Electrical Measurements and Instruments (EX-303) - Unit 4 Short Notes

UNIT 4: ELECTRICAL MEASUREMENTS AND INSTRUMENTS

1.0 FUNDAMENTALS OF MEASUREMENT & ERROR ANALYSIS

1.1 Types of Errors in Measurement

• Gross Errors: Human mistakes (parallax, incorrect reading). Preventable by careful procedure.

• Systematic Errors: Consistent, predictable. Sub-types:

  • Instrumental: Zero error, calibration error, wear & tear.

  • Environmental: Temperature, humidity, stray fields.

  • Observational: Parallax, incorrect sighting.

• Random Errors: Unpredictable fluctuations (noise). Reduced by repeated measurements and statistical analysis.

• Key Quantities:

  • Absolute Error, $$\displaystyle E_A = A_m - A_t $$ (Measured - True)

  • Relative Error, $$\displaystyle \%E_r = \frac{E_A}{A_t} \times 100 $$

  • Correction = $$\displaystyle -E_A $$

[!TIP] Exam often asks to classify errors and suggest remedies for each type.

1.2 Static & Dynamic Characteristics

• Static (Steady-state):

  • Accuracy: Closeness to true value.

  • Precision: Repeatability (consistency).

  • Sensitivity: $$\displaystyle \frac{\text{Output change}}{\text{Input change}} = \frac{\Delta \text{deflection}}{\Delta \text{quantity}} $$.

  • Resolution: Smallest detectable change.

  • Drift: Slow change over time.

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

  • Threshold: Minimum input to cause output change.

  • Linearity: Deviation from straight-line input-output relationship.

• Dynamic (Time-varying input):

  • Speed of Response: How fast instrument follows changes.

  • Fidelity: Accuracy in reproducing waveform.

  • Lag: Delay in response.

  • Overshoot & Oscillation: Stability of response.

1.3 Loading Effect

• Voltmeter Loading: Voltmeter connected in parallel draws current ($$\displaystyle I_V = V/R_V $$), altering circuit current. Effect severe when $$\displaystyle R_V $$ is not $\gg$ circuit resistance.

  • Loading Error % $$\displaystyle \approx \frac{R_{\text{circuit}}}{R_V + R_{\text{circuit}}} \times 100 $$

• Ammeter Loading: Ammeter connected in series adds its internal resistance ($$\displaystyle R_A $$), altering total circuit resistance and current.

• Multimeter Sensitivity: Specified in $\Omega/V$. For a given range, $$\displaystyle R_V = \text{Sensitivity} \times \text{Range} $$. Higher sensitivity $\Rightarrow$ lower loading.

1.4 Damping Mechanisms

• Air Friction Damping: Vane in air chamber. Simple, used in low-sensitivity instruments.

• Eddy Current Damping: Conductive, non-magnetic plate (aluminum) moves in magnetic field of permanent magnet. Induced eddies produce opposing force. Critical for providing quick, stable reading without overshoot. Widely used in wattmeters, energy meters, and analog meters.

  • Principle: $F \propto v$ (damping force proportional to velocity).

  • DiagramCANVAS: Show aluminum disc moving through magnetic field gap of a permanent magnet, with eddy current loops indicated.

2.0 CURRENT TRANSFORMERS (CT) & POTENTIAL TRANSFORMERS (PT)

2.1 Current Transformer (CT)

• Construction:

  • Core: High silicon steel laminations (low loss).

  • Primary: Usually a single turn (bus-bar or toroidal) or few turns. Carries full line current.

  • Secondary: Many turns (100s), rated 5A or 1A. Wound on core, insulated.

  • Insulation: Between primary and secondary, and to ground.

  • Terminals: Polarity marked (P1, S1).

• Principle: $$\displaystyle I_p / I_s = N_s / N_p $$ (ideally). Secondary must never be open when primary is energized.

• Equivalent Circuit & Phasor Diagram:

  • Model includes: $$\displaystyle R_m \parallel jX_m $$ (magnetizing branch), $$\displaystyle R_s $$, $$\displaystyle X_s $$ (secondary impedance), burden $$\displaystyle Z_b $$.

  • Errors:

    • Ratio Error: $$\displaystyle \% \text{ Ratio Error} = \frac{K_n - K}{K} \times 100 $$, where $$\displaystyle K_n = \frac{I_p}{I_s} $$ (actual), $$\displaystyle K = \frac{N_s}{N_p} $$ (nominal).

    • Phase Angle Error: Angle $\delta$ between primary current $$\displaystyle I_p $$ and secondary current reversed $$\displaystyle I_s' $$.

• Factors Affecting Errors:

  • Secondary Burden (Z_b): Increased burden (higher $R$ or low $pf$) increases magnetizing current $\Rightarrow$ larger errors.

  • Core Saturation: High $$\displaystyle I_p $$ or low burden $pf$ saturates core $\Rightarrow$ large ratio & phase errors.

  • Frequency: Deviations from rated frequency affect core losses and reactances.

• Effect of Open Secondary: Primary current becomes exciting current, core saturates, high voltage induced across open secondary terminals (dangerous!).

• Testing & Calibration:

  • Comparison Method: Compare with a standard CT using a zero detector (e.g., differential galvanometer).

  • Ratio & Phase Angle: Measured using a calibrated burden and precision instruments.

2.2 Potential Transformer (PT)

• Construction:

  • Core: Shell-type (toroidal) for low leakage flux.

  • Primary: Many turns, connected across line (high voltage).

  • Secondary: Fewer turns, rated 100V or 110V.

  • Insulation: High-quality, often oil-immersed for high voltage.

• Principle: $$\displaystyle V_p / V_s = N_p / N_s $$ (ideally). Secondary can be open.

• Equivalent Circuit & Phasor Diagram: Similar to CT but with reversed impedance roles. Magnetizing current $$\displaystyle I_m $$ lags $$\displaystyle V_p $$.

• Errors:

  • Ratio Error: $$\displaystyle \% \text{ Ratio Error} = \frac{K_n - K}{K} \times 100 $$, where $$\displaystyle K_n = V_p / V_s $$ (actual), $$\displaystyle K = N_p / N_s $$.

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

• Methods to Minimize Errors:

  • Low loss core (high-grade silicon steel).

  • Low resistance windings (thick conductors).

  • Low burden (use low $VA$ rating instruments).

  • Operate at rated voltage & frequency.

• Testing & Calibration:

  • Ratio Test: Apply rated voltage, measure secondary.

  • Phase Angle Test: Using a standard PT and phase meter.

  • Burden Test: Vary burden, check ratio error within limits.

2.3 Comparison of CT and PT

3.0 ELECTRODYNAMOMETER TYPE INSTRUMENTS

3.1 General Construction & Principle

• Construction:

  • Fixed Coils (Current Coils): Two coils connected in series, carry the current to be measured. Produce magnetic field.

  • Moving Coil (Voltage/Potential Coil): Suspended in field of fixed coils. Carries current proportional to voltage (via series resistor). Connected in series with fixed coils for wattmeter.

  • Control: Hair-springs (provide restoring torque & lead wires).

  • Damping: Air friction (vane) or eddy current.

  • Scale: Non-linear (cramped at ends) due to $\sin \theta$ torque for spring control.

• Torque Equation (DC & AC):

  • Flux $$\displaystyle \phi \propto I_f $$ (fixed coil current).

  • Moving coil current $$\displaystyle I_m \propto I $$ (for ammeter) or $V$ (for voltmeter).

  • Torque, $$\displaystyle T \propto \phi \cdot I_m \cdot \sin \theta \propto I_f \cdot I_m $$.

  • For wattmeter (fixed coils in series with load, moving coil across load): $$\displaystyle T \propto I \cdot V \cdot \cos \phi = \text{Real Power} $$.

  • For ammeter/voltmeter (fixed coils in series with moving coil): $$\displaystyle T \propto I^2 $$ or $$\displaystyle V^2 $$ (since $$\displaystyle I_f = I_m $$). Hence True RMS.

• Nature of Scale: Non-linear ($T \propto \sin \theta$ for spring control). More crowded at low values.

3.2 Electrodynamometer Wattmeter

• Construction Details:

  • Fixed coils: Heavy, low resistance, connected in series with load. Placed to produce uniform field.

  • Moving coil: Light, high resistance, connected in parallel with load via a series resistor (swamping resistor to minimize temperature error). Suspended by ribbon springs.

  • Damping: Air friction (vane) or eddy current (aluminum disc on spindle).

  • Shielding: Iron case to prevent stray field errors.

• Working & Torque Expression for AC:

  • Let fixed coil current $$\displaystyle i_f = \sqrt{2} I_f \sin \omega t $$, moving coil current $$\displaystyle i_m = \sqrt{2} I_m \sin (\omega t - \phi) $$.

  • Instantaneous torque: $$\displaystyle dT \propto i_f \cdot i_m \cdot \sin \theta $$.

  • Average torque: $$\displaystyle \overline{T} \propto I_f I_m \cos \phi = k \cdot V I \cos \phi = k \cdot P $$.

  • Thus, scale is linear for power (P).

• Errors:

  • Stray Magnetic Field Errors: External fields distort main field $\Rightarrow$ incorrect torque. Mitigation: Iron case (magnetic shielding).

  • Eddy Current Errors: Induced in moving coil former by stray fields. Use non-magnetic, high-resistivity material (e.g., aluminum).

  • Friction & Temperature: Spring torque variation with temperature. Use swamping resistor in moving coil circuit.

  • Inductance Error: At low $pf$, phase difference between $$\displaystyle I_f $$ and $V$ causes error. Compensated by compensating coil (connected in series with moving coil, wound to oppose field of fixed coils).

• Special Features for Low Power Factor:

  • Compensating Coil: Reduces phase angle error by making moving coil circuit nearly resistive.

  • High Torque-to-Weight Ratio: To provide sufficient deflection at low power.

• Multi-Range Wattmeters:

  • Two-Element Wattmeter: Two independent wattmeter units on common spindle. Used for single-phase (for redundancy) or three-phase 2-wattmeter method.

  • Three-Element Wattmeter: Three elements, used for three-phase 4-wire (balanced/unbalanced) or three-phase 3-wire (with artificial neutral).

3.3 Electrodynamometer Power Factor Meter (Three-Phase)

• Construction: Two fixed coils (connected in parallel across two phases, e.g., R & Y), moving coil with series resistors/reactors to create phase shift.

• Working: Torque on moving coil depends on phase angle between line voltage and current. Scale calibrated in $pf$ (0.5 lag to 0.5 lead).

3.4 Electrodynamometer Ammeter & Voltmeter

• Ammeter: Fixed coils in series with moving coil (both carry same current $I$). Torque $$\displaystyle T \propto I^2 $$. Scale non-linear.

• Voltmeter: Fixed coils in series with moving coil, both in series with a high resistor. Torque $$\displaystyle T \propto V^2 $$.

• True RMS Measurement: Since $$\displaystyle T \propto I^2 $$ or $$\displaystyle V^2 $$, average torque $\propto$ mean square value $\Rightarrow$ true RMS independent of waveform.

4.0 ENERGY METERS (Watt-hour Meters)

4.1 Single-Phase Induction Type (Energy Meter)

• Detailed Construction:

  • Driving System:

    • Voltage Magnet: High inductance, laminated core, with voltage coil (many turns) across supply. Shunted by a voltage coil resistor (non-inductive) to provide flux in phase with $V$.

    • Current Magnet: Low inductance, laminated core, with current coil (few turns) in series with load. Flux $$\displaystyle \phi_I \propto I $$.

  • Moving System: Light aluminum disc mounted on spindle, rotates in air gap between magnets.

  • Braking System: Permanent magnet (dynamometer type) positioned near disc. Eddy currents in disc produce braking torque $\propto$ disc speed.

  • Registering System: Gear train (worm & wheel) driven by disc spindle, counts revolutions on dials (kWh).

• Working Principle:

  • Driving Torque ($$\displaystyle T_d $$): $$\displaystyle T_d \propto \phi_V \cdot \phi_I \cdot \sin \theta \propto V I \cos \phi = P $$ (since $$\displaystyle \theta \approx 90^\circ $$, $\sin \theta \approx 1$). $$\displaystyle T_d \propto \text{Instantaneous Power} $$.

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

  • Steady Speed: $$\displaystyle T_d = T_b \Rightarrow N \propto P \Rightarrow \text{Revolutions} \propto \text{Energy (kWh)} $$.

• Errors:

  • Frictional Error: Constant torque due to friction in bearings & gear. Causes error at light loads.

  • Creep Error: Disc rotates with voltage applied but no current (should be zero). Caused by residual magnetism, improper shading, or friction. Test: Creep test (voltage only, current zero).

  • Phase Angle Error: Due to imperfect $$\displaystyle 90^\circ $$ shift between $$\displaystyle \phi_V $$ and $$\displaystyle \phi_I $$. Corrected by adjusting shading bands on voltage magnet.

  • Frequency Error: Core losses & reactances vary with frequency.

  • Temperature Error: Resistance changes (voltage coil, disc).

  • Overload Error: Saturation of current magnet at high currents.

• Testing & Calibration:

  • Full Load Test (No-load Test): Apply rated $V$ & $I$ at unity $pf$. Measure disc revolutions $N$ in time $t$.

$$\boxed{K = \frac{3600 \times N}{C \times V \times I \times t}}$$

    where $K$ = % error, $C$ = meter constant (rev/kWh). If $$\displaystyle K > 0 $$, meter fast; $$\displaystyle K < 0 $$, slow.

*   **Phantom Loading:** Use a standard wattmeter and timer. Energize voltage coil from supply, current coil from a separate low-voltage source via a current transformer. Measures error without consuming large power.

*   **Starting Test:** Apply $V$ & $I$ at $0.5$ lag $pf$, disc should start.

*   **Light Load & Creep Tests:** As described.

4.2 Three-Phase Energy Meters

• Two-Element Meter:

  • Construction: Two single-phase meters on common spindle, two discs, one register. Each element has its own voltage & current magnets.

  • Connection for 3-phase 3-wire (2-wattmeter method): Voltage coils across two lines (e.g., R-Y), current coils in two lines (R & Y). Total Energy = Sum of both element energies.

  • Connection for 3-phase 4-wire: Voltage coils across all three lines (R-Y, Y-B, B-R via common voltage coil), current coils in all three lines.

• Three-Element Meter:

  • Construction: Three independent elements, three discs, common register.

  • Application: 3-phase 4-wire systems (balanced/unbalanced loads, with neutral). Each element measures power in one phase.

  • Total Power: $$\displaystyle P_{\text{total}} = P_1 + P_2 + P_3 $$.

4.3 Digital Electronic Energy Meter

• Block Diagram:

  
  Voltage Sensor (Potential Transformer/Resistive Divider)
  
              |
  
              v
  
  [Sample & Hold] ---> [Analog-to-Digital Converter (ADC)] ---> [Microcontroller/Multiplier] ---> [Display & Memory]
  
              |                                          ^
  
              v                                          |
  
  Current Sensor (Current Transformer/Shunt) ------------+
  
  

• Working: Voltage & current signals sampled simultaneously. Multiplied digitally (or by analog multiplier) to get instantaneous power. Integrated over time to get energy (kWh). Stored in non-volatile memory.

4.4 Maximum Demand Meter

• Purpose: Records highest average power consumption over a fixed demand interval (e.g., 15 or 30 min).

• Construction & Working:

  • Thermal Type: Two bimetallic strips (heated by current & voltage coils). The hotter strip (corresponding to peak demand) remains deflected until manually reset.

  • Integrator Type: Uses a rotating disc driven by power. A marking pen records peak position on chart. Reset after each interval.

  • Digital Type: Microcontroller stores maximum of integrated power over sliding interval.

4.5 Energy Calculation Problems

• Basic: $$\displaystyle \text{Energy (kWh)} = \text{Power (kW)} \times \text{Time (h)} $$.

• From Meter Reading: $$\displaystyle \text{Energy} = \frac{\text{Disc Revolutions}}{\text{Meter Constant (rev/kWh)}} $$.

• From Wattmeter & Time: $$\displaystyle \text{Energy (kWh)} = \frac{\text{Wattmeter Reading (W)} \times \text{Time (s)}}{3600 \times 1000} $$.

5.0 GALVANOMETERS & CHARGE MEASUREMENT

5.1 Ballistic Galvanometer (BG)

• Constructional Details vs. d'Arsonval:

  • Heavy Coil & High Inertia: To have long periodic time ($$\displaystyle T_d $$).

  • Low Control Torque: Weak springs (or no springs in some designs).

  • Very Light Damping: Often no damping or very critical damping to avoid affecting first swing.

  • Long Periodic Time: $$\displaystyle T_d $$ typically 5-10 seconds. Ensures most energy goes into first swing.

• Equation of Motion:

  $$\displaystyle I \frac{d^2\theta}{dt^2} + G \frac{d\theta}{dt} + C \theta = T_{\text{emf}} $$

  where $I$ = moment of inertia, $G$ = damping constant, $C$ = control constant, $$\displaystyle T_{\text{emf}} = N I \cdot \text{flux} $$.

  For ballistic (current pulse of short duration $dt$), $$\displaystyle T_{\text{emf}} $$ is an impulse: $$\displaystyle \int T_{\text{emf}} dt = N \cdot \text{charge} \ (Q) \cdot \text{flux} $$.

  Solution for first maximum swing $$\displaystyle \theta_0 $$:

$$\theta_0 = \frac{N \phi Q}{C} \cdot \frac{1}{\sqrt{1 - \zeta^2}}$$

where $$\displaystyle \zeta = \frac{G}{2\sqrt{IC}} $$ (damping ratio). For **light damping** ($\zeta \ll 1$), $$\displaystyle \theta_0 \propto Q $$.

• Proof that Charge ∝ First Swing:

  Under light damping, the first swing $$\displaystyle \theta_{\text{max}} $$ is maximum deflection. The initial kinetic energy imparted by the charge is dissipated by damping and control torque. For $\zeta \le 0.2$, the factor $$\displaystyle \frac{1}{\sqrt{1-\zeta^2}} \approx 1 $$. Hence, $$\displaystyle Q \propto \theta_{\text{max}} $$.

  • Ballistic Constant, $$\displaystyle k_B = \frac{\theta_{\text{max}}}{Q} $$ (determined by calibration).

• Calculation Problems:

  • Logarithmic Decrement: $$\displaystyle \delta = \ln \frac{\theta_1}{\theta_2} $$ (successive swings).

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

  • Periods: Damped $$\displaystyle T_d = \frac{2\pi}{\omega_d} $$, Undamped $$\displaystyle T_n = \frac{2\pi}{\omega_n} $$.

  • $$\displaystyle \omega_d = \omega_n \sqrt{1-\zeta^2} $$, where $$\displaystyle \omega_n = \sqrt{C/I} $$.

  • Given $$\displaystyle \theta_1, \theta_2, T_d $$, find $\zeta$, $$\displaystyle T_n $$, then $C$, $G$, etc.

5.2 Flux Meter

• Construction: Similar to BG but with no controlling spring (or very weak) and high inertia. Damping often provided by a conducting frame (eddy current).

• Comparison with Ballistic Galvanometer:

5.3 Damped Oscillations in Galvanometers

• Logarithmic Decrement: $$\displaystyle \delta = \ln \frac{\theta_n}{\theta_{n+1}} $$ (ratio of successive amplitudes).

• Damping Ratio ($\zeta$): $$\displaystyle \zeta = \frac{G}{2\sqrt{IC}} $$. $$\displaystyle \zeta < 1 $$: underdamped (oscillatory); $$\displaystyle \zeta = 1 $$: critically damped; $$\displaystyle \zeta > 1 $$: overdamped.

• Undamped Natural Frequency: $$\displaystyle \omega_n = \sqrt{\frac{C}{I}} $$ (rad/s), $$\displaystyle T_n = \frac{2\pi}{\omega_n} $$.

• Damped Natural Frequency: $$\displaystyle \omega_d = \omega_n \sqrt{1-\zeta^2} $$, $$\displaystyle T_d = \frac{2\pi}{\omega_d} $$.

6.0 RESISTANCE MEASUREMENT BRIDGES

6.1 Wheatstone Bridge

• Balance Condition & Derivation:

  Bridge: $$\displaystyle R_1, R_2 $$ (ratio arms), $$\displaystyle R_3 $$ (known variable), $$\displaystyle R_x $$ (unknown). Galvanometer between junctions.

  At balance: $$\displaystyle I_g = 0 $$, so $$\displaystyle V_{BC} = V_{DC} $$.

$$\frac{R_2}{R_1} = \frac{R_3}{R_x} \quad \text{or} \quad \boxed{R_x = \frac{R_2}{R_1} \cdot R_3}$$

• Applications: Medium resistances ($1 \Omega$ to $1 \text{M}\Omega$).

• Sources of Error & Minimization:

  • Contact & Lead Resistance: Significant for low $$\displaystyle R_x $$. Use thick, short leads; four-terminal (Kelvin) connections for $$\displaystyle R_x $$.

  • Thermoelectric EMFs: Use AC source (induction regulator) or reverse supply & average readings.

  • Bridge Unbalance Sensitivity: Choose ratio arms $$\displaystyle R_1/R_2 \approx \sqrt{R_x/R_3} $$ for maximum sensitivity.

  • Ratio Arm Errors: Use high-precision, matched ratio arms.

  • Self-Heating: Use low current, measure quickly.

6.2 Kelvin's Double Bridge (Thomson Bridge)

• Why Superior for Low Resistances (<1 Ω)? Eliminates effects of contact resistance ($r$) and lead resistance ($r'$) by using a four-terminal (Kelvin) connection for the unknown and standard resistors.

• Detailed Theory & Derivation:

  DiagramCANVAS: Show Kelvin bridge with outer ratio arms P, Q; inner ratio arms p, q; link r; unknown X (with four terminals); standard S. Current source across A-C, galvanometer between B-D.

  At balance ($$\displaystyle I_g=0 $$), potential at B = D.

$$\frac{P}{Q} = \frac{p}{q} = \frac{X + r}{S + r}$$

But $r$ is unknown. To eliminate $r$, inner ratio arms are chosen such that $$\displaystyle p/q = P/Q $$.

Then, 

$$\boxed{X = \frac{P}{Q} \cdot S}$$

**Guarding Technique:** For very low $X$, a guard terminal on $X$ is connected to the junction of $p$ and $q$ to prevent leakage currents from affecting measurement.

• Calculation Problems: Given $P, Q, p, q, S, r$, compute $X$ using exact balance equation if $p/q \neq P/Q$.

7.0 MOVING IRON INSTRUMENTS

7.1 Construction & Principle

• Construction:

  • Fixed Coil: Carries current to be measured.

  • Moving Iron: Soft iron piece (attraction or repulsion type) pivoted near coil.

  • Control: Hair-springs.

  • Damping: Eddy current (aluminum disc on spindle moving in magnetic field).

  • Scale: Non-linear, cramped at low end.

• Torque Equation Derivation (AC & DC):

  • Flux in coil $\phi \propto I$ (or $i$).

  • Force on iron $$\displaystyle F \propto \frac{d}{dx}(\phi^2) $$ (attraction) or $$\displaystyle \frac{d}{dx}(\phi_1 \phi_2) $$ (repulsion).

  • Deflecting torque $$\displaystyle T_d \propto I^2 $$ (for both AC & DC, since $\phi \propto I$ and $$\displaystyle I^2 $$ average is same).

  • Restoring torque $$\displaystyle T_r \propto \theta $$ (spring).

  • At equilibrium: $$\displaystyle T_d = T_r \Rightarrow \theta \propto I^2 $$.

• Characteristics & Applications:

  • Scale: Non-linear ($$\displaystyle \theta \propto I^2 $$). More cramped at low currents.

  • True RMS Measurement: Since $$\displaystyle T_d \propto I^2 $$, deflection $$\displaystyle \propto I_{\text{rms}}^2 $$ independent of waveform.

  • Use as: Ammeter (low resistance shunt), Voltmeter (high series resistor), Wattmeter (with separate voltage coil).

8.0 MAGNETIC MATERIAL TESTING

8.1 B-H Curve (Hysteresis Loop) Determination

• Method of Reversals (Step-by-Step):

  1. Setup: Specimen (toroidal core or bar) wound with primary (N₁) and secondary (N₂) coils. Primary connected to variable AC supply via voltmeter & ammeter. Secondary open-circuited, connected to ballistic galvanometer (BG).

  2. Procedure:

     • Apply small current $I$, reverse it suddenly. Charge $Q$ flows through BG $\Rightarrow$ first swing $$\displaystyle \theta_1 $$ gives $\Delta B$ (since $Q \propto \Delta B$).

     • Increase $I$ stepwise, reverse at each step. Record $$\displaystyle \theta_1 $$ for each reversal.

     • For each $I$, calculate $$\displaystyle H = \frac{N_1 I}{l} $$ (magnetizing force).

     • For each reversal, $$\displaystyle \Delta B = k_B \cdot \theta_1 $$ (ballistic constant $$\displaystyle k_B $$ from calibration). Cumulative sum gives $B$.

  3. Plotting: $B$ vs $H$ gives hysteresis loop. Repeat for decreasing $H$ to complete loop.

• Six-Point Method:

  Procedure to determine key points: $$\displaystyle B_r $$ (retentivity), $$\displaystyle H_c $$ (coercivity), $$\displaystyle B_s $$ (saturation), initial permeability.

  1. Apply small $I$, note $$\displaystyle \theta_1 $$ $\Rightarrow$ point on initial curve.

  2. Increase $I$ to saturation, note $$\displaystyle \theta_1 $$ $\Rightarrow$ $$\displaystyle B_s $$.

  3. Reduce $I$ to zero, note $$\displaystyle \theta_1 $$ $\Rightarrow$ $$\displaystyle B_r $$.

  4. Reverse $I$ to small value, note $$\displaystyle \theta_1 $$ $\Rightarrow$ point on lower branch.

  5. Increase reversed $I$ to saturation, note $$\displaystyle \theta_1 $$ $\Rightarrow$ $$\displaystyle -B_s $$.

  6. Reduce reversed $I$ to zero, note $$\displaystyle \theta_1 $$ $\Rightarrow$ $$\displaystyle -B_r $$.

  7. Plot all points, draw smooth hysteresis loop.

8.2 Iron Loss Measurement

• Lloyd Fischer Square Method:

  • Construction: Two identical laminated E-core stampings, stacked to form two similar magnetic circuits. Primary winding on one limb, secondary on other. Wattmeter connected in primary circuit.

  • Working:

    1. Total Iron Loss ($$\displaystyle W_t $$): With both cores stacked (low reluctance path), apply rated $V$, read wattmeter $$\displaystyle W_t = P_{\text{core}} + P_{\text{eddy}} $$.

    2. Eddy Current Loss ($$\displaystyle W_e $$): Separate cores (air gap), apply same $V$, read $$\displaystyle W_e $$ (only eddy loss, as hysteresis loss negligible due to low $B$ in air gap).

    3. Hysteresis Loss ($$\displaystyle W_h $$): $$\displaystyle W_h = W_t - W_e $$.

  • Advantage: Separates hysteresis and eddy current losses at rated flux density.

• Wattmeter Method for Strip Steel (Epstein Frame):

  • Standardized frame holding 25 cm × 5 cm strips (1 kg total, 10 strips per layer, 2 layers).

  • Primary & secondary windings around frame.

  • Apply rated voltage, measure total iron loss with wattmeter.

  • Loss per kg = Wattmeter reading / weight (W/kg).

9.0 OTHER SPECIALIZED INSTRUMENTS & METHODS

9.1 Frequency Meter

• Block Diagram & Explanation:

  
  Input Signal (f) --> [Frequency-to-Voltage Converter (FVC)] --> [DC Voltmeter / Digital Display]
  
  

  • FVC Types:

    • Resonant: Tuned circuit (LC) whose voltage varies with $f$.

    • Weston (Synchroscope): Two permanent magnet moving coil meters (one for $f$, one for phase).

    • Digital: Counter (counts cycles in fixed time gate) or period measurement.

9.2 Ratio Meter

• Principle: Measures ratio of two quantities (e.g., $$\displaystyle I_1/I_2 $$ in CT testing, $$\displaystyle V_1/V_2 $$ in PT testing). Often uses a center-zero galvanometer in a balanced bridge circuit. Deflection proportional to ratio deviation.

9.3 Digital Voltmeter (DVM)

• Block Diagram:

  
  Input V --> [Input Amplifier/Attenuator] --> [Sample & Hold Circuit] --> [Analog-to-Digital Converter (ADC)] --> [Digital Display]
  
  

  • Working: Sample & Hold captures instantaneous voltage. ADC converts to digital code (e.g., dual-slope, successive approximation). Display shows value.

9.4 Earth Resistance Measurement

• Methods:

  • Three-Point (Fall-of-Potential):

    1. Earth electrode (E) under test.

    2. Current electrode (C) driven far away (>> electrode depth).

    3. Potential electrode (P) placed midway between E & C.

    4. Measure voltage $$\displaystyle V_{EP} $$ and current $$\displaystyle I_{EC} $$ $$\displaystyle \Rightarrow R_e = V_{EP}/I_{EC} $$.

  • Four-Point (Four-Terminal): Eliminates lead & contact resistances. Two current electrodes, two potential electrodes.

  • Clamp-on (Earth Tester): Uses current transformer principle. Clamp around earth conductor, measures resistance without disconnecting.

• Earth Tester Construction: Hand-cranked or electronic generator (AC voltage, e.g., 25-50 Hz), voltmeter, ammeter (or impedance meter).

9.5 Megger (Insulation Tester)

• Use & Principle: Measures insulation resistance (MΩ, GΩ) of cables, transformers, motors.

• Hand-driven/Electronic: Generates high DC voltage (500V, 1000V, 2500V, 5000V).

• Working: Applies high $$\displaystyle V_{DC} $$ across insulation, measures resulting leakage current $I$. $$\displaystyle R_{\text{insul}} = V_{DC} / I $$.

• Advantage: High voltage stresses insulation to reveal weaknesses.

9.6 Loss of Charge Method

• For Measurement of Insulation Resistance of cables, capacitors, bushings.

• Circuit: Unknown insulation resistance $$\displaystyle R_x $$ in parallel with a known capacitor $C$ of high quality. Initially charged to voltage $$\displaystyle V_0 $$ through a switch. At $$\displaystyle t=0 $$, switch connects $$\displaystyle R_x \parallel C $$ to a voltmeter (or ballistic galvanometer).

• Calculation:

  $$\displaystyle V(t) = V_0 e^{-t/(R_x C)} $$

  Plot $\ln V$ vs $t$, slope $$\displaystyle = -1/(R_x C) $$.

  If voltmeter resistance $$\displaystyle R_v $$ is finite:

$$\boxed{R_x = \frac{R_v \cdot t}{C \cdot \ln(V_0/V) - t}}$$

or using time for voltage to drop from $$\displaystyle V_1 $$ to $$\displaystyle V_2 $$:

$$R_x = \frac{t}{C \ln(V_1/V_2)} - R_v \quad \text{(if $$\displaystyle R_v $$ known)}$$

10.0 RANGE EXTENSION & INSTRUMENT ADAPTATION

10.1 Ammeter Range Extension

• Shunt Connection: Connect low resistance $$\displaystyle R_{sh} $$ in parallel with meter.

• Shunt Calculation:

$$I = I_m + I_{sh}, \quad I_m R_m = I_{sh} R_{sh}$$

$$\boxed{R_{sh} = \frac{I_m R_m}{I - I_m}}$$

where $$\displaystyle I_m, R_m $$ = full-scale meter current & resistance.

10.2 Voltmeter Range Extension

• Series Multiplier Resistance: Connect high resistance $$\displaystyle R_s $$ in series with meter.

• Multiplier Calculation:

$$V = I_m (R_m + R_s)$$

$$\boxed{R_s = \frac{V}{I_m} - R_m}$$

10.3 Multi-Range Voltmeter

• Construction: Multiple series resistors ($$\displaystyle R_{s1}, R_{s2}, ... $$) with a range switch. Switch selects different series combinations to achieve different full-scale voltages $$\displaystyle V_1, V_2, ... $$.

10.4 Adapting a Moving Coil Meter (Example)

• Given: $$\displaystyle I_m = 5 \text{mA} $$, $$\displaystyle R_m = 10 \ \Omega $$.

• As 0-10A Ammeter:

  $$\displaystyle I = 10 \text{A}, I_m = 0.005 \text{A} $$

  $$\displaystyle R_{sh} = \frac{0.005 \times 10}{10 - 0.005} \approx 0.005 \ \Omega $$.

• As 0-100V Voltmeter:

  $$\displaystyle V = 100 \text{V}, I_m = 0.005 \text{A} $$

  $$\displaystyle R_s = \frac{100}{0.005} - 10 = 20000 - 10 = 19990 \ \Omega $$.

11.0 MISCELLANEOUS TOPICS (Recurring Short Notes)

11.1 Testing and Calibration of Instruments

• Purpose: Ensure accuracy, traceability to standards.

• Methods:

  • Comparison: Compare with a standard instrument of higher accuracy.

  • Direct Calibration: Use known sources/loads (e.g., precision voltage source, standard resistor).

  • End-to-End: Apply known input, measure output across full range.

  • For Energy Meters: Full-load test, phantom loading, creep test, light-load test.

  • For CT/PT: Ratio & phase angle tests using standard transformers and precision instruments.

11.2 PMMC Instruments

• Construction: Permanent magnet, moving coil (PMMC), hair-springs, aluminum frame (eddy damping), linear scale.

• Principle: $$\displaystyle T_d = N I \phi \sin \theta $$, $$\displaystyle \phi = \text{constant} $$ (PM) $$\displaystyle \Rightarrow T_d \propto I $$. Spring torque $$\displaystyle T_r \propto \theta $$ $\Rightarrow$ linear scale.

• Advantages: High sensitivity, low power consumption, linear scale, accuracy.

• Limitations: DC only, fragile, expensive.

11.3 Eddy Current Damping (Detailed)

• Principle: When a conductor moves in a magnetic field, eddy currents are induced. By Lenz's law, these currents create a magnetic field opposing the motion $\Rightarrow$ damping force $F \propto v$.

• Application:

  • Wattmeters & Energy Meters: Aluminum disc moves in permanent magnet field.

  • Analog Meters: Aluminum vane or disc in magnet gap.

• Advantage: Smooth, proportional damping, no friction.

11.4 Use of Megger

• Applications:

  • Insulation resistance of cables, transformers, motors, generators.

  • Earth resistance (some models).

  • Continuity testing (low voltage range).

• Procedure: Connect megger terminals to device under test (one to conductor, other to ground/insulation). Hand-crank or electronic switch to generate high $$\displaystyle V_{DC} $$. Read resistance directly.

• Precautions: Ensure device disconnected from supply, discharged. Test at rated voltage.

11.5 Stray Magnetic Field Errors

• In Electrodynamometer Wattmeter: External magnetic fields (from other equipment, earth's field) distort the main field between fixed coils $\Rightarrow$ torque error.

• Mitigation:

  1. Magnetic Shielding: Enclose instrument in iron case (high permeability path for stray flux).

  2. Positioning: Place instrument away from stray fields.

  3. Compensation: Use compensating windings (but shielding is primary method).