Prof.
YingLi
Tian
Sept. 10, 2012
Department of Electrical Engineering
The City College of New York
The City University of New York (CUNY)
Lecture
4: Implementation AND, OR,
NOT
Gates and Compliment
1
EE210: Switching Systems
TA’s Email:
Students who didn’t receive TA’s email,
please send an email to Mr. Zhang, by
putting subject: “EE210 email”
Mr.
Chenyang
Zhang
czhang10@ccny.cuny.edu
Course website:
http://www

ee.ccny.cuny.edu/www/web/yltian/EE2100
.html
2
Outlines
Quick Review of the Last Lecture
AND, OR, NOT Gates
Switching Algebra
Properties of Switching Algebra
Definitions of Algebraic Functions
Implementation AND, OR, NOT Gates
Complement (NOT)
Truth table to algebraic expressions
3
Definition of Switching Algebra
OR

a
+
b
(read
a
OR
b
)
AND

a
∙
b
=
ab
(read
a
AND
b
)
NOT

a
´
(read NOT
a
)
4
SOP and POS
A
sum of products
expression (
often abbreviated SOP
) is one or
more product terms connected by OR operators
.
ab
´
+
bc
´
d
+
a
´
d
+ e
´

??
terms,
??
literals
A
product of sums
expression (
POS
) is one or more sum terms
connected by AND operators.
SOP:
x
´
y
+
xy
´
+
xyz
POS: (
x
+
y
´
)(
x
´
+
y)(x
´
+
z
´
)
A
literal
is the appearance of a variable or its complement.
A
term
is one or more literals connected by
AND, OR
, operators.
Gate Implementation
6
P2b: a(
bc
) = (
ab
) c
These three implementations are equal.
Implementation of functions with
AND, OR, NOT Gates

1
Given function:
f=
x
´
yz
´
+
x
´
yz
+
xy
´
z
´
+
xy
´
z
+
xyz
Two

level circuit
(maximum number
of gates which a signal
must pass from the input
to the output)
7
Implementation of functions with
AND, OR, NOT Gates

2
(1)
x
´
yz
´
+
x
´
yz
+
xy
´
z
´
+
xy
´
z
+
xyz
(
2)
x
´
y
+
xy
´
+
xyz
(
3)
x
´
y
+
xy
´
+
xz
(
4)
x
´
y
+
xy
´
+
yz
Function:
x
´
y
+
xy
´
+
xz
,
when only use
uncomplemented
inputs:
Implementation of functions with
AND, OR, NOT Gates

3
Multi

level circuit
10
Function? (see Page50)
Commonly used terms
DIPs
–
dual in

line pin packages (chips)
ICs
–
integrated circuits
SSI
–
small

scale integration (a few gates)
MSI
–
medium

scale integration (~ 100
gates)
LSI

large

scale integration
VLSI
–
very large

scale integration
GSI
–
giga

scale integration
11
Examples
Need a 3

input OR (or AND), and only 2

input gates are available
Need a 2

input OR (or AND), and only 3

input gates are available
12
Positive and Negative Logic
Use 2 voltages to represent logic 0 and 1
For example:
Low: 0

1.4 Volt;
High: >2.1Volt;
Transition state: 1.4

2.1Volt
Positive logic: High voltage
1, Low voltage
0
Negative logic: Low voltage
1,
High
voltage
0
The Complement (NOT)
DeMorgan
:
P11a
: (a + b)
´
= a
´
b
´
P11b
: (
ab
)
´
= a
´
+ b
´
P11aa
: (a + b + c …)
´
= a
´
b
´
c
´
…
P11bb
: (
abc
…)
´
= a
´
+ b
´
+ c
´
+ …
Note:
(
ab
)
´
≠ a
´
b
´
(a + b)
´
≠ a
´
+ b
´
ab
+
a
´
b
´
≠ 1
14
Find the complement of a given function
Repeatedly apply
DeMorgan’s
theorem
1. Complement each variable (a to a
´
or a
´
to a)
2. Replace 0 by 1 and 1 by 0
3. Replace AND by OR, OR by AND, being
sure to preserve the order of operations
See Example 2.5 (Page53) and Example 2.6
(page 54).
15
Example of Complement
16
f
=
wx
´
y
+
xy
´
+
wxz
f
´
= (
wx
´
y
+
xy
´
+
wxz
)
´
=
(
wx
´
y
)
´
(
xy
´
)
´
(
wxz
)
´
=
(
w
´
+
x+
y
´
)
(
x
´
+
y
)(
w
´
+
x
´
+
z
´
)
f
is 1
if
a
= 0 AND
b
= 1
OR
if
a
= 1 AND
b
= 0
OR
if
a
= 1 AND
b
= 1
f
is 1
if
a
´
= 1 AND
b
= 1
OR
if
a
= 1 AND
b
´
= 1
OR
if
a
= 1 AND
b
= 1
f
is 1
if
a
´
b
= 1
OR if
ab
´
= 1
OR if
ab
= 1
f
=
a
´
b
+
ab
´
+
ab
= a + b (OR)
Truth Table to Algebraic Expressions
f (A, B, C)
= ∑m(1, 2, 3, 4,5)
= A
´
B
´
C
+
A
´
B
C
´
+
A
´
B
C +
AB
´
C
´
+
AB
´
C
f
f
´
0
1
1
0
1
0
1
0
1
0
1
0
0
1
0
1
To obtain f (A, B, C), add
all
minterms
with output
= 1 (SOP):
f
´
(A, B, C)
= ∑m(0, 6, 7) =
A
´
B
´
C
´
+
A
B
C
´
+
AB
C
A
standard product term
,
also
minterm
is a product term that
includes each variable of the problem, either
uncomplemented
or complemented.
f = (f
´
)
´
=
(A + B + C)(A
´
+
B
´
+
C)(
A
´
+
B
´
+
C
´
)
f
f
´
0
1
1
0
1
0
1
0
1
0
1
0
0
1
0
1
A
standard sum term
,
also called a
maxterm
, is a sum term that
includes each variable of the problem, either
uncomplemented
or complemented.
POS:
20
f (A, B, C)
=
A
´
B
´
C
+
A
´
B
C
´
+
A
´
B
C
+
AB
´
C
´
+
AB
´
C
=
A
´
B
´
C
+
A
´
B
+
AB
´
=
A
´
(
B
´
C
+
B)
+
AB
´
=
A
´
C
+
A
´
B +
AB
´
=
B
´
C
+
A
´
B +
AB
´
To simplify:
f
´
(A, B, C)
=
A
´
B
´
C
´
+
A
B
C
´
+
AB
C
=
A
´
B
´
C
´
+
A
B
See page56 for details.
P9a
:
ab
+
ab
´
= a
P10a
: a + a
´
b = a + b
P8a:
a (b + c) =
ab
+ ac
P10a:
B +
C
Truth Table with don’t care
Include them as a separate sum.
21
f (a, b, c)
= ∑m(1, 2, 5) + ∑d(0, 3)
a
b
c
f
0
0
0
X
0
0
1
1
0
1
0
1
0
1
1
X
1
0
0
0
1
0
1
1
1
1
0
0
1
1
1
0
Number of different functions of n variables
Announcement:
Review Chapter 2.3

2.5
HW2 is out today, due
on 9/12.
Next class (Chapter 2.6

2.7):
NAND, NOR, Exclusive

OR (EOR) Gates
Simplification of Algebraic Expressions
23
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