Combinational Logic

Unit 4: Combinational Logic 

1. Design Procedure

Steps to Design Combinational Circuits:

1. Problem Definition: Determine inputs, outputs, and requirements.
2. Truth Table: List all input combinations and corresponding outputs.
3. Boolean Simplification: Use K-maps or algebraic methods to simplify expressions.
4. Logic Diagram: Implement the circuit using gates (AND, OR, NOT, NAND, NOR, XOR).

2. Adders

Half Adder

• Inputs: 2 bits (A, B)
• Outputs: Sum (S) and Carry (C)
• Truth Table:

A	B	S	C
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

Equations: S = A ⊕ B, C = A · B

Full Adder

• Inputs: 3 bits (A, B, C_in)
• Outputs: Sum (S), Carry (C_out)
• Implementation: Two half-adders and an OR gate.

Equation: S = A ⊕ B ⊕ Cin, Cout = (A·B) + (Cin·(A⊕B))

3. Subtractors

Half Subtractor

• Inputs: 2 bits (A, B)
• Outputs: Difference (D), Borrow (B_out)
• Truth Table:

A	B	D	Bout
0	0	0	0
0	1	1	1
1	0	1	0
1	1	0	0

Equations: D = A ⊕ B, Bout = A'·B

Full Subtractor

• Inputs: 3 bits (A, B, B_in)
• Outputs: Difference (D), Borrow (B_out)
• Implementation: Two half-subtractors and an OR gate.

4. Code Conversions

Examples:

• BCD to Excess-3
• Binary to Gray Code

BCD to Excess-3 Converter

• Input: 4-bit BCD
• Output: 4-bit Excess-3 code.
• Method: Add 0011 (3) to BCD input.

BCD	Excess-3
0000	0011
0001	0100
...	...

5. Analysis Procedure

Steps to Analyze Combinational Circuits:

1. Label all gate outputs.
2. Write Boolean expressions for each gate.
3. Derive the output function and simplify.
4. Create a truth table to verify functionality.

6. Multilevel NAND and NOR Circuits

NAND Implementation

• Rule: Replace AND-OR logic with NAND gates.
• Example: F = (A·B) + (C·D) → Use NAND for AND and OR operations.

Circuit Diagram (Equivalent NAND):

A ── NAND ─┐
B ────────┘
C ── NAND ─┐
D ────────┘
    NAND ── Output
        

NOR Implementation

• Rule: Replace OR-AND logic with NOR gates.
• Example: F = (A+B)·(C+D) → Use NOR for OR and AND operations.

7. Exclusive-OR (XOR) Circuits

• Symbol: ⊕
• Equation: A ⊕ B = A·B' + A'·B
• Applications: Parity generation, arithmetic circuits.

XNOR Gate

• Symbol: ⊙
• Equation: A ⊙ B = A·B + A'·B'

Summary

• Combinational circuits have no memory (output depends only on current inputs).
• Adders and subtractors form the basis of arithmetic circuits.
• Code converters use truth tables and logic simplification.
• NAND/NOR gates are universal and used for multilevel implementations.
• XOR/XNOR gates are critical for parity checks and error detection.