Lesson 2. The full adder and 4 bits: the carry runs through the digits

After this lesson a display will show the sum of two 4-bit numbers — from 0 + 0 to 15 + 15 = 30. We build the full adder of one digit FA and put four such digits in a chain — that is ADDER4.

2.1. The idea: column addition

Binary numbers are added the same way as decimal ones at school — in columns, right to left. Each column adds three digits: the two digits of the numbers and the carry from the column on the right. The result is a digit of the sum and a carry onwards:

• A, B — the digits of the numbers, CI (carry in) — the carry that came from the right;
• S — the sum digit of this column, CO (carry out) — the carry into the next column.

Three bits are added by two half adders: the first does A + B, the second adds CI to their sum. A carry onwards is needed if at least one of them gave one: CO = c1 OR c2. Both cannot give a carry at once — the sum of three bits is at most 11 (three).

Four FA in a row are four columns. The CO of each digit goes to the CI of the next, so the carry runs from right to left through all the digits, like a wave. The top carry CO is the fifth digit of the sum: 15 + 15 = 30 does not fit in 4 bits.

2.2. Building

The same project as lesson 1. Delete the lesson 1 setup (switches A, B, the HA chip on the canvas, lamps); keep Power/Ground with the labels VCC/GND. The HA and OR chips stay in the library. Everything on labels (wires are fine too).

If you have FA from the "A Watch and a Game from Gates" kit — use it, the pins are the same: A, B, CI → S, CO.

Step 1. FA — the full adder

Adds three bits: A, B and the carry from the lower digit CI. First A + B with a half adder, then the carry is added to the sum with the second half adder; the carry out CO is set if either of them gave a carry.

Ctrl+G → name FA. Chip pins: inputs A, B, CI, outputs S, CO.

Step 2. ADDER4 — four digits

The carry runs from the lowest digit to the highest: the CO of each FA goes to the CI of the next.

Ctrl+G → name ADDER4. Pins: inputs A0–A3, B0–B3, CI; outputs S0–S3, CO.

Step 3. Canvas: switches, adder, displays

On the canvas the carry CO is the label S4: it is the fifth digit of the sum, so display S shows numbers up to 30. CI is on GND for now (the carry into the lowest digit = 0) — in lesson 3 it becomes the subtract switch.

2.3. Check

In the "Switches" columns the top switch (A3) is on the left and A0 on the right; 1 — on, 0 — off.

If only the high digits of the sum are wrong, check that the carry goes c1 → c2 → c3 and that the CO of the last FA is on the chip's CO.

Lesson 2 paste (Cmd+I)

Needs HA and OR from lesson 1.

# Lesson 2 — the full adder FA and the 4-bit adder ADDER4.chip FA {  portIn(A)  portIn(B)  portIn(CI)  portOut(S)  portOut(CO)  HA(A, B, s1, c1)  HA(s1, CI, S, c2)  OR(c1, c2, CO)} chip ADDER4 {  portIn(A0)  portIn(A1)  portIn(A2)  portIn(A3)  portIn(B0)  portIn(B1)  portIn(B2)  portIn(B3)  portIn(CI)  portOut(S0)  portOut(S1)  portOut(S2)  portOut(S3)  portOut(CO)  FA(A0, B0, CI, S0, c1)  FA(A1, B1, c1, S1, c2)  FA(A2, B2, c2, S2, c3)  FA(A3, B3, c3, S3, CO)} input(A0)input(A1)input(A2)input(A3)input(B0)input(B1)input(B2)input(B3)ADDER4(A0=A0, A1=A1, A2=A2, A3=A3, B0=B0, B1=B1, B2=B2, B3=B3, CI=GND, S0=S0, S1=S1, S2=S2, S3=S3, CO=S4)