1. Unit 7 Multi-Level Gate Circuits / NAND and NOR Gates Ku-Yaw Chang canseco@mail.dyu.edu.tw Assistant Professor, Department of Computer Science and Information Engineering Da-Yeh University

2. 22004/03/01Fundamentals of Logic Design Objectives Design a minimal two-level or multi-level circuit of AND and OR gates to realize a given function. Design or analyze a two-level gate circuit using any one of the eight basic forms. Design or analyze a multi-level NAND- gate or NOR-gate circuit.

3. 32004/03/01Fundamentals of Logic Design Objectives Convert circuits of AND and OR gates to circuits of NAND gates or NOR gates, and conversely, by adding or deleting inversion bubbles. Design a minimal two-level, multiple- output AND-OR, OR-AND, NAND-NAND, or NAND-NOR circuit using Karnaugh maps.

4. 42004/03/01Fundamentals of Logic Design Outline 7.1 Multi-Level Gate Circuits 7.2NAND and NOR Gates 7.3Design of Two-Level Circuits Using NAND and NOR Gates 7.4Design of Multi-Level NAND and NOR Gate Circuits 7.5Circuit Conversion Using Alternative Gate Symbols 7.6Design of Two-Level, Multiple-Output Circuits 7.7Multiple-Output NAND and NOR Circuits

5. 52004/03/01Fundamentals of Logic Design 7.1 Multi-Level Gate Circuits Level Maximum number of gates cascaded in series between a circuit input and the output Maximum number of gates cascaded in series between a circuit input and the output

6. 62004/03/01Fundamentals of Logic Design Four-Level Realization of Z

7. 72004/03/01Fundamentals of Logic Design Terminology AND-OR circuit A two-level circuit composed of a level of AND gates followed by an OR gate at the output A two-level circuit composed of a level of AND gates followed by an OR gate at the output

8. 82004/03/01Fundamentals of Logic Design Terminology OR-AND circuit A two-level circuit composed of a level of OR gates followed by an AND gate at the output A two-level circuit composed of a level of OR gates followed by an AND gate at the output

9. 92004/03/01Fundamentals of Logic Design Terminology OR-AND-OR circuit A three-level circuit composed of a level of OR gates followed by a level of AND gates followed by an OR gate at the output A three-level circuit composed of a level of OR gates followed by a level of AND gates followed by an OR gate at the output

10. 102004/03/01Fundamentals of Logic Design Terminology Circuit of AND and OR gates No particular ordering of the gates No particular ordering of the gates The output gate may be either AND or OR. The output gate may be either AND or OR.

11. 112004/03/01Fundamentals of Logic Design Trade-off Number of Gates Gates Gate inputs Gate inputs Levels Levels Gate delay

12. 122004/03/01Fundamentals of Logic Design Example 7-1 Z = (AB+C) (D+E+FG) + H

13. 132004/03/01Fundamentals of Logic Design Example 7-1 Partially multiplying out Z = (AB+C) (D+E+FG) + H = (AB+C) [(D+E)+FG] + H = AB(D+E) + C(D+E) + ABFG + CFG + H Z = (AB+C) (D+E+FG) + H = (AB+C) [(D+E)+FG] + H = AB(D+E) + C(D+E) + ABFG + CFG + H

14. 142004/03/01Fundamentals of Logic Design Example 7-1

15. 152004/03/01Fundamentals of Logic Design Example 7-1

16. 162004/03/01Fundamentals of Logic Design Problem Find a circuit of AND and OR gates to realize Find a circuit of AND and OR gates to realize f(a, b, c, d) = ∑ m(1, 5, 6, 10,13, 14)

17. 172004/03/01Fundamentals of Logic Design Solution First, simplify f by using a Karnaugh map f(a, b, c, d) = ∑ m(1, 5, 6, 10,13, 14) f(a, b, c, d) = ∑ m(1, 5, 6, 10,13, 14)

18. 182004/03/01Fundamentals of Logic Design Solution This leads directly to a two-level AND-OR gate circuit f = a’c’d + bc’d + bcd’ + acd’

19. 192004/03/01Fundamentals of Logic Design Solution Factoring f = a’c’d + bc’d + bcd’ + acd’ yields f = c’d (a’ + b) + cd’(a + b)

20. 202004/03/01Fundamentals of Logic Design Solution This leads directly to a three-level OR-AND-OR gate circuit f = c’d (a’ + b) + cd’(a + b)

21. 212004/03/01Fundamentals of Logic Design Solution Both solutions have an OR gate at the output Sum of products Sum of products Using 1s on the Karnaugh map Using 1s on the Karnaugh map How about having an AND gate at the output? Product of sums Product of sums Using 0s on the Karnaugh map Using 0s on the Karnaugh map

22. 222004/03/01Fundamentals of Logic Design Solution Secondly, simplify f by using a Karnaugh map (considering 0s) f’ = c’d’ + ab’c’ + cd + a’b’c f = (c+d) (a’+b+c) (c’+d’) (a+b+c’) f’ = c’d’ + ab’c’ + cd + a’b’c f = (c+d) (a’+b+c) (c’+d’) (a+b+c’)

23. 232004/03/01Fundamentals of Logic Design Solution This leads directly to a two-level OR-AND circuit f = (c+d)(a’+b+c)(c’+d’)(a+b+c’)

24. 242004/03/01Fundamentals of Logic Design Solution Partially multiplying out f = (c+d)(a’+b+c)(c’+d’)(a+b+c’) using (X+Y)(X+Z) = X+YZ f = [c + d(a’+b)] [c’ + d’(a+b)] = (c+a’d+bd)(c’+ad’+bd’)

25. 252004/03/01Fundamentals of Logic Design Solution This leads directly to a three-level AND-OR- AND circuit f = (c+a’d+bd)(c’+ad’+bd’)

26. 262004/03/01Fundamentals of Logic Design Comparison f = LevelGates Gate Inputs a’c’d + bc’d + bcd’ + acd’ 2516 c’d(a’+b) + cd’(a+b) 3512 (c+d)(a’+b+c)(c’+d’)(a+b+c’)2514 (c+a’d+bd)(c’+ad’+bd’)3716