Introduction to Digital Systems: Modeling, Synthesis, and Simulation Using VHDL

3.13 NAND AND NOR EQUIVALENT CIRCUIT DESIGN

Through the use of Boolean algebra, it is possible to convert an AND to an OR by inverting the inputs or outputs (DeMorgan's theorem). The same condition holds true for the logical gates NAND and NOR. Figure 3.15 shows equivalencies for AND and OR. Using DeMorgan's theorem, one can generate equivalent logic gates for NAND and NOR gates, as shown in Figure 3.16. The circles at the inputs of the AND and OR gates in Figures 3.15 and 3.16 represent inverters. These inverters are referred to as invert bubbles. Because of the gate equivalency, any logic circuit implemented with NOT, AND, and OR gates could be converted to a logic circuit containing only NAND gates or only NOR gates. This conversion is practical when only one type of gate (NAND or NOR) is available to the designer. In addition, fewer integrated circuits are needed when implementing a logic circuit with NAND or NOR gates.

Often, the final logic circuit is implemented with only NAND gates or only NOR gates. In practice, one usually draws the logic circuit using NOT, AND, and OR gates, then implements a graphical conversion of the gates. During the graphical conversion, invert bubbles are inserted to modify AND and OR gates to NAND or NOR gates, as illustrated in Figure 3.15. Each bubble inserted must be compensated for on the same connecting line. Two consecutive invert bubbles cancel each other out. Single bubbles not canceled must be replaced by inverters. An inverter is replaced ...