Dynamic Characteristics of

Instruments

P M V SubbaraoProfessor

Mechanical Engineering Department

Capability to carry out Transient Measurements….

Cyclic Input: Hysteresis and Backlash

• Careful observation of the output/input relationship of an instrument will sometimes reveal different results as the signals vary in direction of the movement.

• Mechanical systems will often show a small difference in length as the direction of the applied force is reversed.

• The same effect arises as a magnetic field is reversed in a magnetic material.

• This characteristic is called hysteresis• Where this is caused by a mechanism that gives a sharp

change, such as caused by the looseness of a joint in a mechanical joint, it is easy to detect and is known as backlash.

Hysteresis Loop

Dynamic Characteristics of Instrument Systems

• To properly appreciate instrumentation design and its use,• it is necessary to develop insight into the most commonly

encountered types of dynamic loading &• to develop the mathematical modeling basis that allows us

to make concise statements about responses.• The response at the output of an instrument Gresult is

obtained by multiplying the mathematical expression for the input signal Ginput by the transfer function of the instrument under investigation Gresponse

responseinputresult GGG

Standard Forcing Functions

Standard Forcing Functions

Standard Forcing Functions

Characteristic Equation Development

• The behavior of a block that exhibits linear behavior is mathematically represented in the general form of expression given as

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

2

2 txyadtdya

dtyda

Here, the coefficients a2, a1, and a0 are constants dependent on the particular instrument of interest. The left hand side of the equation is known as the characteristic equation. It is specific to the internal properties of the block and is not altered by the way the insturment is used.

• The specific combination of forcing function input and instrument characteristic equation collectively decides the combined output response.

• Solution of the combined behavior is obtained using Laplace transform methods to obtain the output responses in the time or the complex frequency domain.

Behaviour of the Instrument

)(0 txya

)(01 txyadtdya

)(012

2

2 txyadtdya

dtyda

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

2

21

1

1 txyadtdya

dtyda

dtyda

dtyda n

n

nn

n

n

Zero order

First order

Second order

nth order

Behaviour of the Block

• Note that specific names have been given to each order. • The zero-order situation is not usually dealt because it has no

time-dependent term and is thus seen to be trivial. • It is an amplifier (or attenuator) of the forcing function with

gain of a0. • It has infinite bandwidth without change in the amplification

constant.• The highest order usually necessary to consider in first-cut

instrument analysis is the second-order class. • Higher-order systems do occur in.• Computer-aided tools for systems analysis are used to study the

responses of higher order systems.

Solution of ODE

• Define D operator as

dtdyD

The nth order system model:

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

21

1 txyaDyayDayDayDa nn

nn

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

21

1 txyaDaDaDaDa nn

nn

The solution of equations of this type has been put on a systematic basis by using either the classical method of D operators orLaplace Transforms method.

Laplace Transforms: Solution of ODE

• The Laplace transform, is an elegant way for fast and schematic solving of linear differential equations with constant coefficients.

• Instead of solving the differential equation with the initial conditions directly in the original domain, the detour via a mapping into the frequency domain is taken, where only an algebraic equation has to be solved.

• Thus solving differential equations is performed in the following three steps:

• Transformation of the differential equation into the mapped space,

• Solving the algebraic equation in the mapped space, • Back transformation of the solution into the original space.

Schema for solving differential equations using the Laplace transformation

                                                   

Laplace Transforms; nth Order Equation

The nth order system model:

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

21

1 txyaDyayDayDayDa nn

nn

Laplace Transform:

sX

sYassYasYassYassYas nn

nn

........... 012

21

1

sX

sYasaasasas nn

nn

........... 012

21

1

012

21

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

asaasasassXsY

nn

nn

0122

11

1

...........

asaasasassXty

nn

nn

Laplace Transformations for Sensors

sXsYasaasasas nn

nn

012

21

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

012

21

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

asaasasassXsY

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.

Response of the Different Blocks

• Zero-Order Blocks• To investigate the response of a block, multiply its frequency

domain forms of equation for the characteristic equation with that of the chosen forcing function 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 time response. • However, this is not always so because, in systems, design

blocks are often chosen for their ability to change the time shape of signals in a known manner.

Zero Order Instrument: Wire Strain Gauge

: l

lR

RKFactorGauge

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

Strain Gauge

• A strain gauge's conductors are very thin: • if made of round wire, about 1/1000 inch in diameter. • Alternatively, strain gauge conductors may be thin strips of

metallic film deposited on a nonconducting substrate material called the carrier.

• The name "bonded gauge" is given to strain gauges that are glued to a larger structure under stress (called the test specimen).

• The task of bonding strain gauges to test specimens may appear to be very simple, but it is not.

• "Gauging" is a craft in its own right, absolutely essential for obtaining accurate, stable strain measurements.

• It is also possible to use an unmounted gauge wire stretched between two mechanical points to measure tension, but this technique has its limitations.

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.

Type Features and Applications

Capacity Range

Nonlinearity(%RO)

Rated Output(m

V/V)

Compensated Temp.Range

( )℃

HVS High Accuracy type 0.5,..50 MPa 0.2,0.3 1.0,1.5±1 % － 10 to 60

HVU General Purpose type 1,..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 txyadtdya

01

asa

sXsY

01

1

asasXty

• 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.

First‐order instrument step response

b0

0)( 0 tbtx

sbsX 0)(

01

0

01

asas

basa

sXsY

The complex function F(s) must be decomposed into partial fractions in order to use the tables of correspondences. This gives

01

1

0

0

0

0 1 asa

aab

sabsY

1

00

0

0

0 11

aas

ab

sabsY

1

00

0 11

aas

sabsY

1

00

01 11

aas

sabty

1

0

0

0 exp1 a

taabty

constant time:0

1 aa

Factor Gauge:0

0 Kab

tKty exp1

Kty

Dynamic Response of Liquid–in –Glass Thermometer

Liquid in Glass Thermometer

TVV bulbT 1

bulbbulbbulb lrV 2

material volume (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

Thermometer: A First Order Instrument

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

dTUA

CV

bulb

tfbulb

UACV

Time constant

stftf TT

dtdT

s

Tsss stftf

s

Tss stf 1

1

ssTs s

tf

Step Response of Thermometers

1

sT

sTs ss

tf

1s

Ts

Ts sstf

1

11

ssTs stf

1

111

ssTtT stf

tTtT stf exp1

tTtT stf exp1

bulb

tfbulb

UACV

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

dTUA

CV

22max,

max

sstSinTT s

sss

22max,

ssss s

tftf

22max,1

sss s

tf

122max,

ss

s stf

122max,1

sstT s

tf

t

tTeTtT s

t

stf

1

22

max,22max,

tan

sin11

T s,m

ax- T

tf,m

ax