GCE 227

CHAPTER 1

Lecture1: Fundamental Concept of Metrology

Preamble:

Measurement is simply defined as the estimation or determination of extent dimension or capacity, usually in relation to source standard or unit. Metrology on the other hand, is the study of measurement. The comparison must be to standard as indicated below.

Standard

Mesurand Result

(Comparision to a standard)

Comparision

Process

(Measurement)

Fundamental Units Of Measure, SI SystemThe SI system is the latest form of metric system and has two supplementary units viz radian (for solid angle) and steradian (for plane angle) in addition to seven base units of metre, kg, second, ampere, kelvin, candela and mole for length, mass, time, current, thermo dynamic temperature, luminous intensity and amount of substance respectively. All other units can be derived from these quantities. The SI system is a comprehensive, logical and coherent system designed for use in all branches of science, engineering and technology.

The symbols, units and dimension of various mechanical electrical and magnetic quantities are given below:

Quantity Symbol Unit Dimension

Area

Volume

Frequency

Density

Velocity

A

V

F

D

V

M2

M3

Hz

Kg/M3

M/S

L2

L3

T-1

ML-3

LT-1

Quantity Symbol Unit DimensionAcceleration

Force

Work, energy

Power

Surface tension

Charge

Potential, Difference emf

A

F

W, E

P

Q

V, E

M/S2

N(Newton)

J(N-M)

W(J/S)

C(A-S)

V

ML-3

LT-1

LT-2

MLT-2

ML2T-2

TI

ML2T-3I-1

Electric field strength

Resistance

Capacitance

Inductance

Magnetic flux

Magnetic field strength

Magnetic flux density

E

R

C

L

H

B

V/M

(V/A)

F

H

Wb

A/m

T(Wb/M2)

MLT-3I-1

ML2T-3I-2

M-1L-2T4I2

ML2T-2I-2

ML3T-2I-1

IL-1

ML-2I-1

Quantity Symbol Unit Dimension

Magnetic motive force (MMF)

Reluctance

Permeability

Pole-strength

Electric flux

Electric flux density

Permittivity

F

S

µM

D

€

A

A /wb

H/m

Wb

C

C/M2

F/m

I

M-1L-2T2I2

MLT-2I-2

ML2T-2I-1

TL

L-2TI

M-1L-3T4I2

Note: That the dimension for the electrical current quantity is (I).

1.0 INTRODUCTION

The device used for comparing the unknown quality with the unit of measurement or a standard quality is called the measuring instrument. Measurement generally involves the use of an instrument as a means of determining a quantity or a variable. The instrument assists a person to determine the value of an unknown quantity which could not otherwise be measured by this human facilities alone. The value of unknown quantity can be determined by direct method such as measurement of current by ammeter, or by indirect method such as determination of resistance by ammeter voltmeter method. Direct measuring instruments are most widely used in engineering practice since they are the most simple and inexpensive ones and enable the measurements to be made in the shortest possible time.

Measurement work employs the following terms:

i. Instrument: A device for determining the value or magnitude of a variable or quantity.

ii. Input signal: A signal applied to a device, element or system such as pressure applied to the input connection of a pressure transmitter.

iii. Output signal: A signal delivered or generated by a device, element or system.

iv. Accuracy: The degree of conformity or closeness with which an instrument reading approaches the true value of the variable or quantity being measured.

V. Precision: A measure of the reproducibility of the measurement; that is given a value of a variable, precision means the degree to which successive measurements of the same value differ from one another.

Vi. Sensitivity: The ratio of the output signal or response of the instrument to a change of the input or measured value.

ViiResolution: The smallest change in measured value to which the instrument will respond.

1.1 ERRORS IN MEASUREMENT

In practical measurement of quantities, it is not possible to obtain exact or perfect values. No measurement may be made with perfect accuracy, the imperfections of a measurement are due to errors arising in course of practical measurements. Error may come from different sources and are usually classified under three main headings:-

(a) Gross error: These are largely human errors

and include; misreading of instruments, incorrect adjustments (parallax) and improper application of instruments and mistakes in computation.

(b) Systematic Errors: These are errors due to short comings of the instrument. Some systematic errors are:

i. Constructions Errors.

This is found in an instrument as indicated by the manufacturers. The maximum tolerances within which quantities measured by the instrument, should be as indicated e.g. I 1%.

ii. Effect of the environment on the instrument (Ambient condition error) e.g. temperature, attitudes, pressure e.t.c.

iii. Error inherent in method used

Example I

A

IIR R

B

IA~

(r)

V=IRR = IAr

= I RrRtr

IA= I and IR= I

Let I = 5A, R=100R, r=1R

IA= 5 = 4.95A

IR= 5 = 0.05A

Relative error = x100 =1%

Now, let I=5A, R=100 , r=10

IA= 5 =4.545A

IR= 5 =0.45A

Relative error = =9.0%

iv. Error due to ageing and wear of equipment.

Example: consider the measurement of a resistance by the voltmeter/ammeter method; the voltmeter indicates 120volts and the ammeter indicates 2.2A

(120 volts; range:0-200v)

(2.2A; range:0-5A)

RRtr

rRtr

100100+1

1100+1

0.055

100100+10

10100+10

0.455

Each instrument has an error not exceeding 1% maximum. Determine the limit in which the resistance must be:

Soln:

Apparent Resistance, Rapp= = = 54.5%

Maximum voltage; Vmax = 120+1%

= 120+2=122v

Minimum Voltage; Vmin=120-1%

=120-2=118V

Maximum current

Imax =2.2 + 1% = 2.2+0.05 = 2.25A

Minimum current, Imin = 2.2-1%

= 2.2-0.06 = 2.15A

Rmax = = = 56.74

Rmin = = = 52.44

Alternatively a rough guide to the limit can be calculated as follows:

VI

120 2.2

1222.15

VmaxImin

1182.25

VmaxImax

Add up the errors in voltmeter/ammeter1%+1% = 2%

True resistance = Rapp + 2%

= 54.545 + 2%

Rmax = 54.5m + 2.44=56.9452

Rmin = 54.5 – 2.44 = 52.0652

Thus, the more instruments involved in a measurement the greater the uncertainty involved in the value of the measured quantity.

In general the determinant of a quantity may be assumed to be represented mathematically as a product or a sum of the number of quantities. Which are illustrate by the following:

1. Let us look at the sum of 2 or more components, let the measured results be u, v, w and corresponding errors be + u + v + w, and the result Y .

i.e. Y = u+v+w – (1) this is apparent result.

Max. value of Y becomes

Y+y = u+u+v+v+w+w – (2)

Min. value of Y,Y- y= u-u+v-v+w-w – (3)Substract u+v+w from equations (2) and (3)The error in Y is + y = + (u+v+w) or y = + x 100%

Which is equal to

= + % - (4)

2. As the product of two or more factors.

let Y =uv

Then Y+y = (u+u) (v+v)

= uv + uv + vu + uv

The term vv can be neglected

Y+ y =(u+ u) (v+ v)

=uv+uv+vu+uv

The term v v can be neglected

Y+ y=uv+uv+v u

= +

(u+v+w)

Y

(y)

Y

u

Y

u

U

v

Y

(v)

V

(w)

Y

(w)

W

(y)

Y

uv

uv

vu

U

Or = + the relative error add up.

(5)

(3) As the quotient of two factors

let y=u/v

Y+y= u+u

v+v

For max. errors;

Y+y =

Y+y =

Y+ y=

=

Y+y-y = + + -

y = + + -

y =

y

y u

v

v

u

U+u

V-v

Uv+uv+vu+uv

uv+uv+vu

V2-vv+vv-vv

u+u

V-v

v+v

V-v

V2

uv uv vu

V

u

V

(w)

W

V2 V2

u uv v

VV V2

u

V V uv v

VU V2

V

U

Y

y = + y= + - (6)

Again, the Relative errors add up. Therefore for products or quotients, the max. error is (a+b+c+d+…..)% where a, b, c, d, ……..are relative% systematic errors.

(c) Random Error: These are errors due to causes which cannot be directly established because of random variations in the parameter or the system of measurements. That is external influences beyond the control of the operator may have effects on the measurements. Such quantities may be vibration, change in humidity and pressure, electrical and acoustic noise. These quantities are related to the measurement and they give rise to errors which are random in nature.

While they cannot be eliminated, these effects can be reduced statistically by taking a large number of readings and finding the mean value. The deviation of reading about the mean value gives a measure of the amount of random error involves in the

v u

U

V

uU

Y

uv

measurement, ideally this should be very small. One precise method of evaluating the randomness of a set of observations is to calculate the standard deviation of the data.

i.e

1

б = N (Xi – X)2

Where x = mean value

xi = individual value

N = no of values.

Home work

1. If x = a + b + c

and a = 5000 + 0.1%

b = 600 + 0.5%

c = 40 + 1.0%

½N

i = 1

Find the maximum error in X

2. If y = uv + wz

x

Evaluate the maximum error dy

y

The following readings are obtained in a measurement:

X; 5.30, 5.73, 6.77, 5.75, 6.09, 4.33, 5.25, 5.45, 5.64, 5.81

Compute the mean reading and the standard deviation.

1.2 ESSENTIAL OF INDICATING INSTRUMENTS

Indicating instruments consist, essentially of a pointer moving over a calibrated scale and attached to moving system, pivoted on jeweled bearings. For satisfactory working of indicating instruments the torques required are:

i. Deflecting torque: The deflecting torque is produced by making use of the fact that when an

electric current flow along a conductor, the conductor becomes surrounded by a magnetic field causing the moving system of the instrument to move form its zero position when the instrument is connected in an electric circuit to measure the electrical quantity. This property is used in electromechanical instruments to obtain the deflection of a pointer:

(a)By the interaction of the magnetic field around a coil with a permanent magnet;

(b)Between ferromagnetic values in the coil’s magnetic field; or

(c)Through the interaction of the magnetic fields produced by a number of coils.

constraining these forces to form a turning movement produces a deflecting torque = GF (i) Newton metre (NM), which is function of the current in the instrument’s coil and the geometry and type of coil system. To obtain a stable display. It is necessary to equate the deflection tongue with an opposing called the control torque.

ii. Controlling torque: This magnitude of the movement of the moving system would be somewhat indefinite under the influence of deflecting torque unless some controlling torque existed. The magnitude (C) of this control torque must increase with the angular deflection of the pointer and this is arranged by using a control devices. There are two types of controlling devices, these are given below:

(a)Spring Control – In the spring control system, two spiral springs A and B are arranged as shown in figure 1.1

The inner ends of the hair springs are attached to the spindle P. The outer end of hair spring one end of a lever L pivoted at O. the hair springs A and B are also wound in opposite direction as shown, so that when the moving system is deflected, one springs unwinds while the other is winding up so that the controlling tongue is the result of the combination of the torsions of both springs. Since the tensional torque is controlling torque is directly proportional to the angular deflection of the pointer i.e. control torque = C

(b) Gravity Control: In gravity control instruments, a small weight is attached to the moving system in such a way that it produces a controlling torque, when the moving system is in deflected position. The controlling torque can be varied quite easily by adjusting the position of the controlling weight upon the arm. Gravity control is cheap, unaffected by change in temperature and in free form fatigue or deterioration with time but the instrument has to be kept in vertical position always.

iii. Damping Torque: damping force or torque is also necessary to avoid oscillations of the moving system about its final deflected position owing to

The inertia of the moving parts and to bring the moving system to rest in its deflected position quickly. The moving parts of the instruments will have a moment of inertia (J) and when a change in magnitude of deflection takes place an acceleration torque (Jd2 / dt2 Nm) will be present. Since the movable parts are attached to a control spring they combine to form a mass – spring system and in order to prevent excessive oscillations when the magnitude of the electrical input is changed a damping torque (D d /dt Nm) must be provided that will only act it the moving parts are in motion. It is therefore desirable to provide an adequate damping system that enables the pointer to reach its steady position without oscillation, as shown by curve a in fig 1.2

The method by which this damping torque can be obtained may be

(a)Eddy current – where currents induces in a conducting sheet attached to the movement produce a magnetic field opposing any charge in position.

(b) Pneumatic (air / fluid friction Damping) – In this method an aluminium vane is an extension of this principle, a small vane then being constrained to move in a container filled with a suitably viscous fluid

(c)Electromagnetic – movement of a coil is a magnetic field produces a current in the coil which opposes the deflecting current and slows the response of the instrument. The magnitude of the opposing current will be dependent on the resistance of the circuit to which the connected.

J + + C = Gf (ii) – (8)

The above equation will have a steady state solution when

C - Gf (i) – eqn (9)

ddt2

Dddt

And a dynamic or transient solution of the form = Ae>1 + Be>1 - eqn (10)

Where A and B are arbitrary constants and

X1 = ½ - eqn (ii)

And X2 = ½ - eqn (12)

For a particular instrument C and J are fixed in magnitude driving manufacture.

But D (the amount of damping) may be varied. This results in three possible modes of response to a transient as shown in figure 1.2

-D

2J

D2

4J2

-C

J

-D

2J

D2

4J2

-C

J