Electrical Measurements and Instruments (EX-303) - Unit 3 Short Notes

UNIT 3: ELECTRICAL MEASUREMENTS & INSTRUMENTS

I. FUNDAMENTALS OF MEASUREMENT & INSTRUMENT CHARACTERISTICS

Types of Errors

• Systematic: Consistent, predictable, repeatable. Causes: Calibration error, environmental factor, inherent instrument defect. Correctable.

• Random: Unpredictable, statistical variations. Causes: Noise, friction, ambient fluctuations. Reducible by repeated measurements & statistical analysis.

• Gross (Blunders): Human reading/recording errors. Eliminated by care.

Static Characteristics

Dynamic Characteristics

• Speed of Response: Time to reach steady state.

• Fidelity: How accurately output follows input waveform.

• Lag: Delay between input change and output response.

• Oscillation: Undamped overshoots in response.

Loading Effect

• Series Connected (Ammeter): Instrument impedance adds to circuit, reducing current. Minimized by low internal resistance.

• Parallel Connected (Voltmeter): Instrument impedance draws current, reducing voltage across load. Minimized by high internal resistance.

[!TIP] Error Propagation

For $$\displaystyle z = f(x, y) $$, the maximum fractional error is:

$$ \frac{\Delta z}{z} \approx \pm \sqrt{ \left( \frac{\partial z}{\partial x} \frac{\Delta x}{z} \right)^2 + \left( \frac{\partial z}{\partial y} \frac{\Delta y}{z} \right)^2 } $$

Example - Power: $$\displaystyle P = VI $$. If errors in $V$ and $I$ are $\Delta V$ and $\Delta I$,

$$ \frac{\Delta P}{P} = \frac{\Delta V}{V} + \frac{\Delta I}{I} \quad \text{(Worst-case sum)} $$

II. MOVING COIL & MOVING IRON INSTRUMENTS

Permanent Magnet Moving Coil (PMMC)

• Construction: Permanent magnet, moving coil (air-cored) on spindle, hair-springs (control & electrical connection), pointer, scale.

• Principle: $$\displaystyle F = BIl $$ in radial field $\Rightarrow$ Deflecting Torque: $$\displaystyle T_d = BINA = kI $$ (linear scale).

• Range Extension:

  • Ammeter: Shunt $$\displaystyle R_s = \frac{I_m R_m}{I - I_m} $$

  • Voltmeter: Multiplier $$\displaystyle R_m = \frac{V}{I_m} - R_m $$

• Advantages: Linear scale, high sensitivity, low power consumption, good accuracy.

• Limitations: DC only, expensive, fragile.

Moving Iron Instruments

• Types:

  • Attraction Type: Soft iron piece attracted to coil. Torque $$\displaystyle \propto I^2 $$.

  • Repulsion Type: Two iron vanes repelled. Torque $$\displaystyle \propto I^2 $$.

• Torque Equation: $$\displaystyle T_d \propto H^2 \propto I^2 $$. Scale is non-linear (square law), cramped at lower end.

• Applications: AC/DC ammeter, voltmeter, wattmeter (as electrodynamometer).

Damping Methods

III. GALVANOMETERS & CHARGE MEASUREMENT

Ballistic Galvanometer

• Construction vs d'Arsonval: Large moment of inertia (heavy coil/mirror), low control torque, low damping (to see first swing). Used for charge, not current.

• Equation of Motion: $$\displaystyle J\frac{d^2\theta}{dt^2} + G\frac{d\theta}{dt} + K\theta = T_d $$

  For a current pulse $$\displaystyle i = \frac{dq}{dt} $$, $$\displaystyle T_d = NAB \cdot i = NAB \frac{dq}{dt} $$.

  Integrating: $$\displaystyle J\frac{d\omega}{dt} + G\omega + K\theta = NAB q $$

  At first maximum swing ($$\displaystyle \theta = \theta_1 $$, $$\displaystyle \omega=0 $$): $$\displaystyle K\theta_1 = NAB q $$ (if damping small).

  \boxed{q = \frac{K}{NAB} \theta_1 = C \cdot \theta_1} \quad \text{(Ballistic Constant $C$)}

• Proof Charge ∝ First Swing: For critically damped/underdamped, the first deflection $$\displaystyle \theta_1 $$ is proportional to $q$, independent of waveform.

• Calculations:

  • Logarithmic Decrement: $$\displaystyle \delta = \ln\left(\frac{\theta_1}{\theta_2}\right) $$

  • Damping Ratio: $$\displaystyle \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}} $$

  • Period: $$\displaystyle T_d = \frac{2\pi}{\sqrt{\frac{K}{J} - \left(\frac{G}{2J}\right)^2}} $$

Flux Meter

• Construction: Similar to ballistic but no control spring (zero $K$), very high damping.

• Principle: Measures change in flux linkage $$\displaystyle \Delta \phi = \frac{G}{N} \theta_1 $$. Direct reading of $$\displaystyle \theta_1 $$ gives $\Delta \phi$.

• Comparison with Ballistic Galvanometer:

  | Feature | Ballistic Galvanometer | Flux Meter | | :--- | :--- | :--- | | Control Torque ($K$) | Present (spring) | Absent | | Damping | Low | Very High | | Measures | Charge ($$\displaystyle q \propto \theta_1 $$) | Flux Change ($$\displaystyle \Delta\phi \propto \theta_1 $$) | | Time to Read | Wait for first swing | Instantaneous |

Application: Measurement of Inductance (Constant Current Method)

1. Charge capacitor $C$ to voltage $V$ $\Rightarrow$ $$\displaystyle q = CV $$.

2. Discharge through $L$ & galvanometer in series.

3. Measure $$\displaystyle \theta_1 $$ $\Rightarrow$ $$\displaystyle q = C \theta_1 $$ $\Rightarrow$ $$\displaystyle L = \frac{NAB \theta_1}{C} $$ (from $$\displaystyle L\frac{di}{dt} = NAB i $$).

IV. ELECTRODYNAMOMETER TYPE INSTRUMENTS

General Construction & Principle

• Fixed Coils: Carry current $$\displaystyle i_1 $$.

• Moving Coil (Air-cored): Carries current $$\displaystyle i_2 $$, spring-controlled.

• Torque (DC): $$\displaystyle T_d = i_1 i_2 \frac{dM}{d\theta} = k i_1 i_2 $$. For ammeter/voltmeter, $$\displaystyle i_1 \propto i_2 $$ $\Rightarrow$ $$\displaystyle T_d \propto i^2 $$ (non-linear scale).

• Torque (AC - Ferrodynamic): $$\displaystyle T_d = I_1 I_2 \frac{dM}{d\theta} \cos\phi $$ (where $\phi$ is phase between $$\displaystyle i_1, i_2 $$). True RMS response as $$\displaystyle T_d \propto i_1^2 i_2^2 $$.

Electrodynamometer Wattmeter

• Construction:

  DiagramCANVAS: Sketch showing fixed coils (current coil) in series with load, moving coil (potential coil) with high resistance in parallel across load. Pointer on moving coil spindle.

• Torque Derivation (Single-phase):

  $$\displaystyle i_1 = I \cos\omega t $$ (current coil, in-phase w.r.t. $v$)

  $$\displaystyle i_2 = \frac{V}{R_p} \cos\omega t $$ (potential coil, resistive $\Rightarrow$ in-phase w.r.t. $v$)

  $$\displaystyle T_d \propto i_1 i_2 \propto I \cos\omega t \cdot \frac{V}{R_p} \cos\omega t = \frac{VI}{R_p} \cos^2\omega t = \frac{VI}{2R_p}(1+\cos2\omega t) $$

  Average Torque: $$\displaystyle \overline{T_d} \propto \frac{VI}{2R_p} = \frac{P}{2R_p} \propto P $$.

  \boxed{\text{Reading} \propto P = VI \cos\phi}

• Connection: Current coil in series with load, Potential coil across load.

• Errors & Minimization:

  • Stray Magnetic Field Errors: Induces emf in moving coil $\Rightarrow$ error. Minimized by:

    1. Shielding: Cast iron case.

    2. Compensating Coil: Short-circuited turn on moving coil side to nullify stray flux effect.

  • Low Power Factor Errors:

    • Phase Angle Error: $\phi$ small $\Rightarrow$ $\cos\phi \approx 1$, but $\sin\phi$ large $\Rightarrow$ large error in $VI \sin\phi$ component.

    • Lag Adjustment: Add capacitor in parallel with potential coil to make it slightly leading, compensating for inductive burden.

    • Compensating Coil: Series with potential coil to reduce its inductance.

Power Factor Meter (3-phase Electrodynamometer)

• Construction: Two perpendicularly mounted moving coils on same spindle, fixed coils connected to different phases.

• Working: Torque on each coil $$\displaystyle \propto V_L I_L \cos(\theta \pm 30^\circ) $$. Net torque $\propto \cos\phi$. Scale calibrated in $\cos\phi$ or $\phi$.

• Phasor Diagram: Shows torque balance for leading/lagging PF.

Polyphase Wattmeters

• Two-Element (for 3-phase, 3-wire or 4-wire): Two separate dynamometer elements on common spindle. Torques add. Total $$\displaystyle P = W_1 + W_2 $$.

• Three-Element (for 3-phase, 4-wire): One element per phase. $$\displaystyle P = W_A + W_B + W_C $$.

V. CURRENT TRANSFORMERS (CT) & POTENTIAL TRANSFORMERS (PT)

Current Transformer (CT)

• Construction:

  • Bar Type: Primary = 1 turn (bar).

  • Wound Type: Multi-turn primary.

  • Window (Ring) Type: Primary is single turn passing through core window.

• Equivalent Circuit & Phasor Diagram:

  DiagramCANVAS: Phasor diagram showing $$\displaystyle I_p $$, $$\displaystyle I_s $$, $$\displaystyle I_e $$ (exciting), $$\displaystyle I_s' $$ (secondary referred to primary), $$\displaystyle I_0 $$ (exciting). Ratio error = $$\displaystyle (\frac{I_s'}{I_p} - \frac{1}{n}) $$, Phase error = $\delta$.

• Theory:

  • Ratio Error: $$\displaystyle n = \frac{N_p}{N_s} $$ (nominal). Actual ratio $$\displaystyle = \frac{I_p}{I_s} $$. Error due to exciting current $$\displaystyle I_e $$.

  • Phase Angle Error ($\delta$): Angle between $$\displaystyle I_p $$ and $$\displaystyle I_s' $$ (reversed).

• Factors Affecting Errors:

  • Secondary Burden (VA, PF): Higher burden $\uparrow$ $$\displaystyle I_e $$ $\uparrow$ errors.

  • Primary Current: Below rated $\uparrow$ relative $$\displaystyle I_e $$ $\uparrow$ errors.

  • Core Loss (Material, Flux Density).

• Effect of Open-Circuited Secondary (CRITICAL DANGER):

  • $$\displaystyle I_s = 0 $$, $$\displaystyle I_p $$ becomes exciting current.

  • Core flux $\phi$ becomes very high $\Rightarrow$ Core saturation, excessive $$\displaystyle I_e $$, overheating, insulation damage, high voltage across open secondary (dangerous!).

  • Never open CT secondary under primary excitation.

• Testing: Ratio test (compare $$\displaystyle I_p/I_s $$), Phase angle test, Burden test (measure $$\displaystyle Z_b $$).

Potential Transformer (PT)

• Construction: Shell or core type. High insulation, low leakage.

• Equivalent Circuit & Phasor Diagram: Similar to 2-winding transformer. $$\displaystyle Z_b $$ = burden.

• Theory:

  • Ratio Error: Due to voltage drop in internal impedances ($R, X$) and regulation.

  • Phase Angle Error: Angle between primary $$\displaystyle V_p $$ and secondary $$\displaystyle V_s $$ (reversed).

• Factors & Minimization:

  • Use low-loss core (low $$\displaystyle I_e $$).

  • Low leakage reactance (tight coupling, interleaved windings).

  • Keep burden low & PF high.

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

CT vs PT Comparison

VI. ENERGY MEASUREMENT (WATT-HOUR METERS)

Single-Phase Induction (KiloWatt-Hour) Meter

• Construction (4 Systems):

  1. Driving System: Shunt (voltage) magnet (aluminium disc, high resistance) & series (current) magnet (U-shaped, low reluctance). Fluxes $$\displaystyle \phi_v \propto V $$, $$\displaystyle \phi_I \propto I $$.

  2. Moving System: Aluminium disc on spindle, in air-gap between magnets.

  3. Braking System: Permanent magnet (eddy current damping). Torque $$\displaystyle T_b \propto \omega $$.

  4. Counting Mechanism: Register (gear train & dials).

• Operating Principle:

  • Driving Torque: $$\displaystyle T_d \propto \phi_v \phi_I \sin\delta \propto V I \cos\phi = P $$. ($\delta$ = angle between $$\displaystyle \phi_v $$ (lagging $V$) and $$\displaystyle \phi_I $$ (in-phase with $I$)).

  • Braking Torque: $$\displaystyle T_b \propto \omega $$ (disc speed).

  • Steady Speed: $$\displaystyle T_d = T_b \Rightarrow \omega \propto P $$.

  • Energy: $$\displaystyle E = \int P dt \propto $$ No. of revolutions $N$.

  \boxed{E = \frac{N}{K}} \quad \text{(Meter Constant $K$ = rev/kWh)}

• Types of Errors:

  • Frictional: At start/light load.

  • Creep (No-load): Disc rotates with $V$ applied, $$\displaystyle I=0 $$. Due to imperfect shading (phase shift not 90°).

  • Phase Error: $$\displaystyle \delta \neq 90^\circ $$ due to inductive shunt magnet.

  • Frequency Error: $$\displaystyle \phi_v $$ & $$\displaystyle \phi_I $$ phase shift changes with $f$.

  • Temperature: Resistance changes.

  • Stray Load: External magnetic fields.

• Testing & Calibration:

  • Full Load Test: Measure $N$ in time $t$. True energy $$\displaystyle E_t = V I t \cos\phi $$. Meter reading $$\displaystyle E_m = N/K $$.

$$ \% \text{Error} = \frac{E_m - E_t}{E_t} \times 100 $$

*   **No-Load (Creep) Test:** $V$ applied, $$\displaystyle I=0 $$. Disc should not rotate > 1 rev in 10 min.

*   **Light Load Test:** Check friction compensation.

*   **Power Factor & Speed Variation Tests.**

*   **Meter Constant:** $$\displaystyle K = \frac{V I \cos\phi \times t}{N} $$ (at full load).

Three-Phase Energy Meter

• Two-Element (for 3-phase, 3-wire or 4-wire): Two single-phase meters on common shaft. Total $$\displaystyle P = W_1 + W_2 $$. For 3-wire, use 2.5-element (one element for two phases).

• Three-Element (for 3-phase, 4-wire): One element per phase. $$\displaystyle P = W_A + W_B + W_C $$.

• Connection:

  DiagramCANVAS: Connection diagram for two-element meter in 3-phase, 3-wire (using two CTs) and 4-wire (using three CTs or two CTs + neutral).

• Phasor Diagram & Total Power: For balanced load, $$\displaystyle P_{total} = \sqrt{3} V_L I_L \cos\phi = W_1 + W_2 $$.

Maximum Demand Meter

• Construction: Time-Interval Type: Integrates energy over set interval (e.g., 15 min). Moving Demand Indicator: Pointer shows average demand over interval.

• Working: Uses thermal or electromechanical integration. Resets after interval.

Digital Electronic Energy Meter

• Block Diagram:

  DiagramCANVAS: Voltage & Current sensors (shunt/CT) → Sample & Hold → ADC → Microcontroller (multiply, integrate, store) → Display (LCD/LED) & Pulse Output.

• Principle: $V(t)$, $I(t)$ sampled, multiplied instantaneously, integrated over time.

• Advantages: High accuracy, no friction/drift, multiple parameters (kWh, kVA, PF), tamper-proof, remote reading.

VII. RESISTANCE MEASUREMENT BRIDGES

Wheatstone Bridge

• Circuit:

  DiagramCANVAS: $$\displaystyle R_x $$ unknown, $$\displaystyle R_2, R_3 $$ ratio arms, $$\displaystyle R_4 $$ standard. Galvanometer between junctions. Battery across other diagonal.

• Balance Condition: $$\displaystyle \frac{R_x}{R_2} = \frac{R_4}{R_3} $$ or $$\displaystyle R_x = R_4 \frac{R_2}{R_3} $$.

• Use: Medium resistances (1Ω to 1MΩ).

• Errors & Minimization:

  • Lead/Contact Resistance: Use 4-terminal (Kelvin) connections for $$\displaystyle R_x $$.

  • Thermal EMF: Use same metal for all junctions, reverse battery & average.

  • Sensitivity: $$\displaystyle S_G = \frac{dI_g}{dR_x} $$. Maximized when $$\displaystyle R_2/R_3 \approx R_x/R_4 $$.

  • Bridge Unbalance: Use detector with high sensitivity.

Kelvin (Thomson) Double Bridge

• Superiority for Low Resistance (<1Ω): Eliminates effect of lead & contact resistance ($r, r'$) by using 4-terminal sensing.

• Circuit & Construction:

  DiagramCANVAS: $$\displaystyle R_x $$ (unknown) and $$\displaystyle R_s $$ (standard) have **4 terminals**. Outer ratio arms $$\displaystyle R_1, R_2 $$. Inner ratio arms $$\displaystyle R_1', R_2' $$. Link $r$ between $$\displaystyle R_x $$ & $$\displaystyle R_s $$ low-resistance connections. Galvanometer between midpoints of $$\displaystyle R_1-R_2' $$ and $$\displaystyle R_1'-R_2 $$.

• Balance Condition Derivation:

  At balance, $$\displaystyle V_{BC} = V_{AD} $$.

  $$\displaystyle I_1 R_x + (I_1 - I_g)r = I_2 R_s + (I_2 + I_g)r' $$

  For high sensitivity, $$\displaystyle r, r' \ll R_x, R_s, R_1, R_2, R_1', R_2' $$ and $$\displaystyle R_1/R_2 = R_1'/R_2' $$.

  \boxed{R_x = R_s \frac{R_1}{R_2} = R_s \frac{R_1'}{R_2'}} \quad \text{(Independent of $r, r'$)}

• Precautions:

  • Use thick, short leads for $r, r'$.

  • Four-terminal connections for $$\displaystyle R_x $$ & $$\displaystyle R_s $$.

  • Ratio arms $$\displaystyle R_1/R_2 = R_1'/R_2' $$ precisely.

VIII. MAGNETIC CIRCUIT & CORE LOSS MEASUREMENT

B-H Curve Determination

• Method of Reversals (Hopkinson's Method):

  • Setup: Bar specimen in magnetizing coil, fluxmeter, integrating (ballistic) galvanometer, reversing switch, ammeter, voltmeter.

  • Procedure:

    1. Apply DC current $I$, reverse suddenly.

    2. Fluxmeter reads $\Delta \phi$ (change in flux linkage).

    3. $$\displaystyle H = \frac{NI}{l} $$, $$\displaystyle B = \frac{\Delta \phi}{N A} $$.

    4. Repeat for increasing $I$.

  • Graph: Plot $B$ vs $H$. Advantage: No integration error, fast.

• Six-Point Method:

  • Setup: Specimen in primary & secondary coils, ballistic galvanometer, integrator, ammeter, variable DC supply.

  • Procedure: For each $I$, measure:

    1. $$\displaystyle \phi_{max} $$ (with switch closed).

    2. $$\displaystyle \phi_{residual} $$ (after opening switch, reversing).

    3. $$\displaystyle \phi_{reversed} $$ (after reversing & closing).

    Calculate $$\displaystyle B = \frac{\phi}{NA} $$, $$\displaystyle H = \frac{NI}{l} $$ for ascending/descending branches.

  • Graph: Complete hysteresis loop.

Iron Loss Measurement

• Lloyd-Fischer Square Method:

  • Construction: Square frame of magnetic material (test specimen), primary & secondary windings, wattmeter, voltmeter, ammeter, variac.

  • Principle: Primary excited at rated $V$, secondary open $\Rightarrow$ Wattmeter reads core loss ($$\displaystyle P_{fe} = P_h + P_e $$) as primary current is small.

  • Calculation: $$\displaystyle P_{fe} = \text{Wattmeter reading} $$.

• Wattmeter Method for Strip Steel:

  • Testing: Stack of insulated strips (like transformer core) wound with primary & secondary.

  • Procedure: Apply rated $V$ to primary, secondary open. Wattmeter on primary side reads total core loss.

  • Calculation: $$\displaystyle P_{fe} = \text{Wattmeter reading} $$.

IX. SPECIAL MEASUREMENTS & INSTRUMENTS

Earth Resistance Measurement

• Methods:

  • Three-Point (Fall-of-Potential):

    DiagramCANVAS: Earth electrode (E), potential probe (P), current probe (C) at varying distances. Measure $$\displaystyle V_{EC} $$ & $I$, $$\displaystyle R = V/I $$.

  • Four-Point: Eliminates contact resistance.

• Earth Tester: Hand-driven generator (DC), two auxiliary electrodes, voltmeter, current meter. Direct reading.

Frequency Measurement

• Frequency Meter (Wattmeter Type / Resonant):

  • Block Diagram:

    DiagramCANVAS: Input signal → Selective circuit (LC tunable) → Rectifier → DC meter (or moving coil). Tune for max deflection, read calibrated scale.

  • Working: Resonant reed or mechanical vibration type. Pointer indicates frequency.

Ratio Meter

• Construction: Two coils on moving system, fed by two voltages/currents (e.g., from CT/PT). Torque $$\displaystyle \propto V_1 I_1 \cos\phi_1 - V_2 I_2 \cos\phi_2 $$.

• Application: CT/PT ratio testing, phase comparison.

Megger (Insulation Resistance Tester)

• Construction: Hand-driven DC generator (500V/1000V/2500V), Coulomb-movement (two coils 90° apart, one with series resistor).

• Working: Hand crank $\Rightarrow$ generator voltage. One coil in series with test resistance $$\displaystyle R_x $$, other with fixed reference $R$. Deflection $$\displaystyle \propto R_x $$.

• Use: Cable, transformer, motor insulation testing (megohms).

Loss of Charge Method

• For High Insulation Resistance ($$\displaystyle >10^{10}\Omega $$).

• Circuit:

  DiagramCANVAS: Capacitor $C$ charged to $$\displaystyle V_0 $$, connected in parallel with **unknown $$\displaystyle R_x $$** and **known $R$** (via switch). Voltmeter across $C$.

• Calculation: $$\displaystyle V = V_0 e^{-t/(R_x C)} $$ if $$\displaystyle R \gg R_x $$. Measure time for $V$ to drop to $$\displaystyle V_0/2 $$ (half-life $$\displaystyle t_{1/2} $$).

  \boxed{R_x = \frac{t_{1/2}}{C \ln 2}}

  If $R$ comparable, use two-time constant method.

Digital Voltmeter (DVM)

• General Block Diagram:

  DiagramCANVAS: Input → **Sample & Hold** → **ADC** (Ramp, Dual-Slope, Integrating) → **Logic/Controller** → **Display** (7-segment/LCD).

• Types:

  • Ramp Type: Integrate input, measure time to zero.

  • Dual-Slope: Integrate input for fixed time $$\displaystyle T_1 $$, then integrate reference of opposite slope to zero (time $$\displaystyle T_2 \propto V_{in} $$). High noise immunity.

  • Integrating (Double-Integration): Convert $$\displaystyle V_{in} $$ to time, then to digital.

X. INSTRUMENT ACCESSORIES & CALIBRATION

• Range Extension:

  • Ammeter: Shunt $$\displaystyle R_s = \frac{I_f R_f}{I - I_f} $$.

  • Voltmeter: Multiplier $$\displaystyle R_m = R_f \left( \frac{V}{V_f} - 1 \right) $$.

• Calibration Procedures:

  1. Compare with standard instrument (higher accuracy).

  2. Apply known input (calibrator).

  3. Note error at various points.

  4. Plot calibration curve (reading vs true value).

  5. Apply correction (if needed).

• Standard Instruments & Transfer Standards: Primary standards (national labs), secondary standards (calibration labs), working standards (laboratories). Transfer via comparison method.

END OF UNIT 3 NOTES
Focus on derivations (torque, bridge balance), diagrams (wattmeter, CT/PT phasor, bridge circuits), and numerical problems from past papers.