ECE4762011_Lect2 - Lecture 2 Complex Power, Reactive...

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Unformatted text preview: Lecture 2 Complex Power, Reactive Compensation, Three Phase Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS 2 Announcements For lectures 2 through 3 please be reading Chapters 1 and 2 3 Review of Phasors Goal of phasor analysis is to simplify the analysis of constant frequency ac systems v(t) = V max cos( t + v ) i(t) = I max cos( t + I ) Root Mean Square (RMS) voltage of sinusoid 2 max 1 ( ) 2 T V v t dt T = 4 Phasor Representation j ( ) Euler's Identity: e cos sin Phasor notation is developed by rewriting using Euler's identity ( ) 2 cos( ) ( ) 2 Re (Note: is the RMS voltage) V V j t j v t V t v t V e V + = + = + = 5 Phasor Representation, contd The RMS, cosine-referenced voltage phasor is: ( ) Re 2 cos sin cos sin V V j V j j t V V I I V V e V v t Ve e V V j V I I j I = = = = + = + (Note: Some texts use boldface type for complex numbers, or bars on the top) 6 Advantages of Phasor Analysis 2 2 Resistor ( ) ( ) ( ) Inductor ( ) 1 1 Capacitor ( ) (0) C Z = Impedance R = Resistance X = Reactance X Z = =arctan( ) t v t Ri t V RI di t v t L V j LI dt i t dt v V I j C R jX Z R X R = = = = + = = + = + Device Time Analysis Phasor (Note: Z is a complex number but not a phasor) 7 RL Circuit Example 2 2 ( ) 2100cos( 30 ) 60Hz R 4 3 4 3 5 36.9 100 30 5 36.9 20 6.9 Amps i(t) 20 2 cos( 6.9 ) V t t f X L Z V I Z t = + = = = = = + = = = = = - =- 8 Complex Power max max max max ( ) ( ) ( ) v(t) = cos( ) (t) = cos( ) 1 cos cos [cos( ) cos( )] 2 1 ( ) [cos( ) 2 cos(2 )] V I V I V I p t v t i t V t i I t p t V I t = + + =- + + =- + + + Power 9 Complex Power, contd max max max max 1 ( ) [cos( ) cos(2 )] 2 1 ( ) 1 cos( ) 2 cos( ) = = V I V I T avg V I V I V I p t V I t P p t dt T V I V I =- + + + = =- =-- Power Factor Average P Angle ower 10 Complex Power [ ] * cos( ) sin( ) P = Real Power (W, kW, MW)...
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ECE4762011_Lect2 - Lecture 2 Complex Power, Reactive...

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