# Digital Logic cct. Lec. (6)

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1 THE NAND GATE The NAND gate is a popular logic element because it can be used as a universal gate: that is, NAND gates can be used in combination to perform the AND, OR, and inverter operations. The term NAND is a contraction of NOT-AND and implies an AND function with a complemented (inverted) output. The standard logic symbol for a 2-input NAND gate and its equivalency to an AND gate followed by an inverter are shown in Fig.(1), where the symbol means equivalent to. A rectangular outline symbol is shown in part. Operation of a NAND Gate Fig.(1) Standard NAND gate logic symbols. A NAND gate produces a LOW output only when all the inputs are HIGH. When any of the inputs is LOW, the output will be HIGH. For the specific case of a 2-input NAND gate, as shown in Fig.(3-11) with the inputs labelled A and B and the output labelled X, the operation can be stated as follows: For a 2-input NAND gate, output X is LOW only when inputs A and B are HIGH; X is HIGH when either A or B is LOW, or when both A and B are LOW. Note that this operation is opposite that of the AND in terms of the output level. In a NAND gate, the LOW level (0) is the active or asserted output level, as indicated by the bubble on the output. Fig.(2) illustrates the operation of a 2-input NAND gate for all four input combinations, and Table 3-4 is the truth table summarizing the logical operation of the 2- input NAND gate. Page 1

2 The NAND is the same as the AND except the output is inverted. Fig.(2) Operation of a 2-input NAND gate. Table (1) Truth table for a 2-input NAND gate. INPUTS OUTPUT A B Negative-OR Equivalent Operation of a NAND Gate Inherent in a NAND gate's operation is the fact that one or more LOW inputs produce a HIGH output. Table (1) shows that output X is HIGH (1) when any of the inputs, A and B, is LOW (0). From this viewpoint, a NAND gate can be used for an OR operation that requires one or more LOW inputs to produce a HIGH output. This aspect of NAND operation is referred to as negative-or. The term negative in this context mean that the inputs are defined to be in the active or asserted state when LOW. For a 2-input NAND gate performing a negative-or operation, output X is HIGH when either input A or input B is LOW, or when both A and B are LOW. When a NAND gate is used Page 2

3 to detect one or more LOWs on its inputs rather than all HIGHs, it is performing the negative-or operation and is represented by the standard logic symbol shown in Fig.(3). Fig.(3) Standard symbols representing the two equivalent operation of a NAND gate. Logic Expressions for a NAND Gate The Boolean expression for the output of a 2-input NAND gate is X = AB This expression says that the two input variables, A and B, are first ANDed and then complemented, as indicated by the bar over the AND expression. This is a description in equation form of the operation of a NAND gate with two inputs. Evaluating this expression for all possible values of the two input variables, you get the results shown in Table 2. Table 2 Page 3

4 The NOR Gate The NOR gate, like the NAND gate, is a useful logic element because it can also be used as a universal gate; that is, NOR gates can be used in combination to perform the AND, OR, and inverter operations. The term NOR is a contraction of NOT-OR and implies an OR function with an inverted (complemented) output. The standard logic symbol for a 2-input NOR gate and its equivalent OR gate followed by an inverter are shown in Fig.(4). Operation of a NOR Gate Fig.(4) Standard NOR gate logic symbols A NOR gate produces a LOW output when any of its inputs is HIGH. Only when all of its inputs are LOW is the output HIGH. For the specific case of a 2-input NOR gate, as shown in Fig.(4) with the inputs labelled A and B and the output labelled X, the operation can be stated as follows: For a 2-input NOR gate, output X is LOW when either input A or input B is HIGH, or when both A and B are HIGH; X is HIGH only when both A and B are LOW. This operation results in an output level opposite that of the OR gate. In a NOR gate, the LOW output is the active or asserted output level as indicated by the bubble on the output. Fig.(5) illustrates the operation of a 2-input NOR gate for all four possible input combinations, and Table 3 is the truth table for a 2-input NOR gate. Page 4

5 Fig.(5) Operation of a 2-input NOR gate. Table (3) Truth table for a 2-input NOR gate. Negative-AND Equivalent Operation of the NOR Gate A NOR gate, like the NAND, has another aspect of its operation that is inherent in the way it logically functions. Table 3 shows that a HIGH is produced on the gate output only when all of the inputs are LOW. From this viewpoint, a NOR gate can be used for an AND operation that requires all LOW inputs to produce a HIGH output. This aspect of NOR operation is called negative-and. The term negative in this context means that the inputs are defined to be in the active or asserted state when LOW. Page 5

6 For a 2-input NOR gate performing a negative-and operation, output X is HIGH only when both inputs A and B are LOW. When a NOR gate is used to detect all LOWs on its inputs rather than one or more HIGHs, it is performing the negative-and operation and is represented by the standard symbol in Fig.(6). It is important to remember that the two symbols in Fig.(6), represent the same physical gate and serve only to distinguish between the two modes of its operation. Fig.(6) Standard symbols representing the two equivalent operations of a NOR gate Logic Expressions for a NOR Gate The Boolean expression for the output of a 2-input NOR gate can be written as X = A+B This equation says that the two input variables are first ORed and then complemented, as indicated by the bar over the OR expression. Evaluating this expression, you get the results shown in Table 4. The NOR expression can be extended to more than two input variables by including additional letters to represent the other variables. Page 6

7 Table 4 THE EXCLUSIVE-OR AND EXCLUSIVE-NOR GATES Exclusive-OR and exclusive-nor gates are formed by a combination of other gates already discussed. However, because of their fundamental importance in many applications, these gates are often treated as basic logic elements with their own unique symbols. The Exclusive-OR Gate Standard symbol for an exclusive-or (XOR for short) gate is shown in Fig.(7). The XOR gate has only two inputs. Fig.(7) Standard symbol for an exclusive-or. For an exclusive-or gate, output X is HIGH when input A is LOW and input B is HIGH, or when input A is HIGH and input B is LOW: X is LOW when A and B are both HIGH or both LOW. The four possible input combinations and the resulting outputs for an XOR gate are illustrated in Fig.(8). The HIGH level is the active Page 7

8 Digital Logic cct. Lec. (6) or asserted output level and occurs only when the inputs are at opposite levels. The operation of an XOR gate is summarized in the table shown in Table 5. Fig.(8) All possible logic Table 5 Truth table levels for an exclusive-or gate. for an exclusive-or gate. The Exclusive-NOR Gate Standard symbols for an exclusive-nor (XNOR) gate are shown in Fig.(9). Like the XOR gate, an XNOR has only two inputs. The bubble on the output of the XNOR symbol indicates that its outpu is opposite that of the XOR gate. When the two input logic levels are Page 8

9 opposite, the output of the exclusive-nor gate is LOW. The operation can be stated as follows (A and B are inputs, X is the output): For an exclusive-nor gate, output X is LOW when input A is LOW and input B is HIGH, or when A is HIGH and B is LOW; X is HIGH when A and B are both HIGH or both LOW. Fig.(9) Standard logic symbols for the exclusive-nor gate The four possible input combinations and the resulting outputs for an XNOR gate are shown in Fig,(10). The operation of an XNOR gate is summarized in Table 6. Notice that the output is HIGH when the same level is on both inputs. Fig.(10) All possible logic levels for an exclusive-nor gate. Page 9

10 Table (6) Truth table for an exclusive-nor gate. Example Determine the output waveforms for the XOR gate and for the XNOR gate, given the input waveforms, A and B, in Figure below: Page 10

11 THE UNIVERSAL PROPERTY OF NAND AND NOR GATES 1- The NAND Gate as a Universal Logic Element The NAND gate is a universal gate because it can be used to produce the NOT, the AND, the OR, and the NOR functions. An inverter can be made from a NAND gate by connecting all of the inputs together and creating, in effect, a single input, as shown in Fig.(11)(a) for a 2-input gate. An AND function can be generated by the use of NAND gates alone, as shown in Fig.(11)(b). An OR function can be produced with only NAND gates, as illustrated in part (c). Finally; a NOR function is produced as shown in part (d). Fig. (11) Shows how NAND gates are used to form a Inverter,OR, AND and NOR functions Page 11

12 2- The NOR Gate as a Universal Logic Element Like the NAND gate, the NOR gate can be used to produce the NOT, AND. OR and NAND functions. A NOT circuit, or inverter, can be made from a NOR gate by connecting all of the inputs together to effectively create a single input, as shown in Fig.(12)(a) with a 2- input example. Also, an OR gate can be produced from NOR gates, as illustrated in Fig.(12)(b). An AND gate can be constructed by the use of NOR gates, as shown in Fig.(12)(c). Fig.(12) Shows how NOR gates are used to form a Inverter,OR, AND and NAND functions. Page 12

13 Conversion of AND/ OR/ NOT logic to NAND / NOR logic using graphical procedure. Steps for converting to NAND / NOR logic using graphical procedure: 1- Draw AND/ OR logic. 2- If NAND hardware has been chosen, add bubbles on the output of each AND gate, and bubbles on input side to all OR gate. 3- If NOR hardware has been chosen, add bubbles on the output of each OR gate, and bubbles on input side to each AND gate. 4- Add or subtract an inverter on each line that received a bubble in step 2 or Replace bubbled OR by NAND and bubbled AND by NOR. 6- Eliminate double inversions. Page 13

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