Objectives

implement a half-adder as a sub-circuit 
implement a full adder using two half adders 
implement a half subtractor as a subcircuit
use a half-adder to create a half-subtractors
use two half subtractors or a full adder to implement a full subtractor 

Review

When designing combinational circuits. We discussed five steps:

1. Define/Describe/State the problem
2. Determine the number of variables needed and assign letter symbols to the input and output variables
3. Construct the function table that defines the relationship between the inputs and outputs
4. Simplify the boolean expression for each output. This is done with Karnaugh Maps or Boolean algebra
5. Draw the logic diagram and implement the circuit in Logisim

We will look at designing a combinational circuit for a Half-Adder.

Half-Adder

1. State the Problem:

• A Half-adder is a combinational circuit that performs the addition of 2 bits.
• The results of the Half-adder are a sum and carry bit.
• Examples (remember, what is 10 in binary?):
  

          0 + 0 = 00    where sum = 0, carry = 0 
          0 + 1 = 01    where sum = 1, carry = 0 
          1 + 0 = 01    where sum = 1, carry = 0 
  	1 + 1 = 10    where sum = 0, carry = 1

2. Assign Variables:

• Inputs: 2 bits (x and y)
• Outputs: 2 bits (s and c)

3. Construct the Function Table:

You should be able to fill in the boxes for c and s:

4. Simplify

Here is where we cheat a little. We will not use the Karnaugh maps. Because you have a simple truth table (4 rows), you can match the gates with the gates you already know.

Here are the truth tables for the gates you know:

• Which gate will c use?
• Which gate will s use?

We can get the following expressions by observation. There are two inputs, two gates, and two outputs.

        s = x XOR y 
        c = xy 

5. Draw the Logic Diagram and implement it in Logisim

Full-Adder

You can now try and use two half-adders to create a full adder. Again, we will break the problem into steps:

1. State the Problem:

• A Full-adder is a combinational circuit that forms the arithmetic sum of 3 input bits (one is a carry bit).
• The results of the Full-adder are a sum and carry bit.
• Examples (remember, what is 11 in binary?):
  

          0 + 1 + 0 = 01    where  carry = 0, sum = 1, 
          0 + 0 + 1 = 01    where  carry = 0, sum = 1, 
          1 + 0 + 1 = 10    where  carry = 1, sum = 0, 
  	1 + 1 + 1 = 11    where  carry = 1, sum = 1, 

2. Assign Variables:

• Inputs: 3 bits (x, y and c_i)
• Outputs: 2 bits (s and c_i+1)

3. Construct the Function Table:

Fill in the truth table with what you think c_i+1 and s should be. A few have already been filled in for you.

4. Simplify

You can use the Karnaugh Map reduction method to find the simplfied expressions and then draw the full adder circuit.

Here are the functions for your convenience.

	s = x XOR y XOR ci
	ci+1 = xy + ci(x XOR y)

5. Draw Logic Circuit

Assignments