# DIGITAL CIRCUIT LOGIC UNIT 7: MULTI-LEVEL GATE CIRCUITS NAND AND NOR GATES

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1 DIGITAL CIRCUIT LOGIC UNIT 7: MULTI-LEVEL GATE CIRCUITS NAND AND NOR GATES 1

2 iclicker Question 13 Considering the K-Map, f can be simplified as (2 minutes): A) f = b c + a b c B) f = ab d + a b d AB CD 1 1 C) f = b c D) f = b d 1 1 2

3 Learning Objectives 1. Design a minimal two-level or multi-level circuit of AND and OR gates to realize a given function. (Consider both circuits with an OR gate at the output and circuits with an AND gate at the output.) 2. Design or analyze a two-level gate circuit using any one of the eight basic forms (AND-OR, NAND-NAND, OR-NAND, NOR-OR, OR-AND, NOR-NOR, AND-NOR, and NAND-AND). 3. Design or analyze a multi-level NAND-gate or NOR-gate circuit. 4. Convert circuits of AND and OR gates to circuits of NAND gates or NOR gates, and conversely, by adding or deleting inversion bubbles. 5. Design a minimal two-level, multiple-output AND-OR, OR- AND, NAND-NAND, or NOR-NOR circuit using Karnaugh maps. 3

4 Multi-level Gate Circuits Terminology: The maximum number of gates cascaded in series between a circuit input and the output is referred to as the number of levels of gates (not to be confused with voltage levels). AND-OR circuit means a two-level circuit composed of a level of AND gates followed by an OR gate at the output. OR-AND circuit means a two-level circuit composed of a level of OR gates followed by an AND gate at the output. OR-AND-OR circuit means a three-level circuit composed of a level of OR gates followed by a level of AND gates followed by an OR gate at the output. Circuit of AND and OR gates implies no particular ordering of the gates; the output gate may be either AND or OR. 4

5 Multi-level Gate Circuits Different Realizations for the Same Expression, Z: 5

6 Multi-level Gate Circuits Example 1: 6

7 Multi-level Gate Circuits Example 1 (continued): 7

8 Multi-level Gate Circuits Example 1 (continued): 8

9 Multi-level Gate Circuits Example 1 (continued): 9

10 Multi-level Gate Circuits Example 1 (continued): 10

11 Multi-level Gate Circuits Example 1 (continued): 11

12 Multi-level Gate Circuits Example 1 (continued): 12

13 NAND and NOR Gates NAND Gates: A NAND Gate is equivalent to an AND gate followed by an inverter, or AND-NOT gate (as shown in Figure 7-8(b)). The n-input NAND gate s output is: 13

14 NAND and NOR Gates NOR Gates: A NOR Gate is equivalent to an OR gate followed by an inverter, or an OR-NOT gate. The n-inputs NOR gate s output is: 14

15 NAND and NOR Gates Functionally Complete Operations and Gates: A set of logic operations is said to be functionally complete if any Boolean function can be expressed in terms of this set of operations. If a single gate forms a functionally complete set by itself, then any switching function can be realized using only gates of that type, i.e. NAND, NOR. 15

16 NAND and NOR Gates Procedure to Determine if Functionally Complete: 1. Write out a minimum sum-of-products expression for the function realized by each gate. If no complement appears in any of these expressions, then NOT cannot be realized, and the set is not functionally complete. If a complement appears in one of the expressions, then NOT can generally be realized by an appropriate choice of inputs to the corresponding gate. (We will always assume that 0 and 1 are available as gate inputs). 16

17 NAND and NOR Gates Procedure to Determine if Functionally Complete (continued): 2. Next, attempt to realize AND or OR, keeping in mind that NOT is now available. Once AND or OR has been realized, the other one can always be realized using DeMorgan s laws if no more direct procedure is apparent. For example, if OR and NOT are available, AND can be realized by 17

18 Design of Two-Level NAND- and NOR- Gate Circuits Conversion of AND and OR Gate Circuits to NAND and NOR: A two-level circuit composed of AND and OR gates is easily converted to a circuit composed of NAND gates or NOR gates. This conversion is carried out by using F = (F ) and then applying DeMorgan s laws: 18

19 Design of Two-Level NAND- and NOR- Gate Circuits Example 2: 19

20 Design of Two-Level NAND- and NOR- Gate Circuits Example 2 (continued): 20

21 Design of Two-Level NAND- and NOR- Gate Circuits Eight Basic Forms for Two-Level Circuits: 21

22 Design of Two-Level NAND- and NOR- Gate Circuits Eight Basic Forms for Two-Level Circuits (continued): 22

23 iclicker Question 14 What are the output functions of the following logic gates? (50 seconds): A) NOR, NAND B) XOR, XNOR C) NAND, NOR D) [(A+B) ], [(AB) ] E) NOT-OR, NOT-AND 23

24 iclicker Question 14 What are the output functions of the following logic gates? (50 seconds): A) NOR, NAND B) XOR, XNOR C) NAND, NOR D) [(A+B) ], [(AB) ] E) NOT-OR, NOT-AND 24

25 Design of Two-Level NAND- and NOR- Gate Circuits Procedure for Designing a Minimum Two-Level NAND-NAND circuit: 25

26 Design of Two-Level NAND- and NOR- Gate Circuits Procedure for Designing a Minimum Two-Level NOR-NOR circuit: 26

27 Design of Multi-Level NAND- and NOR- Gate Circuits Procedure to Design Multi-Level NAND-Gate Circuits: 27

28 Design of Multi-Level NAND- and NOR- Gate Circuits Example 3: 28

29 Circuit Conversion Using Alternative Gate Symbols Alternative Gate Symbols: Alternate symbol for inverters: The following gate symbols have been derived using DeMorgan s Laws. 29

30 Circuit Conversion Using Alternative Gate Symbols NAND Gate Circuit Conversion: 30

31 Circuit Conversion Using Alternative Gate Symbols NOR Gate Circuit Conversion: 31

32 Circuit Conversion Using Alternative Gate Symbols Conversion of AND-OR Circuit to NAND Gates: 32

33 Circuit Conversion Using Alternative Gate Symbols Procedure to Convert to a NAND (or NOR) Circuit: 1. Convert all AND gates to NAND gates by adding an inversion bubble at the output. Convert all OR gates to NAND gates by adding inversion bubbles at the inputs. (To convert to NOR, add inversion bubbles at all OR-gate outputs and all AND-gate inputs.) 2. Whenever an inverted output drives an inverted input, no further action is needed because the two inversions cancel. 3. Whenever a noninverted gate output drives an inverted gate input or vice versa, insert an inverter so that the bubbles will cancel. (Choose an inverter with the bubble at the input or output as required.) 4. Whenever a variable drives an inverted input, complement the variable (or add an inverter) so the complementation cancels the inversion at the input. 33

34 Circuit Conversion Using Alternative Gate Symbols Defection: Fan-in is the number of inputs a logic gate can handle. For instance the fan-in for the AND gate shown in the figure is 3. Physical logic gates with a large fan-in tend to be slower than those with a small fan-in. This is because the complexity of the input circuitry increases the input capacitance of the device. 34

35 Circuit Conversion Using Alternative Gate Symbols Example 3: Multi-level circuits reduce gate fan-in. See the example below: 35

36 Circuit Conversion Using Alternative Gate Symbols Example 3 (continued): 36

37 Design of Two-Level, Multiple Output Circuits Example 4: 37

38 Design of Two-Level, Multiple Output Circuits Example 4 (continued): 38

39 Design of Two-Level, Multiple Output Circuits Determination of Essential Prime Implicants for Multiple-Output Realization: We can find prime implicants, which are essential to one of the functions and to the multiple-output realization by a modification of the procedure used for the single-output case. In particular, when we check each 1 on the map to see if it is covered by only one prime implicant, we will only check those 1 s which do not appear on the other function maps. 39

40 Design of Two-Level, Multiple Output Circuits Determination of Essential Prime Implicants for Multiple-Output Realization: 40

41 Design of Multiple Output NANDand NOR- Gate Circuits Converting a Two-Output Circuit to NOR Gates: The procedure given in Section 7.4 for design of single-output, multi-level NAND-and NOR-gate circuits also applies to multiple-output circuits. 41

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