Average Power in an AC circuit

P = dH/dt

dH = P.dt

dH = V.I dt                                     

dH = V₀ sin w t. I₀ sin (w t + θ)dt

dH = V₀.I₀ sin w t. sin (w t + θ)dt

= V₀.I₀ sin wt (sin w t cosθ + cos w t sinθ)dt

= V₀.I₀ [sin² w t cosθ + sin w t cos w t sinθ]dt

= V₀.I₀ [((1 – cos2 w t)/2). cosθ + (sin2 w t sinθ)/2]dt

= (V₀.I₀/2) [cosθ – cosθ cos2 w t + sinθ sin2 w t]dt

= [sin2 w t/2 w]^T₀

= (1/2 w) [sin(2 w t) – sin(2 w x 0)]

= (1/2 w) [sin4π – sinθ]

= (1/2w) [0] = 0

= (V₀.I₀/2) [cosθ  T – 0 + 0]

H = (V₀I₀/2) [cosθ. T – 0 + 0]  

H = (V₀.I₀/2) cosθ T --> Heat produced

As Avg. power = H/T

H/T = (V₀I₀/2) cosθ

P_avg = (V₀I₀/2) cosθ

P_avg = (V₀/√2). (I₀/√2) cosθ

P_avg = V_rms. I_rms cosθ

P = V_rms I_rms cosθ

The term cosθ is known as Power Factor and determines the power consumed by AC circuit.

P = VI cosθ V and I are considered to be rms values.

For impedance triangle cosθ = R/Z

And I = V/Z

P = V.I cosθ

= I Z.I = R/Z = I = R