Electrical Measurements and Instruments (EX-303) - Unit 5 Short Notes

UNIT 5: ELECTRICAL MEASUREMENTS AND INSTRUMENTS

I. FUNDAMENTALS OF MEASURING INSTRUMENTS & ERROR ANALYSIS

Static & Dynamic Characteristics

Static Characteristics (describe performance under steady-state):

• Accuracy: Closeness of measured value to true value.

• Precision: Degree of reproducibility (repeatability & reproducibility).

• Sensitivity: Ratio of output scale deflection to input quantity (S = Δθ / Δx).

• Resolution: Smallest detectable change in input.

• Threshold: Minimum input to produce detectable output.

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

• Drift: Slow change in output over time with constant input.

• Staircase effect: Discrete steps in digital instrument output.

• Fidelity: Ability to follow rapid changes (dynamic).

• Lag: Delay in response to input change.

• Speed of response: Time taken to reach steady-state.

[!TIP] Exam Focus: Be ready to define each term and distinguish between accuracy & precision, sensitivity & resolution.

Classification of Errors

1. Gross Errors: Human mistakes (parallax, misreading). Minimized by careful observation.

2. Systematic Errors: Consistent, predictable.

   • Instrumental: Defects in instrument (zero error, calibration).

   • Environmental: Temperature, humidity, magnetic fields.

   • Observational: Parallax, incorrect adjustment.

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

• Limiting Error (Guarantee Error): Specified max error (±a% of FSD or ±b% of reading).

• Probable Error: Error exceeded in 50% of measurements (statistical).

Error Propagation & Calculation

For a measured quantity Z = f(x, y, ...):

• Absolute Error: ΔZ ≈ |∂f/∂x|Δx + |∂f/∂y|Δy + ...

• Relative Error: δZ = ΔZ / Z

• Correction: Correction = -Error

Example: Power P = V * I

If V has error ΔV and I has ΔI:

ΔP ≈ IΔV + VΔI

δP ≈ δV + δI

[!TIP] Common Pitfall: For product/quotient, relative errors add. For power P = I²R or V²/R, error doubles for squared term.

Loading Effect

• Voltmeter Loading: Voltmeter in parallel draws current. Effective resistance R_eff = R_circuit || R_v. If R_v is not >> R_circuit, reading < true voltage.

• Ammeter Loading: Ammeter in series adds its internal resistance R_a. If R_a is not << R_circuit, reading < true current.

Example: Multimeter with sensitivity S Ω/V on V range has R_v = S * V_range. Loading error = (V_true - V_read)/V_true.

II. ANALOG INSTRUMENT TYPES & OPERATING PRINCIPLES

Moving Coil (PMMC) Instruments

Construction: Permanent magnet, moving coil (rectangular, many turns), control spring, damping (eddy current), linear scale. Principle: D'Arsonval movement. Current I in coil in magnetic field B produces torque T_d = B * I * (N * A) = K * I (linear). Applications:

• Ammeter: Shunt resistance R_sh in parallel. I = I_m (1 + R_m/R_sh).

• Voltmeter: Series multiplier R_s. V = I_m (R_m + R_s).

Range Extension Formulas:

• Ammeter: \boxed{R_{sh} = \frac{I_m R_m}{I - I_m}}

• Voltmeter: \boxed{R_s = \frac{V}{I_m} - R_m}

where I_m = full-scale current, R_m = meter resistance.

[!TIP] PMMC works only for DC. For AC, need rectifier (but then measures average, calibrated for RMS of sine).

Moving Iron Instruments

Types: Attraction (single iron vane) & Repulsion (two vanes). Principle: Magnetic field from current-carrying coil magnetizes iron, causing attraction/repulsion. Torque T_d ∝ I² (or V² for voltmeter). Works for AC & DC. Torque Derivation:

T_d ∝ (magnetic flux)² ∝ (N * I)² → T_d = K * I². Scale: Non-linear (cramped at low end). Damping: Air friction (vane in air chamber) or eddy current (aluminum disc).

Electrodynamometer Type Instruments

Construction:

• Fixed coils (current coils, heavy wire, in series).

• Moving coil (voltage/potential coil, many turns, with series resistor).

• C-shaped core for magnetic shielding.

• Control spring, air damping. Torque Equation:

T_d = K * I₁ * I₂ * cos(θ) where I₁ = fixed coil current, I₂ = moving coil current, θ = phase angle.

• For DC: θ=0, T_d = K * I₁ * I₂.

• For AC: I₁ = I₁m sin(ωt), I₂ = I₂m sin(ωt + θ). Average torque T_d(avg) = K * I₁ * I₂ * cos(θ). True RMS Measurement: If moving coil circuit is purely resistive, I₂ ∝ V (voltage across it). With fixed coil in series with load, I₁ = I_load. Then T_d ∝ V * I * cos(θ) = Real Power. For non-sinusoidal, V and I are instantaneous, but torque proportional to instantaneous product v*i. Average torque ∝ average of v*i = true RMS power only if both coils respond to instantaneous values (moving coil has high inductance? No, must be designed with low inductance for true RMS). Actually, for arbitrary waveform, if both coils have same time constant? Standard derivation: T_d ∝ i₁ * i₂. If moving coil circuit is resistive, i₂ ∝ v. Then T_d ∝ i * v. Average T_d ∝ (1/T)∫ v i dt = true average power. So yes, measures true power irrespective of waveform provided moving coil circuit is purely resistive (or appropriately compensated).

Scale: Linear (spring controlled). Applications: Ammeter (fixed coil in series), Voltmeter (fixed coil in series with multiplier), Wattmeter (fixed coil = current coil, moving coil = voltage coil). Errors & Compensation:

1. Stray magnetic field errors: Shield with iron case (magnetic shielding).

2. Eddy current errors: Use non-conductive materials or slit metal parts.

3. Temperature errors: Use manganin for resistors, temperature-compensated springs.

4. Low Power Factor Wattmeter Compensation:

   • Current coil inductance: Causes I to lag V → reading low. Compensation: Add capacitor in parallel with current coil to cancel inductance.

   • Pressure coil inductance & resistance: Pressure coil has inductance L_p and resistance R_p. Effective impedance Z_p = R_p + jωL_p. Current I_p lags voltage V by φ_p = tan⁻¹(ωL_p/R_p). Torque T_d ∝ I * I_p * cos(θ - φ_p). Since φ_p > 0, reading low. Compensation: Use capacitor in series with pressure coil to make its circuit resistive (resonance) OR use swamping resistor (low temp coeff) to make R_p >> ωL_p so φ_p ≈ 0.

III. POWER & ENERGY MEASUREMENT

Single-Phase Electrodynamometer Wattmeter

Construction: As above. Current coil (fixed) in series with load. Voltage coil (moving) with high series resistor across supply. Working Principle: T_d ∝ I_load * V * cos(θ) where θ = phase angle between load I and V. Phasor Diagram:

• V reference.

• I at angle θ (lagging/leading).

• Current through voltage coil I_v in phase with V (if purely resistive).

• T_d ∝ |I| * |I_v| * cos(θ). Torque Derivation (Ferrodynamic):

Instantaneous torque: t_d = K * i₁(t) * i₂(t).

i₁ = √2 I sin(ωt) (load current)

i₂ = √2 I_v sin(ωt) (voltage coil current, in phase with v)

t_d = 2K I I_v sin²(ωt) = K I I_v [1 - cos(2ωt)]

Average torque: T_d = K I I_v (since average of cos(2ωt) = 0).

But I_v = V / Z_v where Z_v is impedance of voltage coil circuit. If resistive, I_v ∝ V. So T_d ∝ V I cos(θ)? Wait, careful: I_v is in phase with V, so I_v = V / R_v (if pure resistance). Then T_d = K (V/R_v) I. But this is V I, not V I cos(θ). Where does cos(θ) come? The torque is proportional to product of instantaneous currents i₁ * i₂. i₁ has phase θ relative to V. i₂ is in phase with V. So i₁ = √2 I sin(ωt + θ), i₂ = √2 I_v sin(ωt). Then:

t_d = 2K I I_v sin(ωt+θ) sin(ωt) = K I I_v [cos(θ) - cos(2ωt+θ)]

Average T_d = K I I_v cos(θ). Yes.

So final: \boxed{T_d = K V I \cos(\theta)} (with K including 1/R_v).

Errors:

• Stray magnetic fields: Shield.

• Inductance of coils: Current coil inductance causes I₁ to lag V; pressure coil inductance causes I_v to lag V. Both reduce cos(θ) term → reading low. Compensate as above.

• Friction & creep: Bearing friction, magnetic creep (non-linear scale at low PF).

• Connection errors: Voltage coil should be connected across supply, independent of current coil connection (for 3-phase, potential coil may be connected to neutral or line).

Three-Phase Power Measurement

Two-Wattmeter Method:

• Connection: Two wattmeters, each current coil in two lines, potential coils between respective line and third line.

• Proof: Total power P_total = Re[V_a I_a* + V_b I_b* + V_c I_c*]. Using symmetrical voltages and I_a + I_b + I_c = 0, can show P_total = W1 + W2.

• Phasor Diagram: For balanced load, W1 = V_L I_L cos(30° - φ), W2 = V_L I_L cos(30° + φ). Sum = √3 V_L I_L cos φ.

• One Wattmeter Negative: When φ > 60° (lagging PF < 0.5) or φ < -60° (leading PF < 0.5). One wattmeter reads negative (need to reverse potential coil connection or note sign).

Three-Wattmeter Method: For 3-phase, 4-wire (with neutral). Each wattmeter between a line and neutral. P_total = W1 + W2 + W3.

Three-Element Dynamometer Wattmeter: For 3-phase, 3-wire. Three separate wattmeter elements on common shaft, each with current coil in one line and potential coil between that line and the other two lines? Actually, for 3-wire, each element's potential coil connected between its line and the virtual neutral? Standard construction: Three units, each current coil in one line, potential coils connected line-to-line? Wait, for 3-wire balanced/unbalanced, three-element meter has each element's current coil in one line, and potential coils connected between that line and the common point of the other two potential coils? Actually, in a three-element integrator type (energy meter), each element is like a single-phase wattmeter with current coil in line and potential coil between that line and neutral? But for 3-wire no neutral, the potential coils are connected in open delta? I think for three-element dynamometer wattmeter (not energy meter), it's used for 3-phase, 4-wire. For 3-wire, two-element is sufficient. But blueprint says "three-element dynamometer wattmeter construction" for Jun 2025. Possibly for 3-phase, 4-wire with neutral current measurement? Actually, three-element wattmeter measures total power in 3-phase, 4-wire by having three separate wattmeters, each between a line and neutral. So construction: three independent wattmeter mechanisms on one shaft.

Single-Phase Induction (Energy) Meter

Construction:

1. Driving System:

   • Voltage Magnet: Series high inductance coil (many turns) across supply. Shunt resistor? Actually, voltage coil is laminated, high inductance, low resistance.

   • Current Magnet: Series low inductance coil (few turns) in line. Has shading band (copper) to create phase shift θ between φ_v and φ_I.

2. Moving System: Aluminum disc on spindle, in air gap of both magnets.

3. Braking System: Permanent magnet (damping) near disc; eddy currents in disc produce braking torque T_b ∝ ω (speed).

4. Registering System: Gear train counting disc revolutions.

Working Principle:

• Flux φ_v from voltage magnet (in phase with V? Actually, voltage coil has high inductance, so I_v lags V by ~90°, but flux φ_v in phase with I_v? In voltage magnet, flux φ_v ∝ I_v. Since I_v lags V by ~90°, φ_v lags V by ~90°.

• Flux φ_I from current magnet (in phase with I).

• Shading band on current magnet causes φ_I to lag I by angle α (typically ~60-70°).

• Net torque: T_d ∝ φ_v * φ_I * cos(β) where β = phase angle between φ_v and φ_I. But φ_v lags V by 90°, φ_I lags I by α. So phase between φ_v and φ_I = (90° + θ) + α? Let's derive properly:

  Let V = V_m sin ωt.

  Current through voltage coil: I_v = (V_m/X_v) sin(ωt - 90°) (since high inductance, X_v >> R_v). So φ_v ∝ I_v ∝ sin(ωt - 90°).

  Load current: I = I_m sin(ωt - θ).

  Current through current magnet: I_c = I (almost, since low resistance). But shading band causes flux φ_I to lag I_c by α. So φ_I ∝ sin(ωt - θ - α).

  Torque T_d ∝ φ_v * φ_I * cos(phase difference)? Actually, torque in induction meter is due to interaction of two fluxes with eddy currents in disc. The net torque is proportional to the product of the two fluxes and the sine of the phase angle between them? Wait, standard formula: T_d ∝ φ_v φ_I sin(δ) where δ is phase angle between fluxes? But for wattmeter, it's cos(θ). For energy meter, the torque is T_d ∝ φ_v φ_I cos(θ - φ)? I recall: T_d = K φ_v φ_I cos(θ - φ) where φ is phase angle of voltage coil circuit? Let's check standard text:

  In single-phase induction meter, the driving torque is proportional to the product of the fluxes and the cosine of the angle between them. But the fluxes are not in phase with V and I. The voltage flux φ_v lags V by nearly 90°, current flux φ_I lags I by α. The angle between φ_v and φ_I is (90° + θ + α)? Then cos(90° + θ + α) = -sin(θ+α). That would give negative torque? Hmm.

  Actually, correct derivation: The torque is proportional to the product of the two fluxes and the sine of the angle between them? No, for a single-phase induction motor, the torque is proportional to φ_f φ_r sin(δ) where δ is the phase difference between main and auxiliary fluxes. But here, both fluxes are alternating. The net torque is the average of the product of the instantaneous torques. The instantaneous torque is proportional to φ_v(t) * φ_I(t) * cos(θ_vI(t))? Better to use standard result:

  T_d = K φ_v φ_I cos(θ - φ) where φ is the phase angle of the voltage coil circuit? I think the standard formula is:

  T_d ∝ φ_v φ_I cos(θ - φ) where φ is the phase angle by which φ_v lags V? Actually, many texts give: T_d ∝ V I cos φ (true power). So the meter is designed so that θ (load power factor angle) appears.

  Let's derive properly:

  Let V = V_m sin ωt.

  Voltage coil current: I_v = (V_m / Z_v) sin(ωt - φ_v) where φ_v is impedance angle of voltage coil circuit (should be ~90° if highly inductive).

  Flux φ_v ∝ I_v = Φ_v sin(ωt - φ_v).

  Load current: I = I_m sin(ωt - θ).

  Current coil flux: φ_I ∝ I = Φ_I sin(ωt - θ - α) where α is lag due to shading band.

  The torque on the disc is due to interaction of φ_v with eddy currents induced by φ_I, and vice versa. The average torque is:

  T_d ∝ Φ_v Φ_I cos(φ_v - (θ+α))? Actually, the angle between φ_v and φ_I is (φ_v) - (θ+α)? But φ_v is lag of voltage flux behind V, and θ+α is lag of current flux behind V. So difference = φ_v - (θ+α). Then T_d ∝ Φ_v Φ_I cos(φ_v - θ - α).

  For correct measurement, we want T_d ∝ V I cos θ. Since Φ_v ∝ V/Z_v and Φ_I ∝ I, we need cos(φ_v - θ - α) ∝ cos θ. This requires φ_v - α = 0 or φ_v = α. So the phase shift α of current flux is made equal to the phase angle φ_v of voltage coil circuit. Typically, φ_v ≈ 90° (highly inductive voltage coil), so α should be 90°? But shading band gives about 60-70°. Actually, in practice, the voltage coil has a power factor of about 0.2-0.3 lagging, so φ_v ≈ 73° (cos⁻¹0.3). Shading band adjusted to give α ≈ φ_v. Then φ_v - α ≈ 0, so T_d ∝ Φ_v Φ_I cos(-θ) = Φ_v Φ_I cos θ. And Φ_v ∝ V, Φ_I ∝ I, so T_d ∝ V I cos θ. Correct.

Braking Torque: T_b ∝ ω (permanent magnet eddy current damping). At steady speed, T_d = T_b → ω ∝ V I cos θ → disc speed proportional to power. Revolutions proportional to energy.

Errors & Adjustments:

1. Friction: Compensated by lightening disc or initial adjustment.

2. Creep: Slow rotation at no-load (due to residual magnetism, friction, stray torque). Adjusted by creep adjustment screw (shading band position).

3. Stray magnetic field: Shielding.

4. Temperature: Use materials with low temp coefficient.

5. Voltage variation: Voltage coil flux φ_v ∝ V. If V changes, φ_v changes, but φ_I also changes? Actually, φ_I ∝ I, but I depends on load. At constant power, if V increases, I decreases. So φ_v φ_I may not be constant. Error: T_d ∝ V * I but for constant power P = V I cos θ, if V changes, I changes inversely. But T_d ∝ V I, not P if cos θ constant? Actually, T_d ∝ V I cos θ ideally. But if φ_v not exactly proportional to V due to saturation? Or if voltage coil resistance causes I_v not exactly proportional to V? Error is small if voltage coil resistance is small compared to inductive reactance. Power factor error: At low PF, I large but cos θ small. The current coil flux φ_I is proportional to I, but the torque depends on cos(φ_v - θ - α). If φ_v - α ≠ 0, then error depends on θ. So low PF error.

6. Adjustments: (i) Light load: Adjust shading band to minimize creep. (ii) Full load: Adjust brake magnet position to correct speed.

Testing & Calibration:

• Full Load Test: Apply rated V and I at unity PF. Measure time t for N revolutions. Theoretical energy E_th = V I t. Meter reading E_m = (N / K) * (3600 / 1000)? Actually, meter constant K = revolutions per kWh. So energy registered E_reg = (N / K) kWh. Percentage error = (E_reg - E_th)/E_th * 100%.

  Example (Jun 2025): V=230V, I=50A, t=37s, N=61 rev, K=520 rev/kWh.

  E_th = 230*50*(37/3600) = 230*50*0.0102778 = 118.0 Wh? Let's compute: 230*50=11500 W, 37s = 37/3600 h = 0.0102778 h, so E_th = 11500 * 0.0102778 = 118.194 Wh = 0.118194 kWh.

  E_reg = 61 / 520 = 0.117308 kWh.

  Error = (0.117308 - 0.118194)/0.118194 * 100 = -0.746%.

• Light Load Test: Apply low current at unity PF, check disc rotation (should be within limits).

• Precision Testing: Account for errors in standard wattmeter (±a%), stopwatch (±b s), human reaction time (±c s). Total error propagation.

Three-Phase Energy Meter

• Two-Element Type: For 3-phase, 3-wire (or 4-wire with neutral connected to one element?). Actually, two-element meter for 3-wire: each element has current coil in two lines? No, each element has current coil in one line and potential coil between that line and the other line? Standard: For 3-wire, two elements, each with current coil in two different lines? Wait, typical two-element energy meter for 3-phase, 3-wire: Each element's current coil is connected in series with one of the two lines? Actually, for 3-wire, there are three lines but no neutral. The two elements are connected such that:

  Element 1: Current coil in line A, potential coil between A and B.

  Element 2: Current coil in line C, potential coil between C and B? Or between C and A? Actually, common connection: Both potential coils connected between line B and line A? I need to recall: In a two-element meter for 3-wire, the potential coils are connected in series across two lines (say A and C), and each current coil is in one of the other two lines (A and C)? That would be wrong. Standard: For 3-wire, the two elements are connected in "two-wattmeter" fashion:

  Wattmeter 1: Current coil in line A, potential coil between A and B.

  Wattmeter 2: Current coil in line C, potential coil between C and B.

  Then total power = W1+W2. The energy meter integrates both.

• Three-Element Type: For 3-phase, 4-wire (with neutral). Each element: current coil in one line, potential coil between that line and neutral. Total energy = sum of three elements.

Phasor Diagram: Similar to two-wattmeter method.

Digital Electronic Energy Meter

Block Diagram:

Voltage Sensor (PT or resistor divider) → ADC → Microprocessor/IC → Display & Memory

Current Sensor (CT or shunt) → ADC ↗

Principle: Sample v(t) and i(t) simultaneously, multiply instantaneously, average over time to get real power. Integrate power over time for energy. Also measures RMS voltage/current, power factor.

Maximum Demand Meter

Construction: Thermal type (bimetallic) or integrating type with reset mechanism. Working: Records maximum average power over a demand interval (e.g., 30 min). Thermal type: Two bimetallic strips, one responds to instantaneous power, other has thermal lag. Pointer indicates max demand. Integrating type: Motor-driven register that resets after interval, holds max reading.

IV. CURRENT & POTENTIAL TRANSFORMERS

Current Transformer (CT)

Construction:

• Primary: 1 turn (bar) or few turns.

• Secondary: Many turns (to keep I_s low, safe).

• Core: Silicon steel laminations.

• CRITICAL: Never open-circuit secondary when primary current flows. Why? I_p fixed, I_s = I_p / n. If secondary open, I_s=0, then N_p I_p - N_s I_s = N_p I_p = large mmf, drives core into deep saturation, induces very high voltage (kV) across open secondary → insulation breakdown, danger. Equivalent Circuit:

          ┌─── R_m ──── jX_m ────┐

Primary ─┤                       ├─ Secondary

          └───────────┘

• R_m, X_m: Magnetizing branch.

• R_e, X_e: Equivalent resistance & leakage reactance Referred to secondary. Errors:

• Ratio Error: n = N_p/N_s (nominal). Actual I_s = I_p / n - I_e (exciting current component). So I_p / I_s = n + error.

• Phase Angle Error: I_e has component I_m (magnetizing) causing phase shift.

• Effect of Secondary Burden (R_b, X_b): Increased burden → larger I_e to supply I_b → larger ratio & phase errors. CT designed for specific burden (e.g., 15 VA).

Testing: Compare with standard CT, measure ratio & phase angle at rated burden.

Potential Transformer (PT)

Construction: Similar to CT but primary many turns, secondary few turns. Insulation high. Equivalent Circuit (referred to secondary):

Primary ── R_p, X_p ──┐

                    ├── R_m, X_m ── Secondary ── R_s, X_s

Secondary ──────────┘

Errors:

• Ratio Error: Due to voltage drop in R_p, X_p and R_s, X_s.

• Phase Angle Error: Due to reactive components. Minimization: Low burden (low VA burden), high accuracy core, low leakage reactance, proper rating. Testing: Comparison with standard PT, ratio test, polarity test (using AC voltage and voltmeter).

CT vs. PT Comparison

V. SPECIAL PURPOSE INSTRUMENTS & MEASUREMENT TECHNIQUES

Ballistic Galvanometer

Construction: High moment of inertia (large coil, heavy pointer), low damping (suspension, minimal damping vane), undamped/slightly damped period T₀. Equation of Motion:

I d²θ/dt² + G dθ/dt + C θ = 0 where I = moment of inertia, G = damping constant, C = control constant.

For ballistic (very low damping, short pulse), solution: θ_max = (Q / (I ω_d)) * e^{-δ π}? Actually, for first swing maximum, if damping very small, θ_max ≈ Q / (I ω₀) where ω₀ = √(C/I), Q = charge. Derivation:

• Torque from coil: T = N A B I (constant for given I? No, I is pulse, but torque T_d = K I where K = N A B.

• Angular momentum: ∫ T_d dt = K ∫ I dt = K Q = I ω_max (since damping negligible for short pulse).

• But ω_max = ω₀ θ_max for small angles? Actually, from energy: ½ I ω_max² = ½ C θ_max² → ω_max = θ_max √(C/I) = θ_max ω₀.

So K Q = I ω_max = I ω₀ θ_max → θ_max = (K Q) / (I ω₀).

But I ω₀ = √(I C). So θ_max ∝ Q. Key: First throw θ_max proportional to charge Q if damping is negligible. With damping, θ_max = (K Q)/(I ω_d) * e^{-δ π}? Actually, for underdamped, the first maximum occurs at t ≈ π/ω_d and θ_max = (K Q)/(I ω_d) e^{-δ π} where δ = G/(2√(I C)) damping ratio, ω_d = ω₀ √(1-δ²). So to find Q, need δ from logarithmic decrement Λ = ln(θ₁/θ₂) ≈ 2πδ for small δ. Use: Measurement of charge, capacitance (Q = C V), inductance (with known capacitor).

Calculation Example (Jun 2024): Given f=125 Hz → T_d=0.008 s, swings 120, 96, 76.8 mm.

Logarithmic decrement δ = (1/n) ln(θ₁/θ_{n+1})? For successive swings: δ = ln(θ₁/θ₂) = ln(120/96)=0.2231. Then damping ratio ζ = δ / √(4π²+δ²) ≈ δ/(2π) = 0.2231/6.2832=0.0355. Period of undamped T₀ = T_d / √(1-ζ²) ≈ T_d. Actually, T_d = T₀ √(1-ζ²). So T₀ = T_d / √(1-ζ²). Compute.

Flux Meter

Similar to ballistic but with higher damping (moving coil in strong magnetic field). Measures change in flux linkage Δψ = N Δφ. Used with search coil: Δψ = N A ΔB. Calibration constant K = 1/(N A).

Comparison: Ballistic Galvanometer vs Flux Meter

Measurement of Magnetic Properties (B-H Curve)

Method of Reversals (Jun 2025, Dec 2024, Jun 2023):

• Circuit: Specimen (bar) as core of a coil (primary) with N₁ turns, connected to variable AC supply via rheostat. Secondary coil with N₂ turns connected to flux meter (or ballistic galvanometer with integrator? Actually, flux meter directly).

• Procedure:

  1. Set current I (hence H = N₁ I / l).

  2. Reverse current suddenly (using reverse switch). Flux changes from +φ_max to -φ_max, so change Δφ = 2 φ_max.

  3. Flux meter deflection θ ∝ Δψ = N₂ Δφ = 2 N₂ φ_max.

  4. So φ_max = θ / (2 N₂ K_f) where K_f is flux meter constant.

  5. B_max = φ_max / A (A = cross-section).

  6. Repeat for increasing I to get B-H curve.

• Advantages over direct method: Eliminates effect of residual magnetism and hysteresis? Actually, direct method uses integrating ballistics? Method of reversals gives maximum B for given H, which is on the hysteresis loop. It's simpler, avoids need for integrating circuit.

Six-Point Method (Jun 2025):

For symmetrical loop, measure at six points: +B_max, +B_{r}, 0, -B_{r}, -B_max, and +B_{r}? Actually, to construct loop, measure B for increasing H from -H_max to +H_max and decreasing. But six-point method: Choose six values of H (including -H_max, -H_{c}, 0, +H_{c}, +H_max)? Not standard. Possibly: For a symmetrical loop, measure B at H=0 (remanent), H=H_c (coercive), H=H_max (saturation) on both sides. That gives key points to sketch loop.

Lloyd Fischer Square (Jun 2024, Dec 2023):

For measuring iron loss (core loss) in specimen (e.g., transformer steel strips).

• Construction: Square frame with primary winding on one limb, secondary on other limb. Specimen strip forms part of magnetic path.

• Method: Connect primary to supply, secondary open. Wattmeter connected in primary circuit measures core loss (hysteresis + eddy) since secondary open → no load current? Actually, primary current is exciting current, which supplies core loss. Wattmeter reading = core loss (since secondary open, no output power). But need to compensate for copper loss? Primary resistance small, so copper loss negligible. So wattmeter reading ≈ core loss.

• Use: Compare losses for different materials, grades.

Earth Resistance Measurement

Three-Point (Fall-of-Potential) Method (most common):

• Setup: Earth electrode (to be tested), two auxiliary electrodes (potential P and current C) driven into earth at suitable distances.

• Procedure:

  1. Connect C far away (to minimize its resistance effect).

  2. Measure voltage V between earth electrode and P with voltmeter.

  3. Measure current I from earth electrode to C with ammeter.

  4. Earth resistance R_e = V/I.

• Distance: Typically, P at 62% of distance between earth electrode and C to minimize error due to P electrode resistance. Four-Point Method: Use separate current and potential electrodes, eliminates need for correction. Earth Tester: Hand-driven generator (or battery) with voltmeter and ammeter in one instrument, reversing switch to eliminate polarization.

Frequency Meter

Types:

1. Resonant (Tuned Circuit) Type: Ferrodynamic movement. Two tuned circuits (L-C) with different natural frequencies. As frequency changes, pointer moves to balance.

2. Weston Type (Dynamometer): Two fixed coils in series with capacitor and resistor, moving coil. Phase difference changes with frequency → torque changes.

3. Digital (Counter) Type:

   • Block Diagram:

     
     Signal → Conditioning (amp, filter) → Schmitt trigger → Counter (gated for fixed time) → Display
     
     

     Counts zero crossings or cycles in fixed time T. Frequency f = N/T.

Megger (Insulation Tester)

Construction: Hand-cranked or battery-operated DC generator (500V, 1000V, 2500V). Two coils (current & pressure) on moving element. Scale in megohms. Working: Apply high DC voltage to insulation. Leakage current measured. Torque T_d ∝ I * V (constant V) → T_d ∝ I → deflection ∝ V/I = resistance. Use: Measure insulation resistance of cables, transformers, motors. Must disconnect equipment from supply.

VI. BRIDGE CIRCUITS FOR RESISTANCE MEASUREMENT

Wheatstone Bridge

Circuit: Four arms: R1, R2 (ratio arms), R3 (known variable), R4 (unknown R_x). Galvanometer between junctions. Balance Condition: R1/R2 = R3/R4 → R_x = R4 = (R2/R1) * R3. Used for: Medium resistances (~1 Ω to 1 MΩ). Errors & Minimization:

1. Thermoelectric EMFs: Use AC source or reverse connections.

2. Contact resistances: Use four-terminal connections for R_x and R3? Actually, for medium resistances, contact resistance in series with R_x can be significant if R_x is low. But Wheatstone not for very low. For medium, ensure good contacts.

3. Lead resistances: Use Kelvin connections? Not typically for Wheatstone; for low resistances use Kelvin bridge.

4. Detector sensitivity: Use sensitive galvanometer.

5. Ratio error: Use high-precision ratio arms.

Kelvin (Thomson) Double Bridge

Construction:

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

• Known standard S, unknown R_x.

• Link resistance r between R_x and S (four-terminal connection).

• Galvanometer between the junction of P-Q and p-q. Circuit Diagram:

     P ────┬──── Q

           │

     R_x ─┼──── S

           │    (with link r)

     p ────┴──── q

Derivation of Balance Equation:

Let current through R_x and S be I₁, through P and Q be I₂, through p and q be I₃.

At balance, galvanometer current = 0 → potential at A = potential at B.

V_A = I₁ R_x + I₃ p

V_B = I₁ (S + r) + I₃ q

Also, I₂ = I₁ - I₃ (from current at node).

V_A = I₂ P = (I₁ - I₃) P

V_B = I₂ Q = (I₁ - I₃) Q

Set V_A = V_B:

I₁ R_x + I₃ p = (I₁ - I₃) P ...(1)

I₁ (S+r) + I₃ q = (I₁ - I₃) Q ...(2)

From (1): I₁ R_x + I₃ p = I₁ P - I₃ P → I₁ (R_x - P) = -I₃ (p + P) → I₁/I₃ = -(p+P)/(R_x - P).

From (2): I₁ (S+r) + I₃ q = I₁ Q - I₃ Q → I₁ (S+r - Q) = -I₃ (q+Q) → I₁/I₃ = -(q+Q)/(S+r - Q).

Equate:

(p+P)/(R_x - P) = (q+Q)/(S+r - Q)

Cross multiply:

(p+P)(S+r - Q) = (q+Q)(R_x - P)

Now, if bridge is designed such that P/Q = p/q (i.e., ratio arms equal: P/Q = p/q = m), then p = m q, P = m Q.

Substitute:

Left: (m q + m Q)(S+r - Q) = m (q+Q)(S+r - Q)

Right: (q+Q)(R_x - m Q)

So m (q+Q)(S+r - Q) = (q+Q)(R_x - m Q)

Cancel (q+Q) (assuming not zero):

m (S+r - Q) = R_x - m Q

m S + m r - m Q = R_x - m Q

m S + m r = R_x

R_x = m (S + r)

But m = P/Q = p/q. So:

\boxed{R_x = \frac{P}{Q} (S + r)}

If r is negligible or known, R_x = (P/Q) S. But r is the link resistance, which is part of the connection. In practice, r is made very small and constant, or the bridge is designed so that r is compensated by making P/Q = p/q and r appears additively with S. Actually, the standard result is:

R_x = \frac{P}{Q} S + \frac{P}{Q} r - \frac{p}{q} r? Wait, from above we got R_x = m(S+r). But m = p/q also. So R_x = (p/q)(S+r). But r is in series with S. However, the link resistance r is between R_x and S, so it affects both. But if P/Q = p/q, then r adds equally? Actually, in the derivation we assumed P/Q = p/q. Then we got R_x = m(S+r). But that means r adds to S. But r is not part of S; it's separate. However, in the balance equation, r appears. But if P/Q = p/q, then r does not cancel; it adds to S. But we want R_x independent of r. How is that achieved? In Kelvin bridge, the four-terminal connection for R_x and S means that the voltage across R_x is measured at a point before the link r. Actually, in the circuit, R_x and S are connected via link r. The potential points for R_x and S are taken at their inner ends (near the link). So the resistance r is outside the measured loop. In the diagram, R_x has two terminals: one connected to A, one connected to C? Wait, standard Kelvin bridge:

     P ────┬──── Q

           │

     R_x ─┼──── S

           │    (link r between C and D)

     p ────┴──── q

Here, R_x and S are four-terminal: current terminals at A and C? Actually, R_x has current terminal at A and potential terminal at B? I'm mixing.

Correct: In Kelvin double bridge, the unknown R_x and standard S are connected in four-terminal (Kelvin) configuration:

• Each has a current terminal and a potential terminal.

• The current from the battery goes into the current terminals of R_x and S.

• The potential (voltage) across R_x and S is measured between their potential terminals.

• The link r connects the potential terminals of R_x and S? Or the current terminals? Actually, in the bridge, the two pairs of ratio arms are connected to the potential terminals of R_x and S. The current terminals of R_x and S are connected together and to the battery via the outer arms? Let's recall standard diagram:

          P

     A ────┼─── B

           │

     R_x ──┼─── S

           │

     C ────┼─── D

          p

Here, A and C are current terminals? Actually, battery connected between A and C? No.

Standard: Battery connected between A and C? I think:

• Outer ratio arms P and Q connected between A and B? Hmm.

Better:

     P

  A─────┼────B

        │

  R_x───┼───S

        │

  C─────┼────D

     p

Where A and C are connected to battery positive? Actually, the bridge has two sources: battery and galvanometer. The battery is connected across the outer terminals A and C. The galvanometer is connected between B and D. The ratio arms: P between A and B, Q between B and C? That would be Wheatstone. For Kelvin double, there are two sets: P and Q from A to B and B to C? And p and q from A to D and D to C? Actually, standard Kelvin double bridge:

     P

  A─────┼────B

        │

  R_x───┼───S

        │

  C─────┼────D

     p

With P between A and B, Q between B and C, p between A and D, q between D and C. Battery between A and C. Galvanometer between B and D. R_x and S are four-terminal: current terminals at A and C? Actually, R_x has one terminal connected to A (current) and one to B (potential)? No, in four-terminal, R_x has two current terminals (for high current) and two potential terminals (for voltage measurement). In the bridge, the current from battery goes into the current terminals of R_x and S. The potential across R_x is measured between its potential terminal and the junction B? I think the standard connection is:

• The current terminals of R_x and S are connected to A and C respectively? Actually, both R_x and S have their current terminals connected to the same points? Wait, in the diagram above, R_x is between A and B, S between B and C? That would be series. But in Kelvin bridge, R_x and S are in series with each other? No, they are in separate arms? Actually, in Wheatstone, R_x and S are adjacent arms. In Kelvin double, R_x and S are still adjacent but with four-terminal connections. The link r is between the potential terminals of R_x and S.

So correct circuit:

     P

  A─────┼────B

        │

  R_x───┼───┐

        │   │

  C─────┼───┼───D

     p   r   q

No, let's look up standard:

     P

  A─────┼────B

        │

  R_x───┼───S

        │

  C─────┼────D

     p

With R_x between A and B, S between B and C? That would make R_x and S in series between A and C. But then r is between B and D? Actually, B and D are connected by galvanometer. The link r is between the potential terminals of R_x and S. In four-terminal R_x, one terminal is current (connected to A), one is potential (connected to B). Similarly, S: current terminal to C? No, if R_x is between A and B, then A is current, B is potential. For S, if between B and C, then B is current? That would short. So actually, R_x and S are not in the same arms as P and Q. The correct topology:

• Outer bridge: P and Q are between A and B and B and C? That's Wheatstone.

For Kelvin double, we have two bridges in parallel? Actually, it's a double bridge: one bridge for current path, one for potential path.

Standard diagram:

          P

     A ────┼─── B

           │

     R_x ──┼─── S

           │

     C ────┼─── D

          p

With P between A and B, Q between B and C, p between A and D, q between D and C. Battery between A and C. Galvanometer between B and D. R_x and S are connected such that:

• R_x has current terminal at A and potential terminal at B? But then R_x is between A and B? That would be in parallel with P. That's not right.

I recall: In Kelvin bridge, R_x and S are in series with each other, and the link r connects the junction between them to the galvanometer point D. Actually:

     P

  A─────┼────B

        │

  R_x───┼───┐

        │   │

  C─────┼───┼───D

     p   r   q

No.

Let's derive from principles: We want to measure R_x without lead and contact resistance. So we use four-terminal: two current leads (thick, carry current) and two potential leads (thin, measure voltage). In the bridge, the potential leads of R_x and S are connected to the inner ratio arms p and q. The current leads are connected to the outer ratio arms P and Q and to the battery.

So:

• Battery positive to A, negative to C.

• P between A and B.

• Q between B and C.

• R_x current terminal to A, potential terminal to B? That would put R_x in parallel with P. Not correct.

Actually, R_x is not in the bridge arms; it's external. The bridge compares the voltage drop across R_x and S. So:

• The current from battery flows through P and Q? No.

Standard: The battery is connected across the outer terminals A and C. The current through R_x and S is supplied from A and C via the outer ratio arms? I think the correct circuit is:

     P

  A─────┼────B

        │

  R_x───┼───┐

        │   │

  C─────┼───┼───D

     p   r   q

Where:

• A and C connected to battery.

• P between A and B.

• Q between B and C.

• R_x has current terminal at A and potential terminal at B? That would make R_x in parallel with P. That is not typical.

After checking memory: In Kelvin double bridge, the unknown R_x and standard S are connected in series between A and C. The junction between R_x and S is connected via a link r to point D. The inner ratio arms p and q are connected between A and D and D and C respectively. The outer ratio arms P and Q are connected between A and B and B and C. The galvanometer is between B and D.

So:

     P

  A─────┼────B

        │

  R_x───┼───┐

        │   │

  C─────┼───┼───D

     p   r   q

But then R_x is between A and the junction? Actually, R_x is between A and the junction point (call it X), and S between X and C. The link r is between X and D. So:

     P

  A─────┼────B

        │

  R_x───X───S

        │

  C─────┼────D

     p   r   q

Yes! That's it. So R_x and S are in series between A and C. The point X (between them) is connected to D via link r. The inner arms p (A-D) and q (D-C). Outer arms P (A-B) and Q (B-C). Galvanometer between B and D. Battery between A and C.

Now, R_x and S are four-terminal: current terminals at A and C? Actually, current flows from A through R_x to X, then through S to C. So current terminals are A and C. Potential terminals are at X (for R_x) and at X (for S)? But we measure voltage across R_x from A to X? But A is current terminal. In four-terminal measurement, we want to measure voltage across R_x only, excluding lead resistances. So the potential leads for R_x should be connected at the ends of R_x itself, not at A and X because A to R_x has lead resistance. In the diagram, A is directly connected to one end of R_x? If we assume R_x is a four-terminal resistor, it has two current terminals (say C1 and C2) and two potential terminals (P1 and P2). In the bridge, the current from battery goes into C1 of R_x and C1 of S. The potential across R_x is measured between P1 and P2. In the circuit above, A is connected to C1 of R_x, and X is connected to C2 of R_x? But then the potential across R_x is between C1 and C2, which includes the link r? No, X is the junction between R_x and S. If R_x is four-terminal, its potential terminals are inside at the actual resistor ends. So in the diagram, R_x is represented as a box with two current terminals and two potential terminals. The current terminal C1 connected to A, current terminal C2 connected to X. The potential terminal P1 connected to B? No, the potential terminals of R_x are connected to the inner ratio arms. Specifically:

• Potential terminal of R_x (say P1) connected to B? Actually, in standard, the potential terminal of R_x is connected to point B via the outer arm? I'm confusing.

Let's derive without diagram: The goal is that the galvanometer balances when the ratio of voltages across R_x and S equals the ratio of the outer arms P/Q. But the voltages across R_x and S are measured at their potential terminals, which are connected to points B and D? Actually, in the bridge, the galvanometer is between B and D. At balance, V_B = V_D. But V_B is the voltage at point B with respect to A? Not exactly.

From standard textbooks: The balance condition is:

(V_A - V_B) / (V_B - V_C) = P/Q (outer ratio)

(V_A - V_D) / (V_D - V_C) = p/q (inner ratio)

And V_B = V_D at balance.

Also, V_A - V_D is voltage across p, V_D - V_C across q.

But V_A - V_B is voltage across P, V_B - V_C across Q.

Now, what is V_A - V_D? That is the voltage from A to D. But D is connected to X via r. So V_A - V_D = (V_A - V_X) + (V_X - V_D) = V_{R_x} + I_r * r? Actually, if R_x is four-terminal, the voltage across R_x is measured between its potential terminals, which are connected to A? No.

I think it's simpler: In the Kelvin bridge, the potential terminals of R_x and S are connected to points B and D respectively? Or both to B and D? Actually, the inner ratio arms p and q are connected directly to the potential terminals of R_x and S. So:

• Potential terminal of R_x → point B?

• Potential terminal of S → point D?

But then p is between A and B, q between D and C? That would make p from A to B (potential terminal of R_x), and q from D (potential terminal of S) to C. Then the galvanometer is between B and D. That makes sense.

So:

     P

  A─────┼────B (pot. term. of R_x)

        │

  R_x───┼───┐

        │   │

  C─────┼───┼───D (pot. term. of S)

     p   r   q

But then R_x current terminal from A to X? Actually, R_x has current terminal connected to A and the other current terminal connected to X (which is connected to S current terminal and to C via S). So R_x is between A and X. S between X and C. The potential terminal of R_x is at B, which is connected to A? That would short. So B must be connected to the potential terminal of R_x which is at the other end of R_x? That would be X. But X is also connected to S. So if potential terminal of R_x is at X, then B is connected to X. Similarly, potential terminal of S is at X? That would be same point. That can't be.

I think I have it: In four-terminal resistor, the potential terminals are separate from current terminals. For R_x, current terminals: C1 connected to A, C2 connected to X. Potential terminals: P1 connected to B, P2 connected to X? But then P2 is at X, same as current terminal C2. That's allowed? Actually, in Kelvin connection, the potential terminal is connected at the resistor itself, near the current terminal. So for R_x, the potential terminal P2 is connected to the same point as C2? That would measure voltage across R_x including the contact between C2 and X. But we want to exclude lead resistance. The idea: The potential leads are connected directly to the resistor ends, not to the current leads. So for R_x, one potential lead is connected to the terminal of R_x where current enters, and the other to the terminal where current exits. But then the voltage measured includes the contact resistance at those terminals? Actually, the four-terminal method: The current is forced through two separate terminals, and the voltage is measured across two other terminals that are in direct contact with the resistor. So the voltage measured is exactly across the resistor, excluding any resistance in the current leads. So for R_x, the potential terminals are connected across the resistor itself, not at the connection points to the bridge. In the bridge circuit, the potential terminals of R_x are connected to points B and X? But X is the junction with S. So if potential terminal of R_x is at X, then voltage across R_x is between A and X? But A is current terminal, not potential. To measure voltage across R_x, we need two potential leads: one at the A-end of R_x and one at the X-end. But the A-end is connected to A via a lead. If we connect potential lead to A, we include lead resistance. So we must connect potential lead directly to the resistor terminal at the A side. That means point A is actually the potential terminal? No, A is the current terminal. So we need a separate point. Hence, in the diagram, R_x is shown with four terminals: two current (connected to A and X) and two potential (connected to B and D?).

Actually, standard Kelvin bridge diagram:

     P

  A─────┼────B

        │

  R_x───┼───┐

        │   │

  C─────┼───┼───D

     p   r   q

Where:

• A and C are connected to battery.

• P between A and B.

• Q between B and C.

• R_x has its current terminals connected to A and X (where X is the junction between R_x and S).

• S has current terminals connected to X and C.

• The potential terminals of R_x are connected to B and X? But then B is connected to A via P. That would measure voltage from A to X plus drop in P? Not good.

I think the correct interpretation: The inner ratio arms p and q are connected directly to the potential terminals of R_x and S. So:

• Potential terminal of R_x → point B.

• Potential terminal of S → point D.

And the link r is between B and D? No, r is the link resistance between the current junction X and the potential terminal of S? Actually, r is the resistance of the link connecting the potential terminal of R_x to the potential terminal of S? That would be r between B and D. But then B and D are connected by galvanometer? No, galvanometer is between B and D. So r is in parallel with galvanometer? That doesn't make sense.

After checking reliable memory: In Kelvin double bridge, the link r is the resistance of the short link connecting the potential terminals of R_x and S. But the potential terminals are connected to the inner ratio arms. So the circuit is:

     P

  A─────┼────B

        │

  R_x───┼───┐

        │   │

  C─────┼───┼───D

     p   r   q

With:

• R_x current terminals: A and X (where X is between R_x and S).

• S current terminals: X and C.

• R_x potential terminal 1: connected to B.

• R_x potential terminal 2: connected to X? But then X is current terminal. That would short? Actually, the potential terminal for R_x at the X end is connected directly to X (so same point as current terminal). That's okay because we want to measure voltage across R_x from A to X, but we can't connect potential lead to A because that's current terminal. So we need a potential lead at the A end of R_x that is separate from the current lead. So point A is the current terminal, but there is a separate point, say A', connected directly to the A-end of R_x (potential terminal). In the diagram, A' is connected to B? That would mean B is the potential terminal for the A-end of R_x. Similarly, for S, the potential terminal at the X end is connected to D? But X is the junction, so potential terminal of S at X end is connected to X? That would be same as current terminal. So we need a separate point at the X-end of S? Actually, for S, potential terminals are at both ends: one at X and one at C. But X is common to R_x and S. So the potential terminal of R_x at X end is connected to X, and potential terminal of S at X end is also connected to X. That means the potential terminal of R_x at X and potential terminal of S at X are the same point X. Then the link r is between X and D? But D is connected to potential terminal of S at C end? I'm tangled.

Given time, I'll state the standard balance equation and principle: Principle: Four-terminal connection for R_x and S eliminates errors due to lead and contact resistances. The potential measuring circuit (inner arms p, q and galvanometer) is connected directly across the resistors, bypassing the link resistance r. Balance Condition (when P/Q = p/q):

\boxed{R_x = \frac{P}{Q} (S + r) - \frac{p}{q} r}? Actually, from many sources: R_x = \frac{P}{Q} S + \frac{P}{Q} r - \frac{p}{q} r. If P/Q = p/q, then R_x = \frac{P}{Q} S. So r cancels.

Yes! That's the key: When the ratio arms are equal (P/Q = p/q), the balance equation simplifies to R_x = (P/Q) S, independent of r.

So the derivation should yield:

R_x = \frac{P}{Q} S + \left( \frac{P}{Q} - \frac{p}{q} \right) r

Thus if P/Q = p/q, R_x = (P/Q) S. Superiority: For low resistances, lead and contact resistances (in series with R_x and S) are significant. In Wheatstone, they are in series with R_x and S and affect balance. In Kelvin, the potential leads are connected directly to the resistor terminals, so lead resistances in potential circuit are negligible (since high impedance of galvanometer). The link r is in the potential circuit but cancels if ratios equal.

VII. MISCELLANEOUS INSTRUMENTS & ACCESSORIES

Power Factor Meter

Electrodynamometer Type:

• Single-phase: Two fixed coils (current) in series, moving coil (voltage) with series resistor. Moving coil mounted at angle to fixed coils. Torque T_d ∝ I * V * cos(θ - 90°?) Actually, fixed coils produce field proportional to I. Moving coil current proportional to V (if resistive). But moving coil is pivoted so that at θ=0 (unity PF), pointer at 0.5 or 1? Scale calibrated in PF.

• Three-phase (Dec 2024): Two elements or single element with two sets of coils? For 3-phase, 3-wire, use two elements like two-wattmeter but with phase shift networks to get single pointer.

Moving Iron Type: Single coil with two iron vanes. Torque proportional to I² but with phase shift network to indicate PF.

Megger

See Section V.

Multi-Range Ammeter & Voltmeter

• Ammeter: Multiple shunts R_sh1, R_sh2, ... with rotary switch. For range I1, R_sh1 = (I_m R_m)/(I1 - I_m).

• Voltmeter: Multiple series multipliers R_s1, R_s2, .... For range V1, R_s1 = V1/I_m - R_m.

Ratio Meter

Measures ratio of two AC quantities. Principle: Two coils at angle, one fed by I1, other by I2. Torque T_d ∝ I1 I2 cos(θ) where θ is phase difference. If θ fixed (e.g., 90°), then T_d ∝ I1 I2. Used in instrument transformers to check ratio and phase angle.

Digital Voltmeter (DVM)

Types:

1. Ramp Type: Integrate input voltage for fixed time, measure time to discharge capacitor.

2. Dual-Slope (Integrating) Type (most common):

   • Block Diagram:

     
     Input V_in → Integrator (with switch) → Comparator → Counter → Display
     
     Control logic: 
     
        Phase 1: Integrate V_in for fixed time T1 → output slope ∝ V_in.
     
        Phase 2: Integrate reference V_ref of opposite polarity until output returns to zero → time T2 ∝ V_in.
     
        Count T2 with clock → reading ∝ T2 ∝ V_in.
     
     

   • Advantage: Rejects noise, high accuracy.

3. Successive Approximation Type: SAR ADC, fast.

Eddy Current Damping

Principle: Conductive disc (aluminum) moving in magnetic field of permanent magnet. Disc cuts flux, induces eddy currents. Eddy currents produce magnetic field opposing motion (Lenz's law), creating braking torque T_b ∝ ω. Used in wattmeters, energy meters, galvanometers.

VIII. INSTRUMENT TESTING, CALIBRATION & LIMITATIONS

Testing & Calibration

• Ammeter/Voltmeter: Compare with standard instrument under various loads. Check linearity, zero, scale.

• Wattmeter: Compare with standard wattmeter at different PFs and loads. Check errors due to inductance.

• Energy Meter:

  • Full load test: As described.

  • Light load test: Check creep.

  • Phase error test: At lagging PF.

  • Starting test: Should start at low current.

• CT/PT: Ratio and phase angle error measurement using comparator set or standard CT/PT.

Limitations & Error Sources

• CT: Open-circuit danger, ratio error with burden, saturation.

• PT: Burden effect, insulation stress.

• Wattmeter: Stray fields, low PF error, frequency sensitivity.

• Energy Meter: Creep, friction, temperature, voltage variation, PF error.

• Bridges: Contact resistance, thermoelectric EMFs, detector sensitivity.

END OF UNIT 5 NOTES