Power Triangle [PDF]

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Power Triangle



Learning Objectives 



Define apparent power.







Calculate apparent power in AC series parallel networks.







Define the power triangle.







Using the power triangle determine relationships between real, reactive and apparent power.







Determine if AC series parallel networks are inductive, capacitive, or purely resistive.



Review AC Power to a Resistive Load P  VRMS I RMS



2 V 2  I RMS R  RMS R



(watts)



AC Power to a Inductive Load QL  I



2 RMS



2 VRMS XL  XL



(VAR)



AC Power to a Capacitive Load QC  I



2 RMS



2 VRMS XC  XC



(VAR CAP )



Review AC Power Summary



Real Power



P = VI (W) P = I2R =V2/R



P = 0 (W)



P = 0 (W)



Reactive Power



Q = 0 (VAR)



Q = VI (VARind) Q = I2XL =V2/XL



Q = VI (VARcap) = I2XC =V2/XC



Resistance Reactance



R



XL = L



XC = 1/C



Apparent Power 







For a load with voltage V and current I, the power that “appears to flow” to the load is VI where V and I are rms values. S = VI (VA) S is called the apparent power and has units of volt-amperes (VA).



Apparent Power 







In terms of load impedance Z, apparent power can be expressed S = I2Z = V2/Z (VA) It is common to see apparent power give in kVA.



Example Problem 1 Determine the real, reactive, and apparent power.



Power Triangle 



The power triangle graphically shows the relationship between real (P), reactive (Q) and apparent power (S). P 



S QL



 P



QC S



Power Triangle 



From the power triangle we can see that



S  P2  Q2 S  P  jQL or



S  P  jQC



S  S  P 



S QL



 P



QC S



Power Triangle 



We can generalize the equations:



P  P0 Q L  jQL QC   jQC I*is complex conjugate of I S  PQ S  VI



P  QC







S



Real and Reactive Power 



The power triangle also shows that we can find real (P) and reactive (Q) power.



P  VI cos  S cos



(W)



Q  VI sin   S sin 



(VAR) P 



S QL



 P



QC S



Example Problem 2 Draw the power triangle for this circuit. Determine if this is an inductive, capacitive, or resistive circuit.



Example Problem 3 Determine the value of R and PT & QT. Draw the power triangle and determine S.