1 Intrductn ME 3200 Mechatrncs Labratry Lab Exercse 8: Operatnal Amplers In ths experment yu wll explre sme the basc prpertes peratnal amplers, better knwn as p-amps. These electrnc devces are very useul n analg crcutry. As ther name mples, they can perrm mathematcal peratns n vltage sgnals: ncludng algebrac and calculus peratns. The Ideal Op-Amp An p-amp s a devce wth tw nputs and ne utput. The tw nputs are knwn as the nn-nvertng nput, v+, and the nvertng nput, v-. The utput the p-amp s dependent upn the ptental derence between the nvertng and nn-nvertng nputs. Op-amps requre an external pwer surce dented cc. The p-amps used n the lab use a ±cc ±12 prvded by the bench pwer supply at each statn. A smpled schematc an deal p-amp s shwn n Fgure 1 belw. It s cmmn t mt the pwer cnnectn n mst p-amp dagrams; therere, Fgure 1 des nt nclude these cnnectns. (a) (b) Fgure 1: (a) Smpled p-amp schematc. (b) Op-amp crcut dagram. The basc eatures deal p-amps and ther representatn n crcut dagrams are shwn n Fgure 1 (a) and (b) respectvely. The tw nputs the p-amp are cnnected thrugh a resstr wth nnte resstance that restrcts current rm lwng nt the p-amp at ether termnal. The utput lead s cnnected t a dependent vltage surce thrugh a resstr zer value, thus causng the utput vltage t be equal t the vltage prvded by the dependent vltage surce. The vltage the dependent surce s prprtnal t the vltage derence between v+ and v- and can be expressed as llws: D ( ) + = A v v (1) where AD s the pen-lp gan the ampler, whch, deally, s nnte.
2 eal Op-amps In realty, hwever, nntely large resstrs, nnte pen-lp gans, and zer-valued resstrs d nt exst. Frtunately, the characterstcs typcal mst p-amps generally allw the use the equatns and assumptns that dene the deal p-amp. Ths act s very useul when desgnng and analyzng p-amp crcuts. The typcal nput resstance an p-amp s n the rder 100 MW whch stll allws very lttle current nt the nput leads. The typcal utput resstance an p-amp s n the rder 10 W. An utput resstance ths lw means that a nn-deal p-amp can prvde a substantal, albet nte, current and that Equatn 1 adequately apprxmates ts vltage utput. Equatn 1, hwever, s nly a gd apprxmatn the nmnal vltage derence between the nputs s small. I ths vltage derence becmes t large the p-amp saturates at a vltage knwn as sat. Saturatn ccurs because the vltage derence between the nputs dctates that must be larger than the supply vltage cc accrdng t Equatn 1. Snce the p-amp cannt prvde mre vltage than t s gven the utput reaches an upper r lwer lmt (±sat). The saturatn vltage an p-amp s always a lttle lwer than ts supply vltage, cc. Fr example, the p-amps used n ths lab cannt supply mre r less than ±12 ; the saturatn vltage r these devces s mre lke ±10. The pen lp gan, AD, the p-amp s clsely related t the saturatn characterstcs a gven p-amp; t denes hw large the derence between v+ and v- can be bere the p-amp saturates. The typcal pen lp, lwrequency gan an p-amp s usually abut 105. Hwever, the pen-lp gan mst p-amps s requency dependent and wll becme smaller and smaller dependng n the requency the nput sgnal(s). A large penlp gan drastcally restrcts the sze the ptental derence between v+ and v-, but t als means that the pamp s very senstve t small changes between v+ and v-. S hw des ne avd saturatn and take advantage the p-amps senstvty? Saturatn s avded by dvertng a prtn the p-amp s utput t ts nvertng termnal by physcally cnnectng the utput thrugh a passve electrnc devce t the nvertng nput. Ths technque s called negatve eedback and s very cmmn n almst all useul p-amp crcuts. Negatve eedback ensures that the derence between v+ and v- wll always be very nearly zer, but that the p-amp s stll senstve t small changes between v+ and v-. The precedng characterstcs real p-amps allw the use the llwng assumptns n p-amp crcut analyss: v = v (2) + = = 0 (3) + where + and - are the currents lwng nt the nn-nvertng and nvertng nputs, respectvely. It s mprtant t nte that Equatn 2 s vald and nly negatve eedback s used n the p-amp crcut and s smetmes reerred t as vrtual equalty. Wth these assumptns, the crcut can then be analyzed usng Krch s current r vltage laws (KCL and KL respectvely). Cnsder the example belw that llustrates hw these assumptns are valdated n the case the smple vltage llwer crcut. Example 1 Determne the utput vltage the p-amp n Fgure 2 as a unctn the nput vltage, gven that the penlp gan AD = 105. Fgure 2: A vltage llwer crcut.
3 The rst step s t denty the vltage values at the nput leads. The nvertng nput s ted t the utput, and the nn-nvertng nput s ted t the nput vltage. v = (4) + v = (5) The next step s t substtute these values and the gven value r the pen-lp gan nt Equatn 1: Ater sme algebra, Equatn 6 becmes = 10, 000( ) (6) 10, 000 = 10, 001 (7) The p-amp s knwn as a vltage llwer, r buer, because the utput vltage s essentally equal t the nput vltage. The Amercan Hertage Dctnary [ denes a buer as Smethng that lessens r absrbs the shck an mpact. Based n the characterstcs a real p-amp, we see that ths p-amp crcut reduces the mpact the current prvded by the vltage nput s that t cannt prpagate t the utput. ecall that current cannt lw between the nput leads. Snce the nput vltage s appled t the nn-nvertng nput, any current cmng rm that nput s blcked rm the utput. The utput s ted t the nvertng nput, and snce there s n current between the nput leads, there s n current n the wre cnnectng the utput t the nvertng nput (ths cnnectn s knwn as the eedback path). Any current that s requred r the devce cnnected t the utput the buer wll be prvded rm the p-amp tsel. Ths eectvely separates the nput devce rm the utput devce, but stll prvdes an utput vltage that s essentally dentcal t the nput vltage. Useul Op-amp Crcuts The Invertng Ampler Fgure 3: An nvertng ampler crcut. Cnsder the nvertng ampler crcut n Fgure 3. The llwng analyss shws hw t determne the utput vltage as a unctn. Wth the nn-nvertng nput cnnected t grund, v + = 0, Equatn 2 states that v - = 0 as well snce the crcut has a negatve eedback lp (thrugh ). The vltage at the nvertng nput s knwn as vrtual grund. The drectns the currents assume that they run tward grund. Applyng Ohm s law results n the llwng equatns: = (8) = (9)
4 Keepng n mnd that the current rm the nvertng termnal - = 0 rm Equatn 3, KCL can be appled at the nde cntanng the nvertng nput. Substtutng Equatns 8 and 9 nt Equatn 10 yelds: Fnally, slvng Equatn 11 r yelds + = 0 (10) + = 0 (11) = (12) The mathematcal peratn perrmed by ths p-amp crcut s multplcatn (.e. amplcatn the nput). Ntce that the sgn the utput s ppste the sgn the nput. Ths sgn nversn s why the crcut s called an nvertng ampler. The Actve Lw-Pass Flter A smple mdcatn can be made t the nvertng ampler s that t can lter ut hgh requency nse a cmmn element und n mst electrnc sgnals. Hgh requency nse s cmmnly caused by external vltage r magnetc surces r rm natural mpurtes n crcut devces r cmpnents. Hgh requency nse s generally characterzed by spradc, hgh requency varatns n the vltage sgnal. The lw pass lter eectvely excludes nput requences hgher than a speced requency called the cut requency. T create an actve lw pass lter, smply cnnect a capactr n parallel wth the eedback resstr the nvertng ampler as shwn n Fgure 5. Fgure 4: An actve lw pass lter Ths crcut perrms tw peratns. Frst, t amples the nput vltage accrdng t Equatn 12 (whch s vald r lw requences), and secnd, t suppresses requences hgher than ts cut requency, where the cut requency, ω 0 n rad/sec, s dened as: 1 ω = (13) 0 C T be exact, the utput ampltude as a unctn nput requency r ths crcut s descrbed by: = ω ( ω ) 0 2 (14)
5 where ω s the requency the nput sgnal. Fgure 5 shws the requency respnse an actve lw pass lter whse resstrs have been chsen t prvde a unty gan, and where the x-axs s the nrmalzed requency, and the y-axs s the abslute value the system gan. Fgure 5: Lw pass lter requency respnse As ndcated n Fgure 5, at ω = 0 (DC vltage nput), the utput vltage s = (15) As nput requency ncreases, ampltude decreases. At the cut requency, ω = ω 0, the magntude the utput s dened by the llwng: = = (16) 2 In rder t decrease the mpact the lw pass lter n the desred utput sgnal, the cut- requency s set as hgh as pssble. The trade- usng a hgh cut requency s that mre nse s permtted t pass thrugh the lter. As a rule thumb, settng the cut requency t at least twce the maxmum expected nput requency and less than ne tenth the expected nse requency s best. A dmnant surce nse n the lab s AC lne nse at 60Hz. Mre sphstcated hgher rder lters can be used t create a mre deal lter. The LM324 Quad Op-Amp Op-amps are readly avalable as nexpensve ntegrated crcuts (ICs). The p-amp IC used n the Mechatrncs lab s the LM324 quad p-amp. Ths s a 14-pn dual n-lne package (DIP) chp that cnssts ur p-amps. When plugged nt an electrcal breadbard and prvded the apprprate ± cc, the chp prvdes ur ndependent p-amps that can be used t rm a wde varety useul crcuts. A schematc the LM324 s ncluded n Fgure 6. Ths type schematc s knwn as a pn ut, and t ndcates hw the pns the DIP crrespnd t the nputs and utputs the ur p-amps. Whenever these types IC devces are used, t s mprtant t reer t the data sheet prvded by the chp manuacturer that descrbes the devce s characterstcs and lmtatns. The data sheet r the LM324 s avalable n the class web page n the lab handuts page.
6 Fgure 6: Schematc the LM324 quad p-amp. Asde rm the applcatns dscussed n ths handut, the p-amp has many ther uses n analg electrncs. Yu are encuraged t explre ther applcatns ths useul devce. Pre-lab Exercses 1. Determne the utput vltage the p-amp n Fgure 1 as a unctn the nput vltage, the resstance, and the capactance C. The methds develped n the llwng backgrund sectn wll prvde yu wth the tls needed t perrm ths task. (ecall that the current thrugh a capactr s prprtnal t the tme dervatve the vltage acrss t.) Fgure 7: Derentatr crcut 2. Calculate the deal utput vltage,, ths crcut the nput vltage s equal t the llwng: = Asn ω t (19) where A s the ampltude the ncmng sgnal n vlts, ω s the requency the ncmng sgnal n rad/sec, and t s tme n secnds.
7 Labratry Exercse equred Materals/Equpment An Electrnc Breadbard An LM324 Quad Op-amp Chp A Sgnal Generatr An Oscllscpe esstrs Capactrs The Benchtp Pwer Supply The 2_channel_scpe.v Labew I used n Lab 2 Sld Cre Wre A Wre Cutter/Strpper Banana Cables BNC t Allgatr Clp Cables Dgtal Multmeters Allgatr Clps BNC t Banana Adapters BNC Cables BNC Tees The crcuts that yu wll buld n ths lab prcedure are based n the nes descrbed n the ntrductry sectns ths lab handut. Nt nly are yu expected t buld the crcuts descrbed belw, but yu are als expected t debug them they d nt wrk apprprately the rst tme. Debuggng crcuts s a very useul skll and requres patence and attentn t detal. The mst cmmn errr n p-amp crcuts s accdentally saturatng the p-amp wth t large a vltage derence at ts nputs. One methd debuggng ths prblem s t use the dgtal multmeters prvded n the lab t check the vltages the nput and utput pns t trace path the crcut rm begnnng t end. Ask yur TA r help the crcut als t unctn ater yur best erts t debug the crcut. Nte: D nt turn n the bench pwer supply untl yur crcut s cmplete. In addtn, whenever any changes are made n yur crcuts turn r unplug the pwer supply rm the crcut. Ths prtects the crcutry and elmnates ptental headaches rm burned ut p-amps. 1. Lcate an electrnc breadbard and an LM324 quad p-amp IC and careully press the chp s pns nt the hles the breadbard s that t straddles the center dvder wthut bendng ts pns. 2. Prvde pwer t the LM324 by cnnectng the bench pwer supply 12 vlts t the cc pn and +12 vlts t the + cc pn. 3. Turn n the sgnal generatr and scllscpe, cnnect the unctn generatr utput t an scllscpe nput, and adjust the utput the sgnal generatr untl the scllscpe shws that yu have a 2 Hz, ±2-vlt sne wave. Dscnnect the scllscpe nput, turn t, and set t asde. 4. Desgn an nvertng ampler that dubles the ampltude the nput sgnal. ecrd the value the resstrs that yu wll use n the space prvded. = = 5. Use a BNC tee t splt the utput the sgnal generatr, and cnnect ne end the tee t the A_CH0 nput the DAQ breakut bx yur lab statn.
8 6. Buld the nvertng ampler yu desgned n the step ur use the grund lead the benchtp pwer supply as yur cmmn grund. Cnnect the utput the sgnal generatr t the nput yur ampler crcut and cnnect the utput the ampler t the A_CH1 the DAQ breakut bx yur lab statn. Nte: Yu wll need t use a BNC t Banana adapter and an allgatr clp t cnnect the utput the sgnal generatr t yur crcut. The black lead the BNC t Banana adapter shuld be cnnected t the grund the benchtp pwer supply. The red lead the BNC t Banana adapter receves the allgatr clp and s cnnected t yur crcut nput thrugh a sld cre wre jumper. When cnnectng the utput yur crcut t the DAQ breakut bx, use a BNC t Allgatr cable and jumper wres t grund (black) and the crcut utput (red). 7. Open the 2_channel_scpe.v rm the Labew lder n yur cmputer s desktp, select the run buttn rm the tlbar, under Oscllscpe Channel Selectn, type 0 and 1 n the bxes belw the ON/OFF swtches, and make sure that they are bth turned On. 8. Turn n the pwer t yur benchtp pwer supply, select apprprate values r the sample rate and the number samples n the Labew I, and press the Acqure Data buttn. 9. Examne the waverm n dsplayed n the Labew I, save yur data t a dsk by pressng the OK buttn under Save Data, and determne the amplcatn actr yur nvertng ampler usng the measurements the Labew I. ecrd ths amplcatn actr belw. Amplcatn = 10. Turn yur benchtp pwer supply. Fgure 8: Buer-lter-ampler-derentatr crcut. 11. Desgn a buer-lter-ampler-derentatr crcut (Fgure 8) that wll derentate a 2 Hz, ±2-vlt sne wave and amply ts ampltude t a 2Hz, ±4.5-vlt (9-vlt peak-t-peak) sne wave. eer t the pre-lab and the actve lw-pass lter sectn ths lab r the necessary methds. Be sure t use an apprprate cut requency. ecrd the requred values r yur electrnc cmpnents n the spaces prvded. = Ω = Ω C = F C = F = Ω 12. Buld the crcut yu just desgned. eer t Fgures 6 and 8 t help wth wrng the crcut payng specal attentn t the pn ut dagram Fgure 6. Use the grund rm the bench pwer supply r the apprprate as the cmmn grund r ths crcut. As bere, all equpment and crcuts shuld be cnnected t that ne grund pnt.
9 13. Send the utput yur crcut t A_CH1 n the DAQ termnal blck and the utput the sgnal generatr t AI_CH0 and t the nput yur crcut as dne when realzng yur nvertng ampler. 14. Turn n the bench pwer supply, chse the apprprate number samples and samplng rate n the Labew I, and press the Acqure Data buttn t cllect the waverm data. Save the data t dsk and examne the waverms. 15. Determne the actual amplcatn acheved by yur buer-lter-ampler-derentatr crcut usng the same technque as bere. Als, determne the phase sht the utput yur crcut usng the data traces n the waverm graph the Labew I. ecrd the amplcatn actr and phase sht belw. Amplcatn = Phase Sht = rad 16. Increase the requency the nput sgnal t 5Hz, chse the apprprate number samples and samplng rate, and push the Acqure Data buttn n the screen. Save the data t dsk and plt the data. Hw has the appearance the waves changed? What s the ampltude and phase derence the utput ths tme? A = Phase Sht = rad Why have the peaks been cut the snusds? (Hnt: reer t the Pre-Lab and the Data Cllectn Labratry.) 17. Increase the requency the nput sgnal t 100Hz, chse the apprprate number samples and samplng rate, and push the Acqure Data buttn n the screen. Save the data t dsk and plt the data. Hw has the appearance the utput changed? Why des the appearance the utput wave change as the nput requency ncreases?
10 Questns 1. What s the mst ecent way t mplement a cmplex p-amp crcut that requres mre than ne p-amp? 2. Descrbe at least tw gd ways debuggng an peratnal ampler crcut. 3. Draw the crcut dagram r an ntegratr-ampler crcut that ntegrates the nput vltage. Slve r the utput vltage as a unctn tme the nput vltage s equal t Equatn 19 abve. 4. Descrbe at least tw practcal applcatns r p-amps r yur rbt prject.