Dr. Kasra Etemadi February 20, 2007

Transcription

1 Dr. Kasra Eeadi February, 7

2 Seady-Sae Sinusidal Analysis Sinusidal Surces: Elecric pwer disribued fr residences and businesses Radi cunicain All signal f pracical ineres are cpsed f sinusidal cpnens Furier Analysis Seady-Sae: Respnse f a newrk has w pars haper 4: he frced respnse and naural respnse. When he frced respnse fr sinusidal surces persis indefiniely, i is called he seady-sae respnse.

3 Sinusidal Wave hree ways lk a he wave: r A a fixed lcain r Snapsh X perid u Phase Phase velciy Wavelengh

4 Sinusidal Wave Wavelengh and frequency f λ u p secnd Wavelengh, [] r 6 3/4 Frequency, f [/s]

5 Sinusidal currens and vlages angular frequency v v cs θ Peak value Phase angle csθ πf

6 unis v cs 6 [Rad/s] r [degrees]? [degrees] sin z cs z 9 v sin 3 cs 6

7 Perid, v cs θ πf π [s] f

8 Pwer and Energy fr Resisances wih D Surces - R Pwer delivered a resisr: p R Energy delivered a resisr: E pd p

9 R R p Pwer delivered a resisr: Energy delivered a resisr: d p E d p E Energy delivered a resisr in ne perid: Average Pwer: d p E f E P avg Pwer and Energy fr Resisances wih A Surces

10 Average Pwer: R d d R d p P avg R P rs avg R P D r-ean-square rs d

11 P avg R rs rs d P avg rs R rs d

12 RMS alue f a Sinusid rs d cs θ rs cs θ d cs z csz rs [ cs θ ] d

13 [ ] rs d cs θ sin sin θ θ rs rs sin θ ake inegral

14 sin sin θ θ rs sin sin4 θ θ π π π rs 7. sin θ

15 rs. 7 v v cs θ P avg R rs

16 & 6 Hz 6. s f 6 67 ax 7 rs ax

17 Many engineering prbles are cas in he fr f linear inegrdifferenial equain, in which exciain frcing funcin varies sinusidally wih ie. negr-differenial equain plex slve sluin Phasr nain back ie dain Linear equain wih n sinusidal funcin; i is sipler slve

18 plex nuber R ez z z x y Real par aginary par

19 hree frs f represenains fr a cplex nuber -Recangular fr z x y -Plar fr phasr nain: z z e θ 3-Graphic Fr z z Rez

20 Relain ang recangular, plar and graphic represenain -Recangular fr x y xy z z csθ sinθ -Plar fr phasr nain: z z e x θ y θ an y / x 3-Graphic Fr Uni ecr θ e csθ sinθ sinθ csθ

21 L.H S sin - R - F Z L L 9 S - R - Z Ω

22 3-Graphic Fr Uni ecr θ z e z csθ z sinθ θ e csθ sinθ csθ sinθ θ x Re z e θ Re e csθ θ e cs θ Re θ Re e Re e Re e cs θ Re e

23 3-Graphic Fr Uni ecr θ z e z csθ z sinθ θ e csθ sinθ csθ sinθ θ x Re z e θ Re e csθ cs θ Re e e θ θ e csθ Re θ e csθ x y sinθ e θ θ Re

24 x cs x Ree cs θ Ree [ ] [ ] [ ] θ Re e θ e Re e e θ θ

25 3-Graphic Fr Uni ecr z e θ z csθ z sinθ θ e csθ sinθ csθ sinθ θ x Re z e θ Re e csθ Real Nuber: e 5 Re 5 [ cs sin ] 5

26 3-Graphic Fr Uni ecr z e θ z csθ z sinθ θ e csθ sinθ csθ sinθ θ x Re z e θ Re e csθ plex Nuber: Re 6 3 e e [ cs3 sin3 ] 6

27 e 3 e 6 5cs 3 5sin3 cs 6 sin 6 [ 5cs3 cs 6 ] [ 5sin3 sin 6 ] [ 3 6 ].5 9.5e.5cs9.5sin e 7 e e.5e e e.5cs [ sin3 ] [ 7 sin 7 ] cs3 cs.5sin.87.49

28 Exaple cs 5sin 6 5cs w way ake his cnversin - sandard rigneric calculain - Phasr calculain 9 cs θ

29 cs 5sin 6 5cs 9 cs 5cs 6 9 5cs 9 cs 5cs 3 5cs 9 Re 3 e 5Re e 9 5Re e Re 3 e Re 5e 9 Re 5e 3 9 Re e 5e 5e

30 3 9 Re e 5e 5e [ e e e ] Re [ e ] Re e 9.9 Re [ ] 9.9 e e 4.54

31 Re [ ] 9.9 e e 4.54 [ ] 9.9 Re4.54e 4.54 cs 9.9 cs 5sin 6 5cs 9

32 4.54 cs 9.9 aginary θ Real -

33 cs 5cs 3 5cs cs Real cs 5cs 3 5 cs cs 9.9

34 L cs - cs θ Re θ θ [ e ] e[ e e ] e[ e ] R R θ

35 plex pedances L sin L L d L d L cs-9 L L cs θ 9 L θ L θ Z L L L θ L L 9 9 L L 9 L L L Z L L

36 cs θ Z L dl d sin θ cs θ 9 θ 9 9 θ 9 /Z

37 R R R R is a real nuber & are in phase.

38 urren lags lage θ L L θ θ 9 Re R θ urren in phase wih lage θ R θ Re θ 9 urren leads lage θ θ Re L cs θ 9 L sin θ d L L L d cs θ θ 9 cs θ θ R R R cs θ cs θ cs θ θ d sin θ d cs θ 9 Pure nducance Pure Resisance Pure apaciance θ 9

39 ircui Analysis wih Phasrs and plex pedance n KL and KL currens,, and vlages,, are replaced by heir phasrs,, respecively. & Phasr currens and lages are relaed by cplex ipedances. Replace inducances and capaciances by heir cplex ipedances. Z L L & Z

40 Exaple 5.4 Series and Parallel binains f plex pedances Given: L.H S sin - R - F Find:.. Phasr curren hrugh each eleen 3. nsruc a phasr diagra shwing he currens and he surce vlage

41 L.H S sin - R - F Z L L 9 S - R - Z Ω

42 Z L L 9 S - R - Z Ω 9 S - Z R R Z -

43 9 S - Z R R Z - Z R R Z S Z L Z R Z R cs 8 cs

44 Z L L cs S - R Z Ω R s Z L Z R R Z R Phasr Diagra S

45 Nde lage Analysis? F sin.h.5cs

46 cs

47 Mesh-urren Analysis See exercise

48 sin Φ Ri i d v s negrdifferenial Equain i?

49 Suary f he five seps Sep : Adp a csine reference Express he frcing funcin as a csine Sep : Express ie-dependen variables as phasrs Any csinusidally ie-varying funcin z can be expressed in he fr Sep 3: Recas he differenial/inegral equain in phasr fr Sep 4: Slve he phasr-dain equain Sep 5: Find he insananeus values

50 Sluin in five seps Sep : Adp a csine reference Express he frcing funcin as a csine Ri i d v s v s v s sin Φ cs Φ π π sin x cs x cs x π Ri i d cs Φ π

51 Re Re Re s e e e R Sep : Express ie-dependen variables as phasrs Any csinusidally ie-varying funcin z can be expressed in he fr Φ cs π d i i R ] Re[ e Z z ] Re[ ] Re[ e e e π π Φ Φ d e d e Re ] Re[ Re e i ie-independen funcin called Phasr nsananeus funcin

52 Sep 3: Recas he differenial/inegral equain in phasr fr Re Re Re s e e e R s R R & are real Re perain is disribuive Phasr Fr Re Re s e e e R

53 Sep 4: Slve he phasr-dain equain s R / R s Φ Φ e R / π Φ s e an R Φ

54 Sep 5: Find he insananeus values Φ Φ e R e i Re cs Φ Φ R i