Active Filters - 1. Active Filters - 2

Transcription

1 PHY35 - Elecronics Laboraory, all Term (K rong) Acie ilers - By combining op-amps wih energy-sorage elemens, circuis can be designed o gie frequency-dependen op-amp responses Acie filers are hose ha use op-amps, so-called because hey proide amplificaion as well as he filering of passie circuis imply replacing resisors R and R wih complex impedances and in our diagrams for he inering and noninering opamps, analogous expressions for he gain can be obained: Closed-loop gain for an inering amplifier: Vou ( jω) V Closed-loop gain for a noninering amplifier: where V, V ou, I, and I are phasors o by choosing suiable raios of o (using filer circuis) he frequency response of an ideal op-amp can be aried V V ou ( jω) Acie ilers - V ~ I I V ou Inering PHY35 - Elecronics Laboraory, all Term (K rong) I I Vou V ~ Noninering (from Rizzoni igure )

2 Low-Pass Op-Amp iler - imples low-pass op-amp filer - uses low-pass RC circui for We know ha he closed-loop gain is: The feedback impedance is: The source impedance is: o he closed-loop gain becomes: R R ALP(jω) jωc R R R ALP(jω) R jωc jωc R C R (from Rizzoni igure ) V _ R V ou PHY35 - Elecronics Laboraory, all Term (K rong) Low-Pass Op-Amp iler - A LP (jω) This expression is he produc of wo erms: a gain facor like ha of an inering amplifier wih no capacior a low-pass filer wih cuoff frequency defined by R and C o he response of his op-amp filer is an amplificaion of he response of a low-pass filer R R jωc R This has he adanages of allowing he gain and bandwidh o be easily adjused by conrolling he raios R /R and /R C, respeciely Noe: he cuoff frequency is: ωo R C PHY35 - Elecronics Laboraory, all Term (K rong)

3 Ampliude raio I is also possible o plo his as decibels (db) s ω The raio of wo ampliudes A and A, is: db log Noe, for wo power leels P and P : Response a cuoff frequency is: db log P (jω ) log R R log where Low-Pass Op-Amp iler - 3 Ampliude response of low-pass acie filer Radian frequency (logarihmic scale) ( ) ALP o db log 3 db or a circui wih R /R and /R C ( A A) ( ) P so ω o is he 3-dB frequency PHY35 - Elecronics Laboraory, all Term (K rong) High-Pass Op-Amp iler - imples high-pass op-amp filer - uses high-pass RC circui for Impedance of he inpu circui: R jωc Impedance of he feedback circui: R Closed-loop gain: R jωcr AHP(jω) R jωc jωcr R _ R V C Vou PHY35 - Elecronics Laboraory, all Term (K rong) (from Rizzoni igure 4)

4 High-Pass Op-Amp iler - Ampliude response of high-pass acie filer Ampliude raio PHY35 - Elecronics Laboraory, all Term (K rong) Radian frequency (logarihmic scale) or a circui wih R /R and /R C As ω, so does A HP (jω) As ω, A HP (jω) approaches a consan: lim A (jω) R R ω HP o aboe some frequency range, he circui behaes as a linear amplifier, as expeced of a high-pass filer Noe: he cuoff frequency is: ωo R C Band-Pass Op-Amp iler - A band-pass op-amp filer can be consruced using a combinaion of he low-pass and high-pass filers, as shown R C _ R V C V ou (from Rizzoni igure 6) PHY35 - Elecronics Laboraory, all Term (K rong)

5 Band-Pass Op-Amp iler - Impedance of he inpu circui: Impedance of he feedback circui: Closed-loop gain: A BP PHY35 - Elecronics Laboraory, all Term (K rong) (jω) R R jωc jωc jωc jωcr jωc ( jωc R )( jωc R ) R R jωc R Each RC produc corresponds o some criical frequency: ω ωlp ωhp R C R C R C When ω HP > ω LP, hen he op-amp filer response is a band-pass Noe ω LP and ω HP are he same as he cuoff frequencies for he low-pass and high-pass op-amp circuis Ampliude raio Band-Pass Op-Amp iler - 3 Normalized response of acie band-pass filer Ampliude response db ampliude response Radian frequency (logarihmic scale) Radian frequency (logarihmic scale) PHY35 - Elecronics Laboraory, all Term (K rong) (from Rizzoni igure 7) db - or a circui wih ω, ω HP, ω LP The band-pass response is superposiion of low- and high-pass The wo 3-dB cuoff frequencies are ω LP and ω HP ω is he frequency a which he response of he filer crosses he -db axis (rising slope) Because db corresponds o a gain of, ω is called he uniy gain frequency

6 Inegraor Circuis - o far we hae looked a he response o sinusoidal inpus of opamp circuis conaining energy-sorage elemens Inegraor and differeniaor op-amp circuis are examples for which he inpu is ime-arying bu no necessarily sinusoidal Consider his inegraor op-amp circui for which s () is an arbirary funcion of ime (eg, pulse rain, riangular wae, square wae) and whose oupu is proporional o he inegral of s () C () _ R i () i () ou () PHY35 - Elecronics Laboraory, all Term (K rong) Inegraor Circuis - () As before: i () i() where i() R d ou() or he capacior: i () C d The source olage is hus a dou() () R funcion of he deriaie of ou (): C d Inegraing gies: ou () (' ) d' RC so he oupu olage is he inegral of he inpu olage This has many applicaions, eg, analog compuers PHY35 - Elecronics Laboraory, all Term (K rong)

7 Example: Inegraion of a quare Wae Deermine he oupu olage for he inegraor circui if he inpu is a square wae of ampliude ±A and period T Known: eedback and source impedances, inpu waeform Gien: T ms, C µ, R kω Assumpions: Ideal op-amp quare wae sars a, so ou () ind: ou () oluion: ar wih equaion ou() (' ) d' R C R C () ou RC PHY35 - Elecronics Laboraory, all Term (K rong) (' ) d' (' (') d' ) d' () A -A T T Inegraion of a quare Wae - The square wae can be inegraed piecewise by noing ha s () A for < T/, and s () -A for T/ < T The resul is a riangular wae, whose saring poin depends on he iniial condiions for he square wae ou ou () RC () ou () (') d' A d' A ou PHY35 - Elecronics Laboraory, all Term (K rong) T ( ) RC T A T T (') d' ( A) d' < T T A A A(T ) T / () ou -5AT T / T T < T

8 Differeniaor Circuis The circui for an ideal differeniaor is shown below Wan o derie he relaionship beween he inpu and oupu olages d () ou() Noe: i () C and i() d R o he oupu olage is hus proporional o he deriaie of he inpu olage: d () ou() RC d In pracice, his circui is less useful han he inegraor because differeniaion amplifies noise presen in a signal R () _ C i () i () ou () PHY35 - Elecronics Laboraory, all Term (K rong) Pracical Limiaions of Op-Amps Ideal op-amps hae infinie inpu resisance, zero oupu resisance, and infinie open-loop olage gain In pracice, op-amps hae seeral limiaions ha are paricularly imporan when designing circuis for high olages and currens, and high-frequency signals We will look a: olage supply limis frequency response limis inpu offse olage inpu bias currens oupu offse adjusmen slew rae limi shor circui oupu curren common-mode rejecion raio (CMRR) PHY35 - Elecronics Laboraory, all Term (K rong)

9 PHY35 - Elecronics Laboraory, all Term (K rong) Volage upply Limis Op-amps are powered by exernal DC olage supplies (V s, V s- ) which are usually symmerical and of he order of ± o ± V A limiing olage supply means ha an op-amp can only amplify signals ha are wihin he range of heir supply olages: i is impossible for an op-amp o generae a olage > V s or < V s- This limiaion can be saed as: V - s < ou < V s or mos op-amps his limi is ~5V less han he supply olages The effec of his limiaion is o clip () ou he peaks of he oupu sine wae which can cause significan errors paricularly serious for inegraors acual response eg, rock guiars oeramplify signal, exceed he olage supply limis, ideal response and cause clipping, which broadens he specral conen of each one and disors he sound requency Response Limis We hae assumed ha he open-loop gain is large Howeer, A V(OL) is acually frequency-dependen wih a low-pass response Typically: A V(OL) (jω) A o ( jω/ ωo ) Here he cuoff frequency for he gain is approximaely he poin where he amplifier response sars o decrease wih frequency This limiaion means ha he assumpion of large open-loop gain becomes increasingly less accurae for higher frequencies This finie bandwidh for real op-amps resuls in a fixed gainbandwidh produc for a gien amplifier V A db Consan K means ha as closedloop gain is increased, he 3-dB bandwidh is reduced, unil in he limi, if he amplifer were used in open-loop mode, he gain A o and 3-dB bandwidh ω o PHY35 - Elecronics Laboraory, all Term (K rong) log A log A log A log scale K A o ω ω ω ω o ω

10 Inpu Offse Volage Real op-amps may hae an offse olage presen a he op-amp inpu, een if here is no exernal inpu This olage is caused by mismaches in he inernal circuiry of he op-amp I appears as a differenial inpu olage beween he inering and noninering inpu erminals R This addiional olage causes a R DC bias error in he op-amp oupu Typical and maximum alues are usually gien in daa shees, so wors-case effecs can be prediced _ V O Usually represened as ±V O PHY35 - Elecronics Laboraory, all Term (K rong) DC offse olage V ou, os A V nominal V os ou PHY35 - Elecronics Laboraory, all Term (K rong) Inpu Bias Currens Real op-amps may hae small inpu bias currens a he inering and noninering erminals These are caused by he inernal consrucion of he inpu sage of he op-amp Typical alues depend on he semiconducor echnology used, eg alues range from na for ET inpu deices o µa for bipolar ransisors Usually represened as I B The inpu offse curren can be more useful: I O I B - I B- This curren causes he op-amp oupu olage o decrease by some alue _ R i V R i I B- I B ou

11 Oupu Offse Adjusmen The offse olage and he inpu offse curren conribue o an oupu offse olage V ou,os I is possible o minimize V ou,os for some op-amps Example: The 8-pin DIP µa74 op-amp proides an offse null connecion for his - i is used o null he oupu offse olage A ariable resisor can be adjused unil ou is a minimum, ideally V This procedure eliminaes he effecs R of boh he inpu offse olage and he curren on he oupu Offse null Inering inpu Noninering inpu _ V No connecion V Oupu Offse null PHY35 - Elecronics Laboraory, all Term (K rong) R offfse null ou 5 V -5 V ariable resisor max PHY35 - Elecronics Laboraory, all Term (K rong) lew Rae Limi Op-amps can only produce a finie rae of change a he oupu This limiing rae is he slew rae Consider an ideal sep inpu: a, he inpu olage swiches from o V ols Expec oupu o swich from o A V ols Bu because ou () can only change a a finie rae, we hae: d ou () o slew rae d max The slew rae limi can also affec sinusoidal signals: () Bsinω d() ωbcos ω d Maximum rae of change occurs a he zero crossings where ω, π, π,, so: d() d ωb o higher ω, B gie larger o

12 hor-circui Oupu Curren Our op-amp model represened he inernal circui by equialen inpu resisance R in and conrolled olage source A V in or real op-amps, he inernal source is no ideal because i canno proide an infinie amoun of curren o he load and/or he feedback connecion The resul is ha he maximum oupu curren of he op-amp is limied by he shor-circui oupu curren I sc : I ou < I C An op-amp needs o proide curren o he feedback pah (o zero he olage differenial a he inpu) and o he load resisance conneced o he oupu: I C is he load curren ha would be proided o a shor circui load (R L ) or a sinusoidal olage, for any load resisance R L < ou, peak /I C, he required load curren > I C This means ha he acual alue of ou, peak does no reach is heoreical maximum PHY35 - Elecronics Laboraory, all Term (K rong) Common-Mode Rejecion Raio The example of an EKG amplifier inroduced differenial-mode and common-mode signals Define A dm as differenial-mode gain and A cm as he commonmode gain of an op-amp Then he op-amp oupu is: ou A dm( ) A cm or ideal condiions, A cm because he differenial amplifier should rejec common-mode signals The deparure from his ideal behaiour is a parameer for a differenial amplifier and is measured using he common-mode rejecion raio (CMRR) which is ofen expressed in db and is defined as: A dm CMRR A cm PHY35 - Elecronics Laboraory, all Term (K rong)