🏠 / measurements / πŸ“Š PT, CT vs PT & Power

MODULE 7 β€” PT, CT vs PT & Power Measurement

πŸ”Ά 7.1 Potential Transformer β€” Complete Formulas

PT Ratio Β· Phase Angle Ξ² Β· Strip Wound Core

Actual Transformation Ratio

$$ \boxed{R = n + \frac{I_m(R_s\cos\Delta + X_s\sin\Delta) + (I_c\cos\Delta - I_m\sin\Delta)r}{V_s}} $$
Simplified: $$ R \approx n + \frac{I_m\cos\delta + I_c\sin\delta}{nI_s \cdot V_s} $$

Phase Angle Ξ²

$$ \boxed{\beta = \frac{1}{nV_s}\left[X_s\cos\Delta - R_s\sin\Delta + \frac{I_m X_p - I_c r}{n}\right]} $$
$$ \beta_{deg} = \frac{180}{\pi}\cdot\beta_{rad} $$

Key Rules

$$ \text{CT: NEVER open secondary!} $$
$$ \text{PT: CAN be open-circuited} $$
Strip wound core β†’ reduces ratio & phase error

Applications of CT & PT

Multiple operation with single device
Isolation from power circuit
Low power consumption
Easy replacement
V₁ nVβ‚‚ Ξ² Im Ic Rs Xs Ξ² = 0.00Β°

πŸ”Ά PT Phase Angle Calculator

Ξ² = β€”Β°

πŸ“‹ 7.2 CT vs PT β€” Complete Comparison

Series vs Parallel Β· Flux Β· Danger Rules

CT Formulas

$$ n_{CT} = \frac{N_s}{N_p},\quad k_n = \frac{I_{1,rated}}{I_{2,rated}} $$
$$ Z_{secondary} \approx 0\;\text{(short circuit)} $$

PT Formulas

$$ n_{PT} = \frac{N_p}{N_s},\quad k_n = \frac{V_{1,rated}}{V_{2,rated}} $$
$$ Z_{secondary} = \infty\;\text{(open circuit)} $$

Danger Rules

$$ \text{CT open} \Rightarrow V_s \to \infty \;\text{⚠️} $$
$$ \text{PT short} \Rightarrow I_s \to \infty \;\text{⚠️} $$
ParameterCTPT
PurposeHigh current measurementHigh voltage measurement
Flux density & Iβ‚€Varies over WIDE rangeVaries over RESTRICTED range
Secondary open circuitNEVER β€” ⚠️ DANGEROUSCAN be open (safe)
Equivalent toSeries transformer (virtual short)Parallel transformer (virtual open)
Primary currentINDEPENDENT of secondaryDEPENDENT on secondary burden
Primary terminal voltageSmall voltage across terminalsFull voltage across terminals
CT β€” Series Connection
Power Line (High I) Ip A βœ“ Safe: Secondary closed
PT β€” Parallel Connection
HV Bus Np Ns V βœ“ Safe: Normal operation

⚑ 7.3 Measurement of Power β€” Overview

DC Β· 1-Ο† AC Β· 3-Ο† AC Β· Y & Ξ”

DC Power

$$ P_{DC} = V_0 \cdot I_0 $$

1-Ο† AC Power

$$ P = V_{rms}I_{rms}\cos\phi \;\text{(W)} $$
$$ Q = V_{rms}I_{rms}\sin\phi \;\text{(VAR)} $$
$$ \boxed{P = \frac{V_m I_m}{2}\cos\phi} $$

3-Ο† AC Power

$$ P = \sqrt{3}\,V_L I_L\cos\phi $$
$$ Q = \sqrt{3}\,V_L I_L\sin\phi $$

Y-connection (Star)

$$ V_L = \sqrt{3}\,V_{ph},\quad I_L = I_{ph} $$

Ξ”-connection (Delta)

$$ V_L = V_{ph},\quad I_L = \sqrt{3}\,I_{ph} $$
P = SΒ·cos Ο† Q = SΒ·sin Ο† S = VΒ·I Ο† P=0 W | Q=0 VAR | S=0 VA

⚑ AC Power Calculator

P = β€” W | Q = β€” VAR | S = β€” VA

πŸ”Œ 7.4 DC Power β€” V-A and A-V Methods

Error Analysis Β· R_critical Β· Method Selection

V-A Method

$$ P_m = P_t + I^2 R_A $$
$$ \boxed{\%\text{Error} = -\frac{R_A}{R_L}\times 100} $$
Best for: $R_L \gg R_A$ (high R load)

A-V Method

$$ P_m = P_t + \frac{V^2}{R_V} $$
$$ \boxed{\%\text{Error} = +\frac{R_L}{R_V}\times 100} $$
Best for: $R_L \ll R_V$ (low R load)

Crossover Point

$$ \boxed{R_{critical} = \sqrt{R_A \cdot R_V}} $$
$R_L > R_c$: V-A | $R_L < R_c$: A-V
ParameterV-A MethodA-V Method
Voltmeter positionNear supply (before A)Near load (after A)
Extra power$I^2 R_A$ (ammeter loss)$V^2/R_V$ (voltmeter)
Error signNegative (overmeasures)Positive (overmeasures)
Error formula$-R_A/R_L \times 100\%$$+R_L/R_V \times 100\%$
Best forHigh R load ($R_L \gg R_A$)Low R load ($R_L \ll R_V$)
V-A Method
V A RL Error: IΒ²RA
A-V Method
A V RL Error: VΒ²/RV
Recommended: β€”

πŸ”Œ DC Power Error Calculator

V-A: β€”% | A-V: β€”% | R_crit = β€”

⚑ 7.5 AC Power β€” Two Wattmeter Method

3-Ο† Power Β· W₁+Wβ‚‚ Β· tan Ο† Β· cos Ο†

Two Wattmeter Method

$$ \boxed{P_{total} = W_1 + W_2} $$
$$ \boxed{\tan\phi = \sqrt{3}\cdot\frac{W_1 - W_2}{W_1 + W_2}} $$

Individual Readings

$$ W_1 = V_L I_L\cos(30Β° - \phi) $$
$$ W_2 = V_L I_L\cos(30Β° + \phi) $$

Special Cases

$$ \phi = 0Β°:\; W_1 = W_2 \;\text{(pf=1)} $$
$$ \phi = 60Β°:\; W_2 = 0 \;\text{(pf=0.5)} $$
$$ \phi > 60Β°:\; W_2 < 0 \;\text{(reverse)} $$

Other Methods

One Wattmeter (balanced 3-Ο†): P = 3W₁
Three Wattmeter: P = W₁ + Wβ‚‚ + W₃
LPF Wattmeter: cos Ο† < 0.5
R Y B W₁ PC Wβ‚‚ PC Load W₁ = 0 W Wβ‚‚ = 0 W
pf = 0.866 Wβ‚‚ > 0

⚑ Two Wattmeter Calculator

P = β€” W | tan Ο† = β€” | cos Ο† = β€”

πŸ“Š 7.6 Power Measurement β€” Summary

Quick Reference Β· All Methods

DC

$$ P = V_0 I_0 $$
V-A: $-R_A/R_L$
A-V: $+R_L/R_V$

1-Ο† AC

$$ S^2 = P^2 + Q^2 $$
$$ P=VI\cos\phi,\;Q=VI\sin\phi $$

3-Ο†

$$ P = \sqrt{3}V_LI_L\cos\phi $$
$$ \text{2W: } P = W_1+W_2 $$
CircuitMethods Available
DCV-A method, A-V method
1-Ο† ACV-A, A-V, EDM wattmeter
3-Ο† balancedOne wattmeter (Γ— 3)
3-Ο† unbalancedThree wattmeter method
3-Ο† (any)Two wattmeter method

πŸ“ 3-Ο† Power Calculator

P = β€” W | Q = β€” VAR | S = β€” VA
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