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.

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 

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

A_t : True value of the quantity.
Relative error: (∈_r) = δA/A_t

Percentage relative error:

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,
  

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,

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,

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.

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.
  

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 A_T.


2. Difference of the Two Quantities

3. Product of the Two Quantities

4. Division of the Two Quantities

5. Power of a factor

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

1. Step input

= A/S

2. Ramp input

3. Parabolic input

4. Impulse input

Response of First Order System to a Unit Step Input

C(t) = 1 - e^-t/τ

e_m(t) = e^-t/τ

ess = lim_{t →∞} e_m (t) = 0

Ramp Response of a First Order System

C(t) = 1 - τ[1 - e^-t/τ]
e_m = τ[1 - e^-t/τ]
e_ss = τ
Impulse Response of a First Order System

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 T_d 
2. Controlling system producing controlling torque T_c 
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 

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 T_d is given by
T_d - NBAI 

Where T_d - 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
T_c = Kθ
Where T_c = controlling torque
K = spring constant, ^Nm/_rad or ^Nm/_deg 
θ = Angular deflection.
For the final steady state position,

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
  Where 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

Operation with AC

T = time period for one complete cycle.
Electrodynamometer Ammeters

Range: upto 100 mA.
Electrodynamometer Voltmeter

(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.

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:

For angular motion:

(d) Rectifier Instruments 
Half wave Rectifier type Instruments

Full wave rectifier type instruments:

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

Instrument constant,

➤ Multiplier with Voltmeter

➤ Shunt for a.c. instruments
Multiplication factor

➤ Multipliers for Moving - Iron Instruments

voltage multiplying factor (m)

Measurement of Resistance 

➤ Measurement of Low Resistance 
Kelvin's Double Bridge is used for the measurement of low resistance as shown in fig

➤ 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

Measured value of resistance,

Where R is the true value of the resistance.

Error = R_a % Error = (R_a/R)

This method is suitable for measurement of high resistance, among the range.

Ammeter - Voltmeter Method

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.

(b) Wheat stone Bridge

Sensitivity of the galvanometer, S_v = θ/e
Where θ = deflection of the galvanometer
e = emf across galvanometer

Sensitivity of galvanometer,

Sensitivity of the Bridge

➤ Measurement of High Resistance 

Loss of charge method

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

1. Power In D.C. Circuits 
In case of fig (a)

Power measured

True value = Measured power - power loss in ammeter
In case of fig (b)

Power measured (P_m2) = V_RI_R + (V²R / R_v)
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,

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 
T_d = i_pi_c (dM /dθ)
Average deflection torque = I_PI cos (Ф) (dM / dθ)
= (V / R_p). I cos (Ф) (dM / dθ)
T_d ∝ VI cos Ф (dM/ dθ) 

∝ power, if (dM / dθ) is constant.

5. Errors in Electro Dynamometer Wattmeter

True Power for lagging pf loads

True Power for leading pf loads

ERROR = tan Ф tan β x true power, Ф = pf angle, β = tan^-1 (X_p / R_P) = 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.

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

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

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

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.
  

At balance

2. Owners Bridge 

• Used for measurement of inductance in terms of capacitance, shown in figure.
  Under balance condition:
  L₄ = R₂ R₃ R₄
  

3. Maxwell Inductance Bridge 

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

At balance

4. Maxwell's Inductance - Capacitance Bridge 

• Here inductance is measured by comparison with standard variable capacitance, shown in figure.
  At balance
  
• 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

At balance

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
  At balance:
  

2. Modifed De Sauty's Bride

At balance

Dissipation factor, D = tan δ₁ = ωC₁r₁
D₂ = tan δ₂ = (ω) C₂r₂

3. Schering Bridge

At balance condition

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

D - Deflection, m

L - distance from centre of deflection plates to screen, m
L_d - effective length of deflection plates, m
E_d - deflection voltage, volts d - separation between the plates, m
E_a- accelerating voltage, volts 

Deflection sensitivity is

Deflection factor G is

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.

T_r = rise time in μ second

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

F_in = nFs

Measurement Of Phase Difference And Frequency 

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

V_x = V_x sin ω_xt    V_y = V_y sin (ωy t - Ф)
By adjusting the values of toy, ω_x, ω_y, V_x, V_y and Ф suitably, various patterns may De obtained on the screen.

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