# How to Design a 4 Bit Arithmetic Circuit Using 4 Full Adders

Written by peter syslo
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A full-adder is a type of integrated circuit that allows two input voltage levels, represented by binary "1" or "0," to be added together. A sum of those two binary numbers is produced at the output of the full-adder, also in the form of a binary 1 or 0. Creating a 4-bit arithmetic circuit means that two 4-bit (four-decimal places) numbers will be added. Each full-adder corresponds to 1-bit therefore, four full-adders are needed to build a 4-bit circuit. Today, 4-bit full-adders are prefabricated, in a single integrated circuit. However, the process of designing the 4-bit circuit is still useful for understanding how a full-adder operates.

Skill level:
Moderate

### Things you need

• Paper
• Pencil
• Binary reference

## Instructions

1. 1

Draw four separate squares, in a horizontal line. Each one represents one full-adder.

2. 2

Label the rightmost full-adder "LSB." This stands for the "Least Significant Bit." For instance, in the binary number 1000, the LSB is the last digit on the right, or 0.

Use a binary reference such as Grinnell College's "The Binary System" (see "Resources" section) for the remainder of this tutorial.

3. 3

Label the leftmost full-adder "MSB." This stands for the "Most Significant Bit." In the binary number 1000, the MSB is the first digit on the left, or 1.

4. 4

Label the inputs and outputs of each full-adder, using a 4-bit full-adder datasheet as a reference. Write "A,""B" and "Cin" at the top of each full-adder and write "E" and "Cout" at the bottom of each full-adder. "A" and "B" stand for the two binary inputs, "Cin" stands for carry input, "E" stands for the sum (main output) and "Cout" stands for carry output. The datasheet lists only one Cin and Cout but in the design stage, each full-adder needs its own Cin and Cout.

5. 5

Label A, B, Cin, E and Cout of each full-adder with a bit number. Write a "1" on the LSB (rightmost) full-adder for bit 1, write "2" on the next full-adder to the left, write "3" on the next full-adder to the left and write "4" on the MSB (leftmost) full-adder. From left to right, the full-adders should be labelled: 4 3 2 1.

6. 6

Write the format of the complete 4-bit numbers, in a space under the full-adders The first 4-bit number, to be added, corresponds to the "A" inputs and will look like this, from left to right: A4 A3 A2 A1. The second 4-bit number to be added, corresponds to the "B" inputs and will look like this: B4 B3 B2 B1. The 4-bit sum, which corresponds to the "E" outputs will look like this: E4 E3 E2 E1. The complete arithmetic for the circuit is: A4 A3 A2 A1 + B4 B3 B2 B1 = E4 E3 E2 E1.

1. 1

Label Cin1 "ground." Electrically, Cin1 (Cin on the datasheet) will be connected to circuit ground because there is no number "carried" into the LSB full-adder. A carry will only go out of this full-adder. For example, when adding 6+6 in decimal, the "2" is placed in the first sum column and the "1" is carried over to the next column. The same principle applies in binary addition.

2. 2

Draw a line from Cout1 to Cin2, draw a line from Cout2 to Cin3 and draw a line from Cout3 to Cin4. In the actual integrated circuit, these connections are made internally and they are designed to pass a carry (binary 1 or 0) along for proper addition.

3. 3

Label Cout4 "Output Bit 5." Because of a carry, the addition of two 4-bit numbers will sometimes result in a 5-bit number. Therefore, there is a total of five possible outputs in a 4-bit arithmetic circuit. At this point, Cout4 (Cout on the datasheet) can be placed alongside the "E" outputs, as follows: Cout4 E4 E3 E2 E1.

4. 4

Assign two 4-bit numbers to be added and separate each 4-bit number into "AB" pairs, for each full-adder. For example, A4 A3 A2 A1 = 1000 and B4 B3 B2 B1 = 1000. A bit number from "A4 A3 A2 A1" will be added to the same bit number from "B4 B3 B2 B1." Write "0+0" next to inputs A1 B1, write "0+0" next to A2 B2, write "0+0" next to A3 B3 and write "1+1" next to A4 B4.

5. 5

Perform the addition of each full-adder, including the carry. For A1 B1, 0+0 = 0 with no carry. For A2 B2, 0+0 = 0 with no carry. For A3 B3, 0+0 = 0 with no carry. For A4 B4, 1+1 = 0 with a carry of 1. That carry of 1 will be the fifth bit that is passed, through Cout4. The 5-bit sum is binary 10000 and the five outputs are as follows, from left to right: Cout4=1, E4=0, E3=0, E2=0, E1=0. This is how the circuit behaves, electrically.

#### Tips and warnings

• For this tutorial, a functional knowledge of binary numbers and binary addition is essential. The datasheet lists only one Cin and Cout for the integrated circuit. The carry bit is passed through each 1-bit full-adder, in the 4-bit combined circuit, but those connections are made internally. The two carry pins reduce the size of the chip, making circuit connections easier. Seeing a Cin/Cout pair for each full-adder provides a logical example of how the binary addition actually works.
• Label everything in the design, with a number and a description. Binary numbers can be confusing and this will help you "retrace your steps," if there is a problem.

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