Section I5: Feedback in Operational Amplifiers

Transcription

1 Sectin I5: eedback in Operatinal mplifiers s discussed earlier, practical p-amps hae a high gain under dc (zer frequency) cnditins and the gain decreases as frequency increases. This frequency dependence is built int the p-amp thrugh the internal cmpensatin netwrk t impre perfrmance and stability. In additin t this built-in cmpensatin, many p-amps are designed t allw the selectin f external cmpensatin netwrks that allw further imprement t perfrmance and are reflected in the shape f the amplifier Bde plts. igure 11.7, reprduced belw, is a straight-line plt f the pen lp gain ersus frequency fr a typical 741 p-amp. Ntice that the cure fllws the frm f a single-ple cmpensatin netwrk, with a 20 db/decade rll ff after the crner frequency (f r ω ). ls nte that the gain bandwidth prduct (GBP) remains cnstant er the peratinal range defined fr this deice. The analytic expressin fr a frequency respnse f this type was deelped in Sectin H7 and is repeated belw: G( s) G 1 + s ω 0. (Equatin and 11.22) Knwing the frm f this relatinship, we can pick ff infrmatin frm the figure abe. The crner frequency is f 10 Hz r, t express in a frm cmpatible with the equatin abe, ω 2π(10)20π radians/secnd. The zer-frequency ltage gain, G, is 100 db r 10 5 V/V.

2 Just as a little heads-up here a lgarithmic plt will neer actually g t zer, but we call the alue f the flat prtin f the plt the zer frequency parameter. S, putting this all tgether, the analytic expressin that describes the specific behair f igure 11.7 is G( s) s 20π. Pre t yurself that this is true. r frequencies mre than a decade belw 20π the magnitude f G(s) is apprximately 10 5 (remember that sjω). t 20π, the magnitude becmes r 3 db, and fr frequencies greater than 20π, the magnitude decreases at a rate f 20 db/decade. T illustrate the effect f feedback n peratinal amplifier circuits, we will begin with the inerting amplifier circuit f igure 11.8a (t the right). T make sure that we are fcusing n errr that may be intrduced due t gain ariatins, all characteristics f the p-amp are cnsidered ideal except fr the ariatin f gain with frequency. T find the effect f the utput ltage n the pamp input, we lk at the rati - /, where - is a functin f in and. We will als assume that the input signal ( in ) is equal t zer since we are interested in nly the prtin f - that is due t. rm Sectin H7, we defined a feedback attenuatin factr γ and the lw-frequency gain f the nn-inerting p-amp t be 1/γ where, in terms f purely resistie cmpnents, γ (Equatin 11.23) This functin represents the fractin f the utput ltage that is fed back t the inerting terminal with in 0. T pre this statement, we can write the KCL equatin at - (remember that in 0) in + 0. (Equatin 11.25)

3 Sling fr -, we btain γ. (Equatin 11.24) Nte: there is sme incnsistency in yur text with the generic terms G and G. The way these hae been defined, and we e been using, is G indicates the pen-lp p-amp gain and G is the clsed-lp gain (i.e., with feedback) that is usually frequency dependent. I hae mdified seeral equatins t reflect these definitins, but if there s any questin, please let me knw. In the figure abe, the nn-inerting terminal is tied t grund s + 0. Using the gain relatinship fr ideal p-amps frm last semester in terms f the pen-lp gain G and the difference seen at the tw input terminals, i.e., G( + ), - can be expressed in terms f G and as G. (Equatin 11.26) Substituting Equatin int Equatin (keeping in this time), rearranging t get in the frm / in, we get the expressin fr the clsed lp ltage gain (expressed in the familiar V ) f the inerting p-amp circuit t be i n (1 + ) / G γg 1 + γg, (Eqns & 11.28) where the feedback attenuatin factr, γ, was incrprated int the last relatinship. s the p-amp gain increases, r we lk at the limit as G, the expressins in Equatin r Equatin ges t the gain expressin fr an ideal inerting p-amp cnfiguratin, r. (Equatin 11.29)

4 This means that as the p-amp gain gets ery large, the clsed-lp gain becmes independent f the alue f G and becmes a functin f the tw resistr alues and (r Z and Z if cmplex impedances are used). similar analysis was perfrmed in Sectin H7 fr the nn-inerting cnfiguratin, r may be deried using the prcedure abe with the result G 1 + Gγ. (Equatin 11.32, Mdified) In the limit that G, the gain f the nn-inerting cnfiguratin becmes γ, (Equatin 11.33) which is the gain expressin fr the ideal p-amp nn-inerting circuit. The tw gain expressins, Equatins and 11.33, can be nrmalized by diiding each clsed lp gain,, by the apprpriate. fter nrmalizing, the same expressin results fr bth the inerting and nn-inerting cnfiguratins: Gγ 1 + Gγ, (Equatin 11.34) where Equatin is in the frm f the generic clsed lp feedback system discussed in Sectin I2 and G γ is the lp gain f Equatin Nte that the lp gain f bth amplifier cnfiguratins is the same. T determine the sensitiity f the clsed-lp gain,, t changes in the lp gain, G γ, we fllw the apprach defined in Sectin I3; i.e., differentiate with respect t G γ, then diide by and rearrange t btain d d( G γ ) ( G γ ) 1 ( + G γ ) ( γ ) 1 d G 1 + G γ G γ. (Equatin 11.36) Equatin applies t bth the inerting and nn-inerting amplifier cnfiguratins and clearly shws the effect f feedback. ny ariatin in the lp gain, G γ, is diided by G γ (1+ G γ) and results in a much smaller ariatin in the clsed lp gain. S just as in the case f discrete

5 cmpnent amplifiers, p-amps exhibit less sensitiity when feedback is used.