Measurement Theory and Devices

Dr Sheila Smith (Module Leader)Dr Peter WallaceDr Geraint Bevan

What is a Measurement?

• A measurement tells us about the property of something.– E.g. how heavy an object is– Or how hot– Or how long

• Measurement gives a number to that property.• Measurements are made using an instrument of some

kind.– Rulers, stopwatch, thermometer, weighing scales are

all measuring instruments.• Result of a measurement is normally in two parts: a

number and a unit of measurement– E.g. ‘How long?’ ‘2 metres’

Measurement Principles• Measurement is the process of gathering information on

physical world variables.

• Instruments are the means of gathering information

• Instrumentation is the study of instruments and their use or alternately the technology of measurement

• The fundamental principles of measurement involve the ability to : perform a measurement, correctly handle the data gathered, make critical judgements on the results, present the results in a coherent and meaningful manner

SI Base UnitsMetre (m) (length)

Kilogram (kg) (mass)

Second (s) (time)

Ampere (A) (current)

Kelvin (K) (temperature)

Candela (cd) (luminous intensity)

Mole (mol) (amount of substance)

Check out the NPL website for definitions of the units: http://www.npl.co.uk/reference/measurement-units/

Derived Units

• Units which are combinations of the base units are known as derived units:

• E.g. Newton [N] is kg m s-2

Criteria of Measurement

• A measurement system is described as having

– an input, I, (i.e. the quantity being measured), and

– an output, O, (i.e. the result of the measurement).

Accuracy• Is the closeness with which the result of a

measurement approaches the true value of a variable.

– The true value is of course unknown in practice

• Usually quoted by a manufacturer in terms of a range of values centred on the maximum value of the measured quantity which the instrument is designed to measure.

– The maximum value of the measured quantity which the instrument is design to measure is often referred to as full scale deflection (f.s.d.)

Precision

• A measure of the reproducibility of the measurements (Repeatability)

High precision, high accuracy

High precision, low accuracy

Low precision, low accuracy

Low precision, high accuracy

Span

• is used to describe both the output and input of a system and is the difference between the maximum and minimum values.

• Input Span: Imax – Imin

• Output Span: Omax - Omin

Resolution

• quantifies the 'fineness' of a measurement and is the smallest variation in the quantity of interest that can be measured. Resolution is a function of the complete measurement system.

• E.g. The resolution of the melting point meter is 1oC

Limit of Detection

• Smallest amount that can be measured on any given instrument

– E.g. Smallest concentration of alcohol that can be measured on a GC.

Linearity

Measured quanity

Ou

tpu

t re

adin

g

• The triangles denote the data points.

• The straight line gives the best fit through the data points and gives the sensitivity of the instrument.

Instrument input / output characteristic

 

Sensitivity

• Rate of change of output with respect to input

Inp ut

Out

put

O

Se nsitivity = O / I

Linearity

 

N(I) represents the non-linear function

Hysteresis

• Range– Specified in terms of maximum and minimum

values of input/output

• Repeatability– Closeness of agreement of a group of output

values for a constant input

• Dead Zone– Largest input change to which transducer fails

to respond

• Drift– Unidirectional variation in transducer output which

is associated with a change in input

• Zero Stability– ability of the transducer to restore its output to

zero when its input returns to zero.

• Monotonicity – a transducer which is subjected to a continuously

increasing input signal its output signal should neither decrease nor skip a value

An Instrumentation SystemMeasurand

Primary sensingelement Transducer

Signalconditioning

Signalprocessing

Signaltransmission

Datapresentation

Recorder Display Processcontrol

Calibration

• A measurement system must be calibrated before it can be used to measure "unknown" values of a measurand.

• Calibration is the process of the determination of the characteristics of a system by measurement of the output for a variety of known input values.

Inputs to a Measurement System

• Wanted input (KI)

• Interfering Input (KI II)

• Modifying Input (KM IMI)

I/P

O/P

I/P

O/P

I/P

O/P

Constant interfering input

Modifying input

Modifying and interfering input

zerodrift

sensitivity drift

Nominal characteristic

Generalised Model of a Measurement System

O = KI + a + N(I) + KMIMI + KIII

Static Characteristic given by:

KM

K I IM M

K IM I

KI

X

K

N( )

Modifying IM

G(s)I

InputKI

N(I)

II Interfering

a

+++ +

+ OOutput

The Measurement Process

Direct Measurement: Mr =0; Result = Difference Output

Null Measurement: Mr = Mu; Result = Reference Output

D iffe re n c e O u tp u t

R e fe r e n c e O u tp u t

M e a s u r a n d S c a la rD if fe re n c e D e te c to r

R e fe r e n c eS ta n d a r d

M - M M

MM

u

u

r r

r

Direct Measurement

V

RE

Meter Loading

• The resistance of the meter and the 1kW resistor is given by:

 

 

• The total resistance of the circuit is therefore 1909.09W.

1

R

1

1000

1

10,000

R10,000

11909.09

• The current, I, is given by:

• VAB = I x RAB = 0.0052 x 909.09 = 4.76V

• The meter therefore reads 4.76V instead of 5V. • The measured output voltage is in error due to the

method of measurement used - loading. • The percentage loading error is defined as:

• In the above example % loading error is given by

IV

R

10

1909.09.0052A 0

%loadingV V

Vx100%ideal meas

ideal

5 0 4 76

5 0100

4 8

. .

.%

. %

x% loading

Null Measurement

REC a lib ra te dvo lta g eso urc e

Nulld e te c to r

Dynamic System Performance

Measurement System

or Mathematical

Operator

I(t)

O(t)

O(t)I(t)

time

sign

al

Dynamic System Performance

• Input can be simplified to box, the order of the system is then determined by n.

00

m

m

mn

n

n dt

xd

dt

ydmn ba

Output Input

• For n = 0, zero order systema0 y = b0 x

No energy is lost, stored or otherwise extracted

• For n = 1, 1st order systema1 dy/dt + a0 y = b0 x

• For n = 2, 2nd order systema2 d2y/dt2 + a1 dy/dt + a0 y = b0 x.

Step change in temperature

• Newton’s Law of Cooling: W = UA(TF – T),

– W is the rate of inflow

– (TF – T) is temperature difference

– U is the overall heat transfer co-efficient in

W m-2 0C-1

– and A is the effective area in m2.

• The heat content is given by

mcDT = mc{T – T(0-)}

– m is the mass of the sensor;– c is the specific heat capacity.

• We can now state that the rate of increase of sensor heat content is mc d{T – T(0-)} /dt.

• Now if DT = T – T(0-) and DTF = TF – TF (0-) then UA(DTF- DT) =mc dDT/dt

• By re-arranging this we get a first order DE

(mc/UA) dDT/dt + DT = DTF

• Here (mc/UA) is the time constant, t, of the system thus

t dDT/dt + DT = DTF

• t is a function of the physical properties of the system and is the factor that determines the speed of response of the system.

• It is the time that it takes for the system to move 63.2% of the way from the initial to the new steady state value.

• Use Laplace transform for this first order case to obtain the transfer function [1/ (1+ts)].