

to a

des

ired

outp

ut/in

put

port

(pre

ss a

nd h

old

the

left

butt

on, d

rag

and

rele

ase)

.N

ote

that

the

poi

nter

cha

nges

to

a cr

oss

hair

s (+

) sha

pe

whi

le o

ver

the

port

To w

eld

a co

nnec

tion

to

an

exis

ting

line

:(D

raw

ing

a B

ranc

h Li

ne)

1.Po

sitio

n th

e m

ouse

poi

nter

on

the

line.

2.D

rag

a lin

e fr

om t

he d

esire

d po

int

usin

g th

e ri

ght

mou

se b

utto

n.

set

sim

ulat

ion

opti

ons

such

as

the

star

t an

d st

op t

ime,

and

the

typ

e of

sol

ver

that

Sim

ulin

kso

ftw

are

uses

to

solv

e th

e m

odel

at

each

ti

me

step

. You

spe

cify

the

se

opti

ons

usin

g th

e C

onfig

urat

ion

Para

met

ers

dial

og b

ox.

Dou

ble-

clic

k th

e Sc

ope

bloc

k in

the

m

odel

win

dow

. The

Sco

pe w

indo

w

disp

lays

the

sim

ulat

ion

resu

lts.

Figure

Scop

eEdito

r

Outpu

tyisdirectlyrelativeto

inpu

tF(t)b

yconstant

gaina 0

Exam

ple:

anelectricalresistancestraingage

aninpu

tstraindirectlycauses

achange

inthe

gage

resistance

DynamicTimeRe

spon

se(zeroorde

r)

39

ForF

(t)b

eing

astep

functio

nof

unitAandy(0)

=y 0

DynamicTimeRe

spon

se(1

storde

r)

40

Dynamicof

aThermom

eter

Thefirstlawof

thermod

ynam

ics

Energy

stored

with

intheliquid=convectivehe

atthansfer

41

Determ

inesystem

timeconstant

bystep

respon

se

Steady

stategain:accuracy

Timeconstant:respo

nsetim

efor

data

measuremen

t(2%

errort

s=

4)

42

ByLaplacetransform

t:tim

econstant

K:system

gain

()

()

1K

Ys

Fs

s

Exam

ple

43

x=5t,T x=10

–0.05

t

Solutio

n

44

/(

)1

tL

es

2

2

2 2

((

)(0

))(

)(

)1

115

((

)10

)(

)10

0.05

11

(15

1)(

)15

010

0.05

150

100.

05(

)(1

51)

(15

1)(1

51)

11.2

510

.75

0.

()

()

()

05(

)(1

51)

rr

x

rr

rx

rr

r

r

r

tt

tt

tt

Lapl sT

st

Ts

Tac

s

sTs

Ts

ss

sT

ss

s

Ts

ss

ss

eTr

ans

s

T

fo s

rm

ss

s

/15 /1

5

()

0.75

10.7

50.

05

=10

0.75

0.05

(15

)

tr

t

tt

et

et

Tim

eA

ltitu

deTe

mp

Tem

p re

adin

gTe

mp

erro

r0

010

10.0

00

0.00

0 10

509.

59.

865

0.36

5 20

100

99.

552

0.55

2 30

150

8.5

9.14

8 0.

648

4020

08

8.69

8 0.

698

5025

07.

58.

223

0.72

3 10

0050

00-4

0-3

9.25

0 0.

750

Respon

seto

Sinu

soidal

Inpu

tForce

(1storde

rsystem)

45

DynamicError

Phaselag:

radiansa

ndph

aselaginsecond

Dynamicerror

whe

re

Dyn

amic

err

ors

are

func

tion

of fr

eque

ncy

2T

22

Syst

em G

ain

46

DynamicErrord

ueto

PhaseLagandGa

inRatio

n

47

Freq

uencyRe

spon

seForsteadystaterespon

seto

asin

usoidalinp

ut(t

)

Magnitude

plot

Phaseplot

11

13 2c

()

tan

tan

tan

c1

48

Freq

uencyRe

spon

seFor

=1,themagnitude

ratio

neq

uals0.70

7(dynam

icerror

of29

%)and

theph

aseshift

equals45

.Themagnitude

ratio

oftenisexpressedinun

itsof

decibe

ls(dB),w

hich

isoriginated

from

power

ratio

Sincepo

wer

isasquaredqu

antity,De

cibe

lisd

efined

as

210

1PBe

llog

P

49

CutoffFreq

uencyandSystem

Band

width

50

Exam

ple

51

Exam

ple

Solutio

n:Assumption:

E(0)

isthevolta

geou

tput

correspo

ndingto

fluidtempe

rature

att=

0,i.e.,

Uni

t :

mV

Uni

t :

52

Exam

ple(con

t.)Dy

namicerror

Timelag

11

13 2c

()

tan

tan

tan

c1

53

DynamicTimeRe

spon

se(2

ndorde

r)

54

TimeRe

spon

seof

2ndorde

rSystems

55

Characteristic

Equatio

nandTheirR

elationshipto

Damping

56

TimeRe

spon

seof

2ndorde

rSystems

57

TimeRe

spon

seof

2ndorde

rSystems

58

Damping

Ratio

=0Nodamping

<1Und

erdamping

=1Criticaldam

ping

>1Overdam

ping

59

Step

Respon

sePerfo

rmance

Specificatio

n

The

desired

perfo

rmance

characteristicsof

atracking

system

may

bespecified

interm

sof

thetransie

ntrespon

seto

aun

itstep

functio

ninpu

t.Somequ

alities

forevaluatin

gsystem

perfo

rmance

aredefin

edas

follows.For

,(1).Peak

overshoo

t(pe

rcento

foversho

otP.O

.)M

p:M

p=10

0%(y

max–y(

))

1)

(y

60

Step

Respon

sePerfo

rmance

Specificatio

n(2

).pe

aktim

e(ti

me

tom

ax.o

vers

hoot

)tp:

y(t p)

=y m

ax

(3).

freq

uenc

yof

osci

llatio

nof

the

trans

ient

d:1/

(2t p)

Hz

(4).

rise

time

t r:(5

).se

ttlin

gtim

et s:

(6).

time

cons

tant

:,

for

%,

2|1

)(

|st

tt

y9.0

)(

and

1.0)

(,

21

12

ty

ty

tt

t r

368

.0|1

)(

|,t

yt

61

TimeRe

spon

seof

a2n

dOrder

System

with

=0.2

Sett

ling

time

of 2

% is

ge

nera

lly u

sed

in c

ontr

ol

syst

em s

tudy

and

set

tling

tim

e of

10%

may

be

used

in

mea

sure

men

t fie

ld.

62

Relatio

nBe

tweenPo

leLocatio

nto

System

Specificatio

n(stand

ard2n

dorde

rSystem

<0.70

7)

t ne

1

t ne

1

Atthetim

eof

2%errort

s

ns

sn

t

t

te

sn

912

.302.0ln

02.0

dn

pt2

1

%10

0

%10

0)1

)1((

..

21

/e

eO

Pp

nt

63

equivalent

timeconstant:

settlingtim

e(2%e s

s):

whe

reiscalledthedamping

(tim

e)constant.

Dampe

d/rin

ging

freq

uency:

rad/sec

peak

time:

Relatio

nBe

tweenPo

leLocatio

nto

System

Specificatio

n(stand

ard2n

dorde

rSystem

<0.70

7)

dn

nn

nnn

jj

ss

sR

sY

-1

-:po

les

2)

()

(2

22

2

nst

491

2.3

21

npt

1

21

nd

64

01

23

45

60123456

X: 2

Y: 2

.594

X: 0

.500

4Y

: 4.8

14

X: 4

Y: 3

.055

X: 1

Y: 1

.897

Relatio

nBe

tweenPo

leLocatio

nto

System

Specificatio

n(stand

ard2n

dorde

rSystem

<0.70

7)

t p=0.5sec,

d=2

rad/sec=1Hz

t s=3.91

2sec(2%error)

P.O.=

60.5%

Timeconstant

=1sec

dn

nn

nnn

jj

ss

sR

sY

-1

-:po

les

2)

()

(2

22

2 40240

)(

3623

.6,

1572

.0

21

,1

21

,11

2

2

2

ss

sG

n

n

nd

n

2832

.61

:lo

catio

npo

lej

65

Respon

seto

Sinu

soidal

Inpu

tForce

(2ndorde

rsystem)

66

Exam

ple

js

bj

sb

as

ba

sb

sY

* 00

2

2

1

1)

(

jej

Gjj

Gj

ss

Gb

j

js

2)(

2)

()

(0

je

jG

jjG

js

sG

bj

js

2)

(2

)(

)(

* 0

22

)(

)(

ss

Gs

YCo

nsider

thesystem

respon

sede

scrib

edby

apartialfractionexpansioncorrespo

ndingto

Y(s)is

67

)si

n()

()(

21

21

tj

Ge

be

bt

yt

at

a

))(

Im(

))(

Re(

tan

1

jG

jG

)si

n()

()

(t

jG

y

)si

n(2

)(

)(

tje

et

jt

j

)(

)(

and

)(

)(

jG

jG

A

for

a st

able

sys

tem

68

Freq

uencyRe

spon

seof

2ndOrder

System

69

Respon

seof

Higher

Order

System

s