Operational Amplifiers

(revision)+VCC (15V)

-VCC (-15V)

output

non-inverting

input

inverting input

+

-

IMPORTANT: Basic assumptions are

• very high voltage gain (2 x 105)

ideal is infinite

• very high input impedance (2 Mohms)

ideal is infinite

•Also an ideal op.amp has infinite bandwidth

( roll-off starts at 10 Hz)

“741” Op. Amp.

top view

1

2

3

4 5

6

7

8offset null

inverting

input

non-inverting

input

-VCC

(-15V)

+VCC

(+15V)

output

offset null

-

+

Consequences of basic

assumptions are:

1. Vout = A(V+ - V-)

or V+ - V- = Vout/A

which means that if A is large

then V+ ≈ V-

The voltage difference at the

inputs tends to be very small

If one of the op. amp. inputs is

earthed then the other input is a

VIRTUAL EARTH.

2. The consequence is that the

inputs draw no (little) current

Op. Amp stability with

feedback

Amplifier

Ao

Feedback

block β

XiXia

-

Xo

Xf

Xf = βXo

Xia = Xi - Xf (negative feedback)

Xo = AoXia

∴Xia = Xi - βXo

∴X0 = AoXi -A0βXo

βo

o

i

o

A

A

X

X

+=

1

What does the feedback equation

approximate to if Ao β is much

greater than 1?

If Aoβ >> 1 then

β

1≅

i

o

X

X

Definitions

• Open-loop gain: The voltage gain

of the op amp without feedback.

• Feedback factor (β): The fraction

of the output voltage fed back to the

input by a (negative) feedback

network.

• Loop gain (Aoβ): The product of

the open-loop gain and the feedback

factor.

• Closed-loop gain (Afb): The

voltage gain of an op amp with

feedback

Ideal case:

ββ

1

1≅

+==∴

∞→

o

o

in

out

fb

o

A

A

V

VA

,A

Loop-gain crossing frequency (fCL):

The frequency at which the closed loop-gain is

equal to the open-loop gain.

Oscillations

An op amp is usually used with

feedback. Negative feedback

gives several advantages:

• increased input impedance

• reduced output impedance

• reduced distortion

• increased stability

(Positive feedback may give

oscillations).

With negative feedback an op

amp circuit may generate an

oscillatory signal at the

frequency at which it has a

phase angle of -180o.

The circuit will then generate an

a.c. output with no input (it

draws dc power from the power

supply).

An op amp negative feedback

circuit will oscillate if there

exists a frequency at which the

magnitude of the loop gain is

greater than unity at a phase-

shift of -180o.

Sinusoidal oscillator

using positive feedback

Use a positive feedback loop

containing a frequency

selective network.

The loop is designed to have a

unity gain at a single frequency

determined by the selective

network.

To determine if a circuit will

oscillate:

(a) Find the magnitude of the

loop-gain at which the phase

angle is -180o

Or

(b) Find the value of the phase angle at

which the magnitude of the loop-gain

is unity.

Phase-margin: The difference

between the phase shift of a signal

through a system and the phase shift

that will cause the system to oscillate.

(usually an extra -180o).

Positive feedback

Ao

β

+Vin Vout

Note that in practice no input is required

for oscillations)

βo

o

fbA

AA

−=

1

Loop-gainNote - sign

At a specific frequency, ωo ,the loop-gain is

unity and Afb is infinite.

At this frequency the circuit will have a finite

output for no input and is by definition an

oscillator.

Wien Bridge Oscillator

Without amplitude stabilisation)

R1

R2

-

+

C R

RC

Vo

Vin

Zs

Zp

Op amp is in non-inverting configuration.

Closed-loop gain is (1 + R2/R1)

Feedback network transfer function is :

( )( )

( )

( )

−+

+

=∴

++

+

=∴

+=

RCRCj

R

R

jL

RCsRCs

R

R

sL

ZZ

Z

sV

sV

sp

p

o

i

ωω

ω1

3

1

13

1

1

2

1

2

Hence loop-gain is a real number (phase is zero)

at ω = 1/RC.

To obtain sustained oscillations at this frequency

set the magnitude of L(jω) to unity by selecting

R2 = 2R1

In practice start oscillations by choosing R2/R1

to be slightly greater than 2.

Note that these oscillations have no amplitude

control.

For oscillations:

L(jωo) = Ao (jωo) β (jωo) = 1

At ωo the phase of the loop-gain is zero, and its

magnitude is 1.

This criterion should be satisfied at one frequency,

ωo, only.

ωo is determined solely by the phase characteristic

of the feedback loop.

If Aoβ becomes less than unity oscillations cease

and if Aoβ becomes greater than unity oscillations

will grow in amplitude.

A non-linear circuit is required for gain control.

Amplitude control

-

+

R2 R

1a b

Vo

R2 is adjusted until oscillations just start to grow.

As oscillations grow the diodes conduct causing

the effective resistance between a and b to decrease.

Equilibrium will be reached at the output amplitude

that causes the loop-gain to be exactly unity.

The output amplitude can be varied by adjusting

R2.

Note that the output at a has lower distortion than at

b.

To ensure that oscillations start design

the circuit so that Aoβ is slightly greater

than 1.

Turn power on and oscillations grow.

At the desired output level the non-linear

network comes into action and causes the

loop-gain to be reduced to exactly 1.

This gives sustained oscillations at the

required amplitude.

If the loop-gain is reduced below 1 the

amplitude of the output will diminish -

which is detected by the non-linear

network which will cause the

loop-gain to increase to exactly 1.

RC Phase-shift oscillator

-

+

C CC

R R R

R f

R1

Inverting configuration gives 180o phase shift.

The three R-C sections give 180o phase shift at

a particular frequency.

Hence oscillations occur at this frequency

if the loop-gain is exactly 1.

Feedback ratio

−+

−

=

C

R

Cj

C

RR

R

ωωω

β2

3322

3

3

615

For 180o phase-shift,

RC

C

R

C

6

1

61 2

33

=⇒

=

ω

ωω

giving

29

1−=β

(the minus sign confirms that the cascade

inverts the feedback at the oscillation frequency)

For unity loop-gain

291

=R

Rf

In practice Rf is made adjustable to allow for

small differences in component values, and to allow

for the loading caused by R1.