Lecture 4: Implementation AND, OR, NOT

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27 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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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

(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