EE210: Switching Systems Lecture 4: Implementation AND, OR, NOT Gates and Complement Prof. YingLi Tian Sept. 14, 2016 Department of Electrical Engineering The City College of New York The City University of New York (CUNY) 1
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 2
3 Gate Implementation P2b: a(bc) = (ab) c P2a: a + (b + c) = (a + b) + c Who want to implement it?
Definition of Switching Algebra OR -- a + b (read a OR b) AND -- a b = ab (read a AND b) NOT -- a (read NOT a) 4
5 Gate Implementation P2b: a(bc) = (ab) c These three implementations are equal. P2a: a + (b + c) = (a + b) + c Need a volunteer to implement it.
Manipulation of Algebraic Functions -- 1 A literal is the appearance of a variable or its complement. ab + bc d + a d + e ---- 8 literals. A product term is one or more literals connected by AND operators. ab + bc d + a d + e ---- 4 product terms (ab, bc d, a d, and e ). A standard product term, also minterm is a product term that includes each variable of the problem, either uncomplemented or complemented. a function of 4 variables, w, x, y, and z, the terms wxyz and w xyz are standard product term.
Manipulation of Algebraic Functions -- 2 A sum of products expression (often abbreviated SOP) is one or more product terms connected by OR operators. ab + bc d + a d + e A canonical sum or sum of standard product terms is just a sum of products expression where all of the terms are standard product terms. x yz + x yz + xy z + xy z + xyz ---- 5 terms, 15 literals
Manipulation of Algebraic Functions -- 3 A minimum sum of products expression is one of those SOP expressions for a function that has the fewest number of product terms. If there is more than one expression with the fewest number of terms, then minimum is defined as one or more of those expressions with the fewest number of literals. (1) x yz + x yz + xy z + xy z + xyz 5 terms, 15 literals (2) x y + xy + xyz 3 terms, 7 literals (3) x y + xy + xz 3 terms, 6 literals (4) x y + xy + yz 3 terms, 6 literals They all are equal. (3) And (4) are minimum sum of products. See page 44 for details.
Manipulation of Algebraic Functions -- 4 A sum term is one or more literals connected by OR operators. A standard sum term, also called a maxterm, is a sum term that includes each variable of the problem, either uncomplemented or complemented. A product of sums expression (POS) is one or more sum terms connected by AND operators. A canonical product or product of standard sum terms is just a product of sums expression where all of the terms are standard sum terms. SOP: x y + xy + xyz POS: (x + y )(x + y)(x + z ) Both: x + y + z or xyz Neither: x(w + yz) or z + wx y + v(xz + w )
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.
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 input to the output) 11
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
Implementation of functions with AND, OR, NOT Gates -- 3 Function: x y + xy + xz, when only use uncomplemented inputs:
14 Multi-level circuit 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 15
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 16
Positive and Negative Logic Use 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 0 Negative 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 18
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 0 3. Replace AND by OR, OR by AND, being sure to preserve the order of operations Practice: Example 2.5 (Page53) and Example 2.6 (page 54). 19
Example of Complement f = wx y + xy + wxz -- SOP f = (wx y + xy + wxz) = (wx y) (xy ) (wxz) = (w +x+y )(x +y)(w +x +z ) -- POS 20
Truth Table to Algebraic Expressions f is 1 f is 1 f is 1 ab = 1 if a = 0 AND b = 1 OR if a = 1 AND b = 0 OR if a = 1 AND b = 1 if a = 1 AND b = 1 OR if a = 1 AND b = 1 OR if a = 1 AND b = 1 if a b = 1 OR if ab = 1 OR if f = a b + ab + ab = a + b (OR)
A standard product term, also minterm is a product term that includes each variable of the problem, either uncomplemented or complemented. To obtain f (A, B, C), add all minterms with output = 1 (SOP): f (A, B, C) = m(1, 2, 3, 4,5) = A B C + A BC + A BC + AB C + AB C f (A, B, C) = m(0, 6, 7) = A B C + ABC + ABC f f 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 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: f = (f ) = (A + B + C)(A +B +C)(A +B +C ) f f 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1
To simplify: 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 f (A, B, C) = A B C + ABC + ABC = A B C + AB P10a: B + C See page56 for details. P8a: a (b + c) = ab + ac P9a: ab + ab = a P10a: a + a b = a + b 24
25 Truth Table with don t care Include them as a separate sum. f (a, b, c) = m(1, 2, 5) + d(0, 3) What is the values of f? a b c f f 0 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 due on 9/21. Next class (Chapter 2.6-2.7): NAND, NOR, Exclusive-OR (EOR) Gates Simplification of Algebraic Expressions 27