6 Karnaugh Maps Theory Support Electronics Digital Electronics

6. Karnaugh Maps Theory Support Electronics - Digital Electronics Karnaugh Maps Topics covered in this presentation: n Karnaugh Maps n Using Karnaugh Maps to Simplify Circuits 1 of 13

6. Karnaugh Maps Theory Support Karnaugh Maps Electronics - Digital Electronics A Karnaugh map is a method of simplifying a Boolean expression that contains at least three variables. The Karnaugh map is a grid showing every possible combination of the variables ANDed together. The grid is arranged so that when moving between adjacent columns or rows, only one variable changes at a time. The grid for a three variable Boolean expression is shown. 2 of 13

6. Karnaugh Maps Theory Support Using a Karnaugh Map Electronics - Digital Electronics The Boolean expression can then be entered into the Karnaugh map. Like a truth table, a 1 is entered into the grid to show when the Boolean expression is true. The following Boolean expression would be entered into the Karnaugh map as shown below. 3 of 13

6. Karnaugh Maps Theory Support Using a Karnaugh Map Electronics - Digital Electronics The next stage is to group 1 s together. Adjacent 1 s can be looped together in horizontal and vertical groups of 2, 4 or 8, but never diagonally. Single 1 s should be looped by themselves. Loops should be drawn starting with the largest possible loop, and additional loops should then be drawn until all the 1 s are included in a loop. Loops should always be made as large as possible, and adjacent 1 s can be included into more than one loop. Since each loop results in a separate term, the smallest number of loops created will result in the simplest expression. 4 of 13

6. Karnaugh Maps Theory Support Electronics - Digital Electronics Using a Karnaugh Maps are designed so that only one variable changes between adjacent rows and columns. This is also true for the edges of the map. Between the left and right columns, and between the top and bottom rows, there is a change to only one variable. This allows loops to be made between the top and bottom rows and left and right columns. In the example opposite, a loop has been drawn grouping together terms at each end of the Karnaugh map. 5 of 13

6. Karnaugh Maps Theory Support Electronics - Digital Electronics Using a Karnaugh Map Each group of 1 s is a separate term in the final simplified expression that can be determined by inspection. The first loop in the example groups together 1 s for both A and C irrespective of the state of B. The expression is therefore A. C The second loop groups together 1 s for both A and B irrespective of the state of C. The expression is therefore A. B The last loop in the example is for a single term, and so cannot be simplified. A. B. C 6 of 13

6. Karnaugh Maps Theory Support Electronics - Digital Electronics Using a Karnaugh Map Each loop on the Karnaugh map equates to one of the terms in the simplified Boolean Expression. The final simplified equation is therefore: 7 of 13

6. Karnaugh Maps Theory Support Karnaugh Maps With More Variables Electronics - Digital Electronics Karnaugh maps can also be used with more than three variables, by expanding the grid. The example below shows a four variable Karnaugh map for the expression: 8 of 13

6. Karnaugh Maps Theory Support Karnaugh Maps With More Variables Electronics - Digital Electronics Groups of 1 s can be grouped together as shown to get the following simplified equation. 9 of 13

6. Karnaugh Maps Theory Support Alternative Karnaugh Map Electronics - Digital Electronics The style of Karnaugh map shown so far is very useful if starting from a Boolean expression. However, a combinational logic problem often starts from a Truth Table. A different grid can be drawn which creates an alternative style of Karnaugh map. It will give exactly the same result but from a different starting point. The Karnaugh map will have a different cell for every line of a truth table. Here is the truth table and Karnaugh map for a simple 2 -input AND. 10 of 13

6. Karnaugh Maps Theory Support Alternative Karnaugh Map Electronics - Digital Electronics You will have noticed that the order of binary bits above the box is not in the standard binary sequence (00, 01, 10, 11). It is arranged so that the values for adjacent columns vary by only a single bit (00, 01, 10). This sequence is known as grey code and it is a key factor in the way Karnaugh maps work. 11 of 13

6. Karnaugh Maps Theory Support Alternative Karnaugh Map Electronics - Digital Electronics Here is the truth table for a 3 -input problem and its associated Karnaugh map. Once the map has been created the process is exactly the same as the first method. The key cells are grouped and expressions for each group of cells derived. 12 of 13

6. Karnaugh Maps Theory Support Alternative Karnaugh Map Electronics - Digital Electronics Here is a four-input problem and its associated Karnaugh map. 13 of 13