3-Lesson_Notes_Lecture_24 - EE 204 Lecture 24...

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EE 204 Lecture 24 Instantaneous, Average and Apparent Powers & Power Factor Average Value of a Periodic Function of Time: The average value of the periodic function () f t is given by: net area under one period period () o o tT t avg ftd t f T + = = Where o t = an arbitrary time instant and T = period of ( ) f t Figure 1 Average Value of the Sinusoidal Function: The general sinusoidal function ( ) cos( ) ft A t ω θ = + is periodic f t has an average value The positive and negative areas under one period T cancel each other perfectly The net area under one period of ( ) f t is zero net area under one period 0 0 period avg f T == = the average value of a general sinusoidal function of time is zero
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t f(t) Figure 2 Instantaneous Power in Sinusoidal Circuits: The current and voltage in any element of a sinusoidal circuit can be expressed as: () cos ( )[ ] i it I t A ω θ =+ & ( ) cos( v vt V t V The instantaneous power absorbed by the element is: ()() cos( )cos( iv p ti t v tI V t t W θω == + + [Note: if ( ) enters from the (-) side of ( ) p t v t = − ] ( )cos ( ) pt IV t t =++ is called the instantaneous power, because it changes from one time instant to another + v(t) i(t) Arbitrary Element Figure 3 Average Power in Sinusoidal Circuits:
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The instantaneous power ( ) cos( )cos( ) vi pt IV t t ω θω θ =++ is actually a periodic function of time. Using the trigonometric identity: 11 cos( )cos( ) cos( ) cos( ) 22 A BA B =− ++ We can express ( ) cos( ) t t as: () cos ( ) ( ) p t IV t t IV t t θωθ ωθωθ =+ + + ( ) (2 ) t + + + ( ) ) p t IV IV t ωθθ + + Where θθ ≡− (the phase difference between voltage and current) The average value of () can then be obtained as follows: avg[ cos( ) cos(2 )] avg v i Pp I V I V t == + + + avg[ cos( )] avg[ cos(2 )] PI V I V t + + 1 avg[ cos(2 )] 0 2 IV t ++ = (why?) avg[ cos( )] cos( ) IV IV = (because 1 cos( ) 2 IV
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This note was uploaded on 01/16/2010 for the course EE ee204 taught by Professor Profosama during the Spring '09 term at King Fahd University of Petroleum & Minerals.

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3-Lesson_Notes_Lecture_24 - EE 204 Lecture 24...

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