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)].
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