Recitation13 - Electric Energy Processing: Fundamentals and...

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1 Electric Energy Processing: Fundamentals and Applications 18-418: Recitation #13 Dr. Nermeen Mahmoud December 2,2009 Final review
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2 Time Domain Representation )) ( ) ( cos( ) ( 2 ) ( )) ( cos( ) ( 2 ) ( t t t t I t i t t t E t e The time-domain representation for AC voltage and current for equilibrium steady state condition : Where E: the rms for voltage. I: the rms for current. : the peak value of voltage. :the peak value of current. : the power factor angle. E 2 I 2 A piece of equipment i(t) i(t) + - e(t)
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3 Instantaneous Power Flow in steady state 1/3 ) sin( ) cos( ) cos( ) sin( ) sin( ) sin( ) sin( ) cos( ) cos( ) cos( )) ( ) ( cos( )) ( cos( ) ( ) ( 2 ) ( )) ( ) ( cos( ) ( 2 ) ( )) ( cos( ) ( 2 ) ( : ) ( ) ( ) ( y x y x y x y x y x y x t t t t t t I t E t p t t t t I t i t t t E t e where t i t e t p A piece of equipment i(t) i(t) + - e(t)
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4 ))] ( cos( * )) ( ))[sin( ( sin( ) ( ) ( 2 ))] ( cos( * )) ( ))[cos( ( cos( ) ( ) ( 2 ) ( ))] ( sin( * )) ( ))[sin( ( cos( ) ( ) ( 2 ))] ( cos( * )) ( ))[cos( ( cos( ) ( ) ( 2 ) ( ))] ( sin( * )) ( sin( )) ( cos( * )) ( ))[cos( ( cos( ) ( ) ( 2 ) ( t t t t t t I t E t t t t t t I t E t p t t t t t t I t E t t t t t t I t E t p t t t t t t t t t I t E t p Instantaneous Power Flow in steady state 2/3
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5 ) ( ) ( ) ( ))} ( ( 2 )){sin ( sin( ) ( ) ( ) ( ))} ( ( 2 cos 1 )){ ( cos( ) ( ) ( ) ( ))} ( ( 2 )){sin ( sin( ) ( ) ( ))} ( ( 2 cos 1 )){ ( cos( ) ( ) ( ) ( t q t t p t t t t I t E t q t t t t I t E t t t t t I t E t t t t I t E t p Instantaneous Power Flow in steady state 3/3 [1]
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6 The Active Power & Energy ) 0 ( )) ( 2 sin 2 1 ( cos ) ( ) ( ))} ( ( 2 cos 1 )){ ( cos( ) ( ) ( ) ( ) ( ) ( 0 0 p t t w t t EI dt t t w t t t t I t E t dt t p t w [1]
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7 The Reactive Power & Energy ) 0 ( ) ( 2 cos sin 2 1 ) ( ) ( ))} ( ( 2 )){sin ( sin( ) ( ) ( ) ( 0 q t q w t EI dt t q t w t t t t I t E t q [1]
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8 Circuit Element
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9 Phasor )) ( cos( 2 ) ( t t E t e E )) ( ) ( cos( ) ( 2 ) ( t t t t I t i I A piece of equipment i(t) i(t) + - e(t)
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10 Phasor )) ( cos( 2 ) ( t t E t e E )) ( ) ( cos( ) ( 2 ) ( t t t t I t i I A piece of equipment i(t) i(t) + - e(t)
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11 Representing Steady state AC Circuits ) cos( ) ( max t V t v Time Domain Phasor Domain rms V V ) sin (cos j V V e V V rms j rms For simplicity, is usually written just as V V Polar Exponential Rectangular 2 max V V rms
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12 Complex Plane • If z= x+ j y where • If z= x- j y 1 j z=x+j y z=x-j y
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13 Impedance and Admittance Impedance Z=R+jX Where: R is the resistance X is the reactance Admittance Y=1/Z= G+jB Where : G is the admittance B is the susceptance In time domain In Phasor domain
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14 Comparison of dc and ac circuit dc circuit Steady state ac Circuit R I I V V P P P I V P I V P IR V V V 2 2 1 2 1 2 2 1 1 2 1 ) ( I V V S S S I V S I V S IZ V V V ) ( 2 1 2 1 2 2 1 1 2 1
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15 Capacitance c X c C j Y 1 The Capacitive susceptance (+ve) The Capacitive reactance (-ve) C j c X 1 V I [3]
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16 Inductance L X L L j Y 1 1 L j X L The Inductive reactance (+ve) The Inductive susceptance (-ve) V I [4]
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17 Complex Power
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This note was uploaded on 09/20/2010 for the course ECE 18418 taught by Professor Illic during the Fall '09 term at Carnegie Mellon.

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Recitation13 - Electric Energy Processing: Fundamentals and...

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