Design a 4-bit parallel Binary Adder using three full adders and one half…

Concept

A half adder adds two one-bit operands and has no carry input. Its outputs are S = A XOR B and C = A AND B.

A full adder adds two operand bits and an incoming carry. Its outputs are S = A XOR B XOR Cin and Cout = (A AND B) OR (Cin AND (A XOR B)). In a ripple-carry parallel adder, each stage passes its carry to the next more significant stage.

Application

1. Write the operands as A3A2A1A0 and B3B2B1B0, with sum bits S3S2S1S0 and inter-stage carries C1, C2, C3, C4.

2. Use the half adder HA0 for the least significant bits A0 and B0 because the question specifies no external input carry. It produces S0 and C1.

3. Connect full adder FA1 to A1, B1, and C1. It produces S1 and C2.

4. Connect full adder FA2 to A2, B2, and C2. It produces S2 and C3.

5. Connect full adder FA3 to A3, B3, and C3. It produces S3 and the final carry C4.

6. Read the complete result as C4S3S2S1S0. If only a 4-bit sum is required, S3S2S1S0 is the sum output and C4 is the carry-out.

Block diagram

The carry output of each block feeds the carry input of the next full adder:

A0, B0 ──► [HA0] ──► S0
                      │
                      └── C1 ──► [FA1] ◄── A1, B1
                                      ├──► S1
                                      └── C2 ──► [FA2] ◄── A2, B2
                                                      ├──► S2
                                                      └── C3 ──► [FA3] ◄── A3, B3
                                                                      ├──► S3
                                                                      └──► C4

Cross-check

For A = 1011 and B = 0110, the stages give:

1. HA0: 1 + 0 gives S0 = 1 and C1 = 0.

2. FA1: 1 + 1 + 0 gives S1 = 0 and C2 = 1.

3. FA2: 0 + 1 + 1 gives S2 = 0 and C3 = 1.

4. FA3: 1 + 0 + 1 gives S3 = 0 and C4 = 1.

Therefore, C4S3S2S1S0 = 10001, which is 17 in decimal and agrees with 11 + 6 = 17. This confirms both the carry chain and the output order.