Implement a function F = A∗B + C∗D with NAND gates.

Q.) Implement a function F = A∗B + C∗D with NAND gates.

 Subject: Digital Logic Design

To implement the function F = A∗B + C∗D using only NAND gates, we need to understand the properties of NAND gates and how they can be used to create other logical operations. A NAND gate is a universal gate, meaning that you can create any other logic gate using just NAND gates.

The truth table for a NAND gate is as follows:

A B A NAND B
0 0 1
0 1 1
1 0 1
1 1 0

The Boolean expression for a NAND gate is: [ \overline{A \cdot B} ]

Now, let's break down the function F = A∗B + C∗D into its components and see how we can implement each using NAND gates.

  1. AND operation (A∗B and C∗D): An AND operation can be implemented using NAND gates by simply negating the output of a NAND gate. The Boolean expression for an AND gate using NAND is: [ A \cdot B = \overline{\overline{A \cdot B}} ]

  2. OR operation (A∗B + C∗D): An OR operation can be implemented using NAND gates by first negating both inputs and then applying them to a NAND gate. The Boolean expression for an OR gate using NAND is: [ A + B = \overline{\overline{A} \cdot \overline{B}} ]

Now, let's implement the function F step by step using NAND gates:

Step 1: Implement the AND operations (A∗B and C∗D)

  • Create the first AND operation by using a NAND gate to get the output ( \overline{A \cdot B} ), and then feed this output into another NAND gate to negate it again, resulting in ( A \cdot B ).
  • Repeat the same process for the second AND operation to get ( C \cdot D ).

Step 2: Implement the OR operation (A∗B + C∗D)

  • Negate the outputs of the AND operations from Step 1 using two additional NAND gates.
  • Finally, use one more NAND gate to combine these negated outputs, which will give us the OR operation.

Here's the schematic using NAND gates:

 A ---|    |---|    |
      |NAND|   |NAND|---|    |
 B ---|    |---|    |   |NAND|--- Output (A * B)
                     |    |
                     |    |---|    |
 C ---|    |---|    |   |NAND|   |
      |NAND|   |NAND|---|    |   |
 D ---|    |---|    |       |    |---|    |
                                 |NAND|--- Output (C * D)
                                 |    |
                                 |    |---|    |
                                 |NAND|   |NAND|--- F = A*B + C*D
                                 |    |---|    |

Step by Step Implementation:

  1. Connect inputs A and B to a NAND gate to get ( \overline{A \cdot B} ).
  2. Connect the output of step 1 to another NAND gate to negate it again, resulting in ( A \cdot B ).
  3. Connect inputs C and D to a NAND gate to get ( \overline{C \cdot D} ).
  4. Connect the output of step 3 to another NAND gate to negate it again, resulting in ( C \cdot D ).
  5. Negate the output of step 2 using a NAND gate to get ( \overline{A \cdot B} ).
  6. Negate the output of step 4 using a NAND gate to get ( \overline{C \cdot D} ).
  7. Connect the outputs of steps 5 and 6 to a final NAND gate. The output of this gate is ( \overline{\overline{A \cdot B} \cdot \overline{C \cdot D}} ), which is equivalent to ( A \cdot B + C \cdot D ), giving us the desired function F.

By following these steps, we have successfully implemented the function F = A∗B + C∗D using only NAND gates.