Combinational Circuits Basic Gates amp Truth Tables Basic Gates AND Gate OR Gate NOT Gate More Gates NAND Gate NOR Gate BUF Gate More Gates XNOR Gate XOR Gate nInput Gates 3Input XOR Gate ID: 618881
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Slide1
Digital Logic Systems
Combinational CircuitsSlide2
Basic Gates
&
Truth TablesSlide3
Basic Gates
AND Gate
OR Gate
NOT GateSlide4
More Gates
NAND Gate
NOR Gate
BUF GateSlide5
More Gates
XNOR Gate
XOR GateSlide6
n-Input Gates
3-Input XOR Gate
5-Input NOR Gate
5-Input AND Gate
4-Input OR GateSlide7
Definitions
AND
It gives a logical output true only if all the inputs are true
OR
It gives a logical output true if any of the inputs is true
XOR
It gives a logical output true only if an odd-number of inputs is true
NOT
It gives a logical output true if the input is false and vice versaSlide8
Truth Table
A truth table is a tabular procedure to express the relationship of the outputs to the inputs of a Logical SystemSlide9
Truth Tables for Gates
a
b
f
AND
0
0
0
0
1
0
1
0
0
1
1
1
a
b
f
OR
0
0
0
0
1
1
1
0
1
1
1
1
a
f
NOT
0
1
1
0
AND Operation
OR Operation
NOT Operation
AND Gate
OR Gate
NOT GateSlide10
Truth Tables for Gates
a
b
f
NAND
0
0
1
0
1
1
1
0
1
1
1
0
a
b
f
NOR
0
0
1
0
1
0
1
0
0
1
1
0
a
f
BUF
0
0
1
1
NAND Operation
NOR Operation
BUF Operation
NAND Gate
NOR Gate
BUF GateSlide11
Truth Tables for Gates
a
b
f
XOR
0
0
0
0
1
1
1
0
1
1
1
0
a
b
f
XNOR
0
0
1
0
1
0
1
0
0
1
1
1
XOR
Operation
XNOR
Operation
XNOR Gate
XOR GateSlide12
A Bubble Implies a Logical Inversion
Bubbles can be replaced by NOT Gates to get logically equivalent circuits
BubblesSlide13Slide14Slide15Slide16Slide17Slide18Slide19Slide20Slide21Slide22
Generate tables for all combinations of bubbles and a XOR gateSlide23
Gate Equivalence
=
=
=Slide24
Gate Equivalence
=
=
?Slide25
Gate Equivalence
=
=Slide26
Switching ExpressionsSlide27
Basic Switching Expressions
AND
f = a . b
OR
f = a + b
NOT
f = a’
f = āSlide28
Is there an expression for XOR operation?Slide29Slide30Slide31
Switching ExpressionsSlide32
Switching ExpressionsSlide33
Switching Expressions
f
1
= a . b’
f
2
= (a + b)’Slide34
Switching ExpressionsSlide35
Switching ExpressionsSlide36
Switching Expressions
f = ? Slide37
Switching Expressions
f = m + n
n = a’ . b
m = a . b’Slide38
Switching Expressions
f = (a . b’) + (a’ . b)
This is the equivalent circuit and equivalent expression for a XOR operation Slide39
From
Digital Design, 5th
Edition
by M. Morris Mano and Michael
Ciletti
Slide40
Switching ExpressionsSlide41
Switching ExpressionsSlide42
Switching Expressions
f
1
= a . b
f
2
= a ^ b
f
2
= (a . b’) + (a’ . b)Slide43
Switching ExpressionsSlide44
x
y
z
p = x ^ y
g = x . y
m = p . z
s = p ^ z
c = m + g
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Slide45
x
y
z
p = x ^ y
g = x . y
m = p . z
s = p ^ z
c = m + g
0
0
0
0
0
0
0
1
0
0
0
1
0
1
0
0
1
1
1
0
1
0
0
1
0
1
0
1
1
0
1
1
0
0
1
1
1
1
0
1
Slide46
x
y
z
p = x ^ y
g = x . y
m = p . z
s = p ^ z
c = m + g
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
1
0
0
0
1
1
1
0
1
1
0
0
1
0
0
1
0
1
1
0
1
1
1
0
0
1
0
1
1
1
0
1
0
Slide47
x
y
z
p = x ^ y
g = x . y
m = p . z
s = p ^ z
c = m + g
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
1
0
0
1
0
0
1
1
1
0
1
0
1
1
0
0
1
0
0
1
0
1
0
1
1
0
1
0
1
1
1
0
0
1
0
0
1
1
1
1
0
1
0
1
1Slide48
x
y
z
s
c
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
1
1
0
0
1
0
1
0
1
0
1
1
1
0
0
1
1
1
1
1
1Slide49
s = s
c = m + g Slide50
s = s
c = m + g
m = p . z
g = g
s = p ^ zSlide51
s = s
c = m + g
m = p . z
g = g
p = x ^ y
g = x . y
s = p ^ zSlide52
s = s
c = m + g
p = x ^ y
g = x . y
m = (x ^ y) . z
g = g
s = (x ^ y) ^ zSlide53
s = (x ^ y) ^ z
c = ((x ^ y) . z)
+ (x . y)
p = x ^ y
g = x . y
m = (x ^ y) . z
g = g
s = (x ^ y) ^ zSlide54
s = (x ^ y) ^ z
c = ((x ^ y) . z) + (x . y)Slide55
s = ((x
. y
’) + (x’ . y)) ^ z
c = (((x . y’) + (x’ . y)) . z) + (x . y)Slide56
s = (((x . y’) + (x’ . y))’ . z) + (((x . y’) + (x’ . y)) . z’)
c = (((x .
y’)
+ (x’ . y)) . z) + (x . y)Slide57
Procedure
To obtain the output functions from a logic diagram, proceed as follows:
Label with arbitrary symbols all gate outputs that are a function of the input variables. Obtain the Boolean Functions for each gate.
Label with other arbitrary symbols those gates that are a function of input variables and/or preciously labeled gates. Find the Boolean functions of these gates.
Repeat the process in step 2 until all the outputs of the circuit are obtained.
By repeated substitution of previously defined functions, obtain the output Boolean functions in terms of input variables only.Slide58Slide59