SWITCHING THEORY AND LOGIC DESIGN PPT

SWITCHING THEORY AND LOGIC DESIGN

1. Switching Theory & Logic Design Unit 2

2. NAND-NAND IMPLEMENTATION: Alternative Representation of a NAND gate: According to De-Morgan’s first theorem NAND ≡ Bubbled OR When all inputs are HIGH, the output is LOW. If any one or both the inputs are LOW, then the output is HIGH. The NAND gate is equivalent to an OR gate with a bubble at its inputs. This alternative representation is used for converting the AND-OR-NOT logic to NAND-NAND logic. ≡ A  B  C A B C Unit 2 , Integral University, Lucknow Unit 2 3 A B C

3. IMPLEMENTATION OF NAND-NAND TRUTH TABLE 1: INPUT OUTPUT A B Y  A . B 1 1 1 1 1 1 1 (i) NOT gate implementation: A A . A  A  A  A Unit 2 , Integral University, Lucknow Unit 2 4 A NOT is obtained from a single input NAND gate behaves as an inverter.

4. (ii) AND Gate Implementation: An AND o p era t ion req u ires two NAND gat e s. (iii) OR Gate Implementation: An OR operation requires 3 NAND gate. For an OR operation the normal input A and B are first complemented by using two single input NAND gates. The complemented variables are then are then fed to another NAND gate producing the normal OR output. A B A . B A . B  A . B A . A  A B . B  B A . B  A  B  A  B Unit 2 , Integral University, Lucknow Unit 2 5 A B

5. Unit 2 , Integral University, Lucknow Unit 2 6 Step to be followed: Simplify the given logical expression and convert it in the SOP form. First Draw the logic diagram using Basic gates ( AND-OR-NOT ) realization. In order to convert AND gate, Apply bubble at the output of the AND gate . In order to convert OR gate, apply bubble at the input of the OR gate . (according to DeMorgan’s Law) . Replacing the every OR gate, by apply bubble at the input of the OR gate , Replace every AND gate by a bubble at the output , replace every NOT gate by a NAND inverter. Draw the circuit using only NAND gates.

6. EXAMPLE 1: Implement the following Boolean function using NAND gates. SOLUTION: Step 1: Draw the AND-OR-NOT realization. (fig.1) Step 2: Replace AND by NAND, OR by bubbled OR and NOT by NAND inverter. (fig.2) Y  A B C  A B C  A B C Y A B C A B C A ≡ B B C C Fig. 1 Fig. 2 A B C A B C A A B C Y A B C A B C A B C Unit 2 , Integral University, Lucknow Unit 2 7

7. The required output is Y  A B C  A B C  A B C Unit 2 , Integral University, Lucknow Unit 2 8    Y     A B C    A B C    A B C  

8. SOLUTION: Step 1: AND/OR Implementation EXAMPLE: Construct a logic circuit for the given Boolean expression, by using NAND gates only. Y  ( A  E ) .  B  C . D  ( A  E ) ( B  C D ) ( B  C D ) ( A  E ) A E Unit 2 , Integral University, Lucknow Unit 2 16 C D B Y

9. Step 2: Substituting equivalent NAND function AND AND OR C D OR B + CD Y A C D B E A  E Unit 2 , Integral University, Lucknow Unit 2 17

10. Step 3: NAND implementation C D B Y ( A  E ) ( B  C D ) A E Unit 2 , Integral University, Lucknow Unit 2 18 1 2 3 4

11. NOR-NOR IMPLEMENTATION: According to De-Morgan’s second theorem NO R ≡ Bu b bled AND ≡ PROCEDURE: Step 1: Simplify the given logical expression and convert it into product of sum (POS) form. Step 2: Draw AND-OR-NOT realization. A B C B C A A  B  C A . B . C Unit 2 , Integral University, Lucknow Unit 2 19

12. Step 3: Replace every OR gate by NOR gate, every AND gate by bubbled input AND gate and every inverter by a NOR inverter. Step 4: Draw the final circuit with NOR gates. EXAMPLE: Implement the following Boolean Function using only NOR gates. F   A  B  C   A  B  D SOLUTION: AND-OR-NOT realization. A B C D A  B  C Unit 2 , Integral University, Lucknow Unit 2 20 A  B F

13. Step 2: Replace OR by NOR, AND by bubbled AND and NOT by NOR inverter. A Unit 2 , Integral University, Lucknow Unit 2 21 B C D F

14. Step 3: F A B C D Unit 2 , Integral University, Lucknow Unit 2 22

15. Unit 2 THA N K YOU