Combinational Logic Circuits: Adders, Subtractors & MUX
Half Adder
Half Adder: A half adder is a basic combinational logic circuit used to add two single-bit binary numbers. It produces two outputs called Sum and Carry. It is called a half adder because it does not consider any carry input from a previous stage.
Inputs and Outputs
- Inputs: A, B
- Outputs: Sum (S), Carry (C)
- Operation performed: A + B
Working Principle
- When both input bits are 0, the Sum and Carry are 0.
- When one input is 1 and the other is 0, the Sum is 1 and Carry is 0.
- When both inputs are 1, the Sum is 0 and Carry is 1.
- Thus, the half adder adds two binary digits correctly but cannot add a carry from a previous addition.
Truth Table
A B Sum (S) Carry (C) 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1
Boolean Expressions
From the truth table:
- Sum: S = A ⊕ B
- Carry: C = A · B
This shows that the Sum is obtained using an XOR gate and Carry using an AND gate.
Logic Circuit
- An XOR gate is used to generate the Sum.
- An AND gate is used to generate the Carry.
- Thus, a half adder requires one XOR gate and one AND gate.
Limitations
- Cannot add carry from a previous stage.
- Not suitable for multi-bit addition.
- Used only as a basic building block.
Applications
- Used in arithmetic circuits.
- Used in calculators.
- Forms the base for full adder design.
- Used in ALU and digital processors.
Full Adder
Full Adder: A full adder is a combinational logic circuit used to add three single-bit binary inputs. These inputs are two significant bits and one carry from the previous stage. It produces two outputs called Sum and Carry. A full adder overcomes the limitation of a half adder by considering the carry input.
Inputs and Outputs
- Inputs: A, B, Carry-in (Cin)
- Outputs: Sum (S), Carry-out (Cout)
- Operation performed: A + B + Cin
Working Principle
- The full adder adds three bits at the same time.
- Sum is 1 when an odd number of inputs are 1.
- Carry is generated when two or more inputs are 1.
- Carry-out is passed to the next higher bit position.
- Thus, a full adder is suitable for multi-bit binary addition.
Truth Table
A B Cin Sum (S) Carry (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
Boolean Expressions
- Sum: S = A ⊕ B ⊕ Cin
- Carry: Cout = AB + B·Cin + A·Cin
Logic Circuit
A full adder can be constructed using two half adders and one OR gate:
- First half adder adds A and B.
- Second half adder adds the Sum and Cin.
- OR gate combines the carry outputs.
Advantages
- Can add carry from previous stage.
- Used for multi-bit addition.
- Accurate and reliable.
Applications
- Binary addition in computers.
- Used in ALU.
- Used in calculators.
Half Subtractor
Half Subtractor: A half subtractor is a combinational logic circuit used to subtract one single-bit binary number from another. It produces two outputs: Difference and Borrow. It is called a half subtractor because it does not consider any borrow input from a previous stage.
Inputs and Outputs
- Inputs: A (Minuend), B (Subtrahend)
- Outputs: Difference (D), Borrow (B)
- Operation performed: A − B
Working Principle
- If both inputs are equal, the Difference is 0 and Borrow is 0.
- If A = 1 and B = 0, Difference is 1 and Borrow is 0.
- If A = 0 and B = 1, Difference is 1 and Borrow is 1.
- Borrow occurs only when A is smaller than B.
- Thus, a half subtractor correctly subtracts two bits but cannot handle borrow from a previous stage.
Truth Table
A B Difference (D) Borrow 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0
Boolean Expressions
From the truth table:
- Difference: D = A ⊕ B
- Borrow: Borrow = A’ · B
These expressions define the logic of the half subtractor.
Logic Circuit
- An XOR gate is used to generate the Difference.
- A NOT gate and an AND gate are used to generate the Borrow.
- Thus, a half subtractor uses XOR, NOT, and AND gates.
Limitations
- Cannot subtract borrow from previous stage.
- Not suitable for multi-bit subtraction.
- Used only as a basic building block.
Applications
- Digital subtraction circuits.
- Arithmetic Logic Unit (ALU).
- Used in designing full subtractors.
- Digital calculators and processors.
Full Subtractor
Full Subtractor: A full subtractor is a combinational logic circuit used to subtract three single-bit binary inputs. These inputs are the minuend (A), subtrahend (B), and borrow from the previous stage (Bin). It produces two outputs called Difference and Borrow-out. A full subtractor removes the limitation of a half subtractor by considering the borrow input.
Inputs and Outputs
- Inputs: A (Minuend), B (Subtrahend), Bin (Borrow-in)
- Outputs: Difference (D), Borrow-out (Bout)
- Operation performed: A − B − Bin
Working Principle
- The full subtractor subtracts three bits at the same time.
- Difference becomes 1 when an odd number of inputs are 1.
- Borrow-out is generated when A is smaller than B or Bin.
- Borrow-out is passed to the next higher bit position.
- Thus, a full subtractor is suitable for multi-bit binary subtraction.
Truth Table
A B Bin Difference (D) Borrow (Bout) 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1
Boolean Expressions
- Difference: D = A ⊕ B ⊕ Bin
- Borrow-out: Bout = A’B + A’Bin + B·Bin
Logic Circuit
- A full subtractor can be designed using two half subtractors and one OR gate.
- First half subtractor subtracts A and B.
- Second half subtractor subtracts Difference and Bin.
- OR gate combines the borrow outputs.
Advantages
- Can handle borrow from previous stage.
- Suitable for multi-bit subtraction.
- More accurate than half subtractor.
Applications
- Binary subtraction in computers.
- Digital calculators.
- Arithmetic Logic Unit (ALU).
Multiplexer (MUX)
Multiplexer (MUX): A multiplexer is a combinational logic circuit that selects one input from multiple input lines and forwards it to a single output line. The selection of a particular input is controlled by a set of select lines. Because it selects one data input at a time, a multiplexer is also called a data selector. Multiplexers play an important role in digital systems to reduce hardware and efficiently route data.
Basic Principle of Multiplexer
- The working of a multiplexer is based on the number of select lines.
- If the number of select lines is n, then the number of input lines is 2ⁿ.
- Only one input is connected to the output at any time.
Main Components of a Multiplexer
A multiplexer consists of the following parts:
- Data input lines (I₀, I₁, I₂, …)
- Select lines (S₀, S₁, S₂, …)
- Single output line (Y)
Select lines decide which input will appear at the output.
Types of Multiplexers
2:1 Multiplexer
A 2:1 multiplexer has inputs A and B, one select line S, and one output Y.
Working:
- When S = 0 → Output Y = A
- When S = 1 → Output Y = B
Boolean Expression: Y = A·S’ + B·S
This is the simplest multiplexer and is used as a basic building block.
4:1 Multiplexer
A 4:1 multiplexer has four inputs: I₀, I₁, I₂, I₃; two select lines: S₀, S₁; and one output.
Working: Depending on the binary value of the select lines, one input is selected.
S1 S0 Output 0 0 I0 0 1 I1 1 0 I2 1 1 I3
This type of multiplexer is widely used in digital circuits and in cascading configurations.
8:1 Multiplexer
An 8:1 multiplexer has eight input lines (I₀ to I₇), three select lines (S₀, S₁, S₂), and one output.
Working: Each combination of select lines chooses one of the eight inputs and connects it to the output.
16:1 Multiplexer
A 16:1 multiplexer has sixteen inputs (I₀ to I₁₅) and four select lines (S₀, S₁, S₂, S₃). Used in complex systems where a large number of inputs need to be selected efficiently.
Cascaded Multiplexer
When a large multiplexer is not available, smaller multiplexers are connected together to form a larger one.
Example: Two 4:1 multiplexers and one 2:1 multiplexer can be used to design an 8:1 multiplexer. This method is called cascading.
Logic Implementation of Multiplexer
Multiplexers are implemented using:
- AND gates
- OR gates
- NOT gates
Each input is ANDed with the appropriate select line combination, and all results are ORed to get the final output.
Applications of Multiplexers
- Data routing in digital systems.
- Used in CPU, ALU, and control units.
- Parallel-to-serial data conversion.
- Communication systems.
- Implementation of Boolean functions.
Advantages of Multiplexers
- Reduces hardware and wiring.
- Saves cost and space.
- Efficient data selection.
- Simplifies digital circuit design.
Limitations of Multiplexers
- Only one input can be selected at a time.
- Delay increases with an increase in the number of inputs.
- Requires additional select lines.