MODULE 8 — Wattmeter Errors & Polyphase Power
⚡ 8.1 Equal Error Condition & EDM Wattmeter Connections
Connection A · Connection B · R_criticalEqual Error Condition
$$ \boxed{R_L = \sqrt{R_{CC} \times R_{PC}}} $$
At $R_L = R_{crit}$: both connections have equal error
Connection A — MC Short
$$ \boxed{P_m = P_t + I^2 R_{CC}} $$
$$ \% \text{Error} = \frac{R_{CC}}{R_L} \times 100 $$
Best for: SMALL load current
Connection B — LC Short
$$ \boxed{P_m = P_t + \frac{V^2}{R_{PC}}} $$
$$ \% \text{Error} = \frac{R_L}{R_{PC}} \times 100 $$
Best for: LARGE load current
Coil Construction
$$ \text{CC (Current/Fixed Coil): } N\downarrow,\; A\uparrow $$
$$ \text{PC (Pressure/Moving Coil): } N\uparrow,\; A\downarrow $$
Power Indicated
$$ \text{Conn A: } P_L + \underbrace{I^2 R_{CC}}_{\text{CC loss}} $$
$$ \text{Conn B: } P_L + \underbrace{V^2/R_{PC}}_{\text{PC loss}} $$
Rcrit = — Ω | Recommended: —
⚡ Wattmeter Connection Error Calculator
Conn A: —% | Conn B: —% | Recommend: —
📐 8.2 Errors in EDM Wattmeter
PC Inductance · PC Capacitance · Connection Error1. PC Inductance Error
$$ \boxed{CF = \frac{\cos\phi}{\cos\beta\cos(\phi+\beta)}} $$
$$ \beta = \tan^{-1}\!\left(\frac{\omega L_{PC}}{R_{PC}}\right) $$
$$ \text{Error} \approx \beta\tan\phi \times P_{true} $$
2. PC Capacitance Error
$$ \text{Inter-turn capacitance} $$
$$ \text{Reads HIGH on lagging PF} $$
$$ \text{Fix: } X_C = X_L \;\text{(cancel reactance)} $$
3. Connection Error
$$ \text{A: } \frac{R_{CC}}{R_L}\times 100\% $$
$$ \text{B: } \frac{R_L}{R_{PC}}\times 100\% $$
Inductance Compensation
$$ \boxed{C = 0.4\frac{L_{PC}}{R_{PC}^2}} $$
Connect C in parallel with RPC → Z purely resistive
True Power Correction
$$ P_{true} = CF \times P_{reading} $$
CF ≈ 1 at high PF, CF ≫ 1 at low PF
Error Direction
Lagging load: reads LOW (−ve error)
Leading load: reads HIGH (+ve error)
Low PF (φ→90°): tan φ → ∞ → LARGE error
Error ≈ 0%
PF OK
📐 PC Inductance Correction Calculator
CF = — | True P = — W | Error ≈ —%
🔧 PC Compensation Calculator
C = — μF
🔵 8.3 Low Power Factor Wattmeter
cos φ < 0.5 · Compensating Coil · Iron Loss TestWhy Standard Fails
$$ P = VI\cos\phi $$
$$ \cos\phi = 0.1 \Rightarrow P = 0.1 \times VI $$
Deflection = 10% of FSD → inaccurate
LPF Modifications
$$ \text{1. Compensating coil added} $$
$$ \text{2. PC resistance reduced} $$
$$ \text{3. Scale: low PF range} $$
Compensating Coil
$$ \text{Without: } P_m = P_L + \frac{V^2}{R_{PC}} $$
$$ \text{With: } P_m = P_L \;\text{✓} $$
Compensating coil cancels PC power loss
Applications
Transformer no-load test (PF ≈ 0.1–0.2)
Capacitive circuit measurement
Synchronous motor no-load power
Standard Wattmeter
LPF Wattmeter
Deflection: —% FSD
🔵 LPF Range Checker
Recommendation: —
🌐 8.4 Blondel's Theorem & Polyphase Power
N-Wire · Two Wattmeter · Three WattmeterBlondel's Theorem
$$ \boxed{\text{N wires, no neutral} \Rightarrow (N\!-\!1) \text{ wattmeters}} $$
$$ \boxed{\text{N wires, with neutral} \Rightarrow N \text{ wattmeters}} $$
Two Wattmeter Method
$$ W_1 = V_L I_L\cos(30°\!-\!\phi) $$
$$ W_2 = V_L I_L\cos(30°\!+\!\phi) $$
$$ \boxed{P = W_1 + W_2} $$
From Two Wattmeters
$$ \boxed{\tan\phi = \sqrt{3}\cdot\frac{W_1 - W_2}{W_1 + W_2}} $$
$$ Q = \sqrt{3}(W_1 - W_2) $$
$$ S = \sqrt{P^2 + Q^2} $$
Special PF Cases
$\phi=0°$ (pf=1): $W_1 = W_2 = P/2$
$\phi=60°$ (pf=0.5): $W_2 = 0$
$\phi=90°$ (pf=0): $W_1 = -W_2$
Other Methods
1 Wattmeter (balanced): $P = 3W_1$
3 Wattmeter (4-wire): $P = W_1+W_2+W_3$
W₂ Reversal
$\phi > 60°$: $W_2 < 0$ → Reverse PC connections
$P = W_1 - |W_2|$ (after reversal)
Blondel's Theorem
Use 2 wattmeters
W₁ = 0
W₂ = 0
P = 0
pf = 1
🔌 Two Wattmeter Calculator
P = — | Q = — | S = — | tan φ = — | cos φ = —
🌐 Blondel's Theorem Calculator
Wattmeters needed: —
📊 8.5 Wattmeter Connection — Summary
Quick Reference · All Methods · PF ZonesKey Formulas
$$ P = W_1 + W_2 $$
$$ Q = \sqrt{3}(W_1 - W_2) $$
$$ \tan\phi = \sqrt{3}\frac{W_1-W_2}{W_1+W_2} $$
PF Markers
pf = 1: $W_1 = W_2$ (equal)
pf = 0.5: $W_2 = 0$ (one zero)
pf < 0.5: $W_2 < 0$ (reverse)
Equal Error
$$ R_{crit} = \sqrt{R_{CC}\cdot R_{PC}} $$
$R_L > R_c$: Conn A | $R_L < R_c$: Conn B
| Parameter | Conn A (MC short) | Conn B (LC short) |
|---|---|---|
| PC connected | Before CC (supply side) | After CC (load side) |
| Wattmeter reads | $P_L + I^2 R_{CC}$ | $P_L + V^2/R_{PC}$ |
| Error type | CC loss (current error) | PC loss (voltage error) |
| Error formula | $R_{CC}/R_L \times 100\%$ | $R_L/R_{PC} \times 100\%$ |
| Best for | Small load current | Large load current |
| Error Source | Correction / Compensation |
|---|---|
| PC inductance | $C = 0.4\,L_{PC}/R_{PC}^2$ in parallel with R |
| PC capacitance | $X_C = X_L$ (add L, make total reactance = 0) |
| Connection method | Choose A or B based on $R_L$ vs $R_{crit}$ |
| Low power factor | Use LPF wattmeter + compensating coil |
| System | Wattmeters Required |
|---|---|
| 1-φ 2-wire | 1 wattmeter |
| 3-φ 3-wire (no N) | 2 wattmeters |
| 3-φ 4-wire (with N) | 3 wattmeters (or 4 for unbalanced) |
| N-wire, no neutral | N − 1 wattmeters |
| N-wire, with neutral | N wattmeters |
📊 Summary Power Calculator
P = — | Q = — | S = — | cos φ = —