# Lecture 4: Implementation AND, OR, NOT

Electronics - Devices

Nov 27, 2013 (4 years and 5 months ago)

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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

a

OR
b
)

AND
--

a

b

=
ab

a

AND
b
)

NOT
--

a
´

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