Design a BCD to Excess-3 code converter.

Designing a BCD (Binary-Coded Decimal) to Excess-3 code converter involves creating a logic circuit that translates a 4-bit BCD input into a 4-bit Excess-3 output. The Excess-3 code is a non-weighted code used for decimal digits which is derived from the corresponding BCD code by adding 3 to each digit.

Here's a step-by-step approach to design the converter:

Step 1: Understand BCD and Excess-3 Codes

BCD is a binary representation of decimal numbers where each digit of the decimal number is represented by its four-bit binary equivalent.

Excess-3 is a self-complementary code and is used in digital systems. To get the Excess-3 code from BCD, simply add 3 to the BCD number.

Step 2: Construct the Truth Table

The truth table for the BCD to Excess-3 conversion is as follows:

BCD Code	Decimal	Excess-3 Code
0000	0	0011
0001	1	0100
0010	2	0101
0011	3	0110
0100	4	0111
0101	5	1000
0110	6	1001
0111	7	1010
1000	8	1011
1001	9	1100

Step 3: Determine the Logic Expressions

From the truth table, we can derive the Karnaugh Maps (K-Maps) for each bit of the Excess-3 code. Let's denote the BCD inputs as A, B, C, and D (where A is the MSB and D is the LSB) and the Excess-3 outputs as W, X, Y, and Z (where W is the MSB and Z is the LSB).

Using K-Maps, we can simplify the logic for each output bit:

• For W (MSB of Excess-3):

  • W = A + BC + BD

• For X:

  • X = A'B + AB' + CD

• For Y:

  • Y = B'C + BC' + AD

• For Z (LSB of Excess-3):

  • Z = A'D + AD' + B'C' + BC

Step 4: Implement the Logic Circuit

Using the logic expressions derived from the K-Maps, we can now draw the logic circuit using AND, OR, and NOT gates.

Here is an example of how the logic circuit would look:

   A ---+---+---+---+
        |   |   |   |
   B ---+---|--|---|---+---(W)
        |   |  |   |   |
   C ---+---|-(AND)--(OR)--(X)
        |   |         |
   D ---+---+--------(OR)--(Y)
            |         |
            +--------(OR)--(Z)

Step 5: Verify the Design

Once the logic circuit is implemented, it's important to verify that it works correctly. You can do this by testing each BCD input and confirming that the output matches the Excess-3 code in the truth table.

Example:

Let's convert the BCD number 0101 (which is 5 in decimal) to Excess-3.

Using the logic expressions:

• W = A + BC + BD = 0 + 0*1 + 0*1 = 0
• X = A'B + AB' + CD = 1*0 + 0*1 + 0*1 = 0
• Y = B'C + BC' + AD = 1*0 + 0*0 + 0*0 = 0
• Z = A'D + AD' + B'C' + BC = 1*1 + 0*0 + 1*1 + 0*0 = 1

So, the Excess-3 code for BCD 0101 is 1000.

By following these steps, you can design a BCD to Excess-3 code converter for any BCD input. The key is to create an accurate truth table, derive the simplest logic expressions using K-Maps, and then implement the logic circuit with the appropriate gates.