What is the Sum and Carry output (Cout) of a full adder circuit having A and B as inputs and Cin as Carry input?
Sum = A ⨁ B ⨁ Cin and Cout = A.B + (A ⨁ B).Cin
Full Adder Circuit Fundamentals
A full adder is a combinational logic circuit that performs the addition of three binary digits. It takes three inputs: two input bits (often denoted as A and B) and a carry-in bit (Cin) from a previous stage. It produces two outputs: a sum bit (Sum) and a carry-out bit (Cout) to be passed to the next stage.
Sum Output of a Full Adder
The Sum output of a full adder represents the least significant bit of the addition of the three input bits. It is essentially the XOR (exclusive OR) operation of all three inputs: input A, input B, and the carry-in Cin.
- The Sum output is 1 if an odd number of inputs (A, B, Cin) are 1.
- The Sum output is 0 if an even number of inputs (A, B, Cin) are 1.
Mathematically, the Boolean expression for the Sum output is:
$$\text{Sum} = A \oplus B \oplus C_{\text{in}}$$
Carry Output (Cout) of a Full Adder
The Carry output (Cout) of a full adder is generated when the sum of the three input bits (A, B, Cin) results in a carry to the next significant bit position. This happens if at least two of the input bits are 1.
The Boolean expression for the Carry output (Cout) can be derived from the full adder's logic. One common and simplified form for Cout that directly relates to the structure of a full adder (often implemented using two half-adders) is:
$$C_{\text{out}} = A.B + (A \oplus B).C_{\text{in}}$$
Let's break down this expression:
- A.B: This term accounts for the carry generated if both A and B are 1, regardless of Cin.
- (A ⨁ B).Cin: This term accounts for the carry generated if either A or B (but not both, due to XOR) is 1, AND the carry-in Cin is also 1. This means if one of A or B is 1, and Cin is 1, then a carry is produced.
Combining these two conditions covers all scenarios where a carry-out is generated in a full adder circuit.
Full Adder Truth Table
To further illustrate the behavior of a full adder circuit, here is its truth table:
| A | B | Cin | Sum | Cout |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 |
Final Full Adder Output Expressions
Based on the definitions and the truth table, the correct Boolean expressions for the Sum and Carry output of a full adder circuit with inputs A, B, and Cin are:
- Sum $$\text{Sum} = A \oplus B \oplus C_{\text{in}}$$
- Carry Output (Cout) $$C_{\text{out}} = A.B + (A \oplus B).C_{\text{in}}$$
These expressions accurately describe the logic behavior of a full adder circuit, which is fundamental in digital electronics for performing binary addition in multi-bit adders.
Important Questions from Combinational Circuits
Carry input is present in which of the following?
The time delay in a look-ahead carry adder is independent of
The characteristics of the combinational circuits are :
A. Output at any time is function of inputs at that time
B. Contains memory elements
C. Do not have feedback paths
D. Clock is used to trigger the circuits to obtain outputs
Choose the correct answer from the options given below :
Match the terms in List - I with the options given in List - II :
List - I
List – II
(a)
Decoder
(i)
1 line to 2 nlines
(b)
Multiplexer
(ii)
n lines to 2 nlines
(c)
De multiplexer
(iii)
2 nlines to 1 line
(iv)
2 nlines to 2 n−1 lines
Which of the following represents the borrow in a half subtraction?
