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UNIT – 2
Logic Gates: Boolean Algebra, Map simplification, Combinational circuits: Half Adder, Full Adder, Decoders, Multiplexers, Sequential Circuits: Flip Flops, SR, JK, D, T flip flop.
Logic Gates :
The logic gates are the main structural part of a digital system. Logic Gates are a block of hardware that produces signals of binary 1 or 0 when input logic requirements are satisfied. Each gate has a distinct graphic symbol, and its operation can be described by means of algebraic expressions. The seven basic logic gates includes: AND, OR, XOR, NOT, NAND, NOR, and XNOR. The relationship between the input-output binary variables for each gate can be represented in tabular form by a truth table. Each gate has one or two binary input variables designated by A and B and one binary output variable designated by x. Basic Logic Gates :
- AND gate
- OR gate
- NOT/ inverter gate AND gate : The AND gate is an electronic circuit which gives a high output only if all its inputs are high. In the AND gate, the output of an AND gate attains state 1 if and only if all the inputs are in state 1. Otherwise, the output is logic 0. The AND operation is represented by a dot (.) sign. The output is logical multiplication of the inputs.
Fig: Three input AND gate OR gate: The OR gate is an electronic circuit which gives a high output if one or more of its inputs are high. When both the inputs are 0, output is 0. The operation performed by an OR gate is represented by a plus (+) sign. The output is logical addition of the inputs.
NAND gate: The NOT-AND (NAND) gate which is equal to an AND gate followed by a NOT gate. The NAND gate gives a high output if any of the inputs are low. The NAND gate is represented by a AND gate with a small circle on the output. The small circle represents inversion. Fig: Three input NAND gate Application of NAND Gate: NAND gates can be used in alarm circuits. They re also used in buzzer and burglar devices NAND logic gates also have application in automatic temperature regulation circuits NAND gates can be used to detect if system has gone low. For e.g. Consider a security system build using NAND gates can monitor signals provided by sensors on doors and windows. If a
single window or door is opened, the security system can output signal “1” and this can be used to trigger an alarm or take some other action. Implication of all gates with NAND gate NOR gate: The NOT-OR (NOR) gate which is equal to a OR gate followed by a NOT gate. The NOR gate gives a low output if any of the inputs are high i.e. it gives output as 1 only if all the inputs are in 0 state. The NOR gate is represented by an OR gate with a small circle on the output. The small circle represents inversion. NOR gate is basically opposite of OR gate.
Implication of all gates with NOR gate Applications of NOR Gate NOR Gate can be used in combinational circuits such as multipliers, multiplexers, half and full adders, and in ripple-carry adders. There are also used in sequential circuits and in shift registers. The sequential circuit is a circuit that has memory unlike combinational circuits and output can vary based on input. They have previous input, output, clock and a memory element. Combinational gates: There are two types of combinational gates: XOR and XNOR gates XOR gate:
The 'Exclusive-OR' gate is a circuit which will give a high output if one of its inputs is high but not both of them. Thus, the output is 1 when, both the inputs are of different types. The XOR operation is represented by an encircled plus sign. An odd number of 1's generate a 1 output and even number of 1's give zero output. Fig: Three input XOR gate XNOR gate: The 'Exclusive-NOR' gate is a circuit that does the inverse operation to the XOR gate. It will give a low output if one of its inputs is high but not both of them. Thus, it produces high output when both the input are at the same logic. The small circle represents inversion.
- Logic gates are also important in data transport, calculation, and data processing. Even transistor-transistor logic and CMOS circuitry make extensive use of logic gates. BOOLEAN ALGEBRA: Boolean algebra is a type of algebra that is created by operating the binary system. In the year 1854, George Boole, an English mathematician, proposed this algebra. This is a variant of Aristotle’s propositional logic that uses the symbols 0 and 1, or True and False. Boolen algebra is concerned with binary variables and logic operations. Boolean Algebra is fundamental in the development of digital electronics systems as they all use the concept of Boolean Algebra to execute commands. A boolean function can be expressed algebrically with binary variables, the logic operation symbols, parenthesis and equal sign. Eg. F = x + y`z Truth Table: The truth table is a table that gives all the possible values of logical variables and the combination of the variables. It is possible to convert the Boolean equation into a truth table. The number of rows in the truth table should be equal to 2n, where “n” is the number of variables in the equation. For example, if a Boolean equation consists of 3 variables, then the number of rows in the truth table is 8. (i.e.,) 2^3 = 8. Logic Diagram: A boolean function can be transformed from an algebraic expression into a logic diagram composed of AND, OR and NOT gates. The purpose of boolean algebra is to facilitate the analysis and design of digital circuits. It provides a convinient tool to :
- Express in algebraic form a truth table relationship between binary variables.
- Express in algebraic form the input – output relationship of logic diagrams.
- Find simpler circuits for the same function. Boolean Operations
- Boolean addition : is equivalent to OR operation. The addition operation of Boolean algebra is similar to the OR operation. In digital circuits, the OR operation is used to calculate the sum term, without using AND operation. A + B, A + B', A + B + C', and A'+ B + C + D' are some of the examples of 'sum term'. The value of the sum term is true
when one or more than one literals are true and false when all the literals are false.
- Boolean multiplication : is equivalent to AND operation. The multiplication operation of Boolean algebra is similar to the AND operation. In digital circuits, the AND operation calculates the product, without using OR operation. AB, AB, ABC, and ABCD are some of the examples of the product term. The value of the product term is true when all the literals are true and false when any one of the literal is false. **Laws of Boolean Algebra
- Commutative Law :** This law states that no matter in which order we use the variables. It means that the order of variables doesn't matter. In Boolean algebra, the OR and the addition operations are similar. In the below diagram, the OR gate display that the order of the input variables does not matter at all. For two variables, the commutative law of addition is written as: A+B = B+A
A B C A+B (A+B)+C
According to this law, no matter in what order the variables are grouped when ANDing more than two variables. In the below diagram, the associative law is applied to 2-input AND gate.
A B C B+C A+(B+C)
A B C BC A(BC)
3) Distributive Law: According to this law, if we perform the OR operation of two or more variables and then perform the AND operation of the result with a single variable, then the result will be similar to performing the AND operation of that single variable with each two or more variable and then perform the OR operation of that product. This law explains the process of factoring. For three variables, the distributive law is written as: A(B + C) = AB + AC
A B C AB (AB)C
A B C B+C A(B+C)
There are the following rules of Boolean algebra, which are mostly used in manipulating and simplifying Boolean expressions. These rules plays an important role in simplifying boolean expressions.
- A+0=A 7. A.A=A
- A+1=1 8. A.A'=
- A.0=0 9. A''=A
- A.1=A 10. A+AB=A
- A+A=A 11. A+A'B=A+B
- A+A'=1 12. (A+B)(A+C)=A+BC Rule 1: A + 0 = A Let's suppose; we have an input variable A whose value is either 0 or 1. When we perform OR operation with 0, the result will be the same as the input variable. So, if the variable value is 1, then the result will be 1, and if the variable value is 0, then the result will be 0. Diagrammatically, this rule can be defined as: Rule 2: (A + 1) = 1 Let's suppose; we have an input variable A whose value is either 0 or 1. When we perform OR operation with 1, the result will always be 1. So, if the variable value is either 1 or 0, then the result will always be 1. Diagrammatically, this rule can be defined as:
Rule 3: (A.0) = 0 Let's suppose; we have an input variable A whose value is either 0 or 1. When we perform the AND operation with 0, the result will always be 0. This rule states that an input variable ANDed with 0 is equal to 0 always. Diagrammatically, this rule can be defined as: Rule 4: (A.1) = A Let's suppose; we have an input variable A whose value is either 0 or 1. When we perform the AND operation with 1, the result will always be equal to the input variable. This rule states that an input variable ANDed with 1 is equal to the input variable always. Diagrammatically, this rule can be defined as: Rule 5: (A + A) = A Let's suppose; we have an input variable A whose value is either 0 or 1. When we perform the OR operation with the same variable, the result will always be equal to the input variable. This rule states an input variable ORed with itself is equal to the input variable always.
this rule can be defined as: Rule 9: A = (A')' This rule states that if we perform the double complement of the variable, the result will be the same as the original variable. So, when we perform the complement of variable A, then the result will be A'. Further if we again perform the complement of A', we will get A, that is the original variable. Rule 10: (A + AB) = A We can prove this rule by using the rule 2, rule 4, and the distributive law as: A + AB = A(1 + B) Factoring (distributive law) A + AB = A.1 Rule 2: (1 + B)= 1 A + AB = A Rule 4: A .1 = A Rule 11: A + A'B = A + B We can prove this rule by using the above rules as:
A + A'B = (A + AB)+ AB Rule 10: A = A + AB A+A'B= (AA + AB)+ AB Rule 7: A = AA A+A'B=AA +AB +AA +AB Rule 8: adding AA = 0 A+A'B= (A + A)(A + B) Factoring A+A'B= 1.(A + B) Rule 6: A + A = 1 A+A'B=A + B Rule 4: drop the 1 Rule 12: (A + B)(A + C) = A + BC We can prove this rule by using the above rules as: (A + B)(A + C)= AA + AC + AB + BC Distributive law (A + B)(A + C)= A + AC + AB + BC Rule 7: AA = A (A + B)(A + C)= A( 1 + C)+ AB + BC Rule 2: 1 + C = 1 (A + B)(A + C)= A.1 + AB + BC Factoring (distributive law) (A + B)(A + C)= A(1 + B)+ BC Rule 2: 1 + B = 1 (A + B)(A + C)= A.1 + BC Rule 4: A .1 = A (A + B)(A + C)= A + BC
