Combinational circuits are digital logic circuits whose outputs depend only on the current inputs. They include half adders, full adders, decoders, encoders, and multiplexers. A half adder performs addition of two bits and produces sum and carry outputs. A full adder can add three bits by including the previous bit's carry. Combinational circuits can be represented through truth tables, Boolean equations, or schematic diagrams. Subtraction circuits like half and full subtractors also exist to perform subtraction operations.

1. COMBINATIONAL
CIRCUITS
Half Adder
Full Adder

2. Combinational Circuits
 Combinational Circuits:
o A combinational circuit is a connected arrangement of logic gates with a set of
inputs and outputs.
o The behavior of a combinational circuit is memoryless i.e. the output depends
solely on its most recent input and is independent of the circuit’s past
history.
o In general term, a combinational circuit consists of n binary inputs and m
binary outputs.
o Combinational Logic circuits are made up from basic logic NAND, NOR or
NOT gates that are combined or connected together to produce more
complicated switching circuits.
o Any combinational circuit can be implemented with only NAND and NOR
gates as these are classified as “universal gates.”

3. Combinational Circuits
o As with logic gates, a combinational circuit
can be defined in three ways:
 Truth Table: For each of the 2n
possible combinations of input
signals, the binary value of each of the
m output signal is listed.
 Graphical Symbols: The
interconnected layout of gates is
depicted.
 Boolean Equations: Each output
signal is expressed as a Boolean
function of its input signals.

4. Representation of Combinational Logic
Circuits
◦ Common combinational circuits made up from individual logic gates that carry
out a desired application include Multiplexer, De-multiplexers, Encoders,
Decoders, Full and Half Adders etc.

6. Half Adder
o The most basic digital arithmetic circuit is the addition of two binary digits.
o A combinational circuit that performs the arithmetic addition of two bits is called a
half-adder.
◦ Half-adder truth table and implementation
◦ S = A`B+AB` = A⊕B C = AB

7. Half Adder
◦ To implement half adder using NAND gates; we require 5 NAND gates.
0+1 = 1 +0 =1  S=1, C= 0
1+1 = 10  S=0, C=1
1+1+1 = 11 S=1, C=1

8. Limitations of Half Adder
oThe reason it’s called half adder is that there is no scope to add
the carry bit from previous bit. This is a major limitation of half
adders when used as binary adders especially in real time
scenarios which involves addition of multiple bits.
oTo overcome this limitation, full adders are developed.

9. Full Adder
◦ A full-adder is a combinational circuit that performs the arithmetic sum of
three input bits.
Block Diagram of Full Adder Logic Diagram of Full Adder

10. Full Adder
◦ The full adder can be thought of as two half adders connected together, with the
first half adder passing its carry to the second half adder.

11. Truth Table for Full Adder
o The Boolean Expression for a full
adder is: Sum = (A⊕B) ⊕Cin
◦C-Out = A.B + Cin(A⊕B) =
AB+ACin+BCin
◦ AB+ACin+BCin
◦ = AB + ACin+BCin(A+A`)
◦ = ABCin+AB+ACin+A`BCin
◦ = AB(1+Cin)+ACin+A`BCin
◦ = AB+ACin+A`BCin
◦ =AB+ACin(B+B`)+A`BCin
◦=ABCin+AB+AB`Cin+A`BCin
◦=AB(1+Cin)+AB`Cin+A`BCin
◦=AB+AB`Cin+A`BCin
◦=AB+Cin(AB`+A`B) =
AB+Cin(A⊕B)

12. Half Subtractor
◦ The half-subtractor is a combinational circuit which is used to perform subtraction of two bits. It has two inputs, A (minuend)
and B (subtrahend) and two outputs Difference and Borrow. The logic symbol and truth table are shown below.

13. Circuit Diagram of Half Subtractor
◦ From the above truth table we can find the boolean expression.
Difference = A ⊕ B
Borrow = A' B

14. Full Subtractor
◦ A full subtractor is a combinational circuit that performs subtraction involving three bits, namely A
(minuend), B (subtrahend), and Bin (borrow-in) . It accepts three inputs: A (minuend), B (subtrahend) and
a Bin (borrow bit) and it produces two outputs: D (difference) and Bout (borrow out). The logic symbol
and truth table are shown below.

15. Truth Table
◦ From the above truth table we can find
the boolean expression.
D = A ⊕ B ⊕ Bin
Bout = A' Bin + A' B + B Bin

16. Circuit Diagram

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