Page 213 - A Practical Guide from Design Planning to Manufacturing

186 Chapter Six

XYZ + XYZ Neighboring products
Differ only by Y input
XZ Can be combined
Not neighboring products
XYZ + XYZ Differ by X and Y inputs
Cannot be combined
Figure 6-9 Neighboring products.

In Karnaugh maps, the cells to the left, right, above, and below any
cell are holding the function’s value for neighboring fundamental prod-
ucts. Cells that are diagonally separated are not neighbors. For exam-
ple in the two-variable map (Fig. 6-10), fundamental product F is a
0
neighbor of the products of F and F , but not F . Being able to graphi-
1
2
3
cally determine logical neighbors allows the minimal sum to be found
more easily than repeated equation transformations.
To find the minimal sum, first the Karnaugh map is filled in with the
function’s values from the truth table. Then all the 1’s in the Karnaugh
map must be circled as part of a set of 1’s. All the 1’s in a set must be
neighbors and the number of 1’s in a set must be a power of 2. The goal
is to include all the 1’s using the smallest number of sets and the largest
sets possible. Each set corresponds to a term in the minimal sum, the
one created by combining all the neighboring products in the set. This
term is the one that includes only the input values, which are the same
for all the members of the set. By examining each set, the terms of the
minimal sum can be written.
In the first example in Fig. 6-11, the two 1’s in the map can be included
in a single set of two cells. The value of input Y is different for the cells
in the set, which means the term corresponding to this set does not include
Y. The value of input Z is 0 for all the cells in the set, so the term corre-
__
sponding to this set will include Z. Because this is the only set needed and
__
Y and Z are the only input variables, the complete minimal sum is Z.
In the second example in Fig. 6-11, the three 1’s in the map require
two sets of two cells. A set of three is not allowed (since three is not a
power of 2). The three 1’s could be covered by one set of two cells and

Y Z F Y
0 0 F 0 0 1
0 1 F 0 F 0 F 2
1
Z
1 0 F 2 1 F 1 F 3
1 1 F 3
Figure 6-10 Two-variable
Karnaugh map.