Welcome to (CRUMB – How to build a full-adder and a 4-bit ripple). This tutorial will demonstrate how to modify a half-adder to produce a full-adder and a 4-bit ripple adder.

# Overview

This guide will reuse the half adder from my previous guide.

I will make the half-adder a full-adder. To make a 4-bit ripple carry, duplicate the full-adder.

Contents:

- Modify the half-adder to full adder
- Logic diagram for full-adder and 1-bitfull-adder
- Adder for Ripple
- Conclusion

*Notes – This guide requires understanding the K-map and the half adder.

Abbreviations:

Cin = Carry in

Cout = Carryout

FA = Full-adder

HA = Half-adder

# Modify the half-adder to full adder

First, we must get the truth table from the previous guide. Below is the truth table for the ** half-adder**. As you can see, the truth table only contains

__2 inputs__A14Y A and) and

__2 outputs__A14Y Sum and Carry e17Y.

However, a ** full adder**Â has

__3 inputs__(A, B & Cin) and

__2 outputs__(Sum and Cout). We need another truth table to complete the full adder. Below is the truth table for a FA.

Next, we will need to find the boolean expression for the Sum and Cout a FA. To solve this problem, we can use a 3-variable (B, Cin) Kmap.

This is an empty 3-variable Kmap. When creating the 3/4 variables Kmap, please be aware of the red circle.

### Sum

The boolean expression for Sum output is

This is the tricky part. We need to simplify the process by applying laws and theorems in boolean algebra.

Step 1: We remove the A and A’

*Important rule (XOR gate): E Y = E’Y + EY’

After removing and A and A’ variables, you can again see the familiar terms (+XY’), which is exclusive-OR (XOR). So X Y = (X’Y+XY’. We can also replace B’C + B’C’ with B’C.

*Important rule (): XY = (XY’ =)’ = EY+ X’Y

Another interesting fact is that if you invert the output of the XOR gate, you get the XNOR gate. We can also replace BC + B’C” to (BC)’

The Sum expression can be simplified.

Sum = A B C

### Cout

Cout’s boolean expression is

Kmap allows you to circle 1,2,4,8 and 16 slots (, assuming up to 4 variables). As you can see, I circled 2 slots three times.

*Note: Always circle as many slots as possible. The simpler the term, the more slots you circle.

How do I get AB, AC, and BC?

After we have circled the terms in Kmap, we need to verify that (A.B.C.) has both 1 and 0 circled. As shown in the image below, both 0 (or 1) of C is circled in the red circle. We can ignore C and only focus on A and B. In this case, A=1, B=1, so it’s AB. The same concept applies to the blue and green circles.

So Cout = AB + AC+ BC

# Logic diagram of Full Adder and 1-bit Full Adder

The boolean expression Sum and Carry out from the FA is, therefore a boolean expression.

Sum = A B Cin

Cout = AB + AC + BC

The logic diagram is now complete.

Source:

http://hyperphysics.phy-astr.gsu.edu/hbase/Electronic/fulladd.html – [gsu.edu]

The FA can be simplified into a 1-bit FA diagram.

# Ripple Carry Adder

The Cout can be connected to the Cin of each bit FA to make it a 4-bit FA. This is also known as the ripple carry adder.

Source:

https://nandland.com/ripple-carry-adder/ – [nandland.com]

To add these 2 numbers, we only need to supply 4-bit inputs A and B (A3A2A1A0. The result will be 5 bits (CoutS3S2S1S0 –

# Conclusion

This guide may seem a bit difficult to beginners. These are some tips to help you get started.

- To see a better explanation and demonstration, go to youtube and search YouTube for vids
- Leave a comment to let us know how we can help you
- To get more familiar with electronics, you can look for beginner circuits.

I hope you enjoy this guide. Let me know if I can help you. =)

*Note: At the moment, there is no plan to adder/subtractor with ripple carry adder. This requires an understanding of the signed and unsigned binary numbers and the 1’s 2â€™s complement. This would require many explanations. If you are interested, you could look for materials to convert your 4-bit adder/subtractor to a 4-bit adder/subtractor. It is easy to do. =)

Thanks.

This is all about CRUMB – How to build a full-adder and a 4-bit ripple; I hope you enjoy reading the Guide! If you feel like we should add more information or we forget/mistake, please let us know via commenting below, and thanks! See you soon!

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