Page 217 - A Practical Guide from Design Planning to Manufacturing

190 Chapter Six

Sum = ABC + ABC + ABC + ABC
Sum = A(B ⊕ C) + A(B ⊕ C)
Sum = A ⊕ B ⊕ C
Figure 6-17 Simplifying with XORs.

Unfortunately no real simplification was possible for the sum output.
Each of the four fundamental products has no neighbors, which means
that Karnaugh maps cannot help to reduce this function. A tactic that
helps in these cases is looking for XOR functions in the minimal sum.
In general, XORs tend to produce fundamental products with few or no
neighbors. Functions that cannot be reduced easily by Karnaugh maps
are often simplified using XORs. This process is shown for the sum func-
tion in Fig. 6-17.
Using the minimal sum for the C out from Fig. 6-16 and the expression
for Sum from Fig. 6-17, we can now draw the logic gate representation
of a full adder. Figure 6-18 shows three full adders connected to form a
3-bit binary adder.
This type of adder is called a ripple carry adder because the carry output
of one full adder becomes an input of the next full adder. By connecting
full adders in this fashion we can create an adder for any number of bits.
Ripple carry adders are typically used only for small adders because the
delay of this adder increases linearly with the number of bits but for
adding small numbers this simple implementation may be the best choice.

A 2 B 2 A 1 B 1 A 0 B 0
A B A B A B

AB

C
C 2 1 C 0 C out AC
C out C C out C C C in
BC

Sum Sum Sum
S 2 S 1 S 0
Figure 6-18 3-bit ripple carry adder.