Chapter 2: Theory of

Measurement
 Introduction
• Measurement is an act on process of comparison between
the unknown physical quantity to its predefined standard in
order to find the value of unknown quantity.
• The measurement is said to be proper if the measured
value is less erroneous.
• To make error free measurement, the user must know the
characteristics and operation of the instrument, basic laws
to handle instrument and causes of errors in measurement.
• Thus we have to study about the characteristics of the
instruments: Error and its analysis, statistical analysis of
error and minimization that of error or controlling of
error, etc.
• Measurement system characteristics can be divided into
two distinct areas:
– Static Characteristics
– Dynamic Characteristics
Static Characteristics
• Some applications involve the measurement of quantities
that are either constant or vary slowly with time.
• Under these circumstances, it is possible to define a set of
criteria that gives a meaningful description quality of
measurement without interfering with dynamic
descriptions that involve the use of differential equations.
• These criteria are called Static Characteristics.
• Important static characteristics are:
– Accuracy
– Precision
– Sensitivity
– Resolution
– Linearity
– Reproducibility
– Drift
– Dead Zone Hysteris
Accuracy
• Accuracy is the closeness with which an instrument reading approaches
the true value of the quantity being measured.
• Thus accuracy of a measurement means conformity to truth.
• It is the degree of closeness with which the reading approaches the true
value of the quantity to be measured.
• The accuracy can be expressed in following ways:
1. Point Accuracy:
– Such accuracy is specified at only one particular point of scale.
– It does not give any information about the accuracy at any other Point on
the scale.
– Point accuracy is stated for one or more points in the range.
2. Accuracy as percentage of scale span:
– When an instrument as uniform scale, its accuracy may be expressed in
terms of scale range.
– Example – if accuracy of a thermometer with range 500oC with ±0.5% then
at 25oC, error will be (500/25*(0.5%)) = 10%, so misleading.
3. Accuracy as percentage of true value:
– The best way to conceive the idea of accuracy is to specify it in terms of the
true value of the quantity being measured.
– Example – if accuracy is ±0.5% of true value then as reading fet smaller so
do that errors, so more informative.
Precision (Precise – clearly or sharply defined)
• It is the measure of consistency or reproducibility i.e., given a fixed
value of a quantity, precision is a measure of the degree to which
successive measurements differ from one another.
• Degree of agreement within a group of measurements.
• The precision is composed of two characteristics: Conformity and
Number of Significant Figures.
• Conformity is necessary but not sufficient for precision because of
lack of significant figures obtained.
• Precision is a necessary but not sufficient condition for accuracy.
• More significant figures in measured value, the grater the precision
of measurement.
• Higher degree of conformity of closeness to the true value of the
measured value guarantees the accuracy.
• Consider a resistor having true value as 2385692 , which is being
measured by an ohmmeter. But the reader can read consistently, a
value as 2.4 M due to the nonavailability of proper scale. The error
created due to the limitation of the scale reading is a precision
error.
Sensitivity
• The sensitivity denotes the smallest change in the
measured variable to which the instrument responds.
• It is defined as the ratio of the changes in the output of
an instrument to a change in the value of the quantity
to be measured.
• Mathematically it is expressed as [see next slide]:
• Thus, if the calibration curve is liner, as shown, the
sensitivity of the instrument is the slope of the
calibration curve.
• If the calibration curve is not linear as shown, then the
sensitivity varies with the input.
• Inverse sensitivity or deflection factor is defined as the
reciprocal of sensitivity.
• Inverse sensitivity or deflection factor = 1/ sensitivity
Resolution
• It is the smallest change in measured value to
which the instrument will respond.
• If the input is slowly increased from some
arbitrary input value, it will again be found that
output does not change at all until a certain
increment is exceeded. This increment is called
resolution.
• Thus it is defined as the smallest change in the
input which results a detectable output.
• Analog Instrument / Meter – It is the significant
of the smallest division in the scale.
• Digital Instrument / Meter – It is the significant
of the least significant bit (LSB).
Linearity
• It means constant sensitivity throughout whole
measurement range.
• It is defined as the ability to produce the input
characteristics symmetrical and this can be expressed as:
y=mx+c
• When the input-output points of the instrument are
plotted on the calibration curve and resulting curve may
not be linear.
• This would be only if the output is proportional to input.
• Linearity is the measure of maximum deviation of these
points from the straight line.
• The departure from the straight line relationship is non-
linearity, but it is expressed as linearity of the instrument.
• This departure from the straight line could be due to non-
linear elements in the measuring system or the elastic after
effects of the mechanical system.
• Linearity is expressed in many different ways:
– Independent Linearity: It is the maximum deviation
from the straight line so placed as to minimize the
maximum deviation.
– Zero based linearity: It is the maximum deviation
from the straight line joining the origin and so placed
as to minimize the maximum deviation.
– Terminal based linearity: It is the maximum deviation
from the straight line joining both the end points of
the curve.
Dynamic Characteristics
• Many measurements are concerned with rapidly
varying quantities and therefore, for such cases we
must examine the dynamic relations which exits
between output and input.
• Normally done with the help of differential equations.
• Performance criteria based upon dynamic relations
constitute the Dynamic Characteristics.
• It is used for study of behaviour of the system between
the time that output value changes and the time the
value has settled down to its steady state value.
• Important dynamic characteristics are: Speed of
Response, Response Time, Measuring Lag, Frequency
Response, Bandwidth, SNR, Damping Factor, Rise
Time, Fall Time, Settling Time, Fidelity, Dynamic Error,
etc.
Speed of Response
• It indicates how fast the input given to the measurement system pr
an instrument brings the output.
• It is defined as the rapidity with which a measurement system
responds to changes in the measured quantity.
Response Time
• It is defined as the time required by the system to settle to its final
steady state position after the application of the input.
Measuring Lag
• It is the retardation or delay in the response of a measurement
system to changes in the measured quantity.
• The measuring lags are of two types:
– Retardation Type: In this case the response of the measurement
system begins immediately after the change in measured quantity has
occurred.
– Time Delay Lag: In this case the response of the measurement system
begins after a dead time after the application of the input. Fidelity: It
is defined as the degree to which a measurement system indicates
changes in the measurand quantity without dynamic error.
Frequency Response
• When the analysis of the measurement system is done by plotting the gain
(output/input) wrt the frequency then the resulting response is called
Frequency Response.
• It is of two types: Gain vs Frequency (Magnitude Response) and Phase
Angle vs Frequency (Phase Response).
• The summation of the both – Bode Plot
Bandwidth
• Range of frequency over which an instrument is designed to operate the
output receiving the input signal or quantity with a constant gain.
• Calculated by magnitude response and unit is Hz.
• The quantity of the measurement system taken in the interval between
those frequencies where the power gain of the system has dropped to one
– half of its maximum value or voltage gain dropped by a factor of 0.707
(1/√2).
Errors in Measurement and Their Statistical Analysis
Error in Measurement
• Error of Measurement can be define difference between
the actual value of a quantity and the value obtained by a
measurement.
• Repeating the measurement will improve (reduce) the
random error (caused by the accuracy limit of the
measuring instrument) but not the systemic error (caused
by incorrect calibration of the measuring instrument).
1. Absolute Error
• Can be defined as the difference between the expected
value (T.V.) of the variable and the measured value (M.V.) of
the variable.
2. Relative Error (Percent of Error)
 TYPE OF ERROR
• Errors are generally categorized under the following three
major types:
1. Gross Error
• This class of errors is generally the fault of the person using
the instruments such as incorrect reading of instruments,
incorrect recording of experimental data or data incorrect
use of instruments.
• As long as human beings are involved, some gross errors
will definitely commit.
• Although complete eliminating of gross errors is probably
impossible, one should try to avoid them.
• The following actions may be necessary to reduce the
effects of gross errors.
– Great care should be taken in reading and recording the data.
– Two or more readings should be taken by different
experimenters
2. Systematic Error
• All the error due to the shortcoming of an instrument
such as less accuracy in the scale calibration, defective
parts and effects of the environment on the equipment or
the users.
• Systematic errors can be divided into four categories:
Instrumental Error
• These errors arise due to main reasons:
– Due to inherit shortcoming in the instruments (may be caused
by the construction, calibration or operation of mechanical
structure in the instruments).
– Due to misuse of the instruments. For example, these may be
caused by failure to adjust zero of the instruments.
– Due to loading effect of the instruments. These errors can be
eliminated or at least reduced by using the following methods:
• The procedure of measurements must be carefully planned.
• Correction factors should be applied after detection of these errors.
• Re-calibration the instrument carefully.
• Use the instrument intelligently.
Observational Errors
• Due to the types on instrument display, whether it is analog or digital.
• Due to parallax (eye should be directly in line with the measurement
point).
• Note: These errors can be eliminated completely by using digital display
instruments.
Environmental Errors
• Due to conditions external to the measuring device such as the area,
surrounding the instrument.
• These conditions may be caused by the changes in pressure, humidity,
dust, vibration or external magnetic or electrostatic fields.
• These errors can be eliminated or reduced by using corrective measure
such as:
– Keep the condition as constant as possible.
– Use instrument/equipment which is immune to these effects.
– Employ technique which eliminates these disturbances.
Simplification Errors
• Due to simplification of a formula: For example: A = B + C + D2.
• If D is too small, then the formula is simplified to: A = B + C.
• There will be a different result between the first and the second equation.
• In high accuracy requirements, a formula should not be simplified to avoid
these types of errors.
3. Random Error
• In some experiments, the results shows variation
from one to another, even after all systematical
and gross errors have been accounted for.
• The cases of these errors are not recognized,
therefore the elimination or reduction of these
errors are not possible.
• When these types of errors are occurred, the best
result can be determined by statistical analysis.
 Statistical Analysis of Experimental Data or
Error in Measurement
• It is important to define some pertinent terms before
discussing some important methods of statistical
analysis of experimental data.
Arithmetic Mean
• When a set of readings of an instrument is taken, the
individual readings will vary somewhat from each
other, and the experimenter is usually concerned with
the mean of all the readings.
• If each reading is denoted by x and there are n
readings, the arithmetic mean is given by
Deviation
Standard Deviation
• It is also called root mean-square deviation.
• It is defined as

Variance
• The square of standard deviation is called variance.
• This is sometimes called the population or biased standard
deviation because it strictly applies only when a large number of
samples is taken to describe the population.

Geometrical Mean
• It is appropriate to use a geometrical mean when studying
phenomena which grow
• in proportion to their size. This would apply to certain biological
processes and
• growth rate in financial resources. The geometrical mean is defined
by
Measurement of Voltage and Current
• Use of Voltmeter and Ammeter.
• May be Analog or Digital type.
• Analog Voltmeter and Ammeter – You Know It!!!
• In case of analog or pointer indicating ammeter
and voltmeter, there are two basic types of
instruments depending upon the operation:
– Moving Iron Instrument / Meter
– Moving Coil Instrument / Meter
 Moving Coil Instrument / Meter
• Very commonly used form of analogue voltmeter because of its
sensitivity, accuracy, and linear scale, although it only responds to
d.c. signals.
• As shown Figure, it consists of a rectangular coil wound round a soft
iron core that is suspended in the field of a permanent magnet.
• The signal being measured is applied to the coil, which produces a
radial magnetic field.
• Interaction between this induced field and the field produced by
the permanent magnet causes torque, which results in rotation of
the coil.
• The amount of rotation of the coil is measured by attaching a
pointer to it that moves past a graduated scale. The theoretical
torque produced is given by: T =BIhwN
where B is the flux density of the radial field, I is the current flowing
in the coil, h is the height of the coil, w is the width of the coil, and
N is the number of turns in the coil.
• If the iron core is cylindrical and the air gap between
the coil and pole faces of the permanent magnet is
uniform, then the flux density B is constant and above
equation can be rewritten as: T=KI
that is, torque is proportional to the coil current and
the instrument scale is linear.
 Moving Iron Instrument / Meter
• As well as measuring d.c. signals, the moving iron meter can also
measure a.c. signals at frequencies up to 125 Hz.
• It is the least expensive form of meter available and, consequently,
this type of meter is also used commonly for measuring voltage
signals.
• The signal to be measured is applied to a stationary coil, and the
associated field produced is often amplified by the presence of an
iron structure associated with the fixed coil.
• The moving element in the instrument consists of an iron vane
suspended within the field of the fixed coil.
• When the fixed coil is excited, the iron vane turns in a direction that
increases the flux through it.
• The majority of moving-iron instruments are either of the
Attraction Type or of the Repulsion Type.
• A few instruments belong to a third Combination Type.
• The Attraction Type, where the iron vane is drawn into the field of
the coil as the current is increased, is shown schematically in Figure
a.
• The alternative Repulsion Type is sketched in Figure b.
• For an excitation current, I, the torque produced that causes the
vane to turn is given by: T = I2dM/2dθ
where M is the mutual inductance and θ is the angular deflection.
• Rotation is opposed by a spring that produces a backwards torque
given by: Ts = K θ
• At equilibrium, T = Ts, and θ is therefore given by: θ = I2dM/2Kdθ
• Has a square-law response where the deflection is proportional to
the square of the signal being measured.
 Measurement of Resistance (Low, Medium and High)
Measurement of Low and Medium Resistance
• This is Ammeter – Voltmeter Method.
• This method (a and b) is very common as voltmeter and ammeter is
available in all labs.
• There are two methods of connecting voltmeter and ammeter for
measurement of resistances as shown in Fig (a) and (b).
• In both the cases the measured value of the unknown resistances is equal
to the reading of voltmeter divided by reading of ammeter.
• Let the reading of voltmeter is 'V and ammeter I, hence measured value of
the resistances = Rm =V/I.
• From Fig ( b), Rm =V/I.
• The reading of ammeter is equal to I= Iv+ Ir
• Let the resistance of voltmeter by ‘Rv’ then I=V/Rv+V/R
• Reading of voltmeter V=Voltage across resistance VR.
• Rm = V/(V/R+V/Rv ) = 1/(1/R+1/Rv )=R/(1+R/Rv)
• R = V/Ia = V/(I – Iv) = V/(I – V/Rv) = V/[I(1 – V/Rv)]
= Rm/(1-Rm/Rv)
• From this expression the true value of resistance R=Rm
only, when resistance of voltmeter is ‘∞’(This is the
ideal case).
• But practically Rv is in medium or high resistance class.
• Hence this method is used for measurement of low
resistance as in this case Rm/Rv will be approximately
equal to zero.
• R = Rm/(1-Rm/Rv) = Rm(1-Rm/Rv)-1/2 = Rm(1+Rm/Rv) if Rm
<< Rv.
 Measurement of Medium Resistance:
Wheatstone Bridge Method
• Bridge circuits are used very commonly as a variable conversion
element in measurement systems and produce an output in the
form of a voltage level that changes as the measured physical
quantity changes.
• They provide an accurate method of measuring resistance,
inductance, and capacitance values and enable the detection of
very small changes in these quantities about a nominal value.
• They are of immense importance in measurement system
technology because so many transducers measuring physical
quantities have an output that is expressed as a change in
resistance, inductance, or capacitance.
• Normally, excitation of the bridge is by a d.c. voltage for resistance
measurement and by an a.c. voltage for inductance or capacitance
measurement.
• Both null and deflection types of bridges exist, and, in a like
manner to instruments in general, null types are employed mainly
for calibration purposes and deflection types are used within closed
loop automatic control schemes.
• The four arms of the bridge consist of the unknown
resistance Ru, two equal value resistors R2 and R3, and
variable resistor Rv (usually a decade resistance box).
• A d.c. voltage Vi is applied across the points AC, and
resistance Rv is varied until the voltage measured
across points BD is zero.
• This null point is usually measured with a high
sensitivity galvanometer.
 Measurement of Inductance : AC
Bridge – Maxwell Bridge
• Bridges with a.c. excitation are used to measure
unknown impedances (capacitances and inductances).
• Both null and deflection types exist.
• As for d.c. bridges, null types are more accurate but
also more tedious to use.
• Therefore, null types are normally reserved for use in
calibration duties and any other application where very
high measurement accuracy is required.
• Otherwise, in all other general applications, deflection
types are preferred.
• A Maxwell bridge is shown in Figure.
• The requirement for a variable inductance box is
avoided by introducing instead a second variable
resistance.
• The circuit requires one standard fixed-value capacitor,
two variable resistance boxes, and one standard fixed-
value resistor, all of which are components that are
readily available and inexpensive.
• Referring to Figure, we have at the null output point:
Measurement of Capacitance: AC
Bridge – Schering Bridge
Let
• C1 = capacitor whose capacitance is to be measured.
• r1 = a series resistance representing the loss in the capacitor C1.
• C2 = a standard capacitor.
• R3 = a non inductive resistance.
• C4 = a variable capacitor.
• R4 = a variable non inductive resistance.

At balance: Z1 Z4 = Z 2 Z4

(r1+1/jωC1)⋅(R4/(jωC4R4+1)) = R3/jωC2......(1)

r1R4−jR4/ωC1 = −jR3/ωC2 + R3R4C4/C2......(2)

Or Equating the real and imaginary terms in equa. (2), we obtain

r1 = R3⋅C4/C2......(3)

C1=R4⋅C2/R3......(4)