ECE 331 – Digital System Design

1. NAND and NOR Circuits, Multi-level Logic Circuits, and Multiple-output Logic Circuits (Lecture #9) ECE 331 – Digital System Design The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

2. ECE 331 - Digital System Design NAND and NOR Circuits

3. ECE 331 - Digital System Design CMOS Logic Gates 2-input AND Inverter 2-input NOR 2-input NAND 2-input OR

4. ECE 331 - Digital System Design CMOS Logic Gates Number of transistors per logic gate: • Thus, in terms of transistor count, it is “cheaper” to design logic circuits using NAND and NOR gates.

5. ECE 331 - Digital System Design The NAND Gate • Any logic function can be realized using only NAND gates. • It is a functionally complete set of gates.

6. ECE 331 - Digital System Design (A+B)' A X X' A+B B A' A (A'+B')' = AB B B' The NOR Gate • Any logic function can be realized using only NOR gates. • It, too, is a functionally complete set of gates.

7. ECE 331 - Digital System Design Alternate Logic Gate Symbols • Digital circuit designers often find it convenient to use more than one representation for a given logic gate.

8. ECE 331 - Digital System Design NAND and NOR Circuits • A two-level circuit composed of AND and OR gates is easily converted to a circuit composed of NAND or NOR gates only. • AND-OR → NAND-NAND • OR-AND → NOR-NOR • To do so algebraically, • First use F = (F')' • Then apply DeMorgan's Theorem

9. ECE 331 - Digital System Design Convert the following SOP expression from the AND-OR form to the NAND-NAND form: F(A,B,C) = A.B' + A'.C' + B.C NAND Circuit: Example

10. ECE 331 - Digital System Design Convert the following POS expression from the OR-AND form to the NOR-NOR form: F(A,B,C) = (A'+B').(A'+C).(B+C') NOR Circuit: Example

11. ECE 331 - Digital System Design Design a NAND Circuit • Find the minimum SOP expression for F. • Draw the corresponding AND-OR circuit. • Replace all gates with NAND gates, leaving the gate interconnection unchanged. • Complement any literals connected directly to the output (OR) gate.

12. ECE 331 - Digital System Design Design the NAND circuit for the following logic function: F(A,B,C) = S m(0, 3, 4, 5, 6, 7) NAND Circuit: Example

13. ECE 331 - Digital System Design Design a NOR Circuit • Find the minimum POS expression for F. • Draw the corresponding OR-AND circuit. • Replace all gates with NOR gates, leaving the gate interconnection unchanged. • Complement any literals connected directly to the output (AND) gate.

14. ECE 331 - Digital System Design Design the NAND circuit for the following logic function: F(A,B,C) = S m(2, 4) NOR Circuit: Example

15. ECE 331 - Digital System Design Multi-level Logic Circuits

16. ECE 331 - Digital System Design Multi-level Logic Circuits • Thus far we have focused on the realization of optimal logic circuits through the derivation of • Minimum Sum of Products (SOP) expressions • Minimum Product of Sums (POS) expressions • Both forms of Boolean expressions are realized as two-level logic circuits • SOP ↔ AND-OR (NAND-NAND) circuit • POS ↔ OR-AND (NOR-NOR) circuit • There are a maximum of two logic gates between every input and the output(s).

17. ECE 331 - Digital System Design Multi-level Logic Circuits • A two-level logic circuit is usually efficient for Boolean expressions of a few variables (i.e. inputs). • However, as the number of inputs increases, a two-level logic circuit may encounter fan-in problems. • Fan-in refers to the number of inputs to a logic gate • Whether fan-in is an issue is dependent upon the technology used to implement the logic circuit. • Standard TTL and CMOS chips • Complex Programmable Logic Device (CPLD) • Field Programmable Gate Array (FPGA)

18. ECE 331 - Digital System Design Multi-level Logic Circuits • A multi-level logic circuit may require fewer logic gates than the logically equivalent two-level logic circuit. • Reduced (silicon) area • Decreased cost • It may require less complex wiring between logic gates • Fewer literals results in fewer interconnecting wires • It will have a greater propagation delay than the logically equivalent two-level logic circuit. • Each additional level adds to the propagation delay • Decreased speed

19. ECE 331 - Digital System Design Design a logic circuit to realize the following logic function: F(A,B,C) = S m(1, 2, 3, 4, 6) Given the following criteria: 1. Use AND and OR gates only 2. Two- or Three-level circuit only 3. Minimize the number of gates 4. Minimize the number of gate inputs Multi-level Logic Circuits: Example

20. ECE 331 - Digital System Design Which design would be possible if the following additional criteria was imposed: 5. Two-input logic gates only Multi-level Logic Circuits: Example

21. ECE 331 - Digital System Design Multi-level Logic Circuits using NAND and NOR gates

22. ECE 331 - Digital System Design Design a Multi-level NAND Circuit • Derive a minimum expression for the logic function. • Design a multi-level circuit using AND and OR gates. • Output gate must be an OR gate. • Gates must alternate: AND, OR, AND, OR, … • Number the levels starting with the output gate. • Replace all gates with NAND gates, leaving the interconnection between gates unchanged. • Leave inputs to gates at the even levels unchanged; complement inputs to gates at the odd levels.

23. ECE 331 - Digital System Design Design a Multi-level NAND Circuit

24. ECE 331 - Digital System Design Design a Multi-level NOR Circuit • Derive a minimum expression for the logic function. • Design a multi-level circuit using AND and OR gates. • Output gate must be an AND gate. • Gates must alternate: OR, AND, OR, AND, … • Number the levels starting with the output gate. • Replace all gates with NOR gates, leaving the interconnection between gates unchanged. • Leave inputs to gates at the even levels unchanged; complement inputs to gates at the odd levels.

25. ECE 331 - Digital System Design Design a Multi-level NOR Circuit

26. ECE 331 - Digital System Design Multiple-output Logic Circuits

27. ECE 331 - Digital System Design Multiple-output Logic Circuits • Thus far, we have focused on designing logic circuits to realize a single logic function. • A logic circuit with a single output. • However, many logic circuits have multiple outputs. • Corresponding to multiple logic functions of the same input variables.

28. ECE 331 - Digital System Design Multiple-output Logic Circuits • An optimal logic circuit may not be realized by simply minimizing each of the logic functions independently. • Rather, it may be necessary to consider which terms (i.e. logic gates), if any, are common to the logic functions to be realized. • Share those logic gates that are in common.

29. ECE 331 - Digital System Design Design an optimal logic circuit to realize the following logic functions: F(A,B,C) = S m(0, 4, 5) G(A,B,C) = S m(0, 2, 6) Multiple-output Circuits: Example Cost = # of logic gates + # of gate inputs

30. ECE 331 - Digital System Design Questions?