Basic of Measurement

24-Feb-16

Measurement & Signal Processing

Objectives

To know the meaning of dimension, unit, measure, consistent, base unit, supplementary unit and derived unit.

To know the seven fundamental dimensions in the SI and Technical English systems.

Units of mass, time, velocity, acceleration, force, pressure, torque, work, energy and power.

To identify the least and most significant digits in a number

To express calculated results with the correct number of significant figures

English and Metric Systems

Unit and measures are developed for effective communication.

Data transmission : 512 kilobyte per second data rate.

Race : a 1600 meter runway.

Driving : 110 kilometer per hour.

A pint of milk, 96 mile per hour fastballs, a 15 stones man

The English system of units originated from Babylonian. It was the Romans who introduced the base of 12 in the English system.

The metric system aroused around the end of 18th century, its development followed a more rational and

scientific approach than anthropometric .

Systems of units

A unit is a particular physical quantity (length, weight, etc), defined and adopted by convention, with which other particular quantities of the same kind are compared to express their value.

The value of a physical quantity is the quantitative expression of a particular physical quantity as the product of a number and a unit, the number being its numerical value.

For example, the circumference of the earth around the equator is given by: Ce = 40,074 × 103 m (meter)

quantity : length

unit: meter

measure(magnitude): 40,074 × 103

Systems of units

Example:

A sheet of European A4 paper is 210 millimeters width by 297 millimeters long.

quantity (dimension): length

unit: millimeter

measures: 210, 297

To avoid misinterpretations the use of the international system of units (Systrme International d'unitrs, SI) is highly recommended when presenting measurement results.

A system units is comprised of based, supplementary and derived units. Based units are dimensionally independent.

Base Units

7 base units: meter, kilogram, second, ampere, Kelvin, mole and candela.

The foundation for the SI system was laid during the French Revolution, with the creation of a metric decimal system of units and of two platinum standards representing the meter and the kilogram (1799). The unit of time was based on the astronomical second (1/86,400 of a mean solar day ).

In 1960 the 11 th Confrrence Grnrrale des Poids et Mesures adopted the ampere as the basic unit for electric current. Two other basic units have been added to the system, for temperature (Kelvin) and light (candela), and in 1971 another for substance (mole).

Meter and Kilogram

The meter (m) is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second [17th General Conference on Weights and Measures CGPM (1983), Res.1]. The speed of light in vacuum at exactly 299,792,458 meters per second.

The kilogram (kg) is the unit of mass; it is equal to the mass of the international prototype of the kilogram [ 1 st CGPM (1889)].

A CGI of the international

prototype kilogram (the inch

ruler is for scale). The prototype

is manufactured from a

platinum–iridium alloy and is

39.17 mm in both diameter and

height, its edges have a four-angle

chamfer to minimize wear

kilogram

http://www.ngtaiwan.com/17618 (to be redefined in 2018)

Second and Ampere

The second (s) 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 (Cs)133 atom [13th CGPM (1967), Res.1].

The ampere (A) 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 [9th CGPM (1948)].

the ampere is a measure of the amount of electric charge

passing a point in an electric circuit per unit time with

6.241 × 1018 electrons, or one coulomb per second

constituting one ampere. Amperes can be viewed as a flow

rate, i.e. number of (charged) particles transiting per unit

time.

Kelvin and Mole

The Kelvin (K), unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water [ 13th CGPM (1967), Res.4].

The mole (mol) 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 [14th CGPM (1971), Res.3].

Lord Kelvin

In chemistry and physics, the Avogadro

constant (symbols: L, NA) is defined as the

number of constituent particles (usually atoms

or molecules) in one mole of a given substance.

It has dimensions of reciprocal mol and its

value is equal to 6.02214129(27)×1023 mol−1

Amedeo Avogadro

Candela

The candela (cd) is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian [16th CGPM (1979), Res.3].

A 100 watt light bulb has the luminous intensity of approximately 135 cd.

The radian (rad) and the steradian (sr) are two SI supplementary units. There are 2p radians over the circumference of a circle and 4p steradians over the surface of a shpere.

A graphical representation of 1 steradian. The

sphere has radius r, and in this case the area of

the patch on the surface is A = r2.

Five system of Units

Physical quantities of different dimensions are related to one another through equations in the form of definition and natural law. Example: 1 newton (N) = 1 kg m / s2

The units within a system are consistent if no numerical factors other than 1 occur in all unit equations.

Example

Conversion factors between unit systems

Example

Assume 100 Cal are expended for each mile jogged

A cup of ice cream contains 400 Cal

1 g of fat produces 9 Cal of energy

To Jog 4 miles

400 Cal/9Cal/g = 44.4 g body fat

Example

Volume of the air 1440 ft3

The density of air at 1 atm and 300K is 1.16kg/m3 = 0.00226 slag/ ft3

The mass of the air 3.25 slugs (TE) = 47.5kg (SI)

The weight of air 3.25 slugs ×32.174 ft/s2 = 104.6 lbf

47.5kg ×9.81 m/s2 =465.4 N

Example

35000 ft/(3.2808 ft/m) =10668 m

g = 0.99666 × g0 = 9.77393 m/s2

Conversion factor

Work, Energy (J, cal)

The small calorie or gram calorie (cal) is

the approximate amount of energy needed to

raise the temperature of one gram of water by

one degree Celsius

1 Cal = 1 kcal = 1000 cal

1cal = 4.184 J

Material properties

The properties of gases, liquids and solids can be repressed in terms of base and supplementary units.

Fluid shear stress (force per unit area) : t (kg m/s2)/m2

Rate of shear strain (strain per unit time) : dq/dt (m/m)/s

Dynamic viscosity m of fluid : (kg/(m s2))/ (1/s) = kg /(m s)

Remarks:

The dynamic viscosity of air and water are strongly affected by temperature as described by Sutherland’s law

d

dt

qt m

Gas constant

N/m2 × m3 = kg (R) K , the base units of R are

N m /(kg K)

m2 /s2 = (R) K

kg (m/s2) m / kg= (R) K, and the base units of R are N m /(kg K)

PV MRT

a RT

Example (Option)

Accuracy of the base units

Other quantities than the base quantities are called derived quantities. They are defined in terms of the seven base quantities via a system of quantity equations.

Decimal prefixes (for most popular cases)

Scientific notation: to avoid writing very large and very small numbers Example: 325000=3.25×105=3.25E+5

The SI preference is to

express a number using a

prefix such that its numerical

value is between 0.1 and

1000. (2.4GHz, 300kW)

Significant Figures

The number of Significant Figures should correspond with the uncertainty in the measurement. (digits up to and including the first uncertain digit).

For example, if the uncertainty in a measurement is ±0.05, then the measurement should be expressed as 2.55 ± 0.05

Rules for rounding off a number

Truncate the number to its desired length

Express the excess digit as a decimal fraction

Round up the least significant digit by 1,

if the fraction > ½ or the fraction is equal to ½ and the least significant digit is odd.

It is left alone, otherwise.

Example: 5.6850×103 become 5.68×103 for three significant digit

Example

Example

Example

54.0℃±0.5=? K 273.15+54.0=327.2K±0.5K for the same level of precision 273.15+54.0=327K for maintaining 3 significant figures

Unit Standard

Length of a foot, defined as the

average of the feet of 12 men

Standard

For each of the basic units a physical materialization should be available that enables a comparison of the associated quantity with other instruments: a standard. A measurement standard should be based on some highly stable and reproducible physical phenomenon representing a particular quantity.

A former standard meter, left:

end view of the British copy

of the International Meter

Bar. (1889 to 1960)

All modern standards are related to fundamental physical constants (such as Planck's constant h, Boltzmann's constant k and the charge of an electron e), except for the kilogram: an alloy cylinder (90% Pt, 10% Ir) of equal height and diameter.

Quantities and properties

Scalar: quantities that have magnitude only (e.g. temperature)

Vector: quantity with magnitude and direction (e.g. velocity).

Pure material properties: properties that are independent of the dimensions or the amount of matter. others have a value that is determined not only by the substance but also by the size or the construction layout.

For instance, the resistivity r(Wm) is a pure material property, whereas the resistance r(W) depends on the material as well as the dimensions of the resistor body.

Variables: characterizing the state of a system are related by physical laws. independent variables and dependent variables.

through-variables and across-variables

The geometric domain

The radian is defined as the plane angle between two radii of a circle which cuts off on the circumference an arc equal in length to the radius. The unit is m/m, so the radian is dimensionless. For clarity we will use the unit radian.

The electrical and magnetic domain

Electrical and magnetic quantities Analogies between electrical

and magnetic expressions

The electrical and magnetic domain

The electric permittivity is the product of the permittivity of free space e0 and the relative permittivity (or dielectric constant) er. the magnetic permeability is the product of the magnetic permeability of free space m0 and the relative permeability mr .

Electrical and magnetic properties

Time signal quantities In the electrical domain, some particular quantities associated with

time apply. The duty cycle is the percentage of one period in which a signal is active:

It varies from 0 (whole period low) to 100% (whole period high).

Time signal quantities

[2]

The thermal domain

The unit of Celsius temperature is the degree Celsius, symbol ℃. The numerical value of the Celsius temperature is given by

T(℃)= T(K) - 273.15.

The interfacial heat transfer coefficient h accounts for the heat flow across the boundary of an object whose temperature is different from that of its (fluidic) environment. Its value strongly depends on the flow type along the object and the surface properties.

Thermal properties

The mechanical domain

Quantities in the mechanical domain describe state properties related to distance, force and motion. A possible categorization of these quantities is a division into:

displacement quantities;

force quantities;

flow quantities.

Mechanical properties

Displacement and force quantities

Force quantities

Displacement quantities

Fluidic quantities

The optical domain

In the optical domain two groups of quantities are distinguished: radiometric and photometric quantities (0.35 < l < 0.77 mm).

Radiometric and photometric quantities

s: light source; d: light detector

End of the Chapter

Review Problems

2.7

Example: Irradiance of a lamp above a table-top

A flat surface can receive radiant energy from its environment, from all directions. The amount of incident power Ed is expressed in terms of W/m 2, and is called the irradiance.

The radiance L of a surface is defined following this idea: it is the emitted radiant energy in a particular direction, per solid angle (to account for the directivity) and per unit of projected area, that is, per unit area when the emitting surface is projected into that direction.

Example: Irradiance of a lamp above a table-top Irradiance of a point

source on a planar

surface

A light bulb has a radiant flux of P = 5 W. The construction of the lamp is such that it emits light isotropically (i.e. independent of orientation). A plane (e.g. a table-top) is located at a distance h of 1 m from the bulb. What is the irradiance at the surface of the plane?

radiant intensity is: I = P/(4zr) = 5/(4zr) = 0.4 W/sr. The normal of a small area dS on the plane at a distance d from the lamp makes an angle Y with respect to the direction of the lamp. Hence,

Relations between quantities in different domains

A sensor performs the exchange of information (thus energy) from one domain to another and therefore it operates on the interface between different physical domains.

The relation between variables is governed by physical effects, by material properties or by a particular system lay-out. It either acts within one physical domain or crosses domain boundaries. Such relations describe the fundamental operation of sensors.

Consistent framework for quantities and properties

Several attempts have been made to set up a consistent framework for quantities and properties. Most of these descriptions are based on energy considerations. For each domain two variables can be defined in such a way that their product equals the energy (or energy density) of the domain type, the conjugate quantities. The product of each domain pair is J/m 3 (energy per unit volume)

mechanical: tension T [N/m2] and deformation S [-];

electrical: field strength E [V/m] and dielectric displacement D [C/m2];

magnetic: magnetic induction B [Wb/m2] and magnetic field H [A/m];

thermal: temperature q [K] and entropy s[J/Km3].

Through-variable and across-variable

The variables in this list belong to either of two classes: through-variables or across variables.

A through-variable is a physical quantity which is the same for both terminals of the lumped element. An across-variable describes the difference with respect to a physical quantity between the terminals. They are also called extensive variables and intensive variables, respectively.

In the above groups of variables, T, E and q are across-variables. On the other hand, S, D and s are through-variables.

If only through-variables affect the system, its internal energy U changes as

dU = T dS + E dD + q ds

If only across-variables affect the energy content, the energy change is written as

dG = -S dT- D dE- s dq

Nine physical effects corresponding to the parameters in equation

Symbols, property names and units of the effects

S(T, E, q), D(T, E, q), and a(T, E, q):

Transducers

A sensor or transducer performs the conversion of information from the physical domain of the measure and to the electrical domain. Signal conditioning may be added or included to protect the sensor from being loaded, to fit the sensor output to the input range of the ADC, or to enhance the S/N (signal-to-noise ratio) prior to further signal processing.

direct (left) and modulating (right) sensors.

Two-port model for a sensor