UNIT 2: Electrical Measurement Systems and Instruments

I. Fundamentals of Measurement

Types of Errors

Static Characteristics

• Accuracy: Closeness to true value. Absolute Error (E) = X_m - X_t.

• Precision: Repeatability of measurements (degree of scatter).

• Sensitivity: Ratio of output change to input change. S = ΔOutput / ΔInput.

• Resolution: Smallest detectable input change.

• Drift: Gradual change in output over time for constant input.

• Hysteresis: Difference in output for increasing vs. decreasing input.

• Dead Zone: Range of input change producing no output.

Dynamic Characteristics

• Fidelity: How accurately instrument follows time-varying input.

• Speed of Response: Time to respond to input change.

• Lag: Delay in response.

• Oscillations: Undamped or overdamped response.

Error Calculation & Propagation

• Absolute Error: E_a = X_m - X_t

• Relative Error: E_r = (X_m - X_t) / X_t

• Percentage Error: E_% = ((X_m - X_t) / X_t) × 100%

• Propagation: For Z = f(x, y), max possible error ΔZ ≈ |∂f/∂x|Δx + |∂f/∂y|Δy.

[!TIP] Exam Focus: Be prepared to calculate combined errors (e.g., in power measurement from voltage & current errors) and distinguish between accuracy & precision.

II. Indicating Instruments

A. Moving Coil Instruments (PMMC)

• Construction: Permanent magnet, moving coil (rectangular, former), control springs (hair-springs), pointer, scale.

• Principle: Torque T_d = BINA sinθ. For radial field, θ=90°, T_d = BINA. T_c = Kθ (spring control). At equilibrium, BINA = Kθ → I ∝ θ.

• Scale: Linear (uniform) due to I ∝ θ.

• Range Extension:

  • Ammeter (Shunt): R_sh = (I_m * R_m) / (I - I_m) where I_m = meter current, R_m = meter resistance.

  • Voltmeter (Series Multiplier): R_s = (V_m / I_m) - R_m.

  • Multi-range Voltmeter: Multiple series resistors switched in or common multiplier with multiple taps.

B. Moving Iron Instruments

• Construction: Fixed coil (produces magnetic field), moving iron (disc or vane), damping (air friction).

• Types:

  • Attraction: Single iron piece inside coil. Always attracted.

  • Repulsion: Two iron vanes inside coil, one fixed, one moving. Like poles repel.

• Principle & Torque: T_d ∝ I² (for both AC/DC). T_d = K I². T_c = K' θ. Equilibrium: θ ∝ I².

• Scale: Non-linear (crowded at lower end).

C. Electrodynamometer Instruments

• Construction: Fixed coils (air-cored, produce field), moving coil (light, many turns), control spring, air damping.

• Torque Equation (AC/DC):

  • Fixed coil current i_f = I_f sin(ωt), Moving coil i_m = I_m sin(ωt - φ).

  • T_d ∝ i_f * i_m = I_f I_m sin(ωt) sin(ωt - φ).

  • Average torque T_d(avg) ∝ (1/2) I_f I_m cos φ. For use as wattmeter, φ is phase angle between voltage & current.

• True RMS Response: For AC voltmeter/ammeter, fixed coil is series with moving coil. T_d(avg) ∝ I_f I_m cos φ. If I_f ∝ V (voltmeter) or I_m ∝ I (ammeter), then T_d(avg) ∝ V_rms I_rms cos(0°) if waveforms are identical. For arbitrary waveforms, T_d(avg) ∝ (1/T)∫ v(t) i(t) dt = true power. Scale calibrated for RMS assuming sinusoidal wave. Not inherently true RMS without additional circuitry (e.g., thermal converters).

• Low Power Factor Feature: Use of compensating coil (connected in series with voltage coil) to reduce phase angle error between voltage coil current & applied voltage.

• Scale: Non-linear (√ scale).

[!TIP] Exam Focus: Derivation of electrodynamometer torque is crucial. Remember: PMMC = DC only, linear scale; Moving Iron = AC/DC, non-linear; Electrodynamometer = AC/DC, non-linear, used as wattmeter.

III. Instrument Transformers

A. Current Transformer (CT)

• Construction: Primary (1- few turns, bar or wound), secondary (many turns), laminated core (silicon steel), window type (bar primary) or wound type.

• Equivalent Circuit & Phasor Diagram:

  • I_p / I_s = N_s / N_p = n (nominal ratio).

  • Exciting current I_e splits into I_m (magnetizing) & I_c (core loss).

  • I_p = n I_s + I_e.

  • Ratio Error: (n I_s - I_p) / I_p × 100%.

  • Phase Angle Error: δ between -I_s & I_p (secondary current reversed).

• Effect of Secondary Burden: Increased burden (Z_b) → larger voltage drop in secondary resistance/reactance → larger I_e → increased ratio & phase errors.

• Danger of Open Secondary: I_s = 0 → I_p = I_e. Core saturates, high flux → dangerously high induced EMF (kV) in secondary, risk of insulation breakdown & core damage. Never open CT secondary under load.

• Testing:

  • Ratio Test: Apply rated primary current, measure secondary.

  • Phase Angle Test: Using zero-phase factor method or wattmeter method.

  • Magnetization Characteristic: Open secondary, apply rising primary voltage, measure I_m vs V.

B. Potential Transformer (PT)

• Construction: Similar to power transformer, high insulation, low flux density, larger core.

• Equivalent Circuit & Phasor Diagram:

  • V_p / V_s = N_p / N_s = n (nominal ratio).

  • Voltage drop in internal impedances: V_p = n V_s + I_s (R_s + jX_s) + I_e (R_e + jX_e) (approx).

  • Ratio Error: (n V_s - V_p) / V_p × 100%.

  • Phase Angle Error: δ between V_p & -n V_s.

• Minimizing Errors:

  • Low resistance & reactance windings.

  • Low flux density → low exciting current.

  • High magnetic material quality.

  • Low burden on secondary.

• Testing: Similar to CT (ratio, phase angle, excitation).

C. Comparison of CT and PT

IV. Power Measurement

A. Single-Phase Wattmeter (Electrodynamometer Type)

• Construction: Fixed coils (current coils, connected in series with load), moving coil (voltage coil, connected in parallel via multiplier), control spring, air damping, scale.

• Torque Equation Derivation:

  • i_f = I_m sin(ωt), i_v = I_v sin(ωt - φ) where φ is phase angle between load current I and voltage V.

  • Instantaneous torque T ∝ i_f * i_v = I_m I_v sin(ωt) sin(ωt - φ).

  • Using trig identity: sin A sin B = ½[cos(A-B) - cos(A+B)].

  • T ∝ ½ I_m I_v [cos φ - cos(2ωt - φ)].

  • Average Torque T_avg ∝ ½ I_m I_v cos φ = k V I cos φ = k P (True Power).

• Errors:

  • Stray Magnetic Field: Shield with iron case or compensate with shorted turns.

  • Frictional & Mechanical: Bearing friction, pivot friction.

  • Eddy Current & Hysteresis: Use high-resistance non-magnetic coil form (e.g., aluminum), laminated core.

• Low Power Factor Use: Voltage coil inductance causes phase lag. Compensation: Use capacitor in parallel with voltage coil or add a compensating winding (connected in series with voltage coil) to cancel inductive effect.

B. Two-Element & Three-Element Dynamometer Wattmeter

• Two-Element: Two independent wattmeter units on same shaft. Used for 3-phase, 3-wire systems (two-wattmeter method).

• Three-Element: Three current coils (each in a line) and three voltage coils (each across a line). Used for 3-phase, 4-wire systems (balanced/unbalanced).

C. Three-Phase Power Measurement

Two-Wattmeter Method

• Circuit: Connect two wattmeters such that:

  • W1: Current coil in R-line, Voltage coil between R & Y.

  • W2: Current coil in B-line, Voltage coil between B & Y.

  • (For sequence R-Y-B).

• Phasor Diagram (Balanced Load, lagging pf):

  • V_RY as reference.

  • I_R lags V_RN by φ.

  • I_B lags V_BN by φ, I_B = I_R ∠-120°.

  • W1 ∝ V_RY I_R cos(30° + φ).

  • W2 ∝ V_BY I_B cos(30° - φ).

• Total Power: P_total = W1 + W2 (Algebraic sum, one may read negative for very low pf).

• Individual Readings:

  • W1 = V_L I_L cos(30° + φ)

  • W2 = V_L I_L cos(30° - φ)

  • For balanced load: P = √3 V_L I_L cos φ.

• Power Factor: tan φ = √3 (W1 - W2) / (W1 + W2).

[!TIP] Exam Focus: Derive total power as sum of two wattmeter readings using phasor diagrams. Remember condition for one wattmeter reading negative (φ > 60°).

D. Maximum Demand Meter

• Construction: Integrating type (like energy meter) with a demand register (reset at fixed intervals, e.g., monthly). Often uses a thermal or moving iron mechanism with a time lag (e.g., 15-30 min) to average out short-duration peaks.

• Working: Measures average power over a preset demand interval (e.g., 30 min). Pointer indicates average demand during that interval. Maximum demand pointer (lagging) records highest average demand since last reset.

V. Energy Measurement

A. Single-Phase Induction Energy Meter (Kapp's Electrodynamic Type)

• Construction:

  1. Driving System: Series electromagnet (current coil, low inductance) & shunt electromagnet (voltage coil, high inductance, with lagging current). Both on laminated core.

  2. Moving System: Light aluminum disc, mounted on spindle.

  3. Braking System: Permanent magnet (eddy current damping) positioned near disc.

  4. Counting Mechanism: Gear train & dials (cyclometer or pointer type).

• Working: T_d ∝ φ_I × φ_V sin δ. φ_I (from current coil) in phase with I. φ_V (from voltage coil) lags V by ~90° due to high inductance. δ ≈ 90°. So T_d ∝ V I cos φ (driving torque ∝ power).

  • Braking Torque T_b ∝ ω (speed of disc) from eddy currents in disc in magnetic field.

  • Equilibrium: T_d = T_b → ω ∝ P (Power). Revolutions N ∝ ∫P dt ∝ Energy.

• Testing & Calibration:

  • Full Load Test: Apply rated V & I, measure time for fixed revs. Compare with theoretical t = (3600 N) / (P * K) where K = meter constant (rev/kWh).

  • Creep Test: Apply rated V, no current. Disc should not rotate (or < 1 rev/hr).

  • Power Factor Variation Test: Test at different pfs (lag/lead) for constant power.

• Errors & Sources:

  • Friction: Bearing friction, pivot friction.

  • Creep: Due to residual magnetism, electrostatic effect, friction.

  • Braking: Incorrect magnet strength, disc conductivity.

  • Phase Errors: φ_V not exactly 90° lagging, φ_I not in phase with I.

• Error Calculation: % Error = [(Meter Rev - True Rev) / True Rev] × 100%. True Rev = (P * t * K_m) / 3600 where K_m = meter constant (rev/kWh).

[!TIP] Past Paper Problem: "A 50A, 230V meter makes 61 rev in 37s. Normal disc speed 520 rev/kWh. Find % error."

• Solution: P_true = V I cosφ? But here no pf given → assume unity pf for test. P = 230 * 50 = 11500 W = 11.5 kW.

• Energy recorded by meter = (61 rev) / (520 rev/kWh) = 0.1173 kWh.

• Actual energy supplied in 37s = 11.5 kW * (37/3600) h = 0.1182 kWh.

• % Error = ((0.1173 - 0.1182)/0.1182)*100 = -0.76% (meter slow).

B. Three-Phase Energy Meter

• Construction: Two or three single-phase meters on common shaft, or integrated 3-element meter.

• Circuit & Phasor Diagram (3-phase, 4-wire, balanced):

  • Each element measures power in one phase.

  • Total energy = sum of three elements.

  • For 3-wire, use two-element meter (two-wattmeter principle for energy integration).

C. Digital Electronic Energy Meter

• Block Diagram:

  
  Voltage Input → Voltage Divider → ADC (or Sample & Hold) → Microcontroller/Processor
  
  Current Input → CT/Shunt → ADC (or Sample & Hold) →           |
  
                                                             → Multiply → Integrate → Display/Output
  
  (Clock/Time Base) ----------------------------------------->|
  
  

• Working: Voltage & current sampled (simultaneously). Multiplier calculates instantaneous power v(t)*i(t). Digital integrator sums over time to get energy. Microcontroller handles calibration, display, tamper detection.

VI. Resistance Measurement

A. Wheatstone Bridge

• Circuit: Four arms: R1, R2 (ratio arms), R3 (known variable), R_x (unknown). Galvanometer between junctions.

• Balance Condition: R1/R2 = R3/R_x → R_x = (R2/R1) * R3.

• Applications: Medium resistances (1 Ω to ~1 MΩ).

• Precision Factors & Minimization:

  • Contact Resistance: Use 4-terminal (Kelvin) connections for R_x if low.

  • Lead Resistance: Use equal length & gauge for ratio arms, or use Kelvin Bridge for low R.

  • Thermal EMF: Use same metal for all joints, keep junction temperatures equal, reverse battery.

  • Bridge Sensitivity: High battery voltage, high galvanometer sensitivity, proper ratio choice.

B. Kelvin Double Bridge (Thomson Bridge)

• Construction: Modification of Wheatstone for low resistances (<1 Ω).

  • Two pairs of ratio arms: P, Q (outer) and p, q (inner).

  • R_x (unknown low R) and R_s (standard low R) have 4-terminal (Kelvin) connections.

  • Link resistance r between R_x & R_s inner terminals.

  • Galvanometer connected between the midpoints of P-Q and p-q arms.

• Theory & Derivation:

  • Balance condition: (P/Q) = (R_x + r') / (R_s + r) where r' is resistance from R_x inner terminal to galvanometer point.

  • To eliminate r & r', make P/Q = p/q (i.e., inner ratio = outer ratio).

  • Then Balance Condition: R_x = (P/Q) * R_s.

• Advantages: Eliminates errors due to contact resistance at current terminals and lead resistance between R_x & R_s because galvanometer is connected to potential (inner) terminals only.

• Balance Calculation: Given P, Q, p, q, R_s, find R_x ensuring P/Q = p/q (or adjust p,q to satisfy).

C. Megger (Insulation Resistance Measurement)

• Construction: Hand-cranked or battery-operated DC generator (500V, 1000V, 2500V), moving coil meter (center zero), two coils (current & pressure) at 90°.

• Working: Generator produces high DC voltage. Test leads: one to equipment under test, other to earth/guard.

  • Current Coil: Connected in series with generator & R_x (insulation resistance). Deflection proportional to current I = V/(R_x + R_g).

  • Pressure Coil: Connected across generator. Deflection proportional to V.

  • Net deflection ∝ V / R_x → directly reads R_x in MΩ.

• Applications: Measuring insulation resistance of cables, transformers, motors, earth resistance (with 3-terminal model).

VII. Special Instruments and Measurements

A. Ballistic Galvanometer

• Construction: Moving coil with large moment of inertia (heavy coil, few turns), very weak control (long period, T > 5s), critical damping (often). No eddy current damping (aluminum frame avoided).

• Difference from d'Arsonval: d'Arsonval has strong spring control, short period, eddy current damping, calibrated for steady current.

• Equation of Motion: J d²θ/dt² + K_d dθ/dt + K θ = N I (for current pulse).

  • J = moment of inertia, K_d = damping constant, K = control constant.

  • For ballistic (single impulse), I is a short pulse (charge q = ∫ I dt). Damping is critical. Initial velocity v_0 proportional to q.

  • First Swing θ_1 ∝ q (proved by integrating equation, neglecting damping after first quarter cycle).

• Damping Characteristics:

  • Logarithmic Decrement δ = ln(θ_1/θ_2).

  • Damped Frequency f_d = f_n √(1 - ζ²) where ζ = damping ratio.

• Use: Measurement of charge (e.g., capacitor discharge, flux linkage change).

B. Flux Meter

• Construction: Similar to ballistic galvo but no control spring (or very weak), high damping (often electromagnetic). Coil rotates freely.

• Working: Measures change in flux linkage Δψ. When flux changes, induced EMF e = -dψ/dt → current through coil → torque → deflection. Deflection ∝ Δψ (independent of resistance & time constant).

• Comparison with Ballistic Galvanometer:

  | Feature | Ballistic Galvanometer | Flux Meter | | :--- | :--- | :--- | | Control | Weak spring | No spring (or very weak) | | Damping | Critical | High (often) | | Measurement | Charge q | Flux change Δψ | | Deflection | ∝ q | ∝ Δψ | | Time Constant | Affects reading | No effect |

C. Frequency Meter (Weston Type)

• Block Diagram:

  
  Input Signal → Selective Network (LC circuits) → Electrodynamometer Movement → Pointer
  
  

• Working (Weston): Two fixed coils (parallel & series with capacitor) form two resonant circuits. Moving coil connected across input. At resonance of each circuit, phase relationship changes, causing pointer deflection proportional to frequency deviation from tuned frequency.

D. Ratio Meter

• Construction: Two coils (fixed & moving) on common shaft, energized by two different AC sources.

• Torque: T ∝ (V1² / Z1) - (V2² / Z2) or for current, T ∝ I1² R1 - I2² R2.

• Application: Measurement of impedance ratio, frequency ratio (if coils are resonant), or as power factor meter (one coil with resistor, other with inductor).

E. Digital Voltmeter (DVM)

• Basic Types:

  1. Successive Approximation: Fast, common. Uses DAC & comparator.

  2. Integrating (Dual-slope): High noise immunity, accuracy. Converts input voltage to time.

  3. Flash (Parallel): Very fast, limited to 8-bit.

• Working Principle (Integrating):

  • Phase 1: Input V_in integrates for fixed time T1 → output V1 ∝ V_in.

  • Phase 2: Reference V_ref (opposite polarity) integrates until output returns to zero → time T2 ∝ V1 ∝ V_in.

  • V_in = (T2/T1) * V_ref. Count T2 with clock → digital readout.

F. Earth Resistance Measurement

• Three-Terminal (Fall-of-Potential) Method:

  • Setup: Earth electrode (E), potential electrode (P), current electrode (C) driven into earth at distances > 20D (D = depth of E).

  • Procedure: Inject current I between E & C. Measure voltage V between E & P. R_e = V/I.

  • Graph: Vary C distance, plot R_e vs distance. Correct value when R_e stabilizes.

• Earth Tester: Hand-driven generator (or battery) with center-zero galvanometer and range resistors. Has three terminals (P, C, E). Uses AC (typically 10-30 Hz) to avoid polarization.

VIII. Magnetic Measurements

A. B-H Curve (Hysteresis Loop) Determination

Method of Reversals

• Circuit: Primary winding on specimen (for H), secondary winding (for flux linkage → charge on ballistic galvo). Reversing switch for primary current.

• Procedure:

  1. Apply max +I → point a on loop.

  2. Reduce to zero → point b (residual flux B_r).

  3. Reverse to -I_max → point c.

  4. Repeat for several I values.

  5. Plot B (from galvo throw ∝ ΔB) vs H (∝ I).

• Advantage: Gives complete loop without needing to demagnetize.

Six-Point Method

• For normal magnetization curve (from demagnetized state).

• Apply 6 specific currents: 0, I1, I2, I3, I_max, then back to 0.

• Measure corresponding B (from induced EMF integration or ballistic galvo).

• Plot points & join smoothly.

B. Iron Loss Measurement

Lloyd Fischer Square Method

• Setup: Square frame (non-magnetic) wound with primary & secondary. Specimen (laminated stack) forms part of magnetic circuit.

• Procedure: Apply rated voltage to primary. Measure:

  • V1, I1, W1 (wattmeter in primary).

  • V2, I2 (open secondary).

• Calculations:

  • Core loss W_core = W1 - I1² R1 (where R1 = primary resistance).

  • Hysteresis loss W_h & Eddy loss W_e separated by testing at two different frequencies f1, f2 (keeping V constant). W_h ∝ f, W_e ∝ f².

  • Solve: W_c1 = a f1 + b f1², W_c2 = a f2 + b f2².

Wattmeter Method for Transformer Steel Strips

• Setup: Single strip wound with primary & secondary, placed in a frame. Similar to Lloyd Fischer.

• Procedure: Apply rated voltage/frequency. Measure total iron loss W_total from wattmeter.

• Separation: Same frequency variation method.

IX. Additional Topics

A. Eddy Current Damping

• Principle: Conductor (aluminum disc) moving in magnetic field → eddy currents → opposing torque (Lenz's law) ∝ speed.

• Application: In moving iron, dynamometer, and induction meters (braking system).

B. Loading Effects of Instruments

• Ammeter (Series): R_A in series → circuit resistance increases → current decreases. % Error = (R_A / R_circuit) × 100%. Minimize by using low R_A (shunt).

• Voltmeter (Parallel): R_V in parallel → circuit resistance decreases → voltage drops. % Error ≈ (R_circuit / R_V) × 100% for high R_V. Minimize by using high R_V (sensitivity in Ω/V).

C. Testing and Calibration of Instruments

• Standards: Use higher accuracy standards (0.1% or better).

• Procedure: Compare instrument reading with standard under known conditions (V, I, pf). Determine error & apply correction.

• Calibration: Adjust instrument (e.g., shunt, series resistor, spring tension) to bring within specified limits.

D. Stray Magnetic Field Errors

• Cause: External magnetic fields deflect moving system (especially in PMMC, dynamometer).

• Shielding: Enclose instrument in high-permeability (mu-metal) case.

• Compensation: Use shorted turns (aluminum ring) on moving coil to generate opposing field.

Final Note: Always draw neat, labeled diagrams for construction questions. For derivations, state assumptions clearly (e.g., "assuming sinusoidal quantities", "neglecting losses"). Practice numerical problems from past papers, especially energy meter errors, CT/PT errors, Kelvin bridge calculations, and two-wattmeter method.