Second-Order Circuits

Transcription

1 //4 econ-orer rcus Inroucon Fnng Inal an Fnal Values The ource-free eres R rcu The ource-free Parallel R rcu ep Response of a eres R rcu ep Response of a Parallel R rcu General econ-orer rcus Dualy pplcaons Inroucon secon-orer crcu s characerze by a secon-orer fferenal equaon. I cons of reors an he equalen of wo energy sorage elemens.

2 //4 Fnng Inal an Fnal Values,, /, /,, an Two key pons: an are efne accorng o he pae sgn conenon. _ + onnuy properes: apacor olage: V -lke Inucor curren: I -lke Example Q : Fn a,, b, c,., ol : a pply c analy for. 4 4 V 4 V

3 //4 on ol : c pply c analy for. V on ol : b To fn : nce he nucor curren canno change abruply. The nucor can be reae as We can easly fn a curren source n hs case. V/s = + 3

4 //4 ol : b To fn : nce he capacor olage canno change abruply. The capacor can be reae as a olage source n hs case. To oban Thus we hae /s. 5 on, applyng KV ges = + The ource-free eres R rcu ume nal conons: I V pplyng KV ges R R 3 To sole 3, s requre. an ges R V a b RI V 4 R Inal conons : I RI V 4

5 //4 R Inal conons : I RI e e s V : an s are consans. s R s s s e se e s R e s s R haracersc s s equaon on R R s s s s R R where Naural frequences R Dampng facor Resonan frequency or unampe naural frequency haracersc equaon : s s s s R where ummary Two soluons f s general soluon : e where e s an he nal conons., s e e s s s : are eermne from Three cases scue Oerampe case snc real roos : > rcally ampe case repeae real roo : = Unerampe case complex-conjugae roos: < 5

6 //4 Oerampe ase > R 4 R Boh s an s are negae e s e s an real. s e s e rcally ampe ase = e s s e 4 R ngle consan can' sasfy wo nal conons! e R e Back o he orgnal fferenal equaon. 3 e f f f e e e e e f e e 6

7 //4 7 rcally ampe ase on e e e Unerampe ase < where sn cos sn cos sn cos sn cos where 4 e B B j B B e B B j B B e j B j B e e B B e e e B B e j s j s R j j j j

8 //4 Unerampe ase on R cos sn e, α e. Fnng The onsans, To eermne we nee an /. I an. KV a ges RI V or RI V, 8

9 //4 onclusons The concep of ampng The graual lo of he nal sore energy Due o he reance R Oscllaory response s poble. The energy s ransferre beween an. Rngng enoes he ampe oscllaon n he unerampe case. Wh he same nal conons, he oerampe case has he longes selng me. The unerampe case has he fases ecay. If a consan s aume. Example Fn. 6+3 < > 9

10 //4 Example on < > e j ω α α s., ω R α b a, sn cos V 6, : conons Inal R - The ource-free Parallel R rcu becomes equaon characersc he, e 3 ges K pplyng b a conons: nal ume s R s e R R V I s R s where,

11 //4 ummary Oerampe case : > s s e e rcally ampe case : = e Unerampe case : < s j, where e cos sn Fnng The onsans, To eermne an, we nee an /.. V. K a ges V I R V RI or R

12 //4 eres R rcu omparsons Parallel R rcu s, R where Inal conons : I V RI s, where R Inal conons : V V RI R Fn for >. = 5 V, = onser hree cases: R =.93 R = 5 R =6.5 Example ase: R.93 α 6, ω R s, e α α ω e 5, 5 Inal conons : 5 R 6 R

13 //4 Example on ase : R 5 α, ω R s, s, α α α ω e cos 6 sn 6 ase 3: R 6.5 α 8, ω R α ω 8 j 6 8 e Inal conons : 5 R R 5 5 Inal conons : 5 R 8 R Example on 3

14 //4 Example Fn. Ge x. Ge x, x/, s,,,. < > Example on > α 5 R ω 354 s, e α 854 α ω 854, 46 e 46 < From he nal conons : V R R

15 //4 ep Response of eres R rcu pplyng KV for, R Bu has V R V he same form as n he source - free case. where : he ransen response : he seay - sae response haracersc Equaon R V ' e V, ' ' ' R The characersc equaon becomes R s s ame as n he source - free case. 5

16 //4 ummary where V s s e e Oerampe e rcally ampe cos sn e Unerampe where are obane from an /., Fn, for >. onser hree cases: R = 5 R = 4 R = Example Ge x. Ge x, x/, s,,,. < > 6

17 //4 ase : R = 5 < > R 5 α.5 ω s, α α ω, 4 e e 4 4 V Inal conons: 4 4, 4 V ase : R = 4 R 4 α ω s α, e 4 V Inal conons : 4 4.8, 4.8 V

18 //4 8 ase 3: R = e j s ω R α, sn.936 cos V, 4 conons : Inal 4 Example on

19 //4 ep Response of Parallel R rcu pplyng K for, R Bu R has I I he same form as n he source - free case. where : he ransen response : he seay - sae response haracersc Equaon I R ' e I, ' ' ' R The characersc equaon becomes s s R ame as n he source - free case. 9

20 //4 where I ummary s s e e Oerampe e rcally ampe cos sn e Unerampe where are obane from an /., General econ-orer rcus eps requre o eermne he sep response. Deermne x, x/, an x. Fn he ransen response x. pply K an KV o oban he fferenal equaon. Deermne he characersc roos s,. Oban x wh wo unknown consans,. Oban he seay-sae response x = x. Use x = x + x o eermne, from he wo nal conons x an x/.

21 //4 Example Fn, for >. Ge x. Ge x, x/, s,,,. < > Example on < > Inal conons : V a b pplyng K a noe a, 6 V/s Fnal alues for : 4 4 V c

22 //4 Example on pplyng K a noe a ges pplyng KV o he lef mesh ges 4 3 ubsung no 3 ges haracersc equaon : s 5 s 6 > s, 3 4 where 3 e e From a an c we oban, 8 can be oban by usng Dualy Dualy means he same characerzng equaons wh ual quanes nerchange. Reance R Inucance Volage Volage source Noe eres pah Open crcu KV Theenn Table for ual pars onucance G apacance urren urren source Mesh Parallel pah hor crcu K Noron

23 //4 3 ase uy KV : n n n f f f K : n n n f f f n -. n + _ Elemen Transformaons Volage urren apacance Inucance onucance I I I V V V R R R

24 //4 Example eres R rcu Parallel R rcu R R s s s s Example 4

25 //4 pplcaon: moohng rcus Oupu from a D/ conerer s 5