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Comments: It should be evident that the original sum-of-products expression, which
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could be implemented with just one AND and one OR gate has a much more ef cient realization In the next section we show a systematic approach to function minimization
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logic circuit of Figure 1320 may be found in the accompanying CD-ROM
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NAND and NOR Gates In addition to the AND and OR gates we have just analyzed, the complementary forms of these gates, called NAND and NOR, are very commonly used in practice In fact, NAND and NOR gates form the basis of most practical logic circuits Figure 1321 depicts these two gates, and illustrates how they can be easily interpreted in terms of AND, OR, and NOT gates by virtue of De Morgan s laws You can readily verify that the logic function implemented by the NAND and NOR gates corresponds, respectively, to AND and OR gates followed by an inverter It is very important to note that, by De Morgan s laws, the NAND gate performs a logical addition on the complements of the inputs, while the NOR gate performs a logical multiplication on the complements of the inputs Functionally, then, any logic function could be implemented with either NOR or NAND gates only
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(A B) = A + B
A B A B
(A + B) = A B
(A B)
(A + B)
A 0 0 1 1
B 0 1 0 1
A 1 1 0 0 NAND gate
B 1 0 1 0
(A B) 1 1 1 0
A 0 0 1 1
B 0 1 0 1
A 1 1 0 0 NOR gate
B 1 0 1 0
(A + B) 1 0 0 0
Figure 1321 Equivalence of NAND and NOR gates with AND and OR gates
In the next section we shall learn how to systematically approach the design of logic functions First, we provide a few examples to illustrate logic design with NAND and NOR gates
EXAMPLE 136 Realizing the AND Function with NAND Gates
Problem
Use a truth table to show that the AND function can be realized using only NAND gates, and show the physical realization
Solution
Known Quantities: AND and NAND truth tables
13
Digital Logic Circuits
A 0 1 A
B(= A) 0 1
AB 0 1
(AB) 1 0
Find: AND realization using NAND gates Assumptions: Consider two-input functions and gates Analysis: The truth table below summarizes the two functions:
NAND
A NOT A
B 0 1 0 1
NAND A B 1 1 1 0
AND A B 0 0 0 1
Figure 1322 NAND gate as an inverters
A B AB
0 0 1 1
NAND
NAND
Clearly, to realize the AND function we need to simply invert the output of a NAND gate This is easily accomplished if we observe that a NAND gate with its inputs tied together acts as an inverter; you can verify this in the above truth table by looking at the NAND output for the input combinations 0-0 and 1-1, or by referring to Figure 1322 The nal realization is shown in Figure 1323
Comments: NAND gates naturally implement functions that contain complemented
products Gates that employ negative logic are a natural consequence of the inverting characteristics of transistor switches (refer to Section 105) Thus, one should expect that NAND (and NOR) gates are very commonly employed in practice
EXAMPLE 137 Realizing the AND Function with NOR Gates
Problem
Show analytically that the AND function can be realized using only NOR gates, and determine the physical realization
A 0 1
B( = A) (A + B) (A + B) 0 0 1 1 1 0
Solution
Known Quantities: AND and NOR functions Find: AND realization using NOR gates Assumptions: Consider two-input functions and gates
Figure 1324 NOR gate as an inverter
A NOR B NOR B (A + B) = AB
Analysis: We can solve this problem using De Morgan s theorem The output of an AND gate can be expressed as f = A B Using De Morgan s theorem we write:
f =f =A B =A+B The above function is implemented very easily if we see that a NOR gate with its input tied together acts as a NOT gate (see Figure 1324) Thus, the logic circuit of Figure 1325 provides the desired answer
Comments: NOR gates naturally implement functions that contain complemented sums
Gates that employ negative logic are a natural consequence of the inverting characteristics of transistor switches (refer to Section 105) Thus, one should expect that NOR (and NAND) gates are very commonly employed in practice
Part II
Electronics
EXAMPLE 138 Realizing a Function with NAND and NOR Gates
Problem
Realize the following function using only NAND and NOR gates: y = (A B) + C
A B C
Solution
Known Quantities: Logical expression for y Find: Realization of y using only NAND and NOR gates Assumptions: Consider two-input functions and gates Analysis: On the basis of the two preceding examples, we see that we can realize the
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