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