Transcript EENG 2710 Chapter 2 - UNT College of Engineering

EENG 2710 Chapter 2
Algebraic Methods For The Analysis
and Synthesis of Logic circuits
1
Chapter 2 Homework
2.1c, 2.2c, 2.3, 2.4, 2.5, 2.6b, 2.7a,
2.10b, 2.16b, 2.18b, 2.25, 2.29a
2
Logic Function or Gate
Representation
• Logic functions or gates can be
represented:
–algebraically
–using truth tables
–using electronic circuits.
3
Basic Logic Functions
• The three basic logic functions are:
– AND Gate
Y = AB
– OR Gate
– NOT Gate
4
Algebraic Representation
• Uses Boolean algebra.
• Boolean variables have two states
(binary).
• Boolean operators include AND, OR, and
NOT gates.
5
Truth Table Representation
• Defines the output of a function for every
possible combination of inputs.
• A system with n inputs has 2n possible
combinations.
6
Electronic Circuit Representation
• Uses logic gates to perform Boolean
algebraic functions.
• Gates can be represented by schematic
symbols.
• Symbols can be either distinctive-shape or
rectangular-outline.
7
Distinctive Shape Schematic
Symbols
• Uses different graphic representations
for different logic functions.
• Uses a bubble (a small circle) to
indicate a logical inversion.
8
Rectangular-Outline Schematic
Symbols
• All functions are shown in rectangular
form with the logic function indicated by
standard notation inside the rectangle.
• The notation specifying the logic
function is called the qualifying symbol.
• Inversion is indicated by a 1/2
arrowhead.
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NOT Function
• One input and one output.
• The output is the opposite logic level of the
input.
• The output is the complement of the input.
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NOT Function Boolean
Representation
• Inversion is indicated by a bar over the
signal to be inverted.
YA
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NOT Function Electronic Circuit
• Called a NOT gate or, more usually, an
INVERTER.
• Distinctive-shape symbol is a triangle with
inversion bubble.
• Rectangular-shape symbol uses “1” and
the inversion 1/2 arrowhead.
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NOT Function Electronic Circuit
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AND Function
• Two or more inputs, one output.
• Output is HIGH only when all of the inputs
are HIGH.
• Output is LOW whenever any input is
LOW.
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AND Function
A
B
Y
0
0
0
0
1
1
1
0
1
0
0
1
15
AND Boolean Representation
• AND symbol is “•” or nothing at all.
Y  A B
Y  AB
16
AND Function Electronic Circuit
• Called an AND gate.
• Distinctive-shape symbol uses AND
designation.
• Rectangular-shape symbol use “&” as
designator.
17
AND Function Electronic Circuit
18
AND Function Electronic Circuit
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AND Function Electronic Circuit
A
B
C
Y
0
0
0
0
0
0
0
1
1
1
1
0
1
1
0
0
1
1
1
0
1
0
1
0
1
0
0
0
0
0
0
1
20
OR Function
• Two or more inputs, one output.
• Output is HIGH whenever one or more
input is HIGH.
• Output is LOW only when all of the inputs
are LOW.
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OR Function
A
B
Y
0
0
0
0
1
1
1
0
1
1
1
1
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OR Boolean Representation
• OR symbol is “+”.
• Y=A+B
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OR Function Electronic Circuit
• Called an OR gate.
• Distinctive-shape symbol uses OR
designation.
• Rectangular-shape symbol uses “” as
designator.
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OR Function Electronic Circuit
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Active Level
• The logic level defined as “ON” for a
circuit.
• When a logic HIGH is “ON”, the signal is
active-HIGH.
• When a logic LOW is “ON”, the signal is
active-LOW.
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NAND Function
• Generated by inverting the output of the
AND function.
• Output is HIGH whenever any input is
LOW.
• Output is LOW only when all inputs are
HIGH.
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NAND Function
A
B
Y
0
0
1
0
1
1
1
0
1
1
1
0
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NAND Boolean Representation
• Uses AND with an inversion overbar.
Y  A B
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NAND Function Electronic Circuit
• Called a NAND gate.
• Uses the AND symbol with inversion on.
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NAND Function Electronic Circuit
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NOR Function
• Generated by inverting the output of the
OR function.
• Output is HIGH only when all inputs are
LOW.
• Outputs is LOW whenever any input is
HIGH.
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NOR Function
A
B
Y
0
0
1
0
1
0
1
0
0
1
1
0
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NOR Boolean Representation
• Uses OR with an inversion overbar.
Y  A B
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NOR Function Electronic Circuit
• Called a NOR gate.
• Uses OR symbol with inversion on the
output.
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NOR Function Electronic Circuit
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3 Input NOR and NAND Function
Truth Tables
• 3 Input NAND:
Y  A B  C
• 3 Input NOR:
Y  A BC
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3 Input NOR and NAND Function
Truth Tables
A BC
A BC
1
1
1
1
1
1
1
0
1
0
0
0
0
0
0
0
A B C
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
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Exclusive OR Gate
• Two inputs, one output.
• Output is HIGH when one, and only one,
input is HIGH.
• Output is LOW when both inputs are equal
– both HIGH or both LOW.
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Exclusive OR Gate
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Exclusive OR Gate
A
B
Y
0
0
0
0
1
1
1
0
1
1
1
0
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Exclusive NOR Gate
• Two inputs, one output.
• Output is HIGH when both inputs are
equal – both HIGH or both LOW.
• Output is LOW when one, and only one,
input is HIGH.
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Exclusive NOR Gate
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Exclusive NOR Gate
A
B
Y
0
0
1
0
1
0
1
0
0
1
1
1
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Gate Equivalence – NAND
• A NAND gate can be represented by an
AND gate with inverted output.
• A NAND gate can be represented by an
OR gate with inverted inputs.
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Gate Equivalence – NAND
46
Gate Equivalence – NOR
• A NOR gate can be represented by an OR
gate with inverted output.
• A NOR gate can be represented by an
AND gate with inverted inputs.
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Gate Equivalence – DeMorgan Forms
• Change an AND function to an OR
function and an OR function to an AND
function.
• Invert the inputs.
• Invert the outputs.
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DeMorgan’s Theorem
• A B  A  B
• A B  A  B
• Break the line and change the sign
49
DeMorgan’s Theorem
• The following are two common errors
associated with DeMorgan’s Theorem:
A B  A  B
A B  A  B
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Active Logic Levels
• Any INPUT or OUTPUT that has a
BUBBLE is considered as active LOW.
• Any INPUT or OUTPUT that has no
BUBBLE is considered as active HIGH.
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Active Logic Levels - NOR
• Y  A B
• At least one input HIGH makes the output
LOW.
• Y  AB
• All inputs LOW make the output HIGH.
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Active Logic Levels - NOR
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Postulates of Boolean Algebra
• P1: + = OR, . = AND
• P2(a): a + 0 = a
b
a
y
0
0
0
0
1
1
1
0
1
1
1
1
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Postulates of Boolean Algebra
• P2(b): = a . 1 = a
b
a
y
0
0
0
0
1
0
1
0
0
1
1
1
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Postulates of Boolean Algebra
• P3(a): a + b = b + a
• P3(b): b + a = a + b
. .a + (b + c) = (a + b) + c
• P4(a):
• P4(a): a(b c) = (a b)c
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Postulates of Boolean Algebra
• P5(a): a + bc = (a + b)(a + c)
a
a + bc
b
c
a
bc
a+b
b
(a + b)(a + c)
c
a+c
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Postulates of Boolean Algebra
• P5(b): a (b + c) = ab + ac
a
b
c
a(b + c)
b+c
a
b
ab +ac
c
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Postulates of Boolean Algebra
• P6(a): a + a’ = 1 where a’ = not a
a’
a
y
0
0
0
0
1
1
1
0
1
1
1
1
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Postulates of Boolean Algebra
• P6(b): a . a’ = 0
a’
a
y
0
0
0
0
1
0
1
0
0
1
1
1
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Postulates of Boolean Algebra
• Postulates not in your book: x  0 = x
A
0
0
1
1
X
0
1
0
1
X=0
X=1
Y=0
Y=1
Y
0
1
1
0
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Postulates of Boolean Algebra
• Postulates not in your book: x  1 = x’
A
0
0
1
1
X
0
1
0
1
X=0
X=1
Y=1
Y=0
Y
0
1
1
0
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Theorems of Boolean Algebra
• T1(a): a + a = a
– Short Proof: If a = 1, 1 + 1 = 1 or If a = 0, 0 + 0 = 0
– Long proof: (shown in book)
63
Theorems of Boolean Algebra
• T1(b): a . a = a
– Short Proof: If a = 1, 1 x 1 = 1 or If a = 0, 0 x 0 = 0
• T2(a): a + 1 = 1
– Short Proof: If a = 1, 1 + 1 = 1 or If a = 0, 0 + 1 = 1
• T2(b): a . 0 = 0
– Short Proof: If a = 1, 1 x 0 = 0 or If a = 0, 0 x 0 = 0
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Theorems of Boolean Algebra
• T3(a): (a’)’ = a
– Short Proof: If a = 1, then a’ =0, Thus (0)’ = 1
If a = 0, then a’ =1, Thus (1)’ = 0
a’
a
(a’)’ = a
• T4(a): a + ab = a
– Proof: a + ab = a(1 + b) = a(1) = a
a
a + ab = a
ab
b
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Theorems of Boolean Algebra
• T4(b): a (a + b) = a
– Proof: a (a + b) = aa + ab) = a + ab
= a(1 + b) = a(1) = a
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Theorems of Boolean Algebra
• T5(a): a + a’b = a + b
– Proof: a + a’b = (a + a’)( a + b) = 1(a + b) = a + b
• T5(b): a (a’+b) = ab
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Theorems of Boolean Algebra
• T6(a): ab + ab’ = a
– Proof: ab + ab’ = a(b + b’) = a(1) = a
• T6(b): (a + b)(a + b’) = a
– Proof: (a + b)(a + b’) = aa + ab’ +ab’ +bb’
= a + ab’ = a(1 + b) = a(1) =a
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Theorems of Boolean Algebra
• T7(a): ab + ab’c = ab + ac
– Proof: ab + ab’c = a(b + b’c) = a (b + b’)( b + c)
= a(1)(b + c) = ab + ac
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Theorems of Boolean Algebra
• T7(b): (a + b)(a + ab’ + c) = (a + b) + (a + c)
– Proof: a + ab’ + c
a+b
aa + aab’ + ac
ab + abb’ + bc
a + ab’ + ac + ab + 0 + bc
a(1 + b’) + ac + ab + bc
a + ac + ab + bc
a(1 + c) + ab + bc
a + ab + bc
a(1 +b) + bc
a + bc = (a + b) + (a + c) Same as P5(a)
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Theorems of Boolean Algebra
(DeMorgan’s Theorem)
• T8(a): (a + b)’ = a’b’
• T8(b): (ab)’ = a’ + b’
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Theorems of Boolean Algebra
• T9(a): ab + a’c + bc = ab + a’c
– Proof: ab + a’c + bc = ab +a’c + (1)(bc)
= ab +a’c + (a + a’)(bc)
= ab +a’c + (abc + a’bc)
= (ab +abc) + (a’c + a’cb)
= ab(1+c) + a’c(1 + b)
= ab + a’c
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Theorems of Boolean Algebra
• T9(b): (a + b)(a’ + c)(b + c) = (a + b)(a’ + c)
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Operations with Logic 0 & 1
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Operations with the Same Variable &
Complement of a Variable
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Simplifying an Expression
Y  AB  AB C
factoringout the common term AB
Y  AB (1  C )
applying theorem 11: (1  x  1)
Y  AB  1
applying theorem 10 : ( x  1  x )
Y  AB
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Simplifying an Expression
Y  AB  (C  A)
applying DeMorgan's theorem 20
Y  AB  (C  A)
applying DeMorgan's theorem 20
Y  A B C  A
factoringout A
Y  A (1  B  C  1)
applying theorem 11 : x  1  1
Y  A (1)
Y A
77
Problem 6a
• Simplify the following switching function:
78
Algebraic Forms of switching functions
• Product term:
– Part of a Boolean expression where one or
more true or complement variables are
ANDed.
• Sum term:
– Part of a Boolean expression where one or
more true or complement variables are
ORed.
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Algebraic Forms of Switching Functions
• Sum-of-products (SOP):
– A Boolean expression where several
product terms are summed (ORed)
together.
• Product-of-sum (POS):
– A Boolean expression where several sum
terms are multiplied (ANDed) together.
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Algebraic Forms of Switching Functions
(Examples of SOP and POS Expressions)
SOP : Y  AB  BC  A D
POS : Y  ( A  B )  (B  C )  ( A  C )
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Algebraic Forms of Switching Functions
• SOP and POS Utility
– SOP and POS formats are used to present a
summary of a switching circuit operation.
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Algebraic Forms of Switching Functions
(Canonical SOP Minterms)
83
Algebraic Forms of Switching Functions
(Canonical SOP Minterms)
84
Algebraic Forms of Switching Functions
(Canonical SOP Minterms)
Example
85
Algebraic Forms of Switching Functions
(Canonical SOP Minterms)
86
Algebraic Forms of Switching Functions
(Canonical POS Maxterms)
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Algebraic Forms of Switching Functions
• Truth Table: POS = 0’s and SOP = 1’s
m2
m3
m6
m7
M0
M1
M4
M5
• M(0,1, 4, 5) = m(2, 3, 6, 7)
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Problem 16a & 18a
• M(1) = m(0, 2,3)
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Problem 18a
(Using Boolean Algebra)
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Universality of NAND/NOR
Gates
• Any logic gate can be implemented using
only NAND or only NOR gates.
91
NOT from NAND
• An inverter can be constructed from a
single NAND gate by connecting both
inputs together.
92
NOT from NAND
93
AND from NAND
• The AND gate is created by inverting the
output of the NAND gate.
Y  A B  AB
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AND from NAND
95
OR and NOR from NAND
OR  X  Y  X  Y
NOR  X  Y  X  Y
96
OR from NAND
97
NOR from NAND
98
NOT from NOR
• An inverter can be constructed from a
single NOR gate by connecting both inputs
together.
99
NOT from NOR
100
OR from NOR
• The OR gate is created by inverting the
output of the NOR gate.
Y  AB  AB
101
OR from NOR
102
AND and NAND from NOR
AND  X  Y  X  Y
NAND  X  Y  X  Y
103
AND from NOR
104
NAND from NOR
105
Simplest Switching Expression From
A Timing Diagram
106
Simplest Switching Expression From
A Timing Diagram
107
Simplest Switching Expression From
A Timing Diagram
108
Simplest Switching Expression From
A Timing Diagram
109
Problem 29b
110
Venn Diagrams
111
Venn Diagrams
112