6.002
CIRCUITS AND
ELECTRONICS
Small Signal 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 11
Review:
„
Small signal notation
vA = VA + va
total operating
point
„
small
signal
vOUT = f (vI )
d
vout =
f (vI )
⋅ vi
dv I
v I =VI
VS
„
vI = VI + vi
vi
VI
RL
vO = VO + vo
+
–
+
–
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 11
Review:
I Graphical view
(using transfer function)
vO
behaves linear
for small
perturbations
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 11
Review:
II Mathematical view
K (vI − VT )
vO = VS −
RL
2
2
⎡V − K v − V 2 R ⎤
( I T ) L⎥
S
⎢
d ⎣
2
⎦
vo =
dvI
⋅ vi
v I =VI
vo = − K (VI − VT ) RL ⋅ vi
gm
related to VI
constant for fixed
DC bias
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 11
How to choose the bias point,
using yet another graphical view
based on the load line
i DS
i DS <
Demo
K 2
vO
2
V S vO
i
=
load line DS R − R
L
L
input signal
response
VI
VO
− 1 + 1 + 2 KR LV S
v I = VT +
KR L
vO
v I = VT
Choosing a bias point:
1. Gain
g m RL ∝ VI
2. Input valid operating range for amp.
3. Bias to select gain and input 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 11
III The Small Signal Circuit View
We can derive small circuit equivalent
models for our devices, and thereby conduct
small signal analysis directly on circuits
e.g. large signal
circuit model
for amp
vI +
–
R
VS
vOUT
K
2
iD = (vI − VT )
2
+
–
1
We can replace large signal models with
small signal circuit models.
Foundations: Section 8.2.1 and also in the
last slide in this lecture.
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 11
Small Signal Circuit Analysis
1
Find operating point using DC bias
inputs using large signal model.
2
Develop small signal (linearized)
models for elements.
3
Replace original elements with small
signal models.
Analyze resulting linearized circuit…
Key: Can use superposition and other
linear circuit tools with linearized
circuit!
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 11
Small Signal Models
A
MOSFET
large
signal
D
vGS
Small signal?
iDS =
K
(vGS − VT )2
2
S
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 11
Small Signal Models
A
MOSFET
large
signal
D
vGS
iDS
Small signal:
K
2
iDS = (vGS − VT )
2
∂
ids =
∂vGS
K
2
= (vGS − VT )
2
S
⎡ K (v − V )2 ⎤
⋅ v gs
T
⎢⎣ 2 GS
⎥⎦
vGS =VGS
ids = K (VGS − VT ) ⋅ v gs
ids is linear in vgs !
gm
D
small
signal
v gs
ids = K (VGS − VT ) v gs
S
ids = g m v gs
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 11
B
DC Supply VS
large
signal
vS = VS
iS
+ vS = VS
–
Small signal
∂VS
vs =
∂iS
is +
vs
–
⋅ is
iS = I S
vs = 0
DC source behaves
as short to 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 11
C
Similarly, R
large
signal
iR +
vR
R
–
v R = R iR
vr =
∂ ( RiR )
⋅ ir
∂iR iR = I R
vr = R ⋅ ir
small
signal
ir +
vr
R
–
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 11
Amplifier example:
Large signal
RL
vO
+ v
– I
iDS
Small signal
RL
vo
+ V
– S
+ vi
–
iDS
K
2
= (vI − VT )
2
ids
ids = K (VI − VT ) ⋅ vi
K
2
vO = VS − (vI − VT ) RL
2
ids RL + vo = 0
vo = −ids RL
vo = − K (VI − VT )RL ⋅ vi
= − g m RL ⋅ vi
Notice, first we need to find operating
point voltages/currents.
Get these from a large signal 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 11
III The Small Signal Circuit View
To find the relationship between the small signal parameters of
a circuit, we can replace large signal device models with
corresponding small signal device models, and then analyze the
resulting small signal circuit.
Foundations: (Also see section 8.2.1 of A&L)
KVL, KCL applied to some circuit C yields:
" + v A + " + vOUT + " + vB + "
1
Replace total variables with
operating point variables plus small signal variables
" + VA + v a " + VOUT + vout + VB + vb + "
Operating point variables themselves satisfy the
same KVL, KCL equations
" + VA
" + VOUT
+ VB
+"
so, we can cancel them out
Leaving
"
+ va "
+ vout
+ vb + "
2
But 2 is the same equation as 1 with small signal
variables replacing total variables, so 2 must reflect same
topology as in C, except that small signal models are used.
Since small signal models are linear, our linear tools will now
apply…
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 11