ELG 35 ELECTONICS I SECOND CHAPTE: OPEATIONAL AMPLIFIES Sesson Wnter 003 Dr. M. YAGOUB
Second Chapter: Operatonal amplfers II - _ After reewng the basc aspects of amplfers, we wll ntroduce a crcut representng the behaor of an deal amplfer, namely the operatonal amplfer that s also commonly called the Op Amp. A INTODUCTION I - Hstory The frst operatonal amplfer was n the form of an ntegrated crcut called the «µa 709». Ths unt was made up a relately large number of transstors and resstors all on the same chp. It s a ery popular crcut because of ts ersatlty (we can do almost anythng wth op amps) as we can see later. II The op amp termnals From a sgnal pont of ew, the op amp has three termnals: two nput termnals and one output termnal (fgure II-). + 3 Fgure II- Moreoer, as amplfers requre dc power to operate, we add two addtonal termnals named «4» and «5» for the poste (V + ) and negate (V - ) dc oltage respectely (fgure II-): V + Fgure II- + 4 5 3 + 4 5 3 V -
Second Chapter: Operatonal amplfers II - 3 _ The two dc power supples as batteres wth a common ground, whch s the reference groundng pont n op-amp crcuts. B IDEAL OPEATIONAL AMPLIFIES The op amp s desgned to sense the dfference between the oltage sgnals and appled at ts two nput termnals {that s the quantty - } multpled by the amplfcaton factor A. The output oltage at the thrd termnal s then o ( ) 3 A () Note: - Quantty means that the oltage s appled between termnal and ground. - If termnal s grounded, we obtan a usual amplfer of nput sgnal and gan A. The deal op amp s assumed to hae any nput current: 0. Ths assumpton mples that the nput mpedance of an deal operatonal amplfer s nfnte. As the output oltage s gen by (), t s ndependent of the output current delered to a load. Then, the output mpedance of an deal operatonal amplfer s supposed to be zero. These conclusons lead to the followng equalent crcut (fgure II-3): Fgure II-3
Second Chapter: Operatonal amplfers II - 4 _ Note: - Output oltage o s n phase wth (the two oltages hae the same sgn) and out of phase wth (opposte sgns). For ths reason, we call: Termnal : Inertng nput termnal (dstngushed by a - sgn) Termnal : Nonnertng nput termnal (dstngushed by a + sgn) - The op amp responds only to the dfference sgnal. Ths property s called the common-mode rejecton. The deal op amp has nfnte common-mode rejecton. 3 - The op amp s then a dfferental-nput, sngle ended-output amplfer 4 - Furthermore, gan A s called the dfferental gan or the open-loop gan. 5 - Gan A remans constant down zero frequency up to nfnte frequency. That means an nfnte bandwdth for the deal op amp. C INVETING CONFIGUATION I Closed loop gan By consderng the followng amplfer confguraton (fgure II-4), Fgure II-4
Second Chapter: Operatonal amplfers II - 5 _ we note that the resstance s connected from the output termnal back to the nertng (or negate) nput termnal. We speak of as applyng negate feedback. In addton, closes the loop around the operatonal amplfer. We hae then a closed-loop gan G: o G () Fgure II-5-a shows the equalent crcut of the nertng confguraton. Fgure II-5
Second Chapter: Operatonal amplfers II - 6 _ If we assume that the output oltage s fnte, then the oltage between the nput termnals should be neglgbly small (because the gan A approaches nfnty) o 0 A (3) We speak then of a rtual short crcut between the two nput termnals or a rtual ground. Here rtual means that there s no physcal shortng wre between and (termnal s rtually not physcally grounded). The current through s then equal to the followng relaton (4) Note: Ths current cannot go to the op amp (nfnte nput mpedance), so t wll flow through to the low mpedance termnal 3. Applyng Ohm s law ges the output oltage o 0 (5) Thus the closed loop gan s (fgure II-5-b) : o G (6) Because of the mnus sgn, ths gan s referred to the nertng confguraton. Note: The gan depends only on external passe components,.e., resstances and. So, we can make the closed-loop gan as accurate as we want. We can start out wth a ery large gan A, and then applyng negate feedback to obtan the predctable gan /.
Second Chapter: Operatonal amplfers II - 7 _ II gorous determnaton of the closed-loop gan We obtaned G usng the assumpton that the open-loop gan A s fnte. Knowng that o (7) A we can hae a more rgorous relaton for equaton (3) o (8) A The current through can now be found from ( / A) o + o / A (9) and o o A / A o + o A (0) The closed-loop gan s then equal to o G / + + ( / )/ A () It s obous that f A s nfnte, G approaches the deal alue expressed n relaton (6). In other words, relaton () can be replaced by relaton (6) f + << A ()
Second Chapter: Operatonal amplfers II - 8 _ III Input and output resstances Assumng an deal op amp wth nfnte open-loop gan, the nput resstance of the closed-loopnertng amplfer s (fgure II-5-a) / (3) Thus to make hgh, we should select a hgh alue for. Howeer, f the requred gan / s also hgh, then could become mpractcably large. Snce the output s { A( - ) } (Fgure II-5-a), the output resstance s zero. Puttng all of the aboe together, we obtan the crcut shown n Fgure II- 6 as the equalent crcut model of the nertng amplfer confguraton. Fgure II-6 IV Alternate crcut to ncrease the nput resstance A soluton to aod a small alue of the nput resstance s to consder the followng crcut (Fgure II-7) Fgure II-7
Second Chapter: Operatonal amplfers II - 9 _ Assumng 0 A o (4) and a fnte output oltage, we can wrte 0 (5) Thus the oltage x at node x s equal to x 0 (6) Ths n turn enables us to fnd the current 3 : x + + + + 3 3 3 3 4 0 (7) and then the output oltage x o + 3 4 4 4 (8) Thus the oltage gan s gen by + 3 4 o (9) whch can be wrtten n the followng form + + 3 4 4 o (0)
Second Chapter: Operatonal amplfers II - 0 _ So f an nput resstance of MΩ s desred, we select MΩ. Then, wth the lmtaton of usng resstors no greater than MΩ, a alue of MΩ ges a rato of for the frst term n the gan expresson. To obtan a gan of 00, 4 MΩ and 3 0. kω could be approprate alues. Ths alue s to compare wth the one obtaned usng the confguraton shown n Fgure II-5-b. In ths case, wth the same alue of G (G -00) and ( MΩ), the desgner should select a feedback resstance 00MΩ whch s not practcal. D OTHE APPLICATIONS OF THE INVETING CONFIGUATION I Confguraton wth general mpedances Let us consder general mpedances Z (s) and Z (s) nstead of resstances and (s jω) as shown n fgure II-8, we hae the closed-loop transfer functon Vo V () t () t Z Z () s () s () Fgure II-8 II Inertng ntegrator By placng a capactor n the feedback path (n place of Z ) and a resstance at the nput (n place of Z ), the crcut realzes the mathematcal operaton of ntegraton (Fgure II-9).
Second Chapter: Operatonal amplfers II - _ Fgure II-9 Let the nput be (t), the rtual ground causes the nput current to be equal to () t () t () Ths current flows through the capactor C. Thus c () t V + () t c t C 0 dt (3) where V C s the ntal oltage on C at t 0. As the output oltage s { c (t)} we hae o () t V () t c t C 0 dt (4) Ths relaton shows that the output s proportonal to the tme-ntegral of the nput, wth V c beng the ntal condton of ntegraton and C the «ntegrator tme constant». Note: Ths crcut s also known as the Mller ntegrator. The operaton of the ntegrator crcut can be descrbed n the frequency doman Vo V () t () t () s Z / sc Z() s sc jω C (5)
Second Chapter: Operatonal amplfers II - _ Magntude and phase of ths expresson are then Vo V ω C Vo V db 0log0 φ + 90 o (6) ω C The bode plot for the ntegrator magntude resposne canbe obtaned by notng that as ω double (ncrease by an octae) the magntude s haled (decreased by 6dB) (Fgure II-0). Fgure II-0 Thus the Bode plot s a straght lne of slope 6dB/octae. Ths lne ntercepts the 0-dB lne at Vo V db Vo 0 db V ω nt (7) C whch s the ntegrator frequency. III Alternate crcut to the ntegrator We can obsere that n dc (zero frequency) the magntude s nfnte. So the op amp s operatng at dc wth an open loop (capactor mpedance s nfnte at dc). Ths s a source of problem because anyt tny dc component nthe nput source wll theortcally produce an nfnte output. Of course, the amplgfer wll saturate at a oltage close to the op amp poste or negate powwer supply.
Second Chapter: Operatonal amplfers II - 3 _ As t s mpossble to predct an nput sgnal wthout any dc part (pure sne waeform) an alternate crcut s requred for the ntegrator. In order to lmt the dc gan, a parallel resstance F (Fgure II-) s connected n parallel wth the capactor. Fgure II- The dc gan wll be then Vo V DC F (8) Unfrtunatly, ntroducng the resstance wll make the ntegrator not deal. The transfer functon s now equalent to that of a low pass flter wth ω3db F C (9) esstance F should be selected as large as possble. IV Dfferentator crcut Interchangng the locaton of the capactor and the resstor of the ntegrator crcut results n the crcut n Fgure II- whch performs the mathematcal functon of dfferentaton. Note: Here the term Dfferentator means the deraton not the dfference.
Second Chapter: Operatonal amplfers II - 4 _ Consderng Fgure II-, we hae o () t d( t ) C dt (30) Fgure II- The frequency doman transfer functon can be found as Vo V () t () t sc jω C (3) Magntude and phase are Vo V Vo ω C 0log0 ( ω C) φ - 90 o (3) V db The Bode plot of the magntude response can show that the magntude doubles for an octae ncrease n ω (Fgure II-3). Fgure II-3
Second Chapter: Operatonal amplfers II - 5 _ The C product s the «dfferentator tme constant». Note I: The dfferentator crcut acts as a hgh pass flter. Note II: As for the deal ntegrator, an deal dfferentator s unstable and mpractcal. Any araton of the nput oltage mples a lne wth a ery hgh slope. An addtonal resstance s requred to reduce the nose magnfer characterstc of an deal dfferentator. V Weghted summer Another applcaton of the nertng confguraton s the summer (Fgure II-4). Fgure II-4 In ths crcut, we hae a feed back resstance f and a number n of nput sgnals each appled to a correspondng resstor, n. The correspondng currents are then () t () t () () n t, K, n t (33) n The nput current s the sum of all these currents () t () t + K+ () t n (34) Thus, the output oltage s o () t () t f () t f 0 (35)
Second Chapter: Operatonal amplfers II - 6 _ Or: o f f f () t () t + () t + L+ () t n n (36) That s, the output oltage s a weghted sum of the nput sgnals,, n. Ths s crcut s the weghted summer where the weghts are the resstances to n. E NON INVETING CONFIGUATION If the nput sgnal s appled drectly to the poste nput termnal of the op amp, we hae the nonnertng confguraton (Fgure II-5). Fgure II-5 I Input-output relatonshp The gan A of the non-nertng confguraton s o 0 A (37) The current flowng through the resstance s (38) whch s the same current for (Fgure II-6).
Second Chapter: Operatonal amplfers II - 7 _ Fgure II-6 Thus: o + (39) and G o + + o (40) We hae a oltage dder. The gan s poste: t s the gan of the non-nertng confguraton. The nput resstance s deally nfnte and the output resstance s zero. The equalent crcut s shown n Fgure II-7. Fgure II-7
Second Chapter: Operatonal amplfers II - 8 _ II gorous determnaton of the closed-loop gan elaton (40) has been obtaned wth the assumpton that the gan A s nfnte. If ths gan s fnte, we hae o G + + + A + + ( / ) + ( / ) A (4) The denomnator s dentcal to that for the case of the nertng confguraton (equaton ()). Ths IS no concdence; t s a result of the fact that both the nertng and the non-nertng confguratons hae the same feed back loop. The numerators are dfferent. The approxmaton between (40) and (4) can be expressed as A >> + (4) F APPLICATIONS OF THE NON-INVETING CONFIGUATION I Voltage follower The property of hgh nput mpedance s a ery desrable feature of the non-nertng confguraton. It enables usng ths crcut as a buffer amplfer to connect a source wth a hgh mpedance to a low mpedance load. Moreoer, by settng 0 and (43) we obtan a unty gan amplfer. Ths crcut s referred to as a oltage follower, snce the output follows the nput wth the propertes o n out 0 (44)
Second Chapter: Operatonal amplfers II - 9 _ Note: Snce the non-nertng confguraton has a gan greater than or equal to unty, dependng on the choce of /, some prefer to call t a follower wth gan. Its confguraton (Fgure II-8-a) and equalent electrcal crcut (Fgure II-8-b) are as follows: Fgure II-8 II Analog Voltmeter Fgure II-9 shows a crcut for an analog oltmeter of ery hgh nput resstance that uses an nexpense mong col meter. Fgure II-9 III Dfference amplfer In order to obtan the dfference between two sgnals (e.g., to compare a sgnal to a reference), we can use a dfference amplfer (Fgure II-0).
Second Chapter: Operatonal amplfers II - 0 _ Fgure II-0 To apply superposton, we frst reduce to zero and then fnd the correspondng output oltage o. Next, we reduce to zero and ealuate o (Fgure II-). Fgure II- Wth 0 (Fgure II--a), we hae o (45) esstances 3 and 4 do not affect the gan expresson snce no current flows through ether of them. Thus, wth 0 (Fgure II--b), we hae 4 + o 3 + 4 (46) Snce 3 and 4 play a oltage dder, we recognze the non-nertng confguraton.
Second Chapter: Operatonal amplfers II - _ The superposton prncple tells o o + o + 4 + + + / 3 + 4 + 3 / 4 (47) thus the crcut s a dfference amplfer because: f 0 we hae o 0 If we select: 4 3 (48) the gan s equal to o ( ) (49) Howeer, for practcal consderatons, the condton (48) ges an alternate crcut (Fgure II-) wth 3 and 4 (50) Fgure II- The nput dfferental resstance s then defned as n (5)
Second Chapter: Operatonal amplfers II - _ Usng the rtual short crcut, the aboe relaton can be changed to + 0 + n (5) Note that f the amplfer s requred to hae a large dfferental gan, then wll be relately small (equaton (49)) and the nput resstance wll be correspondngly small (equaton (5)). Dfference amplfers are used manly n the desgn of nstrumentaton systems. IV Instrumentaton amplfer Let us consder the case of a transducer that exhbts between each of the two wres and ground two close sgnals (for example: V and.00v). In order to compare effcently the small sgnal dfference, a usual crcut s not conenent (t s ery dffcult to detect effcently a mv oltage n a V oltage). The requred crcut, known as the nstrumentaton amplfer, must reject the large nterference sgnal, whch s common to the two wres (.e., V) and amplfy the small dfference (or dfferental) sgnal (Fgure II-3). Fgure II-3 For ths crcut, we hae d Vcm and d Vcm + (53) Ths stuaton denotes the common mode sgnal V cm and the dfferental sgnal V d : V cm (. 00/ )V and d 0. 00V mv (54)
Second Chapter: Operatonal amplfers II - 3 _ V Improed crcut for the nstrumentaton amplfer As the nstrumentaton amplfer s deduced from the dfferentator amplfer confguraton, t presents the same dsadantage namely a low nput resstance and a gan that cannot be ared easly. A much superor nstrumentaton amplfer crcut s shown n Fgure II-4. Fgure II-4
Second Chapter: Operatonal amplfers II - 4 _ Ths crcut conssts of two stages (Fgure II-4-a). The frst stage s formed by deal op amps A and A and ther assocated resstors, and the second stage s formed by deal op amp A 3 together wth ts four asscated resstros. Analyss of the crcut (Fgure II-4-b) shows that the current flowng through s dentcal to the one flowng through. Thus: ( ) ( ) ( ) o o + + ( ) o o + (55) Then, the output oltage of the second stage s ( ) ( ) 3 4 3 4 o o o + (56) Thus, the nstrumentaton amplfer has a dfferental oltage gan 3 4 A o d + (57) Moreoer, f the two nput oltages are dentcal ( d 0), the output oltages of the two frst op amps are equal and ge the followng nput common mode sgnal cm o o cm (58) Thus, f the second stage dfference amplfer s properly balanced t wll produce a zero poutput oltage n response to cm.