Page 1043 - The Mechatronics Handbook

TRUTH TABLE
A B C f(A, B, C)
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
f(A, B, C) = A(B + C) + (A + B)C
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
(a)
P6b → f(A, B, C) = AB + AC + AC + BC P7b → f(A, B, C) = A(A + B + C) + (A + B + C)C
T13a → f(A, B, C) = AB + AC + AC P6b → f(A, B, C) = (A +C)(A + B + C)
(b) (c)
P7a → P7b → f(A, B, C) = (A + C + BB)(A + B + C)
f(A, B, C) = AB(C + C) + A(B + B)C + A(B + B)C
P6a → f(A, B, C) = (A + B + C)(A + B + C)(A + B + C)
P6b →
f(A, B, C) = ABC + ABC + ABC + ABC + ABC + ABC (e)
T8a →
f(A, B, C) = ABC + ABC + ABC + ABC + ABC
(d)
f(A, B, C) = m 111 + m 110 + m 100 + m 011 + m 001 f(A, B, C) = M 000 + M 010 + M 101
f(A, B, C) = ∑m(1, 3, 4, 6, 7) f(A, B, C) = Π(0, 2, 5)
(f) (g)

FIGURE 36.4 Examples of converting boolean functions between forms: (a) given example, (b) conversion to SP
form, (c) conversion to PS form, (d) conversion to canonical SP form, (e) conversion to canonical PS form, (f)
minterm notation/canoncial SP form, (g) maxterm notation/canonical PS form.

Minterms are a special set of functions, none of which can be expressed in terms of the others. Each
minterm has each of the variables in the complemented or the uncomplemented form ANDed together.
An SP expansion in which only minterms appear is a canonical SP expansion. Figure 36.4(d) shows the
development of the canonical SP expansion for the previous example. The canonical SP expansion may
also be simply expressed by enumerating the minterms as shown in Fig. 36.4(f). Comparison of the truth
table with the minterm expansion shows that each function value of 1 represents a minterm of the
function and vice versa. All other function values are 0.
Maxterms are a special set of functions, none of which can be expressed in terms of the others. Each
maxterm has each of the variables in the complemented or the uncomplemented form ORed together. A
PS expansion in which only maxterms appear is a canonical PS expansion. Figure 36.4(e) shows the
development of the canonical PS expansion for the previous example. The canonical PS expansion may
also be simply expressed by enumerating the maxterms as shown in Fig. 36.4(g). Comparison of the truth
table with the maxterm expansion shows that each function value of 0 represents a maxterm of the function
and vice versa. All other function values are 1.

36.13 Realization

The different types of boolean expansions provide different circuits for implementing the generation of the
function. A function expressed in the SP form is directly realized as an AND–OR realization, as illustrated
in Fig. 36.5(a). A function expressed in the PS form is directly realized as an OR–AND realization as
illustrated in Fig. 36.5(b). By using involution and deMorgan’s theorm the SP expansion can be expressed
in terms of NAND–NAND and the PS expansion can be expressed in terms of NOR–NOR, as shown in
Figs. 36.5(c,d). The variable inversions specified in the inputs can be supplied by either NAND or NOR
gates, as shown in Figs. 36.5(g,h), which then provide the NAND–NAND–NAND and the NOR–NOR–NOR
circuits shown in Figs. 36.5(i,j).

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