32 Bit Adder/Subtractor
Explanation of the circuit.
First
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| Full Adder Circuit in Logisim |
We need to
make a Full Adder circuit. Truth-table for the same is:
A
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B
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C
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CARRY
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SUM
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0
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0
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0
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0
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0
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0
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0
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1
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0
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1
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0
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1
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0
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0
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1
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0
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1
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1
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1
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0
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1
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0
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0
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0
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1
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1
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0
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1
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1
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0
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1
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1
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0
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1
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0
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1
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1
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1
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1
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1
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Boolean
Expression in the SOP form is:
1.) For Sum : S = (A+B)C’ + (A XOR B)C
2.) For Carry : CARRY = AC + B(A XNOR C)
Hence, 1-bit
Full Adder circuit is made.
Now we have
to made 32-bit Adder/Subtractor circuit but placing 1-bit FA 32 times is bit
complicated and requires more space.
Alternative
solution for the same is to first, make a 4-bit
Adder using FA, then make 16-bit Adder using ‘4-Bit Adder’ And finally making 32-Bit Adder/Subtractor using 2
“16-Bit Adder” circuit.
· 4-Bit Adder Circuit :-
In the above
4-bit Adder circuit, we are joining Cin input from the 1st
FA to 2nd FA and so on and finally to the Cout. Also,
each FA requires 1-bit input at A and B, So we used Splitter that takes 4-bit input and distribute each bit to the
required FA.
Here “Overflow” happens when at the MSB [4th
digit] summation there will be a carry generated and the whole answer would be
of 5-bit but the output is of 4-bit only. Hence, we have Cout output
at the end so that it “detects Overflow”.
Now, similarly
we make 16-Bit Adder using the above 4-bit circuit.
· 16 – Bit Adder Circuit :-
[Same as the
4-Bit Adder Circuit].
Here A and B
both have the MSB as 1 and the Cout has the value 1 that shows that
the circuit has “Overflow”. Also notice that the 9th digit carry is
shown in the SUM showing that Adder is working fine.
Now Finally.....
·
32-Bit Adder/Subtractor Circuit :-
Now, for making 32-bit Adder/Subtractor we need to make some changes in the circuit. Usually
when we make 4-bit Adder we are getting range of [0,15] numbers in decimal no
system. However when we make a substractor the range changes to [-8,7].
This happens because for making a subtractor, we need
to get the 1’s complement of the B. Example, 8 – 4 = 8 + (-4). Comparing it
with A + (-B), we need the complement of B. For that we add XOR gate to the
input of the B and Cin and the output is given to the FA’s 2nd
output.
The following
figure shows the Circuit in subtraction mode,
Where user
has to set the Cin as 1 to do subtraction of the numbers.
However,
here Cout showing 1 doesn’t mean “Overflow.” Here,
FA2 means
the previous 16-Bit Adder circuit. We are also using “Bit Extender” with sign
mode so that when Cin is at 1 it will give the result at XOR gate as
1 also and vice-versa.
ð Notice that in subtraction, If the
answer is negative than we have to make the 2’s complement of the answer in
order to get the real one.
Q-) What is Overflow?
Ans – The
condition that occurs when a calculation precedes a result that is greater in
the magnitude than that which a storage location cannot store is called
“Overflow”.
Here,
overflow occurs if there are not enough bits to represent the correct result.
Example, For
4-bit Adder we are getting
range of [0,15] numbers in decimal no system. However when we make a
subtractor the range changes to [-8,7].* Thank you visiting this site.
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