This document discusses Boolean algebra and its applications in digital circuits. It begins by introducing Boolean logic gates like AND, OR, and NOT. Combinational logic circuits are described as combinations of these gates. Equivalence between circuits is defined based on whether they produce the same outputs for the same inputs. Switching circuits are introduced as an alternative representation using open and closed switches. Boolean algebra is then discussed, with laws like distribution, De Morgan's laws, and identities. Boolean functions are analogous to algebraic functions, mapping inputs to outputs. Minimizing circuits to reduce complexity is also mentioned.
This document contains information about a laboratory manual for a Linear Integrated Circuits lab class, including the syllabus, list of required equipment, and plans for experiments to be completed over 15 weeks. The experiments include designing inverting and non-inverting amplifiers, differentiators, integrators, and active filters using operational amplifiers. Circuit diagrams and procedures are provided for building the circuits and measuring input/output waveforms on an oscilloscope.
This document provides an overview of logic gates, including:
- The main types of logic gates (NOT, AND, NAND, OR, NOR, EX-OR, EX-NOR)
- Their symbols and truth tables
- How logic gates can be combined to perform more complex logic functions
- How different gate types can be substituted using only NAND or NOR gates
The document discusses fault tolerant and online testability in reversible logic synthesis. It proposes a design for a fault tolerant full adder circuit using reversible logic that is both fault tolerant and online testable. The proposed design uses only 3x3 fault tolerant gates, has a minimum number of garbage outputs of 3, and has lower quantum cost compared to an existing design. Performance analysis shows the proposed design has advantages over the existing design in terms of number of gates, garbage outputs, and quantum cost.
Quantum Cost Calculation of Reversible CircuitSajib Mitra
This document discusses the calculation of quantum cost for reversible circuits. It begins with an overview of reversible logic and quantum computing concepts like quantum gates. It then explains the realization of quantum NOT gate using quantum coin flips. Different quantum gates and their quantum costs are discussed, including Toffoli, Fredkin, Peres and NFT gates. Special cases involving quantum wires that have zero cost are also covered. The document concludes with an assignment to find the cost of additional gates and provides information about the author.
The document describes various logic gates - AND, OR, NOT, NAND, and NOR gates. It provides the truth tables and Boolean expressions for each gate. It also gives examples of how to derive a truth table from a given logic circuit diagram. NAND and NOR gates are shown to function as inverters in some configurations. The document aims to explain the basic concepts and functions of common logic gates.
Logic gates are used to represent binary operations. The main logic gates are AND, OR, NOT, NAND, NOR, XOR, and XNOR. Truth tables define the output of each gate based on all possible combinations of its inputs. A circuit diagram can be translated to a truth table by considering the output for each input combination.
This document describes an experiment to characterize active band-pass and band-stop filters. The objectives are to determine the gain-frequency response, center frequency, bandwidth, quality factor, and phase response. For the band-pass filter, the measured and calculated results for center frequency, gain, bandwidth, and quality factor agree to within 5%. For the band-stop filter, the measured and calculated results for center frequency, gain, bandwidth, and quality factor agree to within 1%. The phase response of the band-pass filter shows the output is approximately 180 degrees out of phase with the input at the center frequency.
1. The document describes an experiment on Fourier theory involving the generation of square waves and triangular waves from a series of sine and cosine waves at different frequencies and amplitudes.
2. Key findings include that a square wave can be produced from odd harmonics of a fundamental sine wave, while a triangular wave can be produced from odd harmonic cosine waves. Eliminating harmonics distorts the output wave shape.
3. The time domain shows voltage over time, while the frequency domain shows amplitude by frequency using a Fourier series. Filtering affects the frequency spectrum and output wave shape.
This document contains information about a laboratory manual for a Linear Integrated Circuits lab class, including the syllabus, list of required equipment, and plans for experiments to be completed over 15 weeks. The experiments include designing inverting and non-inverting amplifiers, differentiators, integrators, and active filters using operational amplifiers. Circuit diagrams and procedures are provided for building the circuits and measuring input/output waveforms on an oscilloscope.
This document provides an overview of logic gates, including:
- The main types of logic gates (NOT, AND, NAND, OR, NOR, EX-OR, EX-NOR)
- Their symbols and truth tables
- How logic gates can be combined to perform more complex logic functions
- How different gate types can be substituted using only NAND or NOR gates
The document discusses fault tolerant and online testability in reversible logic synthesis. It proposes a design for a fault tolerant full adder circuit using reversible logic that is both fault tolerant and online testable. The proposed design uses only 3x3 fault tolerant gates, has a minimum number of garbage outputs of 3, and has lower quantum cost compared to an existing design. Performance analysis shows the proposed design has advantages over the existing design in terms of number of gates, garbage outputs, and quantum cost.
Quantum Cost Calculation of Reversible CircuitSajib Mitra
This document discusses the calculation of quantum cost for reversible circuits. It begins with an overview of reversible logic and quantum computing concepts like quantum gates. It then explains the realization of quantum NOT gate using quantum coin flips. Different quantum gates and their quantum costs are discussed, including Toffoli, Fredkin, Peres and NFT gates. Special cases involving quantum wires that have zero cost are also covered. The document concludes with an assignment to find the cost of additional gates and provides information about the author.
The document describes various logic gates - AND, OR, NOT, NAND, and NOR gates. It provides the truth tables and Boolean expressions for each gate. It also gives examples of how to derive a truth table from a given logic circuit diagram. NAND and NOR gates are shown to function as inverters in some configurations. The document aims to explain the basic concepts and functions of common logic gates.
Logic gates are used to represent binary operations. The main logic gates are AND, OR, NOT, NAND, NOR, XOR, and XNOR. Truth tables define the output of each gate based on all possible combinations of its inputs. A circuit diagram can be translated to a truth table by considering the output for each input combination.
This document describes an experiment to characterize active band-pass and band-stop filters. The objectives are to determine the gain-frequency response, center frequency, bandwidth, quality factor, and phase response. For the band-pass filter, the measured and calculated results for center frequency, gain, bandwidth, and quality factor agree to within 5%. For the band-stop filter, the measured and calculated results for center frequency, gain, bandwidth, and quality factor agree to within 1%. The phase response of the band-pass filter shows the output is approximately 180 degrees out of phase with the input at the center frequency.
1. The document describes an experiment on Fourier theory involving the generation of square waves and triangular waves from a series of sine and cosine waves at different frequencies and amplitudes.
2. Key findings include that a square wave can be produced from odd harmonics of a fundamental sine wave, while a triangular wave can be produced from odd harmonic cosine waves. Eliminating harmonics distorts the output wave shape.
3. The time domain shows voltage over time, while the frequency domain shows amplitude by frequency using a Fourier series. Filtering affects the frequency spectrum and output wave shape.
SPICE MODEL of LM7824 PSpice in SPICE PARK. English Version is http://www.spicepark.net. Japanese Version is http://www.spicepark.com by Bee Technologies.
This document describes an experiment on Fourier theory and its applications in signal processing. The objectives are to: 1) Produce a square wave from sine waves of different frequencies and amplitudes using Fourier theory. 2) Produce a triangular wave similarly using cosine waves. 3) Examine the difference between time domain and frequency domain representations of signals. 4) Analyze periodic pulses with different duty cycles in both domains. 5) Examine the effect of low-pass filtering on pulses as the cutoff frequency varies. The experiment uses function generators, oscilloscopes, spectrum analyzers, and op-amps to generate and analyze signals.
1. The document describes an experiment on Fourier theory and how signals can be represented in both the time domain and frequency domain. Square waves and triangular waves are generated from a series of sine and cosine waves (Fourier series) and plotted in both domains.
2. Low-pass filters are used to remove higher harmonics from signals. This distorts the original waveshape as more harmonics are removed. The bandwidth needed to transmit pulses with minimal distortion depends on the duty cycle.
3. Objectives include learning how square and triangular waves can be produced from Fourier series, comparing time and frequency domain plots, and examining how duty cycle and filtering affect pulses in both domains.
Operational amplifiers (OP amps) can be used to build various electronic circuits. An ideal OP amp has infinite input impedance and zero output impedance, with infinite open-loop gain. Common configurations include inverting and non-inverting amplifiers, which provide negative and positive feedback respectively to control the closed-loop gain. Other circuits like summers, integrators, and differentiators can be built by exploiting the high input impedance and voltage amplification properties of OP amps. However, practical OP amps have non-ideal characteristics that must be accounted for, such as finite gain bandwidth, input bias currents, and output saturation levels.
The document discusses generating square and triangular waves using Fourier series of sine and cosine waves. It also examines signals in the time and frequency domains. Key points:
1) A square wave can be produced from a series of sine waves at different frequencies and amplitudes, with the fundamental and odd harmonics present.
2) A triangular wave results from a series of cosine waves, with the fundamental and odd harmonics.
3) Signals can be viewed in the time domain as voltage over time, or in the frequency domain as the amplitude of sine/cosine waves at different frequencies.
This document discusses the analysis and synthesis of synchronous sequential circuits. It provides an example of analyzing a sequential circuit with two JK flip-flops and one input. The analysis involves deriving the excitation equations, next state equations, output equation, and state table. A state diagram is also shown graphically representing the state transitions. The document explains that the state table contains the present states, next states, and outputs for all possible input combinations according to the circuit equations.
This chapter discusses Boolean algebra and its applications to digital circuits. It introduces Boolean logic gates like AND, OR, and NOT and how they are used to build combinatorial logic circuits. Boolean expressions can be represented by logic gates and switching circuits, with equivalent circuits producing the same outputs for all inputs. Boolean algebra laws and theorems allow simplifying expressions and minimizing circuits. The chapter aims to describe how Boolean logic underlies the functioning of digital circuits used in computers and electronics.
SIGNAL SPECTRA EXPERIMENT 2 - FINALS (for CAUAN)Sarah Krystelle
This document describes Experiment #2 on a class B push-pull power amplifier. The objectives are to determine the dc and ac load lines, observe crossover distortion, measure voltage gain, output power, and efficiency. Sample computations are provided for voltage gain, output power, input power, and efficiency. The theory section describes class B push-pull amplifiers and how biasing the transistors slightly above cutoff can eliminate crossover distortion. Procedures are outlined to simulate and measure the amplifier's input, output, voltage gain, power output, and efficiency.
This document describes experiments performed to characterize active band-pass and band-stop filters, including plotting the gain-frequency response curves to determine cutoff frequencies and bandwidth, calculating quality factors and center frequencies, and comparing measured and expected voltage gains. Procedures are provided to implement and analyze a multiple-feedback band-pass filter and a two-pole Sallen-Key notch filter using op-amps and passive components.
This document describes an experiment on Fourier theory involving the time domain and frequency domain. The objectives are to generate square and triangular waves from Fourier series, examine the difference between time and frequency domain plots, and analyze periodic pulses with different duty cycles in both domains while varying a low-pass filter's cutoff frequency. Procedures generate waves using function generators and measure them on an oscilloscope and spectrum analyzer while eliminating harmonics. The document explains Fourier analysis and how signals can be represented by sine/cosine waves of different frequencies and amplitudes in the frequency domain.
The document contains graphs and tables of experimental data relating the variables M, N, P, and Rx. It also includes equations relating these variables and analyses the relationship between Rx and the variables M, N, and P through linear regression.
The L298 is an integrated circuit that acts as a dual full-bridge driver designed to drive inductive loads such as motors. It can operate at supply voltages up to 46V and source currents up to 4A. It contains two bridge drivers that can be independently enabled. Each bridge can sink or source current depending on the input signals. The circuit includes overtemperature protection and logic-level input tolerance. It is available in Multiwatt and PowerSO packages.
This document contains information about homework help resources and the syllabus for an electronics lab course. The syllabus lists 10 experiments involving digital and analog integrated circuits, including studying logic gates, op-amps, timers, counters, and analog-to-digital converters. Key details are provided on the operation and design of inverting/non-inverting amplifiers, differentiators, integrators, and astable/monostable multivibrators using a 555 timer chip. Circuit diagrams and design procedures are provided for several of the experiments.
The document provides information about the course ECX4238 - Electrical Machines for the 2011/12 academic year at The Open University of Sri Lanka. It outlines the course components which include 3 assignments, laboratory work, 2 continuous assessment tests, 3 day schools, and a final closed book examination. It also provides the eligibility criteria of averaging 40% or higher across 5 of the 6 assessments and a minimum 40% laboratory mark. The document concludes by detailing 4 questions for Assignment 1 covering topics like transformer connections, parallel transformer operation, generator voltage regulation, and shunt generator characteristics.
SPICE MODEL of uPC24A15HF in SPICE PARK. English Version is http://www.spicepark.net. Japanese Version is http://www.spicepark.com by Bee Technologies.
1. The document describes a final project to build an analog PID control circuit using op-amps. It includes objectives, a list of components, and detailed instructions on assembling the circuit and testing it.
2. Key steps include deriving the transfer functions for the proportional, derivative, and integral controllers. Tests are done to observe input-output waveforms for each section alone and for the combined PID controller.
3. Optional tests include modifying the derivative and integral sections, testing with different input signals, closed-loop simulations, and integrating the PID controller into a double integrator plant model.
This document provides the lab manual for the IC Applications lab course for students in the III BTech ECE program. It includes an introduction, a list of 15 experiments to be performed in the lab divided into two parts, general do's and don'ts for the lab, and details on the first experiment - Adder, Subtractor, and Comparator using the IC 741 op-amp. The document provides theory, circuit diagrams, procedures, observation tables and model calculations for the first experiment.
1. The document discusses digital circuits and logic gates. It defines analog and digital signals and introduces the binary number system.
2. Boolean algebra and logic operations such as OR, AND, and NOT are described. The document provides truth tables to define the output of logic gates for all possible input combinations.
3. Common logic gates such as OR, AND, NOT, NOR, and NAND are defined. Their corresponding truth tables and circuit diagrams are given to illustrate how each gate implements Boolean logic operations. The gates can be combined to form other gates like XOR.
There are several types of power supplies that can be used for electronic circuits. A basic power supply consists of a transformer, rectifier, and smoothing capacitor. More advanced supplies also include a voltage regulator. The transformer steps down the high voltage mains power. The rectifier converts AC to DC. Smoothing reduces voltage fluctuations. Regulators ensure a constant output voltage. Some circuits require a dual supply with both positive and negative outputs.
Esol is an ERP consulting firm that works with SAP and Oracle, primarily in the US but also Latin America and Canada. They are seeking a SAP MM Consultant for their Los Angeles office who has at least 5 years of SAP MM experience, including functional skills, solution design, configuration, and implementation experience. The position requires experience with ECC 6.0 and the ability to work in a fast-paced environment.
SPICE MODEL of LM7824 PSpice in SPICE PARK. English Version is http://www.spicepark.net. Japanese Version is http://www.spicepark.com by Bee Technologies.
This document describes an experiment on Fourier theory and its applications in signal processing. The objectives are to: 1) Produce a square wave from sine waves of different frequencies and amplitudes using Fourier theory. 2) Produce a triangular wave similarly using cosine waves. 3) Examine the difference between time domain and frequency domain representations of signals. 4) Analyze periodic pulses with different duty cycles in both domains. 5) Examine the effect of low-pass filtering on pulses as the cutoff frequency varies. The experiment uses function generators, oscilloscopes, spectrum analyzers, and op-amps to generate and analyze signals.
1. The document describes an experiment on Fourier theory and how signals can be represented in both the time domain and frequency domain. Square waves and triangular waves are generated from a series of sine and cosine waves (Fourier series) and plotted in both domains.
2. Low-pass filters are used to remove higher harmonics from signals. This distorts the original waveshape as more harmonics are removed. The bandwidth needed to transmit pulses with minimal distortion depends on the duty cycle.
3. Objectives include learning how square and triangular waves can be produced from Fourier series, comparing time and frequency domain plots, and examining how duty cycle and filtering affect pulses in both domains.
Operational amplifiers (OP amps) can be used to build various electronic circuits. An ideal OP amp has infinite input impedance and zero output impedance, with infinite open-loop gain. Common configurations include inverting and non-inverting amplifiers, which provide negative and positive feedback respectively to control the closed-loop gain. Other circuits like summers, integrators, and differentiators can be built by exploiting the high input impedance and voltage amplification properties of OP amps. However, practical OP amps have non-ideal characteristics that must be accounted for, such as finite gain bandwidth, input bias currents, and output saturation levels.
The document discusses generating square and triangular waves using Fourier series of sine and cosine waves. It also examines signals in the time and frequency domains. Key points:
1) A square wave can be produced from a series of sine waves at different frequencies and amplitudes, with the fundamental and odd harmonics present.
2) A triangular wave results from a series of cosine waves, with the fundamental and odd harmonics.
3) Signals can be viewed in the time domain as voltage over time, or in the frequency domain as the amplitude of sine/cosine waves at different frequencies.
This document discusses the analysis and synthesis of synchronous sequential circuits. It provides an example of analyzing a sequential circuit with two JK flip-flops and one input. The analysis involves deriving the excitation equations, next state equations, output equation, and state table. A state diagram is also shown graphically representing the state transitions. The document explains that the state table contains the present states, next states, and outputs for all possible input combinations according to the circuit equations.
This chapter discusses Boolean algebra and its applications to digital circuits. It introduces Boolean logic gates like AND, OR, and NOT and how they are used to build combinatorial logic circuits. Boolean expressions can be represented by logic gates and switching circuits, with equivalent circuits producing the same outputs for all inputs. Boolean algebra laws and theorems allow simplifying expressions and minimizing circuits. The chapter aims to describe how Boolean logic underlies the functioning of digital circuits used in computers and electronics.
SIGNAL SPECTRA EXPERIMENT 2 - FINALS (for CAUAN)Sarah Krystelle
This document describes Experiment #2 on a class B push-pull power amplifier. The objectives are to determine the dc and ac load lines, observe crossover distortion, measure voltage gain, output power, and efficiency. Sample computations are provided for voltage gain, output power, input power, and efficiency. The theory section describes class B push-pull amplifiers and how biasing the transistors slightly above cutoff can eliminate crossover distortion. Procedures are outlined to simulate and measure the amplifier's input, output, voltage gain, power output, and efficiency.
This document describes experiments performed to characterize active band-pass and band-stop filters, including plotting the gain-frequency response curves to determine cutoff frequencies and bandwidth, calculating quality factors and center frequencies, and comparing measured and expected voltage gains. Procedures are provided to implement and analyze a multiple-feedback band-pass filter and a two-pole Sallen-Key notch filter using op-amps and passive components.
This document describes an experiment on Fourier theory involving the time domain and frequency domain. The objectives are to generate square and triangular waves from Fourier series, examine the difference between time and frequency domain plots, and analyze periodic pulses with different duty cycles in both domains while varying a low-pass filter's cutoff frequency. Procedures generate waves using function generators and measure them on an oscilloscope and spectrum analyzer while eliminating harmonics. The document explains Fourier analysis and how signals can be represented by sine/cosine waves of different frequencies and amplitudes in the frequency domain.
The document contains graphs and tables of experimental data relating the variables M, N, P, and Rx. It also includes equations relating these variables and analyses the relationship between Rx and the variables M, N, and P through linear regression.
The L298 is an integrated circuit that acts as a dual full-bridge driver designed to drive inductive loads such as motors. It can operate at supply voltages up to 46V and source currents up to 4A. It contains two bridge drivers that can be independently enabled. Each bridge can sink or source current depending on the input signals. The circuit includes overtemperature protection and logic-level input tolerance. It is available in Multiwatt and PowerSO packages.
This document contains information about homework help resources and the syllabus for an electronics lab course. The syllabus lists 10 experiments involving digital and analog integrated circuits, including studying logic gates, op-amps, timers, counters, and analog-to-digital converters. Key details are provided on the operation and design of inverting/non-inverting amplifiers, differentiators, integrators, and astable/monostable multivibrators using a 555 timer chip. Circuit diagrams and design procedures are provided for several of the experiments.
The document provides information about the course ECX4238 - Electrical Machines for the 2011/12 academic year at The Open University of Sri Lanka. It outlines the course components which include 3 assignments, laboratory work, 2 continuous assessment tests, 3 day schools, and a final closed book examination. It also provides the eligibility criteria of averaging 40% or higher across 5 of the 6 assessments and a minimum 40% laboratory mark. The document concludes by detailing 4 questions for Assignment 1 covering topics like transformer connections, parallel transformer operation, generator voltage regulation, and shunt generator characteristics.
SPICE MODEL of uPC24A15HF in SPICE PARK. English Version is http://www.spicepark.net. Japanese Version is http://www.spicepark.com by Bee Technologies.
1. The document describes a final project to build an analog PID control circuit using op-amps. It includes objectives, a list of components, and detailed instructions on assembling the circuit and testing it.
2. Key steps include deriving the transfer functions for the proportional, derivative, and integral controllers. Tests are done to observe input-output waveforms for each section alone and for the combined PID controller.
3. Optional tests include modifying the derivative and integral sections, testing with different input signals, closed-loop simulations, and integrating the PID controller into a double integrator plant model.
This document provides the lab manual for the IC Applications lab course for students in the III BTech ECE program. It includes an introduction, a list of 15 experiments to be performed in the lab divided into two parts, general do's and don'ts for the lab, and details on the first experiment - Adder, Subtractor, and Comparator using the IC 741 op-amp. The document provides theory, circuit diagrams, procedures, observation tables and model calculations for the first experiment.
1. The document discusses digital circuits and logic gates. It defines analog and digital signals and introduces the binary number system.
2. Boolean algebra and logic operations such as OR, AND, and NOT are described. The document provides truth tables to define the output of logic gates for all possible input combinations.
3. Common logic gates such as OR, AND, NOT, NOR, and NAND are defined. Their corresponding truth tables and circuit diagrams are given to illustrate how each gate implements Boolean logic operations. The gates can be combined to form other gates like XOR.
There are several types of power supplies that can be used for electronic circuits. A basic power supply consists of a transformer, rectifier, and smoothing capacitor. More advanced supplies also include a voltage regulator. The transformer steps down the high voltage mains power. The rectifier converts AC to DC. Smoothing reduces voltage fluctuations. Regulators ensure a constant output voltage. Some circuits require a dual supply with both positive and negative outputs.
Esol is an ERP consulting firm that works with SAP and Oracle, primarily in the US but also Latin America and Canada. They are seeking a SAP MM Consultant for their Los Angeles office who has at least 5 years of SAP MM experience, including functional skills, solution design, configuration, and implementation experience. The position requires experience with ECC 6.0 and the ability to work in a fast-paced environment.
The document summarizes the Chord peer-to-peer lookup protocol. Chord provides efficient mapping of keys to nodes, scaling logarithmically. Each node maintains routing information for O(log N) other nodes. Lookup requires O(log N) hops, and joins/leaves require O(log^2 N) messages. Chord is decentralized, load balanced, and adapts to node joins, leaves, and failures while maintaining correctness. Simulation shows Chord is robust and lookup latency grows slowly with system size.
La voluntaria Stephanie Harris ha estado trabajando con varias organizaciones locales para mejorar la gestión de residuos sólidos en la comunidad de San Nicolás. Recientemente logró que el servicio de recolección de basura sea permanente cada 15 días, visitando la comunidad los días 27 de mayo y 10 de junio. Además, la voluntaria enseña inglés, liderazgo y educación ambiental a estudiantes de primaria y secundaria en la escuela local, y acompaña al puesto de salud en visitas domiciliarias para promover una mejor
La voluntaria Stephanie Harris está enseñando clases de Vacaciones Útiles a los jóvenes de San Nicolás, incluyendo educación ambiental, inglés, club de libros y club ambiental. Además, está trabajando para hacer permanente el servicio de recolección de basura en la comunidad y coordinando con las autoridades locales para establecer un convenio sobre este servicio. Finalmente, la voluntaria promueve un mejor manejo de recursos y la siembra de árboles en la escuela primaria local.
LinkedIn is the world's largest professional network with over 175 million members. It connects users to their trusted contacts to exchange knowledge, ideas, and opportunities. The basics of a LinkedIn profile include a headline, profile picture, summary, and experience. The headline should showcase your specialty or value proposition in a creative way using important keywords. The profile picture should include a smile. The summary should tell people who you are and why they should do business with you using both industry and layman's terms that express how you provide value and remove pain. Experience listings should provide complete, descriptive explanations to help people learn about you in an interesting way.
Jntuk dap-b.tech(mechanical engineering)- syllabus of b.tech iii year - i sem...Prasad Vejendla
The document provides the course structure for a Bachelor of Technology in Mechanical Engineering at Jawaharlal Nehru Technological University in Kakinada, India from 2010-2011. It includes:
1) A list of subjects covered semester-by-semester over 4 years, including the course credits for each subject.
2) Subjects covered include engineering fundamentals, thermodynamics, mechanics, design, production technology, and electives in specialized areas like robotics and alternative energy.
3) Laboratory sessions are included for subjects like mechanics, thermodynamics, production, and communication skills.
4) To graduate, students must complete a minimum of 200 credits including laboratory and project
This document provides an overview of hydro power plant components and types. It discusses the three types of power houses: surface, semi-underground, and underground. Surface power houses have components on the surface but are limited by topography. Semi-underground power houses combine advantages of surface and underground. Underground power houses are located entirely inside mountains with access tunnels. The document also summarizes the main components of hydro power stations including dams/barrages, water conductor systems, and power houses as well as different types of hydro power projects.
This document discusses steam turbines, including their working principles and different types. It describes how potential energy from steam is converted to kinetic energy and then mechanical energy in a turbine. There are two main types of turbines - impulse turbines and reaction turbines. Impulse turbines expand steam fully in nozzles before it hits moving blades, while reaction turbines feature continuous expansion over fixed and moving blades. The document also discusses methods of compounding turbines to reduce rotor speed, including velocity, pressure, and pressure-velocity compounding.
El documento describe las diferentes clases de triángulos. Explica que un triángulo se forma al unir tres puntos no colineales con segmentos de recta. Luego clasifica los triángulos en dos grupos: por la medida de sus lados (equilátero, isósceles, escaleno) y por la medida de sus ángulos (acutángulo, rectángulo, obtusángulo). Finalmente define los elementos de un triángulo como lados, vértices, ángulos internos, alturas y base, y explica cómo calcular el per
Este documento describe las diferentes clases de triángulos según la medida de sus lados y ángulos. Explica que un triángulo equilátero tiene tres lados iguales y tres ángulos iguales de 60 grados, un triángulo isósceles tiene dos lados iguales y dos ángulos iguales, y un triángulo escaleno no tiene lados ni ángulos iguales. También describe triángulos según la medida de sus ángulos, como los triángulos agudángulo, rectángulo y obtusángulo. Finalmente
Christo Kutrovsky - Maximize Data Warehouse Performance with Parallel QueriesChristo Kutrovsky
Oracle Data Warehouses are typically deployed on servers with very large number of cores, and increasingly on RAC. Making efficient use of all available cores when processing data warehouse workloads is therefore critical in achieving maximal performance. To make efficient use of all cores in a data warehouse system, skilled use of parallel queries is key.
Este documento proporciona información sobre las opciones educativas disponibles después de la ESO en el País Vasco, incluyendo el bachillerato y los ciclos formativos de grado medio y superior. Explica las asignaturas, requisitos de promoción y titulaciones de cada opción, así como las carreras universitarias a las que dan acceso. También incluye recursos para obtener más información sobre estudios posteriores.
O documento discute Ajax, explicando-o como uma técnica que usa Javascript e XML para tornar aplicações web mais interativas e dinâmicas por meio de solicitações assíncronas de informações. Também resume as ferramentas e conceitos necessários para desenvolvimento Ajax, incluindo Firebug, XMLHttpRequest e como Ajax permite que o Javascript busque dados no servidor.
Este documento proporciona información sobre actividades y lugares turísticos en Navarra y los Pirineos Orientales, incluyendo museos, rutas, paseos naturales, balnearios, iglesias y más. Se ofrece información de contacto como teléfonos y páginas web para obtener más detalles sobre horas y visitas. El documento también incluye un mapa de la zona y recomienda disfrutar de las nuevas aventuras de ocio en la naturaleza disponibles este verano.
El documento describe las duras restricciones impuestas a las mujeres afganas bajo el régimen talibán, incluyendo la prohibición de salir solas, trabajar, estudiar o recibir atención médica adecuada. Solo los médicos hombres pueden atender a pacientes en los hospitales, y el 97% de las mujeres dan a luz en casa. El adulterio se castiga con lapidación, en la que la mujer es enterrada hasta la cintura y apedreada hasta la muerte.
This document provides an overview of Boolean algebra. It begins by listing the key objectives of learning Boolean algebra, which include understanding AND, OR, and NOT logic gates, simplifying Boolean expressions, and minimizing circuits. The document then provides examples of logic gates like AND, OR, and NOT gates. It explains how circuits can be represented using Boolean expressions and switching tables. Several laws of Boolean algebra are defined, like distributive, commutative, and De Morgan's laws. The concept of Boolean functions is introduced, where a function defines the relationship between inputs and outputs. Truth tables are used to represent both Boolean functions and to derive a function from a given truth table.
The document provides details about demonstration experiments involving logic gates and transformers.
It describes the basic logic gates - OR, AND, NOT, NOR, NAND, EXOR and EXNOR - and provides their truth tables and circuit designs. It also explains the working of step-down and step-up transformers through circuit diagrams and discusses transformer ratio, efficiency and various energy losses in transformers.
Physics investigatgory project on logic gates class 12appietech
This document describes various logic gates and their workings. It begins with introducing logic gates and their basic components like inputs, outputs, truth tables, and Boolean algebra. It then explains the OR gate, AND gate, NOT gate, NOR gate, NAND gate, EX-OR gate, and EX-NOR gate through their circuit diagrams and truth tables. Each gate is constructed using basic electronic components like diodes, transistors, and resistors. The document concludes that logic gates are fundamental building blocks of modern electronics and digital circuits.
Circuits are designed using logic gates to control the flow of electronic signals, with basic gates including NOT, AND, OR, and XOR gates. More complex circuits can be created by combining logic gates, such as half and full adders used to perform binary addition. Specialized circuits are designed using a process of identifying inputs/outputs, determining necessary logic gates, constructing truth tables, and evaluating circuit design.
Digital logic gates and Boolean algebraSARITHA REDDY
The document discusses digital logic gates and Boolean algebra. It defines logic gates as electronic circuits that make logic decisions. Common logic gates include OR, AND, and NOT gates. Boolean algebra uses truth values of 0 and 1 instead of numbers, and has fundamental laws and operations for AND, OR, and NOT. Boolean algebra can be used to simplify logical expressions and save gates in digital circuit design.
This document provides instructions for laboratory exercises involving digital logic circuits. The exercises include:
1) Studying the operation of logic gates like AND, OR, NOT, NAND, and XOR using integrated circuits and completing truth tables.
2) Verifying Boolean logic laws such as associativity and distributivity using logic gate circuits.
3) Implementing NOT, NAND, NOR, and XOR gates using integrated circuits and observing their truth tables.
4) Demonstrating De Morgan's theorem by connecting logic gate circuits in a specific configuration and completing a truth table.
Chapter 4. logic function and boolean algebraAshish KC
- Boolean algebra is used to analyze and design digital logic circuits and determines logical propositions as either true or false. It uses basic logic gates like AND, OR, and NOT.
- AND gates output 1 only if all inputs are 1, while OR gates output 1 if any input is 1. NOT gates invert the input. More complex gates can be made by combining basic gates, like NAND (AND with output inverted) and NOR (OR with output inverted).
- Boolean algebra has laws like commutative, distributive, complement, identity, and associative laws that define the operations of logical variables and simplify expressions. Together, Boolean algebra and logic gates form the foundation of digital circuit and computer design.
boolean algrebra and logic gates in shortRojin Khadka
The document discusses logic gates and Boolean algebra. It describes the basic logic gates - OR, AND, NOT, NAND, NOR and XOR gates. It explains their symbols, truth tables and functions. Logic gates are electronic circuits that make logic decisions. Boolean algebra uses values of 0 and 1 instead of numbers. It has laws like commutative, associative and distributive laws that define operations on logic values. Logic gates and Boolean algebra are important for designing digital circuits and simplifying logical functions.
The document discusses logic gates and Boolean algebra. It defines key logic gate terms like AND, OR, NAND, NOR, and XOR gates. It provides truth tables that define the output of each gate based on all possible input combinations. Boolean algebra laws and operations are also covered, including addition, multiplication, commutative laws, associative laws, and the distributive law. Methods for converting between Boolean expressions, truth tables, and logic circuits are described. Examples are provided to illustrate how to derive the expression, truth table, or circuit from one of the other representations.
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This document discusses truth tables and Boolean logic. It begins by explaining truth tables and how they represent all possible combinations of variable values in Boolean logic sentences. It then provides examples of truth tables for basic logical operators like AND, OR, and NOT. Finally, it discusses how logic circuits are built from transistors to physically represent Boolean logic operations, and provides an example circuit that implements a specific logical operator.
Physics Investigatory project Class 12 Logic GatesRaghav Rathi
Raghav Rathi, a student of XII Science, completed an investigatory project on logic gates under the guidance of Ma'am Urmila at Bright India Public School during the 2017-2018 academic year. The project report discusses the basic logic gates - OR, AND, NOT, NOR, NAND, EX-OR and EX-NOR - through their truth tables and circuit diagrams. It explains how each gate can be designed using components like diodes, transistors and resistors. The conclusion states that logic gates are essential building blocks of modern electronics and universal gates like NAND and NOR can be used to construct all other basic gates.
Logic gates are small transistor circuits that operate on two voltage levels (0 and 1) to perform logical operations on inputs and determine the output. The main logic gates are AND, OR, NOR, NAND, XOR, and XNOR. AND returns 1 only if all inputs are 1, while OR returns 1 if any input is 1. NAND and NOR are the inversions of AND and OR. XOR returns 1 if only one input is 1, and XNOR returns 1 if both inputs are the same. Logic gates are used to build digital circuits and can be represented by transistors or integrated into logic gate integrated circuits.
This document provides an overview of Boolean algebra, which describes logical relations and operations in digital circuits. It discusses:
1) George Boole's rules that describe logical propositions as either true or false, which can represent digital circuit states of 1 or 0.
2) Basic Boolean operations like AND, OR, and NOT and how they are represented by logic gates. Truth tables show all possible input/output combinations for each gate.
3) Laws of Boolean algebra like commutativity, association, distribution, and others. Karnaugh maps provide a way to simplify Boolean expressions into sum-of-products form.
physics investigatory project class 12 on logic gates ,boolean algebrasukhtej
The document discusses logic gates and their applications. It begins by defining logic gates and their basic components. It then provides details on designing and simulating various logic gate circuits including OR, AND, NOT, NOR, NAND, XOR, XNOR gates. Finally, it discusses some common applications of logic gates such as using OR gates to detect events, AND gates as enable/inhibit gates, XOR/XNOR gates for parity generation/checking, and NOT gates as inverters in oscillators.
Logic gates are electronic circuits that perform basic logical operations and form the building blocks of digital circuits. The document discusses different types of logic gates like AND, OR, NOT, NAND, NOR, XOR and XNOR. It explains their truth tables and Boolean expressions. It also talks about how logic gates are implemented using transistors and their use in basic circuits like flip-flops that form the basis of computer memory.
The document describes various logic gates - OR, AND, NOT, NOR, and NAND. It provides the circuit design and truth tables for each gate. The OR gate can be realized using two diodes and will output 1 if either input is 1. The AND gate uses two diodes and a resistor, and will only output 1 if both inputs are 1. A NOT gate inverts the input and can be made with a transistor. A NOR gate consists of an OR gate followed by a NOT gate, while a NAND gate is an AND gate followed by a NOT.
This document discusses digital circuits and logic gates. It begins with an introduction to analog and digital signals and the binary number system. It then covers Boolean algebra and how it is used to analyze logic gates. The three fundamental logic gates - OR, AND, and NOT - are explained through their truth tables and circuit implementations. More complex gates such as NOR, NAND, and XOR are also introduced and shown to be compositions of the fundamental gates. The document provides detailed explanations of each logic gate's symbol, truth table, and circuit diagram to illustrate their operations.
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1. Chapter 11 Boolean Algebra
11 BOOLEAN
ALGEBRA
Objectives
After studying this chapter you should
• be able to use AND, NOT, OR and NAND gates;
• be able to use combinatorial and switching circuits;
• understand equivalent circuits;
• understand the laws of Boolean algebra;
• be able to simplify Boolean expressions;
• understand Boolean functions;
• be able to minimise circuits;
• understand the significance of half and full adder circuits.
11.0 Introduction
When George Boole (1815-186) developed an algebra for logic,
little did he realise that he was forming an algebra that has
become ideal for the analysis and design of circuits used in
computers, calculators and a host of devices controlled by
microelectronics. Boole's algebra is physically manifested in
electronic circuits and this chapter sets out to describe the
building blocks used in such circuits and the algebra used to
describe the logic of the circuits.
11.1 Combinatorial circuits
The circuits and switching arrangements used in electronics are
very complex but, although this chapter only deals with simple
binary digits
circuits, the functioning of all microchip circuits is based on the
ideas in this chapter. The flow of electrical pulses which
bits
represent the binary digits 0 and 1 (known as bits) is controlled
by combinations of electronic devices. These logic gates act as
switches for the electrical pulses. Special symbols are used to
represent each type of logic gate.
171
2. Chapter 11 Boolean Algebra
NOT gate
The NOT gate is capable of reversing the input pulse. The truth
table for a NOT gate is as follows:
a ~a
Input Output
a ~a This is a NOT gate
0 1
1 0
The NOT gate receives an input, either a pulse (1) or no pulse (0)
and produces an output as follows :
If input a is 1, output is 0;
and if input a is 0, output is 1.
AND gate
The AND gate receives two inputs a and b, and produces an output
denoted by a∧ b . The truth table for an AND gate is as follows :
Input Output
a
a b a∧ b a∧b
b
0 0 0 This is an AND gate
0 1 0
1 0 0
1 1 1
The only way that the output can be 1 is when a AND b are both 1.
In other words there needs to be an electrical pulse at a AND b
before the AND gate will output an electrical pulse.
OR gate
The OR gate receives two inputs a and b, and produces an output
denoted by a∨ b . The truth table for an OR gate is as follows:
Input Output
a b a∨ b
a
0 0 0 a∨b
b
0 1 1 This is an OR gate
1 0 1
1 1 1
The output will be 1 when a or b or both are 1.
172
3. Chapter 11 Boolean Algebra
These three gates, NOT, AND and OR, can be joined together
to form combinatorial circuits to represent Boolean
expressions, as explained in the previous chapter.
Example
Use logic gates to represent
(a) ~ p∨ q
(b) ( x∨ y)∧ ~ x
Draw up the truth table for each circuit
Solution
(a) p q ~p ~ p∨ q
0 0 1 1
∼p
0 1 1 1 p
~p∨q
q
1 0 0 0
1 1 0 1
(b) x y x∨y ~x (x ∨ y) ∧ ~x
~x
0 0 0 1 0
(x∨y)∧ ~x
0 1 1 1 1 x (x∨y)
y
1 0 1 0 0
1 1 1 0 0
Exercise 11A
Use logic gates to represent these expressions and Write down the Boolean expression for each of the
draw up the corresponding truth tables. circuits below.
1. x∧ ( ~ y∨ x) 4. a
b
2. a∨ ( ~ b ∧ c)
c
3. [ a∨ ( ~ b ∨ c)]∧ ~ b
5.
p
q
r
173
4. Chapter 11 Boolean Algebra
11.2 When are circuits
equivalent?
Two circuits are said to be equivalent if each produce the same
outputs when they receive the same inputs.
Example
Are these two combinatorial circuits equivalent?
a
a
b
b
Solution
The truth tables for both circuits will show if they are equivalent :
a b ~ ( a∧ b)
0 0 1
0 1 1 a
b
1 0 1
1 1 0
a b ~ a∨ ~ b
0 0 1
a
0 1 1
1 0 1 b
1 1 0
Work through the values in the truth tables for yourself. Since both
tables give the same results the two circuits are equivalent. Indeed
the two Boolean expressions are equivalent and can be put equal;
i.e. ~ ( a∧ b) = ~ a∨ ~ b
Exercise 11B
Show if these combinatorial circuits are equivalent by
working out the Boolean expression and the truth table
for each circuit.
1. a 2.
b a
b
a
a
b
b
174
5. Chapter 11 Boolean Algebra
3. 4. a
a b
b c
a a
b b
c
11.3 Switching circuits
A network of switches can be used to represent a Boolean
expression and an associated truth table.
Generally the switches are used to control the flow of an electrical
current but you might find it easier to consider a switching circuit as
a series of water pipes with taps or valves at certain points.
One of the reasons for using switching circuits rather than logic
gates is that designers need to move from a combinatorial circuit
(used for working out the logic) towards a design which the
manufacturer can use for the construction of the electronic circuits.
This diagram shows switches A, B and C which can be open or
closed. If a switch is closed it is shown as a 1 in the following table A B
whilst 0 shows that the switch is open. The switching table for this C
circuit is as follows :
A B C Circuit output
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
The table shows that there will be an output (i.e. 1) when A AND B
are 1 OR C is 1. This circuit can therefore be represented as
(A AND B) OR C
ie. (A ∧ B) ∨ C
The circuit just considered is built up of two fundamental circuits:
• a series circuit, often called an AND gate, A ∧ B A B
175
6. Chapter 11 Boolean Algebra
A
• a parallel circuit, often called an OR gate, A ∨ B.
B
The next step is to devise a way of representing negation. The
negation of the truth value 1 is 0 and vice versa, and in
switching circuits the negation of a 'closed' path is an 'open'
path.
A ~A
This circuit will always be 'open' whatever the state of A. In
other words the output will always be 0, irrespective of whether
A is 1 or 0. A
This circuit will always be 'closed' whatever the state of A. The ~A
output will always be 1 irrespective of whether A is 1 or 0.
Example A B
Represent the circuit shown opposite symbolically and give the C ~A
switching table.
Solution
The symbolic representation can be built up by considering
the top line of the circuit (A ∧ B)
the top bottom of the circuit (C ∧ ~A).
Combining these gives the result (A ∧ B) ∨ (C ∧ ~A)
The table is as follows.
A B C ~A A ∧B C∧ ~ A ( A ∧ B ) ∨ (C ∧ ~ A )
0 0 0 1 0 0 0
0 0 1 1 0 1 1
0 1 0 1 0 0 0
0 1 1 1 0 1 1
1 0 0 0 0 0 0
1 0 1 0 0 0 0
1 1 0 0 1 0 1
1 1 1 0 1 0 1
Activity 1 Make your own circuit
Bulb
B
Using a battery, some wire, a bulb and some switches, construct A
the following circuit. C ~B
A simple switch can be made using two drawing pins and a
paper clip which can swivel to close the switch. The pins can be Battery
pushed into a piece of corrugated cardboard or polystyrene.
176
7. Chapter 11 Boolean Algebra
Using the usual notation of 1 representing a closed switch and 0
representing an open switch, you can set the switches to represent
each line of this table:
A B C Output
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Remember that the '~B' switch is always in the opposite state to
the 'B' switch.
Record the output using 1 if the bulb lights up (i.e. circuit is
closed) and 0 if the bulb fails to light (i.e. circuit is open)
Represent the circuit symbolically and draw up another table to see
if you have the same output.
Exercise 11C
Represent the following circuits by Boolean
expressions:
B Draw switching circuits for these Boolean
1. A expressions:
D
~D C 3. A ∨ ( ~ B ∧ C )
4. A ∧ (( ~ B ∧ C ) ∨ ( B∧ ~ C ))
A
2. D
A C B
C
~D
177
8. Chapter 11 Boolean Algebra
11.4 Boolean algebra
A variety of Boolean expressions have been used but George
Boole was responsible for the development of a complete
algebra. In other words, the expressions follow laws similar to
those of the algebra of numbers.
The operators ∧ and ∨ have certain properties similar to those
of the arithmetic operators such as +, −, × and ÷.
(a) Associative laws
(a∨ b) ∨ c= a∨ (b ∨ c)
and (a∧ b) ∧ c= a∧ (b ∧ c)
(b) Commutative laws
a∨ b = b ∨ a
and a∧ b = b ∧ a
(c) Distributive laws
a∧ ( b ∨ c) = ( a∧ b) ∨ ( a∧ c)
and a∨ ( b ∧ c) = ( a∨ b) ∧ ( a∨ c)
These laws enable Boolean expressions to be simplified and
another law developed by an Englishman, Augustus de Morgan
(1806-1871), is useful. He was a contemporary of Boole and
worked in the field of logic and is now known for one important
result bearing his name:-
(d) de Morgan's laws
~ ( a∨ b) = ~ a∧ ~ b
and ~ ( a∧ b) = ~ a∨ ~ b
Note: You have to remember to change the connection,
∧ changes to ∨, ∨ changes to ∧ .
Two more laws complete the range of laws which are included
in the Boolean algebra.
(e) Identity laws
open switch = 0
a∨ 0 = a
a
a
and a∧ 1 = a closed switch = 1
a a
178
9. Chapter 11 Boolean Algebra
(f) Complement laws
a
a∨ ~ a = 1 1
~a
and a∧ ~ a = 0 a ~a 0
The commutative law can be developed to give a further result
which is useful for the simplification of circuits.
a
Consider the expressions a∧ ( a∨ b) and the corresponding a
circuit. ~bb
If switch a is open (a = 0) what can you say about the whole
circuit? What happens when switch a is closed (a = 1)? Does
the switch b have any effect on your answers?
The truth table for the circuit above shows that a∧ ( a∨ b) = a.
a b a∧ (a∨ b)
0 0 0
0 1 0
1 0 1
1 1 1
This result can be extended to more switches. For example
if a∧ ( a∨ b) = a
then a∧ ( a∨ b ∨ c) = a
and a∧ ( a∨ b ∨ c) ∧ ( a∨ b) ∧ ( b ∨ c) = a∧ ( b ∨ c) .
The last of these expressions is represented by this circuit:
a a b
a b
c b c
which can be replaced by the simplified circuit:
b
a
c
Example
c b
Write down a Boolean expression for this circuit. Simplify the
a b c a
expression and draw the corresponding circuit. a d
179
10. Chapter 11 Boolean Algebra
Solution
a∧ b ∧ ( a∨ c) ∧( b ∨ ( c∧ a) ∨ d)
a∧ ( a∨ c) ∧ b ∧ ( b ∨ ( c∧ a) ∨ d)
Since a∧ ( a∨ c) = a
and b ∧ ( b ∨ ( c∧ a) ∨ d) = b ,
an equivalent expression is a∧ b
and the circuit simplifies to a b
Activity 2 Checking with truth tables
Draw truth tables for the example above to check that
a∧ b ∧ ( a∨ c) ∧ ( b ∨ ( c∧ a) ∨ d) = a∧ b
Exercise 11D
Simplify the following and check your answers by 3. Simplify the following circuit:
drawing up truth tables:
a b b a
1. a∨ ( ~ a∧ b)
2. a∧ [ b ∨ ( a∧ b)] ∧ [ a∨ ( ~ a∧ b)]
c c d d
11.5 Boolean functions
In the same way as algebraic functions describe the relationship
between the domain, (a set of inputs) and the range (a set of
outputs), a Boolean function can be described by a Boolean
expression. For example, if
f ( x1 ,x2 ,x3 ) = x1 ∧ ( ~ x2 ∨ x3 )
then f is the Boolean function and
x1 ∧ ( ~ x2 ∨ x3 )
is the Boolean expression.
180
11. Chapter 11 Boolean Algebra
Example
Draw the truth table for the Boolean function defined as
f ( x1 , x2 , x3 ) = x1 ∧ ( ~ x2 ∨ x3 )
Solution
The inputs and outputs of this Boolean function are shown in the
following table:
x1 x2 x3 f ( x1 ,x2 ,x3 )
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
It is sometimes necessary to form a function from a given truth
table. The method of achieving this is described in the
following example.
Example
For the given truth table, form a Boolean function
a b c f ( a, b, c)
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
Solution
The first stage is to look for the places where f (a,b,c) is 1 and
then link them all together with 'OR's. For example, in the last
row f (a, b, c) = 1 and this is the row in which a, b and c are all
true; i.e. when a∧ b ∧ c= 1 .
181
12. Chapter 11 Boolean Algebra
The output is also 1, (i.e. f ( a, b, c) = 1 ) in the 7th row which
leads to the combination
a∧ b∧ ~ c= 1
Similarly, for the 5th row
a∧ ~ b∧ ~ c= 1
and for the 2nd row
~ a∧ ~ b ∧ c= 1
and for the 1st row
~ a∧ ~ b∧ ~ c= 1
All these combinations are joined using the connective ∨ to give
the Boolean expression
( a∧ b ∧ c) ∨ ( a∧ b∧ ~ c) ∨ ( a∧ ~ b∧ ~ c) ∨ ( ~ a∧ ~ b ∧ c) ∨ ( ~ a∧ ~ b∧ ~ c)
If the values of a, b and c are as shown in the 1st, 2nd, 5th, 7th
and 8th row then the value of f ( a, b, c) = 1 in each case, and the
expression above has a value of 1. Similarly if a, b and c are as
shown in the table for which f ( a, b, c) = 0 then the expression
above has the value of 0.
The Boolean function for the truth table is therefore given by
f ( a, b, c) = ( a∧ b ∧ c) ∨ ( a∧ b∧ ~ c) ∨ ( a∧ ~ b∧ ~ c) ∨ ( ~ a∧ ~ b ∧ c) ∨ ( ~ a∧ ~ b∧ ~ c)
This is called the disjunctive normal form of the function f; the
combinations formed by considering the rows with an output
value of 1 are joined by the disjunctive connective, OR.
Exercise 11E
Find the disjunctive normal form of the Boolean
function for these truth tables:
1. a b f (a, b) 3. x y z f (x, y, z)
0 0 1 0 0 0 1
0 1 0 0 0 1 0
1 0 1 0 1 0 0
1 1 0 0 1 1 0
1 0 0 1
2. a b f (a, b)
1 0 1 0
0 0 1 1 1 0 0
0 1 1 1 1 1 1
1 0 0
1 1 1
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13. Chapter 11 Boolean Algebra
11.6 Minimisation with NAND
gates
When designing combinatorial circuits, efficiency is sought by
minimising the number of gates (or switches) in a circuit. Many
computer circuits make use of another gate called a NAND gate
which is used to replace NOT AND, thereby reducing the
number of gates.
The NAND gate receives inputs a and b and the output is
denoted by a↑ b .
The symbol used is
a
a b
b
The truth table for this is
a b a↑ b
0 0 1
0 1 1
1 0 1
1 1 0
The NAND gate is equivalent to
a a∧b
~(a∧b)
b
Note that, by de Morgan's law, ~ ( a∧ b) =~ a∨ ~ b.
Example
Use NAND gates alone to represent the function
f ( a, b, c, d) = ( a∧ b) ∨ ( c∧ d)
Solution
The use of NAND gates implies that there must be negation so
the function is rewritten using de Morgan's Laws:
( a∧ b) ∨ ( c∧ d) = ~ [( ~ a ∨ ~ b) ∧ ( ~ c ∨ ~ d)]
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14. Chapter 11 Boolean Algebra
The circuit consisting of NAND gates is therefore as follows:
a (~a∨~b)
b
f(a, b, c, d,)
c
d
(~c∨~d)
Example
Design combinatorial circuits to represent (a) the negation
function f ( x) =~ x and (b) the OR function f ( x, y) = x∨ y.
Solution
(a) ~ x = ~ ( x∨ x ) x ~x
= ~ x∧ ~ x
= x↑ x
(b) x∨ y = ~ ( ~ x ∧ ~ y) x
= ~ x↑ ~ y y
(
= x↑x ↑ y↑ y) ( )
Exercise 11F
Design circuits for each of the following using only
NAND gates.
1. a∧ b
2. a∧ ~ b
3. ( ~ a∧ ~ b)∨ ~ b
11.7 Full and half adders
Computers turn all forms of data into binary digits , (0 s and 1
s), called bits, which are manipulated mathematically. For
example the number 7 is represented by the binary code
00000111 (8 bits are used because many computers use binary
digits in groups of 8, for example, ASCII code). This section
describes how binary digits can be added using a series of logic
gates. The basic mathematical operation is addition since
subtraction is the addition of negative numbers,
multiplication is repeated addition,
division is repeated subtraction.
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15. Chapter 11 Boolean Algebra
When you are adding two numbers there are two results to note for
each column; the entry in the answer and the carrying figure.
254
178
432
11
When 4 is added to 8 the result is 12, 2 is noted in the answer and the
digit 1 is carried on to the next column.
When adding the second column the carry digit from the first column
is included, i.e. 5 + 7 + 1, giving yet another digit to carry on to the
next column.
Half adder
The half adder is capable of dealing with two inputs, i.e. it can only
add two bits, each bit being either 1 or 0.
a b Carry bit Answer bit
a
0 0 0 0 b
0 1 0 1
1 0 0 1
1 1 1 0 Carry Answer
bit bit
Activity 3 Designing the half adder circuit
The next stage is to design a circuit which will give the results shown
in the table above.
The first part of the circuit is shown opposite; complete the rest of the a
Carry bit
circuit which can be done with a NOT gate, an OR gate and an AND b
?
gate to give the answer bit. ?
Full Adder
A half adder can only add two bits; a full adder circuit is capable of
including the carry bit in the addition and therefore has three inputs.
1 1 1 111
1 1 0 110 +
carry 1 carry 0 carry
Full Full Full 0
adder adder adder
1 1 0 1 1101
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16. Chapter 11 Boolean Algebra
Activity 4 Full adder truth table
Complete this truth table for the full adder.
Inputs
a b c Carry bit Answer bit a b
0 0 0 0 0
Full
0 0 1 0 1 carry bit c
adder
0 1 0
etc ↓ answer bit
1 1 1
The circuit for a full adder is, in effect, a combination of two a
half adders. b
carry bit
half adder
If you think about it, the carry bit of the full adder must be 1 if
either of the two half adders shown gives a carry bit of 1 (and in
fact it is impossible for both those half adders to give a carry bit carry bit c
of 1 at the same time). Therefore the two carry bits from the half adder
half adders are fed into an OR gate to give an output equal to the
carry bit of the full adder.
carry bit answer bit
The circuit for a full adder consists, therefore, of two half adders
with the carry bits feeding into an OR gate as follows:
a
b carry bit
c
answer bit
The dotted lines enclose the two half adders with the whole
circuit representing a full adder.
Activity 5 NAND half adder
Draw up a circuit to represent a half adder using only NAND
gates.
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17. Chapter 11 Boolean Algebra
11.8 Miscellaneous Exercises
1. Use logic gates to represent these expressions 5. Draw the simplest switching circuit
and draw up the corresponding truth tables: represented by this table:
(a) ~ [( a∧ b) ∨ c] P Q R S Output
(b) ( a∧ b)∨ ~ c 1 1 1 1 1
0 1 1 1 1
(c) ~ c∧ [( a∧ b)∨ ~ ( a∧ c)] 1 1 0 1 1
2. Write down the Boolean expression for each of 0 1 0 1 1
these circuits:
(a) 6. A burglar alarm for a house is controlled by a
p
switch. When the switch is on, the alarm sounds
q
if either the front or back doors or both doors are
opened. The alarm will not work if the switch is
r
off. Design a circuit of logic gates for the alarm
and draw up the corresponding truth table.
(b) p *7. A gallery displaying a famous diamond uses a
q special Security Unit to protect access to the
Display Room (D). The diagram below shows
the layout of the system.
r
D D
A B S S
(c) p
Y Z C
q
S S
r D D
The display cabinet (C) is surrounded by a screen
of electronic eyes (S).
3. Write down the Boolean expressions for these Access to the display room is through doors (Y),
circuits : (Z). Boxes (A), (B) are used in the system. The
(a) A following persons are involved in the system :
B Manager,
C C Deputy Manager,
Chief Security Officer.
~B The Display Room is opened as follows :
The Unit must be activated at box A.
(b) R
P Door (Y) is opened by any two of the
P
Q ~P above persons at box A.
S
~Q Box B is activated by the Manager and
Deputy Manager together.
The screen (S) is activated by the Chief
(c) A D
Security Officer alone at box B only.
C
Door (Z) can only be opened once the
B E
screen (S) is activated.
Draw a circuit of logic gates required inside the
4. Draw switching circuits for these Boolean Unit to operate it. Ensure your diagram is
expressions: documented.
(a) ( B ∧ C ) ∨ (C ∧ A ) ∨ ( A ∧ B ) (AEB)
(b) [ A ∧ (( B∧ ~ C ) ∨ ( ~ B ∧ C ))] ∨ ( ~ A ∧ B ∧ C )
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18. Chapter 11 Boolean Algebra
8. Simplify the following expressions and check 10. Design a circuit representing ~ a∨ b using
your answer by drawing up truth tables. NAND gates.
(a) ( a∧ b ∧ c) ∨ ( ~ a∧ b ∧ c) *11. Write a computer program or use a spreadsheet
that outputs a truth table for a given Boolean
(b) a∨ ( ~ a∧ b ∧ c) ∨ ( ~ a∧ b∧ ~ c) expression.
12. (a) Establish a truth table for the Boolean
(c) ( p ∧ q) ∨ ( ~ p∨ ~ q) ∧ ( r∨ s)
( )
function f x1 , x2, x3 = ( ~ x1 ∨ x2 ) ∧ ( ~ x3 ∨ x2 ) .
9. Find the disjunctive normal form of this function; (b) Design a circuit using as few AND, OR and
simplify and draw the combinatorial circuit. NOT gates as possible to model the function
a b c f ( a,b,c) in (a).
0 0 0 0 13. (a) Show, by constructing truth tables or
otherwise, that the following statements are
0 0 1 0 equivalent.
0 1 0 0
p ⇒ q and ~ ( ~ ( p ∧ q) ∧ p) .
0 1 1 1
1 0 0 0 (b) With the aid of (a), or otherwise, construct
combinatorial circuits consisting only of
1 0 1 1 NAND gates to represent the functions
1 1 0 1
f ( x, y) = x ⇒ y and g( x, y) =~ ( x ⇔ y) .
1 1 1 1
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