Karnaugh Maps - Part 1
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KMAP Main. Forgot Password? For help understanding the coming conversion, please click here This Web site is compatible with Microsoft Internet Explorer version 7. User Name. If you have received a PIN letter, you may set up your account now. Log On ID.Table of Contents. Karnaugh map or K-map is a map of a function used in a technique used for minimization or simplification of a Boolean expression.
It results in less number of logic gates and inputs to be used during the fabrication. Booleans expression can be simplified using Boolean algebraic theorems but there are no specific rules to make the most simplified expression. However, K-map can easily minimize the terms of a Boolean function. Unlike an algebraic method, K-map is a pictorial method and it does not need any Boolean algebraic theorems. K-map is basically a diagram made up of squares. Each of these squares represents a min-term of the variables.
K-map is made using the truth table. In fact, it is a special form of the truth table that is folded upon itself like a sphere. Every two adjacent squares of the k-map have a difference of 1-bit including the corners.
The expression produced by K-map may be the most simplified expression but not unique. There can be more than 1 simplified expression for a single function but they all perform the same. In Gray code, every two consecutive number has a difference of 1-bit. As the squares in K-map also differs from its adjacent square by 1-bit which is why the variables in K-map are written in grey code.
The gray code ensures that each cell of K-map is in 1-bit difference with each other. The table for BCD to Gray code is given below. Karnaugh map of 2 to 4 variables is very easy.
However, 5 and 6 variable K-map is a little bit complex. We will discuss one by one in details. Therefore there are 4 cells squares in 2 variable K-map for each minterm. The rows of the columns will be represented by variable B. The square facing the combination of the variable represents that min term as shown in fig below. Grouping in 2 variables K-map is easy as there are few squares. Function F A, B. K-map from Truth table. In the 2 nd group, Variable B is changing and variable A remains unchanged.
Now the simplifies expression will be the sum of these two terms as given below. Compare this expression with the original expression of the function, this expression only uses one gate during its implementation. Note the combination of two variables in either form is written in Gray code.Image to gcode converter software free download
So the min terms will not be in a decimal order. So they can be made into groups.Telemundo online
Some examples of grouping:. They are adjacent as there is only one-bit difference. That is why they can be grouped together.Maurice Karnaugh introduced it in   as a refinement of Edward Veitch 's Veitch chart  which actually was a rediscovery of Allan Marquand 's logical diagram  aka Marquand diagram'  but with a focus now set on its utility for switching circuits.
The Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability. The required Boolean results are transferred from a truth table onto a two-dimensional grid where, in Karnaugh maps, the cells are ordered in Gray code  and each cell position represents one combination of input conditions, while each cell value represents the corresponding output value.
Optimal groups of 1s or 0s are identified, which represent the terms of a canonical form of the logic in the original truth table. Karnaugh maps are used to simplify real-world logic requirements so that they can be implemented using a minimum number of physical logic gates.
Boolean conditions, as used for example in conditional statementscan get very complicated, which makes the code difficult to read and to maintain. Once minimised, canonical sum-of-products and product-of-sums expressions can be implemented directly using AND and OR logic operators.
More systematic methods for minimizing complex expressions began to be developed in the early s, but until the mid to late s the Karnaugh map was the most common used in practice. Karnaugh maps are used to facilitate the simplification of Boolean algebra functions. For example, consider the Boolean function described by the following truth table. Following are two different notations describing the same function in unsimplified Boolean algebra, using the Boolean variables ABCDand their inverses.
Digital Circuits - K-Map Method
In the example above, the four input variables can be combined in 16 different ways, so the truth table has 16 rows, and the Karnaugh map has 16 positions. The row and column indices shown across the top, and down the left side of the Karnaugh map are ordered in Gray code rather than binary numerical order. Gray code ensures that only one variable changes between each pair of adjacent cells. Each cell of the completed Karnaugh map contains a binary digit representing the function's output for that combination of inputs.
After the Karnaugh map has been constructed, it is used to find one of the simplest possible forms — a canonical form — for the information in the truth table. Adjacent 1s in the Karnaugh map represent opportunities to simplify the expression. The minterms 'minimal terms' for the final expression are found by encircling groups of 1s in the map.
Minterm groups must be rectangular and must have an area that is a power of two i. Minterm rectangles should be as large as possible without containing any 0s.Menards 2x6x10 treated
Groups may overlap in order to make each one larger. The optimal groupings in the example below are marked by the green, red and blue lines, and the red and green groups overlap. The cells are often denoted by a shorthand which describes the logical value of the inputs that the cell covers. For example, AD would mean a cell which covers the 2x2 area where A and D are true, i. The grid is toroidally connected, which means that rectangular groups can wrap across the edges see picture.
Cells on the extreme right are actually 'adjacent' to those on the far left, in the sense that the corresponding input values only differ by one bit; similarly, so are those at the very top and those at the bottom. Therefore, A D can be a valid term—it includes cells 12 and 8 at the top, and wraps to the bottom to include cells 10 and 14—as is BDwhich includes the four corners. Once the Karnaugh map has been constructed and the adjacent 1s linked by rectangular and square boxes, the algebraic minterms can be found by examining which variables stay the same within each box.
Thus the first minterm in the Boolean sum-of-products expression is A C. For the green grouping, A and B maintain the same state, while C and D change. B is 0 and has to be negated before it can be included. The second term is therefore A B. Note that it is acceptable that the green grouping overlaps with the red one.In previous chapters, we have simplified the Boolean functions using Boolean postulates and theorems. It is a time consuming process and we have to re-write the simplified expressions after each step.
To overcome this difficulty, Karnaugh introduced a method for simplification of Boolean functions in an easy way. This method is known as Karnaugh map method or K-map method. The adjacent cells are differed only in single bit position. K-Map method is most suitable for minimizing Boolean functions of 2 variables to 5 variables.
Now, let us discuss about the K-Maps for 2 to 5 variables one by one. The number of cells in 2 variable K-map is four, since the number of variables is two. The following figure shows 2 variable K-Map. The number of cells in 3 variable K-map is eight, since the number of variables is three.
The following figure shows 3 variable K-Map. The number of cells in 4 variable K-map is sixteen, since the number of variables is four. The following figure shows 4 variable K-Map. Let R 1R 2R 3 and R 4 represents the min terms of first row, second row, third row and fourth row respectively.
Similarly, C 1C 2C 3 and C 4 represents the min terms of first column, second column, third column and fourth column respectively. The number of cells in 5 variable K-map is thirty-two, since the number of variables is 5.
Karnaugh Maps, Truth Tables, and Boolean Expressions
The following figure shows 5 variable K-Map. There are two possibilities of grouping 16 adjacent min terms. In the above all K-maps, we used exclusively the min terms notation.
Similarly, you can use exclusively the Max terms notation. If the Boolean function is given as sum of min terms form, then place the ones at respective min term cells in the K-map. If the Boolean function is given as sum of products form, then place the ones in all possible cells of K-map for which the given product terms are valid.
Check for the possibilities of grouping maximum number of adjacent ones. It should be powers of two. Start from highest power of two and upto least power of two. Highest power is equal to the number of variables considered in K-map and least power is zero. Each grouping will give either a literal or one product term.
It is known as prime implicant. Note down all the prime implicants and essential prime implicants. The simplified Boolean function contains all essential prime implicants and only the required prime implicants. The given Boolean function is in sum of products form. So, we require 4 variable K-map. The 4 variable K-map with ones corresponding to the given product terms is shown in the following figure.
There are no possibilities of grouping either 16 adjacent ones or 8 adjacent ones.
Introduction of K-Map (Karnaugh Map)
There are three possibilities of grouping 4 adjacent ones.Maurice Karnaugh, a telecommunications engineer, developed the Karnaugh map at Bell Labs in while designing digital logic based telephone switching circuits. Karnaugh maps reduce logic functions more quickly and easily compared to Boolean algebra. By reduce we mean simplify, reducing the number of gates and inputs. We like to simplify logic to a lowest cost form to save costs by elimination of components.
We define lowest cost as being the lowest number of gates with the lowest number of inputs per gate. Given a choice, most students do logic simplification with Karnaugh maps rather than Boolean algebra once they learn this tool. We show five individual items above, which are just different ways of representing the same thing: an arbitrary 2-input digital logic function.
First is relay ladder logic, then logic gates, a truth table, a Karnaugh map, and a Boolean equation. The point is that any of these are equivalent. Two inputs A and B can take on values of either 0 or 1high or low, open or closed, True or False, as the case may be.
This is applicable to all five examples. These four outputs may be observed on a lamp in the relay ladder logic, on a logic probe on the gate diagram. These outputs may be recorded in the truth table, or in the Karnaugh map.
Look at the Karnaugh map as being a rearranged truth table. The Output of the Boolean equation may be computed by the laws of Boolean algebra and transfered to the truth table or Karnaugh map.
Which of the five equivalent logic descriptions should we use? The one which is most useful for the task to be accomplished. The outputs of a truth table correspond on a one-to-one basis to Karnaugh map entries. Below, we show the adjacent 2-cell regions in the 2-variable K-map with the aid of previous rectangular Venn diagram like Boolean regions. Referring to the previous truth table, this is not the case. Which brings us to the whole point of the organizing the K-map into a square array, cells with any Boolean variables in common need to be close to one another so as to present a pattern that jumps out at us.
Compare this to the square Venn diagram above the K-map. Compare the last two maps to the middle square Venn diagram. To summarize, we are looking for commonality of Boolean variables among cells. The Karnaugh map is organized so that we may see that commonality.
The truth table contains two 1 s. Look for adjacent cells, that is, above or to the side of a cell. Diagonal cells are not adjacent. Adjacent cells will have one or more Boolean variables in common. This might be easier to see by comparing to the Venn diagrams to the right, specifically the B column. For the Truth table below, transfer the outputs to the Karnaugh, then write the Boolean expression for the result.
Transfer the 1 s from the locations in the Truth table to the corresponding locations in the K-map. The solution of the K-map in the middle is the simplest or lowest cost solution. A less desirable solution is at far right. After grouping the two 1 s, we make the mistake of forming a group of 1-cell. The reason that this is not desirable is that:. The way to pick up this single 1 is to form a group of two with the 1 to the right of it as shown in the lower line of the middle K-map, even though this 1 has already been included in the column group B.The map is updated Monday, Wednesday and Friday by p.
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The K-map method supports the elimination of potential race conditions and permits the rapid identification. By using Karnaugh map technique, we can reduce the Boolean expression containing any number of variables, such as 2-variable Boolean expression, 3-variable Boolean expression, 4-variable Boolean expression and even 7-variable Boolean expressions, which are complex to solve by using regular Boolean theorems and laws.
It will look like see below image. The possible min terms with 2 variables A and B are A. The following table shows the positions of all the possible outputs of 2-variable Boolean function on a K-map. When we are simplifying a Boolean equation using Karnaugh map, we represent the each cell of K-map containing the conjunction term with 1. After that, we group the adjacent cells with possible sizes as 2 or 4. In case of larger k-maps, we can group the variables in larger sizes like 8 or The groups of variables should be in rectangular shape, that means the groups must be formed by combining adjacent cells either vertically or horizontally.
Diagonal shaped or L-shaped groups are not allowed. The following example demonstrates a K-map simplification of a 2-variable Boolean equation. Here the lower right cell is used in both groups. After grouping the variables, the next step is determining the minimized expression.Linear or nonlinear calculator
By reducing each group, we obtain a conjunction of the minimized expression such as by taking out the common terms from two groups, i. For a 3-variable Boolean function, there is a possibility of 8 output min terms.
The general representation of all the min terms using 3-variables is shown below. A typical plot of a 3-variable K-map is shown below. It can be observed that the positions of columns 10 and 11 are interchanged so that there is only change in one variable across adjacent cells. This modification will allow in minimizing the logic. Up to 8 cells can be grouped in case of a 3-variable K-map with other possibilities being 1,2 and 4. The largest group size will be 8 but we can also form the groups of size 4 and size 2, by possibility.
In the 3 variable Karnaugh map, we consider the left most column of the k-map as the adjacent column of rightmost column. So the size 4 group is formed as shown below. So the group of size 4 is reduced as the conjunction Y.
To consume every cell which has 1 in it, we group the rest of cells to form size 2 group, as shown below. The 2 size group has no common variables, so they are written with their variables and its conjugates. In this equation, no further minimization is possible. There are 16 possible min terms in case of a 4-variable Boolean function.
The general representation of minterms using 4 variables is shown below. A typical 4-variable K-map plot is shown below. It can be observed that both the columns and rows of 10 and 11 are interchanged.
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