6.002
CIRCUITS AND
ELECTRONICS
Operational Amplifier Circuits
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20
Review
„
Operational amplifier abstraction
+
 ∞ input resistance
–
 Gain “A” very large
 0 output resistance
„
Building block for analog systems
„
We will see these examples:
Digital-to-analog converters
Filters
Clock generators
Amplifiers
Adders
Integrators & Differentiators
Reading: Chapter 15.5 & 15.6 of A & L.
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20
Consider this circuit:
R2
i
i
R1
v2 +
–
v1 +
–
R2
+
v = v1
R1 + R2
≈ v−
v2 − v −
i=
R1
R1
v− –
v+ +
R2
+
vOUT
–
vOUT = v − − iR2
v2 − v −
=v −
⋅ R2
R1
−
R2
⎡ R2 ⎤
= v ⎢1 + ⎥ − v2
R1
⎣ R1 ⎦
−
R2
R1 + R2
R2
= v1
⋅
− v2
R1 + R2
R1
R1
=
R2
(v1 − v2 )
R1
subtracts!
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20
Another way of solving —
use superposition
v1 → 0
v2 → 0
R1
R2
R1
v1 +
–
–
v2 +
–
v+ +
R2
–
vOUT2
+
vOUT2
R2
R1
R1 || R2
R2
= − v2
R1
vOUT1
vOUT1
R1 + R2
=v ⋅
R1
+
v1 ⋅ R2 R1 + R2
=
⋅
R1 + R2
R1
= v1
vOUT = vOUT1 + vOUT2
R2
= (v1 − v2 )
R1
R2
R1
Still subtracts!
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20
Let’s build an intergrator…
vI +
–
+
vO
–
∫ dt
Let’s start with the following insight:
i
+
i +
–
C
vO
–
t
1
vO = ∫ i dt
C −∞
vO is related to ∫ i dt
But we need to somehow convert
voltage vI to current.
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20
First try… use resistor
+ vR –
vI +
–
i
+
R
C
vO
vI
→i
R
–
But, vO must be very small compared
to vR, or else
vI
i≠
R
When is vO small compared to vR ?
dv
larger the RC,
RC O + vO = vI
dt
smaller the vO
vR
dvO
when RC
>> vO
for good
dt
integrator
dvO
≈ vI
RC
ωRC >> 1
dt
t
1
or
vO ≈
vI dt
∫
Demo
RC −∞
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20
There’s a better way…
i
Notice
i
–
+
v − ≈ 0V under negative feedback
vI
i=
so,
R
–
R
vI +
–
+
vC
vI
+ –
R
–
vO = −vC
R
+
vI –
+
t
+
vO
1 vI
– vO = − ∫ dt
C −∞ R
We have our integrator.
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20
Now, let’s build a differentiator…
+
vO
–
d
dt
vI +
–
Let’s start with the following insights:
i
vI
+
–
C
dvI
i=C
dt
dvI
i is related to
dt
But we need to somehow convert current
to voltage.
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20
Differentiator…
Recall
i
i
–
+
R
i
–
+
vO
–
+
0V
i
C
vI +
–
+ vC –
Demo
R
–
+
vO = −iR
current
to
voltage
vO
vI = vC
dvI
i=C
dt
vO = − RC
dvI
dt
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.002 Fall 2000
Lecture 20