ECE 3040 - Dr. Alan Doolittle Georgia Tech
Lecture 29
Operational Amplifier frequency Response
Ideal Op Amps Used to Control Frequency Response
Low Pass Filter
+ -R2 R1 Vin Vout + -R2 R1 Vin Vout C2 1 2 R R V V in out = − + − = + − = − = = s C R R R s C R s C R R V V R s C R V V j s If in out in out 2 2 1 2 2 2 2 1 1 2 2 1 1 1 1 1 1 , ϖ
Now put a capacitor in parallel with R2: Previously:
ECE 3040 - Dr. Alan Doolittle Georgia Tech + -R2 R1 Vin Vout 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,
H H in out f C R R s C s C R V V ω π = = − = − ≈ 2 1 1 1 2 2 1 2 2 1
Ideal Op Amps Used to Control Frequency Response
Low Pass Filter
+ − = s C R R R V V in out 2 2 1 2 1 1
dB in expressed gain the is 20 = in out DB V V V Log A
•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).
= 1 2 20 R R Log AV DB Decade dB Slope = −20 / fH Log(f) DB V A
-3dB drop at fH (Vout has dropped in half!)
Ideal Op Amps Used to Control Frequency Response
Low Pass Filter
2 2
2
1
C
R
f
Hπ
=
ECE 3040 - Dr. Alan Doolittle Georgia Tech + -R2 R1 Vin Vout + -R2 R1 Vin Vout C1 1 2 R R V V in out = − s C R s C R V V s C R R V V in out in out 1 1 1 2 1 1 2 1 1 + − = + − =
Ideal Op Amps Used to Control Frequency Response
High Pass Filter
•At DC (s=0), the gain is zero.
•At high frequency, R1C1s>>1, the gain returns to it’s full value, (-R2/R1)
s C R s C R V V in out 1 1 1 2 1+ − =
Ideal Op Amps Used to Control Frequency Response
High Pass Filter
•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).
= 1 2 20 R R Log AV DB Decade dB Slope = +20 / fL Log(f) DB V A -3dB drop at fH 1 1
2
1
C
R
f
Lπ
=
ECE 3040 - Dr. Alan Doolittle Georgia Tech
Ideal Op Amps Used to Control Frequency Response
Band Pass Filter (combination of high and low pass filter)
+ -R2 R1 Vin Vou t C1 C2 + + − = + + − = + − = s C R s C R s C R s C R s C R s C R V V s C R s C R V V in out in out 1 1 1 2 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 1 1 1 1 Low Pass High Pass
Ideal Op Amps Used to Control Frequency Response
Band Pass Filter (combination of high and low pass filter)
+ + − = s C R s C R s C R V V in out 1 1 1 2 2 2 1 1 1 = 1 2 20 R R Log AV DB Decade dB Slopes = −+20 / Log(f) DB V A -3dB drop at fH fL fH 1 1 2 1 C R fL
π
= 2 2 2 1 C R fHπ
= H Lf
f
<<
ECE 3040 - Dr. Alan Doolittle Georgia Tech
Ideal Op Amps Used to Control Frequency Response
Band Pass Filter (combination of high and low pass filter)
+ + − = s C R s C R s C R V V in out 1 1 1 2 2 2 1 1 1 = 1 2 20 R R Log AV DB Log(f) DB V A
More than a -3dB drop at fL and fH
fL fH H L H L
f
and
f
f
f
<
→
Decade dB Slopes = −+20 /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: ( )( )( ) (1 )(1 )(1 )... ... 1 1 3 2 1 2 2 1 s s s s s s A v v p p p z z z in out τ τ τ τ τ τ + + + + + =
Complex Roots of the numerator polynomial are called “zeros” while complex roots of the denominator polynomial are called “poles”
ϖ
in out v v Log 20Each 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)
( ) ( ) ( ) ( ) 1 ( ) 1 ( ) ... 1 ... 1 1 2 3 2 2 2 1 2 3 2 2 1 ϖ τ ϖ τ ϖ τ ϖ τ ϖ τ ϖ τ p p p z z z in out A v v − − − − − = Typically, τ=RC 20 dB /Decad e 20 dB /Decad e 40 d B/D ecad e 0 dB/Decade 0 dB/Decade -20 dB/D ecade z 2 1 τ ϖ = p 1 1 τ ϖ = p 2 1 τ ϖ = p 3 1 τ ϖ = z 3 1 τ ϖ =
ECE 3040 - Dr. Alan Doolittle Georgia Tech
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.
(
frequency where ( ) 1)
frequency gain Unity bandwidth loop Open DC at gain loop Open , ) ( = ≡ ≡ ≡ = + = + = s A A j s where s s A s A OpenLoop T B O B T B B O OpenLoop ϖ ϖ ϖ ϖ ϖ ϖ ϖReal Op Amp Frequency Response 2 2 2 2 1 ) ( ) ( B O OpenLoop B B O OpenLoop A j A A j A ϖ ϖ ϖ ϖ ϖ ϖ ϖ + = + = At Low Frequencies: At High Frequencies: O OpenLoop A A = ϖ ϖ ϖ ϖ B T O OpenLoop A A ≈ =
For most frequencies of interest, ω>>ωB , the product of the gain and frequency is a constant, ωT
(
GBW)
oduct Bandwidth Gain f T T Pr 2 ≡ − = π ϖECE 3040 - Dr. Alan Doolittle Georgia Tech
Real Op Amp Frequency Response
( )
( )
2 1MHz ~ Hz 5 2 ~ 106 000 , 200 ~ Amp, Op "741" For the π ϖ π ϖ T B O dB A =( )
( )
2 0.6 MHz ~ Hz 0.05 2 ~ 141 000 , 000 , 12 ~ Amp, Op 07" Op " For the π ϖ π ϖ T B O dB A =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.
Real Op Amp Frequency Response
Previously, we found that the closed loop gain for the Non-inverting configuration was (for finite open loop gain):
2 1 1 , R R R where , 1+ = + = = β β OpenLoop OpenLoop in out ClosedLoop V A A V V A
Using the frequency dependent open loop gain:
( ) ( ) ( ) ( ) ( ) ( ) ( ) DC ClosedLoop V H O B O O O B O B B O ClosedLoop V O B B O B B O B B O ClosedLoop V OpenLoop OpenLoop in out ClosedLoop V A Bandwith Loop Closed Frequency Cutoff Upper where A s A s A A A s A A A A s A s A s A A A A V V A β ϖ ϖ ϖ β ϖ β β ϖ β ϖ ϖ β ϖ ϖ ϖ ϖ β ϖ ϖ β + = ≡ + = + + + = + + + = + + = + + + = + = = 1 , 1 1 1 1 1 1 1 1 1 1 1 @ , , , , Low Pass
ECE 3040 - Dr. Alan Doolittle Georgia Tech
Real Op Amp Frequency Response
The closed Loop Amplifier has a higher bandwidth than the Open Loop Amplifier The closed Loop
Amplifier has a lower gain than the Open Loop
Amplifier Open LoopGain Closed LoopGain
(
)
DC ClosedLoop V T O B H A A @ , 1 β ϖ ϖ ϖ = + =Closed Loop Bandwidth
OpenLoop OpenLoop ClosedLoop V A A A β + = 1 ,
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 LoopExample: 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:
(
)
(
)
(
Gain)
V V Maximum x Gain x Loop Closed Loop Closed Loop Open 100 000 , 10 5 000 , 200 = =ECE 3040 - Dr. Alan Doolittle Georgia Tech
For the Inverting Configuration:
Real Op Amp Frequency Response
+ -R2 R1 Vin Vout − + = = + = − − = + = + + + = − − − 1 2 , , , 1 2 , , 1 2 2 1 2 2 1 1 1 , , , sup R R A A v v A R R v v A v so A v v but R R v v v R R R v R R R v v erposition By OpenLoop V OpenLoop V in out ClosedLoop V in out OpenLoop V out OpenLoop V out in out in out β β β β β β v
-Inserting the frequency dependent open loop gain:
Real Op Amp Frequency Response
( ) ( ( ) ) ( ) ( ) ( ) + + − + = − + + + + = − + + = − + + = − + + + = − + = 1 1 1 1 1 1 1 1 1 1 2 , 1 2 1 2 , 1 2 1 2 , 1 2 , , , β ϖ β β β ϖ β ϖ β ϖ β ϖ β ϖ β ϖ β ϖ ϖ β ϖ β ϖ ϖ β ϖ ϖ β β O B O O ClosedLoop V O B O B O B B O O B B O ClosedLoop V B O B B O B B O B B O ClosedLoop V OpenLoop V OpenLoop V ClosedLoop V A s R R A A A R R A A s A A R R A s A A R R A s A R R s A s A A R R A A A
ECE 3040 - Dr. Alan Doolittle Georgia Tech
Real Op Amp Frequency Response
(
)
(
)
(
)
DC ClosedLoop V O B ClosedLoop V O B O O ClosedLoop V A A s A A s R R A A A @ , , 1 2 , 1 1 1 1 1 1 + + = + + − + = β ϖ β ϖ β β ( ) DC ClosedLoop V T O B H A A @ , 1 β ϖ ϖ ϖ = + =Closed Loop Bandwidth
− + = 1 2 , , , 1 R R A A A OpenLoop V OpenLoop V ClosedLoop V β β
