Half & Full Adder/Subtractor Circuits

Combinational Circuits

• Designed using multiple interconnected logic gates.
• Output is generated by computing the logical combinations of the present input only.
• No clock pulse is present.

Half Adder

• Combinational logic circuit.
• Designed by connecting one EX-OR gate and one AND gate.
• Has two inputs (A and B)
• Adds those two inputs and generate a carry and a sum.
• Output from the EX-OR gate is the sum.
• Output from the AND gate is the carry.
• There will be no forwarding of carry addition because there is no logic gate to process that.

Logical Expression

• Sum = A XOR B
• Carry = A AND B

Truth Table

Full Adder

• A circuit comprised of two EX-OR gates, two AND gates, and one OR gate.
• Adds three inputs
• Produces two outputs.
• First two inputs are A and B
• The third input is an input carry as C-IN.
• Output carry is designated as C-OUT
• The normal output is designated as S (SUM).
• The equation obtained by the EX-OR gate is the sum of the binary digits.
• The output obtained by AND gate is the carry obtained by addition.

Logical Expression

• SUM = (A XOR B) XOR Cin = (A⊕ B) ⊕ Cin
• CARRY-OUT = A AND B OR Cin(A XOR B) = A.B + Cin(AB)

Truth Table

Subtractor

• A half subtractor is a combinational logic circuit that performs the subtraction of two bits.
• It serves as a foundational element for a full subtractor.
• Used in various digital circuits.

Truth Table for Half Subtractor

• Half subtractor consists of two XOR gates and one AND gate.
• XOR gates are used to calculate the difference between the two input bits
• AND gate is used to calculate the borrow.
• D = A XOR B
• B = A AND B

Applications

• Full subtractors
• Adders
• Comparators
• Decoders
• Encoders
• Multiplexers
• Demultiplexers

Advantages

• Simple in design and implementation
• Inexpensive to manufacture.

Disadvantages

• Can only subtract two bits at a time.
• Subtracting larger numbers requires cascading multiple half subtractors.

Full Subtractor

• Combinational logic circuit

Inputs

• Minuend (A)
• Subtrahend (B)
• Borrow-in (Bin) from the previous stage.

Outputs

• Difference (D)
• Borrow-out (Bout).

Truth Table for Full Subtractor

• Consists of two half subtractors connected in cascade
• First half subtractor subtracts the subtrahend (B) from the minuend (A)
• Second half subtractor subtracts the borrow-in (Bin) from the difference (D) of the first half subtractor.

Outputs

• First half subtractor creates Difference (D) and borrow-out (Bout1)
• Second half subtractor produces the final difference (D) and the final borrow-out (Bout).

Applications

• Arithmetic logic units (ALUs)
• Central processing units (CPUs)
• Graphics processing units (GPUs)
• Digital signal processors (DSPs)
• Essential component of many digital circuits.

Decoder

• Combinational circuit, changes the binary information into 2^N output lines.
• Binary information is given in the form of N input lines.
• The output lines define the 2-bit code for the binary information.
• Performs the reverse operation of the Encoder.
• At a time, only one input line is activated for simplicity.
• The produced 2^N-bit output code is equivalent to the binary information.

Encoder

• Combinational circuit.
• Performs the reverse operation of a Decoder.
• Has a maximum of 2^n input lines and 'n' output lines.
• Encoders encode the information from 2^n inputs into an n-bit code.
• A binary code equivalent to the input, which is active High, will be produced.
• The encoder encodes 2^n input lines with 'n' bits.

Types

• 4 to 2 Encoder
• Octal to Binary Encoder (8 to 3 Encoder)
• Decimal to BCD Encoder
• Priority Encoder

4 to 2 Encoder

• Consists of four inputs (Y3, Y2, Y1, YO)
• Consists of two outputs (A1, A0).
• At any time, only one of these 4 inputs can be '1' in order to get the respective binary code at the output.

Logical Expression

• A1 = Y3 + Y2
• A0 = Y3 + Y1
• A1 and A0 can be implemented using two input OR gates

Multiplexers

• Combinational circuit.
• Has many data inputs and a single output, based on control or select inputs.
• N input lines require log2(N) selection lines, or equivalently, 2^n input lines require n selection lines.
• Multiplexers are also known as "N-to-1 selectors,” parallel-to-serial converters, many-to-one circuits, and universal logic circuits.
• Increase the amount of data that can be delivered over a network within a certain amount of time and bandwidth.

2:1 Multiplexer

• Fundamental circuit also known as 2-to-1 multiplexer.
• Used to choose one signal from two inputs and transmits it to the output.
• Contains two input lines, one output line, and a single selection line.

Truth Table

De-multiplexer

• Combinational circuit with only 1 input line and 2^N output lines.
• Single-input and multi-output combinational circuit.
• The info is received from the single input lines and directed to the output line.
• The input will be connected to one of these outputs, based on the values of the selection lines.
• Opposite to the multiplexer.

Truth Table

Logical Expression

• Y₀ = S₀' . A
• Y₁ = S₀ . A

Sequential Circuit

• Digital circuits that store and use previous state information to determine their next state.
• Commonly used in digital systems to implement state machines, timers, counters, and memory elements.
• Essential components in digital systems design.
• Memory elements in sequential circuits are implemented using flip-flops.
• Flip-flops are circuits that store binary values and maintain their state even when the inputs change.

Flip-Flop

• Electronic component used to store one single bit of information.

SR Flip Flop

• Can be constructed in two ways
• With 2 NOR Gates + 2 AND Gates
• With 4 NAND Gates.

Applications

• Register: SR Flip Flop used to create register.
• Counters: SR Flip Flops used in counters. Counts the number of events that occurs in a digital system.
• Memory: SR Flip Flops used to create memory which are used to store data, when the power is turned off.
• Synchronous System: SR Flip Flop are used in synchronous system which are used to synchronize the operation of different component.

D Flip Flop

• Electronic device known as “delay flip flop" or "data flip flop".
• Stores a single bit of data.
• D flip flops are synchronous or asynchronous.
• Two inputs: data and clock.
• Clock input high: data is transferred to the output of the flip flop.
• Clock input low: the output of the flip flop is held in its previous state.

JK Flip-Flop

• Sequential logic circuit stores binary information in bitwise manner.
• Two inputs: Set(J) & Reset(K)
• Two outputs: Q and Q'.
• Has two modes of operation: synchronous and asynchronous.
• Synchronous mode: The state will be changed with the clock(clk) signal.
• Asynchronous mode: The change of state is independent from its clock signal.

T Flip Flop

• Known as Toggle Flip Flop
• Able to toggle its output depending upon on the input.
• 'T' stands for Toggle
• Toggle basically indicates that the bit will be flipped (1 to 0 or 0 to 1).
• A clock pulse is supplied to operate this flop, hence it is a clocked flip-flop.

Can be constructed in three ways

• By using SR Flip Flops.
• By using D Flip Flops.
• By using JK Flip Flops

Resistor

• Flip flops can be used to store a single bit of binary data (1 or 0).
• In order to store multiple bits of data, multiple flip-flops are needed.
• N flip flops are to be connected in order to store n bits of data

Shift Resistor

• This is a device used to store such information.
• It is a collection of flip-flops linked in series to store several bits of data.
• Shift registers are used to move the data that has been stored inside these registers.

Types of Shift Resistors:

• Serial In Serial Out shift register
• Serial In parallel Out shift register
• Parallel In Serial Out shift register
• Parallel In parallel Out shift register
• Bidirectional Shift Register
• Universal Shift Register
• Shift Register Counter

Bidirectional Shift Register with parallel load

• A binary number shifted to the left by one position is equivalent to multiplying it by 2.
• A binary number shifted to the right by one position is equivalent to dividing the number by 2.
• In order to perform these operations we need a register which can shift the data in either direction.

Binary counter

• A type of sequential logic circuit in digital electronics that can count in binary numbers.
• A binary counter can counter from 0 to 2^(n-1), where n is the total number of bits in the counter.

Asynchronous Counter

• A type of binary counter.
• Flip flops do not receive the same clock pulse at the same time.
• Also known as ripple counter (simplest type of binary counter).
• In asynchronous counters, each flip flop is triggered by the output of the previous flip flop, thus, suffer from propagation delay.

Synchronous Counter

• All the flip flops receive the same clock pulse at the same time.
• Since all flip flops are triggered by the same clock pulse, their outputs change simultaneously.
• Result is little to no propagation delay between the flip flops.

4-bit Binary Synchronous Counter

• Comprises four JK flip-flops
• Represents an M=2 4-bit binary synchronous counter.
• The counting pulse N is simultaneously applied to the clock pulse input CP terminal of each flip-flop.
• When the counting pulse arrives: all the flip-flops are triggered simultaneously and the state of each subsequent flip-flop is determined by the current state of its preceding flip-flop.