BCD ADDER/SUBSTRATOR

Boolean Algebra: Chapter 6

源自於 http://lizarum.com/assignments/boolean_algebra/chapter6.html

Circuits 2011-2012
SACC

Boolean Algebra: Chapter 6

// INTRODUCTION

BCD binary numbers represent Decimal digits 0 to 9. A 4-bit BCD code is used to represent the ten numbers 0 to 9. Since the 4-bit Code allows 16 possibilities, the first 10 4-bit combinations are considered to be valid BCD combinations. The latter six combinations are invalid and do not occur.



You can produce a one-bit sum and one bit carry using 2 boolean functions:

Sum=AB +AB

Carry=AB

Together these functions produce a half-adder, which adds two bits together, but cannot add in a carry from a previous operation.

A full-adder adds 3 1-bit inputs (2-bits and a carry from a previous addition) and produces two outputs (sum and the carry):

Sum=(A BC_in) +(AB C_in)+(AB C_in)+(ABC_in)

C_out=(AB)+AC_in)+BC_in)

// LAB

1. Design a circuit to add 2 2-bit numbers using a 7483 4-bit adder. Use 2 switches on the designer to input 1 value and 2 other switches to input the other value.
   
   
   • The 7483 adds 2 4-bit values. What must be done with the unused inputs?
   • What should be done with the carry input?
   • How many outputs from the adder must be connected to the LEDs to see all the possible outputs?
   
   
   

   ---

2. Design a circuit to display the decimal equivalent of a 4-bit BCD value. Use a 7447 BCD, and a seven-segment common anode display.
   
   

   ---

3. Design a circuit to subtract values. Take the number you want to subtract, invert it, add 1 to it, then add this value to the number you want to subtract from.
   
   

   ---

4. For sums between 10 and 18, you must subtract 10 and produce a carry. Subtract 10 from 1010₂ and add 2s complement >0110₂
   
   For sums >9 you need to add 2's complement of 10₁₀ to the uncorrected result(S₃,S₂,S₁,S₀).
   
   Correction is also needed when a carry out (C₃ is generated (16,17,18)
   
   A decoder is needed to detect when carry out (C_n to the next stage is needed.
   
   K-Map for C_n
   

   	S'1S;0
   S'3S;2		00	01	11	10
   	00	0	1	3	2
   	01	4	5	7	6
   	11
   	10	8	9

   
   C_n=C'₃+S'₃S'₂+S'₃S'₁
   
   To implement a 4-bit BDC adder you need 2 4-bit full adders. One to add 2 4-bit BCD numbers and the other full adder to add the 2's complement of 10₁₀ to the result if C_n=1
   
   You also need 2 AND gates and one OR gate to generate C_n
   
   

   ---

5. BDCD Subtraction
   
   Design a 9's Complement Generator for a BCD Digit 9's complement
   The 9's complement of a decimal number is found by subtracting each digit in the number from 9
   

   Decimal Digit	9's complement
   0	9
   1	8
   2	7
   |	|
   9	0

   
   the 9's complement of 28=99-28=71
   the 9's complement of 562=999-562=437
   
   Subtraction of a smaller decimal number from a larger one can be done by adding the 9's complement of the smaller number to the larger number and then adding the carry to the result (end around carry). When subtracting a larger number from a smaller one there is no carry and the result in the 9's complement form is negative:
   
   86-99=-13
   
   Rules
   • Add 9's complement of B to A
   • If result>9, correct by adding 0110
   • If most significant carry is produced (i.e.=1), then the result is positive and the end around carry must be added
   • If the most significant carry is 0 (no carry) then the result is negative and you get the 9's complement of the result
   
   
   
   
   
   But, unfortunately subtractors are not widely available. Fortunately, you can generate the 9's complement b adding 1010 to the inverted number:
   

   BCD Digit	DIGIT	DIGIT+1010
   0000	1111	1001
   0001	1110	1000
   0010	1101	0111
   0011	1100	0110
   0100	1011	0101
   0101	1010	0100
   0110	1001	0011
   0111	1000	0010
   1000	0111	0001
   1001	0110	0000

   
   Ignore the carry out
   
   

Source: Passafine, John and Michael Douglas, Digital Logic Design