Average power consumed by AC

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Average power consumed by AC


WKT
E = Eo sin(wt)     ... (1) 
& I = Io sin(wt -phie)        ..... (2) 

From (1) and (2) E leads I by phase angle phie

WKT
Electric power = potential × current
P = E × I

P = Eo sin(wt) × Io sin(wt - phie) 
P = EoIo sin(wt) sin(wt - phie) 
P = Eo Io sin(wt) [sin(wt) cos(phie) - sin(phie) coswt]

P = Eo Io [ sin(wt) × sin(wt) × cos(phie) - 2/2(sin(wt) cos(wt) sin(phie) ) ]

P = Eo Io [ sin(wt) × sin(wt) × cos(phie)  - 1/2 sin(2wt) sin(phie) ]

Now average power of AC during 1 cycle

Pav = integration of P with respect to dt where upper limit is T and lower limit is 0 / T

Pav = EoIo/T [ cos(phie) ( ( integration of sin(wt)× sin(wt) with respect to dt where upper limit is T and lower limit is zero) - 1/2 sin(phie)/cos(phie) ( integration of sin(2wt) with respect to dt where upper limit is T and lower limit is 0) ) ]

As integration of sin(2wt) with respect to dt where upper limit is T and lower limit is 0 is equal to 0.

Pav = Eo Io cos(phie) /T × ( integration of sin(wt)× sin(wt) with respect to dt where upper limit is T and lower limit is 0               ..... (3) 

As WKT
Cos(2wt) = 1 - 2sin(wt) × sin(wt) 
2sin(wt) × sin(wt) = 1 - cos(2wt) 

From eq. (3) 

Pav = [Eo Io cos(phie) / T ] [ integration of  ((1-cos2wt) /2  ) with respect to dt where upper limit is T and lower limit is 0

Pav = [Eo Io cos(phie) /2T] [ (integration of 1 with respect to dt where upper limit is T and lower limit is 0) -  ( integration of cos(2wt)  with respect to dt where upper limit is T and lower limit is 0.

Pav = [Eo Io cos(phie) /2T]  × [t] where [t] bracket has upper limit T and lower limit is 0.

Pav = Eo Io cos(phie) × T/2T

Pav = Eo Io cos(phie) /2

Pav = (Eo/(2 of power of 1/2)) × (Io/(2of power of 1/2)) × cos(phie) 

It means, 

[Pav = Erms × Irms × cos(phie) ]
Where, Erms = root mean square value of AC
              Irms = root mean square value of AC
              Cos(phie) = power factor

This is the average power consumed by AC during one complete cycle. 
                          

                               Note

Power factor 

The factor cos(phie) is the Average power consumed by AC is known as power factor. 

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