Measurements & Instrumentation Formula Book

9.1

Two Wattmeter Method: Complete Analysis

For balanced inductive 3-φ load:

W_1 = V_L I_L\cos(30° - \phi)

W_2 = V_L I_L\cos(30° + \phi)

\boxed{P = W_1 + W_2 = \sqrt{3}\,V_L I_L\cos\phi}

\boxed{Q = \sqrt{3}(W_1 - W_2) = \sqrt{3}\,V_L I_L\sin\phi}

\phi = \tan^{-1}\!\left[\frac{\sqrt{3}(W_1-W_2)}{W_1+W_2}\right]

\cos\phi = \cos\!\left[\tan^{-1}\!\left(\frac{\sqrt{3}(W_1-W_2)}{W_1+W_2}\right)\right]

φ	cos φ	W₁ vs W₂
0°	1 (UPF)	W₁ = W₂
30°	0.866 lag	W₁ = 2W₂
60°	0.5 lag	W₂ = 0
0° < φ < 60°	0.5 < cos φ < 1	W₁(+), W₂(+)
90°	0	W₁ = −W₂
60° < φ < 90°	0 < cos φ < 0.5	W₁(+), W₂(−)

φ 30°

W₁ = 0 W W₂ = 0 W ⚠️ Reverse W₂

P = 0 W Q = 0 VAR cosφ = 1

⚡ Two Wattmeter Full Calculator

W₁ (W): W₂ (W):

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9.2

Measurement of Energy

\boxed{E = \int_0^T V\,i\,dt \quad \text{(Joules)}}

1\,\text{kWh} = 3.6\times10^6\,\text{J}

Energy Meter = Integrating type instrument (Induction principle, no controlling torque)

\boxed{K = \frac{N}{P \times t}\;\text{(rev/kWh)}}

E = \frac{N}{K}\;\text{kWh}

n_{disc} = \frac{K \times P}{60}\;\text{RPM}

Three systems: 1. Driving (shunt+series coil) | 2. Moving (Al disc) | 3. Braking (permanent magnet)

Load P (kW) 2

K (rev/kWh)

t = 0 s 0 RPM

🔋 Energy Meter Calculator

K (rev/kWh): P (kW): t (hours):

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9.3

Energy Meter: Working Principle

Two alternating fluxes interact with eddy currents in Al disc → Driving torque

\phi_{sh} \propto V \quad\text{(shunt magnet)}

\phi_{se} \propto I_L \quad\text{(series magnet)}

T_{d} = \phi_{sh}\times I_{m_{se}} - \phi_{se}\times I_{m_{sh}} \propto VI\cos\phi = P

T_{b} = K_b \times \omega_{disc}

At steady state: T_d = T_b → ω ∝ P → N_total = ∫ω dt ∝ ∫P dt = E

\theta = 90° + \phi \quad\text{(ideal condition)}

T_d \propto VI\sin\theta = VI\sin(90°+\phi) = VI\cos\phi = P

Load φ 30°

Speed ∝ VIcosφ θ = 120°

🌀 Disc Speed Calculator

K (rev/kWh): P (kW):

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9.4

Energy Meter: Compensations & Errors

Five Compensations:

1. Lag Compensation — Shading coil → θ = 90°+φ

2. Low Load (Friction) — Shading loop → extra torque at light load

3. Anti-Creep — Holes/slots in disc → disc cannot rotate at no load

4. Overload — Saturable shunt on series magnet → linearize

5. Over Voltage — Saturable shunt magnet → limit φ_sh

📊 Meter Accuracy Test

K (rev/kWh): P (kW): t (hours): N_actual:

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9.5

Energy Meter: Complete Reference

K = \frac{N}{kWh} = \frac{N}{P \times t}

\%\text{Error} = \frac{N_{actual} - N_{expected}}{N_{expected}} \times 100

Two Wattmeter Summary:

P = W_1+W_2,\; Q=\sqrt{3}(W_1-W_2),\; S=\sqrt{P^2+Q^2}

\tan\phi = \frac{\sqrt{3}(W_1-W_2)}{W_1+W_2},\; \cos\phi = P/S

Component	Function
Pressure coil (shunt)	Flux ∝ voltage, highly inductive
Current coil (series)	Flux ∝ load current
Al disc	Rotating element, eddy currents
Braking magnet	Speed control, speed ∝ power
Lag coil	Phase correction, θ = 90°+φ
Shading loop	Low load friction compensation
Disc holes/slots	Anti-creep device
Register/counter	Accumulates rev → kWh

W₁ (W) 2000

W₂ (W) 1000

P = 3000 W Q = 0 VAR S = 0 VA cosφ = 1 tanφ = 0

📐 Reactive Power Calculator

V_L (V): I_L (A): φ (°):

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🎯 W₁ W₂ Relationship Finder

φ (°):

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Module 9 — 3-φ Power, Reactive Power & Energy Meters