Chapter 2 Linear Waveshaping: High-pass Circuits

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1 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Chaptr 2 Linar Wavshaping: High-pass Circuits. A ramp shwn in Fig.2p. is applid t a high-pass circuit. Draw t scal th utput wavfrm fr th cass: (i) T =, (ii) T = 0.2, (iii) T = 5. Fig.2p. A ramp as input Slutin: Frm Eq. (2.64): / v = t T v = t T Th pak f th utput will ccur at t = T. W knw: v (pak) = T (i) Whn T = : = T v (pak) = () = (ii) Whn T = 0.2 : T = 0.2, = 5 T 0.2 v (pak) = (5) = (iii) Whn T = 5 : / / T 200 Drling Kindrsly India Pvt. Ltd

2 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. T = 5 = 0.2 T 5 v (pak) = (0.2) = Fig. Rspns f th high-pass circuit fr ramp input 2. A wavfrm shwn in Fig.2p.2 is applid as input t an high-pass circuit whs tim cnstant is 250 ps. If th maximum utput vltag acrss th rsistr is 50, what is th pak valu f th input wavfrm? Fig.2p.2 Input t th high-pass circuit Slutin: Fr a ramp input v () t ( ) t At t t s v i vi v (max) 000 i Drling Kindrsly India Pvt. Ltd 2

3 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fr t t v 50 t v v A limitd ramp shwn in Fig.2p.3 is applid t an high-pass circuit f Fig.2.2 (a). Th tim cnstant f th circuit is 2 ms. Calculat th maximum valu f utput vltag and th utput at th nd f th input wavfrm. Fig.2p.3 Input t th high-pass circuit Slutin: Fr a ramp input t v ( t) ( ) 3 At t = t = 0.40 s v ( t) 20 ( ) Th pak valu ccurs nly at t = t = 0.40 v (max) Fr t > t t 3 20 v Th vltag at t = 0.4 ms is: 3 s 200 Drling Kindrsly India Pvt. Ltd 3

4 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. v v 0.06 Th vltag at t = 0.4 ms is Th utput wavfrm is shwn in Fig.3. Fig. 3 Output f th high-pass circuit fr th givn input 4. Th pridic wavfrm shwn in Fig.2p.4 is applid t an diffrntiating circuit whs tim cnstant is 0 µs. Sktch th utput and calculat th maximum and minimum valus f th utput vltag with rspct t th grund. Fig.2p.4 Pridic squar wav as an input t th high-pass circuit Slutin: Givn T =00 s, T 2 = s, τ = 0 s Th stady-stat utput wavfrm is drawn by calculating,, 2 and 2. At t 0, v 2 andatt 0, v t 0 t T, v Fr At t = T Fr At 200 Drling Kindrsly India Pvt. Ltd 4

5 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. T 00 0 At t T, v 0 Fr T t ( T T ), v 2 2 T2 T At t T, v Pak-t-pak input is = 00 2 = = T T Fig. 4 Output f th high-pass circuit fr th spcifid input 5. Th pridic ramp vltag as shwn in Fig.2p.5 is applid t a high-pass circuit. Find quatins frm which t dtrmin th stady-stat utput wavfrm whn T =T 2 =. Fig.5 A pridic ramp as input Slutin: / v = t () 200 Drling Kindrsly India Pvt. Ltd 5

6 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. If thr is an initial vltag f n C, Eq.() gts mdifid as fllws: Fr th ramp input, th slp = t / t v =. T Th capacitr chargs frm t 2 in tim T. At t = T +, th capacitr ds nt rspnd fr suddn changs. Hnc, th utput changs t ( 2 ). During T 2, th capacitr blcks th dc. S th capacitr dischargs frm 3 t. Givn T = T 2 = At t = T v () t = 2 Using () T ( ) (2) = T v T ) = 2 ( T2 r ( ) ( ) ( 0.34) = v( T)= 2 = = Drling Kindrsly India Pvt. Ltd 6

7 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fig. 5 Th changs in vltag with tim 6. A squar wav f puls width 2 ms and pak amplitud f 2 as shwn in Fig.2p.6 is applid t high-pass circuit with tim cnstant 4 ms. Plt th first fur cycls f th utput wavfrm. T/2 = 2 ms Fig.2p.6 Symmtric squar wav as an input Slutin: Givn T = T 2 = 0.2 ms, τ = 4 ms (i) Fr t <0, vi = 0, and hnc v = 0 At t = 0, v i jumps t 2 As th vltag acrss capacitr cannt chang instantanusly, v is als qual t 2. At t = 0, v = a = 2. (ii) During th prid 0 < t < 2.0 ms, as th input is cnstant th utput dcays. v = a t t 2 At t = 2.0 ms, v = b = a = 2 4 = 7.27 At t = 2 ms, th input falls by 2. Th utput als falls by 2. = 2 = = 4.73 c b (iii) Fr 2.0 < t < 4.0 v = c ( tt / 2) At t = T=4 ms, v = d = c 4.73 = 2.86 At t = 4 ms, th input riss by 2. Th utput als riss by 2. = d + 00 = = 9.4 (iv) During th prid T < t < 3T/2, that is btwn 4 t 6 ms, th utput dcays. 2 4 At t = 6 ms v = f = = 9.4(0.606) = 5.53 At 6 ms, th input falls by 2. Hnc g = f 2 = 6.47 (v) During 3T/2 < t < 2T, that is, during 6 t 8 ms, th utput dcays At t = 2T=8 ms, v = h = g = 6.47 = j = h + 00 = = Drling Kindrsly India Pvt. Ltd 7

8 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fig.6 Th utput wavfrm 7. A 20-Hz symmtric squar wav rfrncd t 0 vlts and, with a pak-t-pak amplitud f 0, is fd t an amplifir thrugh th cupling ntwrk shwn in Fig. 2p.7. Calculat and plt th utput wavfrm whn th lwr 3-dB frquncy is: (i) 0.6 Hz, (ii) 6 Hz and (iii) 60 Hz. Fig.2p.7 Th givn cupling ntwrk Slutin: Givn = 0 (i) f = 0.6 Hz = = s 2 f 2 (0.6) T = = = 0.05 s. f 20 T = s. 2 = T / 2 0 = / = = T = = 5.25(0.9) = Drling Kindrsly India Pvt. Ltd 8

9 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. = 2 and = 2 = 5.25 = 2 = 4.8 = 2 Th utput in this cas is plttd in Fig.7.. (ii) f = 6 Hz Fig.7.Output whn f = 0.6 Hz = = s 2 f 2 (6) = T / 2 0 = / = 7.20 = T = = 7.20(0.389) = 2.8. = 2 and = 2 =7.20 = 2 = 2.8 = 2 Th utput fr this cnditin is plttd in Fig Drling Kindrsly India Pvt. Ltd 9

10 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. (iii) f = 60 Hz Fig.7.2 Output whn f = 6 Hz = = s 2 f 2 (60) = T / 2 = / =0.0 = T 2 =0( ) = = 2 and = 2 =0.00 = 2 = = 2 Th utput fr this cas is plttd in Fig Drling Kindrsly India Pvt. Ltd 0

11 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fig.7.3 Output whn f = 60 Hz 8. A squar wav is applid as input t an amplifir thrugh a cupling cndnsr f 0 F. Th amplifir has input rsistanc f 0 kω. Dtrmin th lwst frquncy if th tilt is nt t xcd 0 pr cnt. Slutin: W hav P = 0., R = 0 kω and C = 0 F T Pr cnt tilt, P = 00 pr cnt 2 f 50 Hz P f P = 00 pr cnt f f Pf Hz 9. A puls f 0 amplitud and duratin ms is applid t a high-pass circuit with R = 20 k and C = 0.5 F. Plt th utput wavfrm t scal and calculat th pr cnt tilt in th utput. Slutin: 200 Drling Kindrsly India Pvt. Ltd

12 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. τ = = 0 ms Fr 0< t < t p v i =0 v =0 t At t = t p, v = At t = t p+, v = 2 = 3 t = = 0.95 Fr t > t p, v = pr cnt tilt pr cnt 0 Fig. 9 Th utput wavfrm 0. Th input t th high-pass circuit in Fig. 2p.0 is th wavfrm shwn in Fig. 2p.0. Calculat and plt th utput wavfrm t scal, givn that = τ = 0. ms.. Fig.2p.0 Input t th high-pass circuit Slutin: 200 Drling Kindrsly India Pvt. Ltd 2

13 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fr t < 0. ms v = 0, v = 0 (i) At t = 0. ms, th input suddnly falls t 5, and th utput als changs by th sam amunt as th capacitr acts as a shrt circuit. Fr 0. < t <0.2, v i rmains cnstant at 5. Thrf r, v dcays xpnntially with th tim cnstant 0. ms. (ii) At t = 0.2 ms, i v At t = 0.2 ms, th input suddnly riss by 5, v als riss by th sam amunt. v (t = 0.2 ms) = = 3.6 Fr 0.2 ms < t < 0.3 ms, v i rmains at 0. Hnc v dcays xpnntially with th tim cnstant 0. ms (iii) At t = 0.3 ms v At t = 0.3 ms, input suddnly falls by 20. Th utput als changs by th sam amunt. v (t = 0.3 ms) = = 5.6 Fr 0.3 ms < t < 0.4 ms, v i rmains cn stant at 0. Hnc, v will dcay xpnntially with th tim cnstant 0. ms. (iv) At t = 0.4 ms, v Drling Kindrsly India Pvt. Ltd 3

14 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fig.0 Th utput wavfrm. A puls f 0- amplitud with a puls width f 0.5 ms, as shwn in Fig.2p.9, is applid t a high-pass circuit f Fig. 2.(a), having tim cnstant 0 ms. Sktch th utput wavfrm and dtrmin th pr cnt tilt in th utput. Slutin: 0 ms Fr, 0< t < t p v i =0 =0 t 00 v At t = t p-, v = At t = t p+, v = 2 = = = t Fr t > t p, v = pr cnt tilt pr cnt 0 Th utput is als shwn in Fig. 2p.9. Fig.2p.9 Input and utput f th high-pass circuit 2. A high-pass circuit is dsird t pass a 3-ms swp (ramp input) with lss than 0.4 pr cnt transmissin rrr. Calculat th highst pssibl valu f th lwr 3-dB frquncy. Slutin: Cnsidr th circuit in Fig. 2.(a). T = s pr cnt t(max) =0.4 pr cnt r t(max) = Drling Kindrsly India Pvt. Ltd 4

15 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. T t ft f A symmtric squar wav with f = 500 khz shwn f Hz in Fig.2p.3 is fd t an high-pass ntwrk f 3 30 Fig.2.(a). Calculat and plt th transint and th stady-stat rspns if: (i) τ = 5T and (ii) τ = T/20. Fig.2p.3 Input t th cupling ntwrk Slutin: Givn f = 500 Hz, hnc T = 2.0 ms. Cas : Givn, τ = 5T =0 ms. Whn τ is larg, th capacitr chargs and dischargs vry slwly. Th utput has a small tilt. Th vltags ar calculatd t plt th transint rspns. i. Fr t < 0, v = 0, and hnc v = 0 At t = 0, jumps t 50. As th vltag acrss capacitr cann t chang instantanusly, v is als qual t 50. At t = 0 v = a = 50. ii. During th prid 0 < t <.0 ms, as th input is cnstant th utput dcays. v = a v i t i t At t =.0 ms, v = b = a = 50 0 = At t =.0 ms, th input falls by 00. Th utput als falls by 00. = 00 = = c b iii. Fr.0 < t < 2.0, v = c ( tt / 2) At t = T = 2 ms, v = d = c = At t = 2 ms, th input riss by 00. Th utput als riss by 00. = d + 00 = = Drling Kindrsly India Pvt. Ltd 5

16 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. iv. During th prid T < t < 3T/2, that is, btwn 2 t 3 ms, th utput dcays..0 0 At t = 3 ms v = f At 3 ms, th input falls by 00. Hnc g = 00 = 9.73 f = = (0.9048) = v. During 3T /2 < t < 2T, that is, d uring 3 t 4 ms, th utput dcays At t = 2T= 4 ms, v = h = g = 9.73 =7.85. j = h + 00 = =7.85. In a fw cycls, th utput rachs th stady stat. Stady-stat rspns: Undr stady stat, th utput is symmtrical with rspct t zr vlts, sinc th capacitr blcks dc. Thrfr, th dc cmpnnt in th utput is zr. Lt b th vltag at t = 0 t Fr 0 < t < T/2, v = 0. At t = T/2= ms, v = = = = (3) As th input abruptly falls, utput als falls by th sam amunt t 2. (tt / 2) Fr T/2 < t < T v = 2 0. At t = T, v = 2 = 2 = = (4) Fr symmtrical wav = 2 and = 2 (5) 2 = 00 and 2 = 00 (6) Frm (6), w hav 2 = 00 (7) And frm (3), w hav = 2 (8) Substituting (8) in (7), w hav + = 00 (9) Frm (3), w hav = Substituting in (9) = = 00. =52.49 and = = (0.905)(52.49)= Frm (5) as = 2 an d = 2 2 = = W can nw plt th stady-stat rspns as w knw = = Drling Kindrsly India Pvt. Ltd 6

17 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. 2 = = Th transint and stady-stat rspnss ar plttd in Figs.3. and 3.2. Fi g.3. Transint rspns Fig.3.2 Stady-stat rspns Cas 2: Fr vry lw tim cnstant, i.. whn τ = T/20 = 0. ms. Sinc th tim cnstant is vry small, th capacitr chargs and dischargs vry fast. Th input and utput ar shwn in Fig Drling Kindrsly India Pvt. Ltd 7

18 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fig.3.3 Output fr th givn input whn tim cnstant is vry small 4. A currnt puls f amplitud 5 A in Fig.2p. is applid t a paralll cmbinatin shwn in Fig.2p.2. Plt t scal th wavfrms f th currnt flwing thrugh capacitr fr th cass: (i) t p =0., (ii) t p =, (iii) t p = 5 Fig.2p.Th givn input t th circuit Fig. 2p.2 Th givn circuit Slutin: Till t = t p, using Laplac transfrms, th circuit can b drawn as in Fig Drling Kindrsly India Pvt. Ltd 8

19 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fig.4. Circuit in trms f Laplac transfrms Applying KCL, w hav 5 5s 5 IC () s R s s s R s Cs Taking t Laplac invrs, th charging currnt is ic ( t) 5 At t = t p, th currnt suddnly falls frm 5 A t 0. Th vltag acrss th capacitr at t = t p is [ 5 ic ( t p )] R Thrfr fr t t p, [5 ic( tp)] R [5 ic( tp)] IC () s sr ( ) ( s ) Cs Taking Laplacinvrs i ( t) [5 i ( t )] C C p -t-t ( p ) Th circuit that rprsnts th discharg f th cndnsr is prsntd in Fig.4.2. Cas : Fr 0 < t < t i C p ( t) 5 i C dcays xpnntially, Fig.4.2 Circuit that indicats th discharg f th cndnsr t 0. at t t, i ( t ) A p C p Fr t > t p, i C riss xpnntially as 200 Drling Kindrsly India Pvt. Ltd 9

20 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. i C = 5 i C ( t )] [ p ( t t p ) = t t p Cas 2: Fr 0 < t < t p i i C dcays xpnntially, t Fr t > t p i C dcays xpnntially as C ( t) 5 at t tp, Ic( tp) A [5 I ( t )] C p t t p t = 3.6 t p Cas 3: Fr 0 < t < t p i C dcays xpnntially, i a C ( t) 5 t 5 5 t t tp, Ic( tp) A Fr t > t p i C riss xpnntially as t t p t t p [5 i C ( t p )] = Th input and utputs ar plttd in Fig Drling Kindrsly India Pvt. Ltd 20

21 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fig.4.3 Input and utputs fr th givn circuit 5. Draw th utput wavfrm if th wavfrm shwn in Fig.2p.5(a) is applid at th input f th circuit shwn in Fig.2p.5(b). 200 Drling Kindrsly India Pvt. Ltd 2

22 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Fig.2p.5(a) Th input t th high-pass circuit in Fig.2p.5(a) Slutin: Fig.2p.5(b) Th givn high-pass circuit Tim cnstatnt s Tim prid f input wavfrm is T = 4 ms Sinc <<T, th circuit acts as a gd diffrntiatr. dvi 9 dvi Th xprssin fr utput is v 0000 dt dt 00 Fr 0 t 2 ms, vi t, d v 0000 t m 3 3 dt v rmains at 50 m. At t =2 ms, v i falls by 00. Sinc capacitr acts as a shrt circuit, v als falls by th sam amunt. (t = 2 ms) = = v 200 Drling Kindrsly India Pvt. Ltd 22

23 Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G Fr 2 < t < 4 ms, v i ( t 2 0 ) d 00 3 v 0000 ( t 2 0 ) 3 dt 20 9 ( 00) m 3 20 Th utput wavfrm is shwn in Fig. 5.. Fig.5 Output f th high-pass circuit 200 Drling Kindrsly India Pvt. Ltd 23

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