// 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.
// 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
// Example (Cnt d) Applyng KCL at nde, (.5).5 Applyng KCL at nde, x.5 4 4 x 5.5 () () () and () can be put n matrx frm as.5.5 5 8.97 8.4 x.9 98. 7.59 8.4.5 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
// Example (Cnt d) Sl : Applyng KCL at the supernde, 6 r But 6 4 ( ).4 87.8 45 45 5.78 7.48 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
// Example Sl : Applyng KL t mesh, (8 ) ( ) Fr mesh, (4 ) ( ) Fr mesh, 8 8 5 4 4 6. 5. ( ) 9 5 () () 6. 44.78 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
// Example (Cnt d) Sl : Fr mesh, (8 ) ( ) (8 ) Fr mesh, Fr supermesh, (8 4) 8 8 (6 5) 8 4 5 () () () Applyng KCL at nde A gves 4 4 4 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 Example Fnd. = + " ' Example (Cnt d) 44.78 6..59 5.76) (.5.647).5 (.76.647 5 4 4 8 8 () 5 mesh, Fr () ) ( ) ( 4) (4 mesh, Fr () 8) (8 Applyng KL t mesh,, T get.5.5 4.5 4.5 4.5.5 ) (8 ) ( : Sl " ' " " '
// 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
// Example (Cnt d) 4 5 cs t, rad/s H.F L 4 C 5 v ( ) 4 ( 5 4).49.49.498.79.498cs( t.79 ) Example (Cnt d) 9 A sn 5t 9 5 rad/s H.F L C By current dvsn, (9 ) ( 4) v ( ).8 8.4. 8.cs(5 t 8 ) 9
// Surce Transfrmatn Example Fnd x. s 9 s 4 9 4 5 5( 4) s s 5 ( 4) 4 8 4 4(.5.5) 5 By vltage dvsn, x (5 ).5.5 4 5.59 8 s
// Thevenn & rtn Equvalent Crcuts Th Th Example Th Th (8 6) (4 ) 6.48.64 8 (75 ) 8 6 4 7.95.
// Example Example (Cnt d) Applyng KL at ndegves 5.5 Applyng KL t the lp gves Th ( 4).5 ( 4) 5(4 ) 55 55 9 (4 ) Th Set s.5 Applyng KL s Th s s fr smplcty, gves (4 4) (6 ) s (6 )
// 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 5 5 (8 ) 5 (8 ) ( 4).465 8.48 A. () ()
// 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) 6.9 59.4 5 v s cst Example Sl : G s C C C ( C )( C ) f 4.44.6 ( 4)( ) Fnd the clse- lp gan and phase shft. C C F F k rad/s Clsed lp gan : Phase shft :.44.6 4
// 5 Applcatns: Capactance Multpler C C C C ) ( ) ( C C C eq eq where But Applcatns: Oscllatrs v
// 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