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EE210: Switching Systems

Lecture 4: Implementation AND, OR, NOT Gates and Compliment. EE210: Switching Systems. Prof. YingLi Tian Sept . 10 , 2012. Department of Electrical Engineering The City College of New York The City University of New York (CUNY). TA’s Email:.

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EE210: Switching Systems

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  1. Lecture 4: Implementation AND, OR, NOT Gates and Compliment EE210: Switching Systems Prof. YingLiTian Sept. 10, 2012 Department of Electrical EngineeringThe City College of New YorkThe City University of New York (CUNY)

  2. 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

  3. 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

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

  5. SOP and POS • A sum of productsexpression (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 termis one or more literals connected by AND, OR, operators.

  6. Gate Implementation • P2b: a(bc) = (ab) c • These three implementations are equal.

  7. 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)

  8. Implementation of functions with AND, OR, NOT Gates -- 2 x´yz´ + x´yz + xy´z´ + xy´z + xyz (2) x´y+ xy´ + xyz (3) x´y+ xy´ + xz (4) x´y+ xy´ + yz

  9. Implementation of functions with AND, OR, NOT Gates -- 3 Function: x´y + xy´ + xz, when only use uncomplemented inputs:

  10. Multi-level circuit Function? (see Page50)

  11. 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

  12. 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

  13. 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

  14. 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

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

  16. Example of Complement • f ´= (wx´y + xy´ + wxz)´ • = (wx´y)´(xy´)´(wxz)´ • = (w´+x+y´)(x´+y)(w´+x´+z´) f = wx´y + xy´ + wxz

  17. Truth Table to Algebraic Expressions f is 1 if a = 0 AND b = 1 OR if a = 1 AND b = 0 OR if a = 1 AND b = 1 f is 1 if a´ = 1 AND b = 1 OR if a = 1 AND b´ = 1 OR if 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)

  18. A standard product term, also mintermis 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

  19. 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´)

  20. 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´ P10a: B + C f ´(A, B, C) = A´B´C´ + ABC´ + ABC = A´B´C´ + AB See page56 for details. • P8a: a (b + c) = ab + ac P9a: ab + ab´ = a P10a: a + a´ b = a + b

  21. Truth Table with don’t care f (a, b, c) = ∑m(1, 2, 5) + ∑d(0, 3) Include them as a separate sum.

  22. Number of different functions of n variables

  23. 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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