Combinational Circuits

Different logic gates are utilized to build the encoder, multiplexer, decoder, and de-multiplexer in combinational circuits. The output of these circuits largely depends on the levels present at the input terminals at any given time, for example. No memory is present in this circuit. The current state of this circuit is unaffected by the previous state of the input. A combinational circuit has “n” number of inputs and “m” number of outputs.

Adder Circuits

Half Adder

This is one of the basic combinational circuits. Adds two 1-bit independently.It has 2 inputs and 2 outputs.

Half adder Circuit truth table

Half adder circuit diagram

Carry expression for Half Adder: C = A . B

Sum expression for Half Adder: S = A + B

• Only one of them is 1 provided the other is 0 ,and the sum is obtained.

Half adder using NAND gates

Ex:

3     0 0 1 1

+

  2    0 0 1 0

=

  5    0 1 0  1

Half adder application

Half adder has many applications and few of them are listed below

• The computer’s Arithmetic and Logic Unit likes this adder circuit when performing adds on binary bits.
• The Full Adder circuit is created when two half adder circuits are combined.
• Calculators are typically designed with these logic circuits.
• These circuits are preferred for calculating the addresses and tables.
• These circuits can handle a variety of applications in digital circuits instead of just addition. Additionally, this turns into the brain of digital electronics.

Limitation :Half adder adds two 1 bit numbers independently but cannot be used in circuit to add series of numbers,thus we choose full adder.

Full adder

Full Adder is used to add more than 1-bit numbers. It has 3 inputs and 2 outputs.Carry in is given as an input apart from the other two bit inputs.

Carryout expression:  Cout= (A.B) + CinA ⊕ CinB

• If all three inputs are 1.
• If one input is 1 provided all other are 0,these two conditions are preferred to obtain a sum.

Sum expression: S = A + B+ Cin

• Any two of the inputs is 1 irrespective of third one provides a carryout.

Serial Adder

Serial Adder is used when we want to add large numbers we combine the full adders to form a serial adder.Here each adder depends on the previous adders to perform the addition operation.Thus rippling of the carry makes the circuit operation very slow. These combinational circuits is designed very need based.

In serial adder to perform a 4 bit addition the task is completed in 4 cycles and for 8 bits 8 cycles,so on.

Inputs:

A=A₀,A₁,A₂,A₃

B=B₀,B₁,B₂,B₃

C_in

Outputs:

S= S₀ ,S₁ ,S₂ ,S₃

C_out

Here the addition of A₁,B₁ happens only when C₁ is provided,thus this process of addition is slow and consumes more cycle for the operation completion.

Serial adder circuit diagram

Carry Look ahead adder

To improve the performance of the serial adder which operates slowly,alternative method is used to add bits,which is known as carry look ahead adder. Here,The carry is predicted simultaneously.Thus no adder is dependent on previous add,hence addition operation is done simultaneously. 

In order for carry lookahead adders to function, two bits termed Carry Propagate and Carry Generate, denoted by Cp and Cg, are generated. The Cg bit is used to generate the output carry bit, which is independent of the input carry bit, once the Cp bit has been propagated to the following stage. The 4-bit carry lookahead adder architecture is depicted in the image below.

Inputs:

Y=Y₀,Y₁,Y₂,Y₃

X=X₀,X₁,X₂,X₃

C_in

Outputs:

S= S₀ ,S₁ ,S₂ ,S₃

C_out

Let,us see how can the Carry C₁ can be obtained

C₁= X₀ Y₀ + X_o C₀ + Y₃ C_o

C₁= X₀ Y₀ + C₀ (X₃ + Y₃)

C₁    =     g0   +     p0 C_in

              ↓             ↓

      Generate     propagate.

C₂   =  g2  +p2 C₁

C₂   =  g2  +p2 g0 + p2 p0 C_i

Thus we can clearly see that we can obtain the C₂ carry without the addition of first adder.

• In first cycle all the g,p are generated.
• In second cycle all the carries are given to the adder by carry look ahead generator.
• In third cycle the addition in all the adders happen simultaneously
• In 4 bit, 8 bit or 64 bit the addition operation is completed with in 3 cycles,hence this adder is faster than the serial adder,

Adder/Subtractor circuit

Actually there is no subtractor type of combinational circuits in the world. Subtraction is nothing but addition of one number with the other numbers 2’s complement. We can identify if a circuit is an adder or subtractor is by looking at the XOR gate.An adder can perform as a subtractor with the help of the XOR gate.

X + Y (OR) X + (2’s complement of Y) (OR) X + (1’s complement of Y+1) 

Here C₁=C_in

C₂=C_Out

• If S₁ = 0,an additional operation is performed.

S=A+B

• If S_1,= 1,subtraction is performed

S=X + (2’s complement of Y)