Logic gates are basic building blocks of digital circuits that perform logical operations. The seven basic logic gates are AND, OR, NOT, NAND, NOR, XOR, and XNOR. Boolean algebra uses binary numbers (0 and 1) and is used to analyze and simplify digital circuits. Karnaugh maps are a method to simplify Boolean functions by grouping adjacent 1's in a map based on the number of variables. This allows deriving the minimum number of terms to represent the function in sum-of-products form.
Objectives, And or operation, Not operation, Fundamentals laws, Characteristic of logic families, Bipolar families, Diode logic, Resistor, transistor logic.
Most modern devices are made from billions of on /off switches called transistors
We will build a processor in this course!
Transistors made from semiconductor materials:
MOSFET – Metal Oxide Semiconductor Field Effect Transistor
NMOS, PMOS – Negative MOS and Positive MOS
CMOS – Complimentary MOS made from PMOS and NMOS transistors
Transistors used to make logic gates and logic circuits
We can now implement any logic circuit
Can do it efficiently, using Karnaugh maps to find the minimal terms required
Can use either NAND or NOR gates to implement the logic circuit
Can use P- and N-transistors to implement NAND or NOR gates
This topic introduces the numbering systems: decimal, binary, octal and hexadecimal. The topic covers the conversion between numbering systems, binary arithmetic, one's complement, two's complement, signed number and coding system. This topic also covers the digital logic components.
Objectives, And or operation, Not operation, Fundamentals laws, Characteristic of logic families, Bipolar families, Diode logic, Resistor, transistor logic.
Most modern devices are made from billions of on /off switches called transistors
We will build a processor in this course!
Transistors made from semiconductor materials:
MOSFET – Metal Oxide Semiconductor Field Effect Transistor
NMOS, PMOS – Negative MOS and Positive MOS
CMOS – Complimentary MOS made from PMOS and NMOS transistors
Transistors used to make logic gates and logic circuits
We can now implement any logic circuit
Can do it efficiently, using Karnaugh maps to find the minimal terms required
Can use either NAND or NOR gates to implement the logic circuit
Can use P- and N-transistors to implement NAND or NOR gates
This topic introduces the numbering systems: decimal, binary, octal and hexadecimal. The topic covers the conversion between numbering systems, binary arithmetic, one's complement, two's complement, signed number and coding system. This topic also covers the digital logic components.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
2. Logic Gates
• The logic gates are the main structural part of a digital system.
• A logic gate is a device that acts as a building block for digital circuits.
They perform basic logical functions that are fundamental to digital circuits.
• Most electronic devices we use today will have some form of logic gates in
them. For example, logic gates can be used in technologies such as
smartphones, tablets or within memory devices.
• In a circuit, logic gates will make decisions based on a combination of
digital signals coming from its inputs. Most logic gates have two inputs and
one output. Logic gates are based on Boolean algebra.
• Logic gates are commonly used in integrated circuits (IC).
3. Basic logic gates
There are seven basic logic gates: AND, OR, XOR, NOT, NAND, NOR, and
XNOR.
AND GATE:
• The AND gate is an electronic circuit which gives a high output only if
all its inputs are high. The AND operation is represented by a dot (.)
sign.
4. OR GATE:
• The OR gate is an electronic circuit which gives a high output if one or
more of its inputs are high. The operation performed by an OR gate is
represented by a plus (+) sign.
5. NOT gate:
• A logical inverter, sometimes called a NOT gate to differentiate it from
other types of electronic inverter devices, has only one input. It
reverses the logic state. If the input is 1, then the output is 0. If the
input is 0, then the output is 1.
6. 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.
7. NOR GATE:
• The NOT-OR (NOR) gate which is equal to an OR gate followed by a
NOT gate. The NOR gate gives a low output if any of the inputs are
high. The NOR gate is represented by an OR gate with a small circle on
the output. The small circle represents inversion.
8. The ‘Exclusive-OR’ Exclusive-OR/ XOR GATE:
• The 'Exclusive-OR' gategate is a circuit which will give a high output if
one of its inputs is high but not both of them. The XOR operation is
represented by an encircled plus sign.
9. The XNOR (exclusive-NOR) :
The XNOR (exclusive-NOR) :
• The XNOR (exclusive-NOR) gate is a combination XOR gate followed
by an inverter. Its output is "true" if the inputs are the same, and
"false" if the inputs are different.
XNOR gate
Input 1 Input 2 Output
0 0 1
0 1 0
1 0 0
1 1 1
10. Boolean Algebra
• Boolean Algebra is used to analyze and simplify the digital (logic)
circuits.
• It uses only the binary numbers i.e. 0 and 1.
• It is also called as Binary Algebra or logical Algebra.
• It is a convenient way and systematic way of expressing and analyzing
the operation of logic circuits.
• Boolean algebra was invented by George Boole in 1854.
11. Rules in Boolean Algebra
Following are the important rules used in Boolean algebra.
• Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
• Complement of a variable is represented by an overbar (-). Thus, complement of
variable B is represented as B Bar. Thus if B = 0 then B Bar = 1 and B = 1 then B
Bar = 0.
• ORing of the variables is represented by a plus (+) sign between them. For
example ORing of A, B, C is represented as A + B + C.
• Logical ANDing of the two or more variable is represented by writing a dot
between them such as A.B.C. Sometime the dot may be omitted like ABC.
12. Boolean Laws
There are six types of Boolean Laws.
Commutative law:
• Any binary operation which satisfies the following expression is
referred to as commutative operation.
• Commutative law states that changing the sequence of the variables
does not have any effect on the output of a logic circuit.
Associative law
• This law states that the order in which the logic operations are
performed is irrelevant as their effect is the same.
13. Distributive law
• Distributive law states the following condition.
AND law
• These laws use the AND operation. Therefore they are called as AND laws.
OR law
• These laws use the OR operation. Therefore they are called as OR laws.
INVERSION law
• This law uses the NOT operation. The inversion law states that double inversion of a
variable results in the original variable itself.
14. Laws of Boolean algebra
The basic Laws of Boolean Algebra can be stated as follows:
• Commutative Law states that the interchanging of the order of operands in a
Boolean equation does not change its result. For example:
OR operator → A + B = B + A
AND operator → A * B = B * A
• Associative Law of multiplication states that the AND operation are done on
two or more than two variables. For example:
A * (B * C) = (A * B) * C
• Distributive Law states that the multiplication of two variables and adding the
result with a variable will result in the same value as multiplication of addition
of the variable with individual variables. For example:
A + BC = (A + B) (A + C).
15. • Annulment law:
A.0 = 0
A + 1 = 1
• Identity law:
A.1 = A
A + 0 = A
• Idempotent law:
A + A = A
A.A = A
• Complement law:
A + A' = 1
A.A'= 0
• Double negation law:
((A)')' = A
• Absorption law:
A.(A+B) = A
A + AB = A
16. De Morgan's Law
• De Morgan's Law is also known as De Morgan'stheorem, works depending on the
concept of Duality. Duality states that interchanging the operators and variables
in a function, such as replacing 0 with 1 and 1 with 0, AND operator with OR
operatorand OR operator with AND operator.
• De Morgan stated 2 theorems, which will help us in solving the algebraic
problemsin digital electronics. The DeMorgan's statementsare:
• "The negation of a conjunction is the disjunction of the negations", which means
that the complement of the product of 2 variables is equal to the sum of the
compliments ofindividual variables.For example,(A.B)' = A' + B'.
• "The negation of disjunction is the conjunction of the negations", which means
that compliment of the sum of two variables is equal to the product of the
complementof eachvariable. For example,(A + B)' = A'B'.
17.
18. Simplification using Boolean algebra
Let us consider an example of a Boolean function: AB+A (B+C) + B (B+C)
The logic diagram for the Boolean function AB+A (B+C) + B (B+C) can be represented
as
19. Map Simplification
• The Map method involves a simple, straightforward procedure
for simplifying Boolean expressions.
• Map simplification may be regarded as a pictorial arrangement
of the truth table which allows an easy interpretation for
choosing the minimum number of terms needed to express the
function algebraically.
• The map method is also known as Karnaugh map or K-map.
• Each combination of the variables in a truth table is called a
mid-term.
20. K-Maps for 2 to 5 Variables
• The number of cells in 2 variable K-map is four, since the number of
variables is two. The following figure shows 2 variable K-Map.
• There is only one possibility of grouping 4 adjacent min terms.
• The possible combinations of grouping 2 adjacent min terms are {(m0, m1),
(m2, m3), (m0, m2) and (m1, m3)}.
21. 3 Variable K-Map
• The number of cells in 3 variable K-map is eight, since the number of
variables is three. The following figure shows 3 variable K-Map.
• There is only one possibility of grouping 8 adjacent min terms.
• The possible combinations of grouping 4 adjacent min terms are {(m0, m1,
m3, m2), (m4, m5, m7, m6), (m0, m1, m4, m5), (m1, m3, m5, m7), (m3, m2, m7, m6)
and (m2, m0, m6, m4)}.
• The possible combinations of grouping 2 adjacent min terms are {(m0, m1),
(m1, m3), (m3, m2), (m2, m0), (m4, m5), (m5, m7), (m7, m6), (m6, m4), (m0, m4), (m1,
m5), (m3, m7) and (m2, m6)}.
• If x=0, then 3 variable K-map becomes 2 variable K-map.
22. 4 Variable K-Map
• The number of cells in 4 variable K-map is sixteen, since the number of variables is four.
The following figure shows 4 variable K-Map.
• There is only one possibility of grouping 16 adjacent min terms.
• Let R1, R2, R3 and R4 represents the min terms of first row, second row, third row and
fourth row respectively. Similarly, C1, C2, C3 and C4 represents the min terms of first
column, second column, third column and fourth column respectively. The possible
combinations of grouping 8 adjacent min terms are {(R1, R2), (R2, R3), (R3, R4), (R4, R1), (C1,
C2), (C2, C3), (C3, C4), (C4, C1)}.
• If w=0, then 4 variable K-map becomes 3 variable K-map.
23. 5 Variable K-Map
• The number of cells in 5 variable K-map is thirty-two, since the
number of variables is 5. The following figure shows 5 variable
K-Map.
• There is only one possibility of grouping 32 adjacent min terms.
• There are two possibilities of grouping 16 adjacent min terms.
i.e., grouping of min terms from m0 to m15 and m16 to m31.
• If v=0, then 5 variable K-map becomes 4 variable K-map.
24. Minimization of Boolean Functions using K-Maps
• If we consider the combination of inputs for which the Boolean function is ‘1’, then we will get the
Boolean function, which is in standard sum of products form after simplifying the K-map.
• Similarly, if we consider the combination of inputs for which the Boolean function is ‘0’, then we
will get the Boolean function, which is in standard product of sums form after simplifying the K-
map.
• Follow these rules for simplifying K-maps in order to get standard sum of products form.
• Select the respective K-map based on the number of variables present in the Boolean function.
• If the Boolean function is given as sum of min terms form, then place the ones at respective min
term cells in the K-map. If the Boolean function is given as sum of products form, then place the
ones in all possible cells of K-map for which the given product terms are valid.
• Check for the possibilities of grouping maximum number of adjacent ones. It should be powers of
two. Start from highest power of two and upto least power of two. Highest power is equal to the
number of variables considered in K-map and least power is zero.
• Each grouping will give either a literal or one product term. It is known as prime implicant. The
prime implicant is said to be essential prime implicant, if atleast single ‘1’ is not covered with any
other groupings but only that grouping covers.
• Note down all the prime implicants and essential prime implicants. The simplified Boolean
function contains all essential prime implicants and only the required prime implicants.
• Note 1 − If outputs are not defined for some combination of inputs, then those output values will
be represented with don’t care symbol ‘x’. That means, we can consider them as either ‘0’ or ‘1’.
• Note 2 − If don’t care terms also present, then place don’t cares ‘x’ in the respective cells of K-
map. Consider only the don’t cares ‘x’ that are helpful for grouping maximum number of adjacent
ones. In those cases, treat the don’t care value as ‘1’.