EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
The CMOS Inverter: Current Flow during Switching
N: sat
P: sat
VOUT
N: off
P: lin
VDD
VDD
S
G
i
VOUT
A
D
G
C
N: sat
P: lin
D
VIN
i
B
D
E
N: lin
P: sat
S
N: lin
P: off
0
0
VDD
VIN
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
Power Dissipation due to Direct-Path Current
VDD
VDD
vIN:
S
G
D
i
vIN
S
VT
0
Ipeak
vOUT
D
G
VDD-VT
i:
0
tsc
Energy consumed per switching period:
time
Edp  t scVDD I peak
Fall 2001 Lecture 2
EECS 40
W. G. Oldham
Copyright Regents of University of California
N-Channel MOSFET Operation
An NMOSFET is a closed switch when the input is high
A
A
B
B
Y
X
Y
X
Y = X if A and B
Y = X if A or B
NMOSFETs pass a “strong” 0 but a “weak” 1
Fall 2001 Lecture 2
EECS 40
W. G. Oldham
Copyright Regents of University of California
P-Channel MOSFET Operation
A PMOSFET is a closed switch when the input is low
A
A
B
B
Y
X
Y = X if A and B
= (A + B)
Y
X
Y = X if A or B
= (AB)
PMOSFETs pass a “strong” 1 but a “weak” 0
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
Pull-Down and Pull-Up Devices
•
In CMOS logic gates, NMOSFETs are used to connect the
output to GND, whereas PMOSFETs are used to connect the
output to VDD.
– An NMOSFET functions as a pull-down device when it is
turned on (gate voltage = VDD)
– A PMOSFET functions as a pull-up device when it is turned on
(gate voltage = GND)
VDD
A1
input signals A2
AN
A1
A2
AN
Pull-up
network
PMOSFETs only
F(A1, A2, …, AN)
Pull-down
network
NMOSFETs only
EECS 40
Fall 2001 Lecture 2
Copyright Regents of University of California
W. G. Oldham
CMOS NAND Gate
VDD
A
A B
0 0
0 1
1 0
1 1
B
F
A
B
F
1
1
1
0
EECS 40
Fall 2001 Lecture 2
Copyright Regents of University of California
W. G. Oldham
CMOS NOR Gate
VDD
A B
0 0
0 1
1 0
1 1
A
B
F
B
A
F
1
0
0
0
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
CMOS Pass Gate
A
Y
X
A
Y = X if A
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
Logic Gates – From Week 9b
A
A
AND
F=A·B
B
B
A
B
A
B
F=A+B
OR
A
A
NOT
A
B
NAND
F = A B
NOR
F = A B
F  A B
EXCLUSIVE OR
EECS 40
Fall 2001 Lecture 2
Copyright Regents of University of California
W. G. Oldham
Logic Gates – How are they used?
A
AND
C=A·B
B
First of all we must agree on what is high (logical 1) or
low (logical 0). Suppose 1.5 V is 1 and 0V is logical 0.
+
AND
1.5V
+
1.5V
-
C would have the value of
1.5 V (logical 1).
But it would have the
value of 0V (logical 0) if
either one of the inputs
were held at zero V.
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
What are the most basic gates in Digital Electronics?
Not-AND = NAND
A
AB
B
Not-OR = NOR
A
B
A B
A
0
0
1
1
B
0
1
0
1
AB AB
0
1
0
1
0
1
1
0
A
0
0
1
1
B
0
1
0
1
A+B A  B
0
1
1
1
1
0
0
0
Typically use one or the other: “NAND logic” or “NOR logic”
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
How to Combine Gate to Produce a Desired Logic Function?
(This is called Logical Synthesis)
Not-AND = NAND
Logically just
an AND plus
a NOT gate:
A
AB
B
A
B
AND
AB
NOT
A
B
AND
AB
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
How to Combine Gate to Produce a Desired Logic Function?
(More basic Logical Synthesis)
F= A .B
A
AB
B
Again a little shorthand is useful
A
B
A .B
EECS 40
Fall 2001 Lecture 2
Copyright Regents of University of California
W. G. Oldham
How to Combine Gate to Produce a Desired Logic Function?
(More basic Logical Synthesis)
Suppose we are given a truth table (all logic statements
can be represented by a truth table). How can we
implement the function?
Answer: There are lots of ways, but one simple way is
implementation from “sum of products” formulation.
How to do this: 1) Write sum of products expression
from truth table and 2) Implement using standard gates.
(Warning this is probably inefficient – we need to
minimize, or simplify the expression)
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
How to Combine Gate to Produce a Desired Logic Function?
(More basic Logical Synthesis)
Example:
Clearly: F= 1 if
A
B
C
F
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
0
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
0
ABC=1
or
ABC =1
i.e. F= A B C +AB C
EECS 40
Fall 2001 Lecture 2
Copyright Regents of University of California
W. G. Oldham
How to Combine Gate to Produce a Desired Logic Function?
(More basic Logical Synthesis)
Example:
A
B
C
F
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
0
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
0
F= A B C +AB C
A
B
C
A
B
C
F
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
More Logical Synthesis
Example:
Thus
A
B
F
0
0
0
A
0
1
1
B
1
0
0
1
1
1
F
A
B
Clearly
F=A B +AB
But it is easy to show that a simpler valid
expression for F is F = B , hence:
B
(ignore A)
F
EECS 40
Fall 2001 Lecture 2
Copyright Regents of University of California
What circuit of logic gates could produce these (arbitrarily
chosen) outputs F in response to inputs A, B and C?
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
F
0
0
1
1
0
1
0
1
W. G. Oldham
EECS 40
Fall 2001 Lecture 2
A B C
A
W. G. Oldham
Copyright Regents of University of California
A B C
ABC
ABC
B
F
C
ABC
ABC
Example of general purpose circuit to implement the truth
table of Table 22.4. (This solution is NOT minimized.)
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
Rules of boolean algebra. The two entries in the last row
are used frequently and are known as DeMorgan’s theorem.
AND Rules
OR Rules
A•A
=
A
A+A
=
A
A•A
=
0
A+A
=
1
0•A
=
0
0+A
=
A
1•A
=
A
1+A
=
1
A•B
=
B•A
A+B
=
B+A
A(BC) =
(AB)C
A+(B+C) =
(A+B)+C
A(B+C) =
AB+AC
A+BC =
(A+B)(A+C)
A•B
A+B
A+B
A•B
=
=
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
Logical Synthesis
Guided by DeMorgan’s Theorem
Demorgan’s Theorem :

A  BC  A B C

or
A  B  C  A B C
Thus, for example:
F  AB  CD  AB  CD
A
B
F
C
D
Thus any sum of
products
expression can
be immediately
synthesized from
NAND gates
alone
EECS 40
Fall 2001 Lecture 2
•
W. G. Oldham
Copyright Regents of University of California
Karnaugh Maps
Graphical approach to minimizing the number of terms
in a logic expression:
1.
2.
3.
4.
Map the truth table into a Karnaugh map (see below)
For each 1, circle the biggest block that includes that 1
Write the product that corresponds to that block.
Sum all of the products
4-variable Karnaugh Map
2-variable
Karnaugh Map
B
0
1
00
BC
00 01 11 10
0 1
A
CD
00 01 11 10
3-variable
Karnaugh Map
0
A
1
AB
01
11
10
EECS 40
Fall 2001 Lecture 2
W. G. Oldham
Copyright Regents of University of California
Karnaugh Map Example
Input
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
Output
C
0
1
0
1
0
1
0
1
S1
0
0
0
1
0
1
1
1
S0
0
1
1
0
1
0
0
1
Simplification of expression for S1:
BC
A
0
1
BC
00 01 11 10
0 0 1 0
0 1 1 1
AC
AC
AB
S1 = AB + BC + AC
By simplifying we can reduce the
number of gates, transistors, and
the size of the circuits.