Unit 5: Arithmetic and Sequential Circuits

1. Introduction to Arithmetic Circuits

Arithmetic circuits are combinational logic circuits used to perform mathematical operations like addition and subtraction in digital systems. They use logic gates to process binary data bits.

2. Binary Addition: Half Adder and Full Adder

Addition is the most basic arithmetic operation in binary systems.

Half Adder

A Half Adder is a combinational circuit that adds two single-bit binary numbers. It produces two outputs: Sum and Carry.

[Image of Half Adder logic diagram and truth table]

Full Adder

A Full Adder adds three binary bits: two significant bits and an incoming carry bit from a previous stage. It is used for multi-bit binary addition.

3. Binary Subtraction: Half and Full Subtractors

Binary subtraction can be performed using dedicated circuits or by using 1's complement addition.

• Half Subtractor: Subtracts one bit from another and produces two outputs: Difference and Borrow.
• Full Subtractor: Subtracts two bits and a borrow-in from a previous stage.

4. Introduction to Sequential Circuits

Unlike combinational circuits, sequential circuits have outputs that depend not only on current inputs but also on previous states. This is achieved using memory elements.

5. Flip-Flops: SR, JK, and MS-JK

A flip-flop is a basic memory element that can store one bit of binary data.

• SR Flip-Flop: The simplest type, with Set (S) and Reset (R) inputs. It has an invalid state when both inputs are 1.
• JK Flip-Flop: An improvement over the SR flip-flop that eliminates the invalid state. If J=1 and K=1, the output toggles.
• Master-Slave (MS) JK Flip-Flop: Designed to eliminate timing issues like the race-around condition by using two flip-flops in a master-slave configuration.

6. The Race-around Condition

In a JK Flip-Flop, when J=1, K=1, and the clock pulse is too long, the output may toggle multiple times within a single clock cycle. This unpredictable behavior is called the race-around condition. It is solved using edge-triggering or Master-Slave JK flip-flops.

7. Data Processing: Multiplexers and De-multiplexers

These circuits are used to route digital data between multiple sources and destinations.

• Multiplexer (MUX): A "data selector" that picks one of many input signals and sends it to a single output line based on select lines.
• De-multiplexer (DEMUX): A "data distributor" that takes a single input and routes it to one of many output lines.

[Image showing the operation of a 4-to-1 Multiplexer]