Chapter 14, Problem 1. Find the transfer function V o. /V i. of the RC circuit in Fig Express it using ω = 1/RC. Figure For Prob

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1 Chapter 4, Prblem. Fnd the traner unctn V /V the C crcut n Fg Expre t ung /C. Fgure 4.68 Fr Prb. 4.. Chapter 4, Slutn. V H ( ) V H() jc jc jc j 0 j 0, where 0 C H 0 π - H ( ) φ H ( ) tan ( 0 ) 0 Th a hghpa lter. The requency repne the ame a that r P.P.4. except that 0 C. Thu, the ketche H and φ are hwn belw. H /C φ /C POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

2 Chapter 4, Prblem. Obtan the traner unctn V ()/V the crcut n Fg Fgure 4.69 Fr Prb. 4.. Chapter 4, Slutn. V H() V / / 8 8 / 8 / POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

3 Chapter 4, Prblem. Fr the crcut hwn n Fg. 4.70, nd H() V /V (). Fgure 4.70 Fr Prb Chapter 4, Slutn. 0.F 5 jc (0.) 0.F 0 (0.) The crcut becme that hwn belw. V V _ 0 5 V _ et (5 ) 5( ) 0 5 0( ) //(5 ) ( ) ( ) V V 5 V V V V 5 5/ 0( ) V ( ) 0 5 H() V 0( ) ( ) ( ) 0( ) 8 5 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

4 Chapter 4, Prblem 4. Fnd the traner unctn H( ) V O /V the crcut hwn n Fg Fgure 4.7 Fr Prb Chapter 4, Slutn 4. (a) jc jc H ( ) V V jc j jc j( jc) (b) H() H ( ) H() - C j j jc( j) j jc jc( j) - C jc C jc POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

5 Chapter 4, Prblem 5. Fr each the crcut hwn n Fg. 4.7, nd H() V ()/V (). Fgure 4.7 Fr Prb Chapter 4, Slutn 5. (a) et // V V V H() V ( ) (b) et x // C C C C V V V H() V C C C POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

6 Chapter 4, Prblem 6. Fr the crcut hwn n Fg. 4.7, nd H() I ()/I (). Fgure 4.7 Fr Prb Chapter 4, Slutn 6. H j et // We cnvert the current urce t a vltage urce a hwn belw. S I _ V _ ( ) I I V I x I V I I I H() I ( ) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

7 Chapter 4, Prblem 7. Calculate H ( ) H db equal (a) 0.05dB (b) -6. db (c) 04.7 db Chapter 4, Slutn 7. (a).05 0lg 0 0 H lg0 H.5 H (b) (c) lg H 0. lg H H lg0 H 5.5 lg0 H H 5 Chapter 4, Prblem 8. Determne the magntude (n db) and the phae (n degree) H( ) at H ( ) equal 0 j (a) 0.05 db (b) 5 (c) j Chapter 4, Slutn 8. (a) H HdB 0lg , φ 0 (b) H 5 HdB 0lg , φ 0 j0 (c) H () j 0lg , φ 6.4 HdB (d) H ().9 j j j HdB 0lg , φ 4.7 (d) j 6 j POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

8 Chapter 4, Prblem 9. A ladder netwrk ha a vltage gan H( ) ( 0 j )(0 j) Sketch the Bde plt r the gan. Chapter 4, Slutn 9. H ( ) ( j)( j 0) H db -0 lg0 j 0lg0 j/0 - φ - tan ( ) tan ( - /0) The magntude and phae plt are hwn belw. H db lg0-0 j/ lg0 j φ arg j/0 arg j -80 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

9 Chapter 4, Prblem 0. Sketch the Bde magntude and phae plt : 50 H(j ) j (5 j) Chapter 4, Slutn 0. H (j) 50 j(5 j) 0 j j 5 H db lg lg j 00 0lg j 5 φ arg j / 5 arg j -80 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

10 Chapter 4, Prblem. Sketch the Bde plt r 0 j H( ) j( j) Chapter 4, Slutn. 5( j 0) H ( ) j( j ) H db 0lg0 5 0lg0 j 0 0lg0 j 0lg0 j - φ -90 tan 0 tan - The magntude and phae plt are hwn belw. H db φ POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

11 Chapter 4, Prblem. A traner unctn gven by T() ( 0) Sketch the magntude and phae Bde plt. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

12 Chapter 4, Slutn. 0.( j) T ( w), 0lg0. 0 j( j /0) The plt are hwn belw. T (db) arg T POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

13 Chapter 4, Prblem. Cntruct the Bde plt r G(), j ( 0) Chapter 4, Slutn. j ( 0)( j) G ( ) (j) (0 j) (j) ( j 0) G db -0 0lg0 j 40lg0 j 0lg0 - - φ -80 tan tan 0 j 0 The magntude and phae plt are hwn belw. G db φ POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

14 Chapter 4, Prblem 4. Draw the Bde plt r 50( j ) H( ) j( 0 j 5) Chapter 4, Slutn H ( ) 5 j j j0 j 5 5 H db lg 0lg j 0lg j 0lg 0 j 5 (j 5) φ -90 tan tan 5 The magntude and phae plt are hwn belw. H db φ POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

15 Chapter 4, Prblem 5. Cntruct the Bde magntude and phae plt r 40( ) H(), j ( )( 0) Chapter 4, Slutn 5. H ( ) 40( j) ( j)(0 j) ( j) ( j )( j 0) H db 0lg0 0lg0 j 0lg0 j 0lg0 j φ tan tan tan - The magntude and phae plt are hwn belw. 0 H db φ POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

16 Chapter 4, Prblem 6. Sketch Bde magntude and phae plt r H() ( 0, j 6) Chapter 4, Slutn 6. 0 / H ( ) j j j j j j 4 4 j HdB 0lg lg j 0lg j 4 (0lg0, ) The magntude and phae plt are hwn belw. H 0 0 lg (j) lg j j φ tan - 6 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

17 Chapter 4, Prblem 7. Sketch the Bde plt r G(), j ( ) ( ) Chapter 4, Slutn 7. G ( ) ( 4) j ( j)( j ) G db -0lg0 4 0lg0 j 0lg0 j 40lg0 j - φ tan tan - The magntude and phae plt are hwn belw. G db φ POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

18 Chapter 4, Prblem 8. A lnear netwrk ha th traner unctn H() ( 7 4, j 8 4 5) Ue MATAB r equvalent t plt the magntude and phae (n degree) the traner unctn. Take 0. < < 0 rad/. Chapter 4, Slutn 8. The MATAB cde hwn belw. >> wlgpace(-,,00); >> *w; >> h(7*.^4)./(.^8*.^4*5); >> Phaeunwrap(angle(h))*57.; >> emlgx(w,phae) >> grd n H(jw) Phae w POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

19 Nw r the magntude, we need t add the llwng t the abve, >> Hab(h); >> HdB0*lg0(H); >> emlgx(w,hdb); >> grd n HdB w POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

20 Chapter 4, Prblem 9. Sketch the aympttc Bde plt the magntude and phae r 00 H(), j ( 0)( 0)( 40) Chapter 4, Slutn j j / 80 H ( ) ( j 0)( j 0)( j 40) j j j ( )( )( ) j j j HdB 0lg(/ 80) 0lg j / 0lg 0lg 0lg (0lg(/80) -8.06) The magntude and phae plt are hwn belw. H 0 lg j lg j 0 lg φ POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

21 Chapter 4, Prblem 0. Sketch the magntude Bde plt r the traner unctn 0 j H( ) ( j )( j 5) ( j 40) Chapter 4, Slutn 0. 0 j j /00 H ( ) (5)(40)( j )( j / 5) ( j / 40) ( j )( j / 5) ( j / 40) 0lg(/00) -40 The magntude plt hwn belw lg j 0 lg j lg lg j lg j 5 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

22 Chapter 4, Prblem. Sketch the magntude Bde plt r ( 0) H(), j ( )( 60) (400) Chapter 4, Slutn. j( j 0) 0 j( j/ 0) H ( ) ( j )( 60 j 400) j 400( j )( 60 j/ 400 ) j( j/ 0) H ( ) 6j j ( j) 40 0 j HdB 0 lg(0.05) 0lg j 0 lg 0 The magntude plt a ketched belw. H&B 40 0lg j 0 0 lg j 0 lg 0 lg j/0 j6 j lg lg j lg j6 j 40 0 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

23 Chapter 4, Prblem. Fnd the traner unctn H( ) wth the Bde magntude plt hwn n Fg Fgure 4.74 Fr Prb. 4.. Chapter 4, Slutn. 0 0lg0 k k 0 A zer lpe A ple lpe A ple lpe 0 db/ dec at j - 0 db/ dec at - 0 db/ dec at 0 00 j 0 j 00 Hence, H ( ) H() 0( j ) ( j 0)( j 00) 0 4 ( j) (0 j)(00 j) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

24 Chapter 4, Prblem. The Bde magntude plt H( ) hwn n Fg Fnd H( ). Fgure 4.75 Fr Prb. 4.. Chapter 4, Slutn. A zer lpe 0 db/ dec at the rgn j A ple lpe - 0 db/ dec at j A ple lpe - 40 db/ dec at 0 ( j 0) Hence, H ( ) j ( j)( j 0) 00 j H() ( j)(0 j) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

25 Chapter 4, Prblem 4. The magntude plt n Fg repreent the traner unctn a preampler. Fnd H(). Fgure 4.76 Fr Prb Chapter 4, Slutn lg0 K K 00 There a ple at 50 gvng /(j/50) There a zer at 500 gvng ( j/500). There anther ple at gvng /( j/). Thu, r 40 x ( 500) 40( j / 500) H ( ) 500 ( j/ 50)( j/ ) x ( 50)( ) ( 500) H() ( 50)( ) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

26 Chapter 4, Prblem 5. A ere C netwrk ha k Ω, 40 mh, and C µ F. Calculate the mpedance at renance and at ne-urth, ne-hal, twce, and ur tme the renant requency. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

27 Chapter 4, Slutn 5. - C (40 0 )( ( 0 ) ( 0 kω 0 4) j 4 ) 4 C 0 5 krad / ( 0 4) 000 j (5 0 )( 0 ) ( 0 4) 000 j( ) ( 0 4) ( 0 j0.75 kω 0 ) j C 0 (5 0 ) - ( 0 ) 000 j (40 0 ) -6 (5 0 )( 0 ) ( 0 /) 00j(00-000/5) ( 0 ) j0. kω ( 0 ) j 0 C 0 - ( 0 ) 000 j ()(5 0 )(40 0 ) -6 ()(5 0 )( 0 ) ( 0 ) j0. kω (4 0 ) j C 0 - ( 4 0 ) 000 j (4)(5 0 )(40 0 ) -6 (4)(5 0 )( 0 ) ( 4 0 ) j0.75 kω POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

28 Chapter 4, Prblem 6. A cl wth retance Ω and nductance 00 mh cnnected n ere wth a capactr 50 pf, a retr 6 Ω and a gnal generatr that gve 0 V rm at all requence. Calculate, Q, and B at renance the reultant ere C crcut. Chapter 4, Slutn 6. Cnder the crcut a hwn belw. Th a ere C renant crcut. 6 Ω 50 µf Ω _ 00 mh 6 9 Ω 447. krad/ C 00x0 x50x0 447.x0 x00x0 Q x0 B 90 rad/ Q 4969 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

29 Chapter 4, Prblem 7. Degn a ere C renant crcut wth 40 rad/ and B 0 rad/. Chapter 4, Slutn C C 40 B 0 0 I we elect Ω, then /0 0. H and C 6.5 mf x0. Chapter 4, Prblem 8. Degn a ere C crcut wth B 0 rad/ and,000 rad/. Fnd the crcut Q. et 0 Ω. Chapter 4, Slutn 8. et 0 Ω H B 0 µ F (000) (0.5) C 0 0 Q B Therere, 0 Ω then 0.5 H, C µ F, Q 50 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

30 Chapter 4, Prblem 9. et v 0 c(at) V n the crcut Fg Fnd, Q, and B, a een by the capactr. Fgure 4.77 Fr Prb Chapter 4, Slutn 9. We cnvert the vltage urce t a current urce a hwn belw. k 45 k µf 60 mh 0 c t, //45 x45/ kω 4.08 krad/ 6 C 60x0 xx0 B rad/ C x 0 x Q B POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

31 Chapter 4, Prblem 0. A crcut cntng a cl wth nductance 0 mh and retance 0 Ω cnnected n ere wth a capactr and a generatr wth an rm vltage 0 V. Fnd: (a) the value the capactance that wll caue the crcut t be n renance at 5 khz (b) the current thrugh the cl at renance (c) the Q the crcut Chapter 4, Slutn 0. Select 0 Ω H 50 mh 0Q (0)(0) C 0. F 0 (00)(0.05) B 0.5 rad / C (0)(0.) Therere, 0 Ω then 50 mh, C 0. F, B 0.5 rad / Chapter 4, Prblem. Degn a parallel renant C crcut wth 0rad/ and Q 0. Calculate the bandwdth the crcut. et 0Ω. Chapter 4, Slutn. X X B X 6 πx0x0 x5.6x0 40x x0 rad/ POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

32 Chapter 4, Prblem. A parallel C crcut ha the llwng value: 60 Ω, mh, and C 50 µ F. Fnd the qualty actr, the renant requency, and the bandwdth the C crcut. Chapter 4, Slutn krad/ 6 C 0 x50x0 B. rad/ C 6 60 x 50 x Q.4 B. Chapter 4, Prblem. A parallel renant crcut wth qualty actr 0 ha a renant requency rad/. Calculate the bandwdth and hal-pwer requence. Chapter 4, Slutn. Q 80 Q C C π 6 πx5.6x0 x40x0 pf 40x0 Q 4. µh Q 6 π πx5.6x0 x80 Chapter 4, Prblem 4. A parallel C crcut renant at 5.6 MHz, ha a Q 80, and ha a retve branch 40 k Ω. Determne the value and C n the ther tw branche. Chapter 4, Slutn 4. (a) C 6 8x0 x60x0.44 krad/ (b) B C 5x0 x60x0 6. rad/ 6 (c) Q C.44x0 x5x0 x60x POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

33 Chapter 4, Prblem 5. A parallel C crcut ha 5k Ω, 8 mh, and C µ F. Determne: (a) the renant requency (b) the bandwdth (c) the qualty actr Chapter 4, Slutn 5. At renance, Y 40 Ω Y Q 80 Q 0C C 0 µ F 0 (00 0 )(40) µ C C (4 0 )(0 0 ) 0 H B.5 krad / Q 80 B krad/ B krad/ POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

34 Chapter 4, Prblem 6. It expected that a parallel C renant crcut ha a mdband admttance 5 0 S, qualty actr 80, and a renant requency 00 krad/. Calculate the value,, and C. Fnd the bandwdth and the hal-pwer requence. Chapter 4, Slutn 6. 0 C 5000 rad / Y( 0 ) ( 0 ) kω Y ( ) j C 0.5 j8.75 ms 4 0 ( 0 4).4 j5. Ω j Y ( 0 0 ) j C 0.5 j7.5 ms 0 ( 0 ) 8.85 j. 74 Ω j Y (0 ) j j7.5 ms 0C ( 0 ) 8.85 j. 74 Ω Y (40 ) j j8.75 ms 4 0C ( 4 0 ).4 j5. Ω POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

35 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part Chapter 4, Prblem 7. ewrk Prb. 4.5 the element are cnnected n parallel. Chapter 4, Slutn 7. ) C ( ) C j( j C j C j ) C j ( j ) C j //( j ) C (C 0 ) C ( C C Im() Thu, C C

36 Chapter 4, Prblem 8. Fnd the renant requency the crcut n Fg Fgure 4.78 Fr Prb Chapter 4, Slutn 8. Y jc j j jc At renance, Im( Y ) 0,.e C 0 C 0 0 C 484 rad / 0 ( )( ) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

37 Chapter 4, Prblem 9. Fr the tank crcut n Fg. 4.79, nd the renant requency. Fgure 4.79 Fr Prb. 4.9 and 4.9. Chapter 4, Slutn 9. (a) B π( ) π(90 86)x0 π 8 krad / ( ) π(88)x0 76πX0 B C C B 8πx0 xx0 9.89nF (b) µh C 9 C (76πX0 ) x9.89x0 (c ) 76π 55.9krad / (d) B 8π 5.krad / 76π (e) Q B 8π POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

38 Chapter 4, Prblem 40. A parallel renance crcut ha a retance k Ω and hal-pwer requence 86 khz and 90 khz. Determne: (a) the capactance (b) the nductance (c) the renant requency (d) the bandwdth (e) the qualty actr Chapter 4, Slutn 40. (a) mh 0 C 5x0 6 x0x0.857 k rad/ec Q 0C.857x0 x5x0 x0x0 9.8 B rad/ C 6 5x0 0x0 6 (b) T ncreae B by 00% mean that B 4. C B 5x0 x4 0 µf Snce C C C 0µ F and C 0 µf, we then btan C 0 µf. C C Therere, t ncreae the bandwdth, we merely add anther 0 µf n ere wth the rt ne. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

39 Chapter 4, Prblem 4. Fr the crcut hwn n Fg. 4.80, next page: (a) Calculate the renant requency, the qualty actr Q, and the bandwdth B. (b) What value capactance mut be cnnected n ere wth the 0- µ F capactr n rder t duble the bandwdth? Fgure 4.80 Fr Prb Chapter 4, Slutn 4. (a) Th a ere C crcut. 6 8 Ω, H, C 0.4 F 0.58 rad / C 0.4 Q B 8 rad / (b) Th a parallel C crcut. ()(6) µ F and 6 µ F µ F 6 C µ F, kω, 0 mh C ( 0 )(0 0 5 krad / ) (5 0 )(0 0 Q ) B rad/ C ( 0 )( 0 ) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

40 Chapter 4, Prblem 4. Fr the crcut n Fg. 4.8, nd the renant requency, the qualty actr Q, and the bandwdth B. Fgure 4.8 Fr Prb POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

41 Chapter 4, Slutn 4. (a) ( jc) ( j) n n n j jc j C jc j jc ( j)( C jc) ( C) C At renance, Im( n ) 0,.e. 0 0( 0C) 0 C 0 C C C 0 C C (b) (j jc) n n n (j jc) ( C) j jc ( C) jc ( C)[( C) jc] ( C) C At renance, Im( n ) 0,.e. 0 ( C) C C 0 0 C POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

42 Chapter 4, Prblem 4. Calculate the renant requency each the crcut n Fg Fgure 4.8 Fr Prb Chapter 4, Slutn 4. Cnder the crcut belw. /jc n j (a) j) ( jc) n ( n n n n n j j jc j j jc j jc j ( (- j ( j C) j)( j C) C - C j C C j( C j )[ C C j( C)] ( C C ( C) ) C) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

43 At renance, Im( n ) 0,.e. 0 C( C) ( C C ) 0 C C 0 C ( C C ) C C C (0.0)( krad / -6 ) (0.) (9 0-6 ) (b) At 0.57 krad /, - j j(.57 0 )(0 0 ) j47.4 j47.4 j j0.0 j j47.4 jc j(.57 0 )(9 0 ) 6 n ( 0 ) ( j) ( jc) ( j0.0)(0. j47.4) n ( 0 ) ( j0.0) (0. j47.4) n ( 0 ) Ω POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

44 Chapter 4, Prblem 44. * Fr the crcut n Fg. 4.8, nd: (a) the renant requency (b) n ( ) Fgure 4.8 Fr Prb * An aterk ndcate a challengng prblem. Chapter 4, Slutn 44. We nd the nput mpedance the crcut hwn belw. j(/) /j /jc j j jc, jc C jc -j.5 j -j0.5 jc C v (t) and (t) are n phae when purely real,.e. C (C ) r C F C C Ω C V I ( )(0) 0 v (t) 0n(t) V,.e. V 0 V POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

45 Chapter 4, Prblem 45. Fr the crcut hwn n Fg. 4.84, nd, B, and Q, a een by the vltage acr the nductr. Fgure 4.84 Fr Prb Chapter 4, Slutn 45. Cnvert the vltage urce t a current urce a hwn belw. I 0 kω 50 µf 0 mh 50 kω 0//50 0x50/ kω Th a parallel renant crcut rad/ 6 C 0x0 x50x0 B.067 rad/ C x 0 x 50 x Q 49. B.067 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

46 Chapter 4, Prblem 46. Fr the netwrk llutrated n Fg. 4.85, nd (a) the traner unctn H( ) V ( )/I( ), (b) the magntude H at rad/. Fgure 4.85 Fr Prb. 4.46, 4.78, and 4.9. Chapter 4, Slutn 46. (a) Th an C ere crcut. C (πx5x0 ) C x0x0.6nf (b), I V/ 0/0 6 A πx5x0 x0x0 (c ) Q 5π POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

47 Chapter 4, Prblem 47. Shw that a ere crcut a lwpa lter the utput taken acr the retr. Calculate the crner requency c mh and 0 k Ω. Chapter 4, Slutn 47. H ( ) V V j j Hence, H (0) and H ( ) 0 hwng that th crcut a lwpa lter. At the crner requency, H( c ),.e. c r c c π c c c π π khz POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

48 Chapter 4, Prblem 48. Fnd the traner unctn V /V the crcut n Fg Shw that the crcut a lwpa lter. Fgure 4.86 Fr Prb Chapter 4, Slutn 48. H ( ) H ( ) H() jc j jc jc jc jc j jc j C H (0) and H ( ) 0 hwng that th crcut a lwpa lter. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

49 Chapter 4, Prblem 49. Determne the cut requency the lwpa lter decrbed by 4 H( ) j0 Fnd the gan n db and phae H( ) at rad/. Chapter 4, Slutn 49. Hence, 4 At dc, H (0). H ( ) H(0) c c c 0. H() 4 j0 j0 H () In db, 0lg0 H() arg H() -tan POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

50 Chapter 4, Prblem 50. Determne what type lter n Fg Calculate the crner requency c. Fgure 4.87 Fr Prb Chapter 4, Slutn 50. H ( ) V V j j H (0) 0 and H ( ) hwng that th crcut a hghpa lter. H ( r c ) c π c c c c π π Hz POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

51 Chapter 4, Prblem 5. Degn an lwpa lter that ue a 40-mH cl and ha a cut requency 5 khz. Chapter 4, Slutn 5. The lwpa lter hwn belw. v v - - H V V j j / c π c π c πx5x0 x40x0.56kω Chapter 4, Prblem 5. In a hghpa lter wth a cut requency 00 khz, 40 mh. Fnd. Chapter 4, Slutn 5. π c c 5 - π (π)(0 )(40 0 ) 5. kω c POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

52 Chapter 4, Prblem 5. Degn a ere C type bandpa lter wth cut requence 0 khz and khz. Aumng C 80 pf, nd,, and Q. Chapter 4, Slutn 5. π 0π 0 π π 0 B π 0 0 π 0 0 π Q 0.5 B π 0 C 0C.87 - (π 0 ) (80 0 ) H B B (π 0 )(.87) kω Chapter 4, Prblem 54. Degn a pave bandtp lter wth 0 rad/ and Q 0. Chapter 4, Slutn 54. Th an pen-ended prblem wth everal pble lutn. We may che the bandtp lter n Fg C 0.0 C Q 0 0 I we elect H, then 0.5 Ω, and C 0.0/ 0 mf. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

53 Chapter 4, Prblem 55. Determne the range requence that wll be paed by a ere C bandpa lter wth 0 Ω, 5mH, and C 0.4 µ F. Fnd the qualty actr. Chapter 4, Slutn 55. C ( B 0 Q )( krad / ) 0 krad / 9.8 B krad / r.56 khz π 0. B krad / r.6 khz π Therere,.56 khz < <.6 khz POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

54 Chapter 4, Prblem 56. (a) Shw that r a bandpa lter, B H(), jw B 0 where B bandwdth the lter and the center requency. (b) Smlarly, hw that r a bandtp lter, 0 H(), jw B 0 Chapter 4, Slutn 56. (a) Frm Eq 4.54, H () C C C C C Snce B and 0, C B H() B 0 (b) Frm Eq. 4.56, C H () C C C 0 H() B 0 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

55 Chapter 4, Prblem 57. Determne the center requency and bandwdth the bandpa lter n Fg Fgure 4.88 Fr Prb POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

56 Chapter 4, Slutn 57. (a) Cnder the crcut belw. I I /C V /C V () C C C () C( C) C C () C( C) V I C V I I C ( C) C C C V I V C( C) C C C V H () V C C C C H () C C Thu, 0 r rad / C C 0 B C rad / POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

57 (b) Smlarly, ( ) () ( ) () V I, I I V ( ) V I V V H () V Thu, rad / 0 B rad / POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

58 Chapter 4, Prblem 58. The crcut parameter r a ere C bandtp lter are k Ω, 0. H, C 40 pf. Calculate: (a) the center requency (b) the hal-pwer requence (c) the qualty actr Chapter 4, Slutn 58. (a) - C (0.)(40 0 ) rad / (b) 0 4 B Q 5 B 4 0 A a hgh Q crcut, B (50 ) 490 krad / B (50 ) 50 krad / (c) A een n part (b), Q 5 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

59 Chapter 4, Prblem 59. Fnd the bandwdth and center requency the bandtp lter Fg Fgure 4.89 Fr Prb Chapter 4, Slutn 59. Cnder the crcut belw. V V /C () C ( C) C ( C) ( ) C C POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

60 H V V ( C) C C C n ( C) C C n C C C C C j n j C C C jc C n ( C C j C)( C jc) ( C) ( C) Im( n ) 0 mple that C[ C C] C( C) 0 - C C C C 5.8 krad / C ( 0 )(4 0 ) 0 H ( C) j C C C H r max H(0) H max H( ) lm C C j C( ) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

61 At and, H H mzx ( ) ( C) C( ) j C ( C) ( ( )( C) C( )) (96 0 0( -6 ) 4 0 (0-9 ) ) 0 (96 0 0( -6 ) 4 0 (0-9 ) ) ( )( ) (96 0 ) (0 4 0 ) 0 ( )( ) 4 0 ) (96 0 ) ( ( 96 0 ) (0 4 0 ) Hence, 4.65 krad / 7.06 krad / B krad / POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

62 Chapter 4, Prblem 60. Obtan the traner unctn a hghpa lter wth a paband gan 0 and a cut requency 50 rad/. Chapter 4, Slutn 60. jc j H ( ) (rm Eq. 4.5) jc j C Th ha a unty paband gan,.e. H ( ). 50 C c j0 H ^ ( ) 0H ( ) 50 j j0 H() 50 j POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

63 Chapter 4, Prblem 6. Fnd the traner unctn r each the actve lter n Fg Fgure 4.90 Fr Prb. 4.6 and 4.6. Chapter 4, Slutn 6. jc (a) V V jc V V, Snce V V, V V jc H( ) V V jc (b) V V jc V V, Snce V V, jc V V jc V H( ) V jc jc POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

64 Chapter 4, Prblem 6. The lter n Fg. 4.90(b) ha a -db cut requency at khz. I t nput cnnected t a 0-mV varable requency gnal, nd the utput vltage at: (a) 00 Hz (b) khz (c) 0 khz Chapter 4, Slutn 6. Th a hghpa lter. jc H ( ) jc j C H ( ), c π (000) jc C H ( ) j j000 c (a) H ( 00 Hz) j5 V V V 0 mv j5.5 mv (b) H ( khz) j0.5 V V V 0 mv j mv (c) H ( 0 khz) j0. V V V 0 mv j mv POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

65 Chapter 4, Prblem 6. Degn an actve rt-rder hghpa lter wth 00 H() -, j 0 Ue a - µ F capactr. Chapter 4, Slutn 6. Fr an actve hghpa lter, C H() () C But 0 H() () /0 Cmparng () and () lead t: C 0 0 0MΩ C C kΩ C POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

66 Chapter 4, Prblem 64. Obtan the traner unctn the actve lter n Fg. 4.9 n the next page. What knd lter t? Fgure 4.9 Fr Prb Chapter 4, Slutn 64. jc j C jc j C jc Hence, H( ) V V - - j C ( j C )( j C ) Th a bandpa lter. H () mlar t the prduct the traner unctn a lwpa lter and a hghpa lter. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

67 Chapter 4, Prblem 65. A hghpa lter hwn n Fg Shw that the traner unctn H( ) jc jc Fgure 4.9 Fr Prb Chapter 4, Slutn 65. V V jc jc V jc V V Snce V V, V jc V jc H( ) V V jc jc It evdent that a, the gan and that the crner requency. C POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

68 Chapter 4, Prblem 66. A general rt-rder lter hwn n Fg (a) Shw that the traner unctn H() j 4 4 (/ C)[ / / C / ] (b) What cndtn mut be ated r the crcut t perate a a hghpa lter? (c) What cndtn mut be ated r the crcut t perate a a lwpa lter? 4 Fgure 4.9 Fr Prb Chapter 4, Slutn 66. (a) Pr (b) When 4, 4 H () 4 C (c) When, - C H () C POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

69 Chapter 4, Prblem 67. Degn an actve lwpa lter wth dc gan 0.5 and a crner requency 500 Hz. Chapter 4, Slutn 67. DC gan 4 4 Crner requency c π (500) rad / C I we elect 0 kω, then 80 kω and C 5.95 nf (π)(500)(0 0 ) Therere, 0 kω, then 80 kω and C 5.95 nf Chapter 4, Prblem 68. Degn an actve hghpa lter wth a hgh-requency gan 5 and a crner requency 00 Hz. Chapter 4, Slutn 68. Hgh requency gan 5 5 Crner requency c π (00) rad / C I we elect 0 kω, then 00 kω and C 9.8 nf (π)(00)(0 0 ) Therere, 0 kω, then 00 kω and C 9.8 nf POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

70 Chapter 4, Prblem 69. Degn the lter n Fg t meet the llwng requrement: (a) It mut attenuate a gnal at khz by db cmpared wth t value at 0 MHz. (b) It mut prvde a teady-tate utput v (t) 0 n(π 0 8 t 80 ) V r an nput v (t) 4n(π 0 8 t) V. Fgure 4.94 Fr Prb Chapter 4, Slutn 69. Th a hghpa lter wth c khz. c π c C C πc 4π Hz may be regarded a hgh requency. Hence the hgh-requency gan 0 r.5 4 I we let 0 kω, then 4000π 0 5 kω, and C nf. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

71 Chapter 4, Prblem 70. * A ecnd-rder actve lter knwn a a Butterwrth lter hwn n Fg (a) Fnd the traner unctn V /V. (b) Shw that t a lwpa lter. Fgure 4.95 Fr Prb * an aterk ndcate a challengng prblem. Chapter 4, Slutn 70. (a) V () Y Y H () V () Y Y Y4 (Y Y Y ) where Y G, Y G, Y C, Y 4 C. H() G G G G C (G G C ) GG (b) H(0), H ( ) 0 GG hwng that th crcut a lwpa lter. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

72 Chapter 4, Prblem 7. Ue magntude and requency calng n the crcut Fg t btan an equvalent crcut n whch the nductr and capactr have magntude H and F repectvely. Chapter 4, Slutn Ω, 40 mh, C µ F K K m K K m ( ) 5 K K m () C C K K m -6 0 K K m 6 0 K () K m Subttutng () nt (), 6 0 K 5K K K m 5K POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

73 Chapter 4, Prblem 7. What value K m and K wll cale a 4-mH nductr and a 0- µ F capactr t H and F repectvely? Chapter 4, Slutn 7. C C K K C C (4 0 )(0 0 ) 0 ()() - -6 K K 0 K C C m K m C C ()(0 0 ) m 0 ()(4 0 ) -6 K K m Chapter 4, Prblem 7. Calculate the value,, and C that wll reult n k Ω, 40 µ H and C 00 nf repectvely when magntude-caled by 800 and requency-caled by 000. Chapter 4, Slutn 7. m K ()(800 0 ) K m (40 0 ) K MΩ µ F C C K K m pf (800)(000) POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

74 Chapter 4, Prblem 74. A crcut ha Ω, 0 Ω, H and C /0F. Ater the crcut magntude-caled by 00 and requency-caled by 0 6, nd the new value the crcut element. Chapter 4, Slutn 74. ' K m x00 00Ω ' K m 0x00 kω ' K K m () 00 µ H C' C K K m nf Chapter 4, Prblem 75. In an C crcut, 0Ω, 4 H and C F. The crcut magntude-caled by 0 and requency-caled by 0 5. Calculate the new value the element. Chapter 4, Slutn 75. ' K m 0x0 00 Ω ' K K m (4) 400 µ H C C' K mk 5 0x0 µ F POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

75 Chapter 4, Prblem 76. Gven a parallel C crcut wth 5 k Ω, 0 mh, and C 0 µ F, the crcut magntude-caled by K m 500 and requency-caled by K 0 5, nd the reultng value,, and C. Chapter 4, Slutn 76. K x x Ω ' m M Km 500 ' (0 mh) 50 µ H 5 K 0 6 C 0x0 C ' 0.4 pf 5 K K 500x0 m POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

76 Chapter 4, Prblem 77. A ere C crcut ha 0 Ω, 0 40 rad/, and B 5 rad/. Fnd and C when the crcut caled: (a) n magntude by a actr 600, (b) n requency by a actr,000, (c) n magntude by a actr 400 and n requency by a actr 0 5. Chapter 4, Slutn 77. and C are needed bere calng. 0 B H B 5 0 C.5 µ F C (600)() (a) K m (600)() 00 H -4 C.5 0 C µ F K 600 m (b) mh K 0-4 C.5 0 C K K m (400)() (c) 5 8 mh K 0 C K K (400)(0 ) C 5 m nf 7.8 pf POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

77 Chapter 4, Prblem 78. edegn the crcut n Fg that all retve element are caled by a actr,000 and all requency-entve element are requency-caled by a actr 0 4. Chapter 4, Slutn 78. m K (000)() kω K m 0 () 0. H K 0 C 0. µ F K K (0 )(0 ) 4 C 4 m The new crcut hwn belw. kω I kω 0. H 0. µf kω V x POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

78 Chapter 4, Prblem 79. * eer t the netwrk n Fg (a) Fnd n (). (b) Scale the element by K m 0 and K 00. Fnd n () and 0. Fgure 4.96 Fr Prb * An aterk ndcate a challengng prblem. POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

79 Chapter 4, Slutn 79. (a) Inert a -V urce at the nput termnal. I V V /C V V V There a upernde. V V C () But V V V V V V () Al, V V V V C C () Cmbnng () and () C V V V V C V V 4 C (4) Subttutng () and (4) nt () gve V V C V 4 C C 4 C C V V V 4 C 4 C 4 C V 4 C C I V C ( 4 C C) C 4 C n I C n 4 (5) C POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

80 When 5,, C 0., () n At renance, Im( n ) 0 4 C r.8 rad / C (0.)() 0 (b) Ater calng, K m 4 Ω 40 Ω 5 Ω 50 Ω K K m 0 () H C 0. C 0 K K (0)(00) m -4 Frm (5), 4 0 n () C (0.)(0.8 rad / ) 0-4 POPIETAY MATEIA. 007 The McGraw-Hll Cmpane, Inc. All rght reerved. N part

1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits

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