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Error

Gross error Systematic error Random error

Instrumental error

Observational error

Environmental error

Gross error: -

These error occurs due to human mistakes.

Environmental error: -

These error occurs due to environmental conditions like temperature, pressure.

Instrumental error: -

These error occurs due to defective parts like springs, magnet, due to loading effect etc.

Observational error: -

As the name suggests, these types of errors occurs due to wrong observations or reading in the instruments particularly in case of energy meter reading. The wrong observations may be due to PARALLAX. In order to reduce the PARALLAX error highly accurate meters are needed: meters provided with mirror scales.

Random error: -

Random errors are caused by the sudden change in experimental conditions and noise and tiredness in the working persons. These errors are either positive or negative. An example of the random errors is during changes in humidity, unexpected change in temperature and fluctuation in voltage. These errors may be reduced by taking the average of a large number of readings.

Limiting error: -

The limited deviation of the measured value from the true value is known as the limiting error or guarantee error. Such type of error is fixed on the instrument. The magnitude of the limiting error depends on the design, material and the workmanship used for the construction of the instrument.

• The component like resistor, inductor and capacitor uses in the instrument has some rated fixed value. The deviation from their rated value causes the error in the system. The deviation mainly occurs because of the variation in the environmental condition. The manufacturer already knew about the limiting error of the instrument.

• The actual value of the instrument along with the limiting error is expressed as: -

The relative limiting error,

εr = 𝐌𝐞𝐚𝐬𝐮𝐫𝐞𝐝 𝐪𝐮𝐚𝐧𝐭𝐢𝐭𝐲 − 𝐓𝐫𝐮𝐞 𝐪𝐮𝐚𝐧𝐭𝐢𝐭𝐲

𝐓𝐫𝐮𝐞 𝐪𝐮𝐚𝐧𝐭𝐢𝐭𝐲× 𝟏𝟎𝟎

% εr = 𝐀𝐦−𝐀𝐓

𝐀𝐓× 𝟏𝟎𝟎

Error at required value: -

% εr = 𝐅𝐮𝐥𝐥 𝐬𝐜𝐚𝐥𝐞 𝐫𝐞𝐚𝐝𝐢𝐧𝐠×𝐅𝐮𝐥𝐥 𝐬𝐜𝐚𝐥𝐞 𝐞𝐫𝐫𝐨𝐫

𝐑𝐞𝐪𝐮𝐢𝐫𝐞𝐝 𝐯𝐚𝐥𝐮𝐞

Composite errors: -

X1 = a ± εr1

X2 = b ± εr2

X3 = c ± εr3

(1) Addition/subtraction terms: -

X = X1 ± X2 ± X3

% εr =±𝒂

𝒂+𝒃+𝒄ε𝒓𝟏 +

𝒃

𝒂+𝒃+𝒄ε𝒓𝟐 +

𝒄

𝒂+𝒃+𝒄ε𝒓𝟑

(2) Multiplication/division terms: -

X = X1 X2 X3 Or 𝐗𝟏

𝐗𝟐𝐗𝟑

% εr =± ε𝒓𝟏 + ε𝒓𝟐 + ε𝒓𝟑

(3) Power term: -

X = X1m X2

n X3p Or

𝐗𝟏𝐦

𝐗𝟐𝐧𝐗𝟑

𝐩

% εr =± 𝐦× 𝛆𝐫𝟏 + 𝐧 × 𝛆𝐫𝟐 + 𝐩 × 𝛆𝐫𝟑

Standard deviation error: -

m = f(x1, x2……..xn)

σ1, σ2,…….. σn = Standard deviation of x1, x2……..xn.

Standard deviation of m: -

σm = δ𝒎

δ𝒙𝟏

𝟐σ1

𝟐 +δ𝒎

δ𝒙𝟐

𝟐σ2

𝟐…………… .δ𝒎

δ𝒙𝒏

𝟐σn

𝟐

Q: - Power is measure across a resistor by connecting an ammeter in series to the resistance, and voltmeter in parallel to the resistance. This meter measures 200V & 5A and the standard deviation of this meter are 2V and 0.1A. Calculate the standard deviation of the power.

Accuracy: -

It is the closeness of instrument reading with its true value.

𝑨𝒄𝒄𝒖𝒓𝒂𝒄𝒚 = 𝑬𝒓𝒓𝒐𝒓

Precision: -

• It is a measure of reproducibility of the reading.

• Precision does not guarantee for accuracy.

• If the no. of significant figure is increased then precision is increased.

Not accurateNot precise

Accurate butNot precise

AccurateAnd precise

Resolution: -

Smallest change in input able to measure by the instrument is called resolution.

Sensitivity: -

It is the ratio of change in output to change in input.

Q: - The two currents in a parallel branch given as (150± 1) Amp, (250± 2) Amp. Find the total current in the circuit?

Q: - In the above problem if the errors are in the form of standard deviation then find the total current in the circuit.

Types of the torque: -

(1) Damping torque

(2) Deflecting torque

(3) Controlling torque

Damping torque: -

A damping torque is produced by a damping or stopping force which acts on the moving system only when it is moving and always opposes its motion. Such a torque is necessary to bring the pointer to rest quickly. If there is no damping torque, then the pointer will keep moving to and fro about its final deflected position for some time before coming to rest, due to the inertia of the moving system.

• This damping torque acts only when the pointer is in motion and always opposes the motion. The position of the pointer when stationary is, therefore, not affected by damping torque. The degree of damping decides the behavior of the moving system.

• If the instrument is under-damped, the pointer will oscillate about the final position for some time before coming to rest. On the other hand, if the instrument is over damped, the pointer will become slow and lethargic.

• However, if the degree of damping is adjusted to such a value that the pointer comes up to the correct reading quickly without oscillating about it, the instrument is said to be critically damped.

Types of damping torque: -

Air friction damping: -

Used in Moving iron meters and

Electrodynamometer type instrument.

Fluid friction damping: -

• Used in electrostatic meters.

• Jewel bearings are used for reducing wear and tear

of the moving system.

• The friction produced between spindle and

Bearing produces frictional errors so that accuracy of

the instrument will be reduced.

• The instrument which has high torque to weight

ratio has low frictional error so that accuracy is higher.

• Moving system like spindle and pointer are made of aluminium which has low weight.

• The instrument which has low operating torque uses fluid friction damping which reduces frictional error between bearing and spindle.

• At the final steady state position damping are minimized due to friction between fluid and vane(float).

• Used in electrostatic voltmeter type instruments.

• If a core/ former wound with a coil is placed in a magnetic field experiences a force. Due to interaction of current flowing through the coil and magnetic field.

• An emf in the core which produces circulating eddy currents. Which opposes the main torque. This is called eddy current damping which depends on velocity of the moving system if magnetic flux is constant.

• Example: - Used in PMMC.

Electro magnetic damping: -• The opposing flux is produced both in the former and coil wound on the former.• The magnitude depends on current flowing through the coil which intern

depends on resistance of the coil and external circuit, By adjusting this external resistance nearer to critical damped value can be achieved.

• It is used in galvanometer.

Deflecting torque: -By applying different principles like electromagnetic, static, thermal etc, a pointer is deflected proportional to the quantity to be measured and hence deflecting torque is used.

Controlling torque or restoring torque: -Controlling torque is used to keep the pointer at balance position where both controlling and deflecting torques are equal.It brings the pointer to the zero initial position if there is no deflecting torque.

Type of controlling torque: -

(1) Gravity control: -

A small weight is kept at the end of the moving system which produces controlling torque TC.

TC ∝ 𝒘𝒍𝒔𝒊𝒏𝜽

TC ∝ 𝒔𝒊𝒏𝜽

𝒍𝜽

𝒘

𝒘𝒍𝒔𝒊𝒏𝜽

At balance position: -

Td = TC

Td ∝ I

Td = K1I

K1I = 𝒘𝒍𝒔𝒊𝒏𝜽

𝐬𝐢𝐧𝛉 = K1I𝐊𝟐𝐰𝒍

I

𝛉 = 𝐬𝐢𝐧−𝟏K1I𝐊𝟐𝐰𝒍

……………..(1)

Equation (1) shows that gravity control has non-linear scale.

• Gravity control is used in vertically mounting instruments.

• Controlling torque is independent of temperature and time.

• Cost is cheaper.

• Disadvantage is not suitable for horizontal mounted instruments.

Spring control instrument: -

• It is most practically used in all the indicating instruments.

• Spring is made of phosphor bronze which is less affected by temperature.

TC ∝𝐄𝐛𝐭𝟑

𝟏𝟐𝒍𝛉 = K.𝛉

E = Young modulus k = spring/Tortional/restoring/controlling constant

b = width of spring k = 𝐄𝐛𝐭𝟑

𝟏𝟐𝒍Nm/rad.

t = Thickness of spring

l = length of spring

At balance position: -

Td = TC

Td ∝ I

Td = K1I

TC = K𝛉

K𝛉= K1I

• Spring has Linear scale.

• Spring tension will be reduced due to ageing and temperature.

𝛉

𝒕𝛉 ∝ 𝐈

PMMC(Permanent magnet moving coil): -

F = n.B.I.𝒍 Sinαif α = 90°F = nBI𝒍Deflecting torque: -Td = F×bTd = n.B.I.𝒍 ×bTd = n.B.I.ATd = G.IWhere G = nBAn = No. of turnsB = Maximum flux densityI = Current through the coilA = Area of cross section of coil

A = 𝒍 ×b

At balance

T = Td

K𝛉= GI

• Deflecting torque is produced due to interaction of permanent magnet flux and current flows through the coil.

• Spring provides controlling torque.

• If the control spring is failed or snapped then the pointer comes to zero initial position because the current is passing through spring is zero.

• Eddy current damping is used.

𝛉 ∝ 𝐈

𝛉

𝒕

Linear scale

• PMMC measures average quantity of current and voltage.

• For any signal measured average value using :

Io = 𝟏

𝐓𝟎𝐓𝐢 𝐭 𝐝𝐭.

• If the pure AC signal passing through PMMC then pointer vibrates nearer to zero because Iavg = 0.

• If a signal i(t) = Io + I1Sinωt + I2Sinωt……is passing through PMMC. It reads Iavg = Io.

• If a negative value Iavg is present then zero point and center scale is used for measurement of negative average value.

• Torque to weight ratio is higher so that the frictional error is min and accuracy is higher.

• Scale is uniform and linear and it has higher sensitivity.

Enhancement of meters: -(1) Ammeter: -

• For enhancement of the ammeter a shunt resistance is connected in parallel to the meter. This Rsh is made of manganin.

• Which has low value of temperature coefficient of resistance so that the error due to temperature change can be minimized.

• Manganin = α = 0.00015/°C

• Copper = α = 0.00395/°C or 0.004/°C

• Im = I×𝐑𝐬𝐡

𝐑𝐬𝐡+𝐑𝐦⇒

𝐑𝐬𝐡+𝐑𝐦

𝐑𝐬𝐡=

𝐈

𝐈𝐦= m

Rsh = 𝐑𝐦

𝐦−𝟏

Q: - A PMMC ammeter has meter internal resistance of 50 ohm and measures current up to 5 mA. Calculate the shunt resistance and voltage drop across meter at full scale for measuring the currents of: -

(1) 25 mA (2) 250 mA (3) 1 A

Effect of temperature on ammeter: -

Without any compensation: -

Rsh = 𝐑𝐦

𝐦−𝟏

With RSwamp: -

Im = I×𝐑𝐬𝐡𝟐

𝐑𝐬𝐡𝟐+𝐑𝐦+𝐑𝐬𝐰𝐚𝐦𝐩

𝐑𝐬𝐡𝟐+𝐑𝐦+𝐑𝐬𝐰𝐚𝐦𝐩

𝐑𝐬𝐡𝟐=

𝐈

𝐈𝐦= m

1

1

2

𝐑𝐬𝐡𝟐=

𝐑𝐦

𝐦−𝟏+ 𝐑𝐒𝐰𝐚𝐦𝐩

𝐦−𝟏

𝐑𝐬𝐡𝟐> 𝐑𝐬𝐡𝟏

By addition of swamp resistance in series to the meter the error due to change in temperature is compensated.

𝐑𝐬𝐡𝟐=𝐑𝐬𝐡𝟏

+ 𝐑𝐒𝐰𝐚𝐦𝐩

𝐦−𝟏

Effect of frequency on AC ammeters: -

MI and electrodynamometer instrument, are used for measurement of both AC and DC currents, in case of AC measures rms value of the current.

𝐈𝟏 = 𝟏𝟎 𝟐𝐬𝐢𝐧(𝟐𝛑 × 𝟓𝟎𝐭) 𝐈𝐫𝐦𝐬𝟏= 𝟏𝟎𝐀

𝐈𝟐 = 𝟏𝟎 𝟐𝐬𝐢𝐧(𝟐𝛑 × 𝟏𝟎𝟎𝐭) 𝐈𝐫𝐦𝐬𝟐 = 𝟏𝟎𝐀

𝐙𝐬𝐧 = 𝐑𝐬𝐡 + 𝐣𝛚 𝐋𝐬𝐡𝐙𝐦 = 𝐑𝐦 + 𝐣𝛚 𝐋𝐦

𝐈𝐦 =𝐈 .𝐙𝐬𝐡

(𝐙𝐬𝐡+𝐙𝐦)

𝐈𝐬𝐡 =𝐈.𝐙𝐦

(𝐙𝐬𝐡+𝐙𝐦)

RshRm

LshLm

𝐈𝐦

𝐈𝐬𝐡=

𝐙𝐬𝐡

𝐙𝐦

𝐈𝐦

𝐈𝐬𝐡=

𝐑𝐬𝐡𝟐 +𝛚𝟐𝐋𝐬𝐡²

𝐑𝐦𝟐 +𝛚𝟐𝐋𝐦²

𝐈𝐦

𝐈𝐬𝐡=

𝐑𝐬𝐡

𝐑𝐦

𝟏+𝛚𝟐 𝐋𝐬𝐡𝐑𝐬𝐡

𝟐

𝟏+𝛚𝟐 𝐋𝐦𝐑𝐦

²

If 𝐋𝐬𝐡

𝐑𝐬𝐡=

𝐋𝐦

𝐑𝐦

Then 𝐈𝐦

𝐈𝐬𝐡=

𝐑𝐬𝐡

𝐑𝐦

If the time constant of the shunt is equal to the meter time constant then AC ammeter is independent of frequency.

Sensitivity of Ammeter:

Sensitivity is the ratio of change in o/p to the change in i/p.

• For linear/Uniform meter: -

𝐓𝐜 = 𝐓𝐝 𝐤𝛉 = 𝐆 𝐈

𝛉 =𝐆

𝐊𝐈

Static sensitivity S =θ𝐈

θo/p

ΔI1I

Δθ1

Non linear Sensitivity for Non-linear Meter: -

𝛉 ∝ 𝐈²

𝐒 =𝚫 𝛉𝟏

𝚫𝐈𝟏≠

𝚫 𝛉𝟐

𝚫𝐈𝟐

𝐒 =𝐝𝛉

𝐝𝐈=Dynamic Sensitivity

θo/p

I

Linear scale: - Non Linear scale: -

S1 = 𝛉𝟏

𝐈𝟏S1 =

𝛉𝟐

𝐈𝟐

• An ammeter which has minimum full scale value current has high sensitivity.

• With higher sensitivity readability of data is convenient.

0.5

00

0.5

1

1.5

1

𝛉𝟏𝛉𝟐

Cramped scale

Resolution: -

Smallest change in input able to measure by the instrument is called resolution.

Enhancement of voltmeter: -

For enhancement of voltmeter series resistance made of mangnin is connected in series to the meter.

m = 𝐕

𝐕𝐦

Vm = V.𝐑𝐦

(𝐑𝐦+𝐑𝐒)

RS = Rm (m-1)

RS = Series multiplier resistance

IfSRm

Full scale current IfS = Im = 𝐕

(𝐑𝐒+𝐑𝐦)

Voltmeter sensitivity = SV = 𝟏

𝐈𝐟𝐒= 𝐑𝐬+𝐑𝐦

𝐕……Ω/m

Loading effect: -

• In case of voltmeter, which is having higher value of voltmeter sensitivity has more accuracy, because the current passing through the meter is very small and the meter will not work as a load.

• If the voltmeter having lower sensitivity working similar to the load so that error in the measurement of voltage will be more.

Effect of frequency on ac voltmeter: -

if,

• By the addition of compensating capacitor in parallel to RS voltmeter is made independent of frequency.

RSCC

m

m

CCRS = 0.41𝑳𝒎

𝑹𝒔

Applications of PMMC: -

Rectifier meters: -

(a) Half wave rectifier meter: -

V = Vm Sinωt

PMMC reads Iavg

I1 = 𝐈𝐦

𝛑=

𝐕𝐦

𝛑(𝐑𝐬+𝐑𝐦+𝐑𝐟)

Ammeter

Vm = 𝟐Vrms

I1 = 𝟐𝐕𝐫𝐦𝐬

𝛑(𝐑𝐬+𝐑𝐦+𝐑𝐟)

I1 = 𝟎.𝟒𝟓 𝐕𝐫𝐦𝐬

𝛑(𝐑𝐬+𝐑𝐦+𝐑𝐟)

Ideal diode Rf = 0.

With DC input: - Assume VAC = Vrms

I2 = 𝐕𝐫𝐦𝐬

𝛑(𝐑𝐬+𝐑𝐦+𝐑𝐟)

From (1) and (2)

I1 = 0.45 I2

If with DC input

I2 = 1 A

Vrms

(Si)AC = 0.45 (Si)DC

Full wave rectifier meter: -

(1) AC input: -

I1 = 𝟐𝐕𝐦

𝛑=

𝟐𝐕𝐦

(𝐑𝐬+𝐑𝐦+𝟐𝐑𝐟)𝛑

Vm = 𝟐Vrms

I1 = 𝟐𝐕𝐦

𝛑=

𝟐 𝟐𝐕𝐦

(𝐑𝐬+𝐑𝐦+𝟐𝐑𝐟)𝛑

I1 = 𝟎.𝟗𝐕𝐦

(𝐑𝐬+𝐑𝐦+𝟐𝐑𝐟)…………(1)

Ideal diode Rf = 0.

With DC input: -

Assume VDC = Vrms

I1 = 𝐕𝒓𝒎𝒔

(𝐑𝐬+𝐑𝐦+𝟐𝐑𝐟)𝛑…………(2)

From (1) and (2)

I1 = 0.9 I2

(Si)AC = 0.9 (Si)DC

• Rectifier meters used in communication application for calculation of percentage of dc present in the AC signal.

• For reducing the power loss in place of series resistance RS, capacitance can be used then the current measured by the PMMC.

I1 = 0.9𝐕𝐫𝐦𝐬

𝐗𝐂

Calibration errors in PMMC meters: -

Form factor = Ff = 𝐑𝐌𝐒

𝐀𝐯𝐠

For sinusoidal signal F.F. = 1.11 = 𝐑𝐌𝐒

𝐀𝐯𝐠

Rms = 1.11 Avg

For Square Ff = 1 = 𝐑𝐌𝐒

𝐀𝐯𝐠

Rms = avg

% εr = 𝐅.𝐅𝐦𝐞𝐚𝐬𝐮𝐫𝐞𝐝−𝐅.𝐅.𝐓𝐫𝐮𝐞

𝐅.𝐅.𝐓𝐫𝐮𝐞× 𝟏𝟎𝟎

• If the instrument are calibrated for one particular waveform rmsmeasurement, then if the other signal are passing through the instrument producing error in measurement of rms reading. This is called error due to calibration.

Ratio meter (Ohm-meter): -

For coil (C1)

Td1 = N.B.I1.A.Sinθ

For Coil (C2)Td2 = N.B.I2.A.Cosθ

At balance condition

Td1 = Td2

N.B.I1.A.Sinθ = N.B.I2.A.Cosθ

tan θ = 𝐈𝟐

𝐈𝟏

if θ is small then tan θ ≈ θ

θ = 𝐈𝟐

𝐈𝟏

For example: -

θ = 𝐈𝟐

𝐈𝟏= 𝐕/𝐒

𝐕/𝐑= 𝐑

𝐒

if S is known

then R(test)S(known)

C1

V

C2

>> I2I1

θ ∝ R

• Ratio meter is working on the principle of PMMC, used for the measurement of unknown resistance over wide range. This is called ohm meter, measure resistance up to 100 kilo ohm.

Megger: -

• θ = 𝐈𝟐

𝐈𝟏

𝐈𝟏= 𝐕

𝐑𝐓+𝐑𝟏

I2 = 𝐕

𝐑𝟐

θ ∝𝐑𝐓+𝐑𝟏

𝑹𝟐

R1 and R2 = current limiting resistance

RT = test resistance

θ ∝ RT

SSC JE-2019

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Electrical Engineering by ASHISH SIR

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Electrical Engineering by ASHISH SIR

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Electrical Engineering by ASHISH SIR

Electrical Engineering by ASHISH SIR

Electrical Engineering by ASHISH SIR

Electrical Engineering by ASHISH SIR

Electrical Engineering by ASHISH SIR

Electrical Engineering by ASHISH SIR

Electrical Engineering by ASHISH SIR

(28) In a steam power plant…………heats the feed water on its way to the boiler by delivering heat from the flue gases: -

(A) Superheater (B) Economizer

(B) Preheater (D) Turbine

(29) Power generation of thermal power plants is based on: -

(A) Rankine cycle

(B) Otto cycle

(C) Diesel cycle

(D) Carnot cycle