The full adder has 3 inputs and 2 ouputs. The first 2 inputs are the 2 bits a & b to add while the 3rd input c is the carry from the previous significant bit while the outputs are the same: sum S and the carry C. The following table shows the result of different combinations of inputs:
a b c S(sum) C(carry)
0 0 0 0 0
0 1 0 1 0
1 0 0 1 0
1 1 0 0 1
0 0 1 1 0
0 1 1 0 1
1 0 1 0 1
1 1 1 1 1
K-map for the output variable SUM is as follow:
This circuit is a level 3 circuit as we also need inverters at level 1, then we have 4 3-input AND gates at level 2 and 4-input OR gate at level 3. So we need 3 gate delays (3Δ) to get the output for Sum.
K-map for the variable carry is as follow:
When we implement this circuit we see that this is a level 2 circuit as we have AND gates at level 1 and 3-input OR gate at level 2 and hence we need 2 gate delays (2Δ) to get carry output.
Lets now put the equations in different form:
S = ab’c’ + a’b’c + a’bc’ + abc = Σ (1,2,4,7)
= b’ (ac’ + a’c) + b (a’c’ + ac) = b’ (ac’ + a’c) + b (ac’ + a’c)’
=b’ (a xor c) + b (a xor c)’ {We know (ac’ + a’c)’ = a’c’ + ac and a’c + ac’ = a xor c}
= b’z + z’b= b xor z {z= (a xor c)}
= b xor a xor c
S = a xor b xor c
and C = ab + ac + bc = ab(c + c’) + ac (b + b’) + bc (a + a’) = abc + abc’ + abc + ab’c + abc + a’bc = abc +a’bc+ab’c+abc’= Σ(3,5,6,7)
So we can draw the circuits using XOR, NOT, AND & OR gates
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