Section H7: Frequency Response of Op-Amp Circuits

Transcription

1 Sectin H7: requency Repne f Op-mp Circuit In the preiu ectin, we hae lked at the frequency repne f ingle and multi-tage amplifier circuit uing BJT and ET. We e een hw the high frequency cutff i cntrlled by internal deice capacitance, and the lw frequency cutff i determined by external capacitie cmpnent. We re ging t finih up thi ectin f ur tudie by lking at the frequency repne f circuit that utilize ur friendly IC, the peratinal amplifier. Befre we get tarted in earnet, recall the difference between pen lp and cled lp peratin: pen lp peratin mean that there i n feedback between the utput and the input. The gain f the p-amp deice under thi cnditin i defined a the pen lp gain that we e been calling. cled lp n the ther hand, inle me rt f feedback mechanim between the utput and the input. S far, we e nly lked at negatie feedback, but pitie feedback i al ued (but we ll get int that hrtly). ain under cled lp cnditin i dented by the letter in ur text with a ubcript that define the gain type i.e., ltage gain i, current gain i i and pwer gain i p. With that brief (and curry) reiew under ur belt, a typical pen-lp frequency repne plt i hwn in igure.25 and i reprduced (lightly mdified) t the right. In thi figure, ltage gain i ued a an example, with the ertical axi expreed in decibel (db) and the crrepnding cled lp gain nted in blue (i.e., 8dB 4 V/V, etc.). The hrizntal axi in thi figure i frequency in Hertz (cycle per ecnd) uing a lgarithmic cale. r thi firt rder ytem (indicated by the -2 db/decade rllff), the bandwidth i the upper break frequency (al called the crner frequency, half-pwer frequency, ) i defined a the pint at which we make the -3 db crrectin frm the traight line apprximatin.

2 n imprtant characteritic f thi figure i that the gainbandwidth prduct (BP) i cntant, and i 6 fr thi particular example a hwn in Table. f yur text and t the right. Nte that thi example i characteritic f a 74 with an pen lp gain f 5. Since a lgarithmic plt neer ge t zer, and the BP mut remain cntant, the lwet frequency where thi i true ( Hz in thi cae) i cnidered the zer frequency pint. BW BP The repne f igure.5 i typical f a general-purpe ingle rll-ff pamp uch a the 74. Back in Sectin 5, it wa mentined that the 74 include an internal cmpenatin thrugh an n-chip RC netwrk. Thi frm f fixed cmpenatin i a technique that mdifie the pen-lp frequency repne characteritic in rder t impre perfrmance and tability (hang n, we ll get t tability hrtly). Specifically with regard t an RC cmpenatin netwrk, uch a i gien in igure.26 and t the right, the ltage tranfer functin may be written a ( jc ) ( jc ) R jrc jτ ( j) ( j), (Equatin.79) ( j) in where the time cntant, τ, i equal t RC. The RC netwrk ha a ingle ple at a radian frequency f /τ/rc. When the RC cmpenatin netwrk i ued in an p-amp, the gain decreae at with a lpe f 2 db/decade fr frequencie greater than. The 74 ue uch a ingle ple fixed cmpenatin netwrk that i built int the deice and i the caue f the frequency repne illutrated in igure.25. The family f cure in thi figure may be apprximated by ( ), (Equatin.8) where i the dc pen-lp p-amp gain and i the p-amp crner frequency defined by the RC netwrk. Nte: In general, manufacturer data heet will hw the amplitude and phae repne f a particular p-amp under certain perating cnditin.

3 typical nn-inerting p-amp cnfiguratin i hwn in igure.27 and i gien t the right. Nte that we hae changed the purely reitie ntatin f circuit cmpnent t the mre general impedance ntatin fr the feedback and inerting input. Uing ur ideal apprximatin that the current int the pamp terminal are zer, r i i -, we can write the nde equatin at - a fllw: ; r, (Equatin.82) and we can ee by inpectin that in. Uing the gain relatinhip deelped lat emeter and ubtituting in the abe exprein fr and -, we hae ( ) in. inally, rearranging the abe equatin, we can get an exprein fr the utput ltage, in. (Equatin.83) If we define a term (gamma) t be, (Equatin.84) the cled lp gain, / in, becme (fund by rearranging Equatin.83 and uing Equatin.84), (Equatin.85) which apprache / fr lw frequencie if i ery large.

4 If the frequency dependent exprein f Equatin.8 fr () i ubtituted fr in Equatin.85, we get ) (. (Equatin.87) Nte: I will try t be cnitent, but in cae I frget mean () ( i mean i (), etc.) if frequency dependence i inled. If the dc gain,, i ery large we can aume that >> fr reanable alue f and. Equatin.87 may then be reduced t ( ) ) (. (Equatin.88) Thi gain equatin ha a caling factr f / and cntain a ingle ple at C /, (Equatin.9) where R and R wuld replace and if the circuit cmpnent were purely reitie intead f reactie. It can be hwn that the bandwidth the inerting cnfiguratin uing ingle-ple cmpenatin i identical. Mathematically we can hw that, fr large pen-lp gain, the cled-lp gain bandwidth prduct i cntant by ( ) C BP. (Equatin.92, Mdified) S, fr ingle-ple cmpenatin, the gain bandwidth prduct i cntant fr the p-amp. The BP tell u that a the cled-lp gain i increaed, the bandwidth mut decreae that the prduct remain cntant. Thi inere relatinhip between gain and bandwidth may be an imprtant tradeff when deigning p-amp circuit. Phae Shift

5 r an ideal p-amp, there wa n phae hift fr the nn-inerting cnfiguratin and the phae hift between input and utput fr the inerting cnfiguratin wa determined t be 8 (ince c8 -). r a practical p-amp, the phae hift i dependent n the frequency f the input ignal. r example, an inerting cnfiguratin ha a phae difference i 8 at dc. The angle will decreae a the frequency f the input ignal increae due t the cntributin f the ple() f the tranfer functin. t high frequencie, the phae difference apprache zer and a prtin f the utput ignal i fed back t the input in phae with the input. Thi change the feedback mechanim frm negatie t pitie (which we will get int in the next ectin) and the amplifier may becme untable r marginally table (in the marginally table cae, it exhibit behairal characteritic f an cillatr). Thi ptentially undeirable cnditin i aided by internal frequency cmpenatin r by including an external capacitr. In either cae, ince cillatin will nt ccur with pitie feedback a lng a the gain i le than unity, the trategy i t frce the p-amp t hae a gain f le than ne at frequencie where the phae difference between input and utput apprache zer. Slew Rate Ideally, the utput f a nn-inerting p-amp circuit wuld be a perfect tep functin fr a tep input. Hweer, ince the frmatin f a tep input require eentially infinite frequencie, and the practical p-amp ha a repne that i frequency dependent, we cannt realize an ideal utput. Thi characteritic f p-amp, referred t a lew rate limiting, caue ditrtin f the utput ignal and i a figure f merit fr the deice. igure.3a (gien t the right) illutrate the nn-inerting cnfiguratin uing purely reitie cmpnent and atifying the bia balance cnditin. Equatin.95 f yur text i imply a repeat f Equatin.88 and i gien by ( ) ( ), where ( ) in ( )

6 , (Equatin.95) R R R R R that the lw frequency gain, /R /R and the gain bandwidth prduct i a deried earlier. If we apply the unit tep functin, a hwn t the right and in igure.3b, the input t the p-amp may be defined a in Vu(t), where V i the magnitude f the input ignal and the unit tep, u(t) i defined by if t u ( t). if t < Uing Laplace tranfrm methd, the input Vu(t) becme V/, where i the cmplex frequency. Sling Equatin.95 fr () and ubtituting in the exprein fr in (), we btain in( ) V ( ). (Equatin.96) ( ) Uing partial fractin expanin (if yu hae quetin abut thi mathematical tl, let me knw) and cnerting back t an exprein in time, we get t ( e ) V ( t). (Equatin.97) Recalling that the lpe f a functin i the deriatie with repect t the independent ariable (time in thi cae), we can define the initial lpe f the utput waefrm (at t) t be d dt ( t) V t ( ) e t V. (Equatin.98) Nte! Thi reult, which i illutrated in igure.3c (belw and t the left), i alid fr linear peratin nly. Thi mean that the magnitude f the

7 input (V) mut be ufficiently mall that the p-amp de nt aturate. If the magnitude f the input i large enugh t caue the p-amp t aturate, the utput repne will reemble igure.3d (belw and t the right). In igure.3d, nte that the initial lpe f the utput repne cure i le than that predicted by the linear thery. The inability f the p-amp utput t rie a fat a the linear thery predict i defined a lew rate limiting. In igure.3d (abe, right) the p-amp i aid t be lewing, which ccur when the initial lpe f the (t) repne i le than V. When the p-amp i lewing, the initial lpe f the utput repne i defined a the lew rate: Slew rate d ( t) SR ( maximum). (Equatin.99) dt The lew rate fr a particular p-amp i uually pecified in the manufacturer data heet in lt per micrecnd at unity gain. The lew rate i related t the pwer bandwidth, f p, which i defined a the frequency at which a inuidal utput, at rated utput ltage, begin t exhibit ditrtin. Therefre, fr a rated utput ltage V r and a lew rate f SR, the pwer bandwidth i fund by f p SR. (Equatin.3) 2πV r If the utput ltage i le than V r, lew rate ditrtin begin at a frequency that i higher than f p. Nte that all equatin deried fr igure.3 may al be applied t a unity gain buffer by letting R ince thi will make / equal t ne.

8 Deigning mplifier Uing Multiple Op-mp We nw hae all the tl t deign amplifier requiring mre than ne pamp. In additin t the baic electrical characteritic f the p-amp, certain prpertie uch a the input reitance pecified fr each terminal, bandwidth, utput reitance, CMRR, PSRR, and the lew rate may be imprtant in the electin f an apprpriate deice and the ucceful deign f an amplifier ytem. Other cnideratin in the deign f multiple pamp circuit include: The gain per tage. Thi i determined by diiding the gain-bandwidth prduct (BP) f the elected p-amp by the required bandwidth. Thi will al tell yu the minimum number f gain tage required t meet any gain pecificatin. Determine which input are negatie and which are pitie. Thi define whether cnnectin are made t the inerting r nn-inerting terminal f the p-amp. If the required input reitance i larger than the alue f the cupling reitr, the input ltage mut be fed t the nn-inerting terminal f the p-amp, ince the input reitance t the inerting terminal i imply the alue f the cupling reitance ued (ee Sectin fr pecific). t time, ilatin f ariu urce in the circuit and phae relatinhip may becme imprtant.