Digital Design Fundamentals·Combinational Circuits
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The rightmost column of a binary addition has two bits coming in, and , and nothing carried from before. The circuit that adds them is the half adder.
It has two outputs, because does not fit in one bit:
Every possible pair of input bits, and the two-bit total they produce:
| a | b | Carry | Sum | Total |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 2 |
Read carry and sum together as a 2-bit number and it is exactly .
Look down the two output columns and match each to a gate you already know:
Two gates, sharing the same two inputs. That is the entire half adder.
Here is the half adder: one XOR for sum, one AND for carry, both fed by a and b.
Try it:
a = 1, b = 0. sum lights, carry stays off: the total is 1.a = 1, b = 1. sum drops to 0 and carry lights: read together, , which is 2.The half adder handles the first column, but no others. Look at any column past the first in a real addition and three bits meet there: the two input bits and a carry rippling in from the column to its right.
A half adder has only two inputs, so it cannot accept that carry-in. The next lesson fixes this with a circuit that takes all three bits: the full adder.