Tag: 64 bit Carry Look Ahead Adder

Carry Look Ahead Adder | 4-bit Carry Look Ahead Adder

Digital Design

Ripple Carry Adder-

 

Before you go through this article, make sure that you have gone through the previous article on Ripple Carry Adder.

 

In Ripple Carry Adder,

  • Each full adder has to wait for its carry-in from its previous stage full adder.
  • Thus, nth full adder has to wait until all (n-1) full adders have completed their operations.
  • This causes a delay and makes ripple carry adder extremely slow.
  • The situation becomes worst when the value of n becomes very large.
  • To overcome this disadvantage, Carry Look Ahead Adder comes into play.

 

 

In this article, we will discuss about Carry Look Ahead Adder.

 

Carry Look Ahead Adder-

 

  • Carry Look Ahead Adder is an improved version of the ripple carry adder.
  • It generates the carry-in of each full adder simultaneously without causing any delay.
  • The time complexity of carry look ahead adder = Θ (logn).

 

Logic Diagram-

 

The logic diagram for carry look ahead adder is as shown below-

 

 

Carry Look Ahead Adder Working-

 

The working of carry look ahead adder is based on the principle-

The carry-in of any stage full adder is independent of the carry bits generated during intermediate stages.

 

The carry-in of any stage full adder depends only on the following two parameters-

  • Bits being added in the previous stages
  • Carry-in provided in the beginning

 

Now,

  • The above two parameters are always known from the beginning.
  • So, the carry-in of any stage full adder can be evaluated at any instant of time.
  • Thus, any full adder need not wait until its carry-in is generated by its previous stage full adder.

 

Also Read- Full Adder Working

 

4-Bit Carry Look Ahead Adder-

 

Consider two 4-bit binary numbers A3A2A1A0 and B3B2B1B0 are to be added.

Mathematically, the two numbers will be added as-

 

 

From here, we have-

C1 = C0 (A0 ⊕ B0) + A0B0

C2 = C1 (A1 ⊕ B1) + A1B1

C3 = C2 (A2 ⊕ B2) + A2B2

C4 = C3 (A3 ⊕ B3) + A3B3

 

For simplicity, Let-

  • Gi = AiBi where G is called carry generator
  • Pi = Ai ⊕ Bi where P is called carry propagator

 

Then, re-writing the above equations, we have-

C1 = C0P0 + G0 ………….. (1)

C2 = C1P1 + G1 ………….. (2)

C3 = C2P2 + G2 ………….. (3)

C4 = C3P3 + G3 ………….. (4)

 

Now,

  • Clearly, C1, C2 and C3 are intermediate carry bits.
  • So, let’s remove C1, C2 and C3 from RHS of every equation.
  • Substituting (1) in (2), we get C2 in terms of C0.
  • Then, substituting (2) in (3), we get C3 in terms of C0 and so on.

 

Finally, we have the following equations-

  • C1 = C0P0 + G0 
  • C2 = C0P0P1 + G0P1 + G1
  • C3 = C0P0P1P2 + G0P1P2 + G1P2 + G2
  • C4 =C0P0P1P2P3 + G0P1P2P3 + G1P2P3 + G2P3 + G3

 

These equations are important to remember.

 

These equations show that the carry-in of any stage full adder depends only on-

  • Bits being added in the previous stages
  • Carry bit which was provided in the beginning

 

Trick To Memorize Above Equations-

 

As an example, let us consider the equation for generating carry bit C2.

There are three possible reasons for generation of C2 as depicted in the following picture-

 

 

In the similar manner, we can write other equations as well very easily.

 

Implementation Of Carry Generator Circuits-

 

The above carry generator circuits are usually implemented as-

  • Two level combinational circuits.
  • Using AND and OR gates where gates are assumed to have any number of inputs.

 

Implementation Of C1–

 

  • The carry generator circuit for C1 is implemented as shown below.
  • It requires 1 AND gate and 1 OR gate.

 

C1 = C0P0 + G0

 

Implementation Of C2–

 

  • The carry generator circuit for C2 is implemented as shown below.
  • It requires 2 AND gates and 1 OR gate.

 

C2 = C0P0P1 + G0P1 + G1

 

Implementation Of C3 & C4–

 

Similarly, we implement C3 and C4.

  • Implementation of C3 uses 3 AND gates and 1 OR gate.
  • Implementation of C4 uses 4 AND gates and 1 OR gate.

 

Total number of gates required to implement carry generators (provided carry propagators Pi and carry generators Gi) are-

  • Total number of AND gates required for addition of 4-bit numbers = 1 + 2 + 3 + 4 = 10.
  • Total number of OR gates required for addition of 4-bit numbers = 1 + 1 + 1 + 1 = 4.

 

General Formula-

 

The following formula is used to calculate number of gates required for evaluating all carry bits-

 

For a n-bit carry look ahead adder to evaluate all the carry bits, it requires-

  • Number of AND gates = n(n+1) / 2
  • Number of OR gates = n

 

Advantages of Carry Look Ahead Adder-

 

The advantages of carry look ahead adder are-

  • It generates the carry-in for each full adder simultaneously.
  • It reduces the propagation delay.

 

Disadvantages of Carry Look Ahead Adder-

 

The disadvantages of carry look ahead adder are-

  • It involves complex hardware.
  • It is costlier since it involves complex hardware.
  • It gets more complicated as the number of bits increases.

 

To gain better understanding about Carry Look Ahead Adder,

Watch this Video Lecture

 

Next Article- K Maps | Karnaugh Maps

 

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