Prof. YingLi TianSept. 10, 2012

Department of Electrical Engineering The City College of New York

The City University of New York (CUNY)

Lecture 4: Implementation AND, OR, NOT Gates and Compliment

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EE210: Switching Systems

TA’s Email: Students who didn’t receive TA’s email,

please send an email to Mr. Zhang, by putting subject: “EE210 email”

Mr. Chenyang Zhang czhang10@ccny.cuny.edu

Course website:http://www-ee.ccny.cuny.edu/www/web/yltian/EE2100.html

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Outlines Quick Review of the Last Lecture

AND, OR, NOT Gates Switching Algebra Properties of Switching Algebra Definitions of Algebraic Functions

Implementation AND, OR, NOT Gates Complement (NOT) Truth table to algebraic expressions

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Definition of Switching Algebra OR -- a + b (read a OR b) AND -- a · b = ab (read a AND b) NOT -- a´ (read NOT a)

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SOP and POS

A sum of products expression (often abbreviated SOP) is one or more product terms connected by OR operators.

ab´ + bc´d + a´d + e´ ---- ?? terms, ?? literals

A product of sums expression (POS) is one or more sum terms connected by AND operators.

SOP: x´y + xy´ + xyz

POS: (x + y´)(x´ + y)(x´ + z´)

A literal is the appearance of a variable or its complement.

A term is one or more literals connected by AND, OR, operators.

Gate Implementation

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P2b: a(bc) = (ab) c

These three implementations are equal.

Implementation of functions with AND, OR, NOT Gates -- 1 Given function: f= x´yz´ + x´yz + xy´z´ +

xy´z + xyz Two-level circuit

(maximum number of gates which a signal must pass from the inputto the output)

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Implementation of functions with AND, OR, NOT Gates -- 2(1) x´yz´ + x´yz + xy´z´ + xy´z + xyz

(2) x´y + xy´ + xyz

(3) x´y + xy´ + xz

(4) x´y + xy´ + yz

Function: x´y + xy´ + xz,

when only use uncomplemented inputs:

Implementation of functions with AND, OR, NOT Gates -- 3

Multi-level circuit

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Function? (see Page50)

Commonly used terms DIPs – dual in-line pin packages (chips) ICs – integrated circuits SSI – small-scale integration (a few gates) MSI – medium-scale integration (~ 100

gates) LSI -- large-scale integration VLSI – very large-scale integration GSI – giga-scale integration

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Examples Need a 3-input OR (or AND), and only 2-

input gates are available Need a 2-input OR (or AND), and only 3-

input gates are available

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Positive and Negative LogicUse 2 voltages to represent logic 0 and 1 For example:

Low: 0-1.4 Volt; High: >2.1Volt; Transition state: 1.4-2.1Volt

Positive logic: High voltage 1, Low voltage 0Negative logic: Low voltage 1, High voltage 0

The Complement (NOT) DeMorgan:

P11a: (a + b)´ = a´ b´ P11b: (ab)´ = a´ + b´ P11aa: (a + b + c …)´ = a´ b´ c´ … P11bb: (abc…)´ = a´ + b´ + c´ + …

Note: (ab)´ ≠ a´ b´ (a + b)´ ≠ a´ + b´ ab + a´ b´ ≠ 1

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Find the complement of a given function Repeatedly apply DeMorgan’s theorem

1. Complement each variable (a to a´ or a´ to a)2. Replace 0 by 1 and 1 by 03. Replace AND by OR, OR by AND, being

sure to preserve the order of operations

See Example 2.5 (Page53) and Example 2.6 (page 54).

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Example of Complement

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f = wx´y + xy´ + wxz

f ´ = (wx´y + xy´ + wxz)´ = (wx´y)´(xy´)´(wxz)´ = (w´+x+y´)(x´+y)(w´+x´+z´)

f is 1 if a = 0 AND b = 1 ORif a = 1 AND b = 0 ORif a = 1 AND b = 1

f is 1 if a´ = 1 AND b = 1 ORif a = 1 AND b´ = 1 ORif a = 1 AND b = 1

f is 1 if a´b = 1 OR if ab´ = 1 OR if ab = 1

f = a´b + ab´ + ab = a + b (OR)

Truth Table to Algebraic Expressions

f (A, B, C) = ∑m(1, 2, 3, 4,5) = A´B´C + A´BC´ + A´BC + AB´C´+ AB´C

f f´0 11 01 01 01 01 00 10 1

To obtain f (A, B, C), add all minterms with output = 1 (SOP):

f ´(A, B, C) = ∑m(0, 6, 7) = A´B´C´ + ABC´ + ABC

A standard product term, also minterm is a product term that includes each variable of the problem, either uncomplemented or complemented.

f = (f ´ )´= (A + B + C)(A´+B´+C)(A´+B´+C´)

f f´0 11 01 01 01 01 00 10 1

A standard sum term, also called a maxterm, is a sum term that includes each variable of the problem, either uncomplemented or complemented.

POS:

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f (A, B, C) = A´B´C + A´BC´ + A´BC + AB´C´+ AB´C = A´B´C + A´B + AB´ = A´(B´C + B) + AB´

= A´C + A´B + AB´ = B´C + A´B + AB´

To simplify:

f ´(A, B, C) = A´B´C´ + ABC´ + ABC = A´B´C´ + AB

See page56 for details.

P9a: ab + ab´ = a

P10a: a + a´ b = a + b

P8a: a (b + c) = ab + ac

P10a: B + C

Truth Table with don’t care Include them as a separate sum.

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f (a, b, c) = ∑m(1, 2, 5) + ∑d(0, 3) a b c f0 0 0 X

0 0 1 1

0 1 0 1

0 1 1 X

1 0 0 0

1 0 1 1

1 1 0 0

1 1 1 0

Number of different functions of n variables

Announcement: Review Chapter 2.3-2.5 HW2 is out today, due on 9/12. Next class (Chapter 2.6-2.7):

NAND, NOR, Exclusive-OR (EOR) Gates Simplification of Algebraic Expressions

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