Measurements Devices Concepts of Measurements

ANSWER THE FOLLOWING (USING NEAT SKETCHES AS POSSIBLE):

1) What is measurement and what is meant by measurand?

Measurement is the act, or the result, of a quantitative comparison between a give quantity and
a quantity of the same kind chosen as a unit. The result of the measurement is expressed by a
pointer deflection over a predefined scale or a number representing the ratio between the
unknown quantity and the standard.

Measurand: The physical quantity or the characteristic condition which is the object of
measurement in an instrumentation system is variously termed as "measurand", "measurement
variable”, “instrumentation variable” or “process variable”.

2) What is meant by instrumentation?

The technology of using instruments to measure and control the physical and chemical
properties of materials is called "instrumentation."
When the instruments are used for the measurement and control of industrial manufacturing,
conversion or treatment process, the term “process instrumentation“ is used.
When the measuring and controlling instruments are combined so that measurements provide
impulses for remote automatic action, the result is called a control system.

3) Compare between primary, secondary and working standards of measurements.

Primary standards: The highest standard of either a base unit or a derived unit is called a
primary standard. These standards essentially are copies of international prototypes and are kept
throughout the world in national standard laboratories of similar standing. These standardized
units have: quite stability, no independence on environmental conditions, rigidity and accuracy
of machining. The primary standards constitute the ultimate basis of reference and are used for
the purpose of “verification and calibration of secondary standards".

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Secondary standards: are the reference calibrated standards designed and calibrated from the
primary standards. These are sent periodically to the national standard laboratories for their
calibration. These standards are kept by the measurement laboratories and the industrial
organizations to check and calibrate the general tools for their accuracy and precision.
Working standards: These standards have an accuracy of one order lower than that of the
secondary standards. These are the normal standards which are used by the workers and
technicians who actually carry out the measurements.

4) What are the main elements of generalized measurement system, briefly describe their function?

Primary sensing element: it is an element that is sensitive to the measured variable. The
sensing elements sense the condition, state or value of the process variable by extracting a small
part of energy from the measurand, and then produces an output which reflects this condition,
state or value of the measurand.

Variable conversion of transducer element: This element converts the signal from one
physical form to another without changing the information content of the signal.

Manipulation element: This element operates on the signal according to some mathematical
rule without changing the physical nature of the variable.

Data transmission element: This element transmits the signal from one location to another
without changing its information contents.

Data presentation element: This element provides a display record or indication of the output
from the manipulation elements.

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5) Briefly describe the stages of general measurement system.

Stage 1: It senses desired input to exclusion of all others and provides analogous output.

Types and Examples: Mechanical (spring-mass, elastic devices, gyro) - Electrical: (Contacts,
resistance, capacitance, inductance, piezoelectric, thermocouple, semiconductor junction) -
Optical: (Photographic film, photoelectric diodes and transistors, photomultiplier tubes,
holographic plates) - Hydro - pneumatic: (Buoyant float, orifice, venturi, vane, propeller).

Stage 2: it modifies transduced signal into form usable by final stage and usually increases
amplitude and/or power depending on requirement. It may also selectively filter unwanted
component or convert signal into pulsed form.

Types and Examples: Mechanical (Gearing, cranks, slides, connecting links, cams) - Electrical:
(Amplifying or attenuating systems, bridges, filters, telemetering systems) - Optical: (Mirrors,
lenses, optical filters, optical fibers) - Hydro - pneumatic: (Piping, valving, dashpots).

Stage 3: It provides an indication or recording in form that can be evaluated by an unaided

human sense or by a controller and records data digitally on a computer.

Types and Examples: Indicators Displace types: (Moving pointer and scale, moving scale and
index, light beam and scale) - Recorder: (Digital printing, inked pen and chart, direct
photography, magnetic recording) - Processor and computers: (special purpose or general, used
to feed read out/recording devices and /or controlling systems).

6) How can the measurement instruments be classified?

The instruments may be classified as follows:

Absolute and secondary instruments.

Analog and digital instruments.

Mechanical, electrical and electronic instruments.

Manual and automatic instruments.

Self-contained and remote indicating instruments.

Self-operated and power-operated instruments.

Deflection and null output instruments.

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7) What are the main functions of instruments and give examples of their applications?

The main functions of instruments:

Indicating function: The value of the quantity can be read by the movement of the needle on
a calibrated scale provided on the instrument. Examples: The deflection of a pointer of a
‘speedometer' indicates the speed of the automobile at the moment.

Recording function: In several cases the instrument continuously record, with pen and ink, the
value of measured quantity against some other variable or against time. Example: A
’potentiometric type of recorder' used for monitoring temperature records.

Controlling function: Here, the information is used by the instrument system to control the
measured quantity. Examples: Floats for liquid level control.

The applications of instruments:

Monitoring of processes and operations:

In these types of applications, the measuring instruments simply indicate the value or condition
of parameter under study and do not serve any control function. Examples: Water and electric
energy meters installed in homes - An ammeter or a voltmeter indicates the value of current or
voltage.

Control of processes and operations:

There is a strong association between measurements and control. The instruments find a very
useful application in automatic control systems. A common example is the typical refrigeration
system which employs a thermostatic control.

Experimental engineering analysis:

The engineering problems can be solved by theoretical as well as experimental methods; several
applications require the use of both the methods.

Following are the uses of experimental engineering analysis: To determine system parameters,
variable and performance indices - To formulate the generalized empirical relationship in cases
where there is no proper theoretical backing - To test the validity of theoretical predictions - To
solve mathematical relationships with help of analogies.

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8) What is meant by static characteristics of instruments?

Measurement of applications in which parameter of interest is more or less constant; or varies

very slowly with time are called static measurements. A set of criteria (accuracy, drift,
reproducibility, sensitivity … etc.) that provide meaningful description of measurements under
static conditions are called static characteristics.

9) Briefly define and explain each of these terms:

a) Accuracy: is the degree of correctness with which a measuring means yields the "true value"
with reference to accepted engineering standards. Accuracy is determined as the maximum
amount by which the result differs from the true value . It is almost impossible to determine
experimentally the true value.
b) Uncertainty: is the extent to which a reading might be wrong. and is often quoted as a
percentage of the full-scale reading of an instrument.
c) Precision: is a term that describes an instrument’s degree of freedom from random errors. If
a large number of readings are taken of the same quantity by a high precision instrument, then
the spread of readings will be very small.

d) Repeatability: describes the closeness of output readings when the same input is applied
repetitively over a short period of time, with the same measurement conditions, same instrument
and observer, same location and same conditions of use maintained throughout.
e) Reproducibility: describes the closeness of output readings for the same input when there
are changes in the method of measurement, observer, measuring instrument, location,
conditions of use and time of measurement.
f ) Noise: It does not convey any useful information. It is an unwanted signal superimposed
upon the signal of interest thereby causing a deviation of the output from its expected value.
Noise can be classified as: Generated noise – conducted noise – radiated noise.
g) Dead zone: The largest change of input quantity for which there is no output of the instrument
is termed as dead zone. It may occur due to friction in the instrument which does not allow
pointer to move till sufficient driving force is developed to overcome the friction loss.
h) Dead time: the time required by a measurement system to respond to change in measurand.

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i) Tolerance: is a term that is closely related to accuracy and defines the maximum error that is
to be expected in some value.
j) Span or Range: the minimum and maximum values of a quantity that the instrument is
designed to measure, Span represents the algebraic difference between the upper and lower
range values of the instrument.
k) Linearity: It is normally desirable that the output reading of an instrument is linearly
proportional to the quantity being measured, the linearity is simply a measure of maximum
deviation of any of the calibration points from the straight line (drawn by using the method of
least squares). Linearity is considered to be one of the best characteristics of an instrument
because the conversion from a scale reading to the corresponding value of input quantity is very
convenient.

l) Sensitivity: The ratio of the magnitude of output signal to the input signal or response of
measuring system to the measured quantity is called sensitivity, it is represented by the slope of
the calibration curve if the ordinates are expressed in actual units, sensitivity may be linear or
nonlinear.

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m) Threshold: The minimum value below which no output change can be detected when the input
of an instrument is increased gradually from zero is called the threshold of the instrument. Thus,
threshold defines minimum value of input which is necessary to cause a detectable change from
zero output.

n) Sensitivity to environmental changes: is a measure of the magnitude of the change the ambient
conditions like (temperature, pressure…etc.), such environmental changes affect instrument in two
ways, known as zero drift, sensitivity drift.

o) Resolution: defines the smallest change of input for which there will be a change of output, in
case of analog instruments by the observer’s ability to judge the position of a pointer on a scale, in
digital instruments it can be represented by the smallest step of counter.

Hysteresis: it is a phenomenon which shows different output effects when loading and unloading.
It is non-coincidence of loading and unloading curves. Hysteresis results from these factors (Slack
motion in bearings – Mechanical friction – Magnetic and thermal effect).

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10) What is meant by zero drift and sensitivity drift?

Drift is an undesired gradual departure of the instrument output over a period of time that is
unrelated to changes in input, operating conditions or load. The drift may be caused by the
following factors: (High mechanical stresses - Wear & tear - Mechanical vibrations -
Temperature changes - Stray electric and magnetic fields).

Zero drift: a constant error that exists over the full range of measurement of the instrument
caused by a change in ambient condition.

Sensitivity drift (Span drift): Defines the amount by which an instrument’s sensitivity of
measurement varies as ambient conditions change.

Zonal drift: When the drift occurs only over a portion of span of an instrument.

11) What is static calibration and how can it be done?

Static calibration: is a process by which all the static performance characteristics are obtained
in one form or another .

ln general, static calibration refers to a situation in which all inputs, whether desirable,
interfering or modifying except one are kept at some constant values. Then the one input under
study is varied over some range of constant values, which causes the output (s) to vary over
some range of constant values. Thus, an output-input relationship is developed which comprise
a static calibration valid under the stated constant conditions of all the other inputs,

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The following steps are necessary in performing a calibration:

• Examine the construction of the instrument, and identify and list all the possible inputs.
• Decide as best as possible which of the inputs will be significant in the application for
which instrument is to be calibrated.
• Procure apparatus that will allow to vary all the significant inputs over the ranges
considered necessary. Procure standards to measure each input.
• By holding some inputs constant, varying others, and recording the output(s), develop
the desired static input-output relations.

12) What are the types of errors during Measurement process?

Errors and uncertainties are inherent in the process of making any measurement and in the
instrument with which the measurements are made. The study of error is important as a step in
finding ways of reducing them, and also as means of estimating the reliability of final results.

Errors arising during the measurement process can be divided into three groups, known as gross
errors, systematic errors and random errors.

13) Mention the sources of Gross errors and how to reduce these errors?

These errors occur due to human mistakes in reading instruments and recording and calculating
results of measurement. Although it is probably impossible to eliminate the gross error
completely, yet one should try to anticipate and correct them.

The mathematical analysis of gross errors is impossible since these may occur in different
amounts. While some gross errors may be easily detected, others may go unnoticed.

These errors can be avoided by:

• Immense care should be taken while taking the reading and recording the data.
• Two, three, or even more readings should be taken for the quantity being measured.

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14) Mention the sources of systematic errors and how to reduce these errors?

The systematic errors are repeated consistently with the repetition of the experiment; these
errors can be located only by having repeated measurements under different conditions or with
different equipment and where possible by an entirely different method.

Sources of systematic errors:

Instrumental errors :

• Mechanical friction and wear, backlash and hysteresis of elastic members.

• Constructional faults resulting from excessive friction at the mating parts, lost motion.
• Improper selection and poor maintenance of the instrument.
• Assembly errors (Displaced scale - Bent or distorted pointer)
• Misuse of instruments .

Environmental errors: The environmental errors are due to conditions external to the
measuring device (effects of pressure, temperature, humidity, dust, vibrations, external
magnetic or electrostatic fields).

Observational errors: These errors occur due to carelessness of operators. Even when an
instrument has been properly selected, examples (Parallax - Wrong scale reading and wrong
recording of data - inaccurate estimates of average reading - Incorrect conversion of units in
between consecutive readings - Personal bias)

Methods of reducing systematic errors:

• Careful instrument design.

• Method of opposing inputs.
• High-gain feedback (controlled conditions).
• Recalibration for new readings.
• Manual correction of output reading.
• Intelligent instruments.

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15) Mention the sources of random errors and how to reduce these errors?

The random errors are accidental, small and independent, they vary in an unpredictable manner,
the magnitude and direction of these errors cannot be predicted from a knowledge of
measurement system; however, these errors are assumed to follow the law of probabilities.

Sources of random errors:

- Environmental conditions. - Friction in instrument movement.

- Backlash in the movement. - Parallax errors (human observations).

- Hysteresis in elastic members. - Mechanical vibrations.

- Noise (electrical – magnetic fields – induction).

Methods of reducing random errors:

• Statistical analysis (Averaging methods, Normal and Gaussian curves, Least square
methods, Probability tables).
• Careful instrument design.
• Using instruments in controlled conditions.

16) Define each of these terms; Mean value, Median value, standard deviation, Standard
error of the mean, Rouge data points.

Mean value: the average value of a set of measurements, the mean is computed by summing
all the values and dividing by the number of measurements.

Median Value: The median is the middle value when the measurements in the data set are
written down in ascending order of magnitude.

Standard deviation: A measure of the dispersion of a set of data from its mean. The more
spread apart the data, the higher the deviation, it can be defined as is defined as the square root
of the sum of individual deviations squared divided by the number of readings, and as the square
root of variance.

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Standard error from the mean: The error between the mean of a finite data set and the true
measurement value (mean of the infinite data set) is defined as the standard error of the mean,
this is calculated as:

Rouge data points: very large errors that occur at random and unpredictable times, where the
magnitude of the error is much larger than expected random variations, in practice these points
are usually discarded.

SOLVE THE FOLLOWING PROBLEMS (TABULATE THE FINAL ANSWERS):

1) A thermocouple has an output e.m.f as shown in the following table when its hot (measuring)
junction is at the temperatures shown. Determine the sensitivity of measurement for the
thermocouple in mV/°C.

mV 5.42 9.75 14.08 18.41

o
C 200 450 700 950

Required: sensitivity of measurement.

Solution:

Input temperature Vs output voltage

1000

900
18.41, 950
800

700
14.08, 700
600
emf mV

500

400 9.75, 450

300

200
5.42, 200
100

0
0 2 4 6 8 10 12 14 16 18 20

Temperature oC

𝛥𝑒.𝑚.𝑓 9.75−5.42
Slope = = = 0.01732
𝛥𝑇𝑒𝑚𝑝. 450−200

Sensitivity = 0.01732 mV/◦C

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2) A load cell is calibrated in an environment at a temperature of 23°C and has the following
deflection/load characteristic:
Given: the following data table.

23

47

Required:
(a) Sensitivity at 23°C and 47°C.
(b) Total zero drift and sensitivity drift at 47°C.
(c) Zero drift and sensitivity drift coefficients.
Solution:

Input Load Vs output deflection at 23, 47 oC

5
200, 4.6
4.5
4
Deflection mm

150, 3.5 200, 4

3.5
3
150, 3
2.5 100, 2.4

2
50, 1.3 100, 2
1.5
1 Series1 Series2
0, 0.2
50, 1
0.5
0
0 0, 0 50 100 150 200 250
Load Kg
𝛥𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 1.0−0.0
Slope (23◦C) =
𝛥𝑙𝑜𝑎𝑑
=
50−0
= 0.02 mm/kg
𝛥𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 1.3−0.2
Slope (47◦C) =
𝛥𝑙𝑜𝑎𝑑
=
50−0
= 0.022 mm/kg
a) Sensitivity (23◦C) = 0.02 mm/kg, Sensitivity (47◦C) = 0.022 mm/kg.
b) Zero drift 47°C = 0.2 mm
Sensitivity drift at 47°C = Sensitivity (47◦C) - Sensitivity (23◦C) = 0.002 mm/kg
Zero drift 47°C 0.2
c) Zero drift Coefficient = = × 103= 8.33 µm/°C
𝛥𝑇𝑒𝑚𝑝. 47−23
Sensitivity drift 47°C 0.002
Sensitivity drift coefficient = = × 103 = 0.083 µm per kg/°C
𝛥𝑇𝑒𝑚𝑝. 47−23

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3) In a survey of 15 owners of a certain model of car, the following figures for average petrol
consumption were reported.

Calculate the mean value, the median value and the standard deviation of the data set.

Given: the following set of data.

Required:
(a) Mean value.
(b) Median value.
(c) Standard deviation.
Solution:

465.80
a) Xmean =
15
= 31.05
b) Median value = 30.5

25.50 28.90 29.20 31.57 33.42 30.00 30.30 30.50 31.10 31.40 31.70 32.40 33.00 33.30 39.40

c) Standard deviation:
d = X - Xmean, d1 = 25.5 – 31.05 = - 5.55 and so on
owner 1 owner 2 owner 3 owner 4 owner 5 owner 6 owner 7 owner 8
X 25.50 30.30 31.10 29.60 32.40 39.40 28.90 30.00
d -5.55 -0.75 0.05 -1.45 1.35 8.35 -2.15 -1.05
d2 30.84 0.57 0.0025 2.11 1.81 69.67 4.64 1.11

owner 9 owner 10 owner 11 owner 12 owner 13 owner 14 owner 15 Sum

33.30 31.40 29.50 30.50 31.70 33.00 29.20 465.80
2.25 0.35 -1.55 -0.55 0.65 1.95 -1.85
5.05 0.12 2.41 0.31 0.42 3.79 3.43 126.28

126.28
𝝈=√ = 3.003
14

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4) The following measurements were taken with an analogue meter of the current flowing in a
circuit (the circuit was in steady state and therefore, although the measurements varied due to
random errors, the current flowing was actually constant):

Calculate the mean value, the deviations from the mean and the standard deviation.

Given: the following set of data.

Required:
(a) Mean value.
(b) Deviations from the mean.
(c) Standard deviation.
Solution:

218
a) Xmean = = 21.8 mA
10
b) Deviations from the mean:
d = X - Xmean, d1 = 21.5 – 21.8 = - 0.3 and so on

mA M1 M2 M3 M4 M5 M6 M7 M8 M9 M 10 Sum
X 21.5 22.1 21.3 21.7 22 22.2 21.8 21.4 21.9 22.1 218
d -0.3 0.3 -0.5 -0.1 0.2 0.4 0 -0.4 0.1 0.3
d2 0.09 0.09 0.25 0.01 0.04 0.16 0 0.16 0.01 0.09 0.9

c) Standard deviation:

0.9
𝝈 = √ 9 = 0.316 mA

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5) A pressure indicator showed a reading as 42 bar on a scale range of 0-50 bar. If the true value
was 41.4 bar, and the accuracy is specified with ± 2 percent of span, then determine:
a) Static error, b) Static correction. c) Relative static error. d) Maximum static error

Given: Pm = 42 bar, PT = 41.4 bar, Range (0 – 50) bar, accuracy = ±2 % of span.

Required:
(a) Static error. (b) Static correction. (c) Relative static error. (d) Maximum static error.
Solution:
a) Es = 𝑃𝑚 − 𝑃𝑇 = 42 − 41.4 = 0.6 bar.
b) Cs = −𝐸𝑠 = - 0.6 bar.
𝐸𝑠 0.6
c) ER = = × 100% = 1.45 %.
𝑃𝑇 41.4

c) Emax = ±0.02 × 50 = ± 1 bar.

6) Heat transfer from a rod of diameter D immersed in a fluid can be described by the Nusselt
number, Nu = hD/k, where h is the heat-transfer coefficient and k is the thermal conductivity of
the fluid. If h can be measured to within ±7% (95%), estimate the uncertainty in Nu for the
nominal value of h = 150 W/m2-K. Let D = 20 ± 0.5 mm and k = 0.6 ± 2% W/m-K.

Given: h = 150 ± 7% W/m2-K, D = 20 ± 0.5 mm, k = 0.6 ± 2% W/m-K.

Required:
- The nominal value and its uncertainty of Nu number.
Solution:
- The total uncertainty can be calculated as:
2 2 2
𝜕𝑁𝑢 𝜕𝑁𝑢 𝜕𝑁𝑢
𝑈𝑁𝑢 = ±√( 𝑈 ) +( 𝑈 ) +( 𝑈 )
𝜕ℎ ℎ 𝜕𝐷 𝐷 𝜕𝑘 𝑘

2 2 2
𝐷 ℎ −ℎ𝐷
𝑈𝑁𝑢 √
= ± ( 𝑈ℎ ) + ( 𝑈𝐷 ) + ( 2 𝑈𝑘 )
𝑘 𝑘 𝑘
2 2 2
0.02 150 −150 × 0.02
𝑈𝑁𝑢 = ±√ ( (0.07 × 150)) + ( 0.0005) + ( (0.02 × 0.6)) = ± 0.4
0.6 0.6 (0.6)2
- The nominal value of Nu can be calculated as:
ℎ𝐷 150 × 0.02
𝑁𝑢 = = =5
𝐾 0.6
𝑵𝒖 = 𝟓 ± 𝟎. 𝟒 (𝟗𝟓%)

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7) Grashof Number is a non-dimensional parameter that occurs in free convection problems. It is

given by Gr = (gβΔTL)/ν2, where g is the acceleration due to gravity, β is the isobaric
compressibility of the medium, ΔT is a characteristic temperature difference in the problem, L
is a characteristic length dimension and ν is the kinematic viscosity of the medium. In a certain
−1
situation, the following data is given: g = 9.81 m/s2, β = 0.00065K , ΔT = 298 ± 0.5K, L =
0.05 ± 0.001m, ν = (14.1 ± 0.3)10−6 m/s. Determine the nominal value of Gr and its uncertainty.

Given: g = 9.81 m/s2, β = 0.00065 K−1, ΔT = 298 ± 0.5 K, L = 0.05 ± 0.001 m,

ν = (14.1 ± 0.3)10−6 m2/s.

Required:
- The nominal value and its uncertainty of Gr number.
Solution:
- The total uncertainty can be calculated as:
2 2 2
𝜕𝐺𝑟 𝜕𝐺𝑟 𝜕𝐺𝑟
𝑈𝐺𝑟 = ±√( 𝑈 ) + ( 𝑈 ) + ( 𝑈 )
𝜕ΔT ΔT 𝜕L L 𝜕ν ν

2 2 2
gβL gβΔT gβΔTL
𝑈𝐺𝑟 = ±√( 2 𝑈ΔT ) + ( 2 𝑈L ) + (−2 𝑈ν )
𝑣 𝑣 𝑣3
= ±9.8
2 2 2
0.05 298 298 × 0.05
× 0.00065√( 0.5) + ( 0.001) + (−2 0.3 × 10 −6 )
(14.1 × 10−6 )2 (14.1 × 10−6 )2 (14.1 × 10−6 )3

𝑈𝐺𝑟 = ±𝟐𝟎. 𝟑𝟓 × 𝟏𝟎𝟔

- The nominal value of Nu can be calculated as:

𝑔𝛽𝛥𝑇𝐿 9.8 × 0.00065 × 298 × 0.05

𝐺𝑟 = = 2 = 477.4 × 106
𝜈2 −6
(14.1 × 10 )

𝑮𝒓 = (𝟒𝟕𝟕. 𝟒 ± 𝟐𝟎. 𝟑𝟓) × 𝟏𝟎𝟔

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8) A displacement transducer has the following specifications:

The transducer output is to be indicated on a voltmeter having a stated accuracy of ±0.1%

reading with a resolution of 10 mV. The system is to be used at room temperature, which can
vary by ±10oC. Estimate an uncertainty in a nominal displacement of 2 cm at the design stage.
Assume 95% confidence.

Given: Voltmeter of accuracy of ±0.1% reading with a resolution of 10 mV, following transducer
data, ΔT = 10 oC.

Required:
- The nominal value and its uncertainty of Gr number.
Solution:
- The total uncertainty in design stage can be calculated from:

2 2
𝑈𝑑 = ±√(𝑈𝑑𝑇 ) + (𝑈𝑑𝑉 )

- The transducer uncertainty can be calculated as:

𝑈𝑑𝑇 = ±√(𝑈0 )2 + (𝑈𝑐 )2 = √(0.0025 × 2)2 + (0.0005 × 10 × 2)2 + (0.0025 × 2)2

= 0.0122 𝑉
- The voltmeter uncertainty can be calculated as:

𝑈𝑑𝑇 = ±√(𝑈0 )2 + (𝑈𝑐 )2 = √(5 × 10−6 )2 + (0.001 × 2)2 = 0.002 𝑉

- The total uncertainty in design stage can be calculated as:

𝑈𝑑 = ±√(0.0122)2 + (0.002)2 = 0.01236 𝑉

- The uncertainty in in a nominal displacement of 2 cm:
𝑼𝒅 = 𝟎. 𝟎𝟏𝟐𝟑𝟔 𝒄𝒎 𝑎𝑠 𝑡ℎ𝑒 𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = 1 𝑉/𝑐𝑚

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9) A circuit is satisfied by connecting together two resistors of nominal values 250 ohm and 370
ohm. If the first and second resistors have a tolerance of ±2% and ±3% respectively, then
express the total resistance and maximum error of the circuit if the resistors are connected in
series or parallel.

Given: 𝑅1 = 250 ± 2% . 𝑅2 = 370 ± 3%.

Required:
(a) Total resistance in series. (b) Total resistance in parallel.
Solution:
- For series connection of the circuit:
𝑅𝑇 = 𝑅1 + 𝑅2 = (250 + 370) = 620 𝑜ℎ𝑚
The uncertainty of total resistance:

2 2
𝜕 𝑅𝑇 𝜕 𝑅𝑇
𝑈𝑅𝑇 = ±√( 𝑈𝑅1 ) + ( 𝑅2 ) = ±√(1 × 250 × 0.02)2 + (1 × 370 × 0.03)2
𝜕 𝑅1 𝜕 𝑅2

𝑹𝑻 = 𝟔𝟐𝟎 ± 𝟏𝟐. 𝟏𝟕 𝑜ℎ𝑚

- For Parallel connection of the circuit:
𝑅1 𝑅2 250×370
𝑅𝑇 = = = 𝟏𝟒𝟗. 𝟏𝟗 𝒐𝒉𝒎
𝑅1 +𝑅2 250+370
The uncertainty of total resistance:

2 2
𝜕 𝑅𝑇 𝜕 𝑅𝑇
𝑈𝑅𝑇 √
=± ( 𝑈 ) +( 𝑅 )
𝜕 𝑅 1 𝑅1 𝜕 𝑅2 2

𝜕𝑅𝑇 𝑅2 2 3702
= = = 0.3561
𝜕𝑅1 (𝑅1 + 𝑅2 )2 (250 + 370)2

𝜕𝑅𝑇 𝑅1 2 2502
= = = 0.1626
𝜕𝑅2 (𝑅1 + 𝑅2 )2 (250 + 370)2

𝑈𝑅𝑇 = ±√(0.3561 × 250 × 0.02)2 + (0.1626 × 370 × 0.03)2 = 2.535 𝑜ℎ𝑚

𝑹𝑻 = 𝟏𝟒𝟗. 𝟏𝟗 ± 𝟐. 𝟓𝟑𝟓 𝑜ℎ𝑚

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Measurements Devices Concepts of Measurements

10) The power dissipated in a car headlight is calculated by measuring the d.c voltage drop across
it and the current flowing through it (P = V × I) If the possible errors in the measured voltage
and current values are ±1% and ±2% respectively, calculate the likely maximum possible error
in the power value deduced.

Given: 𝑒1 = ±1% . 𝑒2 = ±2%.

Required:
(a) maximum error.
Solution:
𝑃 =𝑉×𝐼
= (V ± 1%) × (I ± 2%)
= (V × I) ± √(0.01)2 + (0.02)2
= (V × I) ± 2.23%
-The maximum possible error:
𝒆𝑚𝑎𝑥 = 𝟐. 𝟐𝟑 %

11) The resistance of a carbon resistor is measured by applying a d.c voltage across it and
measuring the current flowing (R = V / I). If the voltage and current values are measured as 15
± 0.1 V and 214 ± 5 mA respectively, express the value of the carbon resistor.

Given: 𝑉 = 15 ± 0.1 𝑉 . 𝐼 = 214 ± 5 𝑚𝐴.

Required:
(a) The value of the resistor (R).
Solution:
V 15 0.1 2 5 2
R = ± 𝑒𝑚𝑎𝑥 =  √( ) + ( ) = 70.093  2.429% (ohm)
I 214×10−3 15 214
- The value of the resistor (R):
𝑹 = 𝟕𝟎. 𝟎𝟗𝟑  𝟐. 𝟒𝟐𝟗% (ohm)

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Measurements Devices Concepts of Measurements

12) The power usage of a strip heater is to be determined by measuring heater resistance and heater
voltage drop simultaneously. The resistance is to be measured using an ohmmeter having a
resolution of 1 ohm and uncertainty of 1% of reading, and voltage is to be measured using a
voltmeter having a resolution of 1 V and uncertainty of 1% of reading. It is expected that the
heater will have a resistance of 100 ohm and use 100 W of power. Determine the uncertainty
in power determination to be expected with this equipment at the design stage.

Given: 𝑆𝑡𝑟𝑖𝑝 ℎ𝑒𝑎𝑡𝑒𝑟 (𝑅 = 100 𝑜ℎ𝑚. 𝑃 = 100 𝑊 ). 𝑜ℎ𝑚𝑚𝑒𝑡𝑒𝑟 (𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 =

1% 𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = 1 𝑜ℎ𝑚). 𝑉𝑜𝑙𝑡𝑚𝑒𝑡𝑒𝑟(𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 = 1%. 𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = 1𝑣𝑜𝑙𝑡)
Required:
- the uncertainty in power determination at the design stage
Solution:
- The uncertainty in design stage for voltmeter can be calculated as:
𝑈𝑑𝑉 = ±√(𝑈0 )2 + (𝑈𝐶 )2 = ±√(0.5)2 + (0.01 × 100)2 = ±1.118 𝑉𝑜𝑙𝑡
- The uncertainty in design stage for ohmmeter can be calculated as:
𝑈𝑑𝑂 = ±√(𝑈0 )2 + (𝑈𝐶 )2 = ±√(0.5)2 + (0.01 × 100)2 = ±1.118 𝑜ℎ𝑚
- The total uncertainty in design stage can be calculated as:
2 2 2 2
𝜕𝑃 𝜕𝑃 2𝑉 −𝑉 2
√ √
𝑈𝑃 = ± ( 𝑈V ) + ( 𝑈O ) = ± ( 𝑈V ) + ( 2 𝑈O )
𝜕V 𝜕R 𝑅 𝑅

2 2
2 × 100 −10000
√
𝑈𝑃 = ± ( 1.118) + ( 1.118) = 𝟐. 𝟒𝟗𝟗 𝑾
100 10000

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Measurements Devices Concepts of Measurements

13) For a given Linear measurement system, by introducing a different value of the input variable
the output of the system can be represented by this data table:
Input 5 10 15 20 25 30 35 40 45 50
Output 0.12 0.17 0.22 0.33 0.37 0.42 0.45 0.58 0.61 0.64

Given: the following table

Input 5 10 15 20 25 30 35 40 45 50
Output 0.12 0.17 0.22 0.33 0.37 0.42 0.45 0.58 0.61 0.64

Required:
(a) Linear equation (Y = a + b X). (b) Correlation factor (r).
Solution:
- Finding linear equation using linear regression method

X (input) Y (output) X2 XY
5 0.12 25 0.6
10 0.17 100 1.7
15 0.22 225 3.3
20 0.33 400 6.6
25 0.37 625 9.25
30 0.42 900 12.6
35 0.45 1225 15.75
40 0.58 1600 23.2
45 0.61 2025 27.45
50 0.64 2500 32
Sum 275 3.91 9625 132.45

∑𝑥 275 ∑𝑦 3.91
𝑋̅ = = = 27.5 , 𝑌̅ = = = 0.391
𝑛 10 𝑛 10
∑ 𝑥𝑦−𝑛𝑥̅ 𝑦̅ 132.45−10×27.5×0.391
𝑏= = = 0.0121
∑ 𝑥 2 −𝑛𝑥̅ 2 9625−10×27.52
𝑎 = 𝑦̅ − 𝑏𝑥̅ = 0.391 − 0.012 × 27.5 = 0.058
- The linear equation:
Y = 0.058 + 0.0121 X

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Measurements Devices Concepts of Measurements

0.7

0.6

0.5

0.4

0.3
y = 0.0121x + 0.0587
0.2

0.1

0
0 10 20 30 40 50 60

- Finding Correlation factor (r):

dx = X - Xmean, dy = Y - Ymean

X (input) Y (output) dx (dx)2 dy (dy)2 (dx)(dy)

5 0.12 -22.5 506.25 -0.271 0.0734 6.0975
10 0.17 -17.5 306.25 -0.221 0.0488 3.8675
15 0.22 -12.5 156.25 -0.171 0.0292 2.1375
20 0.33 -7.5 56.25 -0.061 0.0037 0.4575
25 0.37 -2.5 6.25 -0.021 0.0004 0.0525
30 0.42 2.5 6.25 0.029 0.0008 0.0725
35 0.45 7.5 56.25 0.059 0.0035 0.4425
40 0.58 12.5 156.25 0.189 0.0357 2.3625
45 0.61 17.5 306.25 0.219 0.0480 3.8325
50 0.64 22.5 506.25 0.249 0.0620 5.6025
Sum 275 3.91 0 2062.5 0 0.3057 24.9250

𝑐𝑜𝑣(𝑥.𝑦)
𝑟=
𝜎𝑥 𝜎𝑦

∑(𝑑𝑥)2 2062.5 ∑(𝑑𝑦)2 0.3057

𝜎𝑥 = √ =√ = 15.14 , 𝜎𝑦 = √ =√ = 0.1843
𝑛−1 9 𝑛−1 9
∑(𝑑𝑥)(𝑑𝑦) 24.925
𝑐𝑜𝑣(𝑥. 𝑦) = = = 2.769
𝑛−1 9
2.769
𝒓= = 𝟎. 𝟗𝟗𝟐𝟑 = 𝟗𝟗. 𝟐𝟑%
15.14×0.1843

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Measurements Devices Concepts of Measurements

14) The following data are expected to follow a linear relation of the form y = ax + b. Obtain the
best fit by least squares. Estimate the value of y at x = 3 and specify an error bar for the same.
Make a plot to compare the data and the fit. Determine the correlation coefficient.

Given: the following table

Required:
(a) Linear equation (Y = a + b X). (b) Correlation factor (r).
Solution:
- Finding linear equation using linear regression method

X (input) Y (output) X2 XY
0.9 1.1 0.81 0.99
2.3 1.6 5.29 3.68
3.3 2.6 10.89 8.58
4.5 3.2 20.25 14.4
5.7 4 32.49 22.8
7.7 6 59.29 46.2
Sum 24.4 18.5 129.02 96.65

∑𝑥 24.4 ∑𝑦 18.5
𝑋̅ = = = 4.066 ̅
,𝑌 = = = 3.083
𝑛 6 𝑛 6
∑ 𝑥𝑦−𝑛𝑥̅ 𝑦̅ 96.65−6×4.066×3.083
𝑏= = = 0.718
∑ 𝑥 2 −𝑛𝑥̅ 2 129.02−6×4.0662

𝑎 = 𝑦̅ − 𝑏𝑥̅ = 3.083 − 4.066 × 0.718 = 0.163

- The linear equation:
Y = 0.149 + 0.721 X

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Measurements Devices Concepts of Measurements

2 y = 0.7188x + 0.16
1

0
0 2 4 6 8 10

- Finding Correlation factor (r):

dx = X - Xmean, dy = Y - Ymean

X (input) Y (output) dx (dx)2 dy (dy)2 (dx)(dy)

0.9 1.1 -3.166 10.024 -1.983 3.932 6.278
2.3 1.6 -1.766 3.119 -1.483 2.199 2.619
3.3 2.6 -0.766 0.587 -0.483 0.233 0.370
4.5 3.2 0.434 0.188 0.117 0.014 0.051
5.7 4 1.634 2.670 0.917 0.841 1.498
7.7 6 3.634 13.206 2.917 8.509 10.600
Sum 24.4 18.5 0 29.79 0 15.73 21.42

𝑐𝑜𝑣(𝑥.𝑦)
𝑟=
𝜎𝑥 𝜎𝑦

∑(𝑑𝑥)2 29.79 ∑(𝑑𝑦)2 15.73

𝜎𝑥 = √ =√ = 2.44 , 𝜎𝑦 = √ =√ = 1.773
𝑛−1 5 𝑛−1 5

∑(𝑑𝑥)(𝑑𝑦) 21.42
𝑐𝑜𝑣(𝑥. 𝑦) = = = 4.284
𝑛−1 5

4.284
𝒓= = 𝟎. 𝟗𝟗𝟎𝟑 = 𝟗𝟗. 𝟎𝟑%
2.44×1.773

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