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**CHAPTER 1 INTRODUCTION TO DIGITAL LOGIC**

PGT 104 ELEKTRONIK DIGIT CHAPTER 1 INTRODUCTION TO DIGITAL LOGIC

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**LOGIC GATES NOT Gate (Inverter) AND Gate OR Gate NAND Gate NOR Gate**

X-OR and X-NOR Gates Fixed-function logic: IC Gates

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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 “¯”

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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

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**OR Gate a) Gate Symbol & Boolean Equation c) Timing Diagram**

b) Truth Table c) Timing Diagram

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**AND Gate a) Gate Symbol & Boolean Equation b) Truth Table**

c) Timing Diagram

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**a) Gate Symbol, Boolean Equation**

NAND Gate a) Gate Symbol, Boolean Equation & Truth Table b) Timing Diagram

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**a) Gate Symbol, Boolean Equation**

NOR Gate a) Gate Symbol, Boolean Equation & Truth Table b) Timing Diagram

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**Exclusive-OR (XOR)Gate a) Gate Symbol, Boolean Equation & Truth Table**

b) Timing Diagram

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**Exclusive-NOR (XNOR)Gate a) Gate Symbol, Boolean Equation**

1 a) Gate Symbol, Boolean Equation

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DIP and SOIC packages

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**Universality of Gates(1)**

NAND Gate

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**Universality of Gates(2)**

NOR Gate

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**Examples : Logic Gates IC**

AND gate NOT gate Note : x is referring to family/technology (eg : AS/ALS/HCT/AC etc.)

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**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

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**Boolean Operations & 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 A X B A X B

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**Laws & Rules of Boolean algebra**

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**Commutative Law of Addition**

A+B = B+A the order of ORing does not matter.

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**Commutative Law of Multiplication**

AB = BA the order of ANDing does not matter.

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**Associative Law of Addition**

A + (B + C) = (A + B) + C The grouping of ORed variables does not matter

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**Associative Law of Multiplication**

A(BC) = (AB)C The grouping of ANDed variables does not matter

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**(A+B)(C+D) = AC + AD + BC + BD**

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

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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

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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

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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

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Boolean Rules(6) 6) A + A = 1 Either A or A must be 1 so A + A =1

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**Boolean Rules(7) 7) A • A = A**

ANDing with itself will give the same result

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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

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**Boolean Rules(10) 10) A + AB = A**

Proof: A + AB = A(1 + B) DISTRIBUTIVE LAW = A∙ RULE 2: (1+B)=1 = A RULE 4: A∙1 = A

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**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

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**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

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