6.002
CIRCUITS AND
ELECTRONICS
Amplifiers --
Small Signal Model
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
10

Review
„
MOSFET amp v
I
V
S
R
L v
O i
DS
„
Saturation discipline — operate
MOSFET only in saturation region
„ Large signal analysis
1. Find v
2. Valid v
O
I
, vs v v
O
I under saturation discipline.
ranges under saturation discipline.
Reading: Small signal model -Chapter 8
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
10

Large Signal Review
1 v
O vs v
I v
O
= V
S
−
K
( v
I
−
1
)
2 R
L
2 valid for v
I
≥ and
V
T v
O
≥ v
I
– V
T i
DS
≤
K
2 v
O
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
10

Large Signal Review
2 Valid operating ranges
5
V corresponding interesting region for v
O
V
S v
O
1 V v
O v
O v
O
>
=
< v
I v
I v
I
−
−
−
V
T
V
T
V
T
V
T v
I
1
V
2 V
“interesting” region for v
I
. Saturation discipline satisfied.
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
10

But…
5
V
V
S v
O v
O
= v
I
− V
T v
O 1 V
V
T
1
V
2
V v
I v
I
Demo v
I v
O
Amplifies alright, but distorts t
Amp is nonlinear … /
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
10

Small Signal Model v
O
~ 5
V V
S
Focus on this line segment
(
V
I
, V
O
)
~ 1
V v
I
V
T
1
V
K
~
( v
I
2 V
− V
T v
O
= V
S
−
2
Amp all right, but nonlinear!
)
2
R
L
Hmmm … So what about our linear amplifier ???
Insight:
But, observe v
I point
(
V
I
,
V
O vs v
O about some
)
… looks quite linear !
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
10

Trick
Δ v
O
V
O
(
V
I
, V
O
) v o v i looks linear
V
I
Δ v
I
™
Operate amp at V
I
, V
O
Æ DC “bias” (good choice: midpoint of input operating range)
™ Superimpose small signal on top of V
I
™ Response to small signal seems to be approximately linear
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
10

Trick
Δ v
O
V
O
(
V
I
, V
O
) v o v i looks linear
V
I
™
Operate amp at V
I
, V
O
Δ v
I
Æ DC “bias” (good choice: midpoint of input operating range)
™ Superimpose small signal on top of V
I
™ Response to small signal seems to be approximately linear
Let’s look at this in more detail —
I graphically
II mathematically
III from a circuit viewpoint next week
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
10

I Graphically
We use a DC bias V signal above V
T
I to “boost” interesting input
, and in fact, well above V
T
.
V
S
R
L interesting input signal
Δ v
I
V
I
+
–
+
–
Offset voltage or bias v
O
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
10

Graphically interesting input signal
Δ v
I
V
I
+
–
+
–
V
S
V
O v
O
V
S
R
L v
O operating point
V
I
,
V
O v
O
= v
I v
I
− V
T
0 V
T
V
I
Good choice for operating point: midpoint of input operating range
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
10

Small Signal Model aka incremental model aka linearized model
Notation —
Input: v
I
=
V
I
+ v i total variable
DC bias small signal (like Δ v
I
) bias voltage aka operating point voltage
V
I
Output: v
O
=
V
O
+ v o
Graphically, v
I v i v
O
V
O v o
0 t
0 t
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
10

v
O
= V
S substituting
−
R
L
K
2 v
I
(
= V
I v
I
− V
T
+ v i
)
2
V
O
= V
S
−
R
L
2
K v i
<< V
I
(
V
I
− V
T
“
) 2 v
O
= V
S
−
R
L
K
2
= V
S
−
R
L
K
2
= V
S
−
R
L
K
2
V
O
+ v o
From “ ,
= V
S
−
R
L
K
2
(
(
(
(
[
V
I
+ v i
] −
[
V
I
[
V
I
V
I
− V
T v
T
)
2
] + v i
)
2
− V
T
]
2 +
2
[
V
I
− v
T
] v i
− V
T
) 2 − R
L
K
(
V
I
+ v i
2
)
− V
T
) v i v o
= − R
L
K
(
V
I
− V
T
) v i g m related to V
I
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
10

Mathematically v o
= − R
L
K
(
V
I
− V
T
) v i g m related to V
I v o
= − g m
R
L v i
For a given DC operating point voltage V
I
V
I
–
V
T is constant. So,
, v o
= − A v i constant w.r.t. v i
In other words, our circuit behaves like a linear amplifier for small signals
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
10

Another way v
O
= V
S
−
R
L
K
2
( v
I
− V
T
)
2 v o
= d dv
I
⎡
⎢ V
S
−
R
L
2
K v
(
I
− V
T
)
2
⎤
⎥ v
I
= V
I
⋅ v i slope at V
I v o
= − R
L
K
(
V
I
− V
T
) ⋅ v i g m
= K
(
V
I
− V
T
)
A = − g m
R
L amp gain
Also, see Figure 8.9 in the course notes for a graphical interpretation of this result
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
10

More next lecture … i
DS
Demo load line input signal response operating point
V
I v
O
V
O
How to choose the bias point:
1. Gain component g m
∝ V
I
2.
v i gets big Æ distortion.
So bias carefully
3. Input valid operating range.
Bias at midpoint of input operating range for maximum swing.
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
10