Hello, readers welcome to the new post. In this post, we will discuss Introduction to Karnaugh Maps. Karnaugh Maps, also famous as K-maps, is considered a significant tool in digital circuit design that helps to make Boolean algebra expressions simple and easy to solve. In this post, we will discuss all details of K-Maps and solve the practical example to have a detailed understanding
Table of Contents
Introduction to Karnaugh Maps
- Karnaugh Mapsare created to denote the Boolean functions that are used in digital circuits to find outputs based on the input provided.
- They are like visual aids which make simple Boolean expressions by defining the groups of ones and zeros which can be joined to make simple expressions.
- These maps are created through the use of squares and rectangles and these squares or rectangles represent a combination of inputs variables
Karnaugh Maps History
- First-time K-Maps was used in 1953 by Maurice Karnaugh when he was working on digital circuit designing at Bell Labs.
- This technique was well improved than older methods to solve the Boolean expression that was based on algebraic manipulation.
- The improved features of K-Maps make it very fast adoption by the designer and engineers and it currently used in different projects
Where to use Karnaugh Maps
- There are different types of K-Maps applications in digital circuit solutions like microprocessor development, circuits memory creation, and other digital operations
- They are very commonly used in Boolean expressions to decrease the number of logic gates used in circuits. They also decrease the power used by the circuits
How to Use Karnaugh Maps
- The use of K-Mpas is somewhat difficult at the start but becomes easy when you have an understanding of it. Here are some steps discussed to solve this map
Step 1:Finds Inputs and Outputs
- The first step is to find the inputs and outputs of circuits that will help us to define the variables that are used in Map
Step 2: Desing Karnaugh Map
- After the identification of variables, we can make K-Map. the number of squares or rectangles in the map depends on the variables of the circuits. Let’s suppose that we have a circuit having 2 variables will be a 2×2 map designed for that circuit and for 4 variables 4×4 map will create
Step 3: Identify the Ones and Zeros
- In this step find the zeros and ones existing in the Boolean expression. Then draw these values on the K-Map adding the one in square or rectangle that is according to input variables
Step 4: Defines Groups
- After plotting the ones and zeros on the map we can start the identification of groups. Thegorupos is a combination of neighboring groups or rectangles that has ones. Groups can be in the vertical or horizontal directions and can wrap around the edges of the Map
Step 5: Simplification of expression
- Now at last we can use groups to simplify Boolean expression every group denotes a term in simplified expression that can be combined with other terms to make a simple expression
Karnaugh Maps Truth Table
- The truth table is used for Boolean algebra to define all combinations of inputs and related output for a given logical function. Table as columns for every input variable and one column for output. The number of rows in tables is 2 raised to the power of numbers of input variables
- Every row in the table denotes a differnt combination of inputs with 1 is true and zero as false. The output column gives the output values of every input combination.
- For example, let’s make logical functions having two input variables A, B, and output C.So truth table will have 4 rows making possible combinations for inputs A and B
| A | B | C |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
- In this example, logical functions denote an XoR gate here the output is 1 or true if any one input is true or 1
- Truth tables are beneficial for the verification of the accuracy of logical functions and can e used as a base for the simplification of a Boolean expression that used K Maps
Steps to solve expression using K-map-
- K-Map is a graphical tool used for the simplification of Boolean expressions. They are the best options for functions that have many inputs and outputs
- Here some steps are exposed to solve the Boolean expression through the use of K-Map
- First of all, write the truth table for the expression
- Utilize ones to adjacent group cells. The groups should be rectangular in shape and as large as possible.
- Use the corresponding Boolean expression to label each group. All of the group’s cells should have the same expression.
- Utilizing the OR operator, combine the expressions for each group to create the simplified Boolean expression.
- Compare the truth tables of the original and simplified expressions to confirm the simplification.
- Let us take an example of a Boolean function:
F(X, Y, Z) = Σ(0, 1, 3, 4, 6)
- These steps involved to solved this fucjtionsTo solve this function using a K-map, we can follow these steps:
- First of all write the truth table for the function:
| X | Y | Z | F |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |
- Utilize ones to adjacent group cells. We can divide it into two groups here.: First having cells (0, 1, 4, 5) and the other with cells (2, 3, 6, 7).
| 0 | 1 | 1 |
|---|
| 1 |
|---|
| 0 | |||
|---|---|---|---|
| 1 |
- Use the corresponding Boolean expression to label each group. The first group is related to the expression X’YZ, and the 2nd group corresponds to XY’Z.
| 0 | 1 | 1 |
|---|
| 1 |
|---|
| 2 | |||
|---|---|---|---|
| 3 |
- Write the simple Boolean expression through a combination of expressions for every group with the use of the OR operator
- F(X, Y, Z) = X’YZ + XY’Z.
- Throug making comparisons verify the truth tables of the original and make simplification of an expression. Here truth tables are same defining that implication results are accurate
5 variable K-Map in Digital Logic
- Now we desing 5 variables K-Map to solve it
- The steps involved in the 5-variable K-Map are explained here
- Mak table has 32 cells (2^5) and then labels rows and columns with binary values with inputs variables X, Y, Z, G, and H.
- Now write truth table of function
- Utilize ones to adjacent group cells. The groups should be rectangular in shape and as large as possible
- Use the corresponding Boolean expression to label each group. All of the group’s cells should have the same expression.
- Utilizing the OR operator, combine the expressions for each group to create the simplified Boolean expression.
- Compare the truth tables of the original and simplified expressions to confirm the simplification.
For example, let us consider the Boolean function
F(X, Y, Z, G, H) = Σ(0, 1, 2, 3, 5, 6, 7, 11, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31)
- For a solution of this function, we use a 5-variable K-Map, we can follow these steps:
- Make table 32 cells and label the rows and columns with binary values for the input variables X, Y, Z,G, and H.
- Now write the table for a function
| X | Y | Z | G | H | F |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 | 0 |
- Utilize ones to group cells that are adjacent. The cells can be grouped into the following rectangular shapes in this instance:
| 2 | 3 | ||||
|---|---|---|---|---|---|
| 1 | 1 | 0 | |||
| 5 | 1 | 1 | 1 | 0 | |
| 4 | 1 | 1 | 1 | 1 | |
| 1 | 1 | 0 | |||
| 1 | 1 |
- Label every group through the use of the Boolean expression. We can write expressions for every group as:
Group 1: Y’Z’G’H’ Group 2: XZ’GH’ Group 3: XY’GH’ Group 4: XY’Z’H’ Group 5: XY’ZG
- Now write the simplified Boolean expression through a combination of expressions for every group with sue of OR operator
F(X, Y, Z, G, H) = B’C’D’E’ + AC’DE’ + AB’DE’ + AB’C’E + AB’CD
- We Now does verification of simplification through a comparison of the truth tables of real and simplified expression.
A 5-variable K-Map can simplify complex Boolean expressions by reducing the number of terms required to represent the function. This can lead to more efficient circuit designs, as fewer logic gates are required to implement the function. However, constructing a K-Map requires careful attention to detail and an understanding of the underlying principles of digital logic. With practice, it becomes easier to construct and use K-Maps to simplify complex Boolean expressions.
Conclusion:
In digital logic, K-Maps are an important tool for simplifying Boolean expressions. They enable us to reduce the number of terms required to represent a function, which results in circuit designs that are more effective. Furthermore, K-Guides are somewhat simple to utilize once you grasp the essential standards of collection contiguous cells and marking bunches with Boolean articulations.
K-Maps, on the other hand, are only useful for a relatively small number of variables, so it’s important to keep this in mind. The K-Map becomes more difficult to use and more complicated with more variables. The Quine-McCluskey algorithm or using a computer program might be better options in these situations.
FAQs
- Are Karnaugh Maps currently used?
- Yes, it is still widely used in different digital circuit design solution
- How K-Maps simplify Boolean expressions?
- It simplifies Boolean expressions through identification groups of ones or zeros that can be combined to create a simpler expression.
- What are the advantages of using Karnaugh Maps in circuit design?
- Using Karnaugh Maps in circuit design we can get efficient designs and fastly because this map helps to reduce the number of logic gates needed and decreases eh power use
- Can Karnaugh Maps be used for circuits with a large number of variables?
- Yes,K Maps can be used for circuits having many variables, through map table will be larger and difficult to work with.
:
Category:Digital Design CircuitsBy HenryApril 1, 20231 Comment
Author:Henry
https://www.theengineeringknowledge.com
I am a professional engineer and graduate from a reputed engineering university also have experience of working as an engineer in different famous industries. I am also a technical content writer my hobby is to explore new things and share with the world. Through this platform, I am also sharing my professional and technical knowledge to engineering students.Follow me on: Twitter and Facebook.
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