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Combinational circuits-Adder and Subtractor circuits
Combinational Circuits
Combinational circuit is a circuit in which we combine the different gates in the circuit, for example encoder, decoder, multiplexer and demultiplexer. Some of the characteristics of combinational circuits are following – The output of combinational circuit at any instant of time, depends only on the levels present at input terminals. The combinational circuit do not use any memory. The previous state of input does not have any effect on the present state of the circuit. A combinational circuit can have an n number of inputs and m number of outputs.
Block diagram
We're going to elaborate few important combinational circuits as follows.
Half Adder
Half adder is a combinational logic circuit with two inputs and two outputs. The half adder circuit is designed to add two single bit binary number A and B. It is the basic building block for addition of two single bit numbers. This circuit has two outputs carry and sum.
Block diagram
Truth Table
Circuit Diagram
Full Adder
Full adder is developed to overcome the drawback of Half Adder circuit. It can add two one-bit numbers A and B, and carry c. The full adder is a three input and two output combinational circuit.
Block diagram Truth Table
N-Bit Parallel Subtractor
The subtraction can be carried out by taking the 1's or 2's complement of the number to be subtracted. For example we can perform the subtraction (A-B) by adding either 1's or 2's complement of B to A. That means we can use a binary adder to perform the binary subtraction.
4 Bit Parallel Subtractor
The number to be subtracted (B) is first passed through inverters to obtain its 1's complement. The 4-bit adder then adds A and 2's complement of B to produce the subtraction. S3 S2 S1 S0 represents the result of binary subtraction (A-B) and carry output Cout represents the polarity of the result. If A > B then Cout = 0 and the result of binary form (A-B) then Cout = 1 and the result is in the 2's complement form.
Block diagram
Half Subtractors
Half subtractor is a combination circuit with two inputs and two outputs (difference and borrow). It produces the difference between the two binary bits at the input and also produces an output (Borrow) to indicate if a 1 has been borrowed. In the subtraction (A-B), A is called as Minuend bit and B is called as Subtrahend bit.
Truth Table Circuit Diagram
Full Subtractors
The disadvantage of a half subtractor is overcome by full subtractor. The full subtractor is a combinational circuit with three inputs A,B,C and two output D and C'. A is the 'minuend', B is 'subtrahend', C is the 'borrow' produced by the previous stage, D is the difference output and C' is the borrow output.
Truth Table Circuit Diagram
Truth Table
Applications of Multiplexers
- Multiplexer is used as data selector device.
- Multiplexers are used in communication systems to increase the efficiency of the system.
- Multiplexers are used in telephone networks for integration of several audio signals on a single transmission line.
- To maintain large amount of data, multiplexers are also used in computer memory systems.
- Multiplexers are also employed in TV broadcasting systems.
- Multiplexers are used in satellite communication and GPS (Global Positioning System).
- Multiplexers are also used in PLC (Programmable Logic Control) systems, etc.
Demultiplexers
A demultiplexer performs the reverse operation of a multiplexer i.e. it receives one input and distributes it over several outputs. It has only one input, n outputs, m select input. At a time only one output line is selected by the select lines and the input is transmitted to the selected output line. A de-multiplexer is equivalent to a single pole multiple way switch as shown in fig. Demultiplexers comes in multiple variations. 1 : 2 demultiplexer 1 : 4 demultiplexer 1 : 16 demultiplexer 1 : 32 demultiplexer
Block diagram Truth Table
Applications of Demultiplexers
- Demultiplexers are used in several input and output devices for data routing.
- Demultiplexers are used in digital control systems to select one signal from a mutual stream of signals.
- Demultiplexers are also employed for data transmission in synchronous systems.
- Demultiplexers are also utilized in data acquisition systems.
- Demultiplexers can be used for generating Boolean functions.
- Demultiplexers can be used in serial to parallel converters.
- Demultiplexers are used for broadcasting of ATM packets.
- Demultiplexers can also be used to design automatic test equipment, etc.
Decoder
A decoder is a combinational circuit. It has n input and to a maximum m = 2n outputs. Decoder is identical to a demultiplexer without any data input. It performs operations which are exactly opposite to those of an encoder.
Block diagram
Applications of decoders
- It is used in code conversion. i.e analog to digital conversion in the analog decoder.
- It may also be used for data distribution.
- In a high-performance memory system, this decode can be used to minimize the effect of system decoding.
- The decoder is used as address decoders in CPU memory location identification.
- It is also be used in electronic circuits to convert instruction into CPU control signals.
- They are mainly used in logical circuits, data transfer.
- They can also be used to create simple other digital logics like half adders and full adders and
- some other digital design also.
- Microprocessor selecting different I/O devices.
- It decoding to binary input to activate the LED segments so that the decimal number can be displayed.
- Microprocessor memory system selecting different banks of memory.
- The decoder can be used as a timing or sequencing signals to turn the device on or off at specific times because when the decoder inputs come from a counter that is being continually pulsed, The decoder output will be activated sequentially.
- The decoder is used whenever an output or a group of output is to be activated only on the occurrence of a specific combination of input signals.
- They can be the application of switching function often with the fewer integrated circuit.
Encoder
Encoder is a combinational circuit which is designed to perform the inverse operation of the decoder. An encoder has n number of input lines and m number of output lines. An encoder produces an m bit binary code corresponding to the digital input number. The encoder accepts an n input digital word and converts it into an m bit another digital word.
Block diagram
Examples of Encoders are following. Priority encoders Decimal to BCD encoder Octal to binary encoder Hexadecimal to binary encoder
Priority Encoder
This is a special type of encoder. Priority is given to the input lines. If two or more input line are 1 at the same time, then the input line with highest priority will be considered. There are four input D0, D1, D2, D3 and two output Y0, Y1. Out of the four input D3 has the highest priority and D0 has the lowest priority. That means if D3 = 1 then Y1 Y1 = 11 irrespective of the other inputs. Similarly if D3 = 0 and D2 = 1 then Y1 Y0 = 10 irrespective of the other inputs.
Block diagram Truth Table
The circuit works by comparing the bits of the two numbers starting from the most significant bit (MSB) and moving toward the least significant bit (LSB). At each bit position, the two corresponding bits of the numbers are compared. If the bit in the first number is greater than the corresponding bit in the second number, the A>B output is set to 1, and the circuit immediately determines that the first number is greater than the second. Similarly, if the bit in the second number is greater than the corresponding bit in the first number, the A<B output is set to 1, and the circuit immediately determines that the first number is less than the second. If the two corresponding bits are equal, the circuit moves to the next bit position and compares the next pair of bits. This process continues until all the bits have been compared. If at any point in the comparison, the circuit determines that the first number is greater or less than the second number, the comparison is terminated, and the appropriate output is generated. If all the bits are equal, the circuit generates an A=B output, indicating that the two numbers are equal. There are different ways to implement a magnitude comparator, such as using a combination of XOR, AND, and OR gates, or by using a cascaded arrangement of full adders. The choice of implementation depends on factors such as speed, complexity, and power consumption. 1-Bit Magnitude Comparator A comparator used to compare two bits is called a single-bit comparator. It consists of two inputs each for two single-bit numbers and three outputs to generate less than, equal to, and greater than between two binary numbers. The truth table for a 1-bit comparator is given below. 1-Bit Magnitude Comparator
From the above truth table logical expressions for each output can be expressed as follows. A>B: AB' A<B: A'B A=B: A'B' + AB From the above expressions, we can derive the following formula. Derivation of 1-Bit Magnitude Comparator By using these Boolean expressions, we can implement a logic circuit for this comparator as given below. Logic Circuit
INPUT OUTPUT
From the above truth table, K-map for each output can be drawn as follows.
Truth Table of Output A>B
Truth Table of Output A=B
Truth Table of Output A<B
4-Bit Magnitude Comparator A comparator used to compare two binary numbers each of four bits is called a 4-bit magnitude comparator. It consists of eight inputs each for two four-bit numbers and three outputs to generate less than, equal to, and greater than between two binary numbers. In a 4-bit comparator, the condition of A>B can be possible in the following four cases.
- If A3 = 1 and B3 = 0
- If A3 = B3 and A2 = 1 and B2 = 0
- If A3 = B3, A2 = B2 and A1 = 1 and B1 = 0
- If A3 = B3, A2 = B2, A1 = B1 and A0 = 1 and B0 = 0 Similarly, the condition for A<B can be possible in the following four cases.
- If A3 = 0 and B3 = 1
- If A3 = B3 and A2 = 0 and B2 = 1
- If A3 = B3, A2 = B2 and A1 = 0 and B1 = 1
- If A3 = B3, A2 = B2, A1 = B1 and A0 = 0 and B0 = 1 The condition of A=B is possible only when all the individual bits of one number exactly coincide with the corresponding bits of another number. From the above statements, logical expressions for each output can be expressed as follows. AA, 831331 r: (A3 Ex-Nor 33)A2132′ a (A3 Ex-Nor 133) (A2 Ex-Nor 132)A131′ a (A3 Ex-Nor
- (A2 Ex-Nor132) (Al Ex-Nor 31)A ,13: A3’03 a (A3 Ex-Nor 33)A211:12 a (A3 Ex-Nor 83) (A2 Ex-Nor 132)Ar131 a (A3 Ex-Nor 33) (A2 Ex-Nor32) (Al Ex-Nor 131)A0N A=B: (A3 Ex-Nor B3) (A2 Ex-Nor 82) (Al Ex-Nor BI) (AO Ex-Nor BO) By using these Boolean expressions, we can implement a logic circuit for this comparator as given below.
4-Bit Magnitude Comparator NOTE: For n- the bit comparator then, the number of combinations for which Total combination=22n Equal combination(A = B) = 2n, Unequal Combination=22n - 2n Greater(A > B) =Less( A < B)combination = (22n - 2n)/ Applications of Comparators
- Comparators are used in central processing units (CPUs) and microcontrollers (MCUs).
- These are used in control applications in which the binary numbers representing physical variables such as temperature, position, etc. are compared with a reference value.
- Comparators are also used as process controllers and for Servo motor control.
- Used in password verification and biometric applications.
