Lecture 29 Operational Amplifier frequency
advertisement
Lecture 29 Operational Amplifier frequency Response Reading: Jaeger 12.1 and Notes Georgia Tech ECE 3040 - Dr. Alan Doolittle Ideal Op Amps Used to Control Frequency Response Low Pass Filter Previously: Vout Vout + Vin - Vin R1 =− R2 R1 R2 Now put a capacitor in parallel with R2: If s = jϖ , + Vout - Vin R1 R2 C2 Georgia Tech Vout =− Vin Vout Vin R2 1 C2 s R1 1 C2 s R 1 =− =− 2 1 R1 R1 R2 + C2 s R2 1 1 + R2 C 2 s ECE 3040 - Dr. Alan Doolittle Ideal Op Amps Used to Control Frequency Response Low Pass Filter + Vout - Vin R1 Vout R =− 2 Vin R1 R2 1 1 + R2 C 2 s C2 •At DC (s=0), the gain remains the same as before (-R2/R1). •At high frequency, R2C2s>>1, the gain dies off with increasing frequency, Vout Vin 1 1 C s = − 2 ≈ − R1 R1C 2 s 1 = 2π f H = ω H R2 C 2 •At high frequencies, more “negative feedback” reduces the overall gain Georgia Tech ECE 3040 - Dr. Alan Doolittle Ideal Op Amps Used to Control Frequency Response Low Pass Filter AV AV DB AV DB R2 = 20 Log R1 DB Vout = 20 Log Vin is the gain expressed in dB -3dB drop at fH (Vout has dropped in half!) Slope = −20 dB / Decade fH fH 1 = 2π R2 C 2 Log(f) •At DC (s=0), the gain remains the same as before (-R2/R1) •At high frequency, R2C2s>>1, the gain dies off with increasing frequency •Implements a “Low Pass Filter”: Lower frequencies are allowed to pass the filter without attenuation. High frequencies are strongly attenuated (do not pass). Georgia Tech ECE 3040 - Dr. Alan Doolittle Ideal Op Amps Used to Control Frequency Response High Pass Filter + Vout - Vin R1 R2 + Vout Vin R1 Vout =− Vin R2 1 R1 + C1 s Vout RCs =− 2 1 Vin 1 + R1C1 s C1 Vout R2 =− Vin R1 R2 •At DC (s=0), the gain is zero. •At high frequency, R1C1s>>1, the gain returns to it’s full value, (-R2/R1) Georgia Tech ECE 3040 - Dr. Alan Doolittle Ideal Op Amps Used to Control Frequency Response High Pass Filter AV DB AV DB R = 20 Log 2 R1 -3dB drop at fH Vout R2 C1 s =− Vin 1 + R1C1 s Slope = +20 dB / Decade fL Log(f) 1 fL = 2π R1C1 •At DC (s=0), the gain is zero. •At high frequency, R1C1s>>1, the gain returns to it’s full value, (-R2/R1) •Implements a “High Pass Filter”: Higher frequencies are allowed to pass the filter without attenuation. Low frequencies are strongly attenuated (do not pass). Georgia Tech ECE 3040 - Dr. Alan Doolittle Ideal Op Amps Used to Control Frequency Response Band Pass Filter (combination of high and low pass filter) + Vou C1 R1 Vin Vout Vin 1 R2 C2 s =− 1 R1 + C1 s Georgia Tech R2 C2 Low Pass High Pass 1 C2 s 1 R2 + R2 C1 s C2 s 1 =− = − 1 1 1 + R C s + R C s 2 2 1 1 R1 + C1 s R2 Vout Vin t ECE 3040 - Dr. Alan Doolittle Ideal Op Amps Used to Control Frequency Response Band Pass Filter (combination of high and low pass filter) R2 C1 s Vout 1 = − 1 1 + Vin + R C s R C s 2 2 1 1 AV DB AV DB R2 = 20 Log R1 1 fL = 2π R1C1 1 fH = 2π R2 C 2 fH fL Georgia Tech Slopes = − 20 dB / Decade + -3dB drop at fH f L << f H Log(f) ECE 3040 - Dr. Alan Doolittle Ideal Op Amps Used to Control Frequency Response Band Pass Filter (combination of high and low pass filter) R2 C1 s Vout 1 = − 1 1 + Vin + R C s R C s 2 2 1 1 AV Slopes = − 20 dB / Decade + DB AV DB R = 20 Log 2 R1 More than a -3dB drop at fL and fH fL < fH fL Georgia Tech fH and fL → fH Log(f) ECE 3040 - Dr. Alan Doolittle General Frequency Response of a Circuit Poles and Zeros Generally, a circuit’s transfer function (frequency dependent gain expression) can be written as the ratio of polynomials: vout (τ 1z s )(1 + τ 2 z s )(1 + τ 2 z s )... ( τ 1zϖ ) 1 − (τ 2 zϖ )2 1 − (τ 3 zϖ )2 ... vout =A =A 2 2 2 (1 + τ 1 p s )(1 + τ 2 p s )(1 + τ 3 p s )... vin vin 1 − (τ ϖ ) 1 − (τ ϖ ) 1 − (τ ϖ ) ... 1p 2p 3p Complex Roots of the numerator polynomial are called “zeros” while complex roots of the denominator polynomial are called “poles” Each zero causes the transfer function to “break to higher gain” (slope increases by 20 dB/decade) Each pole causes the transfer function to “break to lower gain” (slope decreases by 20 dB/decade) -20 dB /D ec a Typically, τ=RC de 0 dB/Decade dB /D eca de 2 0 dB/Decade 40 vout 20 Log vin e B/D d 0 e ca d 2 eca D / B 0d de ϖ= Georgia Tech 1 τ 2z ϖ= 1 τ1p ϖ= 1 τ 2p ϖ= 1 τ 3p ϖ= 1 ϖ τ 3z ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response •To this point we have assumed the open loop gain, AOpen Loop, of the op amp is constant at all frequencies. •Real Op amps have a frequency dependant open loop gain. AOpenLoop ( s ) = AOϖ B ϖT = s +ϖ B s +ϖ B where, s = jϖ AO ≡ Open loop gain at DC ϖ B ≡ Open loop bandwidth ( ) ϖ T ≡ Unity - gain frequency frequency where AOpenLoop ( s ) = 1 Georgia Tech ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response AOpenLoop ( jϖ ) = AOpenLoop ( jϖ ) = AOϖ B ϖ 2 +ϖ B2 AO ϖ2 1+ ϖ B2 At Low Frequencies: AOpenLoop = AO At High Frequencies: AOpenLoop ≈ AOϖ B ϖ = ϖT ϖ For most frequencies of interest, ω>>ωB , the product of the gain and frequency is a constant, ωT ϖ f T = T ≡ Gain − Bandwidth Pr oduct (GBW ) 2π Georgia Tech ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response For the "741" Op Amp, AO ~ 200,000 = 106 dB ϖ B ~ (2π ) 5 Hz ϖ T ~ (2π ) 1 MHz GBW For the " Op 07" Op Amp, AO ~ 12,000,000 = 141 dB ϖ B ~ (2π ) 0.05 Hz ϖ T ~ (2π ) 0.6 MHz GBW If the open loop bandwidth is so small, how can the op amp be useful? The answer to this is found by considering the closed loop gain. Georgia Tech ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response Previously, we found that the closed loop gain for the Noninverting configuration was (for finite open loop gain): AV ,ClosedLoop AOpenLoop Vout R1 = = , where β = Vin 1 + β AOpenLoop R1 + R 2 Using the frequency dependent open loop gain: AV ,ClosedLoop = AV ,ClosedLoop AV ,ClosedLoop AOpenLoop Vout = Vin 1 + β AOpenLoop AOϖ B s +ϖ B = Aϖ 1 + β O B s +ϖ B Low AOϖ B = Pass s + ϖ B (1 + β AO ) AOϖ B AO (1 + β AO ) ϖ B (1 + β AO ) 1 = = = s s s +1 +1 1+ ϖ B (1 + β AO ) ϖ B (1 + β AO ) ϖH A V ,ClosedLoop @ DC where, ϖ H ≡ Upper Cutoff Frequency (Closed Loop Bandwith ) = ϖ B (1 + β AO ) Georgia Tech ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response Open LoopGain The closed Loop Amplifier has a lower gain than the Open Loop Amplifier ϖH Closed Loop Bandwidth ϖT = ϖ B (1 + β AO ) = AV ,ClosedLoop Closed LoopGain @ DC Closed Loop DC Gain AV ,ClosedLoop = Georgia Tech AOpenLoop 1 + β AOpenLoop The closed Loop Amplifier has a higher bandwidth than the Open Loop Amplifier ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response Closed Loop Gain set by feedback network below ωH Closed Loop Gain set Open Loop Gain above ωH (Gain x Bandwidth ) Open Loop = (Gain x Bandwidth ) Closed Loop Example: 741 Op Amp is used as a low pass filter with fL=10kHz. What is the maximum voltage gain possible for this circuit? From before, we can write: (200 , 000 x 5 ) Open (Gain ) Closed Georgia Tech Loop Loop = (Gain x 10 , 000 = 100 V V ) Closed Loop Maximum ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response For the Inverting Configuration: By sup erposition, + vVin R1 Vout R2 v − = vout R1 R2 + vin R1 + R2 R1 + R2 R2 v − = vout β + vin β R1 but , vout = −v − AV ,OpenLoop so, − vout AV ,OpenLoop AV ,ClosedLoop Georgia Tech = vout β + vin β R2 R1 AV ,OpenLoop β vout = = vin 1 + AV ,OpenLoop β R2 − R1 ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response Inserting the frequency dependent open loop gain: AV ,ClosedLoop AV ,OpenLoop β R2 − = 1 + AV ,OpenLoop β R1 AV ,ClosedLoop AOϖ B β s +ϖ B = Aϖ 1 + O B β s +ϖ B AV ,ClosedLoop = AV ,ClosedLoop Georgia Tech R2 AOϖ B β − = R1 s + ϖ B + AOϖ B β AOϖ B β − s + ϖ B (1 + AO β ) R2 − R1 AOϖ B β ϖ B (1 + AO β ) R2 R2 − = R1 s + ϖ B (1 + AO β ) R1 ϖ B (1 + AO β ) AO β R2 − (1 + AO β ) R1 = s +1 ϖ B (1 + AO β ) ECE 3040 - Dr. Alan Doolittle Real Op Amp Frequency Response AV ,ClosedLoop AO β R2 − (1 + AO β ) R1 = s +1 ϖ B (1 + AO β ) AV ,ClosedLoop 1 A = V ,ClosedLoop s 1+ ( ) + ϖ 1 β A B O Closed Loop Bandwidth ϖ H = ϖ B (1 + β AO ) = ϖT AV ,ClosedLoop Closed Loop DC Gain AV ,ClosedLoop @ DC @ DC AV ,OpenLoop β R2 − = 1 + AV ,OpenLoop β R1 The frequency behavior is the same as for the the Non-Inverting case! Georgia Tech ECE 3040 - Dr. Alan Doolittle
