The feedback – technique used to influence the behavior and

properties of a system by the output quantity of the system

the effect produced by a certain cause influences the consequent

action of that cause

fed back to the input a fraction of the output quantity

Feedback circuits

Two types of feedback:

• negative feedback (degenerative) NF. The signal fed back from

the output reduces the effect of the input signal; stabilizing effect

• positive feedback (regenerative) PF. The signal fed back from the

output intensifies the effect of the input signal; leads to instability

General structure of the feedback circuit

xs – source signal, fed from an external signal source;

xo – output signal, fed to the load as well as to the feedback network;

xr – feedback signal, provided by the feedback network

xi – input signal to the basic amplifier, provided by the summing element.

xs – voltage or current;

xo – voltage or current;

• signal flow

diagram

• signals

Feedback

circuit

General structure of the feedback circuit – cont.

• signal flow

diagram

• transmittances

• transmittance of the basic amplifier :

• transmittance of the feedback network:

• transmittance of the feedback circuit:

i

o

x

xa =

o

r

x

xr =

s

o

x

xA =

gain of the basic amplifier; open loop gain

gain of the feedback amplifier

feedback factor; transmittance of the reverse path

Feedback

circuit

Feedback types

Negative

feedback

rsi xxx −=

rsi xxx +=

Positive

feedback

0ar

0ar

5/22

Equations of the ideal feedback

Ideal feedback:

• each block is unilateral (one way signal transmision)

• the r block does not load the output of the a block; that is,

connecting the r block does not change the value of a.

• assume that the source and the load resistance have been included

inside the a circuit

Negative

feedback

rsi xxx −=

0ar 0ar

Negative feedback

rsi vvv −=

0ar

Fundamental

equation of the NF

ar – loop gain

1 + ar amount of feedback

ar > 0;

1+ar > 1

If ar >> 1 (eg. for op amp )→a

The gain of the feedback amplifier is (almost) entirely determined

by the feedback network: accurate, predictable and stable gain.

Negative feedback - cont.

The denominator

is always > 1

0ar

Illustration

Negative feedback: - reduces the gain (of the basic amplifier)

- improves some amplifier properties (stabilizes

the gain, reduces nonlinear distortion, improves the input and output

impedances, extends the bandwidth).

a = 1000, r = 0.009

ar =?

1+ar =?

A=?

a = 10000, r = 0.009

90009.010000 ==ar

919011 =+=+ ar

109.8991

10000

ar1

aA =

+=

111.10.009

1

r

1A == 111.1

0.009

1

r

1A ==

Positive feedback

rsi xxx +=

ar

aA

−=

1

rsi vvv +=

ar > 0; 0 < 1-ar < 1

Amplifier in the

linear domain

aA

But the amplifier

performances get worse

Special case of PF:

→==− Aarar ;1;0)1(

so Axx = xo finite only if xs=0

It is used for sinusoidal oscillator with PF (signal generator)

0 >1-ar; ar>1

• still have PF, but the linear theory can not be used anymore for

circuit analysis;

• specific analyzing method will be used

• multivibrator circuits, bistable circuits

Positive feedback

Feedback topologies input and output signals: voltage and/or currents:

four feedback topologies

• the loop gain ar is dimensionless

• voltage – voltage (series - parallel)

• voltage - current (series - series)

• current - voltage (parallel - parallel)

• current – current (parallel - series)

At the input: in accordance with the method of summing up the signals

voltage: series connected (in a loop)

current: parallel connected (in a node)

At the output: in accordance with the way of taking (measuring) the signals

voltage: in parallel (measurement with the voltmeter)

current: in series (measurement with the ammeter)

Analysis of the feedback amplifier

Ideal feedback:

there are no restrictions on the basic amplifier

the other components of the circuit are made ideal

the feedback network – a controlled source

the signal source is supposed to be ideal (it is assumed that

the source resistance is included inside the basic amplifier)

the load resistance does not load the output of the amplifier

(it is assumed that the load resistance is included inside the

basic amplifier)

Real feedback:

the feedback network loads the input and output of the basic

amplifier

the circuit with real feedback ideal feedback

Ideal feedback, voltage - voltage topology

A=?

Rir=?

Ror=?

Equivalent model

of the basic

amplifier

ar

RR o

or+

=1

)1( arRR iir +=ar

aA

+=

1

Equivalent model

of the feedback

amplifier

+VPS

-VPS

Illustration 741 type op amp.

Typical values of its parameters:

Ω50M;2;000200 === oi RRa

What are the values for the

gain, input and output

resistances for the

feedback amplifier?Solution

091,0101

1

21

1 +

=+

=RR

R

v

vr=

o

r

988.1018201

200000

1==

+ar

aA=

111

1011

1

1

2 =+=+=R

R

rA

20118091.000020011 =+=+ ar

Feedback network:

Illustration – cont.

36.4GΩ36402MΩ18201MΩ2)1( ===+= arRR iir

Compared with the values of resistances in the circuit (kΩ)

one can say that Rir → ∞

2.7mΩ0.0027Ω18201

Ω50

1===

+=

ar

RR o

or

Compared with the values of resistances in the circuit (kΩ)

one can say that Ror → 0

Using NF:

The properties of the feedback amplifier are modified in

respect with the properties of the basic amplifier:

the gain is reduced

the gain sensitivity is reduced

the nonlinear distortion is reduced

the bandwidth is widening

the effect of the noise can be reduced

the input and output impedances are improved

Effects of the negative feedback on the amplifier

Reduction of the gain

ar

aA

+=

1

aA

0ar

If ar >> 1 (e.g. for op amp )→a

Reduction of the nonlinear distortion

Generally, the gain is linear for small-signal, around the operating point

If the output signal increases, the gain becomes nonlinear

ra

aA

1

11

1+=

ra

aA

2

22

1+=

basic amplifier

feedback amplifier

The range of the input signal increases

basic amplifier NF amplifier

];[ 22 ii xx− ];[ 22 ss xx−

1001 =a 502 =a 09,0=r

1009,01001

1001 =

+=A

1,909,0501

502 =

+=A

%91

12 =−

A

AA%50

1

12 =−

a

aa

1

21

21

21

1

1

a

aa

raA

AA −

+=

−

Illustration

Price: reduction

of the gain

Solution: use a

preamplifier

Increases the input signal

range for linear behaviour

Frequency response

Hf

fj

aja

+

=

1

)( 0

Suppose a first order low-pass-type basic amplifier

fRCj

a

RCj

aja

211)( 00

+=

+=

RCf H

2

1=

Typical for op-amp

Around the low-pass-type basic amplifier apply a pure

resistive negative feedback path

The pure resistive feedback (frequency independent

behavior) cannot change the type of feedback amplifier

low-pass-type

Hrf

fj

AjA

+

=

1

)( 0ra

aA

0

00

1+=?0 =A

?=Hrf )1( 0raff HHr +=

Demonstration? ...)(1

)()( =

+=

jar

jajA

basic amplifier

NF amplifier

Widening the bandwidth

)1(1

1)(

0

0

0

raf

fj

ra

a

jA

H ++

+=

Hrf

fj

AjA

+

=

1

)( 0

rja

jajA

)(1

)()(

+=

Low-pass-type amplifier

ra

aA

0

00

1+= )1( 0raff HHr +=

HH f

fj

a

j

aja

+

=

+

=

11

)( 00

basic amplifier

NF amplifier

ra

aA

0

00

1+=

)1( 0raff HHr +=

log 𝐴0 = log𝑎0

1 + 𝑎0𝑟= log 𝑎0 − log(1 + 𝑎0𝑟)

log 𝑓𝐻𝑟 = log 𝑓𝐻(1 + 𝑎0𝑟) = log 𝑓𝐻 + log(1 + 𝑎0𝑟)

log(1 + 𝑎0𝑟)

log(1+𝑎0𝑟)

log 𝑓𝐻

log 𝑓𝐻𝑟

log𝐴0

log𝑎0

Widening the bandwidthLow-pass-type amplifier

basic amplifier

NF amplifier

log(1 + 𝑎0𝑟)

log(1+𝑎0𝑟)

log 𝑓𝐻

log 𝑓𝐻𝑟

log𝐴0

log𝑎0

Consequence: BaBA r 00 = if the high-frequency response is

characterized by a single pole

By introducing NF, the number of times that reduces the gain equals

the number of times that extends the bandwidth.

Gain bandwidth product (GBW) is constant

ra

aA

0

00

1+=

)1( 0raff HHr +=

𝐵𝐵𝑟

BraBr )1( 0+=

Improvement of the input and output resistances

input output

voltage

current

ar

RR i

ir+

=1

ar

RR o

or+

=1

)1( arRR iir +=

)1( arRR oor +=

voltage

current

1 2

3 4

a) What is the relation between vs, vr and vi ?

b) What are the expression and value of the feedback gain, A?

c) What is the feedback topology?

d) If vs(t)=50sinωt[mV] what are the expressions for vo(t), vr(t) and vi(t)?

Plot them.

e) If the input and output resistances of the basic amplifier are Ri=20KΩ,

Ro=5KΩ, compute the input and output resistances of the feedback

amplifier.

f) What is the bandwidth of the feedback amplifier if the basic amplifier

is low-pass-type with 16kHz bandwidth?

Exercise

For the block diagram of a

negative feedback amplifier

we know: a =1000, r = 0.099.