CRUMB – How to build a full-adder and a 4-bit ripple

Warhammer 40000: Darktide – Combat Basics Tips 1 - steamlists.com
Warhammer 40000: Darktide – Combat Basics Tips 1 - steamlists.com

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.

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - BEFA405

However, a full adder 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.

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - 608B92D

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.

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - B906D49

Sum

The boolean expression for Sum output is

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - 5C74A51

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’

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - 493AEAE

*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)’

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - 9114C63

The Sum expression can be simplified.

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - 5B13517

Sum = A B C

Cout

Cout’s boolean expression is

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - 3EE0F84

Kmap allows you to circle 1,2,4,8 and 16 slots (, a*suming 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.

CRUMB - How to build a full-adder and a 4-bit ripple - Modify the half-adder to full-adder - 561FA47

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.

CRUMB - How to build a full-adder and a 4-bit ripple - Logic diagram of Full-Adder and 1-bit Full-Adder - DB7A47D

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

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

CRUMB - How to build a full-adder and a 4-bit ripple - Logic diagram of Full-Adder and 1-bit Full-Adder - C5DCC75

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.

CRUMB - How to build a full-adder and a 4-bit ripple - Ripple Carry Adder - E71AC23

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 2’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.

 

Written by Qahnaarin

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!


Be the first to comment

Leave a Reply

Your email address will not be published.


*