3 ) = 10(1-3t)e -3t A
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1 haper 6, Sluin. d i ( e 6 e ) 0( - )e - A p i 0(-)e - e - 0( - )e -6 W haper 6, Sluin. w w (40)(80 (40)(0) ) ( ) w w w kw haper 6, Sluin. i d x0 480 ma haper 6, Sluin 4. i (0) 6sin cs 4 haper 6, Sluin. i (0) Fr 0 < <, i 4, kv 0x0 () 00 kv
2 Fr < <, i 8-4, (8 4) ( 0x0 00 (4 - - ) 00 kv 6 ) Thus 00 kv, () 00(4 )kv, 0 < < < < haper 6, Sluin 6. d i Fr example, fr 0 < <, 6 0x0 x slpe f he waefrm. d 0 x0 d 6 0 i 0x0 x 0mA x0 Thus he curren i is skeched belw. 0 i() (ma) 4 8 (msec) haper 6, Sluin x0 i ( ) 4x k 0 V 0
3 haper 6, Sluin 8. (a) d i 00Ae 600Be () i( 0) 00A 600B A 6B () ( 0 ) (0 ) 0 A B () Sling () and () leads A6, B- (b) Energy (0) x4x0 x00 J (c ) Frm (), i 00x6x4x0 e 600xx4x0 e 4.4e 6.4e A haper 6, Sluin 9. () 6( e ) 0 ( e ) () ( e - ).6 V p i ( e - ) 6 (-e - ) 7(-e - ) p() 7(-e -4 ) 4.68 W V haper 6, Sluin 0 i d x0 d 6, 6, 64-6, 0 < < µ s < < µ s < < 4µ s
4 6 6x0, 0 < < µ s d 0, < < µ s 6-6x0, < < 4µ s ka, 0 < < µ s i ( ) 0, < < µ s - ka, < < 4µ s haper 6, Sluin. i (0) Fr 0 < <, 40x0 0 kv 6 4x0 () 0 kv Fr < <, () 0kV Fr < <, Thus 4x0-0 0kV 0 kv, () 0kV, 0 0kV, ( 40x0 ) ( ) 6 0 < < < < < < haper 6, Sluin. d x0-0.7e π sin 4π A i x60(4π)( sin 4π ) P i 60(-0.7)π cs 4π sin 4π -.6π sin 8π W W p 8.6πsin 8π.6π / 8 cs8π -.4J 8π
5 haper 6, Sluin. Under dc cnins, he circui becmes ha shwn belw: i 0 Ω i 0 Ω 0 Ω 0 Ω 60V i 0, i 60/(000) A 0i 0V, 60-0i 40V Thus, 0V, 40V haper 6, Sluin 4. (a) eq 4 0 mf (b) eq 4 4 eq 7. mf 0 haper 6, Sluin. In parallel, as in Fig. (a), 00 00V 00V (a) (b)
6 w 0 x0x0 6 x00 0.J w 0 x0x0 6 x00 0.J (b) When hey are cnneced in series as in Fig. (b): 0 V x00 60, 40 0 w 0 x0x0 6 x60 6 mj w 0 x0x0 6 x40 4 mj haper 6, Sluin 6 eq x µ F 80 haper 6, Sluin 7. (a) (b) (c) 4F in series wih F 4 x /(6) F F in parallel wih 6F and F 6 F 4F in series wih F F i.e. eq F eq [6 (4 )] (6 6) 8F F in series wih 6F ( x 6)/9 6F 6 eq eq F
7 haper 6, Sluin 8. Fr he capacirs in parallel µf eq Hence eq 0 eq 0 µf haper 6, Sluin 9. We cmbine 0-, 0-, and 0- µ F capacirs in parallel ge 60 µ F. The 60 - µ F capacir in series wih anher 60- µ F capacir gies 0 µ F µ F, µ F The circui is reduced ha shwn belw µ F capacir in series wih 80 µ F gies (80x0)/ µ F capacir in series wih µ F gies (60x)/7 0 µ F haper 6, Sluin 0. in series wih 6 6x /(9) in parallel wih 4 4 in series wih 4 (4x4)/8 The circui is reduced ha shwn belw: 0 6 8
8 6 in parallel wih 8 8 in series wih in parallel wih in series wih 0 (x0)/ 4 Thus eq 4 mf haper 6, Sluin. 4µF in series wih µf (4x)/6 µf µf in parallel wih µf 6µF 6µF in series wih 6µF µf µf in parallel wih µf µf µf in series wih µf.µf Hence eq.µf haper 6, Sluin. mbining he capacirs in parallel, we bain he equialen circui shwn belw: a b 40 µf 60 µf 0 µf 0 µf mbining he capacirs in series gies eq eq, where eq 0µF Thus eq µf
9 haper 6, Sluin. (a) µf is in series wih 6µF x6/(9) µf 4µF / x 0 60V µf 60V 6µF ( 60) 0V 6 µf V (b) Hence w / w 4µF / x 4 x 0-6 x mJ w µf / x x 0-6 x 600.6mJ w 6µF / x 6 x 0-6 x 400.mJ w µf / x x 0-6 x 600.4mJ haper 6, Sluin 4. 0µF is series wih 80µF 0x80/(00) 6µF 4µF is parallel wih 6µF 0µF (a) 0µF 90V 60µF 0V 4µF 60V 80 0µF x µF V 48V (b) Since w w 0µF / x 0 x 0-6 x 800.mJ w 60µF / x 60 x 0-6 x 900 7mJ w 4µF / x 4 x 0-6 x 600.mJ w 0µF / x 0 x 0-6 x (48).04mJ w 80µF / x 80 x 0-6 x 44.76mJ haper 6, Sluin. (a) Fr he capacirs in series, Q Q
10 s s Similarly, s (b) Fr capacirs in parallel r Q Q Q s Q Q Q Q Q Q Q Qs dq i i, i is is haper 6, Sluin 6. (a) eq µf (b) Q x 0µ 0.7m Q 0 x 0µ.m Q 0 x 0 m (c) w eq xx0 µ J 9.8mJ
11 haper 6, Sluin 7. (a) eq 0 eq µf.87µf (b) Since he capacirs are in series, 0 Q Q Q Q eq x00µv 0.74mV 7 0 (c) w eq x x00 µ J 7.4mJ 7 haper 6, Sluin 8. We may rea his like a resisie circui and apply dela-wye ransfrmain, excep ha is replaced by /. b 0 µf a c 0 µf a 0 40 a µf b µf 00 0
12 c 40 c.7µf 00 4 b in parallel wih 0µF 0 6µF c in series wih 0µF.7µF 6x.7 6µF in series wih.7µf 7.9µ F µF in parallel wih a 7.9.9µF Hence eq.9µf haper 6, Sluin 9. (a) in series wih /() / in parallel wih / x in series wih in parallel wih.6 (b) eq eq eq
13 haper 6, Sluin 0. i i(0) Fr 0 < <, i 60 ma, x0 () 0kV 0 kv Fr < <, i 0-60 ma, 0 (0 60) 6 () x0 [40 0 ] 0kV kv, 0 < < () kV, < < haper 6, Sluin. () i s 0mA, 0mA, 0 0, 0 < < < < < < eq 4 6 0µF i (0) eq Fr 0 < <, 0 0x kv Fr < <, 0 0 kv Fr < <, () ( ) kv 0( ) ()
14 kv kv < < < < < < kv, kv, 0 kv, () d 6x0 d i 6 < < < < < < 0mA, ma, 0 ma, d 4x0 d i 6 < < < < < < 0mA, 8 8mA, 0 8mA, haper 6, Sluin. (a) eq (x60)/7 0 µ F (0) e e e x (0) e e e x (b) A 0.s, , 00 0 e e J 4. (840.) 0 6 x x x w F µ J ) ( x x x w F µ J ) ( x x x µf w
15 haper 6, Sluin Because his is a ally capaciie circui, we can cmbine all he capacirs using he prpery ha capacirs in parallel can be cmbined by jus adng heir alues and we cmbine capacirs in series by adng heir reciprcals. F F F / / / r.f The lage will ide equally acrss he w F capacirs. Therefre, we ge: V Th 7. V, Th. F haper 6, Sluin 4. i 6e -/ -0e -/ mv 0x0 (6) e / () -00e -/ mv mv p i -80e - mw p() -80e - mw -0.8 mw haper 6, Sluin. V 60x0 00 mh i / 0.6 /()
16 haper 6, Sluin 6. x0 4-6 sin mv ()()( sin )V p i -7 sin cs mw Bu sin A cs A sin A p -6 sin 4 mw haper 6, Sluin 7. x0 x4(00) cs cs 00 V p i 4.8 x 4 sin 00 cs sin 00 w / 00 p 9.6 / 00 cs sin 00 J (csπ ) mj 96 mj haper 6, Sluin 8. 40x0 ( e e ) 40( )e mv, > 0
17 haper 6, Sluin 9 i i i(0) 0 i 00x0 0 ( 4) ( 4) 0 i() 0 A haper 6, Sluin 40 0x0 0, 0 < < ms i 0-0, < < ms , < < 4 ms 0x0, 0 < < ms -0x0, < < ms 0x0, < < 4 ms 00 V, 0 < < ms - 00 V, < < ms 00 V, < < 4 ms which is skeched belw. () V (ms)
18 haper 6, Sluin 4. i i(0) 0( ) e 0. 0 e 4. 7A A ls, i e A w i.7j haper 6, Sluin 4. i i(0) () 0 Fr 0 < <, i A 0 Fr < <, i 0 i() A Fr < <, i 0 i() - A Fr < < 4, i 0 i() A Fr 4 < <, i 0 i(4) 4 - A 4 Thus, A, 0< < A, < < i () A, < < A, < < 4, 4< <
19 haper 6, Sluin 4. w i i() i ( ) x80x0 44 µj x( 60x0 ) 0 haper 6, Sluin 44. i ( ) i (4 0cs ) 0.8 sin - haper 6, Sluin 4. i() () i(0) Fr 0 < <, i 0 0x0 0. ka Fr < <, -0 i 0x0 ( 0 ) i() ( 0. ) 0.kA - 0. ka 0. ka, i() 0. ka, 0 < < < <
20 haper 6, Sluin 46. Under dc cnins, he circui is as shwn belw: Ω i A 4 Ω By curren isin, 4 i ( 4 ) A, c 0V w i () J w c c ()() 0J haper 6, Sluin 47. Under dc cnins, he circui is equialen ha shwn belw: i A Ω i 0 (), c i 0
21 w c w i If w c w, c x0 x ( ) 00 x0 x ( ) 80x x (x) x0 x00 ( ) 80 x 0 - Ω haper 6, Sluin 48. Under dc cnins, he circui is as shwn belw: 0V 4 Ω i i 6 Ω 0 i A 4 6 i 6i 8V 0V
22 haper 6, Sluin 49. (a) 6 ( 4 4) 6 eq (b) ( 6 6) 4 eq H (c) 4 ( 6) 4 4 eq H 7H haper 6, Sluin 0. ( 4 6) eq 0 0 ( ) 0.. mh haper 6, Sluin. 0 mh ( 0) eq mh 0x 4 haper 6, Sluin. ////6 H, 4// H Afer he parallel cmbinains, he circui becmes ha shwn belw. H a H ab ()// (4x)/ 0.8 H H b
23 haper 6, Sluin. [ (8 ) 6 (8 4) ] eq (4 4) 6 4 eq 0 mh haper 6, Sluin 4. ( ) eq 4 (9 ) 4 (0 4) 4 eq 7H haper 6, Sluin. (a) // 0., x0. eq // (b) // 0., // // eq // 0. haper 6, Sluin 6. Hence he gien circui is equialen ha shwn belw: / /
24 eq x 8 haper 6, Sluin 7. e eq () 4 () i i i i i i () r (4) and 0 () Incrpraing () and (4) in (), Subsiuing his in () gies mparing his wih (), 67 eq 8.7H 8
25 haper 6, Sluin 8. x slpe f i(). Thus is skeched belw: 6 () (V) (s) haper 6, Sluin 9. (a) s ( ) s, s, s (b) i i i i s ( ) s i s s i
26 i s s i haper 6, Sluin 60 eq eq 8 d // 8 ( 4e ) e i I 0 ( ) i (0) 0 ( ) e.e 0 0..e A haper 6, Sluin 6. (a) i s i i i s ( 0) i(0) i (0) 6 4 i (0) i(0) ma (b) Using curren isin: 0 i ( is 0.4 6e ).4e - ma 0 0 i i.6e - ma s i 0x0 (c) 0 0 mh 0 d 0x0 ( 6e ) x0-0e - µv d x0 ( 6e ) x0-44e - µv 4 6 (d) w x0x0 ( 6e x ) 0mH 0 0.8e 4 µ J 4.6nJ mh x0x0.76e.69nj mh x0x0.96e 7.4 nj 4 6 ( x ) / w ( x ) / w 0 0
27 haper 6, Sluin 6. (a) eq eq 0x60 0 // mh 80 0 i ( ) i(0) e i(0) 0.( e ) i(0) 40x0 eq Using curren isin, 60 i i i, i i i (0) i(0) 0.7i(0) 0.0 i(0) i ( 0.e ) A - e.67 ma 4 i (0).67. ma (b) i - i ( 0.e ) A - 7e e.67 ma 6 ma haper 6, Sluin 6. We apply superpsiin principle and le where and are due i and i respeciely.,, 0 < < < < 6 4, 0, 4, 0 < < < < 4 4 < < 6
28 Adng and gies, which is shwn belw. () V (s) -6 haper 6, Sluin 64. (a) When he swich is in psiin A, i-6 i(0) When he swich is in psiin B, i( ) / 4, τ / / 8 i( ) i( ) [ i(0) i( )] e 9e / ι 8 A (b) - 4i(0) 0, i.e. 4i(0) 6 V (c) A seady sae, he inducr becmes a shr circui s ha 0 V
29 haper 6, Sluin 6. (a) w i xx(4) 40 W w 0 (0)( ) 40 W (b) w w w 0 80 W d 00 i 00 0e x0-0e -00 A (c) ( )( ) i d -00e -00 A 00 ( 0e x ) 0( 00) 0 d i -00e -00 A 00 ( 0e x ) 0( 00) 0 (d) i i i -0e -00 A haper 6, Sluin 66. eq mH i i(0) sin 4 0 ma 60x0 i 0cs 4 0( - cs 4) ma mH 4x0 4.8 sin 4 mv d (0)( cs 4)mV
30 haper 6, Sluin 67. i, 0 x 0 x 0.04 x 0-6 x 0-0 0sin 0 00 cs 0 mv haper 6, Sluin 68. i (0), 0 x 0 x 00 x The p amp will saurae a ± - - 6s haper 6, Sluin x 0 6 x x i i 4 Fr 0 < <, i 0, 0 - mv 4 Fr < <, i 0, 0 ().( ) mv Fr < < 4, i - 0, 0 () ( ) mv Fr 4 < < m, i -0, 0 (4).( 4) mv Fr < < 6, i 0, 0 () ( ) 4-0 mv
31 Thus () is as shwn belw: haper 6, Sluin 70. One pssibiliy is as fllws: 0 e 00 kω, haper 6, Sluin 7. 0.µ F 0x00x0 By cmbining a summer wih an inegrar, we hae he circui belw: Fr he gien prblem, µf, /(4) /(0) /() 00 6 /() 00 kω /(4) 00kΩ/(4) kω /(0) 0 kω
32 haper 6, Sluin 7. The upu f he firs p amp is 00 i i 6 0x0 xx0-0 i ( 0) 6 0x0 x0.x0 00 A.ms, 6 00(.) x0.6 mv haper 6, Sluin 7. nsider he p amp as shwn belw: e a b A nde a, 0-0 () i a b A nde b, i d
33 d () i mbining () and (), r i i d shwing ha he circui is a nninering inegrar. haper 6, Sluin x 0 x 0 - sec d V, V, V, i 0. d 0 < < < < < < 4 msec Thus () is as skeched belw: () (V) (ms) -
34 haper 6, Sluin 7. 6, 0x0 x0x0. 0 d. () -0 mv haper 6, Sluin 76., 0 x 0 x 0 x d 0, 0 < < i 0., < < The inpu is skeched in Fig. (a), while he upu is skeched in Fig. (b). i () (V) () (V) (ms) (ms) (a) -0 (b) haper 6, Sluin 77. i i i i 0 0 F 6 6 F 0 x0 0 d ( 0 )
35 Hence i d Thus i is bained frm as shwn belw: () (V) d ()/ 4 4 (ms) (ms) i () (V) 8 (ms) haper 6, Sluin 78. d d 0sin Thus, by cmbining inegrars wih a summer, we bain he apprpriae analg cmpuer as shwn belw:
36 0 d / -d / d / / d / sin /0 -sin haper 6, Sluin 79. We can wrie he equain as dy f ( ) 4y( ) which is implemened by he circui belw. V 0 /4 dy/ y dy/ f()
37 haper 6, Sluin 80. Frm he gien circui, r d d 000kΩ f () 000kΩ d f () 000kΩ d 00kΩ haper 6, Sluin 8 We can wrie he equain as d f ( ) which is implemened by he circui belw. - / d / - -d/ - d / f() /
38 haper 6, Sluin 8 The circui cnsiss f a summer, an inerer, and an inegrar. Such circui is shwn belw /() - s - haper 6, Sluin 8. Since w 0µF capacirs in series gies µf, raed a 600V, i requires 8 grups in parallel wih each grup cnsising f w capacirs in series, as shwn belw: 600 Answer: 8 grups in parallel wih each grup made up f capacirs in series.
39 haper 6, Sluin 84. q I I x q q 0.6 x 4 x 0-6.4µ 6 q.4x0 0nF (6 0) haper 6, Sluin 8. I is eiden ha ffereniaing i will gie a waefrm similar. Hence, 4,0 < < i 8 4, < < 4,0 < < 4, < < Bu, mv,0 < < mv, < < Thus, 4 x 0 -. mh in a. mh inducr haper 6, Sluin 86. (a) Fr he series-cnneced capacir s... 8 Fr he parallel-cnneced srings, 0s 000 eq 0s 0x µ F 0µF 8
40 (b) T 8 x 00V 800V w 400J 6 ( 0x0 )(800 eq T )
R th is the Thevenin equivalent at the capacitor terminals.
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