Operational Amplifiers

• The three rules of the op amp

• The inverting and the non-inverting configuration.

• The (weighted) summer.

• The difference amplifier. The CMRR (common mode rejection ratio).

• An instrumentation amplifier.

• Frequency effects on open and closed loop amplifiers. The gain-bandwidth theorem.

• The real op amp. Large signal limitations and DC imperfections.

• Integrators and differentiators.

Operational Amplifier is a device that can amplify the difference

between two input signals. Among its pins, there are two signal

inputs (the inverting and the non inverting input), a ground, an output

pin and two dc power supply pins that provide the necessary power

for the op amp IC chip to work.

-

+

V+

1υ

2υ( )12 υυυ −= Ao

V−

Inverting input

Non inverting input The amplification factor A

(or open loop gain) depends

highly on temperature, varies a

lot from op amp to op amp and

depends a lot on frequency.

-

+

V+

1υ

2υ ( )12 υυυ −= Ao

V−

The open loop op amp acts as a voltage

(differential) amplifier. It has huge input and

tiny output resistances.

It can be modeled as:

inR

+

-1υ

2υ

+

-

( )12 υυυ −= Ao

When op amps are being employed in

circuits they are being used almost

exclusively with negative feedback

(closed loops). The gain of the

configuration is reduced but the syste

is much more predictable, stable and

with improved bandwidth.

The closed loop op amp.

When op amps are being employed in circuits they are being used

almost exclusively with negative feedback (closed loops). The gain

of the configuration is reduced but the system is much more

predictable, stable and with improved bandwidth.

Negative feed back means that a portion of the output is being fed

back at the inverting input of the op amp.

When we analyze circuits employing closed loop op amps we are

using the three (approximate) rules of the ideal op amp.

The rules of an ideal op-amp

connected in a closed loop.

1) The voltage gain A of the ideal open loop op-amp is infinitely large.

2) The current through the ideal op-amp is zero. That is, the ideal

op-amp has infinite input resistance.

3) Both terminals of the ideal op-amp are at the same voltage.

(Consequence of rule #2)

The inverting amplifier.

oυiυ

i

2R

1R i

virtual ground

1

2

iR

iR

i

o

=

=−

υ

υA

R

R

i

o =−=1

2

υ

υ

-

+

Input resistance:

1RRiR inini =⇒=υ

may be too small

output resistance 0≈outR

An improved inverting amplifier.

oυiυ

i

2R

1R i

-

+

Input resistance:

1RRin =

can be chosen to be large

4R

3R

++−=

3

4

2

4

1

2 1R

R

R

R

R

R

i

o

υ

υ

amplifier gain:

Choose R1=R

2=R

4=R as large as possible to maximize the input

resistance and minimize its effects on the closed loop voltage gain.

Choose R3

as small as possible to obtain the desired voltage gain.

An improved inverting amplifier.

oυiυ

i

2R

1R i

-

+

4R3R

3ii −

3i

++−=

⇒

++−=⇒−+=−

−==

3

4

2

4

1

2

3

4242422

221

1

)(

,

R

R

R

R

R

R

R

RRRRiRiiiR

iRRiiR

io

oo

i

υυ

υυ

υ

The weighted summer.

oυ

1υfR

1R

-

+

2υ

2R

NυNR

N

N

fff

oR

R

R

R

R

Rυυυυ −−−−= ....2

2

1

1

Effects of finite open loop gain A

on the inverting amplifier.

oυiυ

i

2R

1R i

-

+A

0υ−

A

R

R

R

R

Gi

o

1

2

1

2

1

1

+

+

−==υ

υ

Since1

21R

RA +>>

1

2

R

RG

i

o −≈=υ

υ

The non-inverting amplifier.

oυiυ

2R

1R i

same

voltage ( )21

2

RRi

iR

o

io

+=

=−

υ

υυ

AR

R

i

o =+=1

21υ

υ

-

+

The input resistance

is infinite and the

output resistance is 0.

Effects of finite open loop gain A

on the non-inverting amplifier.oυ

iυ

2R

1R i

-

+

Ai

0υυ −

A

R

R

R

R

Gi

o

1

2

1

2

1

1

1

+

+

+

==υ

υ

Since1

21R

RA +>>

1

21R

RG

i

o +≈=υ

υ

The voltage follower.

iυ

-

+1021 =⇒== GRR

iυ

The voltage follower has infinite input resistance 0 output resistance

and voltage gain unity (the output voltage is equal, follows the input

voltage). It is used as a “buffer” a “matching” element between

different stages of electronic instrumentation.

The difference amplifier.An amplifier that responds to the difference υ

d=υ

2-υ

1 between two

input signals υ1

and υ2. The output voltage υ

o(response) of the

amplifier can be written as:

cmcmddo AA υυυ +=

2

12 υυυ

+=cm

12 υυυ −=dwith the difference input signal

and the common mode input signal

Acm

is the common mode gain and ideally must be 0

Ad

is the difference gain and ideally must be very large.

The common mode rejection ratio CMRR of a difference amplifier

(ideally infinite) is defined as:

( )dBinA

ACMRR

cm

dlog20=

The difference amplifier.

oυ1υ

2R

1R

-

+2υ

4R

3R

43

42

RR

R

+υ

( )( ) 1

21

431

2142

R

R

RRR

RRRo υυυ −

+

+=

straightforward analysis gives:

12 υυυ −=d

2

12 υυυ

+=cm

define

and therefore:2

,2

12d

cmd

cm

υυυ

υυυ −=+=

The output voltage now can be expressed as

a function of the common mode and

difference voltages

The difference amplifier.

oυ1υ

2R

1R

-

+2υ

4R

3R

( )( )

( )( )

−

+

++

+

++=

1

2

431

214

431

214

1

2

2 R

R

RRR

RRR

RRR

RRR

R

Rcm

d

o υυ

υ

By choosing R1

= R3

and R2

= R4

the common mode gain becomes 0.

The difference gain becomes R2/R

1

The difference amplifier.2R

1R

-

+dυ

1

2

R

Rdo υυ =

2R

1R

ii

11 22 RRRi ind =⇒=υ

The input resistance of this difference amplifier

is relatively low (since we want high difference

gain). The instrumentation amplifier is a superior

difference amplifier with infinite input resistance.

Instrumentation amplifier.

oυ

1υ

-

+

2υ

4R

3R

-

+

2R

2R12R-

+

3R

doR

R

R

Rυυ

+=

1

2

3

4 1

A major advantage of this configuration is that the first stage op amps

do not amplify the common mode signal therefore are not prone to

saturation. It is a superior difference amplifier.

4R

1i1i

1i

2i

3i

2i

3i

1

21

2R

Rdυυ −

1

22

2R

Rdυυ +

Frequency dependence of open

loop op amps|)(|log20 fA

f

dB3

Hzfb 10=

The open loop gain of an op amp has a very strong

dependence on frequency. It reaches its 3dB

attenuation very quickly at 10 Hz very low value.

bf

fj

AfA

+

=

1

)( 0

We can model this behavior with the function:

A0

is the dc open loop gain and fb

is the 3 dB frequency. The product

ft=A

0 fb

is called the “unity-gain bandwidth” and it is specified in the

data sheet by the manufacturer of the op amp.

Frequency dependence of closed

loop op amps

Let’s examine the following question. How does the external feedback

affect the frequency response of the gain of an inverting amplifier?

+

++

+

−=

+

+

−=

1

2

0

1

2

1

2

1

2

1

2

1

11

)(

1

1

)(

RR

f

fj

A

RR

RR

fA

R

R

R

R

fG

t

where A0

is very large

The gain bandwidth theorem

Since is A0

very large we can write

+

+

−≈

1

2

1

2

1

1

)(

RR

f

fj

RR

fG

t

We see that the new 3dB frequency of the closed loop is:

( ) .1

13

1

2

3 constfKf

R

R

ff tdB

t

dB ==+⇒

+

=This is the gain bandwidth

theorem. It can also be

proven for a non inverting

amplifier.

Frequency response of closed loop

op amps

|)(|log20 fG

f

dB3

dBf3

Amplifier bandwidth

io fG υυ |)(|=

Op Amp large signal limitations.

Output voltage saturation. The output voltage of an op amp cannot be larger than a specified

value, the rated output voltage of the op amp.

Maximum output current. The output current supplied by an op amp is limited to a specified

maximum value. (about 20 mA for an741C op amp).

Slew Rate (SR).There is a maximum rate of change of the output voltage that an op

amp can handle. It is called the slew rate (SR) of the op amp.

max

=

dt

dSR oυ

Op Amp full power bandwidth

For a sinusoidial input signal to an op amp. )sin()( tVt ii ωυ =

)cos()( tVtdt

di

i ωωυ

=

The full power bandwidth fM

of an op amp is defined by the relationship:

SR=ωM

Vomax

=2π fM

Vomax

where Vomax

is the rated output voltage of the op amp (the maximum

output voltage the op amp can deliver without clipping).

The physical meaning of the op amp’s full power bandwidth fM

is the

maximum frequency a harmonic input to the op amp can have when the

output amplitude is equal to the op amps rated voltage. If the frequency

is increased the output signal is distorted due to slew rate limitation of

the op amp. The full power bandwidth fM

of an op amp is specified by

the manufacturer in the date sheet of the device.

DC imperfections of an op amp

Offset voltageIf the op amp is supplied with two equal voltages at its inputs the

output voltage is not zero. The non ideal op amp is equivalent to

an ideal op amp with a small dc offset voltage connected to one

of its terminals.

Input bias and offset currents.In order for a non ideal op amp to operate its two input terminals

are supplied with two small input bias currents IB1

and IB2

Op Amp Offset Voltage

In order to account for the offset voltage effect a non ideal op amp

can be modeled as shown

-

++-

VOS

The small dc voltage at the positive

terminal of the op amp is called the

“offset voltage” of the op amp and

it is specified by the manufacturer.

Op Amp Offset Voltage

The offset voltage of the op amp appears amplified at its output.

Consider the inverting amplifier below where both its terminals

are connected to ground. What is its output voltage?

-

++-

VOS

2R

1R

+=

1

21R

RVOSoυ

The offset dc voltage of the op amp

appears amplified at the output.

Op Amp Offset Voltage

By connecting a capacitor at the input of the inverting amplifier

the offset voltage appears with unity gain at the output.

Useful, but only if we want to amplify ac signals.

-

++-

VOS

2R

1RC

OSOSo VR

V =

∞+= 21υ

Op Amp Offset VoltageSome op amps have two additional terminals to which a specified

circuit should be connected to trim the output voltage due to VOS

1V+

2V−

Op Amp input bias and offset currents

In order for a non ideal op amp to operate its two input terminals

are supplied with two small input bias currents IB1

and IB2

Input offset current 21 BBOS III −=

Input bias current

2

21 BB

B

III

+=

typical value 10 nA

typical value 100 nA

A non ideal op amp with input bias and offset currents is modeled as

2BI

1BI

3

1

2

Effect of op amp input offset and bias

currents to a closed loop amplifier

2BI

1BI

-

2R

1R

+

0 V

1BI

221 RIRI BBo ≈=υ

By introducing the appropriate resistance R3

to the non inverting input

of the op amp we can fix the dc output voltage due to the bias current

Effect of op amp input offset and bias

currents to a closed loop amplifier

2BI

1BI

-

2R

1R

+

We can get a more meaningful

expression for the output voltage

if we do the substitution:

2BI

3R

32 RI B−

1

32

R

RI B

1

3

21R

RII BB −

+−=

1

23221 1

R

RRIRI BBoυ

21

OS

BB

III +=

22

OSBB

III −=

Effect of op amp input offset and bias

currents to a closed loop amplifier

2BI

1BI

-

2R

1R

+2BI

3R

32 RI B−

1

32

R

RI B

1

3

21R

RII BB −

( )

( )

+++

+

+−=

21

2132

21

2132

12

1

RR

RRRRI

RR

RRRRI

OS

Boυ

By choosing21

21

3RR

RRR

+= we get 2RIOSo ≈υ

Op Amp differentiators

-

++-

VOS

R

C

oυ

iυ( )

( ) ( )tdt

dRCt

dt

dqRtiR

C

q

i

O

oo

i

υυ

υυ

υ

−=

⇒−=⇒=−

=

Op Amp integrators

-

+

R

C ( ) ( )

( ) ( ) tdtRC

tC

q

tdtR

tqdt

dq

RI

t

ioo

t

ii

′′−=⇒=−

′′=⇒==

∫

∫

0

0

1

1

υυυ

υυ

oυiυ

Effect of op-amp offset voltage

on integrators

-

++-

VOS

R

CV

OS

oυI

I

C

qV

dt

dqI

IRV

OSo

OS

=−

=

=

υ

+=

=⇒=

RC

tV

tR

Vq

R

V

dt

dq

OSO

OSOS

1υ

Because of the offset voltage the integrator

will saturate soon after its power on.

A remedy for the offset voltage

effect on integrators

-

++-

VOS

R

CV

OS

oυI

FR

1I

FIThe presence of a small R

Fwill

limit the output voltage at an

asymptotic value a little larger

than VOS

.

Effect of op-amp offset current on

integrators

-

+I

B2

R

C0

oυI

B1

IB1

tC

I

C

q

tIqIdt

dq

BO

BB

1

11

==

=⇒=

υ

Remedy of op-amp offset current

effect on integrators

-

+

IB2

R

C

-IB2

R

oυI

B1

IB2

-IB1

R

IB2