# Analysis of Combinational Circuit Notes Tutorial with Examples

**Combinational Circuit**

Combinational circuits are Logic circuits that perform arithmetic functions (e.g., addition, subtraction, multiplication and division).

These circuits don’t have memory and the output depends only on input_{s} provided. Combinational circuit is a logic circuit containing only logic gates.

**Key Points**

- Logic gates are not always required because simple logic functions can be performed with switches or diodes.

(a) **Switches in series (AND function) (b) Switches in parallel (OR function)**

**(c) Combining IC outputs with diodes (OR function)**

**Basic Logic Gates**

**NOT Gate**

- One input, one output
- Whatever logical state (1, 0) is applied to the input the opposite state (0,1 respectively) will appear at output.
- Also known as inverter.

**OR Gate **

The output will be high / true / 1, if any or all of its inputs are high / true / 1. The output will be low / zero / false only if all of inputs are low / zero / false.

**AND Gate**

output will be high / true / 1, if all of its inputs are high / true / 1. If any of its inputs is low/false/ 0, then output will be low / false / 0.

To change the type of gate, such as changing OR to AND, you must do three things ,

- Invert (NOT) each input.
- Change the gate type (OR to AND or AND to OR) .
- Invert (NOT) the output.

**Universal Gates**

Any function can be implemented with the help of these (NAND, NOR) gates.

**NAND Gate**

This is an AND gate with the output inverted or we can say (AND + NOT). Therefore, the output expression of the two input NAND gate is X = *(A . B)’.*

**NOR ****G****ate**

**This **is an OR gate with the output inverted or we can say (OR + NOT). °re, the output expression of the two input NOR gate is X = *(A + B) . **we ref *X (NOT *(A *OR *B))*

**Other Types of Gates**

**Exclusive OR (EX-OR / XOR) Gate**

The XOR gate provides 1 as an output only if its two inputs are different. If the inputs are same, the output will be a ‘0’. Unlike standard OR / NOR and AND / NAND functions the XOR function always has exactly two inputs.

*X=A+B=A•B+A•B*

If *A *and *B *are different, then the output will be high.

** Exclusive NOR (EX- NOR / XNOR) Gate**

The XNOR gate produces output 1 only if the inputs are same. If the inputs are different the output will be a zero 0.

* X=A•B=A•A+A•B*

If *A *and *B *are different, then the output will be low or zero.

**Su****bstituting one Type of Gate for Another**

**Any**** Logic Gate can be built from NAND or**

**NOR Gates**

NAND or NOR gates can be combined to create any type of gate. This enables a circuit to be built from just one type of gate, either NAND or NOR. e.g, an AND gate is a NAND gate, then a NOT gate (to undo the inverting function).

**Key Points**

- AND and OR gate can’t be used to create other gates because they lack the inverting (NOT) function.

e.g., an OR gate can be built from NOTed inputs fed into a NAND ( AND + NOT) gate.

**Combinational Circuit for ****Arithmetic Operations**

**Half Adder**

Addition of 2 binary digits requires 2 inputs and 2 outputs, one for result and one for carry.

**Full Adder**

A full adder is a combinational circuit that forms the arithmetic sum of 3 input bits. It consists of 3 inputs and 2 outputs. Two of the input variables are the 2 significant bits to be added. The 3 rd input represents the carry from previous lower significant position.

** S = Z ED (X @ y) = z’ (xy’ + x’ y) + z (xy’ + x’ y)’ **

*S = z’ (xy’ + x’ y) + z (xy + x’ y’ )*

*S = xy’ z’ + x’ yz’ + xyz + x’ y’ z*

**and the carry output is C = z (xy’ + x’ y) + xy = xy’ z + x’ yz + xy**

** Positive and Negative Logic**

- If the signal that activates the circuit (the 1 state) has a voltage level that is more positive that the 0 state, then the logic polarity is considered to be positive. Thus, in positive logic 1 is considered as high value and 0 is considered as low value.
- If the signal that activates the circuit (the 1 state) has a voltage level that is more negative than the 0 state, then the logic polarity is considered to be negative. Thus in negative logic 1 is considered as low value and 0 is considered as high value.

**Decoders**

A decoder is a combinational circuit that converts binary information from *n *input lines of a maximum of 2″ unique output lines. If the n bit decoded information has unused or don’t care combinations, the decoded output will have less than *2″ *outputs. The decoders are also called as *n *to *m *line

**decoders where m < 2″.**

**Truth Table of 3 Bit Binary to Decimal Decoder**

This decoder takes binary values as input and produces decimal value output. Suppose, if D_{3} is high it means the binary combination of 3 that_{;} 011, means x = 0, y = land *z *= 1.

*x *x’* *y y’ *z z’*

## Encoder

An encoder is a digital function that produces a reverse operations from to a decoder. An encoder has 2 (or less) input lines and *n *output lines. The output lines generate the binary code for 2″ input variables,

From the above truth table, we get

*x *= *D4 + D5 + D6 + **D _{7 }*

**y = D2 + D3 4^{–} D6 + D _{7}**

Z = 01 + 03 + D_{5} + D

# Sorting in Design and Analysis of Algorithm Study Notes with Example

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