Measurement of Electrical Quantities GATE Quick Notes

Basic Measurement Concepts

Fundamental Definitions

  • Measurement: Process of comparing unknown quantity with standard

  • Accuracy: Closeness of measured value to true value

  • Precision: Repeatability of measurements

  • Resolution: Smallest change that can be detected

  • Sensitivity: Change in output per unit change in input

  • Linearity: Proportional relationship between input and output

  • Hysteresis: Different output for same input (loading vs unloading)

Types of Errors

  • Systematic Errors: Constant deviation (calibration, environmental)

  • Random Errors: Unpredictable variations

  • Gross Errors: Human mistakes, equipment failure

  • Absolute Error: \(\Delta A = A_m - A_t\)

  • Relative Error: \(\dfrac{\Delta A}{A_t} \times 100\%\)

  • Limiting Error: Maximum possible error

  • Percentage Error: \(\dfrac{\Delta A}{A_t} \times 100\%\)

Static Characteristics

  • Range: Minimum to maximum input values

  • Span: Algebraic difference between maximum and minimum

  • Threshold: Minimum input to produce output change

  • Dead Zone: Range where output remains constant

  • Drift: Change in output over time with constant input

  • Repeatability: Closeness of successive measurements

  • Reproducibility: Ability to reproduce results under different conditions

DC Measurements

DC Voltage Measurement

  • Moving Coil Voltmeter: High series resistance

  • Multiplier: \(R_m = R_g(n-1)\) where \(n = \dfrac{V}{V_g}\)

  • Digital Voltmeter: High input impedance, accurate

  • Potentiometer: Null deflection method, high accuracy

  • Loading Effect: Meter resistance affects circuit

  • Voltage Divider: For range extension

  • Input Impedance: Should be much higher than source resistance

DC Current Measurement

  • Moving Coil Ammeter: Low series resistance

  • Shunt: \(R_s = \dfrac{R_g}{n-1}\) where \(n = \dfrac{I}{I_g}\)

  • Current Transformer: For high currents, isolation

  • Hall Effect Sensor: Non-contact measurement

  • Shunt Types: Manganin (low temp coefficient)

  • Shunt Box: Multiple range selection

  • Burden: Voltage drop across shunt

Resistance Measurement

  • Ammeter-Voltmeter Method: \(R = \dfrac{V}{I}\)

  • Wheatstone Bridge: \(R_x = \dfrac{R_2 R_3}{R_1}\) (balanced condition)

  • Kelvin Bridge: For low resistance (\(< 1\Omega\))

  • Megger: For high resistance and insulation

  • Ohmmeter: Self-contained, less accurate

  • Substitution Method: Replace unknown with known standard

  • Comparison Method: Direct comparison with standard

Resistance Measurement Methods

  • High Resistance: Guard ring method, megohmmeter

  • Medium Resistance: Wheatstone bridge (1\(\Omega\) to 1M\(\Omega\))

  • Low Resistance: Kelvin bridge, four-terminal method

  • Micro-ohmmeter: For very low resistance

  • Insulation Resistance: Megger, guard terminal

  • Earth Resistance: Fall-of-potential method

AC Measurements

AC Voltage & Current

  • Moving Iron Instruments: RMS response, AC/DC

  • Rectifier Type: Average responding, RMS calibrated

  • True RMS: Thermal converters, accurate for distorted waves

  • Peak Reading: Diode peak detector circuits

  • Form Factor: \(\dfrac{\text{RMS}}{\text{Average}} = 1.11\) (sine wave)

  • Crest Factor: \(\dfrac{\text{Peak}}{\text{RMS}} = 1.414\) (sine wave)

  • Peak-to-Peak: \(V_{p-p} = 2V_{peak}\)

AC Instrument Types

  • Attraction Type: Iron piece attracted to coil

  • Repulsion Type: Two iron pieces repel each other

  • Dynamometer: Two coils, one fixed, one movable

  • Induction Type: Rotating magnetic field principle

  • Thermocouple Type: Heat generated by current

  • Electrostatic: Electrostatic force between plates

Power Measurement

  • Wattmeter: \(P = VI\cos\phi\) (single phase)

  • Two Wattmeter Method: \(P = W_1 + W_2\) (3-phase)

  • Reactive Power: \(Q = \sqrt{3}VI\sin\phi\)

  • Power Factor: \(\cos\phi = \dfrac{P}{VI}\)

  • Energy Meter: Induction type, kWh measurement

  • Apparent Power: \(S = VI\) (VA)

  • Power Triangle: \(S^2 = P^2 + Q^2\)

Single Phase Power Measurement

  • Direct Method: One wattmeter for resistive load

  • Wattmeter Connections: Voltage and current coils

  • Compensation: For wattmeter losses

  • Low Power Factor: Special connections required

  • Reactive Power: \(Q = VI\sin\phi\)

  • Power Factor Meter: Direct reading instrument

Three Phase Power

  • Balanced Load: \(P = \sqrt{3}V_L I_L \cos\phi\)

  • Two Wattmeter Method:

    • \(P_1 = V_L I_L \cos(30^{\circ} - \phi)\)

    • \(P_2 = V_L I_L \cos(30^{\circ} + \phi)\)

    • \(P_{total} = P_1 + P_2\)

  • Reactive Power: \(Q = \sqrt{3}(P_1 - P_2)\)

  • Power Factor: \(\tan\phi = \dfrac{\sqrt{3}(P_1 - P_2)}{P_1 + P_2}\)

Three Phase Power (Continued)

  • Three Wattmeter Method: For unbalanced loads

  • Star Connection: Neutral point available

  • Delta Connection: No neutral point

  • Blondel’s Theorem: (n-1) wattmeters for n-wire system

  • Power Factor Limits:

    • \(\cos\phi > 0.5\): Both wattmeters positive

    • \(\cos\phi = 0.5\): One wattmeter reads zero

    • \(\cos\phi < 0.5\): One wattmeter negative

Inductance and Capacitance

Inductance Measurement

  • Maxwell’s Bridge: \(L_x = \dfrac{R_2 R_3 C_1}{1}\), \(R_x = \dfrac{R_2 R_3}{R_1}\)

  • Hay’s Bridge: For high Q coils

  • Anderson Bridge: Most accurate for medium Q

  • Owen Bridge: For high Q inductors

  • Resonance Method: \(L = \dfrac{1}{\omega^2 C}\)

  • Q-Factor: \(Q = \dfrac{\omega L}{R}\)

Capacitance Measurement

  • Schering Bridge: \(C_x = \dfrac{C_1 R_3}{R_2}\), \(\tan\delta = \omega C_1 R_1\)

  • Wien Bridge: For capacitance and frequency

  • De Sauty Bridge: For perfect capacitors

  • Resonance Method: \(C = \dfrac{1}{\omega^2 L}\)

  • Loss Angle: \(\delta\) represents dielectric loss

  • Dissipation Factor: \(D = \tan\delta\)

Impedance Measurement

AC Bridge Fundamentals

  • Balance Condition: \(Z_1 Z_3 = Z_2 Z_4\)

  • Magnitude Balance: \(|Z_1||Z_3| = |Z_2||Z_4|\)

  • Phase Balance: \(\phi_1 + \phi_3 = \phi_2 + \phi_4\)

  • Convergence: Bridge should balance quickly

  • Sensitivity: Change in detector for small unbalance

  • Wagner Earth: Eliminates stray capacitance effects

Common AC Bridges

  • Maxwell: Inductance measurement (medium Q)

  • Hay: Inductance measurement (high Q)

  • Anderson: Inductance measurement (most accurate)

  • Owen: Inductance measurement (variable frequency)

  • Schering: Capacitance and loss angle

  • Wien: Frequency measurement

  • Robinson: Capacitance measurement

Energy Measurement

Energy Meters

  • Induction Type: Most common for AC energy

  • Rotating Disc: Speed proportional to power

  • Energy: \(E = \int P \, dt\) (kWh)

  • Meter Constant: Energy per revolution

  • Creep: Slow rotation at no load

  • Friction: Compensation for bearing friction

  • Digital Meters: Electronic energy measurement

Energy Meter Testing

  • Calibration: Against standard wattmeter

  • Accuracy Class: 0.1, 0.2, 0.5, 1.0, 2.0

  • Light Load: Performance at 10% of full load

  • Full Load: Performance at 100% of full load

  • Unity Power Factor: Most accurate condition

  • Lagging Power Factor: Typical industrial condition

  • Temperature Effect: Compensation required

Important Formulas

Key Formulas - Part 1

  • Multiplier: \(R_m = R_g(n-1)\)

  • Shunt: \(R_s = \dfrac{R_g}{n-1}\)

  • Wheatstone Bridge: \(R_x = \dfrac{R_2 R_3}{R_1}\)

  • Power (3-phase): \(P = \sqrt{3}V_L I_L \cos\phi\)

  • Two Wattmeter: \(\tan\phi = \dfrac{\sqrt{3}(W_1 - W_2)}{W_1 + W_2}\)

  • Maxwell Bridge: \(L_x = \dfrac{R_2 R_3 C_1}{1}\)

Key Formulas - Part 2

  • Form Factor: \(F.F = \dfrac{RMS}{Average}\)

  • Crest Factor: \(C.F = \dfrac{Peak}{RMS}\)

  • Schering Bridge: \(C_x = \dfrac{C_1 R_3}{R_2}\)

  • Q-Factor: \(Q = \dfrac{\omega L}{R}\)

  • Power Triangle: \(S^2 = P^2 + Q^2\)

  • Reactive Power: \(Q = \sqrt{3}(P_1 - P_2)\)

GATE Tips

GATE Exam Tips

  • Common Topics: Bridge circuits, power measurement, AC/DC instruments

  • Numerical Problems: Shunt/multiplier calculations, bridge balance

  • Conceptual Questions: Instrument selection, measurement principles

  • Practice Areas: Two wattmeter method, AC bridges, energy meters

  • Time Management: Quick formula recall essential

  • Units: Always check units in calculations

Common GATE Questions

  • Bridge Calculations: Balance conditions, unknown component values

  • Power Measurement: Three-phase power, power factor calculation

  • Instrument Selection: Appropriate instrument for specific measurement

  • Range Extension: Shunt and multiplier calculations

  • Energy Measurement: Meter constants, calibration

  • AC Quantities: RMS, average, peak relationships