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Power_Formulas2 (2)

# Power_Formulas2 (2) - POWER FORMULAS I Time Domain...

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POWER FORMULAS I. Time Domain Functions If: v ( t ) = V M · cos( ωt + Θ V ) i ( t ) = I M · cos( ωt + Θ I ) then: p ( t ) = V M I M cos( ωt + Θ V ) cos( ωt + Θ I ) = V M I M 2 [cos(Θ V - Θ I ) + cos(2 ωt + Θ V + Θ I )] and since Θ Z = Θ V - Θ I , p ( t ) = V M I M 2 [cos(Θ Z ) + cos(2 ωt + Θ V + Θ I )] (in Watts). II. Average Power (aka. “real power”) (A) Average Power : P = Z t 0 + T t 0 p ( t ) dt = V M I M 2 cos(Θ Z ) , where T = 1 f (in Watts) (B) For pure resistances , Θ Z = 0, so: P = V M I M 2 = I 2 M Z M 2 = V 2 M 2 Z M (C) For pure reactances , Θ Z ∈ {- 90 , 90 } , so cos(Θ Z ) = 0, thus: P = 0 (average power). III. Root Mean Square (A) For any time domain function K ( t ): K RMS = s 1 T · Z t 0 + T t 0 ( K ( t )) 2 dt (B) In the case where K ( t ) is a sinusoid of magnitude K M : K RMS = K M 2 (C) For power (in Watts): P = V RMS I RMS cos(Θ Z ) = ( I RMS ) 2 · R = ( V RMS ) 2 R 1

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IV. Power Factor (A) Power Factor pf : pf = cos(Θ V - Θ I ) = cos(Θ Z ) (No units.) (B) Apparent Power P A : P A = P pf (Average power over power factor) = | V RMS | · | I RMS | (C) For pure resistance/reactance: For pure resistances : pf = 1 For pure reactances : pf = 0 .
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