6.002
CIRCUITS AND
ELECTRONICS
Incremental Analysis
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 7
Review
Nonlinear Analysis
X Analytical method
X Graphical method
Today
X Incremental analysis
Reading: Section 4.5
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 7
Method 3: Incremental Analysis
Motivation: music over a light beam
Can we pull this off?
iD
+
vD LED
light
intensity
I D ∝ iD
vI music signal
iR
vI (t ) +
–
t
vI (t )
iD (t )
light
AMP
iR ∝ I R
light intensity IR
in photoreceiver
LED: Light
Emitting
expoDweep ☺
iR (t )
sound
nonlinear
linear
problem! will result in distortion
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 7
Problem:
The LED is nonlinear
distortion
iD
iD
vD
vD = vI
t
vD
t
iD
vD
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 7
Insight:
iD
small region
looks linear
(about VD , ID)
ID
VD
vD
DC offset
or DC bias
Trick:
vi (t ) +
–
vI
VI
+
–
iD = I D + id
+
vD LED
vD = VD + vd
VI
vi
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 7
Result
iD
id
ID
vD
VD
vd
very small
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 7
Result
vD = vI
vd
vD
VD
t
iD
id
iD
~linear!
ID
t
Demo
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 7
The incremental method:
(or small signal method)
1. Operate at some DC offset
or bias point VD, ID .
2. Superimpose small signal vd
(music) on top of VD .
3. Response id to small signal vd
is approximately linear.
Notation:
iD = I D + id
total
DC
small
variable offset superimposed
signal
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 7
What does this mean
mathematically?
Or, why is the small signal response
linear?
nonlinear
iD = f (vD )
We replaced
vD = VD + ΔvD
large DC
vd
increment
about VD
using Taylor’s Expansion to expand
f(vD) near vD=VD :
iD = f (VD ) +
+
df (vD )
⋅ ΔvD
dvD vD =VD
1 d 2 f (v D )
2! dvD 2 v
2
⋅ ΔvD + "
D =VD
neglect higher order terms
because ΔvD is small
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 7
iD ≈ f (VD ) +
constant
w.r.t. ΔvD
d f (v D )
⋅ ΔvD
d vD vD =VD
constant w.r.t. ΔvD
slope at VD, ID
We can write
X : I D + ΔiD ≈ f (VD ) +
d f (v D )
⋅ Δ vD
d vD vD =VD
equating DC and time-varying parts,
I D = f (VD )
operating point
d f (v D )
ΔiD =
⋅ ΔvD
d vD vD =VD
constant w.r.t. ΔvD
so, Δ iD ∝ ΔvD
By notation,
Δ iD = id
Δ v D = vd
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 7
In our example,
iD = a e
bv D
From X : I D + id ≈ a e bVD + a e bVD ⋅ b ⋅ vd
Equate DC and incremental terms,
I D = a ebVD
operating point
aka bias pt.
aka DC offset
id = a ebVD ⋅ b ⋅ vd
id = I D ⋅ b ⋅ vd
constant
small signal
behavior
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 7
Graphical interpretation
operating point
I D = a ebVD
id = I D ⋅ b ⋅ vd
A
slope at
VD, ID
iD
ID
id
B
VD
operating
point
vd
vD
we are
approximating
A with B
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 7
graphically
mathematically
now, circuit
We saw the small signal
Large signal circuit:
ID
VI
+
LED VD
-
+
–
I D = a ebVD
Small signal reponse: id = I D b vd
+ vd -
behaves like:
id
R=
small signal circuit:
1
ID b
id
vi
+
–
+
vd
-
1
I Db
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 7