Dr. Kasra Eeadi February, 7
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.
Sinusidal Wave hree ways lk a he wave: r A a fixed lcain r Snapsh X perid u Phase Phase velciy Wavelengh
Sinusidal Wave Wavelengh and frequency f λ u p secnd Wavelengh, [] r 6 3/4 Frequency, f [/s]
Sinusidal currens and vlages angular frequency v v cs θ Peak value Phase angle csθ πf
unis v cs 6 [Rad/s] r [degrees]? [degrees] sin z cs z 9 v sin 3 cs 6
Perid, v cs θ πf π [s] f
Pwer and Energy fr Resisances wih D Surces - R Pwer delivered a resisr: p R Energy delivered a resisr: E pd p
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
Average Pwer: R d d R d p P avg R P rs avg R P D r-ean-square rs d
P avg R rs rs d P avg rs R rs d
RMS alue f a Sinusid rs d cs θ rs cs θ d cs z csz rs [ cs θ ] d
[ ] rs d cs θ sin sin θ θ rs rs sin θ ake inegral
sin sin θ θ rs sin sin4 θ θ π π π rs 7. sin θ
rs. 7 v v cs θ P avg R rs
& 6 Hz 6. s f 6 67 ax 7 rs ax
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
plex nuber R ez z z x y Real par aginary par
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
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
L.H S sin - R - F Z L L 9 S - R - Z Ω
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
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
x cs x Ree cs θ Ree [ ] [ ] [ ] θ Re e θ e Re e e θ θ
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: 5 5 5 e 5 Re 5 [ cs sin ] 5
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: 3.868 3.86 8 3.86 Re 6 3 e 6 3 3.86 8 8 e [ cs3 sin3 ] 6
5 3 6 5e 3 e 6 5cs 3 5sin3 cs 6 sin 6 [ 5cs3 cs 6 ] [ 5sin3 sin 6 ] 9.33 6.6 9.33 6.6 5 3 7 5 9 [ 3 6 ].5 9.5e.5cs9.5sin9. 5 3 5e 7 e 3 3 5 3 7.5 e.5e 7 7 5 e e.5cs [ sin3 ].87.5.5 [ 7 sin 7 ].34. 94 5 cs3 cs.5sin.87.49
Exaple cs 5sin 6 5cs w way ake his cnversin - sandard rigneric calculain - Phasr calculain 9 cs θ
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
3 9 Re e 5e 5e [ e e e ] 3 9 5 5 Re [ e ] 5 3 5 9 Re 5 3 5 9 4.33.5 5 4.33.5 4.54 9.9 4.54e 9.9 Re [ ] 9.9 e e 4.54
Re [ ] 9.9 e e 4.54 [ ] 9.9 Re4.54e 4.54 cs 9.9 cs 5sin 6 5cs 9
4.54 cs 9.9 aginary θ Real -
cs 5cs 3 5cs 9 4.54 cs 9.9 9.9 Real cs 5cs 3 5 cs 9 4.54 cs 9.9
L cs - cs θ Re θ θ [ e ] e[ e e ] e[ e ] R R θ
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
cs θ Z L dl d sin θ cs θ 9 θ 9 9 θ 9 /Z
R R R R is a real nuber & are in phase.
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 9 9 9 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
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
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
L.H S sin - R - F Z L L 9 S - R - Z Ω
Z L L 9 S - R - Z Ω 9 S - Z R R Z -
9 S - Z R R Z - Z R R Z.. 7.7 45 5 5.44 45 S Z L Z R Z R 9 9 7.7 45 7.7 45 7.7 45 9 5 5 8 7.7 45 5 5 cs 8 cs
Z L L cs 8 8 9 S - R Z Ω R s 9 9 9 Z L Z R 5 5 5 5 7.7 45 8. 8 R 8 8. 9 Z 9.44 35 R Phasr Diagra S
Nde lage Analysis? F sin.h.5cs -5 9.5
-5 9.5 5 5 9.5......5 6.cs 9.7 6. 9. 7
Mesh-urren Analysis See exercise 5.9-5.
sin Φ Ri i d v s negrdifferenial Equain i?
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
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
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
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
Sep 4: Slve he phasr-dain equain s R / R s Φ Φ e R / π Φ s e an R Φ
Sep 5: Find he insananeus values Φ Φ e R e i Re cs Φ Φ R i