ELECTRICAL CIRCUITS 7. NONLINEAR COMPARATOR OSCILLATORS


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1 87 ELETIAL IUITS 7. NONLEA OMPAATO OSILLATOS Inroducion A linear oscillaor is a basic feedback conrol sysem ha has been made deliberaely unsable a he frequency of oscillaion. The linear oscillaor sysem analysis was explored in deail in a separae handou. A nonlinear oscillaor is a 2 sae sysem where neiher sae is a sable sae. The developmen herein will inroduce comparaors and describe wih an example of a comparaor oscillaor how he condiions of nonlinear oscillaion are obained. inally, he 555 imer circui will be examined for he nonlinear oscillaor. The omparaor The comparaor is an elecronic device ha has an opamp fron end (signal inpus) and a logic gae oupu. An opamp wihou feedback acs like a comparaor. The oupu of he comparaor will be High or Low depending upon which of 2 signals a he invering or noninvering inpu was he bigger. The symbol for a comparaor is he same as an opamp (igure ). igure symbol for a comparaor Table illusraes he ruh able for he comparaor. V inver V noninver V ou Bigger Smaller V low Smaller Bigger V high Equal Equal Indeerminae Table comparaor ruh able If an opamp is used as a comparaor V low is V PS and V high is + V PS. Where V PS is he power supply volage. Devices ha are comparaors can have V low and V high be almos anyhing. On some devices V low and V high are TTL signals or hey could be MOS signals. or our purpose we choose o use he LM convenion wih he oupu an open collecor NPN ransisor. Where V high is he ransisor is O (collecor high) and V low is he ransisor is ON (collecor low). The open collecor oupu requires for
2 88 many applicaions ha here mus be a pullup resisor ied o he oupu. igure 2 illusraes his. igure 2. LM comparaor wih pullup resisor The k pullup resisor is a ypical value and wha he designer acually chooses depends upon he applicaion. I should be noed for any applicaions of comparaors ha he oupu can no be assumed a perfec volage source. As seen in igure 2, he source impedance depends upon he oupu sae. When he oupu is high i is k and when he oupu is low i is a sauraed ransisor, exremely low. omparaor Hyseresis A very imporan applicaion of comparaors is o deermine which of 2 analog signals is he larger. This applicaion is basically a bi analog o digial converer. Alarm sysems ha are par of many conrol and insrumenaion sysems all use hese comparaor circuis. A design issue for hese applicaions is indecision oscillaions. Observe in able ha he oupu sae is indeerminae when he inpus are equal. In pracice if he inpus are very close hen noise a he inpus, i s always here, will cause he oupu o oscillae a a very high frequency and do an excellen job of disruping any insrumenaion and conrol sysem. A comparaor design can be made resisan o his issue. onsider a ypical alarm scenario wih he circui of igure. igure he comparaor hyseresis circui or his analysis we assume an ideal comparaor wih he oupu considered an ideal volage source and he inpus o be a perfec opamp fron end. To include source impedance in he design, jus make he pullup resisor (LM case) much less han any of he oher resisors in he circui. To include an imperfec opamp fron end refer o he
3 89 handou Operaional Amplifiers Par II D Errors. The ypical alarm applicaion is for he oupu o go o VHIGH when V exceeds he alarm se poin V E. Addiionally, he circui will be designed o be resisan o comparaor oscillaion. The comparaor oupu will be in one of 2 saes. We can obain an expression for he volage VX a he noninvering inpu for each of hese 2 cases. V V V XHI HIGH () V V V XLO (2) Examining he circui of igure 2 if VX is equal o VE hen he comparaor will change saes. Equaions and 2 can be rewrien o reflec his. VE V V HIGH () VE V V (4) Where: V is he value ha V mus fall o when he oupu is in he high sae and will swich o a low sae. V is he value ha V mus rise o when he oupu is in a low sae and will swich o a high sae. Upon examining Equaions and 4 i is observed ha here is a deadband for he inpu signal V and if his deadband is made in excess of sysem noise hen immuniy from comparaor oscillaion is realized. The dead band is: ( V V ). Equaions and 4 can be solved for he deadband. V V VHIGH V (5) Equaion 5 gives he comparaor hyseresis and his along wih noise filering will eliminae he endency of comparaor oscillaion. Design of a omparaor Oscillaor The comparaor oscillaor is a comparaor circui ha is designed o generae a pulse rain as i s oupu a a specified frequency and pulse duy cycle. Addiionally, if he duy cycle
4 90 is made 50:50, hen wih he addiion of a high Q bandpass filer, he oupu can be a prey good sine wave. onsider he circui of igure 4 he comparaor oscillaor. igure 4 he comparaor pulse oscillaor Equaions and 2 can be applied o igure 4 and hus Equaions 6 and 7 give he volage a he noninvering inpu. V V V XHI E HIGH (6) V V V XLO E (7) An undersanding of he circui operaion will make he analysis easier. onsider ha he circui oupu jus swiched low, V. This means ha he capacior volage on he invering inpu had charged up o where i jus equaled V XHI. Because he circui oupu has swiched o V, he volage a he noninvering inpu has becomev XLO and he capacior volage is now discharging. The capacior circui hinks ha i is going o discharge all he way o V, bu i won make i. When he capacior volage a he invering inpu discharges o V XLO, he oupu will swich back o VHIGH and his will cause he volage a he noninvering inpu o become V XHI. This causes he capacior o sar charging up and i hinks ha i is going o charge o V HIGH, bu again i won make i. When i his VXHI he oupu will swich low and he cycle will sar over again. igure 5 illusraes a iming diagram of he various waveforms. Upon examining igure 5 i is obvious ha he sawooh characerisics of rise ime rise and fall ime fall carry informaion on oscillaion frequency fosc and pulse duy cycle P D. f osc rise fall (8)
5 9 P D rise rise fall (9) igure 5 comparaor oscillaor iming diagram We use Equaion 5 from he handou lassical Soluion of s and 2 nd Order DEQ s Obained via ircui Analysis o obain expressions for charge up capacior volage, VHAGE and he discharge capacior volage, V DISHAGE. AP V V V V e (0) HAGE HIGH XLO HIGH AP V V V V e () DISHAGE XHI The rise ime rise, can be obained from Equaion 0 by seing VHAGE equal o VXHI and solving for he rise ime rise. In Equaion, se VDISHAGE equal o VXLO hen solve for he fall ime fall. VHIGH VXHI rise APln VHIGH VXLO (2)
6 92 VXLO V fall APln VXHI V () or a 50:50 duy cycle in igure 4 pick resisors and VHIGH V average beween V HIGH and V : VE equal and pick VE. When hese consrains are o be he 2 plugged ino Equaions 6, 7, 2 and hen, he rise ime and he fall ime are he same as given by Equaion 4. rise / fall AP ln (4) The frequency of oscillaion fosc becomes: osc 2 ln AP f (5) Designing a omparaor Oscillaor wih a LM or LM comparaor designs, he general guideline is ha he pullup resisor mus be much smaller han any of he oher resisors. One mus keep in mind ha a limiing consrain on small is he maximum curren sinking ha can be allowed in he open collecor oupu. onsider a 50:50 duy cycle oscillaor. igure 6 illusraes he general schemaic for a LM o his applicaion. igure 6 LM applicaion for a 50:50 D comparaor oscillaor PU = k hen le = AP = 20k. Then, wih PU << and he oscillaion If we se frequency specified, we can use Equaion 5 o solve for.
7 9 2 f ln() (6) OS AP This circui could be expeced o oscillae up o 00 s of khz. To go higher, all he resisors would have o be reduced. A V =5V, could fall o 250 ohms. This would drive he collecor curren for he oupu ransisor o 20ma. or V a higher volages, he collecor curren and LM power dissipaion would need o be considered. The 555 Timer The 555 Timer is an inegraed circui ha is he embodimen of he comparaor oscillaor and much more. igure 7 illusraes a sysem level schemaic of he 555. There are a myriad of oher applicaions for his device. Applicaions from One Sho pulse generaors o phase locked loops and more have been designed. However, his handou will only consider he 555 imer as a nonlinear oscillaor. PU igure imer as a pulse oscillaor The circui of igure 7 operaes as follows: When power is firs urned on, comparaor U 2 forces he flipflop o be rese which in urn holds he ransisor Q urned off (Noe ha he se and rese signals o he flipflop are a low ). The capacior now charges up hrough a and b. When he capacior volage V AP, reaches / V, he rese signal from U 2 o he flipflop ceases. When he capacior finally charges up o 2/ V, he comparaor U rips low and ses he flipflop. This causes he ransisor Q o urn on and discharge he capacior hrough b. As he capacior volage falls away from 2/
8 94 V, he se signal from U ceases. When he discharging capacior his / V comparaor U 2 goes low and reses he flipflop and he cycle sars over again. learly, his operaion is very similar o he LM comparaor oscillaor. Noe ha he power supply volage V, ranges from 5 VD down o 5 VD. The LM can also operae wih a single supply over he same range. The charge and discharge for he capacior volage V AP are given by Equaions 7 and 8., 2 V V V V e (7) AP HAGE a b VAP DISHAGE 0 V 0, e b (8) Equaions 7 and 8 can be solved for he capacior volage rise ime and fall ime. 2 V, V V V e a b ln, a b 2 V 0 2 0, V e b ln, b 2 (9) (20) Equaion 2 gives he frequency of oscillaion f OS. f (2) OS Observe ha in igure 7 and Equaions 9 and 20 he duy cycle can approach 50:50 by making a bbecause he charge and discharge ime consans are differen. or many oher uses for he 555 imer he reader should refer o he Philips applicaion noe AN70 which can easily be down loaded from he WEB by going o GOOGLE and searching on 555 imer.
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