23-Steady State AC 2

23-Steady State AC 2 - Steady-State AC Analysis II...

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Steady-State AC Analysis II Sinusoids & Phasors: &S Chp 9- 0+Transient A&S Chp 9 10+Transient Response © Fred Terry Winter, 2008 1
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AC Steady State Analysis • Real Sinusoidal Input –V in =V 0 cos( ω t) • Assume Input is Complex j [ ] 00 () cos( ) s in ( ) e jt in Vt V e V t j t l Input Signal ω == + S iti ALL Ci it Q titi f j Re () Re in in al Input Signal ⎡⎤ = ⎣⎦ • By Superposition, ALL Circuit Quantities of Interest (voltages, currents) are given by taking the Real Part of the Response to the Complex Signal 2
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Taylor Series Expansions Around x=0 (MacLaurin Series) & Euler’s Relation ( ) ( ) 23 0 0 0 0 0 nk xx x x x f x f f f f f ′′ ++ + + + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 00 2! 3! ! ! 1 ! k k x k fx f x ex k = = = = =+ + + + = …… () () () () 4 2 4 6 2 0 357 12 cos cos 0 sin 0 cos 0 sin 0 cos 0 1 1 4! 6! ! n 1 k k k k even k k x x x x x k xxx x x = =− + + + = + + = − + − + = ( ) ( ) 0 sin 3! 5! 7! ! k k = =− + − + = 23456 3456 1 kodd j xxxx ≡− 45 6 1 3! 5! 1 3! jx xxxxx ej x j j j j j x jx j j = + ++++++ =+ − − + + − + 3 () () 246 35 1 cos sin jx xj x ⎛⎞ + + + + + ⎜⎟ ⎝⎠ =+
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AC Steady State Analysis II • For a Linear Circuit: If Input Signal is at frequency ω, All Circuit Responses are at the me frequency same frequency ω – Outputs are Different only in Magnitude and Phase, not Frequency on inearities Produce Harmonics (2 3 etc ) – Non-Linearities Produce Harmonics (2 ω , 3 ω , etc.) • Multiple Frequency Problems Are Handled by Superposition • When Working a Single Frequency Problem, Issue is Finding Magnitudes and Phases not Frequency oncept of Phasor 4 Concept of Phasor
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asic Complex Numbers: Basic Complex Numbers: Polar/Rectangular Representation Im(x) b r = | x Re(x) a x θ = ( 5
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Basic Complex Math ( ) ( ) ( ) () cos sin 1 jt et j t e ith ωφ + + =+ + + = lb Im unit Example Polar Plot of z 1 1 2 11 1 22 2 , j j If are with rx j y r e j y r e ddition φ == + = + = ( ( 12 1 2 z zc o m p lex numbers z z Re circle 1 1 : , : Addition x x j y y Any EasyCombinationof r r Subtraction xx j yy A n y E a s y +=++ +≠ −=−+ −≠ (( zz 2 2 , Combinationof r r l e n g t h = φ=−60° 1 1
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23-Steady State AC 2 - Steady-State AC Analysis II...

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