# Chap-2 Boolean Algebra

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1 Chap-2 Boolean Algebra Contents: My name Outline: My position, contact Basic information theorem and postulate of Boolean Algebra. or project description Boolean Algebra. Canonical and Standard form. Digital Logic Gates. Integrated circuit. Prepared By- Mohammed Abdul Kader Assistant Prof, Dept. of EEE, IIUC

2 Basic theorem and Properties of Boolean Algebra

3 Basic theorem and Properties of Boolean Algebra (Cont.)

4 Basic theorem and Properties of Boolean Algebra (Cont.)

5 Basic theorem and Properties of Boolean Algebra (Cont.) Proof of By Truth Table Proof of By Truth Table

6 Basic theorem and Properties of Boolean Algebra (Cont.) Venn Diagram Venn Diagram for two variable Venn Diagram illustration of distributive law

7 Boolean Function A binary variable can take a value of 0 or, 1. A boolean function is an expression formed with binary variables, the two binary operators OR and AND, unary operator NOT, parenthesis and an equal sign. For a given value of variables, the function either can 0 or 1.

8 Boolean Function Truth Table of Boolean functions Here, F3 and F4 are same. Two functions of n binary variables are said to be equal if they have same values for all possible 2^n combinations of the n variables.

9 Boolean Function: Implementation Implementation of Boolean functions with logic gate

10 Boolean Function: Algebraic Manipulation When a Boolean function is implemented with logic gates, each literal in the function designates an input to a gate and each term is implemented with a gate. The minimization of the number of literals and the number of terms results is a circuit with less equipment.

11 Boolean Function: Complement of a function Generalized theorems for finding complement-

12 Boolean Function: Complement of a function

13 Canonical and Standard forms: Minterms and Maxterms Minterms or standard product: Each row of a truth table can be associated with a minterm, which is a product (AND) of all variables in the function, in direct or complemented form. A minterm has the property that it is equal to 1 on exactly one row of the truth table. Maxterms or standard sums : Each row of a truth table is also associated with a maxterm, which is a sum (OR) of all the variables in the function, in direct or complemented form. A maxterm has the property that it is equal to 0 on exactly one row of the truth table.

14 Canonical and Standard forms: Minterms and Maxterms Expressing Boolean function by sum of minterms A boolean function may be expressed algebraically from a given truth table by forming a minterm for each combination of the variables which produces a 1 in the function and then taking the OR of all those terms.

15 Canonical and Standard forms: Minterms and Maxterms Expressing Boolean function by sum of minterms A boolean function may be expressed algebraically from a given truth table by forming a minterm for each combination of the variables which produces a 1 in the function and then taking the OR of all those terms.

16 Canonical and Standard forms: Minterms and Maxterms Finding Complement of Boolean function by sum of minterms The complement of a Boolean function can be obtained from the truth table by forming a minterm of each combination that produces a 0 in the function and then Oring those terms. The complement of f1 is written as-

17 Canonical and Standard forms: Minterms and Maxterms Expressing Boolean function by product of maxterms A boolean function may be expressed algebraically from a given truth table by forming a maxterm for each combination of the variables which produces a 0 in the function and then taking the AND of all those terms.

18 Canonical and Standard forms: Minterms and Maxterms Canonical and Standard forms Boolean functions expressed as a sum of minterms or product of maxterms are said to be in canonical form. The two canonical forms of Boolean algebra are basic forms that one obtain from reading a function from the truth table. These forms are very seldom the ones with least number of literals, because each minterm or maxterm must contain, by defination, all the variables either complemented or uncomplemented. Another way to express Boolean functions is in standard form. In this configuration, the terms that form the function may contain one, two or any number of literal. There are two types of standard forms: the sum of products and product of sums. Sum of products: Product of sums:

19 Canonical and Standard forms: Minterms and Maxterms Sum of Minterms (Example 2-4) Solution Inclusion of variable B Inclusion of variable C The function has three variables, the first term A is missing two variables B and C The second term missing one variable Inclusion of variable A Combining all terms-

20 Canonical and Standard forms: Minterms and Maxterms Product of Maxterms (Example 2-5) Solution Converting the function into OR terms using distributive law- Including missing with each term- Combining and avoiding the repeated terms-

21 Conversion between canonical forms Considering a function- Taking the complement of F Taking the complement of F Similarly,

22 Digital Logic Gates * A buffer produces the transfer function bust does not produce any particular logic operation, since the binary value of the output is equal to the binary value of the input. The circuit is used merely for power amplification of the signal and is equivalent to two inverters connected is cascade.

23 Digital Logic Gates The NAND and NOR gates are extensively used as standard logic gates and are in fact more popular than the AND and OR gates. This is because NAND and NOR gates are easily constructed with transistor circuits and because boolean functions can easily implemented with them.

24 Integrated Circuit: Levels of Integration Small Scale Integration (SSI) devices contain several independent gates in a single package. The inputs and outputs of the gates are connected directly to the pins in the package. The number of gates is usually fewer than 10 and is limited by number of pins available in the IC. Medium-scale integration (MSI) devices have a complexity of approximately 10 to 100 gates in a single package. They usually perform specific elementary digital operations such as decoders, adders or multiplexers. Large-scale integration (LSI) devices contain between 100 and a few thousand gates in a single package. They include digital systems such as processor, memory chips and programmable logic devices. Very large-scale integration (VLSI) devices contain thousands of gates within a single package. Examples are large memory arrays and complex microcomputer chips. Because of their small size and low cost, VLSI devices have revolutionized the computer system design technology, giving the designer the capabilities to create structures that previously were uneconomical

25 IC digital Logic Families

26 Positive and Negative Logic Many different logic families of digital IC s have been introduced commercially.

27 Special Characteristics The characteristics that describe the performance of IC digital logic families are: Fan-out, power dissipation, propagation delay and noise margin. Fan-out specifies the number of standard loads that the output of a gate can drive without impairing its normal operation. A standard load is usually defined as the amount of current needed by an input of another logic gate in the same IC family. Sometimes the term loading is used instead of fan-out. This term is derived from the fact that the output pin of a gate can supply limited current, above which it ceases to operate properly and is said to be overloaded. Power dissipation is the supplied power required to operate the gate. This parameter is expressed in milliwatts (mw) and represents the actual power dissipated in the gate. Propagation delay is the average transition delay time for a signal to propagate from input to output when the binary signals change in value. Propagation delay is expressed in nanoseconds (ns).

28 Special Characteristics Noise margin is the maximum noise voltage added to the input signal of a digital circuit that does not cause an undesirable change in the circuit output. There are two types of noise to be considered. DC noise is caused by a drift in the voltage level of a signal. AC noise is a random pulse that may be created by other switching signals. Noise margin is expressed in volts (V) and represent the maximum noise signal that can be tolerated by the gate. Table: Typical Characteristics of IC Logic Family

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