ee177 supplement 2

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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