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Instrument Characteristics 1.1 Introduction This chapter
concentrates on 'how well an instrument performs its various
functions'. That determine how closely the instrument output
reflects the value of the variable that is being measured.
Instrument performance is described by means of quantitative
qualities which are referred to as characteristics : the two realms
being the static and the dynamic. The static characteristics
pertain to a system where quantities to be measured are constant or
vary slowly with time. When the instrument is required to measure a
time-varying process variable, one has to be concerned with dynamic
characteristics which quantify the dynamic relation between the
instrument input and output. 2.2 Static terms and characteristics
2.2.1 Range and span : The region between the limits within which
an instrument is designed to operate for measuring, indicating or
recording a physical quantity is called the range of the
instrument. The range is expressed by stating the lower and upper
values. Span represents the algebraic differences between the upper
and lower range values of the instrument. For example, Range 10C to
80C ; Span 90C Range 0 volt to 75 volt; Span 75 volt 2.2.2.
Accuracy, errors and correction : No instrument gives an exact
value of what is being measured. There is always some uncertainty
in the measured value. This uncertainty is expressed in terms of
accuracy and error. Accuracy of an indicated (measured) value may
be defined as conformity with or closeness to an accepted standard
value (true value). Accuracy of the measured signal depends upon
the intrinsic accuracy of the instrument itself, variation of the
signal being measured, accuracy of the observer and whether or not
the quantity is being truly impressed upon the instrument. For
example, the accuracy of a micrometer depends upon factors like
error in screw, anvil shape, temperature difference, and the
applied torque variations etc In general, the result of any
measurement differs somewhat from the true value of the quantity
being measured. The difference between the measured value (Vm) and
the true value (Vt) of the quantity represents static error or
absolute error of measurement (Es), i.e.
Es = Vm - Vt The error may be either positive or negative. For
positive static errors the instrument reads high and for negative
static errors the instrument reads low. From experimentalist's view
point, static correction or simply correction (Cs) is more
important than the static error The static correction is defined as
the difference between the true value and the measured value of a
quantity.
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Cs = Vt - Vm The correction of the instrument reading is of the
same magnitude as the error, but opposite in sign, i.e., Cs= -Es
Error specification or representation : (i) Point accuracy wherein
the accuracy of an instrument is stated for one or more points in
its range. For example a given thermometer may he stated to read
within +0.5C between 100 oC and 200 oC. Likewise a scale of length
may be read within 0.025 cm. (ii) Percentage of true value or the
relative error wherein the absolute error of measurement is
expressed as a percentage of true value of the unknown
quantity.
percent 100 x V
VV
percent 100 x value true
value truevalue measurederror
t
tm
=
=
The percentage error stated in this way is the maximum for any
point in the range of the instrument. The size of the error,
however, diminishes with a drop in the true value. (iii) Percentage
of full scale deflection where in the error is calculated on the
basis of maximum value of the scale. The accuracy specified above
refers to the intrinsic accuracy of the instrument itself and does
not include procedural or personal performance.
Solve Example 2.1-5 2.2.3 Calibration. The magnitude of the
error and consequently the correction to be applied is determined
by making a periodic comparison of the instrument with standards
which are known to be constant. The entire procedure laid down for
making, adjusting, or checking a scale so that readings of an
instrument or measurement system conform to an accepted standard is
called the calibration. The graphical representation of the
calibration record is called calibration curve and this curve
relates standard values of input or measurand to actual values of
output throughout the operating range of the instrument. A
comparison of the instrument reading may be made with either (i) a
primary standard, (ii) a secondary standard of accuracy greater
than the instrument to be calibrated or (iii) a known input source.
For example, we may calibrate a flow meter by comparing it with a
standard flow measurement facility at the National Bureau of
Standards ; by comparing it with another flow meter (a secondary
standard) which has already been compared with a primary standard;
or by direct comparison with a primary measurement such as weighing
a certain amount of water in a tank and recording the time elapsed
for this quantity to flow through the meter. The calibration
standards, along with their typical accuracies, for certain
physical parameters have been given in Table 2.1. The calibration
standard should be at least an order more accurate than the
instrument being calibrated.
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In a typical calibration curve (Fig. 2.1) ABC represents the
readings obtained while ascending the scale; DBF represents the
readings during descending; KLM represents the median and is
commonly accepted as the calibration curve. The term median refers
to the mean of a series of up and down readings. Quite often, the
indicated values are plotted as abscissa and the ordinate
represents the variation of the median from the true values.
A faired curve through the experimental points then represents
the correction curve. This type of deviation presentation
facilitates a rapid visual assessment of the accuracy of the
instrument. The user looks along the abscissa for the value
indicated by the instrument and then reads the correction to be
applied.
A properly prepared calibration/correction curve gives
information about the absolute static errors of the measuring
device, the extent of the instrument's linearity or conformity, and
the hysterises and repeatability of the instrument.
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Solve Example 2.6
2.2.4 Hysterises dead zone: From the instrument calibration
curve (Fig. 2.1), it would be noted that the magnitude of output
for a given input depends upon the direction of the change of
input. This dependence upon previous inputs in called Hysterises.
Hysterises is the maximum difference for the same measured quantity
(input signal) between the upscale and downscale readings during a
full range traverse in each direction. Maximum difference is
frequently specified as a percentage of full scale. Hysterises
results from the presence of irreversible phenomenon such as
mechanical friction, slack motion in bearings and gear, elastic
deformation, magnetic and thermal effects. Hysterises may also
occur in electronic systems due to heating and cooling effects
which occur differentially under conditions of rising and falling
input. Dead zone is the largest range through which an input signal
can be varied without initiating any response from the indicating
instrument. Friction or play is the direct cause of dead zone or
band 2.2.5 Drift: It 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. Wear and tear, high
stress developing at some parts and contamination of primary
sensing elements cause drift. It may occur in obstruction flow
meters because of wear and erosion of the orifice plate, nozzle or
venturimeter. Drift occurs in thermocouples and resistance
thermometers due to the contamination of the metal and a change in
its atomic or metallurgical structure. Drift occurs very slowly and
can be checked only by periodic inspection and maintenance of the
instrument. 2.2.6 Sensitivity : Sensitivity of an instrument or an
instrumentation system is the ratio of the magnitude of the
response (output signal) to the magnitude of the quantity being
measured (input signal), i.e.,
signal input of Changesignal output of Change(k)itivity
Staticsens =
Sensitivity is represented by the slope of the calibration curve
it the ordinates are expected in the actual units. With a linear
calibration curve, the sensitivity is constant. However, if the
calibration curve in non-linear, the static sensitivity is not
constant and must be specified in terms of the input values as
shown in Fig. 2.4
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Sensitivity has a wide range of units, and these depend upon the
instrument or measurement system being investigated. For example,
the operation of a resistance thermometer depends upon a change in
resistance (output) to change in temperature (input) and as such it
sensitivity will have units of ohms/C. Sensitivity of an instrument
system is usually required to be as high as possible because then
it becomes easier to take the measurement (read the output).
Let the different elements comprising a measurement system have
static sensitivities of K1, K2, K3, .etc. When these elements are
connected in series or cascades (Fig. 2.5), then the overall
sensitivity is worked out from the following relations:
321
2
o
1
2
i
1
i
o
2
o1
1
22
i
11
K x K x K
; x
x
(K)y sensitivit Overall
;K ;
K ;
K
==
=
===
The above relation is based upon the assumption that no
variation occurs in the values of individual sensitivities K1, K2,
K3, .etc. due to loading effects. When the input to and output from
the measurement system used with electrical/electronic equipment
have the same form, the term gain is used rather than sensitivity.
Likewise an increase in displacement with the optical and
mechanical instruments is described by the term amplification.
Apparently the terms sensitivity, gain and magnification all mean
the same; and they describe the relationship between output and
input. Further when the input or output signal is changing with
time, the term transfer function or transfer operator is used other
than sensitivity, gain or amplification. 2.2.7 Threshold and
resolution : The smallest increment of quantity being measured
which can be detected with certainty by an instrument represents
the threshold and resolution of the instrument. When the input
signal to an instrument is gradually increased from zero, there
will be some minimum value input before which the instrument will
not detect any output change. This minimum value is called the
threshold of the instrument. Thus threshold defines the minimum
value of input which is necessary to cause a detectable change from
zero output. In a digital system, it is the input signal necessary
to cause one least significant digit of the output reading to
change. Threshold may be caused by backlash or internal noise.
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When the input signal is increased from non-zero value, one
observes that the instrument output does not change until a certain
input increment is exceeded. This increment is termed resolution or
discrimination. Thus resolution defines the smallest change of
input for which there will be a change of output. With analogue
instruments, the resolution is determined by the ability of the
observer to judge the position of a pointer on a scale, e.g. the
level of mercury in a glass tube. Resolution is usually reckoned to
be no better than about 0.2 of the smallest scale division. With
digital instruments, resolution is determined by the number of neon
tubes taken to show the measured value. For example, if there are
four neon tubes to represent voltage measurement on a 1 volt range,
one tube will be taken by the decimal point and the others by
digits to show readings up to a maximum of 0.999 volt. Thus the
third digit shows or resolves millivolts, and consequently the
resolution is 1 mV. Threshold and resolution may be expressed as an
actual value or as a fraction or percentage of full scale value.
2.2.8. Precision, repeatability and reproducibility. These terms
refer to the closeness of agreement among several measurements of
the same true value under the same operating conditions. Proper
checking and maintenance of instrument should be carried out to
ensure its reproducibility. Let us differentiate between accuracy
and precision as applied to the realms of measurements. Accuracy
refers to the closeness or conformity to the true value of the
quantity under measurement. Precision refers to the degree of
agreement within a group measurements, i.e., it prescribes the
ability of the instrument to reproduce its readings over and over
again for a constant input signal. This distinction can be
elaborated by considering the following two examples : The
difference between accuracy and precision has been illustrated in
Fig. 2.6. The arrangement may be thought to correspond to the game
of darts where one is asked to strike a target represented by
centre circle. The centre circle then represents the true value,
and the result achieved by (he striker has been indicated by the
mark 'X'.
Two further terms used to define reproducibility are : Stability
refers to the reproducibility of the mean reading of an instrument,
repeated on
different occasions separated by intervals of time which are
long compared with the time of taking a reading. The conditions of
use of the instrument remain unchanged.
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Constancy refers to the reproducibility of the mean reading of
an instrument when a constant input is presented continuously and
the conditions of test are allowed lo vary within specified limits.
This variation may be due to some changer in the external
environmental conditions.
The above discussion also points out that it is possible to
obtain high precision with poor accuracy, but not high accuracy
with low precision. In other words precision is a necessary
prerequisite to accuracy but it does not guarantee accuracy. 2.2.9.
Linearity : The working range of most of the instruments provides a
linear relationship between the output (reading taken from the
scale of the instrument) and input (measurand, signal presented to
the measuring system). This aspect tends to facilitate a more
accurate data reduction. Linearity is defined as the ability to
reproduce the input characteristics symmetrically, and this can be
expressed by the straight line equation
y = mx + c where y is the output, x the input, m the slope and c
the intercept. Apparently the closeness of the calibration curve to
a specified straight line is the linearity of the instrument. Any
departure from the straight line relationship is non-linearity. The
non-linearity may be due to non-linear elements in the measurement
device, mechanical hysterises, viscous flow or creep, and elastic
after-effects in the mechanical system. In a nominally linear
measurement device, the non-linearity may taken different forms as
illustrated in Fig. 2.7.
(i) Theoretical slope linearity : Maximum departure a from the
theoretical straight line OA passing through the origin. The line
OA refers to the straight line between the theoretical end points,
and it is drawn will out regard to any experimentally determined
values. (ii) End point linearity : Maximum departure b from the
straight line OB passing through the through the terminal readings
(experimental end pointszero and full scale position)
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(iii) Least square linearity : Maximum departure c from the best
fit straight line OC determined by the least square technique. In
roost instruments, the linearity is taken to be the maximum
deviation from a linear relationship between input and output,
i.e., from a constant sensitivity and is often expressed as a
percentage of full scale. The calculation of measurement error
requires numerical values of accuracy, resolution, linearity etc.
for the instrument being used. For the majority of laboratory
instruments, this data is given in a manufacturer's hand book.
However for some instruments such as micrometers, vernier calipers,
thermometers and materials testing equipment, the data is given in
the standards maintained by the country. 2.2.10 Some other terms
associated with the static performance of an instrument are :
Tolerance : Range of inaccuracy which can be tolerated in
measurements : it is the
maximum permissible error. For example, the tolerance would be
1% when an inaccuracy of 1 bar can be tolerated for 100 bar value
of pressure.
Readability and least count : The term readability indicates the
closeness with which the scale of the instrument may be read. The
term least count represents the smallest difference that can be
detected on the instrument scale. Both readability and least count
are dependent on length scale, spacing of graduations, size of the
pointer and parallax effect.
Back lash : The maximum distance or angle through which any part
of a mechanical system may be moved in one direction without
applying appreciable force or motion to the next part in a
mechanical system.
Zero stability : A measure of the ability of the instrument to
restore to zero reading after the measurand has returned to zero,
and other variations (temperature, pressure, humidity, vibration
etc.) have been removed.
2.3 Dynamic terms and characteristics When the instruments are
required to measure au input which is varying with time, the
dynamic or transient behavior of the instrument becomes as
important as the state behavior. The signals cannot be impressed
upon instantaneously and the mass add capacitances (thermal,
electrical, or fluid) introduce slowness or sluggishness in the
measurement system. A pure time delay may also he encountered when
the instrument has to wait for some reactions to take place.
Consequently the system does not settle to its equilibrium steady
state condition immediately after the application of input signal;
it does so only after passing through a transient period. Certain
terms used with dynamic systems are defined below : 2.3.1 Speed of
response and measuring lag. In a measuring instrument the speed of
response or responsiveness is defined as the rapidity with which an
instrument responds to a change in the value of the quantity being
measured. Measuring lag refers to retardation or delay in the
response of an instrument to a change in the input signal. The lag
is caused by conditions such as capacitance, inertia, or
resistance.
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2.3.2 Fidelity and dynamic error : Fidelity of an
instrumentation system is defined as the degree of closeness with
which the system indicates or records the signal which is impressed
upon it. It refers to the ability of the system to reproduce the
output in the same form as the input. If the input is a sine wave
then for 100 per cent fidelity, the output should also be a sine
wave. The difference between the indicated quantity and the true
value of the time varying quantity is the dynamic error, here
static error of the instrument is assumed to be zero. 2.3.3
Overshoot. Because of mass and inertia, a moving part, i.e., the
pointer of the instrument does not immediately come to rest in the
final deflected position. The pointer goes beyond the steady state
i.e., it overshoots (Fig. 2.8).
The overshoot is defined as the maximum amount by which the
pointer moves beyond the steady state. 2.3 4 Dead time and dead
zone : Dead time is defined as the time required for an instrument
to begin to respond to a change in the measured quantity. It
represents the time before the instrument begins to respond after
the measured quantity has been altered. Dead zone defines the
largest change of the measurand to which the instrument does not
respond. Dead zone is the result of friction, backlash or
hysterises in the instrument.
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Some of the dynamic terms are graphically shown in Fig. 2.9
where the measured quantity and the instrument readings arc plotted
as function of time. 2.3.5 Frequency response : Maximum frequency
of the measured variable that an instrument is capable of following
without error. The usual requirement is that the frequency of
measurand should not exceed 60 percent of the natural frequency of
the measuring instrument. 2.4 Standard test-Inputs The dynamic
performance of both measuring and control systems is determined by
applying some known and predetermined input signal to its primary
sensing element and then studying the behavior of the output
signal. The most common standard inputs used for dynamic analysis
have been illustrated in Fig 2.10, and these are ;
(i) Step function which is a sudden change from one steady value
to another. The step input is mathematically represented by the
relationship
i = 0 at t < 0 i = o at t 0
where o is a constant value of the input signal i. The capacity
of the system to cope with changes in the input signal is indicated
by the transient response.
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(ii) Ramp or linear function wherein the input varies linearly
with time. The ramp input is mathematically represented as :
i = 0 at t < 0 i = t at t 0
where is the slope, of the input versus time relationship. The
ramp-response becomes indicative of the steady state error in
following the changes in the input signal. (iii) Sinusoidal or sine
wave function where the input has a cyclic variation ; (he input
varies sinusoidal with a constant maximum amplitude.
Mathematically, it may be represented as :
i = A sin t where A is the amplitude and is the frequency in
rad/s. The frequency or harmonic response is a measure of the
capability of the system to respond to inputs of cyclic nature. A
general measurement system can be mathematically described by the
.following differential equation :
(An Dn + An-1 Dn-1 +...............+ A1 D +Ao) o = (Bm Dm + Bm-1
Dm-1 +...............+ B1 D +Bo) i where the A's and B's are
constants depending upon the physical parameters of the system, Dk
is the operative derivative of the order K, o is the information
out of the measurement system and i is the input information. The
time factor in the input or driving function may correspond to step
input, ramp input, sinusoidal input or any combination of these.
The order of the measurement system is generally classified by the
value of the power of n * Zero order system : n=0 and A1, A2, .....
An = 0 * First order system : n=1 and A2, A3, ..... An = 0 * Second
order system : n=2 and A3, A4, ..... An = 0 2.5 Zero, first and
second order systems 2.5.1 Zero order systems : Consider an ideal
measuring system, i.e., a system whose output is directly
proportional to input; no matter how the input varies. The output
is a faithful reproduction of input without any distortion or time
lag. The mathematical equation relating output to input is of the
form
o = K i where K is the sensitivity of the system. This equation
of the zero order system is obtained when the power of n is set
equal to zero in the general equation for a measurement system.
That gives :
Ao o = Bo iOR o = (BBo/ Ao) i
= K i
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The static sensitivity is the only parameter which characterizes
a zero order system and its value can be obtained through the
process of static calibration. A block diagram representing
zero-order system has been shown in Fig. 2.11 (a).
Some examples of zero-order system are : mechanical levers,
amplifiers, and a linear electrical potentiometer which gives an
output voltage proportional to the displacement of the wiper. 2.5.2
First-order systems : The behavior of a first-order system is
represented by a first-order differential equation of the form
A1 D o + Ao o = Bo i (obtained by substituting n=1 in the
general equation , D is the del operator = d/dt) This equation may
be manipulated to rewrite in the following standard form :
ioo
io
oo
o
o
Kdtd
AB
dtd
AA
=+
=+1
where is the time constant ( = (A1 / Ao ) and K is the static
sensitivity (K = (Bo / Ao ) In terms of D-operator where
D = d/dt , D2 = d2/dt2 We have:
D o + o = K i( D + 1) o = K i
1+= DK
i
o
The above equation represents the standard form of transfer
operator for the first-order system ; its block diagram has been
indicated in Fig.2.11(b).
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Some examples of first order system are : Temperature
measurement by mercury-in-glass thermometers, thermocouples and
thermistor, build-up of air pressure in bellows, network of
resistance capacitance, velocity of a free falling mass. 2.5.3
Second-order systems : The input/output relationship of a second
order system is described by a differential equation of the form
:
oooooo BA
dtdA
dtdA =++ 12
2
2
(obtained by substituting n=2 in the general equation) Dividing
both sides by Ao and letting
2
0
AA
n = = undamped natural frequency, rad/s
2
1
2 AAA
o
= = damping ratio, dimensionless
o
o
ABK = = static sensitivity or steady state gain
we obtain :
ooo
n
o
n
Kdtd
dtd
=++
212
2
2
OR in terms of the D-operator we can write
)12(
)12(
2
2
2
2
++=
=++
DDK
KDD
nn
o
o
oonn
The block diagram of a second-order system or instrument is
given in Fig. 2.11 (c).
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Some examples of second-order instruments are : *spring-mass
system employed for acceleration and force measurements. *piezo
electric pickups. *U.V. galvanometer and pen control system on X-Y
plotters. Most of the mechanical instruments invariably consist of
a spring and a moving mass, and their combination provides a system
which will oscillate naturally at a given frequency. The amplitude
of the oscillation is affected by damping which .is a means of
dissipating energy in the system. Damping may occur naturally (e.g.
hysterises in materials, viscous friction at bearings etc ) or may
be introduced intentionally (e.g. dashpot similar to the automobile
damper). The damping force opposes motion and is taken proportional
to the linear/angular velocity. Examples to be solved : Example
2.1. A thermometer reads 73.5C and the true value of the
temperature is 73.15C. Determine the error and the correction for
the given thermometer. Example 2.2. A temperature transducer has a
range of 0C to 100C and an accuracy of 0'5 percent of full scale
value. Find the error in a leading of 55C. Example 2.3 A pressure
gauge of range 0.20 bar is said to have an error of 0.25 bar when
calibrated by the manufacturer. Calculate the percentage error on
the basis of maximum scale value. What would be the possible error
as a percentage of the indicated value when a reading of 5 bar is
obtained in a test ? Example 2.4 A pressure gauge having a range of
1000 kN/m2 has a guaranteed accuracy of 1 percent of full scale
deflection (i) What would be the possible readings for a true value
of 100 kN/m1 ? (ii) Estimate the possible readings if the
instrument has an error of 1 % of the true value. Example 2.5 The
pressure at a remote point has been measured by a system comprising
a transmitter, a relay and a receiver element. The specified
accuracy limits are :
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Transmitter : within 0.2% Relay : within 1.1% Receiver : within
0.7% Estimate the maximum possible error and the root-square
accuracy of the measurement system. Example 2.6 : Following data is
taken while calibrating a bourdon gauge with a dead weight gauge
tester : Actual Pressure Kgf/cm2 5 10 15 20 25 30 25 20 15 10 5
Gauge Reading Kgf/cm2 4.5 9.6 14.2 18.0 22.5 28.0 26.0 21.0 16.2
11.4 7.0 Draw the calibration, the error and the correction curves.
Make suitable comments on your results. Example 2.7 A spring scale
requires a change of 15kgf in the applied weight to produce a 2 cm
change in the deflection of the spring scale. Determine the static
sensitivity. Example 2.8 Explain the following statements: (i) A
galvanometer has a sensitivity specified of 15 mm/A. (ii) An
automatic balance has a quoted sensitivity of 1 vernier
division/0.1 mg. Example 2.9 A measuring system consists of a
transducer, an amplifier and a recorder, and their individual
sensitivities are stated as follows: Transfer sensitivity K1 = 0.25
mV/ C Amplifier gain K2 = 2.5 V/mV Recorder sensitivity K3 = 4 mm/V
What would be the overall sensitivity of the measuring system?
Example 2.10 A pressure measuring system consists of a
piezoelectric transducer, a charge amplifier and a ultra violet
charge recorder. The sensitivities of these elements are stated as
follows : Piezoelectric transducer, K1 = 8.5 pC/bar Charge
amplifier, K2 = 0.004 V/pC Ultraviolet charge recorder, K3 = 20
mm/V What would be the deflection on the chart due to a pressure
change of 30 bar? Example 2.11 How resolution is reckoned for the
analogue and digital read out devices? A force transducer measures
a range of 0150 N with a resolution of 0.1 percent of full scale.
Find the smallest change which can be measured. Example 2.12.
Distinguish between threshold and resolution (or discrimination).
The pointer scale of a thermometer has 100 uniform divisions, full
scale reading is 200C and 1/10th of a scale division can be
estimated with a fair decree of accuracy. Determine the resolution
of the instrument. Example 2.13 When a step input of 100 kgf/cm2 is
applied to a pressure gauge, the pointer swings to pressure of
102.5 kgf/cm2 and finally comes to rest at 101.3 kgf/cm2. Determine
the
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overshoot of the gauge reading and express it as a percentage of
the final reading. Also calculate the percentage error of the
gauge. Example 2.14 The dynamic performance of a thermocouple in a
protective sheath has been described by the following differential
equation :
ioo x
dtd 51025.15.225 =+
where o is the output volts and i is the input temperature in
oC. Determine the time constant and the static sensitivity of the
thermocouple.