smith ch02

1S. C. Lin, EE National Chin-Yi University of Technology

Sedra/SmithMicroelectronic Circuits 6/E

Chapter 2: Operational Amplifiers

2S. C. Lin, EE National Chin-Yi University of Technology

【Outline】

2.1 The Ideal OP Amp

2.2 The Inverting Configuration

2.3 The Noninverting Configuration

2.4 Difference Amplifiers

2.5 Integrators and Differentiators

2.6 DC Imperfections

2.7 Effect of Finite Open-Loop Gain and Bandwidth on Circuit Performance

2.8 Large-Signal Operation of Op Amp

3S. C. Lin, EE National Chin-Yi University of Technology

2-1 The ideal op amp

CCV

CCV

CCV

CCV

4S. C. Lin, EE National Chin-Yi University of Technology

The properties associated with an ideal Amplifier are:

1. infinite voltage gain ( )

2. Infinite input impedance ( )

3. Zero output impedance( )

4. Output voltage when input voltages

5. Infinite bandwidth ( no delay of the signal through the amplifier)

inZ

vA

1 2V V0outZ

0outV

Low Outputstages

outZ OutputInverting input Differential

input stageIntermediate Amplifier stagesNoninverting

input

Figure 1. Block diagram of an operational amplifier

5S. C. Lin, EE National Chin-Yi University of Technology

2.1.2 Function and Characteristics of the ideal Op Amp

2v

1 0i

2 0i

1v

(virtual short circuit: 0, 0, )i i inv I R

6S. C. Lin, EE National Chin-Yi University of Technology

Some Specifications1. Open loop gain ( ): Usually several thousand.2. Input offset voltage ( ): Small, usually a few millivolts.3. Input offset current ( ): Usually between a few and several

hundred nanoamps.

4. Input resistance ( ):Typically greater than one megohm, but it can be as high as several hundred megohms.

5. Output resistance ( ): Usually less than a few hundred ohms.6. Slew rate ( S ): The maximum rate of output voltage change

given in volts per microsecond.

olA

osIosV

outR

inR

7. CMRR d

cm

AA

1BI

2BI1 2os B BI I I

7S. C. Lin, EE National Chin-Yi University of Technology

2.1.3 Differential and Common-Mode Signal

icmv

/ 2idv

/ 2idv

2v

1v

2 1

2 1

1

2

The difference input signal : The Common-mode input signal : / 2

/ 2

/ 2

id

id

icm

icm

icm id

icm id

vv v v

vv v v

v v vv v v

8S. C. Lin, EE National Chin-Yi University of Technology

2.2 The inverting configuration

11

1 1 1

1 1 1

1

1

2 1 0

(virtual short circuit: 0, 0, )

0

0

i i i

o f f

o

i i in

f

if

fo

i

v v

v

v viR R R

v v i R v i R

v RR

Rv

v vA

v I R

Gv R

fR

1R

1i

fi

iv

ov

2 1v v

2 1A v v

fR

1R

1i

fi

iv

1v

ov

9S. C. Lin, EE National Chin-Yi University of Technology

2.2.2 Effect of Finite Open-Loop Gain

fR

1R

1iiv

ovA

0

ov

11

1

1

1

1

1

1

( / )

/

/

/

1 (1 / )(2.

/

whe

5)

n , /

i o

i o

oo f

o i of

fo

i f

f

v v AiR

v v AR

vv i RAv v v A RA R

R RvGv R R A

A G R R

CCV

CCV

iv

ov

1G /fR R

CCVG

CCVG

10S. C. Lin, EE National Chin-Yi University of Technology

/o iv v2

1 2 21

2 1

23

3 1 3

24 2 3

1 1 3

0 2

0 4

4 4

1

4

22 4

1 1

2

1 3

3

0

/0

1

ix

i

xi

ii

i

i

x

i i

v Rv v i RR

v R Rv Ri v

R R Rv Ri i i vR R R

v v i R

v v RR v RR R R R

v R R Rv R R R

Example2.2: Assuming the op amp to be ideal, derive an expression for closed-loop gain =？

11

1 1

2 11

0 (virtual short)

i i

i

o oi

v v viR

v

Rvi

A

R

vv

i

1R

1iiv

2i3i

4i2R 4R

3R

xxv

1v

Solution

11S. C. Lin, EE National Chin-Yi University of Technology

1 21 2

1 2

1 21 2 3

1

1 2 1 2

2

1 21 2

, , ,

( )

If The

n

)

( )

(2.7

n on f

n f

o nf

f n

f f fo

n

nn

f o n

v vv vi i I IR R R R

v vv vI I I IR R R R

R R R R

R R Rv v v v

Rv v v

R Rv

2.2.4 An Important Application - The Weighted Summer

2R

fR

fi

1R

1i

ni

2i

1v

2v

nv

ov

nR

12S. C. Lin, EE National Chin-Yi University of Technology

2.3 The Noninverting ConfigurationfR

1R

1i

fi

iv

ov

11

11 1

(

,

1 )

i o if

f

if o i

o fi

f

v v vi iR R

v v vi iR

Rv v

RR

CCV

CCV

iv

1G fR / R

CCVG

CCVG

13S. C. Lin, EE National Chin-Yi University of Technology

11

11

1 1

1

1

1

( / )( / ) /( / ) , ,

/( / )

( ) (1 )

1 ( / ) (2. 1 ( / )

1

when , 1

)

(

1

/ )

1

i o

o i o o i oi of

f f

o i oi of

f

i f o f

fo

fi

f

v v v Av v v A v v v Av v Ai i

R R R

v v v Av v Ai iR R

v A R R v A R R

R RvG R RvA

A G R R

=

2.3.3 Effect of Finite Open-Loop Gain

14S. C. Lin, EE National Chin-Yi University of Technology

2.3.4 The Voltage Follower

CCV

CCV

iv

ov

( ) 1Ｇ

CCV

CCV

ov

, , 0o i in outv v R R

15S. C. Lin, EE National Chin-Yi University of Technology

2.4 Difference Amplifiers

icmv

/ 2idv

/ 2idv

1 2id i iv v v

1 2 / 2icm i iv v v

1 / 2i icm idv v v

2 / 2i icm idv v v

11 1CMRR

CMRR , CMR (2.14R 20log )

cm icmo d id cm icm d id d id

d id

cm icm icmd id d id

d id id

d ddB

cm cm

A vv A v A v A v A vA v

A v vA v A vA v v

A AA A

16S. C. Lin, EE National Chin-Yi University of Technology

2.4.1 A Single Op-Amp Difference Amplifier

4R

2R

3R

1R

ov

1iv

2iv

4

2

21

1 3

2 22 1

3 4 1 1

2 4

1

2 1

' ''

(2)Assume 0,

then ''

If ,

then

1

i

O i

o

a b

b bo i i id

a a

bd

i

a

O O

i

v v vR R Rv v

R R

R R R R R RR Rv v v vR R

RAR

R R

vRv vR

(2 .17)

1

2

1

4 22

3 4 1

(1)Assume 0,

then ' 1

1

i

O

i

v

Rv vR

R RvR R R

17S. C. Lin, EE National Chin-Yi University of Technology

A common-mode signal applied at the input

4R

2R

3R

1R

4

3 4icm

R vR R

2i

1i

icmv

ov4

2 23 4

34 2

3 4 1 3 4

34 2

3 4 1 4

34 2

3 4 1 4

1 3 2 4If ,

1

1 ,

.

(2.1

)0

9

o icm

icm icm

icm

ocm

ic

c

m

m

Rv v i RR R

RR Rv vR R R R R

RR R vR R R R

v RR RAv R R R R

R R R R A

41 2

1 3 4

3

1 3 4

1

1

icm icm

icm

Ri i v vR R R

R vR R R

18S. C. Lin, EE National Chin-Yi University of Technology

1 1 1

1 (2 2.20)

idid

i

id i

id

vRi

v R i R iR R

2R

1R

2i

1i

idvov

1R

2R1i

idR

19S. C. Lin, EE National Chin-Yi University of Technology

4R

4 22 1

3 1

21 ( )oR Rv v vR R

1R

3R

2R

1v

2v

3R

4R

1 2v v 1 2 1( ) /v v R

1 2 1( ) /v v R

21 2

1

2( ) 1 Rv vR

1ov

2ov

1R

1fR 1vR

2R

0V

1 2 1( ) /v v R

0V

0A

0A

ov

2.4.2 A Superior Circuit－The Instrumentation Amplifier

20S. C. Lin, EE National Chin-Yi University of Technology

2.5 Integrators and Differentiators

iV

oV

1Z

2Z

2

1

o

i

V ZV Z

21S. C. Lin, EE National Chin-Yi University of Technology

Example 2.4For the circuit in Fig.2.23, (a) derive the transfer function. ( ) / ( )o iv s v s

Solution:

(a)1

2 1

2

2

1 2

2

02 2

2

( ) 1( ) ( ) ( )

1

/

1

11

o

i

v sv s z s Y s

R sCR

R RsC R

ωC R

2R

1R

2C

iv

ov

22S. C. Lin, EE National Chin-Yi University of Technology

02 2

(c) Evaluate 3-dB frequency

the 3-dB frequency 1ωC R

2

1

(b) find the dc gain

The dc gain K RR

2 1

(d) design the circuit to obtain a dc gain of 40 dB, a 3-dB frequency of 1 kHz, and input resistance of 1 k .

In order to obtain a dc gain of 40 dB, we select R /R 100.

Solu

For

tion:

an in

1

2 0 2

0 2 3 32 2 0 2

put resistance of 1 k , we select R 1 k , and thus R 100k , for a 3-dB frequency 1kHz, we select C from

1 1 1 1.59 F. 2 1 10 100 10

f

ω C nC R ω R π

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2.5.2 Inverting Integrator

RC

ov

1i

1i

iv

Cv

10

10

1( ) ( )

1( ) ( ) ( )

t

C C

t

o C C

v t V i tC

v t v t V i tC

24S. C. Lin, EE National Chin-Yi University of Technology

12

We can be described alternatively in the frequency domain 1by subtituting ( ) , and ( )

( )Z s R Y s sC

Z s

1

1

1

1

1 ,

(

/ 90

the unity gai

)

n frequency as1

1( )

( ) 1

( )

o

o

oo to

t

t

v

v v vv

sv s sRC

v jv j j RC

RC

RC

1RC

20dB/decade

(dB)o

i

VV

25S. C. Lin, EE National Chin-Yi University of Technology

1

2

1

( )1( ) 1 1

( ) /1( ) 1

1the Corner frequency as ,

the dc gain as /F

F

F

F

F

F

o FF

F

Z s RRZ ssR

R

CscR

Rv s R RsR Cv s C

R

sR

R

R

C

Fig2.42 The Miller integrator with a large resistance RF connectedin parallel with C in order to provide negative feedback and

hence finite gain at dc

FR

1R

C

( )iv t

( )ov t

26S. C. Lin, EE National Chin-Yi University of Technology

i oi C C

o

R

i oR C

i

dv vdQQ Cv , i i C , Idt dt R

dv vi

dvv RC

dti C

dt R

iv

R

C

ov

Ri

Ci

2.5.3 The Op-Amp Differentiator

27S. C. Lin, EE National Chin-Yi University of Technology

1 2

The frequency domain transfer function of the differentiator circuit 1can be found by subtituting ( ) , and ( )Z s Z s R

sC

1

1

1

1

,

/ 90the unity gain frequency as

1

( ) ( )( ) ( )

o

t

oo

t

o

t

o

v R

v s v jsRC j RCv s v

Cv

v v

RC

j

1/ RC

20dB/decade

/ (dB)o iV V

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2.6.1 Offset Voltage

fR

1R

osV

Offset-free op amp

2

1

1o osRV VR

oV

2.6 DC Imperfections

+V

V

29S. C. Lin, EE National Chin-Yi University of Technology

2R

1RC

( )iv t

( )ov t

2R

o osV V

osV

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2.6.2 Input Bais and Offset Currents

1BI

2BI

1 2

1 2

2B B

B

OS B B

I II

I I I

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2R

1R

1BI

2BI

0

1BI

1 2o BV I R

1 2 2O B BV I R I R

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2R

1R

3R

1BI

2BI

31 2

1B B

RI IR

2 3

1

BI RR

2BIoV

2 3BI R

2 3 2 1 2 3 1

1 2

2 3 2 1

3

2 1 23 1 2

2 1 1 2

/Consider first case , which results in

1 /

Thus we can reduce to zero by selecting such that

//1 /

O B B B

B B B

O B

O

V I R R I I R RI I I

V I R R R R

V RR R RR R RR R R R

33S. C. Lin, EE National Chin-Yi University of Technology

2.7 Effect of Finite Open-Loop Gain and Bandwidth on Circuit Performance

2.7.1 Frequency Dependence of the Open-loop Gain

0 0 0

0

t 0

t t

( ) ( ) ( )1 / 1 /

( )

unity gain frequency

( ) ( )

b

b b

b

b

t

A A A ωA s A jω A jωs ω jω ω jω

A ωA jωω

ω A ωω ω fA jω A jωjω ω f

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210 310 410 610 710510

0A( )A dB

bf tf

6dB/Octave or

20dB/decade

3dB

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2.7.2 Frequency Response of Closed-loop Amplifiers

1 1

1 1

1

1

0

1

1

0

0

/ /( )( )1 ( / ) 1 ( / )

1 1

/1 ( / ) 1 ( / )

1

/

1 ( /

( ) / 1 ( / )

)1

f fo o

i if f

f

f b

f

f

bA s A

R R R Rv v sv v sR R R R

R RR R s

AR R

R

s

RA

10

0

1

where 1, ( / ) 1 ( / )

f

b fs R RA

RA

R

The inverting amplifier transfer function

36S. C. Lin, EE National Chin-Yi University of Technology

1

1

1

1

3dB1

1 /

1 /1

( )

1 /( )( ) 1

1 /

1 /

fo

i f

f t

f

o

i

t

f

R Rvv R R

A s

R Rv ssv s

R R

R R

Similarly, the noninverting amplifier transfer function:

13dB

1 1

0

0

/,

1 ( / ) 1 ( / )1

fo t t

fi f

b

b

R RvR Rv R RA

As

37S. C. Lin, EE National Chin-Yi University of Technology

2.8 Large-Signal Operation of Op AmpExample 2.7: Consider the noninverting amplifier circuit shown in Fig. below. The op amp is specified to have output saturation voltages of ±13V, And output current limits of ±20mA.

(a) Find VP =1V and RL=1k,specify the signal resulting at the output of the amplifier.

2

1

Sol:

1+ =10

10V 10mA1kΩ

the feedback current will be 10V =1mA

(9+1)kΩthe total output current is 11mA,well under its limit of 20 mA.

L

F

RGR

i

i

2 9kR

1 1kR

LR

1I

fI

iv

ov

0PV

oI LI

38S. C. Lin, EE National Chin-Yi University of Technology

(b) Find VP =1.5V and RL=1k, specify the signal resulting at the output of the amplifier.

Sol: is increased to 1.5V , will saturate at 13V

13V 13V 13mA, =1.3mA1kΩ (9+1)kΩ

= 14.3 mA, well under its limit of 20 mA.

p o

L F

o

V V

i i

i

ov

ccV

ccV

13V

15V

13V

15V

39S. C. Lin, EE National Chin-Yi University of Technology

(d) Find Vp=1V, what is the lowest value of RL for which an undistorted sine-wave output is obtained？

(max)min

min

Sol: 1.5V ,

10V 10V 20mA=9k +1k

=526 .

p

oL

L

V

iR

R

Sol: The maximum value of for undistoted sine-wave output 1.3V. The output will be a 13-V peak sine-wave. The op-amp output current at peak will be 14.3mA.

PV

(c) Find RL=1k, what is the maximum value of VP for which an undistorted sine-wave output is obtained？

40S. C. Lin, EE National Chin-Yi University of Technology

2.8.3 Slew Rate

iv

ov

iv

t

t

ov

t

ov

V

V

V

Slop V SRt

Slop SR

( )d

( )b

( )cmax

11 /

( ) (1 )t

o

o

i tω t

o

dvSRdt

vv s ω

v t V e

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2.8.4 Full-Power Bandwidth

max

max

max

ˆ sin

ˆ cos

2

The Maximum amplitude of the undistortedoutput sinusoild is given by

i i

ii

M o

Mo

Mo o

v V tdv V tdtSR V

SRfV

V V

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