4 Pulse Waeforms and he R-C Response 4.1 INRODUCION Our analysis hus far has been limied o alernaing waeforms ha ary in a sinusoidal manner. his chaper will inroduce he basic erminology associaed wih he pulse waeform and will examine he response of an R-C circui o a square-wae inpu. he imporance of he pulse waeform o he elecrical/elecronics indusry canno be oersaed. A as array of insrumenaion, communicaion sysems, compuers, radar sysems, and so on, all employ pulse signals o conrol operaion, ransmi daa, and display informaion in a ariey of formas. he response of he neworks described hus far o a pulse signal is quie differen from ha obained for sinusoidal signals. In fac, we will be reurning o he dc chaper on capaciors (Chaper 1) o reriee a few fundamenal conceps and equaions ha will help us in he analysis o follow. he conen of his chaper is quie inroducory in naure, designed simply o proide he fundamenals ha will be helpful when he pulse waeform is encounered in specific areas of applicaion. 4. IDEAL VERSUS ACUAL he ideal pulse of Fig. 4.1 has erical sides, sharp corners, and a fla peak characerisic; i sars insananeously a 1 and ends jus as abruply a. he waeform of Fig. 4.1 will be applied in he analysis o follow in his chaper and will probably appear in he iniial inesigaion of areas of applicaion beyond he scope of his ex. Once he fundamenal operaion of a deice, package, or sysem is clearly undersood using ideal characerisics, he effec of an acual (or rue or pracical) pulse mus be considered. If an aemp were made o inroduce all he differences beween an ideal and acual pulse in a single figure, he resul would probably be complex and confusing. A number of waeforms will herefore be used o define he criical parameers.
194 PULSE WAVEFORMS AND HE R-C RESPONSE Rising or leading edge Ideal pulse Falling or railing edge Ampliude 1 p (pulse widh) FIG. 4.1 Ideal pulse waeform. he reacie elemens of a nework, in heir effor o preen insananeous changes in olage (capacior) and curren (inducor), esablish a slope o boh edges of he pulse waeform, as shown in Fig. 4.. he rising edge of he waeform of Fig. 4. is defined as he edge ha increases from a lower o a higher leel. V 1.5V 1 Rising or leading edge Falling or railing edge Ampliude p (pulse widh) FIG. 4. Acual pulse waeform. he falling edge is defined by he region or edge where he waeform decreases from a higher o a lower leel. Since he rising edge is he firs o be encounered (closes o s), i is also called he leading edge. he falling edge always follows he leading edge and is herefore ofen called he railing edge. Boh regions are defined in Figs. 4.1 and 4.. Ampliude For mos applicaions, he ampliude of a pulse waeform is defined as he peak-o-peak alue. Of course, if he waeforms all sar and reurn o he zero-ol leel, hen he peak and peak-o-peak alues are synonymous. For he purposes of his ex, he ampliude of a pulse waeform is he peak-o-peak alue, as illusraed in Figs. 4.1 and 4..
IDEAL VERSUS ACUAL 195 Pulse Widh he pulse widh ( p ), or pulse duraion, is defined by a pulse leel equal o 5% of he peak alue. For he ideal pulse of Fig. 4.1, he pulse widh is he same a any leel, whereas p for he waeform of Fig. 4. is a ery specific alue. Base-Line Volage he base-line olage (V b ) is he olage leel from which he pulse is iniiaed. he waeforms of Figs. 4.1 and 4. boh hae a -V base-line olage. In Fig. 4.3(a) he base-line olage is 1 V, whereas in Fig. 4.3(b) he base-line olage is 4 V. 5 V Ampliude = 4 V V b = 1 V (a) 1 V V b = 4 V (b) Ampliude = 6 V FIG. 4.3 Defining he base-line olage. Posiie-Going and Negaie-Going Pulses A posiie-going pulse increases posiiely from he base-line olage, whereas a negaie-going pulse increases in he negaie direcion from he base-line olage. he waeform of Fig. 4.3(a) is a posiie-going pulse, whereas he waeform of Fig. 4.3(b) is a negaie-going pulse. Een hough he base-line olage of Fig. 4.4 is negaie, he waeform is posiie-going (wih an ampliude of 1 V) since he olage increased in he posiie direcion from he base-line olage. 9 V Ampliude = 1 V 1 V V b p FIG. 4.4 Posiie-going pulse.
196 PULSE WAVEFORMS AND HE R-C RESPONSE Rise ime ( r ) and Fall ime ( f ) Of paricular imporance is he ime required for he pulse o shif from one leel o anoher. he rounding (defined in Fig. 4.5) ha occurs a he beginning and end of each ransiion makes i difficul o define he exac poin a which he rise ime should be iniiaed and erminaed. For his reason, he rise ime and he fall ime are defined by he 1% and 9% leels, as indicaed in Fig. 4.5. Noe ha here is no requiremen ha r equal f. V 1 (9%).9V 1 (1%).1V 1 r f FIG. 4.5 Defining r and f. il V 1 V il V An undesirable bu common disorion normally occurring due o a poor low-frequency response characerisic of he sysem hrough which a pulse has passed appears in Fig. 4.6. he drop in peak alue is called il, droop, or sag. he percenage il is defined by % il V 1 V 1% V (4.1) where V is he aerage alue of he peak ampliude as deermined by FIG. 4.6 Defining il. Oershoo Ringing V V 1 V (4.) Naurally, he less he percenage il or sag, he more ideal he pulse. Due o rounding, i may be difficul o define he alues of V 1 and V. I is hen necessary only o approximae he sloping region by a sraighline approximaion and use he resuling alues of V 1 and V. Oher disorions include he preshoo and oershoo appearing in Fig. 4.7, normally due o pronounced high-frequency effecs of a sysem, and ringing, due o he ineracion beween he capaciie and inducie elemens of a nework a heir naural or resonan frequency. Preshoo FIG. 4.7 Defining preshoo, oershoo, and ringing. EXAMPLE 4.1 Deermine he following for he pulse waeform of Fig. 4.8: a. posiie- or negaie-going? b. base-line olage
PULSE REPEIION RAE AND DUY CYCLE 197 (V) 8 7 1 3 4 5 6 7 8 9 1111 131415 (ms) 4 FIG. 4.8 Example 4.1. c. pulse widh d. maximum ampliude e. il Soluions: a. posiie-going b. V b 4 V c. p (1 7) ms 5 ms d. V max 8 V 4 V 1 V V 1 V 1 V 11 V 3 V e. V 11.5 V V 1 V 1 V 11 V % il 1% 1% 8.696% V 11.5 V (Remember, V is defined by he aerage alue of he peak ampliude.) EXAMPLE 4. Deermine he following for he pulse waeform of Fig. 4.9: a. posiie- or negaie-going? b. base-line olage c. il d. ampliude e. p f. r and f 9% 1% p Soluions: a. posiie-going b. V b V c. % il % d. ampliude (4 di.)(1 mv/di.) 4 mv e. p (3. di.)(5 ms/di.) 16 ms f. r (.4 di.)(5 ms/di.) ms f (.8 di.)(5 ms/di.) 4 ms r f Verical sensiiiy = 1 mv/di. Horizonal sensiiiy = 5 ms/di. FIG. 4.9 Example 4.. 4.3 PULSE REPEIION RAE AND DUY CYCLE A series of pulses such as hose appearing in Fig. 4.1 is called a pulse rain. he arying widhs and heighs may conain informaion ha can be decoded a he receiing end.
198 PULSE WAVEFORMS AND HE R-C RESPONSE FIG. 4.1 Pulse rain. If he paern repeas iself in a periodic manner as shown in Fig. 4.11(a) and (b), he resul is called a periodic pulse rain. he period () of he pulse rain is defined as he ime differenial beween any wo similar poins on he pulse rain, as shown in Figs. 4.11(a) and (b). he pulse repeiion frequency (prf), or pulse repeiion rae (prr), is defined by prf (or prr) 1 (Hz or pulses/s) (4.3) Applying Eq. (4.3) o each waeform of Fig. 4.11 will resul in he same pulse repeiion frequency since he periods are he same. he resul clearly reeals ha he shape of he periodic pulse does no affec he deerminaion of he pulse repeiion frequency. p p (1 ms) ( ms) 3 (3 ms). (1 ms) ( ms) 3 (3 ms) (a) (b) FIG. 4.11 Periodic pulse rains. he pulse repeiion frequency is deermined solely by he period of he repeaing pulse. he facor ha will reeal how much of he period is encompassed by he pulse is called he duy cycle, defined as follows: Duy cycle pulse widh period 1% or Duy cycle p 1% (4.4) For Fig. 4.11(a) (a square-wae paern),.5 Duy cycle 1% 5% and for Fig. 4.11(b),. Duy cycle 1% % he aboe resuls clearly reeal ha he duy cycle proides a percenage indicaion of he porion of he oal period encompassed by he pulse waeform.
PULSE REPEIION RAE AND DUY CYCLE 199 EXAMPLE 4.3 Deermine he pulse repeiion frequency and he duy cycle for he periodic pulse waeform of Fig. 4.1. V b = 3 mv (mv) 5 1 15 5 3 (ms) 1 FIG. 4.1 Example 4.3. Soluion: (15 6) ms 9 ms 1 1 prf 111.11 khz 9 ms p Duy cycle 1% (8 6) ms 1% 9 ms 1%.% 9 EXAMPLE 4.4 Deermine he pulse repeiion frequency and he duy cycle for he oscilloscope paern of Fig. 4.13 haing he indicaed sensiiiies. Verical sensiiiy =. V/di. Horizonal sensiiiy = 1 ms/di. di. FIG. 4.13 Example 4.4. Soluion: (3. di.)(1 ms/di.) 3. ms p (.8 di.)(1 ms/di.).8 ms 1 1 prf 31.5 Hz 3. ms p.8 ms Duy cycle 1% 1% 5% 3. ms
11 PULSE WAVEFORMS AND HE R-C RESPONSE EXAMPLE 4.5 Deermine he pulse repeiion rae and duy cycle for he rigger waeform of Fig. 4.14. V.5 V di. Horizonal sensiiiy = 1 s/di. FIG. 4.14 Example 4.5. Soluion: (.6 di.)(1 ms/di.) 6 ms 1 1 prf 38,46 khz 6 ms p (. di.)(1 ms/di.) ms p ms Duy cycle 1% 1% 7.69% 6 ms 4.4 AVERAGE VALUE he aerage alue of a pulse waeform can be deermined using one of wo mehods. he firs is he procedure oulined in Secion 13.6, which can be applied o any alernaing waeform. he second can be applied only o pulse waeforms since i uilizes erms specifically relaed o pulse waeforms; ha is, V a (duy cycle)(peak alue) (1 duy cycle)(v b ) (4.5) In Eq. (4.5), he peak alue is he maximum deiaion from he reference or zero-ol leel, and he duy cycle is in decimal form. Equaion (4.5) does no include he effec of any il pulse waeforms wih sloping sides. EXAMPLE 4.6 Deermine he aerage alue for he periodic pulse waeform of Fig. 4.15. Soluion: By he mehod of Secion 13.6, area under cure G (1 ) ms 1 ms (8 mv)(4 ms) ( mv)(6 ms) G 1 ms 44 1 9 4.4 mv 1 1 6 3 1 9 1 1 9 1 1 6
AVERAGE VALUE 111 (mv) 8 7 6 5 4 3 1 V a 5 1 15 ( s) FIG. 4.15 Example 4.6. By Eq. (4.5), V b mv p (6 ) ms 4 Duy cycle.4 (decimal form) 1 ms 1 Peak alue (from -V reference) 8 mv V a (duy cycle)(peak alue) (1 duy cycle)(v b ) (.4)(8 mv) (1.4)( mv) 3. mv 1. mv 4.4 mv as obained aboe. EXAMPLE 4.7 Gien a periodic pulse waeform wih a duy cycle of 8%, a peak alue of 7 V, and a base-line olage of 3 V: a. Deermine he aerage alue. b. Skech he waeform. c. Verify he resul of par (a) using he mehod of Secion 13.6. Soluions: a. By Eq. (4.5), V a (duy cycle)(peak alue) (1 duy cycle)(v b ) (.8)(7 V) (1.8)( 3 V) 1.96 V (.16 V). V b. See Fig. 4.16. (7 V)(.8) (3 V)(.7) c. G 1.96 V.16 V. V as obained aboe. 7 V 3 V.8 FIG. 4.16 Soluion o par (b) of Example 4.7. di. di. Verical sensiiiy = 5 mv/di. Horizonal sensiiiy = 5 s/di. (a) ac mode Verical sensiiiy = 5 mv/di. Horizonal sensiiiy = 5 s/di. Insrumenaion he aerage alue (dc alue) of any waeform can be easily deermined using he oscilloscope. If he mode swich of he scope is se in he ac posiion, he aerage or dc componen of he applied waeform will be blocked by an inernal capacior from reaching he screen. he paern can be adjused o esablish he display of Fig. 4.17(a). If he mode swich is hen placed in he dc posiion, he erical shif (posiie or (b) V a = 4 mv dc mode FIG. 4.17 Deermining he aerage alue of a pulse waeform using an oscilloscope.
11 PULSE WAVEFORMS AND HE R-C RESPONSE negaie) will reeal he aerage or dc leel of he inpu signal, as shown in Fig. 4.17(b). 4.5 RANSIEN R-C NEWORKS In Chaper 1 he general soluion for he ransien behaior of an R-C nework wih or wihou iniial alues was deeloped. he resuling equaion for he olage across a capacior is repeaed below for conenience. C V f (V i V f )e /RC (4.6) V i V f C 5 V f V i FIG. 4.18 Defining he parameers of Eq. (4.6). C 5 V.44 V V 5 FIG. 4.19 Example of he use of Eq. (4.6). Recall ha V i is he iniial olage across he capacior when he ransien phase is iniiaed as shown in Fig. 4.18. he olage V f is he seady-sae (resing) alue of he olage across he capacior when he ransien phase has ended. he ransien period is approximaed as 5, where is he ime consan of he nework and is equal o he produc RC. For he siuaion where he iniial olage is zero ols, he equaion reduces o he following familiar form, where V f is ofen he applied olage: C V f (1 e /RC ) V i V (4.7) For he case of Fig. 4.19, V i V, V f 5 V, and C V i (V f V i )(1 e /RC ) V [5 V ( V)](1 e /RC ) C V 7 V(1 e /RC ) For he case where RC, C V 7 V(1 e / ) V 7 V(1 e 1 ) V 7 V(1.368) V 7 V(.63) C.44 V as erified by Fig. 4.19. E 8 V R 1 k C 1 mf V EXAMPLE 4.8 he capacior of Fig. 4. is iniially charged o Vbefore he swich is closed. he swich is hen closed. a. Deermine he mahemaical expression for C. b. Deermine he mahemaical expression for i C. c. Skech he waeforms of C and i C. Soluions: a. V i V V f (afer 5) E 8 V RC (1 k )(1 mf) 1 ms By Eq. (4.6), C V f (V i V f )e /RC 8 V ( V 8 V)e / FIG. 4. Example 4.8. and C 8 V 6 V e /
RANSIEN R-C NEWORKS 113 b. When he swich is firs closed, he olage across he capacior canno change insananeously, and V R E V i 8 V V 6 V. he curren herefore jumps o a leel deermined by Ohm s law: 8 C (V) V R 6 V I Rmax R.6 ma 1 k he curren will hen decay o zero amperes wih he same ime consan calculaed in par (a), and i C.6 mae / c. See Fig. 4.1..1..3.4.5.6.7.8 5 i C (ma).1 (s) EXAMPLE 4.9 Skech C for he sep inpu shown in Fig. 4.. Assume ha he 4 mv has been presen for a period of ime in excess of fie ime consans of he nework. hen deermine when C V if he sep changes leels a s..6.1..3.4.5.6.7.8 (s) i 1 mv 4 mv i R 1 k 4 mv C.1 mf C FIG. 4.1 C and i C for he nework of Fig. 4.. FIG. 4. Example 4.9. Soluion: V i 4 mv V f 1 mv RC (1 k )(.1 mf) 1 ms By Eq. (4.6), C V f (V i V f )e /RC 1 mv ( 4 mv 1 mv)e /1ms and C 1 mv 14 mv e /1ms he waeform appears in Fig. 4.3. 1 C (mv) = 3.37 ms 1 3 4 5 6 7 8 (ms) 4 5 FIG. 4.3 C for he nework of Fig. 4..
114 PULSE WAVEFORMS AND HE R-C RESPONSE and Subsiuing C V ino he aboe equaion yields C V 1 mv 14 mv e /1ms 1 mv 14 mv e /1ms or bu.714 e /1ms log e.714 log e (e /1ms ) 1 ms and (1 ms)log e.714 (1 ms)(.377) 3.37 ms as indicaed in Fig. 4.3. 4.6 R-C RESPONSE O SQUARE-WAVE INPUS he square wae of Fig. 4.4 is a paricular form of pulse waeform. I has a duy cycle of 5% and an aerage alue of zero ols, as calculaed below: p / Duy cycle 1% 1% 5% (V 1 )(/) ( V 1 )(/) V a V V 1 3 V 1 FIG. 4.4 Periodic square wae. he applicaion of a dc olage V 1 in series wih he square wae of Fig. 4.4 can raise he base-line olage from V 1 o zero ols and he aerage alue o V 1 ols. If a square wae such as deeloped in Fig. 4.5 is applied o an V 1 V 1 FIG. 4.5 Raising he base-line olage of a square wae o zero ols.
R-C RESPONSE O SQUARE-WAVE INPUS 115 R-C circui as shown in Fig. 4.6, he period of he square wae can hae a pronounced effec on he resuling waeform for C. For he analysis o follow, we will assume ha seady-sae condiions will be esablished afer a period of fie ime consans has passed. he ypes of waeforms deeloped across he capacior can hen be separaed ino hree fundamenal ypes: / 5; / 5; and / 5. i R V i C C FIG. 4.6 Applying a periodic square-wae pulse rain o an R-C nework. / > 5 he condiion / > 5, or > 1, esablishes a siuaion where he capacior can charge o is seady-sae alue in adance of /. he resuling waeforms for C and i C will appear as shown in Fig. 4.7. Noe how closely he olage C shadows he applied waeform and how i C is nohing more han a series of ery sharp spikes. Noe also ha he change of V i from V o zero ols during he railing edge simply resuls in a rapid discharge of C o zero ols. In essence, when V i, he capacior and resisor are in parallel and he capacior simply discharges hrough R wih a ime consan equal o ha encounered during he charging phase bu wih a direcion of charge flow (curren) opposie o ha esablished during he charging phase. C 5 > 5 V 5 (a) V R V R i C 5 5 (b) FIG. 4.7 C and i C for / > 5. / 5 If he frequency of he square wae is chosen such ha / 5 or 1, he olage C will reach is final alue jus before beginning is discharge phase, as shown in Fig. 4.8. he olage C no longer resembles he square-wae inpu and, in fac, has some of he characerisics of a riangular waeform. he increased ime consan has resuled in a more rounded C, and i C has increased subsanially in widh o reeal he longer charging period.
116 PULSE WAVEFORMS AND HE R-C RESPONSE C 5 V = 5 5 (a) V R i C 5 V 5 R (b) FIG. 4.8 C and i C for / 5. / < 5 If / < 5 or < 1, he olage C will no reach is final alue during he firs pulse (Fig. 4.9), and he discharge cycle will no reurn o zero ols. In fac, he iniial alue for each succeeding pulse will change unil seady-sae condiions are reached. In mos insances, i is a good approximaion o assume ha seady-sae condiions hae been esablished in fie cycles of he applied waeform. V C 5 < 5 V R i C 5 3 (a) V R 3 (b) FIG. 4.9 C and i C for / < 5. As he frequency increases and he period decreases, here will be a flaening of he response for C unil a paern like ha in Fig. 4.3 resuls. Figure 4.3 begins o reeal an imporan conclusion regarding he response cure for C : Under seady-sae condiions, he aerage alue of C will equal he aerage alue of he applied square wae. C << 5 V 3 FIG. 4.3 C for / K 5 or K 1.
R-C RESPONSE O SQUARE-WAVE INPUS 117 Noe in Figs. 4.9 and 4.3 ha he waeform for C approaches an aerage alue of V/. EXAMPLE 4.1 he 1-Hz square wae of Fig. 4.31 is applied o he R-C circui of he same figure. a. Compare he pulse widh of he square wae o he ime consan of he circui. b. Skech C. c. Skech i C. i f = 1 Hz V = 1 mv i R 5 k C i C.1 mf C FIG. 4.31 Example 4.1. Soluions: 1 1 a. 1 ms f 1 p.5 ms p RC (5 1 3 )(.1 1 6 F).5 ms.5 ms.5 ms 1 and C p 1 he resul reeals ha C will charge o is final alue in half he pulse widh. b. For he charging phase, V i V and V f 1 mv, and C V f (V i V f )e /RC 1 mv ( 1 mv)e / and C 1 mv(1 e / ) For he discharge phase, V i 1 mv and V f V, and C V f (V i V f )e / V (1 mv V)e / and C 1 mve / he waeform for C appears in Fig. 4.3. c. For he charging phase a s, V R V and I Rmax V/R 1 mv/5 k ma, and i C I max e / mae / For he discharge phase, he curren will hae he same mahemaical formulaion bu he opposie direcion, as shown in Fig. 4.33. 1 mv 5 p = 1 FIG. 4.3 C for he R-C nework of Fig. 4.31. i C ma 5 ma FIG. 4.33 i C for he R-C nework of Fig. 4.31.
118 PULSE WAVEFORMS AND HE R-C RESPONSE EXAMPLE 4.11 Repea Example 4.1 for f 1 khz. Soluion: 1 1.1 ms f 1 khz and.5 ms wih p.5 ms In oher words, he pulse widh is exacly equal o he ime consan of he nework. he olage C will no reach he final alue before he firs pulse of he square-wae inpu reurns o zero ols. For in he range o /, V i V and V f 1 mv, and C 1 mv(1 e / ) We recall from Chaper 1 ha a, C 63.% of he final alue. Subsiuing ino he equaion aboe yields C (1 mv)(1 e 1 ) (1 mv)(1.368) (1 mv)(.63) 6.3 mv as shown in Fig. 4.34. C V = 1 mv 6.3 mv 7.18 mv 7.9 mv 7.31 mv 7.31 mv.69 mv.33 mv.64 mv.68 mv.69 mv 3 4 () (3) (4) (5) (6) (7) (8) (9) () FIG. 4.34 C response for p /. For he discharge phase beween / and, V i 6.3 mv and V f V, and C V f (V i V f )e / V (6.3 mv V)e / C 6.3 mve / wih now being measured from / in Fig. 4.34. In oher words, for each ineral of Fig. 4.34, he beginning of he ransien waeform is defined as s. he alue of C a is herefore deermined by subsiuing ino he aboe equaion, and no as defined by Fig. 4.34. Subsiuing, C (6.3 mv)(e 1 ) (6.3 mv)(.368).33 mv as shown in Fig. 4.34.
R-C RESPONSE O SQUARE-WAVE INPUS 119 For he nex ineral, V i.33 mv and V f 1 mv, and C V f (V i V f )e / 1 mv (.33 mv 1 mv)e / C 1 mv 7.67 mv e / A (since is now s for his ineral), C 1 mv 7.67 mve 1 1 mv.8 mv C 7.18 mv as shown in Fig. 4.34. For he discharge ineral, V i 7.18 mv and V f V, and C V f (V i V f )e / V (7.18 mv )e / C 7.18 mve / A (measured from 3 of Fig. 4.34), C (7.18 mv)(e 1 ) (7.18 mv)(.368).64 mv as shown in Fig. 4.34. Coninuing in he same manner, he remaining waeform for C will be generaed as depiced in Fig. 4.34. Noe ha repeiion occurs afer 8, and he waeform has essenially reached seady-sae condiions in a period of ime less han 1, or fie cycles of he applied square wae. A closer look will reeal ha boh he peak and he lower leels coninued o increase unil seady-sae condiions were esablished. Since he exponenial waeforms beween 4 and 5 hae he same ime consan, he aerage alue of C can be deermined from he seady-sae 7.31-mV and.69-mv leels as follows: 7.31 mv.69 mv 1 mv V a 5 mv which equals he aerage alue of he applied signal as saed earlier in his secion. We can use he resuls of Fig. 4.34 o plo i C. A any insan of ime, i R C or R i C and i R i C i C R A, C V, and i R i 1 mv R C ma 5 k as shown in Fig. 4.35. As he charging process proceeds, he curren i C will decay a a rae deermined by i C mae / A, i C ( ma)(e / ) ( ma)(e 1 ) ( ma)(.368).736 ma as shown in Fig. 4.35.
111 PULSE WAVEFORMS AND HE R-C RESPONSE i C ma 1.534 ma 1.47 ma 1.464 ma 1.46 ma.736 ma.565 ma.54 ma.539 ma.538 ma 3 4 () (3) (4) (5) (6) (7 ) (8) (9) ().465 ma.58 ma.537 ma.538 ma.538 ma 1.64 ma 1.436 ma 1.458 ma 1.46 ma 1.46 ma FIG. 4.35 i C response for p /. For he railing edge of he firs pulse, he olage across he capacior canno change insananeously, resuling in he following when i drops o zero ols: i C i R i 6.3 mv R C 1.64 ma 5 k as illusraed in Fig. 4.35. he curren will hen decay as deermined by i C 1.64 mae / and a (acually in Fig. 4.35), i C ( 1.64 ma)(e / ) ( 1.64 ma)(e 1 ) ( 1.64 ma)(.368).465 ma as shown in Fig. 4.35. A ( ), C.33 mv, and i reurns o 1 mv, resuling in i C i R i 1 mv.33 mv R C 1.534 ma 5 k he equaion for he decaying curren is now i C 1.534 mae / and a (acually 3 in Fig. 4.35), i C (1.534 ma)(.368).565 ma he process will coninue unil seady-sae condiions are reached a he same ime hey were aained for C. Noe in Fig. 4.35 ha he posiie peak curren decreased oward seady-sae condiions while he negaie peak became more negaie. I is also ineresing and imporan o realize ha he curren waeform becomes symmerical abou he axis when seady-sae condiions are esablished. he resul is ha he ne aerage curren oer one cycle is zero, as i should be in a series R-C circui. Recall from Chaper 1 ha he capacior under dc seady-sae condiions can be replaced by an open-circui equialen, resuling in I C A.
OSCILLOSCOPE AENUAOR AND COMPENSAING PROBE 1111 Alhough boh examples proided aboe sared wih an uncharged capacior, here is no reason ha he same approach canno be used effeciely for iniial condiions. Simply subsiue he iniial olage on he capacior as V i in Eq. (4.6) and proceed as aboe. 4.7 OSCILLOSCOPE AENUAOR AND COMPENSAING PROBE he 1 aenuaor probe employed wih oscilloscopes is designed o reduce he magniude of he inpu olage by a facor of 1. If he inpu impedance o a scope is 1 M, he 1 aenuaor probe will hae an inernal resisance of 9 M, as shown in Fig. 4.36. Scope V V Probe R p 9 M Verical R s 1 M V V FIG. 4.36 1 aenuaor probe. Applying he olage diider rule, (1 M )(V V scope i ) 1 V i 1 M 9 M 1 In addiion o he inpu resisance, oscilloscopes hae some inernal inpu capaciance, and he probe will add an addiional capaciance in parallel wih he oscilloscope capaciance, as shown in Fig. 4.37. he probe capaciance is ypically abou 1 pf for a 1-m (3.3-f) cable, reaching abou 15 pf for a 3-m (9.9-f) cable. he oal inpu capaciance is herefore he sum of he wo capaciie elemens, resuling in he equialen nework of Fig. 4.38. V i R p 9 M Probe C c Cable 1 pf (1 meer cable) Scope C s pf V scope R s = 1 M V i R p 9 M R s 1 M V scope C i = C c C s = 3 pf FIG. 4.37 Capaciie elemens presen in an aenuaor probe arrangemen. FIG. 4.38 Equialen nework of Fig. 4.37. For he analysis o follow, le us deermine he héenin equialen circui for he capacior C i : (1 M )(V E h i ) 1 V i 1 M 9 M 1 and R h 9 M 1 M.9 MQ
111 PULSE WAVEFORMS AND HE R-C RESPONSE R h.9 M E i C i 3 pf h.1 C FIG. 4.39 héenin equialen for C i of Fig. 4.38. s =.1 i scope V C = scope 1 s 17 s FIG. 4.4 he scope paern for he condiions of Fig. 4.38 wih i V peak. he héenin nework is shown in Fig. 4.39. For i V (peak), E h.1 i V (peak) and for C, V f V and V i V, wih RC (.9 1 6 )(3 1 1 F) 7 ms For an applied frequency of 5 khz, 1. ms and.1 ms 1 ms f wih 5 135 ms > 1 ms, as shown in Fig. 4.4, clearly producing a seere rounding disorion of he square wae and a poor represenaion of he applied signal. o improe maers, a ariable capacior is ofen added in parallel wih he resisance of he aenuaor, resuling in a compensaed aenuaor probe such as he one shown in Fig. 4.41. In Chaper 1, i was demonsraed ha a square wae can be generaed by a summaion of sinusoidal signals of paricular frequency and ampliude. If we herefore design a nework such as he one shown in Fig. 4.4 ha will ensure ha V scope is.1 i for any frequency, hen he rounding disorion will be remoed, and V scope will hae he same appearance as i. Applying he olage diider rule o he nework of Fig. 4.4, ZsVi V scope Zs Z p (4.8) If he parameers are chosen or adjused such ha FIG. 4.41 Commercial compensaed 1 :1 aenuaor probe. (Couresy of ekronix, Inc.) V i Z p C p R p 9 M Probe R s 1M C i V scope FIG. 4.4 Compensaed aenuaor and inpu impedance o a scope, including he cable capaciance. Z s (4.9) he phase angle of Z s and Z p will be he same, and Equaion (4.8) will reduce o (4.1) which is insensiie o frequency since he capaciie elemens hae dropped ou of he relaionship. In he laboraory, simply adjus he probe capaciance using a sandard or known square-wae signal unil he desired sharp corners of he square wae are obained. If you aoid he calibraion sep, you may make a rounded signal look square since you assumed a square wae a he poin of measuremen. oo much capaciance will resul in an oershoo effec, whereas oo lile will coninue o show he rounding effec. 4.8 APPLICAION R p C p R s C s RsVi V scope Rs R p V Remoe he V remoe, or clicker, is so much a par of our modern-day liing ha we mus all hae wondered a one ime or anoher how i looks inside or how i works. In many ways i is similar o he garage door
APPLICAION 1113 opener or he car alarm ransmier in ha here is no isible connecion beween he ransmier and he receier, and each ransmier is linked o is receier wih a special code. he only major difference beween he V remoe and he oher conrols is ha he V remoe uses an infrared frequency while he oher wo use a much lower radio frequency. he V remoe of Fig. 4.43(a) has been opened o reeal he inernal consrucion of is key pad and face in Fig. 4.43(b). he hree componens of Fig. 4.43(b) were placed a a leel ha would permi maching he holes in he coer wih he acual keys in he swich membrane and wih he locaion ha each buon on he key pad would hi on face of he prined circui board. Noe on he prined circui board ha here (a) (b) (c) (d) FIG. 4.43 V remoe: (a) exernal appearance; (b) inernal consrucion; (c) carbon key pads; (d) enlarged iew of S31 key pad.
1114 PULSE WAVEFORMS AND HE R-C RESPONSE Crysal (crysal oscillaor) Capacior IR LED Resisor Swich-marixencoder IC is a black pad o mach each key on he membrane. he back side of he swich membrane is shown in Fig. 4.43(c) o show he sof carbon conacs ha will make conac wih he carbon conacs on he prined board when he buons are depressed. An enlarged iew of one of he conacs (S31) of Fig. 4.43(c) is shown in Fig. 4.43(d) o illusrae he separaion beween circuis and he paern used o ensure coninuiy when he solid round carbon pad a he boom of he key is pu in place. All he connecions esablished when a key is pressed are passed on o a relaiely large swich-marix-encoder IC chip appearing on he back side of he prined circui board as shown in Fig. 4.44. For he pad (S31) of Fig. 4.43(d), hree wires of he marix appearing in Fig. 4.43(b) will be conneced when he corresponding key (number 5) is pressed. he encoder will hen reac o his combinaion and send ou he appropriae signal as an infrared (IR) signal from he IR LED appearing a he end of he remoe conrol, as shown in Fig. 4.43(b) and Fig. 4.44. he second smaller LED (red on acual uni) appearing a he op of Fig. 4.43(b) blinks during ransmission. Once he baeries are insered, he CMOS elecronic circuiry ha conrols he operaion of he remoe is always on. his is possible only because of he ery low power drain of CMOS circuiry. he power (PWR) buon is used only o urn he V on and aciae he receier. he signal sen ou by he majoriy of remoes is one of he wo ypes appearing in Fig. 4.45. In each case here is a key pulse o iniiae he signal sequence and o inform he receier ha he coded signal is abou o arrie. In Fig. 4.45(a), a 4-bi binary-coded signal is ransmied using pulses in specific locaions o represen he ones and using he absence of a pulse o represen he zeros. ha coded signal can hen be inerpreed by he receier uni and he proper operaion performed. In Fig. 4.45(b), he signal is frequency conrolled. Each key will hae a differen frequency associaed wih i. he resul is ha each key will hae a specific ransmission frequency. Since each V receier will respond o a differen pulse rain, a remoe mus be coded for he V under conrol. here are fixed program remoes ha can be used wih FIG. 4.44 Back side of V remoe of Fig. 4.43. V Key pulse 1 1 1 1 1 ON CHANNEL OFF (a) V Key pulse ON High frequency (b) OFF INCREASE VOLUME Low frequency Mid-frequency FIG. 4.45 Signal ransmission: (a) pulse rain; (b) ariaion.
COMPUER ANALYSIS 1115 only one V. hen here are smar remoes ha are preprogrammed inernally wih a number of remoe conrol codes. Remoes of his ype simply need o be old which V is inoled using a hree-digi coding sysem, and hey will adap accordingly. Learning remoes are hose ha can use he old remoe o learn he code and hen sore i for fuure use. In his case, one remoe is se direcly in fron of he oher, and he informaion is ransferred from one o he oher when boh are energized. Remoes are also aailable ha are a combinaion of he las wo. he remoe of Fig. 4.43 uses four AAA baeries in series for a oal of 6 V. I has is own local crysal oscillaor separae from he IC as shown by he discree elemens o he op righ and midlef of he prined circui board of Fig. 4.43(c). he crysal iself, which is relaiely large compared o he oher elemens, appears on he oher side of he board jus aboe he elecrolyic capacior in Fig. 4.44. I is he responsibiliy of he oscillaor o generae he pulse signal required for proper IC operaion. Noe how flush mos of he discree elemens are in Fig. 4.43(b), and noe he raher large elecrolyic capacior on he back of he prined circui board in Fig. 4.44. he specificaions on he uni gie i a range conrol of 5 f wih a 3 coerage arc as shown in Fig. 4.46. he arc coerage of your uni can easily be esed by simply poining i direcly a he V and hen moing i in any direcion unil i no longer conrols he V. Clicker 3 FIG. 4.46 Range and coerage arc for V remoe of Fig. 4.43. 5 4.9 COMPUER ANALYSIS PSpice R-C Response Our analysis will begin wih a erificaion of he resuls of Example 4.1 which examined he response of he series R- C circui appearing on he schemaic of Fig. 4.47. he source is one used in Chapers 1 and 1 o replicae he acion of a swich in series wih a dc source. he defining aribues for he pulse waeform are repeaed for conenience in Fig. 4.48. Recall ha he PW was made long enough so ha he full ransien period could be examined. In his analysis he pulse widh will be adjused o permi iewing he ransien behaior of an R-C nework beween changing leels of he applied pulse. Iniially he PW will be se a 1 imes he ime consan of he nework so ha he full ransien response can occur beween changes in olage leel. he ime consan of he nework is RC (5 k )(.1 mf).5 ms, resuling in a PW of.5 ms in Fig. 4.47. o esablish a square-wae appearance, he period was chosen as wice he pulse widh or 1 ms as shown in he VPulse lising. In he Simulaion Seings dialog box, ime Domain(ransien) is seleced because we wan a response ersus ime. he Run o ime is seleced as ms so ha wo full cycles will resul. he Sar saing daa afer was lef on he defaul alue of s, and he Maximum sep size was se a ms/1 ms. Afer simulaion, race-add race- I(C)-OK, he boom plo of Fig. 4.49 was he resul. Noe ha he maximum curren is ma as deermined by I Cmax 1 mv/5 k, and he full ransien response appears wihin each pulse. Noe also ha he curren dropped below he axis o reeal a change in direcion when he applied olage dropped from he 1-mV leel o V. hrough Plo- Add Plo o Window-race-Add race-v(vpulse: )-OK-race- Add race-v(c:1)-ok, he plos of he applied olage and he olage across he capacior can be displayed in he upper graph of Fig. 4.49.
1116 PULSE WAVEFORMS AND HE R-C RESPONSE FIG. 4.47 Using PSpice o erify he resuls of Example 4.1. V1 V D R V PW F PER Firs we selec he upper graph of Fig. 4.49 o which o moe he SEL>>, and hen we selec he oggle cursor key. Now we can lefclick on V(C:1) a he boom righ of he graph and se a cursor on he graph wih a lef click of he mouse. Seing he cursor a fie ime consans reeals ha he ransien olage has reached 9.935 mv. Seing he righ-click cursor a en ime consans reeals ha V C has essenially reached he 1-mV leel. FIG. 4.48 Defining he PSpice Vpulse parameers. Seing p / he parameers of he source will now be modified by changing he frequency of he pulse waeform o 1 khz wih a period of.1 ms and a pulse widh of.5 ms. For Vpulse he changes are PW.5 ms and PER.1 ms. he ime consan of he nework remains he same a.5 ms, so we hae a siuaion where he pulse widh equals he ime consan of he circui. he resul is ha i will ake a number of pulses before he olage across he capacior will reach is final alue of 1 mv. Under he Simulaion Seings, he Run o ime will be changed o.5 ms 5 ms orfie cycles of he applied olage. he Maximum sep size will be changed o 5 ms/1 5 ns.5 ms. Under he SCHEMAIC1 window, race-add race-v(c:1)-ok is seleced o obain he ransien olage across he capacior, while race-add race-v(vpulse: ))-OK will place he applied olage on he same screen. Noe in he resuling plos of Fig. 4.5 ha he olage builds up from V unil i appears o reach a fairly seady sae afer 4 ms. A 4 ms, a lef cursor (A1) was used o find he minimum poin wih.71 mv resuling a close mach wih he longhand calculaion of Example 4.11 a.69 mv. A 45 ms, he righ-click cursor (A) proided a leel of 7.9 mv which is again a close mach wih he calculaed leel of 7.31 mv.
COMPUER ANALYSIS 1117 FIG. 4.49 Plo of pulse, C, and i C for he circui of Fig. 4.47. FIG. 4.5 Plo of C for he circui of Fig. 4.47 wih p /.
1118 PULSE WAVEFORMS AND HE R-C RESPONSE PROBLEMS SECION 4. Ideal ersus Acual 1. Deermine he following for he pulse waeform of Fig. 4.51: a. posiie- or negaie-going? b. base-line olage c. pulse widh d. ampliude e. % il. Repea Problem 1 for he pulse waeform of Fig. 4.5. (V) (mv) 8 7 7.5..4 1.8. 3.4 3.6 (ms) 1 4 7 1 15 3 (ms) FIG. 4.51 Problems 1, 8, and 1. FIG. 4.5 Problems and 9. di. Verical sensiiiy = 1 mv/di. Horizonal sensiiiy = ms/di. FIG. 4.53 Problems 3, 4, 1, and 13. 3. Repea Problem 1 for he pulse waeform of Fig. 4.53. 4. Deermine he rise and fall imes for he waeform of Fig. 4.53. 5. Skech a pulse waeform ha has a base-line olage of 5 mv, a pulse widh of ms, an ampliude of 15 mv, a 1% il, a period of 1 ms, and erical sides, and ha is posiie-going. 6. For he waeform of Fig. 4.54, esablished by sraighline approximaions of he original waeform: a. Deermine he rise ime. b. Find he fall ime. c. Find he pulse widh. d. Calculae he frequency. 7. For he waeform of Fig. 4.55: a. Deermine he period. b. Find he frequency. c. Find he maximum and minimum ampliudes. mv 6 1 6 3 (ms) FIG. 4.54 Problems 6 and 14. Verical sensiiiy =. V/di. Horizonal sensiiiy = 5 ms/di. FIG. 4.55 Problems 7 and 15.
PROBLEMS 1119 SECION 4.3 Pulse Repeiion Rae and Duy Cycle 8. Deermine he pulse repeiion frequency and duy cycle for he waeform of Fig. 4.51. 9. Deermine he pulse repeiion frequency and duy cycle for he waeform of Fig. 4.5. 1. Deermine he pulse repeiion frequency and duy cycle for he waeform of Fig. 4.53. SECION 4.4 Aerage Value 11. For he waeform of Fig. 4.56, deermine he a. period. b. pulse widh. c. pulse repeiion frequency. d. aerage alue. e. effecie alue. 1. Deermine he aerage alue of he periodic pulse waeform of Fig. 4.51. 13. o he bes accuracy possible, deermine he aerage alue of he waeform of Fig. 4.53. 14. Deermine he aerage alue of he waeform of Fig. 4.54. 15. Deermine he aerage alue of he periodic pulse rain of Fig. 4.55. SECION 4.5 ransien R-C Neworks 16. he capacior of Fig. 4.57 is iniially charged o 5 V, wih he polariy indicaed in he figure. he swich is hen closed a s. a. Wha is he mahemaical expression for he olage C? b. Skech C ersus. c. Wha is he mahemaical expression for he curren i C? d. Skech i C ersus. 17. For he inpu olage i appearing in Fig. 4.58, skech he waeform for o. Assume ha seady-sae condiions were esablished wih i 8 V. 6 (mv) E 1 3 9 11 17 19 R V FIG. 4.56 Problem 11. R 1 k C C FIG. 4.57 Problem 16. i C. mf 5 V (ms) 8 V i 4 V i k C 1 mf o FIG. 4.58 Problem 17. 18. he swich of Fig. 4.59 is in posiion 1 unil seady-sae condiions are esablished. hen he swich is moed (a s) o posiion. Skech he waeform for he olage C. 19. Skech he waeform for i C for Problem 18. 1 R 1 k i C 1 V C C 1 mf V FIG. 4.59 Problems 18 and 19.
11 PULSE WAVEFORMS AND HE R-C RESPONSE SECION 4.6 R-C Response o Square-Wae Inpus. Skech he olage C for he nework of Fig. 4.6 due o he square-wae inpu of he same figure wih a frequency of a. 5 Hz. b. 1 Hz. c. 5 Hz. R i V i 5 k C i c.4 mf c FIG. 4.6 Problems, 1, 3, 4, 7, and 8. 1. Skech he curren i C for each frequency of Problem.. Skech he response C of he nework of Fig. 4.6 o he square-wae inpu of Fig. 4.61. 3. If he capacior of Fig. 4.6 is iniially charged o V, skech he response C o he same inpu signal (of Fig. 4.6) a a frequency of 5 Hz. 4. Repea Problem 3 if he capacior is iniially charged o 1 V. i V V f = 5 Hz FIG. 4.61 Problem. SECION 4.7 Oscilloscope Aenuaor and Compensaing Probe 5. Gien he nework of Fig. 4.4 wih R p 9M and R s 1 M, find V scope in polar form if C p 3 pf, C s 18 pf, C c 9 pf, and i (1) sin p1,. ha is, deermine Z s and Z p, subsiue ino Eq. (4.8), and compare he resuls obained wih Eq. (4.1). Is i erified ha he phase angle of Z s and Z p is he same under he condiion R p C p R s C s? 6. Repea Problem 5 a q 1 5 rad/s. SECION 4.9 Compuer Analysis PSpice or Elecronics Workbench 7. Using schemaics, obain he waeforms for C and i C for he nework of Fig. 4.6 for a frequency of 1 khz. *8. Using schemaics, place he waeforms of i, C, and i C on he same prinou for he nework of Fig. 4.6 a a frequency of khz. *9. Using schemaics, obain he waeform appearing on he scope of Fig. 4.37 wih a -V pulse inpu a a frequency of 5 khz. *3. Place a capacior in parallel wih R p in Fig. 4.37 ha will esablish an in-phase relaionship beween scope and i. Using schemaics, obain he waeform appearing on he scope of Fig. 4.37 wih a -V pulse inpu a a frequency of 5 khz.
GLOSSARY 111 Programming Language (C, QBASIC, Pascal, ec.) 31. Gien a periodic pulse rain such as ha in Fig. 4.11, wrie a program o deermine he aerage alue, gien he base-line olage, peak alue, and duy cycle. 3. Gien he iniial and final alues and he nework parameers R and C, wrie a program o abulae he alues of C a each ime consan (of he firs fie) of he ransien phase. 33. For he case of / < 5, as defined by Fig. 4.9, wrie a program o deermine he alues of C a each halfperiod of he applied square wae. es he soluion by enering he condiions of Example 4.1. GLOSSARY Acual (rue, pracical) pulse A pulse waeform haing a leading edge and a railing edge ha are no erical, along wih oher disorion effecs such as il, ringing, or oershoo. Ampliude of a pulse waeform he peak-o-peak alue of a pulse waeform. Aenuaor probe A scope probe ha will reduce he srengh of he signal applied o he erical channel of a scope. Base-line olage he olage leel from which a pulse is iniiaed. Compensaed aenuaor probe A scope probe ha can reduce he applied signal and balance he effecs of he inpu capaciance of a scope on he signal o be displayed. Duy cycle Facor ha reeals how much of a period is encompassed by he pulse waeform. Fall ime ( f ) he ime required for he railing edge of a pulse waeform o drop from he 9% o he 1% leel. Ideal pulse A pulse waeform characerized as haing erical sides, sharp corners, and a fla peak response. Negaie-going pulse A pulse ha increases in he negaie direcion from he base-line olage. Periodic pulse rain A sequence of pulses ha repeas iself afer a specific period of ime. Posiie-going pulse A pulse ha increases in he posiie direcion from he base-line olage. Pulse repeiion frequency (pulse repeiion rae) he frequency of a periodic pulse rain. Pulse rain A series of pulses ha may hae arying heighs and widhs. Pulse widh ( p ) he pulse widh defined by he 5% olage leel. Rise ime ( r ) he ime required for he leading edge of a pulse waeform o rael from he 1% o he 9% leel. Square wae A periodic pulse waeform wih a 5% duy cycle. il (droop, sag) he drop in peak alue across he pulse widh of a pulse waeform.