Power in Three Phase Circuit

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 = e₁ I₁ + e₂ I₂ + e₃ I₃ , but I₁ + I₂ + I₃ = 0  then , I₂ = -I₁ - I₃

P_ins = e₁ I₁ + e₂ ( -I₁ - I₃ ) + e₃ I₃

P_ins = ( e₁ - e₂ ) I₁ + (e₃ - e₂) I₃

       Wattmeter, "W₁." measures the power which the current coil is I_R and the voltage coil is V_RS ( line voltage ) , or its power is ( e₁ - e₁ ) I₁

       Wattmeter "W₂" measures the power which the current coil is I_T and the voltage coil is V_TS ( line voltage), or its power is ( e₃ - e₂ ) I₃

       For the instantaneuos value of R.M.S values : 

V_RS = e₁ - e₂ , V_TS = e₃ - e₂

I₁ = I_R , I₃ = I_T , I₂ = 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.

W₁ = I_R x V_RS x cosθ  

Where θ is the angle between current I_R  and voltage V_RS.

       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 is 

W₁ = 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 

  θ2 = ( 30° - Φ )                                       then W2 = IT VTS cos( 30° - Φ )
 

       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 )

                          = 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 = IS
        the 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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