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Chapter 7
Units of Measurement
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Objective of a measurement
– to assign a number to the quantity being measured
• a magnitude
– to assign a unit of measurement to the magnitude
The units distinguish one physical quantity from another.
Example inch units distinguish a length measurement from a volume measurement
using gallon units.
The present system of using units originated in ancient times:
– Greeks
– Chinese
– Indians
Created to satisfy the need for
– quantifying weights and measures
– standardization
• trade issues
Our particular system began in 1790
– The French Academy of Science
– Called the ‘Metric System’
In 1875, 18 countries agreed to a treaty called the ‘Convention du Metre’
– A set of weights and measures that would be maintained in Paris
By 1960 after years of refining and improving the SI system was agreed upon
– Système International d’Unités
The SI system
• The SI system is a set of definitions
• It defines the units that shall be used for reporting all scientific or non scientific
measurements.
• Other systems exist
– mks - meter/kilogram/second
•based on artifacts or physical objects
– foot/pound/gallon
•The SI system defines
– 7 Fundamental or Base units
– 2 Supplementary units
• late additions
– 27 Derived units
• arrived at by combining the fundamental and supplementary units WITHOUT any numerical
factors
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7 Base Units
• LENGTH
– The Metre
– The meter is the length of the path travelled by light in vacuum during a time
interval of 1/299 792 458 of a second
– requires that the speed of light is a universal constant ( in vacuum)
• 299 792 458 m/s
•TIME
– The Second
The second is the duration of 9 192 631 770 periods of the radiation corresponding to the
transition between the two hyperfine levels of the ground state of the cesium 133 atom.
• ELECTRIC CURRENT
– The Ampere
• The ampere is that constant current which, if maintained in two straight parallel
conductors of infinite length, of negligible circular cross-section, and placed 1 meter
apart in vacuum, would produce between these conductors a force equal to 2 x 10-7
newton per meter of length.
• THERMODYNAMIC TEMPERATURE
– The Kelvin
• The Kelvin, the unit of thermodynamic temperature, is the fraction 1/273.16 of the
thermodynamic temperature of the triple point of water.
• LUMINOUS INTENSITY
– The Candela
• The candela is the luminous intensity, in a given direction, of a source that emits
monochromatic radiation of frequency 540 x 1012 Hz and that has a radiant intensity
in that direction of 1/683 watt per steradian.
• AMOUNT OF SUBSTANCE
– The Mole
– The mole is the amount of substance of a system which contains as many
elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol
is "mol."
– When the mole is used, the elementary entities must be specified and may be
atoms, molecules, ions, electrons, other particles, or specified groups of such
particles.
• MASS
– The Kilogram
– The kilogram is the unit of mass; it is equal to the mass of the international
prototype of the kilogram.
– The only fundamental unit which is based on an artifact
– The other 6 are based on universal constants for example: speed of light, magnetic
permeability etc
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Supplementary Units
– The Radian
• A measure of plane angle
– The Steradian
• A measure of solid angle
Derived Units
– Expressed directly in terms of the fundamental and supplementary units without
any numerical quotients
– this is called coherence.
Examples:
– Volume
m3
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Density
kg m-3
– Frequency
s-1 (Hz)
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Force
kg.m.s-2
Of special interest to us in electrical engineering are:
– The VOLT
– The OHM
– The FARAD
– The HENRY
The Volt
– 1 Volt is the potential difference that exists between 2 points when 1 joule of work
is done in moving 1 Coulomb of charge between the two points.
1V 
1J 1N  m 1kg  m  s  2  m
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1C 1A  s
1A  s
1V  1kg  m 2  s 3  A1
– Note that there are no constants
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• COHERENT
Resistance
– From Ohm’s Law
1 
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Capacitance
– The Farad
1F 
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1V
 1kg  m 2  s 3  A  2
1A
1C
 1kg 1  m  2  s 4  A 2
1V
Inductance
– The Henry
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1H 
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1Wb
 1kg  m 2  s  2  A 2
1A
Standards
The SI system is a set of definitions
– Exact statements describing a base unit
– Coherent
– Uniform
• based on presumably unchanging constants of nature
• exception the kilogram
– Unified
• Measurements in dynamics, electrodynamics and thermodynamics can be compared in such a
way to observe the laws of conservation of mass and energy
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The definition is a set of words
Following from this is the realisation
– creation of a physical object or physical behavior whose observed attributes
closely match the definition
For example the definition of time
– A clock based on Cs133 is created
– the radiation produced has attributes that allow the definition to be realised
• creates a Primary Standard
Measurement labs around the world conduct experiments to realise the various SI
definitions ( via Primary Standards)
– National Physical Laboratory (NPL) - UK
– National Institute of Standards and Technology (NIST) - USA
– National Research Council (NRC) - Canada
– Bureau International des Poids et Mesures (BIPM) - France
• The values obtained from the realization are transferred to devices which represent
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the SI definition
The representations do not involve continuous recreation of the SI base unit
They are:
– relatively low cost
– easily maintained
– stable
– rugged
– called Secondary Standards
The overall picture is thus:
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SI Definition
International Primary
Standard
Realisation
Secondary
Standard
Representation
Working
Standard
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We will examine those International Primary Standards related to electrical
engineering
– The Ampere
– The Volt
– The Ohm
The Ampere
• Recall the SI definition
• Note the extreme restrictions
– infinite length
– negligible CSA
– Vacuum
• In practice the Ampere is realized using a current balance
• Experiment is complex
• Best Uncertainty about 15ppm
• Because of discoveries in Quantum Physics the ampere is now realised by using the
ratio of Volt to Resistance
• Volt
– realised with the Josephson Junction
• Ohm
– realised using the Quantum Hall Effect
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7.3 THE JOSEPHSON JUNCTION
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The Primary standard for voltage is based on the Josephson effect
This occurs across 2 semiconductors separated by a thin insulator
– A Josephson Junction
• When cooled in liquid helium the semiconductors become superconductors
– exhibit zero resistance
The following diagram illustrates the device.
Niobium semiconductor
Niobium semiconductor
Oxidised Aluminium insulator
The Josephson Junction
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The junction is biased with a DC current
The junction is exposed to high frequency microwave radiation (GHz)
A DC voltage is produced across the junction
– voltage is directly proportional to frequency
•It was discovered that the following I-V characteristic was produced
Bias
Current
DC Voltage across junction
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As the bias current was increased the voltage increased in a number of equal steps of
constant voltage
The voltage at a particular step is given by
Vn 
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f nh
2e
where:
f = the microwave radiation frequency (Hz)
n = the number of the particular step
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e = the fundamental charge on an electron
h = Plank’s Constant
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In 1990 the ratio 2e/h, called the Josephson Constant Kj, was agreed internationally
to be:
Kj = 483 597.9 GHz/V
Note that the output is based on constants of nature
– Anywhere in the universe Kj should be the same
Thus
f n
Kj
Vn 
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For example if f =75GHz:
Vj = 155.1V/step
This is an incredibly small voltage
Modern systems contain several thousand junctions combined in series
Typical Uncertainty is 0.2 ppb
– parts per billion
– 0.2 parts in 1000 000 000
7.4
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The Quantum Hall Effect
To realise the ampere we have created half of the story
– the volt
We now examine the other half
– the ohm
The primary standard for resistance is based on the Hall effect
Recall the principle
– A thin semiconductor bar carries a DC current
– The bar is subjected to a magnetic field perpendicular to it
– a Voltage develops across the bar perpendicular to the direction of the current flow
• This is the Hall Voltage
– The ratio of the Hall Voltage to the DC current is called the Hall Resistance RH of
the bar.
In 1980 it was discovered that by:
– cooling the bar in liquid helium
• semiconductor becomes a superconductor; and
– greatly increasing the magnetic field
the Hall resistance increased in discrete steps
At each step the resistance remained extremely constant
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This effect was called the Quantum (Quantized) Hall Effect
Hall Resistance RH
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Magnetic Field
QUANTUM HALL EFFECT
• The more remarkable discovery was the equation describing RH
RH 
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h
 n  RK  n
2
e
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Where:
n = the number of the particular step
e = the fundamental charge on an electron
h = Plank’s Constant
Since 1990 it was agreed that the value of RK was:
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RK is called the Von Klitzing Constant after the discoverer of the effect
Typical uncertainty is ±0.2 ppm
Again we have realised a standard based on universal constants
The standards for the volt and ohm allow us to realise the ampere to about 0.2 ppm
RK  25812 .807  0.005
7.5
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Secondary and Working Standards
The Josephson Junctions and the QHE devices are:
– expensive to operate
– difficult to use
They realize the fundamental units each time
– they are primary standards
For practical applications, the fundamental units must be readily available for use
We need a representation of the primary standard
Representation occurs at 2 levels
– Secondary standards
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– Working standards
Secondary Standards
• Do not recreate the fundamental units
• Produce a value which is compared to the primary standard
– called calibration
• The value is created by any practical engineering method
• Typical Secondary standards in electrical engineering are:
– The Zener reference cell; and
– The Weston cell
• both for voltage
– The standard resistor
• for resistance
• There are no inexpensive secondary current standards
– A precision voltage is converted by a device called a transconductance amplifier
into a precise current
• Secondary standards are still too delicate for everyday use
• Typically used only a few times per year and have to be maintained under constant
temperature and humidity conditions
• For more frequent use
– like commercial calibration
working standards are used
• Typical working standards in Electrical Engineering are:
– Multifunction calibrators
• Contain precision sources for voltage, current and resistance
– Decade boxes or individual devices for:
• Resistance
• Capacitance
• Inductance
– Precision power supplies
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