eeg383 measurement - chapter 2 - characteristics of measuring instruments
TRANSCRIPT
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Prof Fawzy IbrahimEEG383 Ch.2 Charst of Instrms1 of 26
EEG 383 Electrical Measurements andInstrumentations
CHAPTER 2
Characteristics of Measuring Instruments
Prof. Fawzy Ibrahim
Electronics and Communication Department
Misr International University (MIU)
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Chapter Contents
2.1 Static characteristics of instruments
2.1.1 Accuracy and inaccuracy (measurement uncertainty)2.1.2 Precision/repeatability/reproducibility
2.1.3 Tolerance
2.1.4 Range or span
2.1.5 Linearity
2.1.6 Sensitivity of measurement
2.1.7 Threshold2.1.8 Resolut ion
2.1.9 Sensitiv ity to disturbance
2.1.10 Hysteresis effects
2.1.11 Dead space
2.2 Dynamic characteristics of Instruments2.2.1 Zero order instrument
2.2.2 First order instrument
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2.1 Static characteristics of Instruments If we have a thermometer in a room and its reading shows a temperature of
20C, then it does not really matter whether the true temperature of the room is
19.5C or 20.5C. Such small variations around 20C are too small to affectwhether we feel warm enough or not. Our bodies cannot discriminate between
such close levels of temperature and therefore a thermometer with an
inaccuracy of 0.5C is perfectly adequate.
If we had to measure the temperature of certain chemical processes, however,
a variation of 0.5C might have a significant effect on the rate of reaction or
even the products of a process.
A measurement inaccuracy much less than 0.5C is therefore clearly required. Accuracy of measurement is thus one consideration in the choice of instrument
for a particular application.
Other parameters such as sensitivity, linearity and the reaction to ambient
temperature changes are further considerations.
These attributes are collectively known as the static characteristics of
instruments, and are given in the data sheet for a particular instrument. It isimportant to note that the values quoted for instrument characteristics in such a
data sheet only apply when the instrument is used under specified standard
calibration conditions.
The static characteristics of measuring instruments are connected only with the
static reading that the instrument settles down to.
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2.1 Static characteristics of Instruments
Definitions: Measurand:the physical quantity being measured is called the measurand.
Measuring system is the assembly that is used to measure the measurand
(Method and Instruments).
Is-Value:The real measured value (XIS).
Should-Value:The value that it should be (VSH), reference value.
Error is the A deviation between the Is-Value and the Should-Value of a
measurand or Also it can be defined as the difference between the measured
and the true value (as per standard). Arithmetic mean (or simply mean):Sum of all values divided by the number
of quantities and is given by.
n
xxxX n
.....21
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2.1 Static characteristics of Instruments (Continued)
2.1.1 Accuracy and inaccuracy (measurement uncertainty)
The accuracyof an instrument is a measure of how close the output reading of theinstrument is to the correct value or How close is the mean measurement of a
series of trials to the true value?
Where xnis the nth measured value .
In practice, it is more usual to quote the inaccuracy figure rather than theaccuracy figure for an instrument.
Inaccuracyis the extent to which a reading might be wrong, and is often quoted as
a percentage of the full-scale (f.s.) reading of an instrument.
If, for example, a pressure gauge of range 010 bar has a quoted inaccuracy of
1.0% f.s. ( 1% of full-scale reading), then the maximum error to be expected
in any reading is 0.1 bar. This means that when the instrument is reading 1.0bar, the possible error is 10% of this value.
Thus, if we were measuring pressures with expected values between 0 and 1
bar, we would not use an instrument with a range of 010 bar. The term
measurement uncertainty is frequently used in place of inaccuracy.
%100)(
1
sh
shn
X
XxAccuracy
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2.1 Static characteristics of Instruments (Continued)2.1.2 Precision/repeatability/reproducibilityPrecisionHow much do the measurements vary from trial to trial?
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.
Precision is often, though incorrectly, confused with accuracy. High precision
does not imply anything about measurement accuracy. A precise system is notnecessarily an accurate system and vice versa.
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.
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.
Both terms thus describe the spread of output readings for the same input. This
spread is referred to as repeatability if the measurement conditions are
constant and as reproducibility if the measurement conditions vary.
valuemeantheisX
X
Xxecision n
%100
)(1Pr
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2.1 Static characteristics of Instruments (Continued)2.1.2 Precision/repeatability/reproducibility Figure 2.1 shows the results of tests on three industrial robots that were
programmed to place components at the center of the concentric circles shown,
and the black dots represent the points where each robot actually deposited
components at each attempt.
Both the accuracy and precision of Robot 1 are shown to be low in this trial.
Robot 2 consistently puts the component down at approximately the same
place but this is the wrong point. Therefore, it has high precision but low
accuracy. Finally, Robot 3 has both high precision and high accuracy, becauseit consistently places the component at the correct target position.
Fig. 2.1 Comparison of accuracy and precision (a) Robot 1; (b) Robot 2 and (c) Robot 3.
(a) Low precision, low accuracy (b) High precision, low accuracy(c) High precision, high accuracy
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First digit (A)
Second digit (B)
Multiplier (C)
Tolerance (D)
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2.1 Static characteristics of Instruments (Continued)2.1.3 Tolerance Tolerance describes the maximum deviation of a manufactured component
from some specified value.
For instance, crankshafts are machined with a diameter tolerance quoted as so
many microns (10-6 m).
Another example, electric circuit components such as resistors have tolerances
of perhaps 5%. One resistor chosen at random from a batch having a nominal
value 1000W and tolerance 5% might have an actual value anywhere between
950W and 1050 W. The colored bands that are found on a resistor as shown in Fig. 2.2 can be
used to determine its resistance. The first and second bands of the resistor give
the first two digits of the resistance, and the third band is the multiplier which
represents the power of ten of the resistance
value. The final band indicates what tolerance
value (in %) the resistor possesses. The
resistance value written in equation form is:
Fig. 2.2 The resistors
representation
D%10AB C (2.1)
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2.1 Static characteristics of Instruments (Continued)2.1.2 Tolerance
Fig. 2.2 (b) The color code for resistors.
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2.1 Static characteristics of Instruments (Continued)2.1.4 Range or span
The range or span of an instrument
defines the minimum and maximum
values of a quantity that the
instrument is designed to measure.
2.1.5 Linearity Linear device is the one in which the
relation between its output reading is
linearly proportional to the quantitybeing measured as shown Fig. 2.3 .
The Xs marked on Fig. 2.3 show a
plot of the typical output readings of
an instrument when a sequence of
input quantities are applied to it.
The non-linearity is then defined asthe maximum deviation of any of the
output readings marked X from this
straight line.
Non-linearity is usually expressed as a
percentage of full-scale reading.
Fig. 2.3 Instrument output characteristic.
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2.1.6 Sensitivity of measurement The sensitivity of measurement is a measure of the change in instrument output
that occurs when the quantity being measured changes by a given amount.Thus, sensitivity is the ratio:
The sensitivity of measurement is therefore the slope of the straight line drawn
on Fig. 2.3.
Example 2.1The following resistance values of a platinum resistance thermometer were
measured at a range of temperatures. Determine the measurement sensitivity
of the instrument in ohms/C.
Solution
If these values are plotted on a graph,
the straight-line relationship betweenresistance change and temperature
change is obvious, so it is a linear device.
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2.1 Static characteristics of Instruments (Continued)
valuemeasuredinChange
readingoutputinChangeySensitivit
CySensitivit o/30
7
200230
307314
(2.2)
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2.1 Static characteristics of Instruments (Continued)2.1.7 Threshold If the input to an instrument is gradually increased from zero, the input will have
to reach a certain minimum level before the change in the instrument outputreading is of a large enough magnitude to be detectable. This minimum level of
input is known as the threshold of the instrument.
Manufacturers may specify threshold for instruments as absolute values or as a
percentage of full-scale readings.
As an illustration, a car speedometer typically has a threshold of about 5 km/h.
This means that, if the vehicle starts from rest and accelerates, no outputreading is observed on the speedometer until the speed reaches 5 km/h.
2.1.8 Resolution How finely can we and/or the instrument separate one value from another thats
close to it?
Resolution is specified as an absolute value or a percentage of f.s. deflection.
One of the major factors influencing the resolution of an instrument is how finely
its output scale is divided into subdivisions.
For instance, if we have 4-bit and 8-bit multimeters to show the range from 0-
10V, the best resolution for both of them:
for 4-bit the resolution is: for 8-bit8
1039.0625 .
2mVmV625
2
104
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2.1 Static characteristics of Instruments (Continued) Using a car speedometer as an example again, this has subdivisions of
typically 20 km/h. This means that when the needle is between the scale
markings, we cannot estimate speed more accurately than to the nearest 4km/h. This figure of 4 km/h thus represents the resolution of the instrument.
2.1.9 Sensitivity to disturbance All calibrations and specifications of an instrument are only valid under
controlled conditions of temperature, pressure etc. These standard ambient
conditions are usually defined in the instrument specification.
As variations occur in the ambient temperature, pressure, etc., the sensitivity todisturbance is a measure of the magnitude of this change.
Such environmental changes affect instruments in two main ways, known as
zero drift and sensitivity drift.
1. Zero drift or biasdescribes the effect where the zero reading of an instrument
is modified by a change in ambient conditions. This causes a constant error
that exists over the full range of measurement of the instrument. The mechanical form of bathroom scale is a common example of an instrument
that is prone to bias. It is quite usual to find that there is a reading of perhaps
1 kg with no one stood on the scale.
If someone of known weight 70 kg were to get on the scale, the reading would
be 71 kg, and if someone of weight 100 kg, the reading would be 101 kg.
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2.1 Static characteristics of Instruments (Continued)
2.1.9 Sensitivity to disturbance
1. Zero dr ift or bias If a thumbwheel is usually provided that can be turned until the reading is zero
with the scales unloaded, thus removing the bias.
Zero drift is also commonly found in instruments like voltmeters that are
affected by ambient temperature changes. Typical units by which such zero drift
is measured are volts/C.
A typical change in the output characteristic of an instrument subject to zero
drift is shown in Figure 2.4(a).
2. Sensitivity drift(also known as scale factor drift) defines the amount by which
an instruments sensitivity of measurement varies as ambient conditions
change.
Figure 2.4(b) shows what effect sensitivity drift can have on the output
characteristic of an instrument. Sensitivity drift is measured in units of the form(angular degree/bar)/C.
If an instrument suffers both zero drift and sensitivity drift at the same time, then
the typical modification of the output characteristic is shown in Figure 2.4(c).
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2.1 Static characteristics of Instruments (Continued)
2.1.9 Sensitivity to disturbance
Fig. 2.4 Effects of disturbance: (a) zero drift; (b) sensitivity drift; (c) Both drifts.
(a)
(b)
(c)
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Example 2.2
The A spring balance is calibrated in an environment at a temperature of 20Cand has the deflection/load characteristic (Table 1).
It is then used in an environment at a temperature of 30C and the following
deflection/ load characteristic is measured.
Determine per C change in ambient temperature the following:
(a) The zero drift. (b) The sensitivity drift.
Solution
At 20C, deflection/load characteristic is a straight line. Sensitivity = 20 mm/kg.
At 30C, deflection/load characteristic is still a straight line. Sensitivity = 22
mm/kg.
Bias (zero drift) = 5mm (the no-load deflection)
Sensitivity drift = 2 mm/kg
(a) Zero drift/C = 5/10 = 0.5 mm/C
(b) Sensitivity drift/C = 2/10 = 0.2 (mm per kg)/C
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2.1 Static characteristics of Instruments (Continued)
2.1.9 Sensitivity to disturbance
Table 1 at 20C Table 2 at 30C
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2.1 Static characteristics of Instruments (Continued)2.1.10 Hysteresis effects If the input measured quantity to the instrument is steadily increased from a
negative value, the output reading varies in the manner shown in curve (a) ofFig. 2.5.
If the input variable is then steadily decreased, the output varies in the manner
shown in curve (b) of Fig. 2.5.
The non-coincidence between
these loading and unloading
curves is known as hysteresis. Two quantities are defined,
maximum input hysteresis and
maximum output hysteresis,
as shown in Fig. 2.5. These
are normally expressed as a
percentage of the full-scale
input or output reading
respectively.
Fig. 2.5 Instrument characteristic with hysteresis.
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2.1 Static characteristics of Instruments (Continued)2.1.11 Dead space Dead space is defined as the range of different input values over which there is
no change in output value. Any instrument that exhibits hysteresis also displaysdead space, as marked on Figure 2.6.
Some instruments that do not
suffer from any significant
hysteresis can still exhibit a
dead space in their output
characteristics.
Fig. 2.6 Instrument characteristic with dead
space.
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2.2 Dynamic characteristics of Instruments The dynamic characteristics of a measuring instrument describe its behavior
between the time a measured quantity changes value and the time when the
instrument output attains a steady value in response. As with static characteristics, any values for dynamic characteristics quoted in
instrument data sheets only apply when the instrument is used under specified
environmental conditions.
Outside these calibration conditions, some variation in the dynamic parameters
can be expected.
In any linear, time-invariant measuring system, the following general relation
can be written between input (x) and output (y) for time t > 0:
where x is the measured quantity, y is the output reading and a0. . . an, b0. . .
bmare constants.
If we limit consideration to that of step changes in the measured quantity only,
then equation (2.3) reduces to:
xbdt
dxb
td
xdb
td
xdb
td
xdb
yadt
dya
td
yda
td
yda
td
yda
m
m
mm
m
mm
m
m
n
n
nn
n
nn
n
n
012
2
21
1
1
012
2
21
1
1
....
....
(2.3)
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2.2 Dynamic characteristics of Instruments (Continued)
2.2.1 Zero Order Instrument If all the coefficients a1. . . another than a0in equation (2.4) are assumed zero,
then:
where K is a constant known as the instrument sensitivity as defined earlier. Any instrument that behaves according to equation (2.5) is said to be of zero
order type.
Following a step change in the measured quantity at time t, the instrument
output moves immediately to a new value at the same time instant t, as shown
in Figure 2.7.
A potentiometer, which measures motion, is a good example of such aninstrument, where the output voltage changes instantaneously as the slider is
displaced along the potentiometer track.
xbya
dt
dya
td
yda
td
yda
td
yda
n
n
nn
n
nn
n
n 0012
2
21
1
1 ....
(2.4)
kxyxa
byorxbya
0
000
(2.5)
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2.2 Dynamic characteristics of Instruments (Continued)
Fig. 2.7 Zero order instrument characteristic. Fig. 2.8 First order instrument characteristic
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2.2 Dynamic characteristics of Instruments (Continued)2.2.1 First Order Instrument If all the coefficients a2 . . . an except for a0 and a1 are assumed zero in
equation (2.4) then:
Any instrument that behaves according to equation (2.6) is known as a first
order instrument.
If equation (2.6) is solved analytically, the output quantity y in response to a
step change in x at time t varies with time in the manner shown in Fig. 2.8. Thetime constant of the step response is the time taken for the output quantity, y to
reach 63% of its final value.
A large number of other instruments also belong to this first order class: this is
of particular importance in control systems where it is necessary to take
account of the time lag that occurs between a measured quantity changing in
value and the measuring instrument indicating the change.
Fortunately, the time constant of many first order instruments is small relative to
the dynamics of the process being measured, and so no serious problems are
created.
(2.6)xbyadt
dya 001
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2.2 Dynamic characteristics of Instruments (Continued)2.2.1 First Order Instrument
Example 2.3
A balloon is equipped with temperature and altitude measuring instruments andhas radio equipment that can transmit the output readings of these instruments
back to ground. The balloon is initially anchored to the ground with the
instrument output readings in steady state. The altitude-measuring instrument
is approximately zero order and the temperature transducer first order with a
time constant, of 15 seconds. The temperature on the ground, T0, is 10C and
the temperature Txat an altitude of x meters is given by the relation:
(a) If the balloon is released at time zero, and thereafter rises upwards at a
velocity of V = 5 meters/second, draw a table showing the temperature and
altitude measurements reported at intervals of 10 seconds over the first 50
seconds of travel. Show also in the table the error in each temperature reading.
(b) What temperature does the balloon report at an altitude of 5000 meters?
Solution
(a) The transducer is assumed to be a first order differential. Let the temperature
reported by the balloon at some general time t be Tr. Then Tx is related to Trby
the relation:
xTTx 01.00 (2.7)
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2.2 Dynamic characteristics of Instruments (Continued)2.2.1 First Order Instrument
Example 2.3 Solution
Where in Eq. (2.6) the output, y = Tr(the reported reading), the input, x = Tx(the temperature Txat an altitude of x ), the constants, a1= , a0= 1, and b0= 1.
Substitute for Txfrom Eq.(2.7), we get:
The solution of the differential (2.9) consists of two parts: The transient or
complementary function part (TrCF) when Tx= 0, and the particular integral part
(TrPI) in the form:
The transient or complementary function part of the solution of Eq. (2.8) whenTx= 0 is given by:
xrr TT
dtdT (2.8)
ttxTdt
dTr
r
05.010)5(01.01001.010 (2.9)
0 rr T
dt
dT
15// tt
rCF AeAeT
rPIrCFr TTT (2.10)
(2.11)
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2.2 Dynamic characteristics of Instruments (Continued)2.2.1 First Order Instrument
Example 2.3 Solution
Thus, the whole solution is given by:
Using Eq. (2.15) to calculate Trfor various values of t, the following table can be
constructed:
(b) At 5000 m, t = 1000 seconds. Calculating Tr from Eq. (2.15) we get:
(2.15)teTTT t
rPIrCFr 05.075.1075.0 15/
CxeT o
r 25.39100005.075.1075.0 15/1000