1. ECE 331 – Digital System Design
Multilevel Logic Circuits
and
Logic Circuit Analysis
(Lecture #8)

2. Multilevel Logic Circuits
ECE Digital System Design

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

4. Multilevel Logic Circuits
A two-level logic circuit is usually efficient for Boolean expressions of a few variables.
However, as the number of inputs increases, a two- level logic circuit may result in 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
Field Programmable Gate Array (FPGA)
Complex Programmable Logic Device (CPLD)
ECE Digital System Design

5. Multilevel Logic Circuits
Example:
Realize the following Boolean expression using only 2-input AND gates and 2-input OR gates.
F(A,B,C) = S m(0, 5, 6)
ECE Digital System Design

6. Multilevel Logic Circuits
Often requires fewer logic gates than the logically equivalent two-level logic circuit.
Reduced (silicon) area
Decreased cost
Requires less complex wiring between logic gates
Fewer literals results in fewer interconnecting wires
Has a greater propagation delay than the logically equivalent two-level logic circuit.
Each additional level adds to the propagation delay
Decreased speed
ECE Digital System Design

7. Multilevel Logic Circuits
Objectives:
1. Design logic circuits that meet the fan-in requirements of the chosen technology.
2. Design a minimum-cost logic circuit.
ECE Digital System Design

8. Multilevel Logic Circuits
Two techniques that can be used to realize multilevel logic circuits:
1. Factoring
2. Functional Decomposition
ECE Digital System Design

9. Factoring
Example:
Realize a logic circuit that has a maximum fan-in of two for the following Boolean expression.
F(A..G) = ACF' + ADEF' + BCG + BDEG
ECE Digital System Design

10. ECE 331 - Digital System Design
Factoring
Example:
Design the minimum-cost logic circuit that implements the following Boolean expressions.
F1(A,B,C,D) = S m(1,2,3,7,11,15)
F2(A,B,C,D) = P M(0,1,2,3,4,8,12)
ECE Digital System Design

11. Functional Decomposition
Example:
Design a minimum-cost logic circuit to implement the following Boolean expression.
F(A,B,C,D) = A'BC + AB'C + ABD + A'B'D
ECE Digital System Design

12. ECE 331 - Digital System Design
NAND and NOR Circuits
As with two-level circuits, multilevel circuits can be realized using NAND or NOR gates only.
ECE Digital System Design

13. Example: Realizing a NAND Circuit
NAND and NOR Circuits
Example:
Realizing a NAND Circuit
ECE Digital System Design

14. ECE 331 - Digital System Design
NAND and NOR Circuits x 2 1 3 4 5 6 7 f
ECE Digital System Design

15. ECE 331 - Digital System Design
NAND and NOR Circuits x 2 1 3 4 5 6 7 f
ECE Digital System Design

16. ECE 331 - Digital System Design
NAND and NOR Circuits x 2 1 3 4 5 6 7 f
ECE Digital System Design

17. Example: Realizing a NOR Circuit
NAND and NOR Circuits
Example:
Realizing a NOR Circuit
ECE Digital System Design

18. ECE 331 - Digital System Design
NAND and NOR Circuits x 2 1 3 4 5 6 7 f
ECE Digital System Design

19. ECE 331 - Digital System Design
NAND and NOR Circuits x 2 1 3 4 5 6 7 f
ECE Digital System Design

20. ECE 331 - Digital System Design
NAND and NOR Circuits x 2 1 3 4 5 6 7 f
ECE Digital System Design

21. Logic Circuit Analysis
ECE Digital System Design

22. Logic Circuit Analysis
Determining the behavior of a system given its description.
The description of the system is often provided
in the form of a circuit diagram.
ECE Digital System Design

23. Logic Circuit Analysis
For two-level circuits, the analysis process is simple.
The Boolean expression representing the circuit can often be written by inspection.
For multilevel circuits, the analysis process is much more complicated.
Cannot write a Boolean expression by inspection.
Must follow a procedure to implement the analysis.
ECE Digital System Design

24. Logic Circuit Analysis
Identify inputs and outputs
Track circuit behavior from input to output
Determine Boolean expression for output(s)
Determine Truth Table
Examine circuit timing, power dissipation, etc.
ECE Digital System Design

25. Logic Circuit Analysisx 1 2 5 4 f 3 P 6 8 9 10 7
ECE Digital System Design