Download 1E6 Electricity and Magnetism

5. The Principle of Negative Feedback
Concept:
It has been seen that the gain of a simple transistor amplifier varies considerably
because of the tolerances associated with the manufacturing process. The variation
is further exasperated with changes in operating temperature. Often, the variation
also applies to input and output resistance and other properties of the amplifier.
This makes it impossible to design amplifiers with any precision unless a means of
overcoming the effects of these variations is found. This comes in the form of the
principle of negative feedback.
+
input
signal
sourc
e
Vi
Σ
_
AO = VO / Ve
error
signal Ve
output
signal
β =Vf /VO
Vf
feedback
signal
Fig. 1 General Structure of a Negative Feedback Amplifier
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VO
A block diagram of the general structure of a feedback amplifier is shown in Fig. 1.
This consists of an amplifier having a voltage gain, AO , as before. The amplifier
will be taken as having a very high input resistance and a very low output
resistance so that the effects of these can be neglected.
In this structure the input signal is not applied directly to the input of the amplifier
but to a summing unit, as shown. The output signal of the amplifier is fed into a
feedback unit. This unit simply senses the output signal of the amplifier and
generates a feedback signal which is a fraction, β, of the output ( β is taken as real
and < 1), i.e. Vf = βVO. The feedback signal is then fed into a second input of the
summing unit where it is first inverted, so that it is actually subtracted from the
input signal in the summing unit. This is why the feedback is called negative. The
error signal, Ve , at the output of the summing unit is then the difference between
the input and feedback signals. This error signal is then fed into the amplifier
where it is amplified by the gain, AO , to generate the output signal, VO.
Consider changes in the amplifier properties such that the gain varies.
If the input signal level is constant but the gain of the amplifier increases, then the
output signal level will tend to increase. This in turn means that the feedback signal
level will increase proportionately. The increased feedback signal level is then
subtracted in the summing unit from the constant input signal level to give a
reduced error signal. This reduced error signal appears at the input of the
amplifier and when passed through the amplifier will tend to reduce the output
signal level to counteract its original tendency to increase.
Alternatively, if the input signal level is constant but the gain of the amplifier
decreases, then the output signal level will tend to decrease. This in turn means that
the feedback signal level will decrease proportionately. The reduced feedback
signal level is then subtracted in the summing unit from the constant input signal
level to give an increased error signal. This increased error signal appears at the
input of the amplifier and when passed through the amplifier will tend to increase
the output signal level to counteract its original tendency to decrease.
Consequently the change in output signal level caused by the variation in the gain
of the amplifier is much diminished compared to that which exists when no
negative feedback is present. Therefore, negative feedback reduces the effects of
parameters variations in the amplifier.
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Analysis:
For the amplifier:
VO  AO Ve
But the error signal is:
Ve  Vi  Vf
Substituting:
VO  AO Vi  Vf 
But the feedback signal is:
Vf  VO
Then:
so that:
VO  AO Vi  VO 
VO  A O Vi  A O VO
VO 1  A O    A O Vi
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Then the voltage gain of the feedback amplifier is:
VO
AO
AV 

Vi 1  A O  
If
A O   1
then:
VO
AO
1
AV 


Vi
AO  
This means that when negative feedback is used the actual voltage gain of the
feedback amplifier becomes independent of the voltage gain of the amplifier itself,
AO , and depends only on the feedback factor, β , which is determined by external
components. If external components which have low manufacturing tolerances and
temperature coefficients, such as good quality resistors, are use to determine the
feedback factor, β, then precise amplifier gain can be accomplished.
Definitions:
Open Loop Voltage Gain:
Closed Loop Voltage Gain:
The Feedback Factor:
AO 
VO
Ve
AV 

VO
Vi
Vf
VO
Loop Gain:
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AO 
The Operational Amplifier:
The operational amplifier is an integrated amplifier which is manufactured
specifically for the purposes of designing negative feedback amplifiers as shown in
Fig. 2. It is a differential amplifier having two input terminals so that two separate
input voltages can be applied to it.
+
Ve
AO
_
V+
VO = AOVe
-
V
Fig. 2 An Operational Amplifier
The ‘positive’ terminal is a non-inverting input so that it influences the output in
the same sense, i.e. an increase in the voltage V+ causes an increase in the output
voltage. The ‘negative’ terminal is an inverting input so that it influences the
output in the opposite sense, i.e. an increase in the voltage V- causes a decrease in
the output voltage.
The output voltage is therefore the open-loop gain times the differential voltage
seen between the non-inverting and inverting inputs, i.e. VO = AO ( V+ - V- ).
This means that the subtracting operation of the negative feedback amplifier can
be performed by the operational amplifier if the feedback, βVO , is applied to the
inverting input of the op-amp, as shown in Fig. 2.
The open-loop, AO , is deliberately made extremely high in order that the loop-gain,
AOβ , can be made much greater than unity for finite and even small values of β.
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error
signal Ve
+
AO = VO / Ve
_
input
signal
sourc
e
Vi
output
signal
β =Vf /VO
VO
Vf
feedback
signal
Fig. 3 A Negative Feedback Amplifier Composed of an Operational Amplifier
Precisely the same analysis of the closed loop voltage gain as carried out above can
be applied to the op-amp based circuit with exactly the same result:
VO
AO
1
AV 


Vi
AO  
Consider another analysis on the circuit of Fig. 3:
Ve  V   V Then substituting:
Ve  Vi  Vf
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With:
Vf  VO
Ve  Vi  VO
But taking:
AO
VO 
Vi
1  A O  
Then
AO
Ve  Vi  
Vi
1  AO  

AO 
Ve  Vi 1  

1  AO  

If AOβ >> 1
 AO  
Ve  Vi 1 
  Vi 1  1  0
 AO  
This means that if the loop gain is very high the differential error signal present
between the terminals of the op-amp, Ve= V+ - V- → 0. This principle can be used in
later circuit analysis of op-amp applications.
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