Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering PDF Download

1. Basics of Measurements and Error Analysis (Static & Dynamic Characteristics of Measuring Instrument)

Fundamental Units

These include the following along with dimension and unit symbol.
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Performance Characteristics 

  • The performance characteristics of an instrument are mainly divided into two categories:
    (i) Static characteristics
    (ii) Dynamic characteristics 
  • Set of criteria defined for the measurements, which are used to measure the quantities, which are slowly varying with time or almost constant, i,e do not vary with time, are called static characteristics 
  • When the quantity under measurement changes rapidly with time, the relation existing between input and output are generally expressed with the help of differential equations and are called dynamic characteristics 
  • The various performance characteristics are obtained in one form or another by a process called calibration 
Static Characteristics
1. Accuracy
It is the degree of closeness with which the instrument reading approaches the true value of the quantity. 

2. Static error
It is the difference between the measured value and true value of the quantity Mathematically δA = Am - At ........eq (1.1)
δA : absolute static error
Am: Measured value of the quantity.

At : True value of the quantity.
Relative error: (∈r) = δA/At
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Percentage relative error:
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

From relative percentage error, accuracy is expressed as A = 1 - | ∈r|
Where A: relative accuracy and a = A x 100%
where a = percentage Accuracy.

  • error can also be expressed as percentage of full scale reading (FSD) as,
    Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

3. Precision
It is the measure of degree of agreement within a group of measurements. 

  • High degree of precision does not guarantee accuracy.
    Precision is composed of two characteristics
    (i) Conformity.
    (ii) Number of significant figures. 

4. Significant Figures 

  • Precision of the measurement is obtained from the number of significant figures, in which the reading is expressed. 
  • Significant figures convey the actual information about the magnitude and measurement precision of the quantity. 

5. Sensitivity 
The sensitivity denotes the smallest change in the measured variable to which the instrument responds.
It is defined as the ratio of the changes in the output of an instrument to a change in the value of the quantity to be measured.
Mathematically it is expressed as,

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

6. Resolution 
Resolution is the smallest measurable input change.

7. Threshold 
If the input quantity is slowly varied from zero onwards, the output does not change until some minimum value of the input is exceeded. This minimum value of the input is called threshold.
Resolution is the smallest measurable input change while the threshold is the smallest measurable input.

8. Linearity 
Linearity is the ability to reproduce the input characteristics symmetrically and linearly. Graphically such relationship between input and output is represented by a straight line.

The graph of output against the input is called the calibration curve, The linearity property indicates the straight line nature of the calibration curve.
Thus, the linearity is defined as,
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

9. Zero Drift 

The drift is the gradual shift of the instrument indication, over an extended period during which the value of the input variable does not change. 

10. Reproducibility 

It is the degree of closeness with which a given value may be repeatedly measured. It may be specified interms of units for a given period of time. 

11. Repeatability 
Repeatability is defined as variation of scale reading and is random in nature. Both reproducibility and the repeatability are a measure of the closeness with which a given input may be measured again and again. The Fig shows the input and output relationship with positive and negative repeatability.
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

12. Stability 
The ability of an instrument to retain its performance throughout its specified operating life and the storage life is defined as its stability

13. Tolerance
The maximum allowable error in the measurement is specified interms of some value which is called tolerance. This is closely related to the accuracy. 

14. Range or Span 

The minimum and maximum values of a quantity for which an instrument is designed to measure is called its range or span. Sometimes the accuracy is specified interms of range or span of an instrument.

Limiting Errors/ Relative Limiting Error 

Guarantee Errors: The limits of deviations from the specified value are defined as limiting errors or guarantee errors.
Actual value of quantity = A= An ± δa; δa: limiting error or tolerance
An: specified or rated value

  • It is also called as fractional error. It is the ratio of the error to the specified magnitude of a quantity.
    Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Combination of Quantities with Limiting Errors

1. Sum of the Two Quantities: Let a1 and a2 be the two quantities which are to be added to obtain the result as AT.
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
2. Difference of the Two Quantities
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
3. Product of the Two Quantities
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
4. Division of the Two Quantities
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

5. Power of a factor
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Types of Errors 

The static error may arise due to number of reasons. The static errors are classified as

  1. Gross errors
  2. Systematic errors
  3. Random errors
Time Domain test signals

1. Step input
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
= A/S
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

2. Ramp input
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

3. Parabolic input
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

4. Impulse input

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Response of First Order System to a Unit Step Input

C(t) = 1 - e-t/τ

em(t) = e-t/τ

ess = limt →∞ em (t) = 0

Ramp Response of a First Order System

C(t) = 1 - τ[1 - e-t/τ]
em = τ[1 - e-t/τ]
ess = τ
Impulse Response of a First Order System


Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

2. Measurements of Basic Electrical Quantities 1 (Current Voltage, Resistance)

Indicating Instruments

Analog Instruments 

Analog instruments are classified in one way as
(a) Indicating
(b) Recording
(c) Integrating Instruments.

Essential Requirements of an Instrument 

For satisfactory operation of any indicating instrument, following systems must be present in an instrument. 

1. Deflecting system producing deflecting toque Td 
2. Controlling system producing controlling torque Tc 
3. Damping system producing damping torque.

The deflecting system uses one of the following effects produced by current or voltage, to produce deflecting torque.
1. Magnetic Effect: When a current carrying conductor is placed in uniform magnetic field, it experiences a force which causes it to move. This effect is mostly used in many instruments like moving iron attraction and repulsion type, permanent magnet moving coil instruments etc.
2. Thermal Effect: The current to be measured is passed through a small element which heats it to cause rise in temperature which is converted to an e.m.f. by a thermocouple attached to it.
When two dissimilar metals are connected end to end, to form a closed loop and the two junctions formed are maintained at different temperatures, then e.m.f. is induced which causes the flow of current through the closed circuit which is called a thermocouple.
3. Electrostatic Effect: When two plates are charged, there is a force exerted between them, which moves one of the plates. This effect is used in electrostatic instruments which are normally voltmeters.
4. Induction Effect: When a non-magnetic conducting disc is placed in a magnetic field produced by electromagnets which are excited by alternating currents, an e.m.f. is induced in it.
If a closed path is provided, there is a flow of current in the disc. The interaction between induced currents and the alternating magnetic fields exerts a force on the disc which causes to move it. This interaction is called an induction effect. This principle is mainly used in energymeters.
5. Hall Effect: If a semiconductor material is placed in uniform magnetic field and if it carries current, then an e.m.f. is produced between two edges of conductor. The magnitude of this e.m.f. depends on flux density of magnetic field, current passing through the conducing bar and hall effect co-efficient which is constant for a given semiconductor. This effect is mainly used in flux-meters.

Controlling System

It produces a force equal and opposite to the deflecting force in order to make the deflection of pointer at a definite magnitude.

Damping System

The quickness with which the moving system settles to the final steady position depends on relative damping.
Three types of damping exists
1. Critically damped
2. Under damped
3. Over damped
Effect of damping on deflection Effect of damping on deflection 

The following methods are used to produce damping torque.
1. Air friction damping
2. Fluid friction damping
3. Eddy current damping.

Measurement of Voltage and Current 

Analog Ammeter and Voltmeters 
The Instruments used for measurement of voltage and current can be classified as:
(a) Moving coil instruments

  • Permanent magnet type
  • Dynamometer type

(b) Moving iron instruments
(c) Electrostatic Instruments
(d) Rectifier instruments
(e) Induction instruments
(f) Thermal instruments

  • Hot - wire type
  • Thermocouple type

(a) Moving Coil Instruments
(i) Permanent Magnet Type:
It works on the principle of magnetic effect 

Torque equation:
The deflecting torque Td is given by
Td - NBAI 

Where Td - deflecting torque in N - m
B - Flux density in air gap Wb/m2.
N - Numbers of turns of the coil
A - effective coil area m2 (l x b)
l - length; b - breadth of the coil
I - current in the moving coil, amperes 

The controlling torque is provided by springs and is proportional to angular deflection of the pointer
Tc = Kθ
Where Tc = controlling torque
K = spring constant, Nm/rad or Nm/deg 
θ = Angular deflection.
For the final steady state position,
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
G = NBA
* Thus we get a linear relation between current and deflection angle. 

• Damping used in this type of instrument is eddy current damping. 

(ii) Dynamometer Type 

  • Dynamometer instrument uses the current under measurement to produce the magnetic field. 
  • Deflecting Torque
    Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringWhere i1 = instantaneous value of current in fixed coils; A
    i2 = Instantaneous value of current in moving coil; A
    M = Mutual inductance between fixed and moving coil; H

Operation with D,C

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Operation with AC
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
T = time period for one complete cycle.
Electrodynamometer Ammeters
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Range: upto 100 mA.
Electrodynamometer Voltmeter
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

(b) Moving Iron Instruments

Moving iron instruments depend for their indication upon the movement of a piece of soft iron in the field of a coil produced by the current to be measured.
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Where I is the current through the coil and L is the inductance.
Linearization of Scale: Compensation towards frequency errors can be done by connecting a capacitor across a part of series resistance in Mi voltmeter, C = 0.41 x (L/R2)

(c) Electrostatic Instruments
For linear motion:
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
For angular motion:
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

(d) Rectifier Instruments 
Half wave Rectifier type Instruments
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Full wave rectifier type instruments:
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Shunts and Multipliers

Shunts and multipliers are the resistance connected in shunt or series with ammeter and voltmeters to enhance their measuring capacity.

➤ Shunt with ammeter
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Instrument constant,
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Multiplier with Voltmeter
Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringChapter 7 - Measurements | Additional Study Material for Mechanical Engineering

➤ Shunt for a.c. instruments
Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringMultiplication factor
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

➤ Multipliers for Moving - Iron Instruments
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

voltage multiplying factor (m)
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Measurement of Resistance 

➤ Measurement of Low Resistance 
Kelvin's Double Bridge is used for the measurement of low resistance as shown in fig
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

➤ Measurement of Medium Resistance
Two wires are required to represent a medium resistance:
This can be measured by:
(a) Ammeter voltmeter method
(b) Wheatstone bridge method
(c) Ohm meter

(a) Voltmeter - Ammeter Method

From fig
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Measured value of resistance,
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Where R is the true value of the resistance.

Error = Ra % Error = (Ra/R)

This method is suitable for measurement of high resistance, among the range.
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Ammeter - Voltmeter Method

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering 

This method is suitable for measurement of low resistance among the range.
The resistance where both the methods give same error is obtained by equating the two errors.
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

(b) Wheat stone Bridge
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Sensitivity of the galvanometer, Sv = θ/e
Where θ = deflection of the galvanometer
e = emf across galvanometer
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Sensitivity of galvanometer,

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Sensitivity of the Bridge
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

➤ Measurement of High Resistance 

Loss of charge method
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

3. Measurements of Basic Electrical Quantities 2 (Power and Energy, Instrument Transformers)

1. Power In D.C. Circuits 
In case of fig (a)
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Power measured
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

True value = Measured power - power loss in ammeter
In case of fig (b)
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Power measured (Pm2) = VRIR + (V2R / Rv)
True power = Measured power - power loss in voltmeter

2. Power In A.C. Circuits
Instantaneous power = VI
Average power = VI cos (Ф)
Where V and I are r.m.s values of voltage and current and cos Ф is the power factor of the load.

3. Electro Dynamometer Wattmeter: This type of wattmeter is mostly used to measure power. The deflecting torque in electrodynamometer instruments is given by,
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Many watt meters are compensated for errors caused by inductance of pressure coil by means of a capacitor connected in parallel with a portion of multiplier.

Capacitance C = (L/r2)

4. Low Power Factor Wattmeter 
Td = ipic (dM /dθ)
Average deflection torque = IPI cos (Ф) (dM / dθ)
= (V / Rp). I cos (Ф) (dM / dθ)
Td ∝ VI cos Ф (dM/ dθ) 

∝ power, if (dM / dθ) is constant.

5. Errors in Electro Dynamometer Wattmeter

True Power for lagging pf loads

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
True Power for leading pf loads
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
ERROR = tan Ф tan β x true power, Ф = pf angle, β = tan-1 (Xp / RP) = VI sin Ф tan β
% ERROR = tan Ф tan β x 100
β → is the angle between PC current and voltage.

6. Measurement of Power in Three Phase Circuits
(a) Three watt meter Method: The figure depicts three wattmeter method to determine the power in 3 - Ф, 4 wire system.
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Sum of the instantaneous readings of watt meters
= P = P1 + P2 + P3
= V1i1 + V2 i2 + V3 i3 
i3 Instantaneous power of load = V1 i1 + V2 i2 + V3i3 
Hence the summation of readings of three watt meters gives the total power of load.
(a) Two Wattmeter Method

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Sum of reading of two wattmeters = 3 VI cos Φ 

Difference of readings of two watt meters = √3 VI sin Φ 

∴ Reactive power consumed by load = √3 (Difference of two wattmeter readings) = √3 (P1 - P2)
Power factor cos Φ
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

7. Measurement of Reactive Power in Three Phase Circuits 

Reading of wattmeter = V23 i1 cos (angle between i1 and V23)
= V23 i1 cos (90 - Φ)
= √3 V I sin Φ
Total reactive power of the circuit
= √3 (watt meter reading).

4. Electronic Measuring Instruments 1 (Analog, Digital Meters & Bridges, ADC type DVM)

The general ac bridge circuit is as follows
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Under balanced condition
Z1Z4∠(θ1 + θ4) = Z2Z3∠(θ2 + θ3)
Equating the magnitudes and angles,
Z1 Z4 = Z2 Z3 
θ1 + θ4 = θ2 + θ3 

Measurement of Self Inductance

1. Hay's Bridge 

  • Used for measurement of high Q coils, shown in figure.
    Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

At balance

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

2. Owners Bridge 

  • Used for measurement of inductance in terms of capacitance, shown in figure.
    Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringUnder balance condition:
    L4 = R2 R3 R4
    Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

3. Maxwell Inductance Bridge 

  • This bridge measures inductance by comparison with a variable standard self - inductance, shown in figure

Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringAt balance
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

4. Maxwell's Inductance - Capacitance Bridge 

  • Here inductance is measured by comparison with standard variable capacitance, shown in figure.
    Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringAt balance
    Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
  • Useful for measurement of low Q coils (1 < Q < 10)

5. Anderson's Bridge: 

  • In this method, self - Inductance is measured in terms of a standard capacitor, shown in fig below 
  • Applicable for precise measurement of self - inductance over a very wide range of values

Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringAt balance
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
L1 = C R3 [r (R4 + R2) + R2 R4]R4.

Measurement of Capacitance 

1. De Sauty's Bridge 

  • It measures the unknown capacitance by comparing with a standard capacitor, shown in figure
    Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringAt balance:
    Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

2. Modifed De Sauty's Bride

Chapter 7 - Measurements | Additional Study Material for Mechanical EngineeringAt balance
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Dissipation factor, D = tan δ1 = ωC1r1
D2 = tan δ2 = (ω) C2r2
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

3. Schering Bridge
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

At balance condition

Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
C1 = C2 (R2 / R3)
Dissipation factors D1 = tan δ1 = ωC1r1 = C4r3.

Digital Voltmeters

Type of DVM's 
1. Ramp Type DVM: The operating principle is to measure the time that a linear ramp voltage takes to change from level of input voltage to zero voltage or vice - versa.
2. Integrating Type Digital Voltmeter: The frequency of the saw tooth wave (E0) is a function of the value of Es, the voltage being measured. The number of pulses produced in a given time interval and hence the frequency of saw tooth wave is an indication of the value of voltage being measured.
3. Potentiometric Type DVM: A potentiometric type of DVM, employs voltage comparison technique. In this DVM the unknown voltage is compared with a reference voltage whose value is fixed by the setting of the calibrated potentiometer.

5. Electronic Measuring Instruments 2 (C.R.O., RF Meters, Special Meters, Q meter)

Cathode Ray Tube(CRO)

For electrostatic deflection
Deflection
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

D - Deflection, m

L - distance from centre of deflection plates to screen, m
Ld - effective length of deflection plates, m
Ed - deflection voltage, volts d - separation between the plates, m
Ea- accelerating voltage, volts 

Deflection sensitivity is
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Deflection factor G is
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering

Oscilloscope Specifications

1. Sensitivity: It means the vertical sensitivity. It refers to smallest deflection factor G = (1 / s) and expressed, as mv / div. The alternator of the vertical amplifier is calibrated in mv / div.
2. Bandwidth: It is the range of frequencies between ± 3 dB of centre frequency.
3. Rise Time: Rise time is the time taken by the pulse to rise from 10% to 90% of its amplitude.
BW = 1/2πRC = band width in MHz

90% of amplitude is normally reached in 2.2 RG or 2.2 time constants.
Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering
Tr = rise time in μ second

Synchronization means the frequency of vertical signal input as an integral multiple of the sweep frequency.

Fin = nFs

Measurement Of Phase Difference And Frequency 

If Vx and Vy be the instantaneous values and of voltages applied to the deflection plates x and y and let them be expressed as

Vx = Vx sin ωxt    Vy = Vy sin (ωy t - Ф)
By adjusting the values of toy, ωx, ωy, Vx, Vy and Ф suitably, various patterns may De obtained on the screen.

1. When ωx = ωy = ω, Ф = 0, then (Vx / Vy) = (Vx / Vy) = K is an equation of straight line passing through origin and making an angle of tan θ = (Vy / Vx) with horizontal.
2. ωx = ωy = ωy, Ф = (π/4) ⇒ an ellipse whose major axis has a slope of (Vx / Vy)
3. ωx = ωy = ωy, Ф = (π/2) radians ⇒ a circle.
4. When ωx = 2ωy, we get Fig (i). When ωy = 2ωx we get Fig (ii).

Q - Meter

The Q meter is an instrument which is designed to measure the value of the circuit Q directly and as such is very useful in measuring the characteristics of coil and capacitors.

The storage factor Q of a Q network is equal to
Q = ω0L/R where,
ω0 = resonant angular freq
L = inductance of coil
R = effective resistance of coil

Shielding 

Electromagnetic shielding is the process of limiting the penetration of electromagneticflelds into a space, by blocking them with a barrier made of conductive material.

Grounding 

Grounding electrically interconnects conductive objects to keep voltages between them safe, even if equipment fails.

The document Chapter 7 - Measurements | Additional Study Material for Mechanical Engineering is a part of the Mechanical Engineering Course Additional Study Material for Mechanical Engineering.
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