Download Slew Rate of Op Amp

Chapter 8
Operational Amplifier as A Black
Box
 8.1 General Considerations
 8.2 Op-Amp-Based Circuits
 8.3 Nonlinear Functions
 8.4 Op-Amp Nonidealities
 8.5 Design Examples
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Chapter Outline
CH8 Operational Amplifier as A Black Box
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Basic Op Amp
Vout  A0 Vin1  Vin 2 
 Op amp is a circuit that has two inputs and one output.
 It amplifies the difference between the two inputs.
CH8 Operational Amplifier as A Black Box
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Inverting and Non-inverting Op Amp
 If the negative input is grounded, the gain is positive.
 If the positive input is grounded, the gain is negative.
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Ideal Op Amp
 Infinite gain
 Infinite input impedance
 Zero output impedance
 Infinite speed
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Virtual Short
Vin1
Vin2
Vout  A0 (Vin1  Vin 2 )
Vout  finite, A 0  수천
(Vin1  Vin 2 ) 
Vout
0
A0
 Due to infinite gain of op amp, the circuit forces Vin2 to be
close to Vin1, thus creating a virtual short.
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Unity Gain Amplifier
Vout  A0 (Vin  Vout )
Vout
A0

Vin 1  A0
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Op Amp with Supply Rails
 To explicitly show the supply voltages, VCC and VEE are
shown.
 Ex) VCC=+5V, VEE=-5V; VCC=+5V, VEE=0V (general)
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Noninverting Amplifier (Infinite A0)
Vout
R1
 1
Vin
R2
 A noninverting amplifier returns a fraction of output signal
thru a resistor divider to the negative input.
 With a high Ao, Vout/Vin depends only on ratio of resistors,
which is very precise.
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Noninverting Amplifier (Finite A0)
Ex 8.3
Vin1  Vin ,
Vout  Ao (Vin1  Vin 2 )
Vin 2  Vout
R2
R1  R2
Vout

Vin
1

A0
R 
1
 1  1 
R2
R2  1  R1  R2 1
A0 
R1  R2
R2
A0

R1   
R1  1 
 1  1 

 1 

R
R
2 
2  A0 


 The error term indicates the larger the closed-loop gain, the
less accurate the circuit becomes.
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Extreme Cases of R2 (Infinite A0)
 If R2 is zero, the loop is open and Vout /Vin is equal to the
intrinsic gain of the op amp.
 If R2 is infinite, the circuit becomes a unity-gain amplifier
and Vout /Vin becomes equal to one.
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Inverting Amplifier
0  Vout Vin

R1
R2
Virtual ground: + 와 –node 의 전 압 이
같아서 ground(0V)로 보인다는 말임
Vout  R1

Vin
R2
 Infinite A0 forces the negative input to be a virtual ground.
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Another View of Inverting Amplifier
Inverting
CH8 Operational Amplifier as A Black Box
Noninverting
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Gain Error Due to Finite A0
Vin1  0,
Vout  Ao (Vin1  Vin 2 )   AoV X
Vin  VX
V  Vout
 X
R2
R1
Vout
R
 1
Vin
R2

R1
R2
1
1 
R1 
1
1

A0 
R2 



1 
R1  
1

1




A
R
0
2



 The larger the closed loop gain, the more inaccurate the
circuit is.
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Complex Impedances Around the Op Amp
Vout
Z1

Vin
Z2
 The closed-loop gain is still equal to the ratio of two
impedances.
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Integrator
Vout
1

Vin
R1C1s
CH8 Operational Amplifier as A Black Box
Vin
dV
1
 C1 out  Vout  
Vin dt

R1
dt
R1C1
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Integrator with Pulse Input

1
V1
Vout  
Vin dt  
t 0  t  Tb
R1C1
R1C1
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Comparison of Integrator and RC Lowpass Filter
 The RC low-pass filter is actually a “passive” approximation
to an integrator.
 With the RC time constant large enough, the RC filter
output approaches a ramp.
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Lossy Integrator
Vout
1

Vin 1  1 
 1   R1C1s
A0  A0 
 When finite op amp gain is considered, the integrator
becomes lossy as the pole moves from the origin to 1/[(1+A0)R1C1].
 It can be approximated as an RC circuit with C boosted by a
factor of A0+1.
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Differentiator
Vout
dVin
  R1C1
dt
CH8 Operational Amplifier as A Black Box
Vout
R1

  R1C1s
1
Vin
C1s
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Differentiator with Pulse Input
Vout   R1C1V1 (t )
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Comparison of Differentiator and High-Pass Filter
 The RC high-pass filter is actually a passive approximation
to the differentiator.
 When the RC time constant is small enough, the RC filter
approximates a differentiator.
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Lossy Differentiator
Vout
 R1C1s

Vin 1  1  R1C1s
A0
A0
 Finite op amp gain의 경우, 미분기는 원점 zero에서 –
(A0+1)/R1C1의 pole을 추가
 RC HPF와 비교하여 R  R/(A0+1)로 근사화 가능.
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Op Amp with General Impedances
Vout
Z1
 1
Vin
Z2
 This circuit cannot operate as ideal integrator or
differentiator.
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Voltage Adder
Vout
Ao
Vout
 V1 V2 
  RF   
 R1 R2 
 RF

V1  V2 
R
If R1 = R2=R
 If Ao is infinite, X is pinned at ground, currents proportional
to V1 and V2 will flow to X and then across RF to produce an
output proportional to the sum of two voltages.
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Precision Rectifier
OP  AMP
1)Vin  0
D1  순방향  FB loop(on)  Vin1  Vin 2 ,VY  VD,on  Vout
2)Vin  0
D1  역방향  FB loop(off)  VY  A0 (Vin1  Vin 2 )
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Inverting Precision Rectifier (input=Vin,output=Vx)
OP  AMP
1)Vin  0
D1  F.B.   FB loop(O)   Vin1  Vin 2  0  V X
2)Vin  0
D1  R.B.   FB loop(X)   VX  Vin ,VY   A0VX
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Logarithmic Amplifier
VX  0
I C  I s eVBE / VT
VBE  Vout
Vout  VT ln
Vin
R1 I S
 By inserting a bipolar transistor in the loop, an amplifier
with logarithmic characteristic can be constructed.
 This is because the current to voltage conversion of a
bipolar transistor is a natural logarithm.
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Square-Root Amplifier
VX  0
1
W
 nCox VGS  VTH 2
2
L
 Vout
I DS 
VGS
Vout  
2Vin
 VTH
W
 nCox R1
L
 By replacing the bipolar transistor with a MOSFET, an
amplifier with a square-root characteristic can be built.
 This is because the current to voltage conversion of a
MOSFET is square-root.
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Op Amp Nonidealities: DC Offsets by mismatch
Wafer 제작과정에서 생길 수 있는
length, oxide thickness등등의 차이
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Op Amp Nonidealities: DC Offsets
 Offsets in an op amp that arise from input stage mismatch
cause the input-output characteristic to shift in either the
positive or negative direction (the plot displays positive
direction).
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Effects of DC Offsets
Vout
 R1 
 1  Vin  Vos 
 R2 
 As it can be seen, the op amp amplifies the input as well as
the offset, thus creating errors.
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Saturation Due to DC Offsets
 Since the offset will be amplified just like the input signal,
output of the first stage may drive the second stage into
saturation.
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Offset in Integrator
t
Vos
1
Vout  Vos 
Vos dt  Vos 
t

R1C1 0
R1C1
시간이 지나면 출력전압이 계속 커져서 전원전압까지 올라감
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Offset in Integrator
출력전압이 계속 커지는 것을 막아줌
 R2 
Vout  Vos 1  
 R1 
Vout
R2
1

Vin
R1 R2C1s  1
 A resistor can be placed in parallel with the capacitor to
“absorb” the offset. However, this means the closed-loop
transfer function no longer has a pole at origin.
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Input Bias Current
 The effect of bipolar base currents can be modeled as
current sources tied from the input to ground.
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Effects of Input Bias Current on Noninverting
Amplifier
 R1 
Vout   R2 I B 2     R1 I B 2
 R2 
 It turns out that IB1 has no effect on the output and IB2
affects the output by producing a voltage drop across R1.
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Input Bias Current Cancellation
 R1 
Vout  Vcorr 1    I B 2 R1
 R2 
 We can cancel the effect of input bias current by inserting a
correction voltage in series with the positive terminal.
 In order to produce a zero output, Vcorr=-IB2(R1||R2).
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Correction for  Variation
I B1  I B 2
Current offset이 없는 경우에 위와 같이 전류가 흐른는 경우에도
출력으로 DC offset이 생기게 되는데, OP-AMP의 +node에서 바라본
저항과 –node에서 바라본 저항이 같아지도록 하면 제거된다.
 Since the correction voltage is dependent upon , and 
varies with process, we insert a parallel resistor
combination in series with the positive input. As long as
IB1= IB2, the correction voltage can track the  variation.
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Effects of Input Bias Currents on Integrator
Vout
1

R1C1

 I B 2 R1 dt
 Input bias current will be integrated by the integrator and
eventually saturate the amplifier.
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Integrator’s Input Bias Current Cancellation
앞과 같은 방법으로 저항 추가
 By placing a resistor in series with the positive input,
integrator input bias current can be cancelled.
 However, the output still saturates due to other effects such
as input mismatch, etc.
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Speed Limitation
Vout
A0
s  
s
Vin1  Vin 2
1
1
Open loop Gain(feedback 회로를 연결하지 않은)의 주파수 특성
 Due to internal capacitances, the gain of op amps begins to
roll off.
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Continued : Gain X bandwidth = constant
Closed loop Gain(feedback 회로를 연결된)의 주파수 특성
Vout

Vin
1
A0
Ao
, A0 ( s ) 
R2
1  1 / 1
A0
R1  R2
Vout
( s) 
Vin
1
A0 ( s )
A0

R2
s
R2
A0 ( s )

A 1
R1  R2
1 R1  R2 0
A0
R2
A0  1
 R2

A0
R  R2
R1  R2

, Gain( DC ) 
 1
, BW  1 
A0  1
s
R2
R2
 R1  R2

1
A0  1
R1  R2
 R2

1 
A0  1
 R1  R2

Gain  BW  A01
OP-AMP를 이용해서 만든 회로는 OPAMP의 한계를 넘어서는 특성을 절대로
보일 수 없다.
43
Bandwidth and Gain Tradeoff
 Having a loop around the op amp (inverting, noninverting,
etc) helps to increase its bandwidth. However, it also
decreases the low frequency gain.
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Slew Rate of Op Amp
# step 입력에 대해서 출력이 step출력이 나오지 않게 됨: OP-AMP가 가지고 있는 최대 단위
시간당 전압의 변화를 slew rate라고 함.
1)step입력에 대해서 M2가 off되고, M4가 공급해 주는 전류에 의해서, Vout이 High로 올라
간다. 출력전압은 capacitor에 공급되는 전류에 의해서 High로 올라가기 때문에, linear하게
커지는 출력전압이 나온다.
2)step입력에 대해서 M1이 off되고, M2가 빼내는 전류에 의해서, Vout이 Low로 떨어진다.
출력전압은 capacitor에서 빼내는 전류에 의해서 Low로 떨어지기 때문에, linear하게 떨어지는
출력전압이 나온다.
# Slew rate한계를 가지는 OP-AMP를 사용한 회로: 출력이 단위시간당 전압의 변화가 OPAMP의 slew rate를 넘게 되는 경우에 출력 파형이 입력파형과 다르게 나옴.
A) 출력 전압의 큰 변화가 생기는 경우, B) 주파수가 높은 경우
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Slew Rate of Op Amp: 전압의 큰 변화
 In the linear region, when the input doubles, the output and
the output slope also double. However, when the input is
large, the op amp slews so the output slope is fixed by a
constant current source charging a capacitor.
 This further limits the speed of the op amp.
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Comparison of Settling with and without Slew Rate
 As it can be seen, the settling speed is faster without slew
rate (as determined by the closed-loop time constant).
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Slew Rate Limit on Sinusoidal Signals: 주파수가 높은 경우

dVout
R1 
 V0 1   cos t
dt
 R2 
 As long as the output slope is less than the slew rate, the
op amp can avoid slewing.
 However, as operating frequency and/or amplitude is
increased, the slew rate becomes insufficient and the
output becomes distorted.
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Maximum Op Amp Swing
Vout
Vmax  Vmin
Vmax  Vmin

sin t 
2
2
 FP
SR

Vmax  Vmin
2
 To determine the maximum frequency before op amp slews,
first determine the maximum swing the op amp can have
and divide the slew rate by it.
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Nonzero Output Resistance
A0 
Rout
R1
vout
R1

vin
R2 1  Rout  A  R1
0
R2
R2
 In practical op amps, the output resistance is not zero.
 It can be seen from the closed loop gain that the nonzero
output resistance increases the gain error.
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Design Examples
 Many design problems are presented at the end of
the chapter to study the effects of finite loop gain,
restrictions on peak to peak swing to avoid
slewing, and how to design for a certain gain error.
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