Op amp output
Electronics
Negative Feedback in Operational Amplifiers
Terry Sturtevant
Wilfrid Laurier University
July 29, 2013
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
The equation for the output of an op amp is always true as long as
the output is not saturated
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
The equation for the output of an op amp is always true as long as
the output is not saturated
V+
Vo
V−
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
The equation for the output of an op amp is always true as long as
the output is not saturated
V+
Vo
V−
Vo = A (V+ − V− )
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For negative feedback, make the voltage at V− be some
portion of Vo .
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For negative feedback, make the voltage at V− be some
portion of Vo .
This ratio of V− to Vo is called the feedback factor, β.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output with negative feedback
Example: non-inverting amplifier
Op amp output
For negative feedback, make the voltage at V− be some
portion of Vo .
This ratio of V− to Vo is called the feedback factor, β.
V−
β
6
Vo
V+
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For instance, with just a wire from V− to Vo , the feedabck
factor is 1.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For instance, with just a wire from V− to Vo , the feedabck
factor is 1.
This is the case for a voltage follower.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For instance, with just a wire from V− to Vo , the feedabck
factor is 1.
This is the case for a voltage follower.
β=1
V−
Vo
V+
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For feedback
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For feedback
V− = βVo
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For feedback
V− = βVo
where
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For feedback
V− = βVo
where
β ∈ [0, 1]
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For feedback
V− = βVo
where
β ∈ [0, 1]
So the op amp equation
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For feedback
V− = βVo
where
β ∈ [0, 1]
So the op amp equation
Vo = A (V+ − V− )
becomes
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For feedback
V− = βVo
where
β ∈ [0, 1]
So the op amp equation
Vo = A (V+ − V− )
becomes
Vo = A (V+ − βVo )
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
For feedback
V− = βVo
where
β ∈ [0, 1]
So the op amp equation
Vo = A (V+ − V− )
becomes
Vo = A (V+ − βVo )
Vo = AV + − βAVo
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Vo + βAVo = AV+
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Vo + βAVo = AV+
Vo (1 + βA) = AV+
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Vo + βAVo = AV+
Vo (1 + βA) = AV+
Vo (1 + βA) = AV+
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output with negative feedback
Example: non-inverting amplifier
Op amp output
Vo + βAVo = AV+
Vo (1 + βA) = AV+
Vo (1 + βA) = AV+
Vo = V+
Terry Sturtevant
A
1 + βA
Electronics Negative Feedback in Operational Amplifiers
Op amp output with negative feedback
Example: non-inverting amplifier
Op amp output
Vo + βAVo = AV+
Vo (1 + βA) = AV+
Vo (1 + βA) = AV+
Vo = V+
A
1 + βA
Vo = V+
1
+β
Terry Sturtevant
1
A
!
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If the gain, A, is much greater than 1, then
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If the gain, A, is much greater than 1, then
Aβ
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If the gain, A, is much greater than 1, then
Aβ
and so
1
β
A
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If the gain, A, is much greater than 1, then
Aβ
and so
1
β
A
Vo = V+
Terry Sturtevant
1
!
1
A + β
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If the gain, A, is much greater than 1, then
Aβ
and so
1
β
A
Vo = V+
thus
1
!
1
A + β
1
β
Vo ≈ V+
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If the gain, A, is much greater than 1, then
Aβ
and so
1
β
A
Vo = V+
thus
1
!
1
A + β
1
β
Vo ≈ V+
so Vo only depends on β.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
1
β
Vo ≈ V+
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
1
β
Vo ≈ V+
So for a voltage follower, where β = 1,
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
1
β
Vo ≈ V+
So for a voltage follower, where β = 1,
Vo ≈ V+
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
1
β
Vo ≈ V+
So for a voltage follower, where β = 1,
Vo ≈ V+
as expected.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Non-inverting amplifier
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Non-inverting amplifier
Rf
V−
-
V+
-
Vo
Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
This can be redrawn
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
This can be redrawn
Rf
V−
V+
Vo
Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
This can be redrawn
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
This can be redrawn
Rf
V−
V+
Vo
Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Rf and Ri form a voltage divider, so the voltage at V− is a
fraction of Vo
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Rf and Ri form a voltage divider, so the voltage at V− is a
fraction of Vo
By definition, this is the feedback factor, β.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Rf and Ri form a voltage divider, so the voltage at V− is a
fraction of Vo
By definition, this is the feedback factor, β.
Thus
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Rf and Ri form a voltage divider, so the voltage at V− is a
fraction of Vo
By definition, this is the feedback factor, β.
Thus
β=
Ri
Rf +Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Rf and Ri form a voltage divider, so the voltage at V− is a
fraction of Vo
By definition, this is the feedback factor, β.
Thus
β=
Ri
Rf +Ri
so
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Rf and Ri form a voltage divider, so the voltage at V− is a
fraction of Vo
By definition, this is the feedback factor, β.
Thus
β=
Ri
Rf +Ri
so
gain =
1
β
=
Rf +Ri
Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Rf and Ri form a voltage divider, so the voltage at V− is a
fraction of Vo
By definition, this is the feedback factor, β.
Thus
β=
Ri
Rf +Ri
so
gain =
1
β
=
Rf +Ri
Ri
as expected
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
This can be redrawn
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
This can be redrawn
Rf
VB
VA
Vi
Ri Vo
R
@
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
This can be redrawn
Rf
VB
VA
Vi
Ri Vo
R
@
Vo
Vi
=
VA
VB
=1+
Rf
Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If Ri is not grounded, then it looks like this.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If Ri is not grounded, then it looks like this.
Rf
VB
Vi
VA
Ri
Vo
R
@
VC
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If Ri is not grounded, then it looks like this.
Rf
VB
Vi
VA
Ri
Vo
R
@
VC
V0 is VA - VC
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If Ri is not grounded, then it looks like this.
Rf
VB
Vi
VA
Ri
Vo
R
@
VC
V0 is VA - VC
Vi is VB - VC
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
If Ri is not grounded, then it looks like this.
Rf
VB
Vi
VA
Ri
Vo
R
@
VC
V0 is VA - VC
Vi is VB - VC
so
Vo
Vi
=
VA −VC
VB −VC
=1+
Rf
Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
We can redraw the circuit slightly differently.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
We can redraw the circuit slightly differently.
Rf
VC
Ri
VB
Terry Sturtevant
VA
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
We can redraw the circuit slightly differently.
Rf
VC
Ri
VB
Since
VA −VC
VB −VC
=1+
VA
Rf
Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output with negative feedback
Example: non-inverting amplifier
Op amp output
We can redraw the circuit slightly differently.
Rf
VC
Ri
VB
Since
VA −VC
VB −VC
=1+
VA
Rf
Ri
This can be rewritten VA − 1 +
Terry Sturtevant
Rf
Ri
VB = − RRfi VC
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
The derivation goes like this:
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
The derivation goes like this:
VA − VC
Rf
=1+
VB − VC
Ri
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
The derivation goes like this:
VA − VC
Rf
=1+
VB − VC
Ri
VA − VC = 1 +
Terry Sturtevant
Rf
Ri
(VB − VC )
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
The derivation goes like this:
VA − VC
Rf
=1+
VB − VC
Ri
VA − VC = 1 +
VA − V
C =
Rf
1+
Ri
Terry Sturtevant
Rf
Ri
(VB − VC )
VB − V
C −
Rf
VC
Ri
Electronics Negative Feedback in Operational Amplifiers
Op amp output with negative feedback
Example: non-inverting amplifier
Op amp output
The derivation goes like this:
VA − VC
Rf
=1+
VB − VC
Ri
Rf
Ri
VA − VC = 1 +
VA − V
C =
Rf
1+
Ri
Rf
VA − 1 +
Ri
Terry Sturtevant
(VB − VC )
VB − V
C −
VB = −
Rf
VC
Ri
Rf
VC
Ri
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Finally, we can set VB to ground.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Finally, we can set VB to ground.
Rf
VC
Ri
VA
VB = 0
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Finally, we can set VB to ground.
Rf
VC
Ri
VA
VB = 0
So if VA − 1 +
Rf
Ri
VB = − RRfi VC
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Finally, we can set VB to ground.
Rf
VC
Ri
VA
VB = 0
So if VA − 1 +
Rf
Ri
VB = − RRfi VC
and VB = 0
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Finally, we can set VB to ground.
Rf
VC
Ri
VA
VB = 0
So if VA − 1 +
Rf
Ri
VB = − RRfi VC
and VB = 0
Then VA = − RRfi VC
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Finally, we can set VB to ground.
Rf
VC
Ri
VA
VB = 0
So if VA − 1 +
Rf
Ri
VB = − RRfi VC
and VB = 0
Then VA = − RRfi VC which is an inverting amplifier!
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers
Op amp output
Op amp output with negative feedback
Example: non-inverting amplifier
Finally, we can set VB to ground.
Rf
VC
Ri
VA
VB = 0
So if VA − 1 +
Rf
Ri
VB = − RRfi VC
and VB = 0
Then VA = − RRfi VC which is an inverting amplifier!
So the feedback factor, β, is the same as for a non-inverting
amplifier.
Terry Sturtevant
Electronics Negative Feedback in Operational Amplifiers