These include the following along with dimension and unit symbol.
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.
3. Precision
It is the measure of degree of agreement within a group of measurements.
4. Significant Figures
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.
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
1. Sum of the Two Quantities: Let a_{1} and a_{2} 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
The static error may arise due to number of reasons. The static errors are classified as
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
Analog Instruments
Analog instruments are classified in one way as
(a) Indicating
(b) Recording
(c) Integrating Instruments.
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 nonmagnetic 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 coefficient which is constant for a given semiconductor. This effect is mainly used in fluxmeters.
It produces a force equal and opposite to the deflecting force in order to make the deflection of pointer at a definite magnitude.
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.
Analog Ammeter and Voltmeters
The Instruments used for measurement of voltage and current can be classified as:
(a) Moving coil instruments
(b) Moving iron instruments
(c) Electrostatic Instruments
(d) Rectifier instruments
(e) Induction instruments
(f) Thermal instruments
(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/m^{2}.
N  Numbers of turns of the coil
A  effective coil area m^{2} (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
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/R^{2})
(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 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 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
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_{R}I_{R} + (V^{2}R / 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/r^{2})
4. Low Power Factor Wattmeter
T_{d} = i_{p}i_{c} (dM /dθ)
Average deflection torque = I_{P}I 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 = P_{1} + P_{2} + P_{3}
= V_{1}i_{1} + V_{2} i_{2} + V_{3} i_{3}
i_{3} Instantaneous power of load = V_{1} i_{1} + V_{2} i_{2} + V_{3}i_{3}
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 (P_{1}  P_{2})
Power factor cos Φ
7. Measurement of Reactive Power in Three Phase Circuits
Reading of wattmeter = V_{23} i_{1} cos (angle between i_{1} and V_{23})
= V_{23} i_{1} cos (90  Φ)
= √3 V I sin Φ
Total reactive power of the circuit
= √3 (watt meter reading).
The general ac bridge circuit is as follows
Under balanced condition
Z_{1}Z_{4}∠(θ_{1} + θ_{4}) = Z_{2}Z_{3}∠(θ_{2} + θ_{3})
Equating the magnitudes and angles,
Z_{1} Z_{4} = Z_{2} Z_{3}
θ_{1} + θ_{4} = θ_{2} + θ_{3}
1. Hay's Bridge
At balance
2. Owners Bridge
3. Maxwell Inductance Bridge
At balance
4. Maxwell's Inductance  Capacitance Bridge
5. Anderson's Bridge:
At balance
L_{1} = C R_{3} [r (R_{4} + R_{2}) + R_{2} R_{4}]R_{4}.
1. De Sauty's Bridge
2. Modifed De Sauty's Bride
At balance
Dissipation factor, D = tan δ_{1} = ωC_{1}r_{1}
D_{2} = tan δ_{2} = (ω) C_{2}r_{2}
3. Schering Bridge
At balance condition
C_{1} = C_{2} (R_{2} / R_{3})
Dissipation factors D_{1} = tan δ_{1} = ωC_{1}r_{1} = C_{4}r_{3}.
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.
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
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
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_{x}t 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).
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 = ω_{0}L/R where,
ω_{0} = resonant angular freq
L = inductance of coil
R = effective resistance of coil
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 electrically interconnects conductive objects to keep voltages between them safe, even if equipment fails.
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