Combinational Circuits

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Combinational Circuits

• ECE-331, Digital Design
• Dr. Ron Hayne
• Electrical and Computer Engineering

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Equivalent Representations of Logic Functions

• Equivalent, Not Necessarily Equal
• Four Formalisms
• Truth Table
• Schematic symbols
• Boolean (Switching) Algebra
• Hardware Description Language

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First Equivalent Representation

• Truth Tables
• Enumerate all possible inputs
• Determine output by applying binary operators
• Identical I/O relationships show equivalent
  functions

Inputs Inputs A AND B
A B C
0 0 0
0 1 0
1 0 0
1 1 1
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Circuit Diagram Symbols

• AND
• OR
• NOT

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Circuit Diagram Symbols

• NAND
• NOR
• XOR

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2nd Equivalent Representation

• Circuit Diagrams
• Implement logic functions using representations
  of physical circuits comprised of logic gates

S
I1
F1
E
Z
I0
F2
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Binary Logic Operations

• AND
• a AND b c
• a b c
• OR
• a OR b c
• a b c

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Unary Logic Operation

• INVERT
• NOT ( a ) a pronounced a bar
• 1s Complement

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3rd Equivalent Representation

• Boolean Algebra
• Math. description of logic circuit behavior
• Apply PT to put expression in canonical form

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4th Equivalent Representation

• (V)HDL
• VHSIC Hardware Description Language
• Two Components
• Entities specify the interface
• Architecture(s) specify the implementation
• Hierarchical
• Simplest architecture is behavioral description
• Structural architecture is composed of entities

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Two Processes

• Analysis
• Circuit Boolean Function TT
• Design
• TT Boolean Function Circuit

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Analysis of Parity Generator

• Circuit

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Analysis of Parity Generator

• Boolean Algebraic Representation

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Analysis of Parity Generator

• Truth Table

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Algebras

• Algebras are Comprised of two Components
• Set of elements
• Binary Operator(s)
• a b c multiplication operator
• a b c addition operator
• Binary because there are two operands, not
  because the operands take on binary values
• Groups, Rings, and Fields Are Algebras

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Boolean Algebra

• George Boole
• 19th century English mathematician
• Book on propositional calculus
• Claude Shannon
• 20th century American
• Applied Boolean algebra to a network of switches
• Switching algebra

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Boolean Algebra

• E. Huntington
• 20th century
• Developed 10 postulates of Boolean algebra
  (axioms) which are assumed to be true
• All theorems of switching algebra can be derived
  from these postulates

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Huntingtons Postulates

• Commutative
• Distributive

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Huntingtons Postulates

• Identity
• Complement

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Complete Set

• All Boolean Algebraic Functions Can Be Generated
  From Either of Two Sets of Two Fundamental
  Operators
• AND and INVERT
• OR and INVERT
• Common Implementation Gates
• NAND
• NOR

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DeMorgans Theorem

• Example Implement the following function using
  only NAND gates

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Canonical Forms

• Non-minimal, Standard Forms for Writing Algebraic
  Expressions
• SOP
• Sum of Products
• Minterms
• POS
• Product of Sums
• Maxterms

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Canonical SOP Form

• a.k.a., Disjunctive Normal Form
• SOP
• Algebraic Expression with ANDed terms ORed
  together
• Canonical
• Each product term contains each variable in
  either complemented or uncomplemented form
• Product terms are known as minterms
• Represent 1s of the function

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Minterms

• The Order of the Variables in Each Term Is As
  Specified in the Function, e.g.,

MSB
LSB
minterm ( 11 )2 ( 3 )10
minterm ( 01 )2 ( 1 )10
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Minterm Example
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Equivalent Specifications

• Algebra
• Circuit Diagram

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Canonical POS Form

• a.k.a., Conjunctive Normal Form
• POS
• Algebraic Expression with ORed terms ANDed
  together
• Canonical
• Each sum term contains each variable in either
  complemented or uncomplemented form
• Sum terms are known as Maxterms
• Represent 0s of the function

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Maxterm Example
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Algebraic Manipulation
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Canonical SOP Form Example
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TT from SOP Form
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TT from SOP Form
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TT from Canonical SOP (fill in)
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Language of Digital

• A Complemented Variable is CalledX-BAR
• An Asserted Signal Is One That Is True
• Asserted as a 1 Positive logic
• Asserted as a 0 Negative logic
• Crossing Wires in a Schematic Are NOT Connected,
  ANSI Drafting Standard

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Timing Diagrams
Tested Circuit OR3 port map (AgtInput(2),
BgtInput(1), CgtInput(0), ZgtZ)
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End of Lecture

• Equivalence
• Boolean Algebra
• Minterms Maxterms
• SOP/POS
• Canonical Forms
• Dont Cares
• Timing Diagrams