Adders and Subtractors in Digital Logic

Adders and subtractors are digital circuits used to perform arithmetic operations on binary numbers. An adder adds two binary values to produce a sum and often including a carry bit for values that exceed a single digit. A subtractor calculates the difference between two binary numbers, sometimes using methods like two’s complement to handle negative results. These circuits form the core of arithmetic logic units (ALUs) in processors and enable efficient calculation and data processing.

Half Adder

• It is a arithmetic combinational logic circuit designed to perform addition of two single bits.
• It contain two inputs and produces two outputs.
• Inputs are called Augend and Added bits and Outputs are called Sum and Carry.

Let us observe the addition of single bits,

0+0=0
0+1=1
1+0=1
1+1=10

Since 1+1=10, the result must be two bit output. So, Above can be rewritten as,

0+0=00
0+1=01
1+0=01
1+1=10

The result of 1+1 is 10, where '1' is carry-output (C_out) and '0' is Sum-output (Normal Output). 

Truth Table of Half Adder:

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Next Step is to draw the Logic Diagram. To draw Logic Diagram, We need Boolean Expression, which can be obtained using K-map (karnaugh map). Since there are two output variables 'S' and 'C', we need to define K-map for each output variable. 

K-map for output variable Sum 'S':

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 K-map is of Sum of products form. The equation obtained is

S = AB' + A'B

which can be logically written as,

S = A ⊕ B

K-map for output variable Carry 'C':

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The equation obtained from K-map is,

C = AB

Using the Boolean Expression, we can draw logic diagram as follows..

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Limitations: Adding of Carry is not possible in Half adder. 

Read more about Half Adder.

Full Adder

• To overcome the above limitation faced with Half adders, Full Adders are implemented.
• It is a arithmetic combinational logic circuit that performs addition of three single bits.
• It contains three inputs (A, B, C_in) and produces two outputs (Sum and C_out).
• Where, C_in -> Carry In and C_out -> Carry Out

Truth table of Full Adder:

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K-map Simplification for output variable Sum 'S' :

The equation obtained is,

S = A'B'C_in + AB'C_in' + ABC + A'BC_in'

The equation can be simplified as,

S = B'(A'C_in+AC_in') + B(AC + A'C_in')
S = B'(A xor C_in) + B (A xor C_in)'
S = A xor B xor C_in

K-map Simplification for output variable 'C_out'

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The equation obtained is,

C_out = BC_in + AB + AC_in

Logic Diagram of Full Adder:👁 Image

Read more about Full Adder.

Half Subtractor

• It is a combinational logic circuit designed to perform the subtraction of two single bits.
• It contains two inputs (A and B) and produces two outputs (Difference and Borrow-output).

Truth Table of Half Subtractor:

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K-map Simplification for output variable 'D':

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The equation obtained is,

D = A'B + AB'

which can be logically written as,

D = A xor B

K-map Simplification for output variable 'B_out' :

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The equation obtained from above K-map is,

B_out = A'B

Logic Diagram of Half Subtractor:

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Read more about Half Subtractor.

Full Subtractor

• It is a Combinational logic circuit designed to perform subtraction of three single bits.
• It contains three inputs(A, B, B_in) and produces two outputs (D, B_out).
• Where, A and B are called Minuend and Subtrahend bits.
• And, B_in -> Borrow-In and B_out -> Borrow-Out

Truth Table of Full Subtractor:

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K-map Simplification for output variable 'D' :👁 Image
The equation obtained from above K-map is,

D = A'B'B_in + AB'B_in' + ABB_in + A'BB_in'

which can be simplified as,

D = B'(A'B_in + AB_in') + B(AB_in + A'B_in')
D = B'(A xor B_in) + B(A xor B_in)'
D = A xor B xor B_in

K-map Simplification for output variable 'B_out' :👁 Image

The equation obtained is,

B_out = BB_in + A'B + A'B_in

Logic Diagram of Full Subtractor:👁 Image
Applications:

1. For performing arithmetic calculations in electronic calculators and other digital devices.
2. In Timers and Program Counters.
3. Useful in Digital Signal Processing.

Read more about Full Subtractor.