296 71 5MB
English Pages [419] Year 2007
6.002
CIRCUITS AND ELECTRONICS
Introduction and Lumped Circuit Abstraction
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 1
ADMINISTRIVIA Lecturer: Prof. Anant Agarwal Textbook: Agarwal and Lang (A&L) Readings are important! Handout no. 3 Web site —
http://web.mit.edu/6.002/www/fall00
Assignments — Homework exercises Labs Quizzes Final exam
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 1
Two homework assignments can be missed (except HW11). Collaboration policy Homework You may collaborate with others, but do your own writeup. Lab You may work in a team of two, but do you own writeup. Info handout Reading for today — Chapter 1 of the book
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 1
What is engineering? Purposeful use of science
What is 6.002 about? Gainful employment of Maxwell’s equations From electrons to digital gates and opamps
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 1
6.002
Nature as observed in experiments V
3
6
9
12
…
I
0.1
0.2
0.3
0.4
…
Physics laws or “abstractions” Maxwell’s abstraction for Ohm’s tables of data V=RI Lumped circuit abstraction +– R V C L M Simple amplifier abstraction Digital abstraction Operational amplifier abstraction abstraction Combinational logic +

S
f
Filters
Clocked digital abstraction
Analog system components: Modulators, oscillators, RF amps, power supplies 6.061
Instruction set abstraction Pentium, MIPS 6.004 Programming languages Java, C++, Matlab 6.001 Software systems 6.033 Operating systems, Browsers
Mice, toasters, sonar, stereos, doom, space shuttle 6.455 6.170
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 1
Lumped Circuit Abstraction
Consider
The Big Jump from physics to EECS I
V
? Suppose we wish to answer this question: What is the current through the bulb?
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 1
We could do it the Hard Way… Apply Maxwell’s Differential form ∂B Faraday’s ∇× E = − ∂t ∂ρ Continuity ∇ ⋅ J = − ∂t Others
ρ ∇⋅E = ε0
Integral form ∂φ B ∫ E ⋅ dl = − ∂t ∂q J ⋅ dS = − ∫ ∂t q E ⋅ dS = ∫
ε0
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 1
Instead, there is an Easy Way… First, let us build some insight: Analogy F
a?
I ask you: What is the acceleration? You quickly ask me: What is the mass? I tell you:
m
F You respond: a = m Done !! !
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 1
Instead, there is an Easy Way… First, let us build some insight: Analogy
F a? In doing so, you ignored the object’s shape its temperature its color point of force application Pointmass discretization
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 1
The Easy Way… Consider the filament of the light bulb. A B We do not care about how current flows inside the filament its temperature, shape, orientation, etc. Then, we can replace the bulb with a
discrete resistor
for the purpose of calculating the 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 1
The Easy Way… A B Replace the bulb with a
discrete resistor
for the purpose of calculating the current. A I V + and I = V R R – B
In EE, we do things the easy way…
R represents the only property of interest! Like with pointmass: replace objects F with their mass m to find a = m 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 1
The Easy Way…
A + V –
I R
and
B
I=
V R
In EE, we do things the easy way…
R represents the only property of interest! R relates element v and i
V I= R
called element vi relationship
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 1
R is a lumped element abstraction for the bulb.
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 1
R is a lumped element abstraction for the bulb. Not so fast, though … I A + S A
V B
–
SB
black box Although we will take the easy way using lumped abstractions for the rest of this course, we must make sure (at least the first time) that our abstraction is reasonable. In this case, ensuring that V I are defined for the element 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 1
A
+
I SA
V
V B
–
I
must be defined for the element
SB
black box
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 1
I
must be defined. True when
= I out of S B ∂q True only when = 0 in the filament! ∂t ∫ J ⋅ dS I into S A
SA
∫ J ⋅ dS SB
∫ J ⋅ dS − ∫ J ⋅ dS = SA
from ell w x a M
SB
IA
∂q ∂t
IB
∂q =0 I A = I B only if ∂t So let’s assume this
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 1
V
Must also be defined. see A&L
So let’s assume this too
∂φ B =0 ∂t outside elements
VAB defined when
So
VAB = ∫AB E ⋅ dl
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 1
Lumped Matter Discipline (LMD) Or self imposed constraints:
More in Chapter 1 of A & L
∂φ B = 0 outside ∂t ∂q = 0 inside elements ∂t bulb, wire, battery
Lumped circuit abstraction applies when elements adhere to the lumped matter discipline.
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 1
Demo only for the sorts of questions we as EEs would like to ask!
Demo
Lumped element examples whose behavior is completely captured by their V–I relationship.
Exploding resistor demo can’t predict that! Pickle demo can’t predict light, smell
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 1
So, what does this buy us? Replace the differential equations with simple algebra using lumped circuit abstraction (LCA). For example —
a R1
V
+ –
b
R3
R4
d R2
R5
c What can we say about voltages in a loop under the lumped matter discipline?
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 1
What can we say about voltages in a loop under LMD?
a R1
V
b
+ –
R4
R3
d R2
R5
c ∂φ B under DMD ∫ E ⋅ dl = − ∂t 0 ∫ E ⋅ dl + ∫ E ⋅ dl + ∫ E ⋅ dl = 0 ca
ab
bc
+ Vca + Vab + Vbc
= 0
Kirchhoff’s Voltage Law (KVL): The sum of the voltages in a loop is 0. 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 1
What can we say about currents? Consider
I ca
S a
I da
I ba
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 1
What can we say about currents? I ca
S a
I da
I ba ∂q ∫S J ⋅ dS = − ∂t
under LMD 0
I ca + I da + I ba = 0 Kirchhoff’s Current Law (KCL): The sum of the currents into a node is 0. simply conservation of charge
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6.002 Fall 2000
Lecture 1
KVL and KCL Summary KVL:
∑ jν j = 0 loop
KCL:
∑jij = 0 node
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 1
6.002
CIRCUITS AND ELECTRONICS
Basic Circuit Analysis Method (KVL and KCL method)
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 2
Review Lumped Matter Discipline LMD:
Constraints we impose on ourselves to simplify our analysis
∂φ B =0 ∂t ∂q =0 ∂t
Outside elements Inside elements wires resistors sources
Allows us to create the lumped circuit abstraction
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 2
Review
LMD allows us to create the lumped circuit abstraction i
+
v
Lumped circuit element
power consumed by element = 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 2
Review Review Maxwell’s equations simplify to algebraic KVL and KCL under LMD! KVL:
∑ jν j = 0 loop
KCL:
∑jij = 0 node
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 2
Review a R1
+ –
b
R4
R3
R2
d R5
c
DEMO
vca + vab + vbc = 0
KVL
ica + ida + iba = 0
KCL
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 2
Method 1: Basic KVL, KCL method of Circuit analysis Goal: Find all element v’s and i’s 1. write element vi relationships (from lumped circuit abstraction) 2. write KCL for all nodes 3. write KVL for all loops
lots of unknowns lots of equations lots of fun solve
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 2
Method 1: Basic KVL, KCL method of Circuit analysis
Element Relationships For R,
V = IR
For voltage source, V = V0
R +–
V0 For current source, I = I 0 J Io 3 lumped circuit elements
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 2
KVL, KCL Example a +
ν1 +
ν 0 = V0 –
R3
b +
ν2 –
ν4
R1
–
+ –
+
+ν 3 – R2
–
R4
d +
ν5 –
R5
c The Demo 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 2
Associated variables discipline i
+ ν
Element e
Current is taken to be positive going into the positive voltage terminal
Then power consumed by element e
= νi is positive
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 2
KVL, KCL Example a +
+
ν 0 = V0 –
ν1
i0
L1
+ –
–
+
ν2 –
i4 i1 L 2 + R1 ν 4 R4 – R3 b i3 d +ν 3 – i2 i5 + R2 ν 5 R5 L3 –
c The Demo Circuit
L4
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 2
Analyze ν 0 …ν 5 ,ι0 …ι5 1. Element relationships (v, i ) given v3 = i3 R3 v0 = V0 v4 = i4 R4 v1 = i1 R1 v5 = i5 R5 v2 = i2 R2
12 unknowns 6 equations
2. KCL at the nodes a: i0 + i1 + i4 = 0 3 independent b: i2 + i3 − i1 = 0 equations d: i5 − i3 − i4 = 0 e: − i0 − i2 − i5 = 0 redundant 3. KVL for loops L1: − v0 + v1 + v2 = 0 3 independent equations L2: v1 + v3 − v4 = 0 s L3: v3 + v5 − v2 = 0 n o L4: − v0 + v4 + v5 = 0 redundant ati s n w u no k eq n u 2 2 1 1
/
ugh @#! 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 2
Other Analysis Methods Method 2— Apply element combination rules
A B
C
R1
R2 R3
G1
G2
V1
V2
+–
+–
…
RN
GN
⇔
+ RN
+ GN
V1 + V2 +–
J
J
J
I2
G1 + G2
1 Gi = Ri
⇔
D I1
⇔
⇔
R1 + R2 +
I1 + I 2
Surprisingly, these rules (along with superposition, which you will learn about later) can solve the circuit on page 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 2
Other Analysis Methods Method 2— Apply element combination rules
I =?
Example
R1
V + –
R3
R2
I
I V + –
R1
V + –
R2 R3 R2 + R3 R = R1 +
R R2 R3 R2 + R3
V I= 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 2
Method 3—Node analysis Particular application of KVL, KCL method 1. Select reference node ( ground) from which voltages are measured. 2. Label voltages of remaining nodes with respect to ground. These are the primary unknowns. 3. Write KCL for all but the ground node, substituting device laws and KVL. 4. Solve for node voltages. 5. Back solve for branch voltages and currents (i.e., the secondary unknowns)
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 2
Example: Old Faithful plus current source
V0
R2
R5
J
e2
+ V e1 – 0
Step 1
R4
R1 R 3
I1
Step 2
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 2
Example: Old Faithful plus current source
V0
+ V e1 – 0 R2
R4 e2
R5
J
R1 R 3
for I1 convenience, write 1 Gi = Ri
KCL at e1 (e1 − V0 )G1 + (e1 − e2 )G3 + (e1 )G2 = 0
KCL at e2 (e2 − e1 )G3 + (e2 − V0 )G4 + (e2 )G5 − I1 = 0 Step 3
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 2
Example: Old Faithful plus current source
V0
e2
R2
R5
+ V e1 – 0
J
R4
R1 R 3
I1
1 Gi = Ri
KCL at e1 (e1 − V0 )G1 + (e1 − e2 )G3 + (e1 )G2 = 0
KCL at l2 (e2 − e1 )G3 + (e2 − V0 )G4 + (e2 )G5 − I1 = 0 move constant terms to RHS & collect unknowns
e1 (G1 + G2 + G3 ) + e2 (−G3 ) = V0 (G1 ) e1 (−G3 ) + e2 (G3 + G4 + G5 ) = V0 (G4 ) + I1 2 equations, 2 unknowns (compare units)
Solve for e’s Step 4
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 2
In matrix form: − G3 ⎡ G1V0 ⎤ ⎡G1 + G2 + G3 ⎤ ⎡ e1 ⎤ = ⎢G V + I ⎥ ⎢ G3 + G4 + G5 ⎥⎦ ⎢⎣e2 ⎥⎦ − G3 ⎣ 4 0 1⎦ ⎣
conductivity matrix
sources
unknown node voltages
Solve G3 ⎡G3 + G4 + G5 ⎤ ⎡ G1V0 ⎤ G3 G1 + G2 + G3 ⎥⎦ ⎢⎣G4V0 + I1 ⎥⎦ ⎡ e1 ⎤ ⎢⎣ ⎢e ⎥ = (G1 + G2 + G3 )(G3 + G4 + G5 ) − G3 2 ⎣ 2⎦
(
)(
) ( )(
)
G +G +G G V + G G V + I 3 4 5 1 0 3 4 0 1 e = 1 G G +G G +G G +G G +G G +G G +G 2 +G G +G G 1 3 1 4 1 5 2 3 2 4 2 5 3 3 4 3 5 e2 =
(G3 )(G1V0 ) + (G1 + G2 + G3 )(G4V0 + I 1 ) G1G3 + G1G4 + G1G5 + G2G3 + G2G4 + G2 G5 + G3 + G3G4 + G3G5 2
(same denominator)
Notice: linear in V0 , I1 , no negatives in denominator 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 2
Solve, given G1 ⎫ 1 = ⎬ G5 ⎭ 8.2 K
G2 ⎫ 1 ⎬= G4 ⎭ 3.9 K
1 G3 = 1.5 K
I1 = 0
(
)(
)
G G V + G +G +G G V + I e = 3 10 1 2 3 40 1 2 G + G + G + G + G + G −G 2 1 2 3 3 4 5 3 1 1 1 G +G +G = + + =1 1 2 3 8.2 3.9 1.5
(
G3 + G4 + G5 =
)(
)
1 1 1 + + =1 1.5 3.9 8.2
1 1 1 × + 1× 3.9 V e2 = 8.2 1.5 0 1 1− 2 1.5
Check out the DEMO
e2 = 0.6V0
If V0 = 3V , then e2 = 1.8V0 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 2
6.002
CIRCUITS AND ELECTRONICS
Superposition, Thévenin and Norton
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 3
Review Circuit Analysis Methods z KVL:
∑Vi = 0
loop
KCL: ∑ Ii = 0
VI
node
z Circuit composition rules z Node method – the workhorse of 6.002
KCL at nodes using V ’s referenced from ground (KVL implicit in “ (ei − e j ) G ”)
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 3
Linearity R1
V
+ –
R2
e J
Consider
I
Write node equations –
e −V e + −I =0 R1 R2 Notice: linear in e,V , I No eV ,VI terms
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 3
Linearity
R1
Consider
V
+ –
J
R2
Write node equations e −V e + −I =0 R1 R2 Rearrange ⎡1 1⎤ ⎢ R + R ⎥e ⎣ 1 2⎦ conductance matrix
G
=
I
linear in e,V , I
V + I R1
node linear sum voltages of sources
e
=
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 3
Linearity Write node equations e −V e + −I =0 R1 R2 Rearrange ⎡1 1⎤ ⎢ R + R ⎥e ⎣ 1 2⎦ conductance matrix
G or
=
linear in e,V , I
V + I R1
node linear sum voltages of sources
e
=
S
R2 R1 R2 e= V+ I R1 + R2 R1 + R2 e = a1V1 + a2V2 + … + b1 I1 + b2 I 2 + …
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 3
Linearity
⇒
Homogeneity Superposition
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 3
Linearity
⇒
Homogeneity Superposition
Homogeneity x1 x2 . ..
y
⇓ αx1 αx2 .. .
αy
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 3
Linearity
⇒
Homogeneity Superposition
Superposition
x1a x2 a . ..
ya
x1b x2 b . ..
yb
⇓ x1a + x1b x2 a + x2 b . ..
y a + yb
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 3
Linearity
⇒
Homogeneity Superposition
Specific superposition example:
V1 0
0 V2
y1
y2
⇓ V1 + 0 0 + V2
y1 + y2
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 3
Method 4: Superposition method The output of a circuit is determined by summing the responses to each source acting alone. s e c r u so t n e nd e p e i nd only
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 3
i
V =0 + –
i + v
+ v

short
I =0
J
i
i + v
+ v


open
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6.002 Fall 2000
Lecture 3
Back to the example Use superposition method
V
+ –
e
R2
J
R1
I
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6.002 Fall 2000
Lecture 3
Back to the example Use superposition method V acting alone e
R1
V
I = 0 eV =
R2
+ –
I acting alone
R2 V R1 + R2
e
R2
V =0
sum
J
R1
I
R1 R2 eI = I R1 + R2
superposition
R2 R1 R2 e = eV + eI = V+ I R1 + R2 R1 + R2
Voilà ! 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 3
Demo salt water
constant + –
?
+ –
output shows superposition
sinusoid
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 3
Yet another method…
Consider
By superposition v =
∑ α mVm + ∑ β n I n + Ri m
n
no resistance units units By setting ∀n I n = 0, ∀mVm = 0, i = 0 i = 0
i
+ v 
J
y network r a r t i N Arb resistors Vm In + – J
i
also independent of external excitement & behaves like a resistor
All ∀n I n = 0, ∀mVm = 0
independent of external excitation and behaves like a voltage “ vTH ” 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 3
Or
v = vTH + RTH i
As far as the external world is concerned (for the purpose of IV relation), “Arbitrary network N” is indistinguishable from: RTH Thévenin equivalent network
vTH RTH
+ vTH –
+ v
J
N
i

open circuit voltage at terminal pair (a.k.a. port) resistance of network seen from port ( Vm ’s, I n ’s set to 0)
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 3
Method 4: The Thévenin Method J
i
N + –
+ –
+ v 
E
+ v
E
Thévenin equivalent RTH
+ vTH –
i

Replace network N with its Thévenin equivalent, then solve external network E. 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 3
Example: + V –
R2
J
i1 R1
I
i1 R1 RTH
+ V –
VTH
i1 =
+ I –
V − VTH R1 + RTH
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 3
VTH : VTH = IR2
RTH : RTH = R2
+ VTH 
R2
+ RTH 
R2
J
Example:
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 3
Graphically,
v = vTH + RTH i
i
1 RTH
v vTH
“V ” OC
− I SC
Open circuit (i ≡ 0)
v = vTH
Short circuit (v ≡ 0)
− vTH i = RTH
VOC − I SC
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 3
in recitation, see text
Method 5:
The Norton Method
J + –
+ –
+ v 
IN
J
i
RTH = RN
Norton equivalent
IN =
VTH RTH
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 3
Summary Discretize matter LMD Physics
LCA EE
R, I, V
Linear networks
Analysis methods (linear) KVL, KCL, I — V Combination rules Node method Superposition Thévenin Norton
Next Nonlinear analysis Discretize voltage
…
101100
…
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 3
6.002
CIRCUITS AND ELECTRONICS
The Digital Abstraction
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 4
Review z Discretize matter by agreeing to
observe the lumped matter discipline
Lumped Circuit Abstraction zAnalysis tool kit: KVL/KCL, node method, superposition, Thévenin, Norton (remember superposition, Thévenin, Norton apply only for linear 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 4
Today
Discretize value
Digital abstraction
Interestingly, we will see shortly that the tools learned in the previous three lectures are sufficient to analyze simple digital circuits
Reading: Chapter 5 of Agarwal & Lang 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 4
But first, why digital? In the past … Analog signal processing R1 V0
R2
V1 + –
and V2 might represent the outputs of two sensors, for example.
V1
+ –
V2
By superposition, V0 =
R2 R1 V1 + V2 R1 + R2 R1 + R2
If R1 = R 2 ,
V1 + V2 V0 = 2
The above is an “adder” 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 4
Noise Problem
t
add noise on this wire
Receiver: huh?
…
noise hampers our ability to distinguish between small differences in value — e.g. between 3.1V and 3.2V.
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 4
Value Discretization Restrict values to be one of two HIGH
LOW
5V
0V
TRUE
FALSE
1
0
…like two digits
0 and 1
Why is this discretization useful? (Remember, numbers larger than 1 can be represented using multiple binary digits and coding, much like using multiple decimal digits to represent numbers greater than 9. E.g., the binary number 101 has decimal value 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 4
Digital System sender
noise VN
VS
VR
VN = 0V
receiver
VS
VR
5V “0” “1” “0” HIGH
“0” “1” “0” 5V
t
2.5V
0V
LOW
0V
t
2.5V
With noise
VS
VN = 0.2V
“0” “1” “0” 5V
“0” “1” “0”
0.2V
t t
2.5V
VS
2.5V
t
0V 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 4
Digital System
Better noise immunity Lots of “noise margin” For “1”: noise margin 5V to 2.5V = 2.5V For “0”: noise margin 0V to 2.5V = 2.5V
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 4
Voltage Thresholds and Logic Values
5V
1
1
sender 0
1 2.5V receiver
0
0 0V
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 4
But, but, but … What about 2.5V? Hmmm… create “no man’s land” or forbidden region For example, 5V
1 sender
3V 2V
0
1
VH
forbidden region
receiver
VL
0
0V
“1”
V
“0”
0V
H
5V V
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 4
But, but, but …
Where’s the noise margin? What if the sender sent 1:
VH ?
Hold the sender to tougher standards! 5V 1
V 0H
1 V IH
sender
V IL
0
receiver 0
V 0L
0V
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 4
But, but, but …
Where’s the noise margin? What if the sender sent 1:
VH ?
Hold the sender to tougher standards! 5V 1
V 0H
1
sender
Noise margins
V IH
receiver
V IL
0
0
V 0L
0V “1” noise margin: V
 V IH 0H “0” noise margin: VIL  V 0L
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 4
5V V 0H V IH V IL V 0L 0V
5V V 0H V IH V IL V 0L 0V
0
1
0
1
sender
t
0
1
0
1
receiver
t
Digital systems follow static discipline: if inputs to the digital system meet valid input thresholds, then the system guarantees its outputs will meet valid output thresholds. 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 4
Processing digital signals Recall, we have only two values —
1,0
Map naturally to logic: T, F Can also represent numbers
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 4
Processing digital signals Boolean Logic If X is true and Y is true Then Z is true else Z is false. Z = X AND Y
X, Y, Z are digital signals “0” , “1”
Z = X • Y Boolean equation X Y
AND gate
Z
Truth table representation: X Y Z 0 0 1 1
0 1 0 1
0 0 0 1
Enumerate all input combinations 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 4
Combinational gate abstraction Adheres to static discipline Outputs are a function of
inputs alone.
Digital logic designers do not have to care about what is inside a gate.
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 4
Demo
X
Y
Z Noise X Y
Z
Z = X • Y 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 4
Examples for recitation X
t Y
t Z
t Z = X • Y 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 4
In recitation… Another example of a gate If (A is true) OR (B is true) then C is true else C is false C = A + B A B
Boolean equation OR C
OR gate
More gates B
B Inverter
X Y
Z NAND
Z = X • Y 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 4
Boolean Identities X X X X
• 1 = X • 0 = X + 1 = 1 +0 = X
1 = 0 0 = 1 AB + AC = A • (B + C)
Digital Circuits Implement: B C
output = A + B • C B•C output
A
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 4
6.002
CIRCUITS AND ELECTRONICS
Inside the Digital Gate
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 5
Review The Digital Abstraction z Discretize value 0, 1 z Static discipline
meet voltage thresholds sender VOH VOL
receiver VIH VIL
forbidden region
Specifies how gates must be designed
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 5
Review Combinational gate abstraction outputs function of input alone satisfies static discipline
A B
C NAND
A 0 0 1 1
B 0 1 0 1
C 1 1 1 0
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 5
For example: a digital circuit
Demo
A⋅ B
A B
D
C D = (C ⋅ (A ⋅ B )) 3 gates here
A Pentium III class microprocessor is a circuit with over 4 million gates !! The RAW chip (http://www.cag.lcs.mit.edu/raw) being built at the Lab for Computer Science at MIT has about 3 million gates. 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 5
How to build a digital gate Analogy l ike power supply
A
(li taps
s) e h c t i ke sw
B C
if A=ON AND B=ON C has H20 else C has no H20 Use this insight to build an AND gate.
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 5
How to build a digital gate
OR gate
A C 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 5
Electrical Analogy C B
A
V + –
Bulb C is ON if A AND B are ON, else C is off Key: “switch” device
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 5
Electrical Analogy equivalent ckt
Key: “switch” device
control
in
C =0
in
out
C
in
out
C=1
3Terminal device if C = 0 else
out
short circuit between in and out open circuit between in and out
For mechanical switch, control mechanical pressure 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 5
Consider VS RL
RL
VOUT
+ VS –
VOUT
IN
C
VS =
C
“1”
OUT
VS
VOUT C =0
Truth table for C VOUT 0 1 1 0
VS
VOUT C =1 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 5
What about? VS
Truth table for c1 c2 VO 0 0 1 0 1 1 1 0 1 1 1 0
VOUT
c1 c2
Truth table for
VS
VOUT c1
c2
c1 c2 VO 0 0 1 0 1 0 1 0 0 1 1 0
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 5
What about? can also build compound gates
VS D A
C
D = (A ⋅ B) + C
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 5
The MOSFET Device MetalOxide Semiconductor FieldEffect Transistor
drain D
G gate
≡ S source
3 terminal lumped element behaves like a switch
G : control terminal D, S : behave in a symmetric manner (for our needs) 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 5
The MOSFET Device Understand its operation by viewing it as a twoport element —
out k k c e Ch extboo l the t s interna for it ture. iG c u r t s
D
iDS
G
+ vGS –
vDS S
–
D off
G vGS < VT
G vGS ≥ VT
S
+
D iDS on S
VT ≈ 1V typically
“Switch” model (S model) of the MOSFET 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 5
Demo
Check the MOS device on a scope. i DS
+ vDS
+ vGS –
–
iDS vGS ≥ VT
vGS < VT
vDS
iDS vs vDS 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 5
A MOSFET Inverter VS = 5V RL vOUT
A
B
IN
A
B
Note the power of abstraction. The abstract inverter gate representation hides the internal details such as power supply connections, RL, GND, etc. (When we build digital circuits, the and are common across all gates!) 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 5
Example vOUT
5V
vOUT
vIN
v IN
0V V T =1V
5V
The T1000 model laptop desires gates that satisfy the static discipline with voltage thresholds. Does out inverter qualify?
1:
VOL = 0.5V
VIL = 0.9V
VOH = 4.5V
VIH = 4.1V
sender 5 4.5 V OH
receiver
5 4.1
0.9 0.5 VOL 0: 0 0 Our inverter satisfies this.
1 VIH VIL
0
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 5
E.g.: Does our inverter satisfy the static discipline for these thresholds: VOL = 0.2V
VIL = 0.5V
VOH = 4.8V
VIH = 4.5V
yes
x VOL = 0.5V
VIL = 1.5V
VOH = 4.5V
VIH = 3.5V
no
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 5
Switch resistor (SR) model of MOSFET …more accurate MOS model D
D
G
G
G S
D
vGS < VT
S
RON
vGS ≥ VT S e.g. RON = 5 KΩ
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 5
SR Model of MOSFET D
D
G
G
G
vGS < VT
S
S
MOSFET S model
RON
vGS ≥ VT S
MOSFET SR model
vGS ≥ VT
vGS ≥ VT iDS
D
iDS vGS < VT
1 RON
vGS < VT
vDS
vDS
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 5
Using the SR model VS RL
RL
vOUT
+ VS –
vOUT
IN
C
VS =
C
“1”
OUT
Truth table for
VS RL
vOUT
C VOUT 0 1 1 0
RON C =0
VS
RL C =1
vGS ≥ VT
vOUT
RON
Choose RL, RON, VS such that: V R v = S ON ≤ V OL OUT R +R L ON
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 5
6.002
CIRCUITS AND ELECTRONICS
Nonlinear 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 6
Review Discretize matter t LCA m1 X KVL, KCL, iv m2 X Composition rules m3 X Node method m4 X Superposition m5 X Thévenin, Norton
any circuit linear 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 6
Review Discretize value t Digital abstraction X Subcircuits for given “switch” setting are linear! So, all 5 methods (m1 – m5) can be applied
VS
VS
A =1 B =1
RL
RL
C A
C RON
B
RON
SR MOSFET 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 6
Today Nonlinear Analysis X Analytical method based on m1, m2, m3 X Graphical method X Introduction to 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 6
How do we analyze nonlinear circuits, for example: Hypothetical nonlinear D device (Expo Dweeb ☺) iD
V
+ vD 
+ –
+ vD 
D iD iD
iD = aebvD
a vD
0,0
(Curiously, the device supplies power when vD is negative) 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 6
Method 1: Analytical Method Using the node method,
(remember the node method applies for linear or nonlinear circuits)
vD − V + iD = 0 R iD = aebvD
2 unknowns
1 2
2 equations
Solve the equation by trial and error numerical methods
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 6
Method 2: Graphical Method Notice: the solution satisfies equations 1 and 2 iD
2
iD = aebvD
a vD
iD
V vD 1 iD = − R R
V R
1 slope = − R vD
V 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 6
Combine the two constraints iD
V 1 R ~ 0 .4 a ¼
called “loadline” for reasons you will see later vD
~ 0.5
e.g.
V =1 R =1 1 a= 4 b =1
V 1 vD = 0.5V iD = 0.4 A
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 6
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 6
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 6
If only it were linear … iD
iD
vD
vD t
it would’ve been ok.
What do we do? Zen is the answer … next 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 6
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 timevarying 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
6.002
CIRCUITS AND ELECTRONICS
Dependent Sources and Amplifiers
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 2002: Lecture 8
Review
Nonlinear circuits — can use the node method
Small signal trick resulted in linear response
Today
Dependent sources
Amplifiers
Reading: Chapter 7.1, 7.2
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 2002: Lecture 8
Dependent sources Seen previously + i
Resistor Independent Current source
+ i
v
–
R v – I
v i= R
i=I
2terminal 1port devices New type of device: Dependent source iI i O
+ control port
vI
f ( vI )
–
+
vO
output port
– 2port device
E.g., Voltage Controlled Current Source Current at output port is a function of voltage at the input port
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 2002: Lecture 8
Dependent Sources: Examples
Example 1: Find V + R V –
independent current source
I = I0
V = I0R
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 2002: Lecture 8
Dependent Sources: Examples Example 2: Find V voltage controled current source
+ R V –
K I = f (V ) = V
iI +
+ R V –
f (vI ) =
K vI
iO +
vI
vO
–
–
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6.002 – Fall 2002: Lecture 8
Dependent Sources: Examples Example 2: Find V voltage controled current source
+ R V –
K I = f (V ) = V e.g. K = 103 Amp·Volt R = 1kΩ
K V = IR = R V or V 2 = KR or V = KR = 10 −3 ⋅ 10 3 = 1 Volt
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 2002: Lecture 8
Another dependent source example
RL iIN
vI + –
iD
+
+
vIN
vO
–
–
e.g.
VS + –
iD = f (vIN ) iD = f (vIN ) K 2 = (vIN − 1) for vIN ≥ 1 2 iD = 0 otherwise
Find vO as a function of 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 2002: Lecture 8
Another dependent source example VS RL iIN
vI + –
iD
+
+
vIN
vO
–
–
iD = f (vIN ) e.g.
iD = f (vIN ) K 2 = (vIN − 1) for vIN ≥ 1 2 iD = 0 otherwise
Find vO as a function of vI .
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6.002 – Fall 2002: Lecture 8
Another dependent source example VS RL vI vI
+ –
vO K 2 iD = (vIN − 1) for vIN ≥ 1 2 iD = 0 otherwise
Find vO as a function of vI .
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6.002 – Fall 2002: Lecture 8
Another dependent source example VS RL vI vI
+ –
vO K 2 iD = (vIN − 1) for vIN ≥ 1 2 iD = 0 otherwise
KVL
− VS + iD RL + vO = 0 vO = VS − iD RL K 2 vO = VS − (vI − 1) RL 2 vO = VS
for vI ≥ 1 for vI < 1
Hold that thought 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 2002: Lecture 8
Next, Amplifiers
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 2002: Lecture 8
Why amplify? Signal amplification key to both analog and digital processing. Analog: AMP IN
Input Port
OUT
Output Port
Besides the obvious advantages of being heard farther away, amplification is key to noise tolerance during communcation
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 2002: Lecture 8
Why amplify? Amplification is key to noise tolerance during communcation No amplification
useful signal
1 mV
e nois
10 mV
huh?
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 2002: Lecture 8
Try amplification e nois
AMP
not bad!
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 2002: Lecture 8
Why amplify? Digital: Valid region 5V
5V
VIH IN VIL 0V
5V
OUT Digital System
IN
5V
VOL
OUT
V OH
VIH VIL
0V
0V
VOH
t
V OL
0V
t
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6.002 – Fall 2002: Lecture 8
Why amplify? Digital:
Static discipline requires amplification! Minimum amplification needed: VIH VIL
VOH VOL
VOH − VOL VIH − VIL
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 2002: Lecture 8
An amplifier is a 3ported device, actually Power port Input port
iO
iI
+v – I
Amplifier
+ v Output – O port
We often don’t show the power port. Also, for convenience we commonly observe “the common ground discipline.” In other words, all ports often share a common reference point called “ground.”
POWER IN OUT
How do we build one? 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 2002: Lecture 8
Remember? VS RL vI vI
+ –
vO K 2 iD = (vIN − 1) for vIN ≥ 1 2 iD = 0 otherwise
KVL
− VS + iD RL + vO = 0 vO = VS − iD RL K 2 vO = VS − (vI − 1) RL 2 vO = VS
for vI ≥ 1 for vI < 1
Claim: This is an amplifier 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 2002: Lecture 8
So, where’s the amplification? Let’s look at the vO versus vI curve. mA e.g. VS = 10V , K = 2 2 , RL = 5 kΩ V K 2 vO = VS − RL (vI − 1) 2 2 −3 2 3 = 10 − ⋅10 ⋅ 5 ⋅ 10 (vI − 1) 2 vO = 10 − 5 (vI − 1) vO VS
2
ΔvO
1 ΔvO >1 Δv I
ΔvI
vI
amplification
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 2002: Lecture 8
Plot vO versus vI vO = 10 − 5 (vI − 1)
2
0.1 change in vI
Demo
vI
vO
0.0 1.0 1.5 2.0 2.1 2.2 2.3 2.4
10.00 10.00 8.75 5.00 4.00 2.80 1.50 ~ 0.00
1V change in vO
Gain!
Measure 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 2002: Lecture 8
One nit … vO
What happens here? 1
vI
Mathematically, K 2 vO = VS − RL (vI − 1) 2 So
is mathematically predicted behavior
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 2002: Lecture 8
One nit … vO
K 2 vO = VS − RL (vI − 1) 2 What happens here? vI
1 However, from
K 2 iD = (vI − 1) 2 VS
for vI ≥ 1
RL vO
VCCS
iD
For vO>0, VCCS consumes power: vO iD For vO vI − VT vO = vI − VT vO < vI − VT
1V
vI
VT
1V
2V
“interesting” region for vI . 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… 5V
VS
vO
vO = vI − VT vO 1V
vI VT 1V
Demo
vI
2V
Amplifies alright, but distorts
vI vO 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 vO
~ 5V VS
Focus on this line segment
(VI , VO )
~ 1V vI
VT 1V
~ 2V 2 K (vI − VT ) vO = VS − RL 2 Amp all right, but nonlinear! Hmmm … So what about our linear amplifier ???
Insight: But, observe vI vs vO about some point (VI , VO) … 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 ΔvO
vo VO
vi
(VI ,VO )
looks linear
VI ΔvI
Operate amp at VI , VO Æ DC “bias” (good choice: midpoint of input operating range)
Superimpose small signal on top of VI 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 ΔvO
vo VO
vi
(VI ,VO )
looks linear
VI ΔvI
Operate amp at VI , VO Æ DC “bias” (good choice: midpoint of input operating range)
Superimpose small signal on top of VI 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 VI to “boost” interesting input signal above VT, and in fact, well above VT .
VS RL
interesting input signal
ΔvI + – VI + –
vO
Offset voltage or 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 10
Graphically
VS RL
interesting input signal
vO
ΔvI + – VI + –
VS
vO
operating point
VO
0
VI , VO
vO = vI − VT vI
VT
Good choice for operating point: midpoint of input operating range
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 10
Small Signal Model aka incremental model aka linearized model
Notation — Input:
vI = VI + vi
total DC small variable bias signal (like ΔvI) bias voltage aka operating point voltage Output: vO = VO + vo Graphically, vI
vO
vi
vo
VI
VO
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
II Mathematically
(… watch my fingers)
RL K 2 vO = VS − (vI − VT ) VO = VS − RL K (VI − VT )2 2 2 substituting vI = VI + vi vi