Formulation of Characteristic Equations for Instruments

P M V SubbaraoProfessor

Mechanical Engineering Department

Clues for Design of an Instrument to carry out Transient Measurements….

Characteristic Equation of A General Instrument

The characteristic equation of nth order system:

)(........... 012

21

1 txyaDyayDayDayDa nn

nn

)(....... 012

2

21

1

1 txyadt

dya

dt

yda

dt

yda

dt

yda

n

n

nn

n

n

Laplace Transformations for Instruments

sXsYasaasasas nn

nn

012

21

1 ...........

012

21

1 ...........

1

asaasasassX

sY

nn

nn

0

1 :

asX

sYOrderZero

01

1 :

asasX

sYOrderFirst

012

2

1 :

asaassX

sYOrderSecond

Generalized Instrument System : A combination of Blocks

The response analysis can be carried out to each independent block.

A True Instrument

• A true instrument should consist of only Zero-Order Blocks

• To investigate the response of a block, multiply its frequency domain form of characteristic equation with that of the chosen input equation.

• This is an interesting case because Equation shows that the zero-order block has no frequency dependent term, so the output for all given inputs can only be of the same time form as the input.

• What can be changed is the amplitude given as the coefficient a0.

• A shift in time (phase shift) of the output waveform with the input also does not occur.

• This is the response often desired in instruments because it means that the block does not alter the dynamics of input signal.

• The final destination of research in instrumentation is to find a true zero order instrument for every measurement.

Description of Turbulent Flow : Lessons from Nature

A Hurricane : THE VORTEX

Experimental Research in Turbulence

The Structure of Vortex

The vortical structures visualized by iso-surfaces of vorticity and Laplacianof pressure.

The size of energy-containing (largest) eddies may beEstimated as:

2

2

21

.

uul eddyenergetic

where, ω = ×u is the vorticity.∇

Distribution of Vortices in a Turbulent Field

Iso-surface representations of vortical structures

|ω| p2

True Structure of Turbulent Flow

1/size of vortex

Non

-dim

ensi

onal

Ene

rgy

of e

ddy

Only zero order probes can measure true Turbulence !!!

Zero Order Instrument: Wire Strain Gauge

:

ll

RR

KFactorGauge

This is the response often desired in instruments because it means that the block does not alter the time response. All instruments behave as zero order instruments when they give a static output in response to a static input.

Wire Strain Gauge

Wire Strain Gauge Pressure Transducers

In comparison with other types of pressure transducers, the strain gage type pressure transducer is of higher accuraciy, higher stability and of higher responsibility.The strain gage type pressure transducers are widely used as the high accuracy force detection means in the hydraulic testing machines.

TypeFeatures and

ApplicationsCapacity

RangeNonlinearity(%

RO)

Rated Output(m

V/V)

Compensated Temp.Range

( )℃

HVSHigh Accuracy

type0.5,..50 MPa 0.2,0.3 1.0,1.5±1 % － 10 to 60

HVUGeneral Purpose

type1,..50 MPa 0.3,0.5 1.5,2.0±1 % － 10 to 60

HVJSSmall & High

Temperature type

1,..50 MPa 0.5 1.0,1.5±20 % － 10 to 150

HVJS-JS

Small & High Temperature type,Vibratio

n-proof

1,..10 MPa 0.5 1.0,1.5±20 % － 10 to 150

Micro Sensor Technology Tokyo

First Order Instruments

• A first order linear instrument has an output which is given by a non-homogeneous first order linear differential equation

)(01 txyadt

dya

01

asa

sXsY

01

1

asa

sXty

• In these instruments there is a time delay in their response to changes of input.

• The time constant is a measure of the time delay.

• Thermometers for measuring temperature are first-order instruments.

• The time constant of a measurement of temperature is determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.

• A cup anemometer for measuring wind speed is also a first order instrument.

• The time constant depends on the anemometer's moment of inertia.

Development of Characteristic Equation

for Liquid–in –Glass Thermometer

Liquid in Glass Thermometer

TVV bulbT 1

bulbbulbbulb lrV 2

materialvolume

 (10−6 K−1)alcohol, ethyl 1120gasoline 950jet fuel, kerosene 990mercury 181water, liquid (1 )℃ −50water, liquid (4 )℃ 0water, liquid (10 )℃ 88water, liquid (20 )℃ 207water, liquid (30 )℃ 303water, liquid (40 )℃ 385water, liquid (50 )℃ 457water, liquid (60 )℃ 522water, liquid (70 )℃ 582water, liquid (80 )℃ 640water, liquid (90 )℃ 695

Development of Characteristic Equation for a Thermometer

Conservation of Energy during a time dt

Heat in – heat out = Change in energy of thermometer

Assume no losses from the stem.

Heat in = Change in energy of thermometer

System theof eTemperatur ousInstantanetTs

tTs tTtf

fluid ric thermometof eTemperatur ousInstantanetTtf

Rs Rcond Rtf

Ts(t) Ttf(t)

tfconds RRRU

1

dtTTUAinheat tfsbulb

Change in energy of thermometer: tftfbulb dTCV

tftfbulbtfsbulb TCVdtTTUA

sbulbtfbulbtf

tfbulb TUATUAdt

dTCV

stftf

bulb

tfbulb TTdt

dT

UA

CV

bulb

tfbulb

UA

CV Define Time constant

stftf TT

dt

dT

s

Tsss s

tftf

s

Tss s

tf 1 1

ss

Ts s

tf

Step Response of Thermometers

1

s

T

s

Ts ss

tf

1

s

T

s

Ts ss

tf

1

11

ss

Ts stf

1

111

ss

TtT stf

t

TtT stf exp1

t

TtT stf exp1

bulb

tfbulb

UA

CV

Response of Thermometers: Periodic Loading

• If the input is a sine-wave, the output response is quite different;

• but again, it will be found that there is a general solution for all situations of this kind.

stftf

bulb

tfbulb TTdt

dT

UA

CV

22

max,max

s

stSinTT ssss

22

max,

ssss s

tftf

22

max,1

sss s

tf

122

max,

ss

s stf

122

max,1

sstT s

tf

t

tTe

TtT s

t

stf

1

22

max,

22max,

tan

sin11

Ts,

max

- T

tf,m

ax