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Chapter 2 Professor Brendan Morris, SEB 3216, - - PowerPoint PPT Presentation

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Chapter 2 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Section 1004: Dr. Morris Combinational Logic Design Chapter 2 <1> Chapter 2 :: Topics

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Chapter 2 <1>

Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/

Chapter 2

CPE100: Digital Logic Design I

Section 1004: Dr. Morris Combinational Logic Design

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SLIDE 2

Chapter 2 <2>

  • Introduction
  • Boolean Equations
  • Boolean Algebra
  • From Logic to Gates
  • Multilevel Combinational Logic
  • X’s and Z’s, Oh My
  • Karnaugh Maps
  • Combinational Building Blocks
  • Timing

Chapter 2 :: Topics

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SLIDE 3

Chapter 2 <3>

A logic circuit is composed of:

  • Inputs
  • Outputs
  • Functional specification
  • Timing specification

inputs

  • utputs

functional spec timing spec

Introduction

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Chapter 2 <4>

  • Nodes
  • Inputs: A, B, C
  • Outputs: Y, Z
  • Internal: n1
  • Circuit elements
  • E1, E2, E3
  • Each a circuit

A E1 E2 E3 B C n1 Y Z

Circuits

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SLIDE 5

Chapter 2 <5>

  • Combinational Logic (Ch 2)
  • Memoryless
  • Outputs determined by current values of inputs
  • Sequential Logic (Ch 3)
  • Has memory
  • Outputs determined by previous and current values
  • f inputs

inputs

  • utputs

functional spec timing spec

Types of Logic Circuits

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SLIDE 6

Chapter 2 <6>

  • Every element is combinational
  • Every node is either an input or connects

to exactly one output

  • The circuit contains no cyclic paths

– E.g. no connection from output to internal node

  • Example:

Rules of Combinational Composition

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SLIDE 7

Chapter 2 <7>

  • Functional specification of outputs in terms
  • f inputs
  • Example: S

= F(A, B, Cin) Cout = F(A, B, Cin)

A S S = A  B  Cin Cout = AB + ACin + BCin B Cin C L Cout

Boolean Equations

A B Cin S Cout

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SLIDE 8

Chapter 2 <8>

Goals:

  • Systematically express logical functions using

Boolean equations

  • To simplify Boolean equations

Functional specification

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Chapter 2 <9>

  • Complement: variable with a bar over it

A, B, C

  • Literal: variable or its complement

A, A, B, B, C, C

  • Implicant: product (AND) of literals

ABC, AC, BC

  • Minterm: product that includes all input

variables ABC, ABC, ABC

  • Maxterm: sum (OR) that includes all input

variables (A+B+C), (A+B+C), (A+B+C)

Some Definitions

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SLIDE 10

Chapter 2 <10>

  • All equations can be written in SOP form
  • Each row has a minterm
  • A minterm is a product (AND) of literals
  • Each minterm is TRUE for that row (and only that row)

A B Y 1 1 1 1 1 1 minterm A B A B A B A B minterm name m0 m1 m2 m3

Canonical Sum-of-Products (SOP) Form

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SLIDE 11

Chapter 2 <11>

Y = F(A, B) =

  • All equations can be written in SOP form
  • Each row has a minterm
  • A minterm is a product (AND) of literals
  • Each minterm is TRUE for that row (and only that row)
  • Form function by ORing minterms where the output is TRUE

A B Y 1 1 1 1 1 1 minterm A B A B A B A B minterm name m0 m1 m2 m3

Canonical Sum-of-Products (SOP) Form

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SLIDE 12

Chapter 2 <12>

Y = F(A, B) = AB + AB = Σ(m1, m3)

Canonical Sum-of-Products (SOP) Form

  • All equations can be written in SOP form
  • Each row has a minterm
  • A minterm is a product (AND) of literals
  • Each minterm is TRUE for that row (and only that row)
  • Form function by ORing minterms where the output is TRUE
  • Thus, a sum (OR) of products (AND terms)

A B Y 1 1 1 1 1 1 minterm A B A B A B A B minterm name m0 m1 m2 m3

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Chapter 2 <13>

Y = F(A, B) =

SOP Example

  • Steps:
  • Find minterms that result in Y=1
  • Sum “TRUE” minterms

A B Y 1 1 1 1 1 1

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SLIDE 14

Chapter 2 <14>

Aside: Precedence

  • AND has precedence over OR
  • In other words:
  • AND is performed before OR
  • Example:
  • 𝑍 =

𝐵 ⋅ 𝐶 + 𝐵 ⋅ 𝐶

  • Equivalent to:
  • 𝑍 =

𝐵𝐶 + (𝐵𝐶)

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SLIDE 15

Chapter 2 <15>

  • All Boolean equations can be written in POS form
  • Each row has a maxterm
  • A maxterm is a sum (OR) of literals
  • Each maxterm is FALSE for that row (and only that row)

Canonical Product-of-Sums (POS) Form

A + B A B Y 1 1 1 1 1 1 maxterm A + B A + B A + B maxterm name M0 M1 M2 M3

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SLIDE 16

Chapter 2 <16>

  • All Boolean equations can be written in POS form
  • Each row has a maxterm
  • A maxterm is a sum (OR) of literals
  • Each maxterm is FALSE for that row (and only that row)
  • Form function by ANDing the maxterms for which the
  • utput is FALSE
  • Thus, a product (AND) of sums (OR terms)

Canonical Product-of-Sums (POS) Form

A + B A B Y 1 1 1 1 1 1 maxterm A + B A + B A + B maxterm name M0 M1 M2 M3

𝑍 = 𝑁0 ⋅ 𝑁2 = 𝐵 + 𝐶 ⋅ ( 𝐵 + 𝐶)

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SLIDE 17

Chapter 2 <17>

  • Sum of Products (SOP)
  • Implement the “ones” of the output
  • Sum all “one” terms  OR results in “one”
  • Product of Sums (POS)
  • Implement the “zeros” of the output
  • Multiply “zero” terms  AND results in “zero”

SOP and POS Comparison

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Chapter 2 <18>

  • You are going to the cafeteria for lunch

– You will eat lunch (E=1) – If it’s open (O=1) and – If they’re not serving corndogs (C=0)

  • Write a truth table for determining if you

will eat lunch (E).

O C E 1 1 1 1

Boolean Equations Example

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SLIDE 19

Chapter 2 <19>

  • You are going to the cafeteria for lunch

– You will eat lunch (E=1) – If it’s open (O=1) and – If they’re not serving corndogs (C=0)

  • Write a truth table for determining if you

will eat lunch (E).

O C E 1 1 1 1 1

Boolean Equations Example

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SLIDE 20

Chapter 2 <20>

  • SOP – sum-of-products
  • POS – product-of-sums

O C E 1 1 1 1 minterm O C O C O C O C O + C O C E 1 1 1 1 maxterm O + C O + C O + C

SOP & POS Form

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SLIDE 21

Chapter 2 <21>

  • SOP – sum-of-products
  • POS – product-of-sums

O + C O C E 1 1 1 1 1 maxterm O + C O + C O + C

O C E 1 1 1 1 1 minterm O C O C O C O C

E = (O + C)(O + C)(O + C) = Π(M0, M1, M3) E = OC = Σ(m2)

SOP & POS Form

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SLIDE 22

Chapter 2 <22>

  • SOP – sum-of-products
  • POS – product-of-sums

O + C O C E 1 1 1 1 1 maxterm O + C O + C O + C

O C E 1 1 1 1 1 minterm O C O C O C O C

E = OC = Σ(m2)

SOP & POS Form

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SLIDE 23

Chapter 2 <23>

  • SOP – sum-of-products
  • POS – product-of-sums

O + C O C E 1 1 1 1 1 maxterm O + C O + C O + C

O C E 1 1 1 1 1 minterm O C O C O C O C

E = (O + C)(O + C)(O + C) = Π(M0, M1, M3) E = OC = Σ(m2)

SOP & POS Form

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SLIDE 24

Chapter 2 <24>

  • Axioms and theorems to simplify Boolean

equations

  • Like regular algebra, but simpler: variables

have only two values (1 or 0)

  • Duality in axioms and theorems:

– ANDs and ORs, 0’s and 1’s interchanged

Boolean Algebra

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SLIDE 25

Chapter 2 <25>

Boolean Axioms

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SLIDE 26

Chapter 2 <26>

Duality in Boolean axioms and theorems:

– ANDs and ORs, 0’s and 1’s interchanged

Duality

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SLIDE 27

Chapter 2 <27>

Boolean Axioms

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SLIDE 28

Chapter 2 <28>

Boolean Axioms

Dual: Exchange:• and + 0 and 1

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SLIDE 29

Chapter 2 <29>

Boolean Axioms

Dual: Exchange:• and + 0 and 1

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SLIDE 30

Chapter 2 <30>

Basic Boolean Theorems

B = B

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SLIDE 31

Chapter 2 <31>

Basic Boolean Theorems: Duals

Dual: Exchange:• and + 0 and 1