Unit - 1

Measurement and Error

Q1) Define the following?

(i)                Accuracy

(ii)             Precision

(iii)          Significant figures

A1)

Accuracy: It is the closeness with which the instrument reading approaches the true value of the quantity being measured. Thus, accuracy of measurement means conformity to truth.

Precision: It is a measure of reproducibility of the measurements that is given a fixed value of quantity precision is a measure of degree of agreement within a group of measurements. The term precise means clearly or sharply defined.

Significant figures convey actual information regarding the magnitude and the measurement precision of quantity. The more the significant figures the greater the precision of measurement.

Q2) Explain the types of errors?

A2) Errors may rise from different sources and are classified as:

• Gross Errors
• Systematic errors
• Random Errors

Gross Errors: This class of errors covers human mistakes in reading instruments and recording and calculating measurement results. The responsibility of the mistake normally lies with the experimenter. The experimenter may grossly misread the scale.

Example

Due to some reasons, he might have read the temperature as 31.5 while the actual reading may be 21.5. He may transpose the reading while recording. For example, he may read 25.8 as 25.8. If human beings are involved gross errors will be committed.

Systematic Errors

These types of errors are divided into three categories:

• Instrument errors
• Environmental errors
• Observational errors

Instrumental Errors

These errors arise due to three main reasons:

(i)                Due to inherent shortcomings in the instrument

(ii)              Due to misuse of the instrument

(iii)           Due to loading effects of instruments.

(1) Inherent short comings of instrument because of their mechanical structure. They may be due to construction, calibration or operation of the instruments or measuring device.

While making precision measurements we recognize the possibility of such errors as it is often possible to eliminate them by using the following methods:

(1)  The procedure of measurement must be carefully planned. Substitution methods or calibration against standards may be used for the purpose.

(2)  Correct factors should be applied after determining the instrumental errors.

(3)  The instruments may be re-calibrated carefully.

(2) Misuse of instrument:

A good instrument used in an unintelligent way may give erroneous results.

Examples which may be cited for this misuse of instrument may be failure to adjust the zero of instruments, poor initial adjustments using leads too high a resistance and so on.

(3) Loading effects: One of the most common errors committed by beginners is the improper use of an instrument for measurement work. For example, a well calibrated voltmeter may give a misleading voltage reading when connected across high resistance circuit. However, when the same voltmeter is connected in low resistance circuit it may give dependable reading. These examples illustrate that the voltmeter has loading effect on the circuit.

Q3) Write a short note on standards of measurement?

A3) Standards of Measurement are classified into the following categories:

(i) International Standards (ii) Primary Standards

(iii) Secondary Standards (iv) Working standards

Classification of Standards

The international standards are defined on the basis of international agreement. They represent the units of measurements which are closest to the possible accuracy attainable with present day technological and scientific methods .

International standards are checked and evaluated regularly against absolute measurements in terms of fundamental units.

The international standards are maintained at the International Bureau of Wieghts and Measures and are not available to ordinary user of measuring instruments for purpose of calibration or comparison.

Primary standards:

Primary standards are absolute standards of such high accuracy that can be used as the ultimate reference standards. These standards are maintained by national standards laboratories in different parts of the world. One of the main functions of primary standards is the verification and calibration of secondary standards.

The following points must be taken into consideration when a primary standard is built :

(i)                The material should have long time stability

(ii)              The temperature co-efficient of the material should be as small as possible.

(iii)           The deterioration of materials caused by moisture and other environmental conditions should be eliminated as far as possible.

(iv)            The machining of parts should be accurate

(v)              The measurement of physical dimension on which the accuracy of the standard depends predominantly should be done with most sosphisticated techniques available.

(vi)            The rigidity of the construction should be insured.

Secondary Standard:

The secondary standard are the basic reference standard used in industrial measurement laboratories. The responsibilty of maintainence and calibration of these standards lies with the particular industry involved. These standards are checked locally against reference standards available in the area.

Secondary standards are normally sent periodically to the national standards laboraties for calibration and comparison against primary standards. The secondary standards are sent back to the industry by national laboratories with ceritification as regards their measured vallues in terms of primary standards.

Working Standards

The working standards are the major tools of measurement laboratory. These standards are used to check and calibrate general laboratory instruments for their accuracy and performance.

For example a manufacturer of precision resistances may use a Standard Resistance in quality control department for checking the values of resistors that are being manufactured.

Q4) Write a short note on PMMC?

A4) The permanent magnet moving coil instrument is the most accurate type for dc measurements. The working principle is same as that of d’Arsonval type of galvanometer the difference is that direct reading instrument is provided with pointer and scale.

Figure.  PMMC

The three important torque involved in this instrument are:

Deflecting torque:

The force F which will be perpendicular to both the direction of the current flow and the direction of the magnetic field as per Fleming’s left hand rule can be written as

F = NBIL

Where N: turns of wire on the coil

B: flux density in the air gap

I: current in the movable coil

L: vertical length of the coil

Theoretically, the torque that is electro-magnetically torque is equal to the multiplication of force with distance to the point of suspension.

Hence Torque on the left side of the cylinder TL = NBIL x W/2 and torque on right side of the cylinder TR = NBIL x W/2

Therefore, the total torque will be = TL + TR

T = NBILW or NBIA where A is an effective area (A= LxW).

Controlling Torque

This torque is produced by the spring action and opposes the deflection torque so as the pointer can come to rest at the point where these two torques are equal that is Electromagnetic torque = control spring torque. The value of control torque depends on the mechanical design of spiral springs and strip suspensions.

The controlling torque is directly proportional to the angle of deflection of the coil.

Control torque Ct =Cθ where,

θ = deflection angle in radians and C = spring constant Nm /rad

Damping torque

This torque ensures the pointer comes to an equilibrium position that is rest in the scale without oscillating to give an accurate reading. In PMMC as the coil moves in the magnetic field, eddy current sets up in a metal former or core on which the coil is wound or in the circuit of the coil itself which opposes the motion of the coil resulting in the slow swing of a pointer and then come to rest quickly with little oscillation.

Q5) Explain the working of moving iron attraction and repulsion type?

A5)

General Torque Equation:

An expression for the torque of moving iron instrument may be derived by considering the energy relations when there is small increment in current supplied to the instrument. When this happens there will be small deflection d and some mechanical work.

Let Td = deflecting torque

Mechanical work done = Td . d

Along side there will be change in the energy stored owing to inductance.

Suppose initial current is I, the instrument inductance L and deflection If the current increases by dI then the deflection changes by d and inductance by dL. To increment dl in the current there must be an increase in the applied voltage given by

e = d/dt(LI) = I dL/dt + L dI/dt

The electrical energy supplied eIdt = I2 dL + IL dI

The stored energy changes from ½ I2 L to ½ (I +dI) 2 (L +dL)

Hence the change in stored energy = ½ ( I2 + 2IdI + dI2) (L+dL) – ½ I2 dL

From the principle of conservation of energy

Electrical energy supplied = increased in stored energy + mechanical work done.

I2 dL + IL dI = IL dI = ½ I2 dL + Td d

Td d = ½ I2 dL

Or Deflecting torque Td = ½ I2 dL/d

T is in newton metre I in ampere L is in henry and in radian .

The moving system is provided with control springs and it turns the deflecting torque Td is balanced by controlling torque Tc.

Controlling torque Tc = K      where K = control spring constant Nm/rad

= deflection

At equilibrium or final steady position Tc = Td

Or k = ½ I 2 dL/d  or deflection = ½ I2 /K dL/d

Hence deflection is proportional to the square of the rms value of the operating current.

Attraction type:

Figure. Attraction type

The coil is flat and has narrow slot like opening. The moving iron is flat disc or sector eccentrically mounted. When the current flows through the coil magnetic field is produced and the moving iron moved from the weaker filed outside the coil to stronger field inside it or moving iron is attracted in. The controlling torque is provided by springs, but gravity control can be used for panel type instruments which are vertically mounted. Damping is provided by air friction usually by vane moving in a sector shaped chamber.

Repulsion type:

Figure. Repulsion type

In repulsion type there are two vanes inside the coil one fixed and the other movable. These are similarly magnetized when the current flows through the coil and there is force of repulsion between the two vanes resulting in the movement of the moving vane.

Q6) Write a short note on Electrodynamometer?

A6)

Figure. Electrodynamometer

Construction: A fixed coil is divided in to two equal- half. The moving coil is placed between the two- half of the fixed coil. Both the fixed and moving coils are air cored. So that the hysteresis effect will be zero. The pointer is attached with the spindle. In a non- metallic former the moving coil is wounded.

Control: Spring control is used.

Damping: Air friction damping is used.

Principle of operation: When the current flows through the fixed coil, it produced a magnetic field, whose flux density is proportional to the current through the fixed coil. The moving coil is kept in between the fixed coil. When the current passes through the moving coil, a magnetic field is produced by this coil. The magnetic poles are produced in such a way that the torque produced on the moving coil deflects the pointer over the calibrated scale. This instrument works on AC and DC. When AC voltage is applied, alternating current flows through the fixed coil and moving coil. When the current in the fixed coil reverses, the current in the moving coil also reverses. Torque remains in the same direction. Since the current i1 and i2 reverse simultaneously. This is because the fixed and moving coils are either connected in series or parallel.

Torque Equation:

Let i1 = instantaneous value of current in the fixed coils.

       i2 = instantaneous value of current in moving coils.

       L1 = self-inductance of fixed coils

       L2 = self -inductance of moving coils.

       M = mutual inductance between fixed and moving coils.

Inductance

Flux linkages of coil 1, λ1= L1i1+ Mi2

Flux linkages of coil 2, λ2 = L2i2+ Mi1

Electrical input energy

e1i1dt + e2i2dt= i1dλ1+i2dλ2

As e1= dλ1/dt and e2=dλ2/dt

= i1d(L1i1+Mi2) +i2d(L2i2+Mi2)

= i1L1di1+i1i2dM+ i12dL1+i1Mdi2+i2L2id2+i2 2 dL2 + i1i2dM+i2Mdi1-----(1)

Energy stored in the magnetic field is = ½ i12 L1 + ½ i2 2 L2 + i1 i2 M

Change in energy stored = d(1/2 i1 2 L1 + ½ i 2 2 L2 + i1 i2 M)

= i1L1di1+(i12/2 ) dL1 + i2L2dI2 + i2 2 /2 dL2 + i1MdI2 + i2 Mdi1 + i1i2dM ---(2)

From principle of conservation of energy,

Total electrical input energy = change in energy stored + mechanical energy

The mechanical energy can be obtained by subtracting (ii) from (1)

Mechanical energy = ½ i12 dL1 + ½ i2 dL2 + i1i2dM

Now the self- inductances L1 and L2 are constants and therefore dL1 are dL2 are both equal to zero.

Hence mechanical energy = i1i2dm

Suppose T1 is the instantaneous deflecting torque and d is the change in deflection then

Work done = Ti d

Thus, we have

Ti d = i1i2dM or Ti= i1i2dM/d

Q7) Write a short note on Energy Meter?

A7)

Figure. Single Phase Energy Meter

There are four main parts of the operating mechanism.

(i)                Driving system (ii) Moving system

(ii)              Braking system (iv) Registering system

Driving System:

The driving system of the meter consists of two electromagnets. The electromagnets are made up of silicon steel laminations. The coil of the electromagnet is excited by load current. This coil is current coil. The coil of the second electromagnet is connected across the supply and therefore carries current proportional to supply volts. This coil is called pressure coil. These two electromagnets are known as series and shunt magnets respectively.

Copper shading bands are provided on the central limb. The function of these bands to bring the flux produced by the shunt magnet exactly in quadrature with the applied voltage.

Moving System:

It consists of aluminium disc mounted on light alloy shaft. The disc is positioned in the air gap between series and shunt magnets. The moving system floats without touching either bearing surface and the only contact with the movement is that of the gear connecting to the shaft with gear of the train thus the friction is drastically reduced.

Braking System:

A permanent magnet positioned near the edge of the aluminium disc forms the braking system. The aluminium disc moves in the field of this magnet and thus provides a braking torque. The position of permanent magnet is adjustable and therefore braking torque can be adjusted by shifting the permanent magnet to different radial positions.

Registering Mechanism:

The function of registering or counting mechanism is to record continuously a number which is proportional to the revolution made by the moving system.

Q8) Write a short note on Induction type wattmeter?

A8)

Figure. Induction type Wattmeter.

Principle of Operation:

The shunt and series magnets produce alternating fluxes. These fluxes produce eddy emfs in disc and in turn cause eddy currents to flow. These eddy currents interact with the fluxes to produce deflecting torque.

Theory:

The theory and operation of induction type wattmeter is similar to that of an induction type wattmeter. The only difference is that in induction wattmeter a control spring restricts the motion of the disc by exerting a controlling torque while an induction watthour energy meter the disc is allowed to rotate continuously.

Deflecting torque Td = K1VI cos ɸ = K1 P

Where P=VI cos ɸ = power in the circuit

Controlling torque Tc = K

Deflection = K1/K P = K’ VI cos ɸ where K1 and K’ are constants.

Q9) Explain Single phase Induction meter?

A9)

Single phase induction type energy meter consists of four important systems which are written as follows:

Driving System:

Driving system consists of two electromagnets on which pressure coil and current coils are wounded, as shown in the diagram.

The coil which consisted of load current is called current coil while coil which is in parallel with the supply voltage is called pressure coil. Shading bands are wounded on as shown above in the diagram to make angle between flux and applied voltage equal to 90 degrees.

Moving System:

To reduce friction to greater extent floating shaft energy meter is used, the friction is reduced to greater extinct because the rotating disc which is made up of light material like aluminium is not in contact with any of the surface. It floats in the air.

Braking System:

A permanent magnet is used to produce breaking torque in single phase induction energy meters which are positioned near the corner of the aluminium disc.

Counting System:

Numbers marked on the meter are proportion to the revolutions made by the aluminium disc, the main function of this system is to record the number of revolutions made by the aluminium disc.

Figure. Single Phase Induction Meter

The pressure coil is highly inductive in nature and consists of  large number of turns. The current flowing in the pressure coil is Ip which lags voltage by an angle of 90 degrees. This current produce flux F. F is divided into two parts Fg and Fp.

Q10) Write a short note on Vibrating reed type frequency meter?

A10)

Vibrating reed type

Construction:

This meter consists of several thin steels called reeds. These reeds are placed in arow alongside and close to an electromagnet as shown in figure. The electromagnet has a laminated iron core, and its coil is connected in series with the resistance across the supply whose frequency is to be measured.

The natural frequency of the reeds depends on the weights and dimensions. Since the reeds have different weight and size their natural frequency of vibration are different.

Operation:

When the frequency meter is connected across the supply whose frequency is to be measured the coil of electromagnet carries I which alternates at the supply frequency. The force of attraction between the reeds and the electromagnet is proportional to i2 and therefore this force varies twice the supply frequency.

Thus, the force is exerted on the reeds every half cycle. All the reeds will tend to vibrate but the reed whose natural frequency is equal to twice the frequency of supply will be in resonance and vibrate the most. The tuning in these meters is so sharp that as the excitation frequency departs from the resonant frequency the amplitude of vibration decreases rapidly becoming negligible.

When the 50Hz reed is vibrating with maximum amplitude some vibrations of 49.5 Hz and 50.5 Hz reeds in fig(b). For frequency midway between the reeds will vibrate with amplitudes which are equal in magnitude. Fig(c) shows the condition of vibrating reeds when the frequency is exactly midway between 49.5Hz and 50Hz.

Figure. Indication of vibrating reeds.

Q11) Write a short note on power factor frequency meter?

A11)

The power factor meter measures the power factor of a transmission system. The power factor is the cosine of the angle between the voltage and current.

Figure. Single phase Electrodynamometer Power meter

The construction of the single -phase electrodynamometer is shown in the figure below. The meter has fixed coil which acts as a current coil. This coil is split into two parts and carry the current under test. The magnetic field of the coil is directly proportional to the current flow through the coil.

The meter has two identical pressure coils A and B. Both the coils are pivoted on the spindle. The pressure coil A has no inductive resistance connected in series with the circuit, and the coil B has highly inductive coil connected in series with the circuit.

The current in the coil A is in phase with the circuit while the current in the coil B lag by the voltage nearly equal to 90º. The connection of the moving coil is made through silver or gold ligaments which minimize the controlling torque of the moving system.

The meter has two deflecting torque one acting on the coil A, and the other is on coil B. The windings are so arranged that they are opposite in directions. The pointer is in equilibrium when the torques are equal.

Deflecting torque acting on the coil A is given as

TA = KVIM cosɸ sin

θ – angular deflection from the plane of reference.

Mmax – maximum value of mutual inductance between the coils.

The deflecting torque acting on coil B is expressed as

IB = KVIMmax cos(90-ɸ) sin (90+ɸ)

IB = KVIMmax cos ɸ sin

The deflecting torque is acting on the clockwise direction.

The value of maximum mutual inductance  is same between both the deflecting equations.

TA = TB

KVIM cosɸ sin = KVIMmax cos ɸ sin

This torque acts on anti-clockwise direction. The above equation shows that the deflecting torque is equal to the phase angle of the circuit.