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9/16/2015
1
INTRODUCTION TO DIGITAL LOGIC
LOGIC GATES NOT Gate (Inverter) AND Gate OR Gate NAND Gate NOR Gate X‐OR and X‐NOR Gates Fixed‐function logic: IC Gates
Introduction(1) All Logic circuit and functions are made from basic logic gates Three basic logic gates:
AND gate – expressed by “ . ” OR gate – expressed by “+” sign (NOTE: it is not an ordinary addition) NOT gate – expressed by “ ’ “ or “ˉ”
Introduction(2)
Think about these logic gates as bricks in a structure. Individuals bricks can be arranged to form various type of buildings, and bricks can be used to build fireplaces, steps, walls, walkways and floor.
Likewise, individual logic gates are arranged and interconnected to form various function in a digital system
There are several type of logic gates, just as there may be several shapes/sizes of bricks in a structure.
By: Thomas L. Floyd & David M. Buchla
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NOT Gate (Inverter)
a) Gate Symbol & Boolean Equation
b) Truth Table c) Timing Diagram
OR Gate
a) Gate Symbol & Boolean Equation
b) Truth Table c) Timing Diagram
AND Gate
a) Gate Symbol & Boolean Equation
b) Truth Table c) Timing Diagram
NAND Gate
a) Gate Symbol, Boolean Equation
& Truth Table
b) Timing Diagram
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NOR Gate
a) Gate Symbol, Boolean Equation & Truth Table
b) Timing Diagram
Exclusive-OR (XOR)Gate
BABABA
a) Gate Symbol, Boolean Equation & Truth Table b) Timing Diagram
Exclusive-NOR (XNOR)Gate
BABABA
XNOR
a) Gate Symbol, Boolean Equation
1 0 0 1
DIP and SOIC packages
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Universality of Gates(1) NAND Gate
Universality of Gates(2) NOR Gate
Examples : Logic Gates IC
NOT gate AND gate Note : x is referring to family/technology (eg : AS/ALS/HCT/AC etc.)
Performance Characteristics and Parameters
Propagation delay Time
High-speed logic has a short pdt.
DC Supply Voltage (VCC)
Power Dissipation
Lower power diss. means less current from dc supply
Input and Output (I/O) Logic Levels
Speed-Power product
Fan-Out and Loading
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BOOLEAN ALGEBRA Boolean Operations & expression Laws & rules of Boolean algebra DeMorgan’s Theorems Boolean analysis of logic circuits Simplification using Boolean Algebra Standard forms of Boolean Expressions Boolean Expressions & truth tables The Karnaugh Map (K‐Map) – SOP, POS, 5 Variables Programmable Logic
Boolean Operations & expression Expression: Variable: a symbol used to represent logical quantities (1 or 0)
Eg.: A, B,..used as variable
Complement: inverse of variable and indicated by bar over variable Eg.: Ā
Operation: Boolean Addition – equivalent to the OR operation
Eg.: X = A + B
Boolean Multiplication – equivalent to the AND operation
Eg.: X = A∙B
X
X
A B
A B
Laws & Rules of Boolean algebra
Commutative Law of Addition
Commutative law of addition, A+B = B+A
the order of ORing does not matter.
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Commutative Law of Multiplication
Commutative law of Multiplication AB = BA
the order of ANDing does not matter.
Associative Law of Addition
Associative law of addition A + (B + C) = (A + B) + C
The grouping of ORed variables does not matter
Associative Law of Multiplication
Associative law of multiplication A(BC) = (AB)C
The grouping of ANDed variables does not matter
Distributive Law
A(B + C) = AB + AC
(A+B)(C+D) = AC + AD + BC + BD
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Boolean Rules (1)
1) A + 0 = A
Mathematically if you add O you have changed nothing In Boolean Algebra ORing with 0 changes nothing
Boolean Rules (2)
2) A + 1 = 1 ORing with 1 must give a 1 since if any input is 1 an OR gate will give a 1
Boolean Rules (3)
3) A • 0 = 0 In math if 0 is multiplied with anything you get 0. If you AND anything with 0 you get 0
Boolean Rules (4)
4) A • 1 = A ANDing anything with 1 will yield the anything
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Boolean Rules (5)
5) A + A = A
ORing with itself will give the same result
Boolean Rules(6)
6) A + A = 1 Either A or A must be 1 so A + A =1
Boolean Rules(7)
7) A • A = A ANDing with itself will give the same result
Boolean Rules(8)
8) A • A = 0 In digital Logic 1 =0 and 0 =1, so AA=0 since one of the inputs must be 0.
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Boolean Rules(9)
9) A = A If you NOT something twice you are back to the beginning
Boolean Rules(10)
10) A + AB = A Proof:
A + AB = A(1 + B) DISTRIBUTIVE LAW
= A·1 RULE 2: (1+B)=1
= A RULE 4: A·1 = A
Boolean Rules(11)
11) A + AB = A + B If A is 1 the output is 1 , If A is 0 the output is B
Proof :
A + AB = (A + AB) + AB RULE 10
= (AA +AB) + AB RULE 7
= AA + AB + AA +AB RULE 8
= (A + A)(A + B) FACTORING
= 1·(A + B) RULE 6
= A + B RULE 4
Boolean Rules(12)
12) (A + B)(A + C) = A + BC Proof :
(A + B)(A +C) = AA + AC +AB +BC DISTRIBUTIVE LAW
= A + AC + AB + BC RULE 7
= A(1 + C) +AB + BC FACTORING
= A.1 + AB + BC RULE 2
= A(1 + B) + BC FACTORING
= A.1 + BC RULE 2
= A + BC RULE 4
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END OF BOOLEAN THEOREM