INME_Module2

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Module 2: Characteristics of Instruments
Chapter Objectives
At the end of this chapter, you will be able to:
• Calculate the accuracy of measured variables
• List the probable measurement errors
Active and Passive Instruments
3
 Passive: output produced 
completely by quantity 
measured
 No external power source is 
required
• Active: output signal is 
modulated magnitude of 
external power source
• Quantity measured can be 
amplified for better 
resolution and reliability 
Passive pressure gauge
Float-type 
tank level 
gauge
Example: Level Measurements
4
 Passive
• Active
Null-Type and Deflection-Type
5
 Null: quantity measured by putting 
external value equivalent. 
 Better accuracy as external value 
can be easily chosen
• Deflection: quantity 
measured using equivalent 
motion. 
• Accuracy depends on the 
linearity and calibration of 
pointer spring
Passive pressure gauge
Dead-weight pressure gauge
Example: Wheatstone Bridge
6
Potentiometer VS. Wheatstone 
Bridge
 Potentiometer - measures an 
unknown voltage by opposing with 
a known voltage, without drawing 
current from the voltage source 
being measured
 Wheatstone bridge - circuit used 
to measure unknown electrical 
resistance by balancing two legs of 
a bridge circuit - one leg includes 
the unknown component –
indicated by zero deflection on 
meter
Wheatstone bridge
Analog and Digital
7
 Analog: displayed 
continuous proportional 
changes to actual 
measurements
• Digital: quantity 
displayed in terms of 
discrete equivalent 
values after processing
Exhibit A: Bourdon Tube Pressure Gauge
8
Exhibit B: Bourdon Tube Pressure Gauge
9
Example: Electrodynamic Displacement 
Instruments
10
Instrument Characteristics
11
 Static properties: 
characteristics when 
measurement remains 
constant or has reached 
steady-state
• Dynamic properties: 
relationship between 
input and output when 
measured quantity varies
Static Characteristics
12
 Accuracy
 Precision
 Sensitivity
 Resolution
 Threshold
 Drift
 Error
 Repeatability
 Reproducibility
 Dead Zone
 Backlash
 True Value
 Hysteresis
 Linearity
 Range (Span)
 Bias
 Tolerance
 Stability
Accuracy of Measurement
13
 Accuracy: 
closeness/conformity to 
the true value of a 
quantity under 
measurement
 Precision: reproducibility 
of the measurement 
(measure / difference of 
successive 
measurements) 
Accuracy of Measurement
14
 Sensitivity: ratio of instrument output over response to 
change of input/measured variable
 Resolution: smallest change in measured value 
instrument able to record
 Significant Figures: precision of measurements and 
reported result
 Error: deviation from the true value of measured variable
variable measured Δ
output instrument Δ
ySensitivit 
Example: Pressure Measurement
15
Pressure Gauge: measurement range 0 -10 bar
%0.1fsFull-scale reading precision:
bar0.1
bar

 10%0.1maxeMaximum error:
%10100
1
1)1.01(



bar
barbar 
bar 1e
Measurement error for 
reading 1 bar (in %):
%100


value true
value truevalue measured
error %
Example: Tolerance
16
Resistors:
pack of resistors with R = 1000 W
%5tolerance:
WW 950%51000minRMinimum value:
Maximum value: WW 1050%51000maxR
Linearity and Measurement Sensitivity
17
Linearity
 instrument output is 
proportional to measured 
quantity
Sensitivity:
Slope
Value
deflection Scale
ySensitivit


Sensitivity to Disturbance
19
 Instrument specifications are described for controlled 
conditions (ambient conditions) eg. pressure, temperature.
 Instrument static properties vary as ambient conditions 
change, described in zero drift and sensitivity drift.
Sensitivity to Disturbance
20
Zero Drift
 AKA bias
 Affects zero reading when 
condition changes, constant 
error across full range
Sensitivity Drift
 AKA scale factor drift
 Instrument sensitivity changes 
when condition changes
Example 1: Measurement Sensitivity
21
Platinum resistance thermometers:
resistivity measured at varying temperatures
Measurement sensitivity:
C/233.0
C30
7
sens


W
W
R
R T
307 200
314 230
321 260
328 290
305
310
315
320
325
330
150 200 250 300
R
e
s
is
ta
n
c
e
 (
W
)
Temperature (C)
Resistance vs. Temperature
Example 2: Measurement Sensitivity
22
Spring Balance:
calibrated at 20C
Measurement sensitivity:
kg/mm20
kg1
mm20
)20(sens CD 
Load (kg) 0 1 2 3
Deflect (mm) 0 20 40 60
used at 30C
Load (kg) 0 1 2 3
Deflect (mm) 5 27 49 71
0
20
40
60
80
0 1 2 3
D
e
fl
e
c
ti
o
n
 (
m
m
)
Load (kg)
Sensitivity
Deflection (mm) Deflection (mm)
kg/mm22
kg1
mm22
)30(sens CD 
Zero drift = 5 mm
Sensitivity drift = 2 mm/kg
Sensitivity drift coefficient = 2 (mm/kg)/10 C 
= 0.2 (mm/kg) /C
Zero drift coefficient = 5 mm/10 C = 0.5 mm /C
Statistical Analysis of Measurement
23
 Arithmetic mean (average) of readings:
where x = measured value,
n = number of reading / 
measurement
 Average reading the most likely value for measured variable
 Deviation, d : 
 Average deviation
 Average amount of measurement error
n
x
n
xxx
x
n
i in  

 121 
xxdxxdxxd nn  2211
n
xx
n
ddd
D
n
i
n
 


 121 
Statistical Analysis of Measurement
24
 Standard deviation,  :
 Probable error:
 random errors that lie scattered within 50% probability 
region around mean
n
d
n
ddd
n
i
i
n



 1
2
22
2
2
1 

6745.0r
Example: Statistical Analysis
25
Normal Error Distribution
26
Probable error:
6745.0r
Deviation
σ
Fraction Area
0.6745 0.5000
1.0 0.6828
2.0 0.9546
3.0 0.9972
 
 dDeDDDP
exF
D
D
D
mx





2
1
22
22
2/
21
2/)(
2
1
)(
2
1
)(




Example 3: Statistical Analysis
27
Reading Deviation
R (Ω) d d2
100.2 0.2 0.04
100.3 0.3 0.09
99.8 -0.2 0.04
100.5 0.5 0.25
99.3 0.7 0.49
100.4 0.4 0.16
100.1 0.1 0.01
99.5 -0.5 0.25
99.7 0.3 0.09
99.7 0.3 0.09
W
  0.100
10
5.999
10
10
1
ave
i iR
R
W 2.13.995.100minmaxRange RR
Determine:
a) measurement range
b) average reading
c) Deviation, d
- as shown in table -
Precision:
ONE decimal
Example 3: Statistical Analysis
28
Reading Deviation
R (Ω) d d2
100.2 0.2 0.04
100.3 0.3 0.09
99.8 -0.2 0.04
100.5 0.5 0.25
99.3 0.7 0.49
100.4 0.4 0.16
100.1 0.1 0.01
99.5 -0.5 0.25
99.7 0.3 0.09
99.7 0.3 0.09
W





 4.0
110
51.1
1
1
2
n
d
n
i
i

Determine:
d) standard deviation
e) probable error
W 2763.06745.0 r error, Probable
Example 4: Statistical Analysis
29
1370
409
11
2 


deviation
reading (mean) Average
 t,measuremen of Number n
89.76745.0
7.11137
137
10
1370
1
2









r
n
 error, Probable
 deviation, Standard
deviation
 Variance, 2
Hysteresis
30
Hysteresis:
 Different increasing and reducing trend
 Hysteresis in magnetic element/spring : non-
coincidence between loading and unloading.
Dead Space
31
 Also known as Dead Zone, is the range of input values 
over which there is no change in the output
 Example: Backlash in gears used to measure rotational 
velocity
Types of Errors
32
 Gross Error: human error due to incorrect use of equipment, wrong 
observation, carelessness etc.
 Systematic Errors
 Instrumental: inherent to measuring eqpt.
 Static: limitation of device
 Dynamic: inability to respond to change in measured variable 
 Environmental: due to change in external conditions 
(temperature, pressure, humidity, magnetic / electric fields) 
 Random Errors: due to unknown 
causes
Gross Errors
33
Caused by human error due to incorrect use of equipment, wrong 
observation, carelessness etc.
Other 
Examples
Estimation
Reduction 
Methods
• Erroneous 
calculations
• Improper choice of 
instruments
• Incorrect adjustment
• Neglectof loading 
effects
• Not possible to 
estimate
• Careful attention and 
observation
• Awareness of 
instrument limitations
• Taking at least 3 
readings
• >1 observer to observe 
critical data
Systematic Errors
34
Can come from 2 sources: 
1. Instrumental: inherent to measuring equipment
 Static: limitation of device
 Dynamic: inability to respond to change in measured variable 
2. Environmental: due to change in external conditions (temperature, 
pressure, humidity, magnetic/electric fields) 
Systematic Errors
35
1. Instrumental: inherent to measuring equipment
Examples
Estimation
Reduction 
Methods
• Bearing friction
• Nonlinearities
• Calibration errors
• Damaged eqpt
• Loss during 
transmission
• Compare to more 
accurate standard
• Check if error is 
constant or 
proportional
• Careful calibration
• Inspection of eqpt
• Applying correction 
factors
• High gain feedback –
reduce error
• Intelligent instruments
Systematic Errors
36
2. Environmental: due to change in external conditions
Estimation
Reduction 
Methods
• Hermetically seal eqpt
and components
• Signal filtering
• Maintain constant 
temperature and 
humidity
• Shield eqpt from stray 
magnetic fields
• Use eqpt not affected 
greatly by 
environmental 
changes
• Careful monitoring 
of changes
• Calculate expected 
changes / drifts
Random Errors
37
• Unknown events that causes small variations in measurements.
• Random and unexplainable
• Estimate: Take many readings and conduct statistical analysis
• Methods of reduction:
1. Careful design of eqpt to reduce unwanted interference
2. Statistical analysis to determine best estimate and/or outlier values
Error Reduction
39
 Inspection and Care: ensure measurement integrity
 Calibration: correct measurement drift and scaling factor
 Method of opposing input or manual adjustment: compensate 
environmental bias in measurement
 High gain feedback: eliminate error
 Signal filtering: reduce noise
 Intelligent instruments: attenuate error 
and amplify signals
Example 5: Maximum and Likely Errors
40
THREE separate sources of error are identified:
• system loading: 1.2%
• environmental changes: 0.8%
• calibration error: 0.5%
%5.2)%5.08.02.1(
error possible Maximum


%53.1
%5.08.02.1
errorLikely 
222



Aggregated Errors (1)
41
 When a measurement system is made up of several components -
each with its error estimate – the combined error needs to be
aggregated
 For 2 outputs y and z of separate components with maximum errors of
ay and bz respectively, - where a and b are errors fractions - the sum
/ difference S is
22 )()(
)(
bzaye
ezyS


 error,likely where
Error in a sum / difference
Note: Here e is 
the absolute 
error.
Example 6: Errors in a Sum
42
The total resistance of 2 resistors (to 3 significant figures) each with a 
tolerance of:
Solution:
THREE significant figures
%5.6123.46
%0.13.99
2
1


 
 
R
R
%2.2145
158.3423.145
)(
158.3988.8986.0
)123.46065.0()3.9901.0(
21
22
W
W



 
 
eRRR
e
T
Note: a = 0.01, b = 0.065
error fractions
Example 7: Errors in a Difference
43
The difference between 2 voltage measurements, VD:
Solution:
The maximum likely error, 
%6.2020
123.420
)(
123.4161
)8005.0()10001.0(
21
22





 V 
V 
eVVV
e
T
Note: The 
percentage error 
increases for 
difference between 
measurements.
5%V
1%V


80
100
2
1
V
V
Exercise 1
44
A fluid flow rate is calculated from the difference in pressure
measured on both sides of an orifice plate. The pressure
measurements are 10.0 and 9.5 bar and the error in the
pressure measuring instruments is specified as 0.1%.
Find the difference in pressure.
Aggregated Errors (2)
45
 For 2 outputs y and z of separate components with maximum errors of
ay and bz respectively, the product P is
and the quotient Q is
e
z
y
Q 
Error in a product / quotient
eyzP 
22 bae  where
Note: Here e is the 
fraction / percentage 
error.
Example 8: Errors in a Product
46
If the density of a substance is calculated from measurements of its mass 
and volume, where the respective errors are 2 and 3%, find the maximum 
likely error in the density value.
Solution:
Since density is mass per unit volume , then the likely error is
V
m

036.00013.0
03.002.0 22

e
Example 9: Errors in a Product
47
The current through RT is measured at 3.15 mA using a 0 – 10 mA 
ammeter with full-scale accuracy of  0.1%. Determine the voltage across 
RT = 2kW with tolerance 5%.
To calculate likely relative error in 
voltage V
In absolute error

%01.5%5317.0 22 e
Solution: Voltage, V = IR
Maximum error for I,
Therefore at reading of 3.15 mA, 
mA 10.0)mA 10%1.0(max e
%317.0OR 1017.3
mA 15.3
mA 15.3mA)01.015.3(
3
mA15.3




e V 316.0V 3.6%01.5 e
eeV W V 3.6k 2mA 15.3
%01.5V 3.6
V 316.0 3.6

V
To calculate likely relative error in 
power P (2 variables in product, 1 in 
ratio)
In absolute error

Example 10: Errors in a Ratio
48
Power calculation using following resistance value and voltage measured 
across resistor:
scale-full %20 inaccuracy , voltmeterV 10-0on V 4.5
10%820
.V
R

W
%02.10%1044.02 22 e
e
R
V
P 
2
Solution: Power, 
Maximum error for V,
Therefore at reading of 4.5 V, 
 V02.0)10%2.0(max e
%44.0OR 104.4
5.4
5.4)02.05.4(
3
5.4




Ve
mW 47.2mW 7.24%02.10 e
eeP  mW 7.24820/5.4 2
%02.10mW 24.7
mW 47.2 7.24

P
Dynamic Inputs
51
Periodic input
Transient input Random signal
Dynamic Characteristics 
52
 Describes instrument behavior from the time measured 
quantity changes until the time instrument output reaches 
steady value 
 Numbered and categorized according to order of the 
derivative
xb
dt
dx
b
dt
xd
bya
dt
dy
a
dt
yd
a
dt
yd
a
m
m
mn
n
nn
n
n 01011
1
1 


 
where x = input
y = output
a, b = coefficient
Dynamic Characteristics 
53
Zero order
• If all coefficients a1…an other than a0 is zero, then
where K is the instrument sensitivity.
• Instrument output, y changes immediately at the 
same time, t as change in measured variable, x
• Example: Potentiometer, where slider motion/rotation 
changes resistance instantaneously.
Kxx
a
b
yxbya 
0
0
00 or 
Dynamic Characteristics 
54
 First Order
If all coefficients a2…an other than a0 and a1 is zero then
xbya
dt
dy
a 001 
01
0
)(
)(
)(
asa
b
sX
sY
sG


In transfer function
(Laplace transform)
or
0
1
0
0
1
)(
a
a
a
b
K
s
K
sG





 and where
 
Dynamic Characteristics 
55
First Order
 Instrument output, y changes in time monotonously in response to 
step change in measured variable, x
 Time constant,  is the time taken for output to reach 63% of final 
value
 Time lag has to be considered when taking measurement
 Example: Thermocouple, which 
output e.m.f does not change 
immediately with temperature change
Dynamic Characteristics 
56
Second Order
 Response of a 2nd order instrument depends on damping factor, 
  = 0.707 is preferable as it provides critical damping
A = undamped, constant 
oscillation
B = underdamped
C = critically damped
D = damped
E = overdamped
Dynamic Characteristics 
57
Second Order
 If all coefficients a3…an other than a0, a1 and a2 is zero then
 Example: Accelerometer which 
has a damping factor between 
0.6 – 0.8.
xbya
dt
dy
a
dt
yd
a 0012
2
2 
In transfer 
function
0
1
20
1
0
0
2
2
2
2
,
1
2
)(
a
a
aa
a
a
b
K
ss
K
sG
n
nn
















 where
 
K = sensitivity, 
 = damping factor, 
n = undamped natural 
frequency
Example 10: Seismic Motion Transducer
58
Second Order
 Attached toobject whose motion is 
measured
xyzxyKxy
dt
d
c
dt
yd
m  0)()(
2
2