Electronic Instruments (EI)

Lecturer: Dr. Cheng-Kai Lu

Phone: (02)7749-3554

Office: TD302/BAIR Lab

Email: cklu@ntnu.edu.tw

Chapter Objectives

2

At the end of this chapter, you will be able to:

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• Discuss the principle of measurement system

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• Describe the standard of measurements

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• Calculate the accuracy of measured variables

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• List the probable measurement errors

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3

Introduction

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Measurement

Measurement is the process of observing and recording the observations that are collected as part of a research effort.

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5

Measurement

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Measurement: use of instruments to determine present state, condition or characteristic of system in quantitative terms (quantity of physical variable)

Instrument: device to determine value/magnitude of quantity/variable that includes

the sensing device, transducer element, data processing and representation

Example: DC current meter

6

Elements of Measuring System

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Transducers: a combination of sensor(s) which responds to physical state/condition

to provide output as a function of the measurand(s) and variable conversion elements

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Transmitters: combining signal processing, amplification and transmission capabilities

7

Application of Measurement System

• Measurement of System Parameters
• Control of Process or Operations
• Simulation of Systems Conditions
• Experimental Design Studies
• Perform System Manipulations
• Material Testing and Standards Specifications
• Verification of Theories
• Quality Control
• Measurement for commerce

Feedback Control Systems

8

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• Control variable measurements
• Comparator
• Control

Example : Temperature Control

9

Units of Measurement

10

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• Physical property measured described in terms of type (unit) and magnitude
• The International System of Units (SI)

- Fundamental units/base units: mass, length, time

- Derived units: area, speed, acceleration etc.

Derived Units

11

Derived units: originate from physical law

Supplementary unit

12

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• Dimensionless derived units: the radian (rad) and the steradian (sr).

Example: Electromotive Force

13

⎢ ⎥

⎣ ⎦

3

As

^⎡kg.m^{2 ⎤}

volt =

20

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Emf (volt) is defined as energy required to move and electron charge

_{emf [volt]=} workdone [Joule]

charge [Coulomb]

₌ Force [N]×distance [m] ₌ mass [kg]×acceleration [m/s² ]×distance [m] current [A]× time [s] current [A]× time [s]

Standards of Measurement

14

• International Standards: units of measurement closest possible accuracy production/technology can produce, maintained by International Bureau of Weight and Measures

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• Primary Standards: representing fundamental and some derived units, maintained by national standards libraries (eg. National measurement laboratory R.O.C: NML, Phsikalisch-Technische Reicshsanstalt Germany, Underwriters Laboratories (UL), SIRIM), independently calibrated by absolute measures.

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• Secondary Standards: reference standard used by industry, checked locally against other reference standards.

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• Working Standards: principal tools used by measurement laboratory to check and calibrate general laboratory equipment

Centimetre–Gram–Second system of units(CGSe) Absolute System

15

Q(statcoulomb) = cm^{3/ 2}g^{1/ 2}s⁻¹

₌ Q₁Q₂

_r 2

F = k ^Q1Q2

εr ²

• Meter (Length): ten-millionth part of distance from pole to equator measured along meridian through Paris
• Gram (Mass): mass of cubic centimeter of distilled water at 4°C at normal atmospheric pressure (760 mm Hg)
• Second (Time): 1/86400 of mean solar day

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• Statcoulomb (electric charge): derived CGSe units of electric charge

SI Units of Measurements

16

Mass and Length

17

• Mass (kilogram): defined as mass of a cubic decimeter of water, materially represented by an International Prototype Kilogram (platinum-iridium cylinder), preserved at International Bureau of Weights and Measures in Paris.

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• Length (meter): defined as 1/10⁴ part of meridional quadrant through Paris, redefined as distance of light travel in 1/299,792,458 sec, represented as distance between two lines on a platinum-iridium plate

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• Matter (mole): Number of atoms in an 0.012-kg mass of carbon 12

Time and Frequency

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• Time (second): originally defined as 1/86,400 of the mean solar day. Ephemeris Time (ET) based on rotation of the moon around the earth as 1/31,556,925.9747 of the tropical year.

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• Frequency (1/second): atomic resonator used as the constant clock based originally on cesium atom (f = 9,192,631,770 Hz).

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• Time/Frequency Calibration: standard transmitted between standard lab using HF transmission (after correcting for doppler effect). Loran-C is a low frequency navigation system transmitting shaped pulses of 100kHz carrier frequency with 20kHz bandwidth is the best method for worldwide calibration of time. Currently, TV and digital communication network provides world time calibration.

Basic Definitions

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v [ ms⁻¹ ]= ^{∂x [m]}

∂t [s]

• Acceleration (m/s²): rate of change of velocity wrt time

_{a [ ms−2 ]=} ∂v [m/s]

dt[s]

_{−2 ] =} ∂(mv)[kg ms⁻¹ ]

F [N = kg ms

• Speed/velocity (m/s): rate of change of distance wrt time

∂t [s]

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• Momentum: mass x velocity

• Force (N): rate of change of momentum with time

p [N = kg ms

_{−2\1 ] =}

m [ kg]

× v [ ms⁻¹ ]

Basic Definitions

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0

x

2 -2 _] -2 _]

⋅ x [m]

= _∫ Fdx = F [kg ms

E [J = kg m s

• Power (Watt): rate of work done

∂t [s]

p [watt = J s⁻¹ ]= ^{∂E [J]}

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• Energy (Joule): distance integral of force, work done

Facts

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• Gravitational force

• Electron Charge:

• Electron Volts:

= 1.602 ×10⁻¹⁹ C

1

6.24 ×10¹⁸

Q =

J

1

6.24 ×10¹⁸

1 eV =

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g = 9.81 m/s²

Electrical Standards

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• Current (Ampere): fundamental unit defined as flow or charge in two straight parallel conductors of infinite length/negligible cross section placed 1 meter apart in vacuum to produce 2×10^-7 newton per meter length.

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Ampere balance at the US National Bureau of Standards

Coils

Ohm's Law

Electrical Standards

23

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• Resistance (Ohm): a coil of manganin alloy wire with electrical resistivity and temperature coefficient mounted in double walled sealed container.

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• Voltage: originally based on saturated standard cell.

New standard based on thin-film junction irradiated with microwave energy to develop the junction voltage:

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Basic Definitions

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2 -1

emf V

_{-2 ]=} workdone [J] ₌ workdone [kg m² s^-1 ]

charge [C] charge [As]

[ = kg m A s

• Resistance (Ohm): rate of resistance to current flow

=

∂I [A] ∂I [A]

∂V [V] ∂V [kg m²A^-1s^-3 ]

R [Ω]=

• Electric Field (V/m): potential per meter

=

∂x [m] ∂x [m]

∂V [V] ∂V [kg m²A^-1s^-3 ]

E [V/m]=

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• Current (Ampere): rate of flow of charge

_{I [A]=} ∂q [C] ₌ ∂q [A - s]

∂t [s] ∂t [s]

• Potential (volt): work done to bring unit charge from infinity to same point

Electrical Standards

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• Capacitance (Farad): measured with a Maxwell DC commutated bridge. Standard capacitor constructed from interleaved metal plates with air dielectric.

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• Inductance (Henry): primary standard derived from ohm and farad as the limitation of the large geometrical construction required.

Basic Definitions

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• Permittivity (farad/m): strength of electric field line per meter

ε [Fs^-1 ]= [kg^-1 m⁻³ A²s⁴ ]

• Capacitance (farad):

-1 -2 2 4

d [m] V [V]

_{−1 ]} A [m² ] ₌ Q [C]

C [F = kg m A sec ]= ε [Fm

• Inductance (Henry):

et[kg m²A^-1s^-2 ]

I [A]

L =

∂t [s]

emf [V]= −L ^{∂i [A]}

• Permeability (H/m): potential per meter

H [A m^-1 ]

μ [H/m]= ^{B [T]}

Basic Definitions

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[ ] [ ]

l [m]

μ kg m A^-2sec^-2 .A m²

S [S = kg^-1m^-2A²sec² ]=

• Magnetic Flux (Weber):

S [S]

φ [W = kg m²A^-1s^-2 ]= −L [H] ^{NI [A]}

• Frequency (Hz): cycles per second

1

T [s]

f [Hz]=

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• Reluctance (S): magnetic resistance to magnetic field lines in the same material

Temperature and Luminous Intensity

28

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• Temperature (Kelvin): fundamental scale defining the thermodynamic temperature of the triple point of water (temperature equivalent of ice, liquid and vapour) at exactly 273.16 K. International Practical Scale of Temperature is based on two fundamental fixed point: boiling point at 100 °C and triple point

at 0.01 °C.

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• Luminous Intensity (Candela): one-sixtieth of the luminous intensity per cm² of the full radiator (Planckian radiator). Secondary standard is based on special tungsten filament lamps.

Heat vs Temperature

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• Heat is energy to do work
• Temperature is arbitrary scale indicating direction of heat flow
• Heat is measured using calorimeter and NOT using thermometer
• Heat is measured in Joules (not Celcius)
• Temperature is measured with a thermometer (in degree Celcius, Fehrenheit)

IEEE Standards

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Measurement methods

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• Direct measurement method

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• Indirect measurement method

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• Comparison method

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• Absolute measurement method

Active and Passive Instruments

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• 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

Null-Type and Deflection-Type

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• Null: quantity measured by putting external value equivalent.
• Better accuracy as external value can be easily chosen

Passive pressure gauge

Dead-weight pressure gauge

• Deflection: quantity measured using equivalent motion.
• Accuracy depends on the linearity and calibration of pointer spring

Analog and Digital

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• Analog: displayed continuous proportional changes to actual measurements

• Digital: quantity displayed in terms of discrete equivalent values after processing

Display methods

Instrument Characteristics

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• Static properties: characteristics when measurement remains constant

• Dynamic properties: relationship between input and output when measured quantity varies

Properties of Instruments

Static Characteristics

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• Accuracy
• Precision
• Sensitivity
• Resolution
• Threshold
• Drift
• Error
• Repeatability
• Reproducibility

• Dead Zone
• Backlash
• True Value
• Hysteresis
• Linearity
• Range (Span)
• Bias
• Tolerance
• Stability

Accuracy of Measurement

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• 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

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• 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

_{Sensitivit y =} Δ instrument output

Δ measured variable

Example: Pressure Measurement

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Pressure Gauge:

measurement range 0 -10 bar

fs ±1.0%

Full‐scale reading precision:

e_max = ±1.0% ×10 bar = ±0.1bar

Maximum error:

1 bar

₌ (1± 0.1 bar ) −1 bar _{×100 = ±10%}

1bar

e

Measurement error for reading 1 bar (in %):

_{% error =} measured value − true value _×100%

true value

Example: Tolerance

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Resistors:

pack of resistors with R = 1000 Ω

± 5%

tolerance:

Minimum value:

Maximum value:

R_min = 1000 Ω − 5% = 950 Ω

R_max = 1000 Ω + 5% = 1050 Ω

Linearity and Measurement Sensitivity

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Linearity

• instrument output is proportional to measured quantity

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Sensitivity:

Sensitivity = ^{Scale deflection} = Slope

Value

Linearity and Deviations

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Sensitivity to Disturbance

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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

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• 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

Example 1: Measurement Sensitivity

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Platinum resistance thermometers:

resistivity measured at varying temperatures

Measurement sensitivity:

30^o C

= 0.233 Ω /^o C

7Ω

^Rsens ⁼

305

310

315

330

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325

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320

0

50

100

150

200

250

300

350

Resistance

Example 2: Measurement Sensitivity

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Measurement sensitivity:

1 kg

20 mm

sens

= 20 mm / kg

D (20 C) =

0

Spring Balance:

calibrated at 20°C

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used at 30°C

20

10

0

70

60

50

40

30

80

0

1

2

3

Sensitivity

Deflection (mm)

Deflection (mm)

1 kg

sens

D (30⁰ C) = ^{22 mm} = 22 mm / kg

Zero drift = 5 mm Sensitivity drift = 2 mm/kg

Zero drift coefficient = 5 mm/10 °C = 0.5 mm /°C Sensitivity drift coefficient = 2 (mm/kg)/10 °C

= 0.2 (mm/kg) /°C

Statistical Analysis of Measurement

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Statistical Analysis of Measurement

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• Arithmetic mean (average) of readings:

where x = measured value,

n = number of reading / measurement

0 Average reading the most likely value for measured variable

• Deviation, d :

• Average deviation

0 Average amount of measurement error

x

n

n

i

∑

=

x =

i=1

n

x₁ + x₂ +·+ x_n

d₁ = x₁ − x d₂ = x₂ − x · d_n = x_n − x

Statistical Analysis of Measurement

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• Standard deviation, σ :

• Probable error:

0 random errors that lie scattered within 50% probability region around mean; A range within one probable error on either side of the mean will include 50% of the data values

(✓)

Normal Error Distribution

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Probable error:

r = ±0.6745σ

D₂

D₁

1

P(D₁ ≤ D ≤ D₂ ) =

1

F(x) =

_e[−D² / 2σ ² ]_dD

_e[−( x−m)² / 2σ ² ]

2π

∫ _σ

σ 2π

Example 3: Statistical Analysis

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=

∑

= 100.0 Ω

999.5

10 10

10

ave

i=1

i

R

R =

Determine:

1. measurement range

Range = R_max − R_min = 100.5 − 99.3 = 1.2 Ω

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2. average reading

c) Deviation, d

- as shown in table -

Precision: ONE decimal

Example 3: Statistical Analysis

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= 0.4 Ω

1.51

2

n −1 10 −1

d

n

i=1 ₌

^∑ i

σ =

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Determine:

d) standard deviation

e) probable error

Probable error, r = ±0.6745σ = ±0.2763 Ω

Example 4: Statistical Analysis

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Number of measuremen t, n = 11

Average (mean) reading = 409

_∑deviation² = 1370

10

1370

137 = 11.7

= 137

=

n −1

Probable error, r = ±0.6745σ = ±7.89

Standard deviation, σ =

_∑deviation ²

Variance, σ ² =

Hysteresis

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Hysteresis:

• Different increasing and reducing trend
• Hysteresis in magnetic element/spring : non- coincidence between loading and unloading.

Dead Space

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• 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

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• Gross Error: human error due to incorrect use of equipment, wrong observation, carelessness etc.
• Systematic Errors

0 Instrumental: inherent to measuring eqpt.

x Static: limitation of device

x Dynamic: inability to respond to change in measured variable

0 Environmental: due to change in external conditions (temperature, pressure, humidity, magnetic/electric fields)

• Random Errors: due to unknown causes

Gross Errors

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Other Examples

Reduction Methods

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Caused by human error due to incorrect use of equipment, wrong observation, carelessness etc.

• Erroneous

calculations

• Improper choice of instruments
• Incorrect adjustment
• Neglect of loading effects

• Careful attention and observation
• Awareness of instrument limitations
• Taking at least 3 readings

• >1 observer to observe critical data

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Estimation

• Not possible to estimate

Systematic Errors

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Can come from 2 sources:

1. Instrumental: inherent to measuring equipment

x Static: limitation of device

x 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

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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

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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

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• Unknown events that causes small variations in measurements.
• Random and unexplainable

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• Estimate: Take many readings and conduct statistical analysis

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• Methods of reduction:
• Careful design of eqpt to reduce unwanted interference
• Statistical analysis to determine best estimate and/or outlier values

Sources of Errors

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• Disturbances: external/environmental factors that modify inputs or outputs.
• Measurement:

inaccurate/incorrect methods.

• Dynamic: changes occurring in the measurement systems.
• Instrument: resistance or other types of characteristics inherent to the devices used

Error Reduction

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• Inspection and Care: ensure measurement integrity.
• Calibration: correct measurement drift and scaling factor
• Method of opposing input: compensate environmental bias in measurement
• High gain feedback: eliminate error
• Signal filtering: reduce noise
• Manual adjustment: compensate environmental bias in measurement
• Intelligent instruments: attenuate error and amplify signals

Maximum and Likely Errors

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THREE separate sources of error are identified:

• system loading: ±1.2%
• environmental changes: ±0.8%
• calibration error: ±0.5%

Maximum possible error =

± (1.2 + 0.8 + 0.5)% = ±2.5%

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Likely error =

± 1.2² + 0.8² + 0.5² %

= ±1.53%

Aggregated Errors (1)

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• 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

S = ( y ± z) ± e

where likely error, e = (ay)² + (bz)²

Error in a sum / difference

Note: Here e is the absolute error.

Example 5: Errors in a Sum

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The total resistance of 2 resistors (to 3 significant figures) each with a tolerance of:

Solution:

THREE significant figures

R₁ = 99.3 ±1.0%

R₂ = 46.123 ± 6.5%

= 145.423 ± 3.158 Ω

= 145 Ω ± 2.2%

= (R₁ + R₂ ) ± e

e = ± (0.01× 99.3)² + (0.065× 46.123)²

= ± 0.986 + 8.988 = ±3.158

∴ R_T

Note: a = 0.01, b = 0.065

error fractions

Example 6 : Errors in a Difference

e = ± (0.01×100)² + (0.05× 80)²

= ± 1+16 = ±4.123

∴V_T = (V₁ −V₂ ) ± e

= 20 ± 4.123 V

= 20 V ± 20.6%

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Solution:

The maximum likely error,

Note: The percentage error increases for difference between measurements.

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The difference between 2 voltage measurements, V_D:

V₁ = 100 V ± 1%

V₂ = 80 V ± 5%

Aggregated Errors (2)

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• 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

Q = ^y ± e

z

Error in a product / quotient

P = yz ± e

where e = a² × b²

Note: Here e is the fraction/percentage error.

Example 7: Errors in a Product

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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.

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Solution:

V

Since density is mass per unit volume ^{ρ = m} , then the likely error is

e = 0.02² × 0.03²

= ± 0.0013 = ±0.036

Propagation of Uncertainties

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n

i

i n

∂x

∂x ∂x

∂x

dy = ^∂y dx + ^∂y dx +· + ^∂y dx +· ^∂y dx

1 2

1 2

y = f (x₁ , x₂ ,., x_i , ,., x_n )

Consider: General equations and difference

2

2

2

2

2

1

1 2

⎥_⎦

_⎤1/ 2

⎟

_⎞2

⎛ _∂

_⎞2

⎛ _∂

_⎞2

⎛ _∂

⎟

_⎞2

⎡_{⎛ ∂}

xn

xi

x 2

x1

y

xn

xi

i n

x 2

x1

y max

xn

xi

i n

x 2

y x1

(U ) ^⎥

⎝ ^∂xn ⎠

y

⎝ ^∂xi ⎠

y

⎝ ^∂x2 ⎠

y

(U ) + ^{⎜ ⎟}

^⎢_⎣⎝ ^∂x1 ⎠

y

U = ^⎢⎜

∂x

∂x

∂x

) + ^∂y (U

∂x

= ^∂y (U

U

∂x

∂x

∂x ∂x

U = ^∂y (U ) + ^∂y (U

(U ) +· + ^⎜

(U ) +· + ^{⎜ ⎟}

) +· + ^∂y (U ) +·+ ^∂y (U )

) +· + ^∂y (U ) +· + ^∂y (U )

Uncertainties: Due to difference in each components

Dynamic Characteristics

70

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• 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

dy

d ⁿ y ^an dt ⁿ

_d n−1 _y

• ^an−1 _{dt n−1}

d ^m x ₊ dx ₊

_{dt m} · + b_{1 dt} b₀ x

+· + a_{1 dt} + a₀ y = b_m

where

x = input

y = output

a, b = coefficient

Dynamic Inputs

71

Periodic input

Transient input

Random signal

Periodic Signals

72

Periodic input

∞

f (t) = a₀ / 2 + _∑ a_n cos nωt + b_n sin nωt

n=1

∫

T

n

T

0

a = ²

f (t) cos nωtdt

∫

T

n

T

0

a = ²

f (t) cos nωtdt

Dynamic Characteristics

73

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Zero order

• If all coefficients a₁…a_n 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.

a

0

b

a₀ y = b₀ x or y = ⁰ x = Kx

Frequency Response

74

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Response to sinusoidal input

• Amplitude and phase will shift depending on frequency

Frequency Response

75

Magnitude (log dB)

Frequency (log)

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Response to sinusoidal input

• Amplitude and phase plots against frequency

76

END