**Instrument Transformer and Power Management (P1) Course**

**Chapter (5) : Power Measurement in 3-ph Circuits**

__5.1 Power in Three Phase Circuits :____Blondel's theorem :__

In any system whatsoever, of "N" wires, the true power may be measured by connecting a wattmeter in each line but one ( N-1 wattmeters), the current coil being connected in the line and the potential coil connected between that line and the line which contain no current coil. The total power for any load is the algebraic sum of the wattmeter readings. This is known as Blondel's theorem.

Applying the above theorem to any three phase, three wire circuit, true power may be measured by with tow wattmeters under all condition of loads, balance and P.F the instantaneous power is :

P

_{ins }= e_{1}I_{1}+ e_{2}I_{2}+ e_{3}I_{3}, but I_{1}+ I_{2}+ I_{3}= 0 then , I_{2}= -I_{1}- I_{3}P

_{ins}= e_{1}I_{1}+ e_{2}( -I_{1}- I_{3}) + e_{3}I_{3} Wattmeter, "W

_{1}." measures the power which the current coil is I_{R}and the voltage coil is V_{RS}( line voltage ) , or its power is ( e_{1}- e_{1}) I_{1} Wattmeter "W

_{2}" measures the power which the current coil is I_{T}and the voltage coil is V_{TS}( line voltage), or its power is ( e_{3}- e_{2}) I_{3} For the instantaneuos value of R.M.S values :

V

_{RS}= e_{1}- e_{2}, V_{TS}= e_{3}- e_{2}I

_{1}= I_{R}, I_{3}= I_{T}, I_{2}= I_{S} The wattmeter reading must multiply the current which pass through the current coil by the voltage which applied on the voltage by cosine the angle between voltage and current.

W

_{1}= I_{R}x V_{RS}x cosθ Assume a balanced 3-phase system is applied to a 3-phase star connected load if the load is known to be inductive load so the current I

_{R}must be lag the phase voltage V_{R}with angle . The line voltage V_{RS}lead the phase voltage V_{R}with 30. So, the total angle between I_{R}and V_{RS}isW

_{1}= I_{R}x V_{RS}x cos ( Φ + 30° ) To calculate the power in the other wattmeter..W2 ;

W2 = I

_{T}x V_{TS}cosθ2 When is the angle between current I and voltage V from the diagram

In the balanced 3 phase, 3wire circuit : VRS = VTS = VL and IR = IS = IT = IL .

the total power is :

P1 = W1 + W2 = VL LL cos ( 30° + Φ ) + VL IL cos ( 30° - Φ )

the total power is :

P1 = W1 + W2 = VL LL cos ( 30° + Φ ) + VL IL cos ( 30° - Φ )

_{ }= VL IL cos ( 30° + Φ ) + cos ( 30° - Φ )

= VL IL (cos 30° cosΦ - sin30° sinΦ + cos 30V cosΦ + sin30° sinΦ

P = VL IL x (√3/2 ) x 2 cosΦ

P = (√3 VL IL cosΦ )

When two wattmeters are used to measure the total power, then by the vector analysis the wattmeter readings at different power factors are as following :

a) Reading of W1 = W2 at unity power factor (Resistive load ).b) Reading of W1 > W2 ( Capacitive load).

c) Reading of W1 < W2 (Inductive load).

Calculation of reactive power as a different between the tow reading of wattmeters :

√3( W2 - W1 ) = √3 VTS IT cos (30° - Φ ) - √3 VRS x IR cos (30° + Φ )

= √3 VL IL cos30° cosΦ + sin30° sinΦ - cos30° cosΦ + sin30° sinΦ

√3( W2 - W1 ) = VL IL x 0.5 (sin Φ) x 2 = √3 VL IL sin Φ

__Using a single wattmeter to measure the reactive power in a 3-phase balanced system load :__

The current coil of the wattmeter is connected in one line and the voltage coil is connected across the other tow lines as shown in the Fig.

the current through the current coil = ISthe voltage across the voltage coil = VRT

And the reading of wattmeter = VRT IS (cosΦ + 90 ) = - VL IL sin Φ

then, the total reactive power volt, ampere of the circuit :

QT = 3 V Φ IL sinΦ

QT = √3 VL IL sinΦ

QT = √3 x (reading of wattmeter)

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