Half adders and full adders

A binary adder is the circuit that adds two binary numbers. Every calculator and every processor has one.

It adds the way you add on paper: one column at a time, and a carry moves to the next column. Two small circuits, the half adder and the full adder, do that for one column.

Add the bits in the right-hand column first. 1 + 1 is 2, which is 10 in binary. So the column keeps a 0 and sends a 1 to the left.

That 1 is the carry. The 0 that stays is the sum. The next column must add its own two bits and the carry that arrived.

A half adder adds two bits. Its sum is a XOR b, because the sum is 1 when exactly one input is 1. Its carry is a AND b, because the carry is 1 only when both are 1.

It is called half because it has no place for a carry coming in. It can only do the right-hand column.

# A half adder: two bits in, a sum bit and a carry bit out.
def half_add(a, b):
    return [a ^ b, a & b]

for a in [0, 1]:
    for b in [0, 1]:
        print(a, "+", b, "-> sum, carry =", half_add(a, b))
# Add a line that prints half_add(1, 1)[0] and half_add(1, 1)[1] on their own.

Every column except the first has a carry arriving. So its circuit adds three bits: a, b and the carry in. It gives a sum and a carry out.

A full adder can be built from two half adders and one more gate. Chain full adders, each carry out into the next carry in, and you can add numbers of any width.

Working out that last gate is the first task of the lab.

You can also write x + y and let the tools build the adder. Here x and y are 4 bits wide and the answer has 5 bits.

The fifth bit is the carry out of the top column. Three Runs write a testbench, compile it and add 9 + 5 and then 15 + 1. The files go in /tmp.

echo 'module add_tb; reg [3:0] x, y; wire [4:0] total; assign total = x + y; initial begin x = 9; y = 5; #1 $display("%0d + %0d = %0d, binary %b", x, y, total, total); x = 15; y = 1; #1 $display("%0d + %0d = %0d, binary %b", x, y, total, total); end endmodule' > /tmp/add_tb.v

iverilog -o /tmp/add /tmp/add_tb.v

vvp /tmp/add