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Active, Reactive, Apparent and Complex Power with Formulas.

Active, Reactive, Apparent and Complex Power

Active Power: (P)

Active Power is the actual power which is really transferred to the load such as transformer, induction motors, generators etc and dissipated in the circuit.

Alternative words used for Real Power (Actual Power, True Power, Watt-full Power, Useful Power, Real Power, and Active Power) and denoted by (P) and measured in units of Watts (W) i.e. The unit of Real or Active power is Watt where 1W = 1V x 1 A. .

Active Power in DC Circuits:

In DC Circuits, power supply to the DC load is simply the product of Voltage across the load and Current flowing through it i.e., P = V I because in DC Circuits, there is no concept of phase angle between current and voltage. In other words, there is no frequency (f) or Power factor in DC Circuits.

Active Power in AC Circuits:

But the situation in Sinusoidal or AC Circuits is more complex because of phase difference (θ) between Current and Voltage. Therefore average value of power (Real Power) is P = VI Cosθ is in fact supplied to the load.

In AC circuits, When circuit is pure resistive, then the same formula used for power as used in DC as P = V I.

You may also read about Power Formulas in DC, AC Single Phase and and AC Three Phase Circuits.

Active Power Formulas:

  • P = V x I                                         (In DC circuits)
  • P = V x I x Cosθ                            (in Single phase AC Circuits)
  • P = √3 x VLx IL x Cosθ         or      (in Three Phase AC Circuits)
  • P = 3 x VPh x IPhx Cosθ
  • P = √ (S2 – Q2)or
  • P =√ (VA2 – VAR2) or

Real or True Power or Active Power = √ (Apparent Power2 – Reactive Power2) or

kW = √ (kVA2 – kVAR2)

Where:

  • P = Power in Watts
  • V = Voltages in Volts
  • I = Current in Amperes
  • Cosθ = Power Factor (Phase angle Difference)
  • V= Line Voltage
  • IL = Line Current
  • S = Apparent Power in VA (Volt Ampere)
  • Q = Reactive Power in VAR (Volt Ampere Reactive)

Reactive Power: (Q)  

Also known as (Use-less Power, Watt less Power)

The powers that continuously bounce back and forth between source and load is known as reactive Power (Q)

Power merely absorbed and returned in load due to its reactive properties is referred to as reactive power.

Reactive Power represent that the energy is first stored and then released in the form of magnetic field or electrostatic field in case of inductor and capacitor respectively.

Reactive power is given by Q = V I Sinθ which can be positive (+ve) for inductive loads and negative (-ve) for capacitive load.

The unit of Reactive Power is Volt-Ampere reactive i.e. VAR where 1 VAR = 1V x 1A.

In more simple words, in Inductor or Capacitor, how much magnetic or electric field produced by 1A x 1V is known as  the unit of Reactive Power.

Reactive Power Formulas:

  • Q = V I Sinθ
  • Reactive Power = √ (Apparent Power2– True power2)
  • VAR = √ (VA2 – P2)
  • kVAR = √ (kVA2 – kW2)

Where:

  • θ = Phase angle

Apparent Power: (S)

The Product of voltage and current if and only if the phase angle differences between current and voltage are ignored.

Total power in an AC circuit, both dissipated and absorbed/returned is referred to as apparent power

The combination of reactive power and true power is called apparent power

In an AC circuit, the product of the r.m.s voltage and the r.m.s current is called apparent power which is denoted by (S) and measured in units of Volt-amp (VA).

It is the product of Voltage and Current without phase angle.

The unit of Apparent power (S) VA i.e. 1VA = 1V x 1A.

When the circuit is pure resistive, then apparent power is equal to real or true power, but in inductive or capacitive circuit, (when Reactances exist) then apparent power is greater than real or true power.

Apparent Power Formulas:

  • S = V I
  • S = √ (P + Q2)
  • Apparent Power = √ (True power2 + Reactive Power2)
  • kVA = √kW2 + kVAR2

Complex Power: (S = P+jQ or S=VI*)

The Complex sum of Real Power (P) and Reactive Power (Q) is known as Complex Power which can be expressed like S = P+jQ and measured in terms of Volt Amps Reactive (generally in kVAR).

It may also be expressed as S=VI* where “I*” is the conjugate of the complex current I. This current “I” flows through a reactive load Z caused by the Voltage.

Complex Power Formulas:

Complex Power in Capacitive Loads
  • Z = R – jXC
  • I = IP + jIQ
  • Cosθ = R / |Z| (leading)
  • I* = IP – jIQ
  • S = P – jQ

A Capacitive Load provide Leading VARS (i.e. it eliminates VARS and improves the overall power factor of the system). That’s why capacitors are used to correct and improve the power factor.

Complex Power in Inductive Loads
  • Z = R + jXL
  • I = IP – jIQ
  • Cosθ = R / |Z| (lagging)
  • I* = IP + jIQ
  • S = P + jQ

Where:

  • Z = Impedance
  • R = Resistance
  • XL = Inductive Reactance
  • XC = Capacitive Reactance
  • Cosθ = Power Factor
  • P = Active Power
  • S = Apparent Power
  • Q = Reactive Power

An Inductive Load provide lagging VARS (i.e. it added VARS and decrease the overall power factor.)

Complex power can also be expressed by the following formula.

Complex Power Formulas

Good to know:

Resistor absorbs the real power and dissipates in the form of heat and light.

Inductor absorbs the reactive power and dissipates in the form of magnetic field

Capacitor absorbs the reactive power and dissipates in the form of electric or electrostatic filed

Power Triangle

∴ Active, Reactive, Apparent Power and Power factor are trigonometrically related to each other as shown in below figure (Power Triangle).Power Triangle - Active Reactive Apparent and Complex Power

For easy explanation, all the related quantities can be easily understand by the funny Lays Chips  and Beer Analogy for Real or True or Active Power, Reactive Power , Apparent Power and power factor.

Active, Reactive, Apparent and Complex Power
Lays Chips  Analogy of Active, Reactive, Apparent power & power factor
Beer Analogy of Active power, Reactive power, Apparent Power and Power factor
Beer Analogy of Active power, Reactive power, Apparent Power and Power factor

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28 Comments

  1. The analogy with the beers and the chips are simply awesome!!!! specially the one with the beers. Can I use the beer image for one of my presentations? how can I get this image? thanks in advance.<br />

  2. Electrical Technology says:

    You are Welcome dear

    1. why electrical engineering is much more complex than other technologies

    2. thank alot .complex power explain .pls

  3. Anonymous says:

    The beer and chip diagrams are deceptive. They imply True Power and Reactive Power when added together are Apparent Power. It does not. It is the the square of true power and the square of reactive power which equals the square of apparent power.

    1. It’s Vector Addition…

  4. Actually u people made electrical is very easy…hatsoff to all ur people efforts…once upon a time i was very confused about this subject..but by d grace of allah nw all my confusion was gone…if i go good position in my carrier one of d credit goes to u people also..thanks a lot..keep it up..

  5. Anonymous says:

    No this is very simple farmula but using to mind so all heard also question simple but some solution these are hord, I can do this question but all people some many help me…?

  6. Please help me to understand s=vi* i am seeing this in many places. Why we are putting * near current. I know it is conjugate but I want to know what is the need for that

    1. same confusion if got answer plz cnfrm me

      Please help me to understand s=vi* i am seeing this in many places. Why we are putting * near current. I know it is conjugate but I want to know what is the need for that

  7. vishwa prathap shetty says:

    plz say me in uae all use std for power factor pf=0.9….? can yu explain me properly

  8. STANLEY ADJEI says:

    how can i compute for values of R,L, and C in an AC circuit

  9. jaimon james says:

    pls help active power and reactive power

  10. Article is really awesome and full of information. You can also see information about active reactive and apparent power here.

  11. Explanation with lays example is really awesome..it helped me a lot thank you

  12. Thanks,
    Really very simple to understand. thanks a lot

  13. sopno bilashi ismail says:

    book. Q. c,r,r

  14. Cammy Christmas says:

    Practical commentary . I am thankful for the insight ! Does anyone know where I can get ahold of a template UK VT79 form to fill in ?

  15. awesome explanation ……. it helps us to understand very well and also help to .. remember in future …thanks

  16. Can I use this in star and delta to fund out real ,apprentice and reactive power

  17. IF S is known than how can we calculate values of R and Q ????

  18. This is the most simple explanation! The beer and chips example is very humorous and it cleared my doubts! Thanks.

  19. Eng. Abdolgabar Ahmed Mahmood says:

    HHHHHHHHHHHH I like the LAYS example it represents the reality

  20. How does a negative power factor look when it is graphed? How will this impact apparent power versus true power versus reactive power? And how do we interpret this as per power dissipated versus power absorbed/returned to the system?

  21. Donald Livesay says:

    Quote; Inductor absorbs the reactive power and dissipates in the form of magnetic field”
    Absolutely disagree with this statement.
    Inductors absorbs real power ( Kinetic Energy) in the form of a magnetic field then dissipates in the form of reactive potential. this reactive potential is then a completely separate energy quanta not being associated with the original energy quanta that created the field in the first place.

    1. I think both statements are true cos inductors are just coils in a motor or an alternator/generator and they perform opposite functions……

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