4-bit binary Adder-Subtractor

In Digital Circuits, A Binary Adder-Subtractor is can do both the addition and subtraction of binary numbers in one circuit itself. The operation is performed depending on the binary value the control signal holds. It is one of the components of the ALU (Arithmetic Logic Unit). 

What is Binary Adder?

A Binary Adder is an electronic circuit that sums two binary numbers. The most basic types are half adders and full adders. A half adder takes in two one-bit numbers, producing a sum and a carry bit. Full adder have three input bits-two actual bits and an incoming carry from the preceding operation. This makes the Full Adder to add multi-bit binary numbers as the carry can propagate through a number of Full Adders in a ripple-carry adder configuration.

What is Binary Subtractor?

A Binary Subtractor is used to perform the subtraction of two binary numbers. Just as in the case of addition, this also can be done using an array of logic gates. The subtraction is done by using two's complement, in which the circuit first negates the binary number to obtain its complement and then adds one to the result before imputing it into a binary adder circuit together with the original number. Both addition and subtraction are handled by this type of circuit.

This Circuit Requires prerequisite knowledge of Xor Gate, Binary Addition and Subtraction, and Full Adder. 
Let's consider two 4-bit binary numbers A and B as inputs to the Digital Circuit for the operation with digits

A0 A1 A2 A3 for A
B0 B1 B2 B3 for B 

The circuit consists of 4 full adders since we are performing operations on 4-bit numbers. There is a control line K that holds a binary value of either 0 or 1 which determines that the operation is carried out is addition or subtraction. 

As shown in the figure, the first full adder has a control line directly as its input(input carry Cin), The input A0 (The least significant bit of A) is directly input in the full adder. The third input is the exor of B0 and K. The two outputs produced are Sum/Difference (S0) and Carry (C0). 

If the value of K (Control line) is 1, the output of B0(exor)K=B0'(Complement B0). Thus the operation would be A+(B0'). Now 2's complement subtraction for two numbers A and B is given by A+B'+Cin. This suggests that when K=1, the operation being performed on the four-bit numbers is subtraction. 

Similarly If the Value of K=0, B0 (exor) K=B0. The operation is A+B which is simple binary addition. This suggests that When K=0, the operation is performed on the four-bit numbers in addition. 

Then C0 is serially passed to the second full adder as one of it's outputs. The sum/difference S0 is recorded as the least significant bit of the sum/difference. A1, A2, A3 are direct inputs to the second, third and fourth full adders. Then the third input is the B1, B2, B3 EXORed with K to the second, third and fourth full adder respectively. The carry C1, C2 are serially passed to the successive full adder as one of the inputs. C3 becomes the total carry to the sum/difference. S1, S2, S3 are recorded to form the result with S0. 

For an n-bit binary adder-subtractor, we use n number of full adders. 
Example: Assume that we have two 3-bit numbers, i.e., X=100 and Y=011, and feed them in Full-Adder as an input.

X0 = 0     X1 = 0     X2 = 1
Y0 = 1     Y1 = 1    & Y2 = 0

For K=0: 
Y0⨁K=Y0 and Cin=K=0
So, from first Full-Adder

S0 = X0+Y0+Cin
S0= 0+1+0
S0=1
C0=0

Similarly,

S1 = X1+Y1+C0
S1 = 0+1+0
S1=1 and C1=0

Similarly,

S2 = X2+Y2+C1
S2 = 1+0+0
S2=1 and C2=0

Thus,

X= 100 =4
Y = 011 = 3
Sum = 0111 = 7

For K=1

Y0⨁K=Y0' and Cin=k=1

So,

S0 = X0+Y0'+Cin
S0 = 0+0+1
S0=1 and C0=0

Similarly,

S1 = X1+Y1'+C0
S1 = 0+0+0
S1=0 and C1=0

Similarly,

S2 = X2+Y2'+C1
S2 = 1+1+0
S2=0 and C2=0

Thus,

X = 010 = 4
Y = 011 = 3

Difference = 001 = 1

Advantage of Binary Adder and Subtractor

• Low Design Complexity: Both binary adder and subtractor circuits are easy to design using logic gates like XOR, AND, and OR.
• High-Speed Operations: Binary adders such as parallel adders-and subtractors can do their operations at a high speed.
• Versatility: It allows the same hardware : to add ,to subtract. Saving on redundancy of separate components and designs for different functions.

Disadvantages of Binary Adder and Subtractor

• Carry Propagation Delay: In a simple ripple carry adder, the carry has to ripple through all stages of the adder; this increases the computation time as more bits are added. This will be more for a larger number of bits in binary.
• Hardware Complexity: The more the number of bits, the more complicated the circuit gets due to more gates. To eliminate the problem of delays, higher versions of adders such as carry lookahead adders are included which increases complexity.

Conclusion

In general, the most important of these in binary adders and subtractors is to form the backbone of any arithmetic performed by computers and calculators or any other digital system. The design is relatively easy for small numbers but gets complex as the size of the numbers increases resulting in carry propagation delay.

Considering two's complement, among other methods, it can be done both addition and subtraction without additional hardware. In Reality, it happens exactly the opposite: designers manage to maintain balance between speed, complexity, and power consumption by not implementing these circuits in real life.