Sinusoidal Steady-State Analysis
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- James Dalton
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1 // Snusdal Steady-State Analyss ntrductn dal Analyss Mesh Analyss Superpstn Therem Surce Transfrmatn Thevenn and rtn Equvalent Crcuts OP-amp AC Crcuts Applcatns ntrductn Steps t Analyze ac Crcuts: The natural respnse (due t ntal cndtns) s gnred. Transfrm the crcut t the phasr (frequency) dman. Slve the prblem usng crcut technques (ndal/mesh analyss, superpstn, etc.). Transfrm the resultng phasr t the tme dman.
2 // dal Analyss arables = de ltages. Applyng KCL t each nde gves each ndependent equatn. Supernde f superndes ncluded, Applyng KCL t each supernde gves equatn. Applyng KL at each supernde gves mre equatn. Fnd x. Example cs4t 4 rad / s L L C C H 4.5 H.F.5
3 // Example (Cnt d) Applyng KCL at nde, (.5).5 Applyng KCL at nde, x x 5.5 () () () and () can be put n matrx frm as x x 7.59 cs(4t 8.4 ) Applyng KCL fr the supernde gves equatns. Applyng KL at the supernde gves equatns. varables slved by equatns. Example
4 // Example (Cnt d) Sl : Applyng KCL at the supernde, 6 r But 6 4 ( ) Mesh Analyss arables = Mesh Currents. Applyng KL t each mesh gves each ndependent equatn. f supermeshes ncluded, Applyng KL t each Supermesh supermesh gves equatn. Applyng KCL at each supernde gves mre equatn. S Excluded 4
5 // Example Sl : Applyng KL t mesh, (8 ) ( ) Fr mesh, (4 ) ( ) Fr mesh, ( ) 9 5 () () Fnd. Applyng KL fr mesh & gves equatns. Applyng KL fr the supermesh gves equatns. Applyng KCL at nde A gves equatns. 4 varables slved by 4 equatns Example Fnd. 5
6 // Example (Cnt d) Sl : Fr mesh, (8 ) ( ) (8 ) Fr mesh, Fr supermesh, (8 4) 8 8 (6 5) () () () Applyng KCL at nde A gves varables can be slved by usng 4 equatns. ( ) (4) Superpstn Therem Snce ac crcuts are lnear, the superpstn therem apples t ac crcuts as t apples t dc crcuts. The therem becmes mprtant f the crcut has surces peratng at dfferent frequences. Dfferent frequency-dman crcut fr each frequency. Ttal respnse = summatn f ndvdual respnses n the tme dman. Ttal respnse summatn f ndvdual respnses n the phasr dman. 6
7 // 7 Example Fnd. = + " ' Example (Cnt d) ) (.5.647).5 ( () 5 mesh, Fr () ) ( ) ( 4) (4 mesh, Fr () 8) (8 Applyng KL t mesh,, T get ) (8 ) ( : Sl " ' " " '
8 // Example v v v v v s due t the where v s due t the v s due t the 5- dc vltage surce cs t vltage surce sn 5t current surce Example (Cnt d) Snce, L C shrt - crcut pen crcut By vltagedvsn, v (5) 4 8
9 // Example (Cnt d) 4 5 cs t, rad/s H.F L 4 C 5 v ( ) 4 ( 5 4) cs( t.79 ) Example (Cnt d) 9 A sn 5t 9 5 rad/s H.F L C By current dvsn, (9 ) ( 4) v ( ) cs(5 t 8 ) 9
10 // Surce Transfrmatn Example Fnd x. s 9 s ( 4) s s 5 ( 4) (.5.5) 5 By vltage dvsn, x (5 ) s
11 // Thevenn & rtn Equvalent Crcuts Th Th Example Th Th (8 6) (4 ) (75 )
12 // Example Example (Cnt d) Applyng KL at ndegves 5.5 Applyng KL t the lp gves Th ( 4).5 ( 4) 5(4 ) (4 ) Th Set s.5 Applyng KL s Th s s fr smplcty, gves (4 4) (6 ) s (6 )
13 // Example By current dvsn, ( 5) Example (Cnt d) () () Apply mesh analyss t get Applyng KL fr meshgves 4 (8 ) Applyng KL fr the supermesh gves ( ) can be fund easly, Applyng KCL at nde a gves ( 4) () (), (), and () gve (8 ) 5 (8 ) ( 4) A. () ()
14 // OP AMP AC Crcuts: Example deal p amps wth negatve feedback assumed. er nput current & zer dfferental nput vltage. Applyng KCL at nde, 5 6 (5 4) - Applyng KCL at nde, () and () gve () () v ( t) 9cs(. t 59. 4) v s cst Example Sl : G s C C C ( C )( C ) f ( 4)( ) Fnd the clse- lp gan and phase shft. C C F F k rad/s Clsed lp gan : Phase shft :
15 // 5 Applcatns: Capactance Multpler C C C C ) ( ) ( C C C eq eq where But Applcatns: Oscllatrs v
16 // Oscllatn Cndtns Barkhausen crtera must be meet fr scllatrs.. Overall phase shft (nput-t-utput-t-nput) = AH = The scllatn frequency can be determned.. Overall gan f the scllatr = AH Lses must be cmpensated by an amplfyng devce. The mnmum gan requrement can be determned. + + A v A v H v f H C C f and C C C, C C ( C C ) Example The phase requrement gves C C fr The gan requrement gves f f g g 6
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