Unit 5 Karnaugh Maps Fundamentals of Logic Design by Roth and Kinney

5.1 Minimum forms of Switching Functions Cost is related to the number of gates in a realization. Karnaugh Maps: can be used to find the minimum cost two level circuits using AND, OR, and NOT gates. SOP—AND gates feed an OR gate. POS—OR gates feed an AND gate.

5.1 (cont.) Minimum SOPs are not always unique. Steps: –1. Combine terms using the uniting theorem. –2. Eliminate redundant terms using the consensus them or other theorems.

5.1 Cont. Definitions: –Minimum SOPs Has a minimum number of terms and Of all the expressions which have the same minimum number of terms, also has a minimum number of literals. –Minimum POSs Has a minimum number of terms and A minimum number of literals. Examples on page 135 and 136 illustrate the process.

5.2 Two and Three-Variable Karnaugh Maps Karnaugh Maps specify the value of the function for every combination of values of the independent variables. –Figure 5.1—2 value Karnaugh Map –Figure 5.2—3 value Karnaugh Map –Figure 5.3—Illustrates binary and decimal notation.

5.2 (cont.) Procedure for Karnaugh Maps 1. Plot the minterm expansion on the map by placing a 1 and 0 in the appropriate boxes. (see Figure 5-4, page 138) 2.Group the 1’s into pairs and groups of 4 (see Figure 5.5, page 139). 3. Assign product terms to the groups (again see Figure 5.5)

5.2(cont.) F’ can also be found using K maps (substitute 1’s for 0’s and 0’s for 1’s.) After finding the minimum minterm expansion for F’, then the complement can be taken to find the minimum Maxterm expansion (see page 140.)

5.3 Four-Variable Karnaugh See figure 5.10—minterms are located next to the four terms with which each could be combined. Example: f(a,b,c,d) = acd+ a’b + d’ –Figure 5-11, 5-12 –Incompletely Specified Function—Fig

5.4 Determination of Minimum Expressions Using Essential Prime Implicants Implicant of F: any single or group of 1’s which can be combined. Prime Implicant of F: product term implicant which cannot be combined with another term to eliminate a variable.

5.4 (cont.) A second 1 on a map is a prime implicant if it is not adjacent to another 1. A group of two is a prime implicant if it is not contained in a group of 4. Four adjacent 1’s form a prime implicant if they are not in a goup of 8. Figure 5-16 illustrates prime implicants and non-prime implicants.

5.4 (cont.) Procedure for finding a minimum sum. –Page 147 shows steps. –Flow chart is shown on page 148.

5.5 Five Variable Karnaugh Maps 3 dimensions are used. Section 5.5 illustrates the process.

5.6 Other Uses of Karnaugh Maps Can be used to list minterms, similar to truth tables. Can be useful to factor expressions.

5.7 Other Forms of Karnaugh Maps See Figure 5.27 and 5.28