This document summarizes a workshop on applications of derivatives in telecommunications engineering. It begins with introductions to derivatives, optimization of functions, and instantaneous rates of change. It then outlines objectives and theoretical foundations, discussing concepts like potential difference, Faraday's law, Lenz's law, and mutual inductance. Examples are given of derivatives in circuit analysis, electric current, power, and harmonic waves. The document provides two examples of using derivatives to find voltage and energy stored in an inductor. In summary, it examines the important role that calculus, and specifically derivatives, play in understanding and applying concepts in telecommunications.
This document presents a new technique for high voltage power transmission using a beam of conductors. It describes developing a mathematical model and numerical simulation to analyze the complex dynamics when such a beam is subjected to electromagnetic forces during a short circuit. These can cause the sub-conductors in the beam to choke and impact each other. The summary develops a finite element model incorporating the electrical connections between sub-conductors and nonlinear contact mechanics during impacts. Software is developed using this model to simulate beam structures and validate results against experimental data.
This document provides instructions for navigating a presentation on electric circuits. It begins with an overview slide and table of contents. It then covers topics like schematic diagrams, components of electric circuits, and calculating equivalent resistances and currents in series and parallel circuits. Examples are provided, such as calculating the equivalent resistance and current in a complex circuit.
NanoScale TiO2 based Memory Storage Circuit Element:- MemristorAM Publications
This document discusses the memristor, a fourth fundamental circuit element predicted by Leon Chua in 1971. In 2008, HP Laboratories created the first physical model of a memristor using a thin film of titanium dioxide sandwiched between platinum electrodes. This memristor exhibits variable resistance depending on the charge that has passed through it, fulfilling Chua's prediction. While this initial design had limitations like low speed and heat dissipation, memristors show potential for applications like non-volatile memory, neuromorphic computing, and new circuit designs. Further research is still needed to improve memristor models and address challenges for practical applications.
The document provides instructions for viewing a presentation in slideshow mode using a computer. It explains how to advance slides, access resources and lessons from the menu, and exit the slideshow. The table of contents lists the sections and objectives covered in an electric forces and fields chapter.
Verifying Faraday’s Law of Induction using an electric generatorVijayan thanasekaran
1) An experiment was conducted to verify Faraday's law of induction using a hand-crank generator with different coil configurations.
2) The results showed that increasing the number of coil loops or the rotation speed increased the induced electromotive force (EMF), as predicted.
3) However, inconsistencies in measuring the magnetic field introduced errors, so the data was not considered accurate enough to conclusively prove the theory. Improving the magnetic field measurements would strengthen future experiments.
This document summarizes a method for calculating the sensitivity matrix that defines the linear relationship between circuit parameters and poles/response of an RLC network. The sensitivity matrix enables efficient statistical analysis and yield predictions. It is obtained by taking derivatives of the poles and transfer function, which are calculated from the eigenvalues and eigenvectors of the network's state equation. An example RLC circuit demonstrates calculating the sensitivity matrix and using it to predict yield based on Monte Carlo simulations.
This document provides instructions for viewing a presentation as a slideshow and navigating between its slides and sections. It can be viewed as a slideshow by selecting "View" and "Slide Show" from the menu bar. Clicking the right arrow or space bar advances the slides. Clicking on resources from the resources slide or lessons from the Chapter menu screen goes directly to those sections. The Esc key exits the slideshow.
Finite Element Method for Designing and Analysis of the Transformer – A Retro...idescitation
Finite Element Analysis (FEA) using Finite Element Method (FEM) was
developed over 70 years to solve the complex elasticity and structural analysis problem in
civil and aeronautical engineering. Application of FEA is being expanded to simulation in
electrical engineering also to solve the complex design problems. The circuit theory models
for designing transformers are not much accurate in determining the transformer
parameters such as winding impedance, leakage inductance, hot spot temperature etc. The
physical realization of these parameters is needed on a prototype unit. The finite element
method can play a vital role in deriving these parameters without any physical verification.
An effort has been made in this paper to show the effectiveness of finite element method in
determining the above said parameters while designing the transformers - both oil cooled as
well as dry type - for power and distribution sectors as well as to analyze and detect the
internal faults in the transformer.
This document presents a new technique for high voltage power transmission using a beam of conductors. It describes developing a mathematical model and numerical simulation to analyze the complex dynamics when such a beam is subjected to electromagnetic forces during a short circuit. These can cause the sub-conductors in the beam to choke and impact each other. The summary develops a finite element model incorporating the electrical connections between sub-conductors and nonlinear contact mechanics during impacts. Software is developed using this model to simulate beam structures and validate results against experimental data.
This document provides instructions for navigating a presentation on electric circuits. It begins with an overview slide and table of contents. It then covers topics like schematic diagrams, components of electric circuits, and calculating equivalent resistances and currents in series and parallel circuits. Examples are provided, such as calculating the equivalent resistance and current in a complex circuit.
NanoScale TiO2 based Memory Storage Circuit Element:- MemristorAM Publications
This document discusses the memristor, a fourth fundamental circuit element predicted by Leon Chua in 1971. In 2008, HP Laboratories created the first physical model of a memristor using a thin film of titanium dioxide sandwiched between platinum electrodes. This memristor exhibits variable resistance depending on the charge that has passed through it, fulfilling Chua's prediction. While this initial design had limitations like low speed and heat dissipation, memristors show potential for applications like non-volatile memory, neuromorphic computing, and new circuit designs. Further research is still needed to improve memristor models and address challenges for practical applications.
The document provides instructions for viewing a presentation in slideshow mode using a computer. It explains how to advance slides, access resources and lessons from the menu, and exit the slideshow. The table of contents lists the sections and objectives covered in an electric forces and fields chapter.
Verifying Faraday’s Law of Induction using an electric generatorVijayan thanasekaran
1) An experiment was conducted to verify Faraday's law of induction using a hand-crank generator with different coil configurations.
2) The results showed that increasing the number of coil loops or the rotation speed increased the induced electromotive force (EMF), as predicted.
3) However, inconsistencies in measuring the magnetic field introduced errors, so the data was not considered accurate enough to conclusively prove the theory. Improving the magnetic field measurements would strengthen future experiments.
This document summarizes a method for calculating the sensitivity matrix that defines the linear relationship between circuit parameters and poles/response of an RLC network. The sensitivity matrix enables efficient statistical analysis and yield predictions. It is obtained by taking derivatives of the poles and transfer function, which are calculated from the eigenvalues and eigenvectors of the network's state equation. An example RLC circuit demonstrates calculating the sensitivity matrix and using it to predict yield based on Monte Carlo simulations.
This document provides instructions for viewing a presentation as a slideshow and navigating between its slides and sections. It can be viewed as a slideshow by selecting "View" and "Slide Show" from the menu bar. Clicking the right arrow or space bar advances the slides. Clicking on resources from the resources slide or lessons from the Chapter menu screen goes directly to those sections. The Esc key exits the slideshow.
Finite Element Method for Designing and Analysis of the Transformer – A Retro...idescitation
Finite Element Analysis (FEA) using Finite Element Method (FEM) was
developed over 70 years to solve the complex elasticity and structural analysis problem in
civil and aeronautical engineering. Application of FEA is being expanded to simulation in
electrical engineering also to solve the complex design problems. The circuit theory models
for designing transformers are not much accurate in determining the transformer
parameters such as winding impedance, leakage inductance, hot spot temperature etc. The
physical realization of these parameters is needed on a prototype unit. The finite element
method can play a vital role in deriving these parameters without any physical verification.
An effort has been made in this paper to show the effectiveness of finite element method in
determining the above said parameters while designing the transformers - both oil cooled as
well as dry type - for power and distribution sectors as well as to analyze and detect the
internal faults in the transformer.
This document contains a 30 question physics exam with multiple choice answers. The exam covers topics in mechanics, electromagnetism, optics, modern physics, and waves. For each question, there is a short problem statement followed by 4 possible answer choices. The full solutions to the exam questions are available online at the given website.
1. The document discusses the derivation of the permittivity constant (ε0) and permeability constant (μ0) from first principles, which has never been done before.
2. It shows that ε0 and μ0 can be derived from dimensional analysis and have dimensions of meters per second, corresponding to the speed of light.
3. The product of ε0 and μ0 is inversely proportional to the square of the speed of light, allowing the speed of light to be calculated from Maxwell's equations.
This document provides an overview of key concepts in electrostatics, including:
- Electric charge can be positive, negative, or neutral depending on the balance of protons and electrons in an object. Charging methods include contact and induction.
- Coulomb's law quantitatively describes the electric force between two charges. It is analogous to the gravitational force formula.
- Electric fields are described by field lines and show the influence of electric forces. The strength of an electric field can be determined from its effect on a small test charge.
- Electric potential, or voltage, describes the electric potential energy per unit charge. It is measured in volts. Equipotential lines represent positions of equal electric potential
This document discusses Gauss's law and related electromagnetic theory concepts taught in a course. It provides:
1) An overview of Gauss's law, which states that the total outward electric flux through a closed surface is equal to the total charge enclosed divided by the permittivity of the medium.
2) Derivations and proofs of Gauss's law using calculus theorems like divergence theorem.
3) Applications of Gauss's law to problems involving charge distributions and electric field calculations.
4) Discussions of related concepts like Gauss's law in polarized media, current continuity equation, relaxation time, and Poisson's equation.
5) Example problems demonstrating the application of these electromagnetic theory principles.
This document summarizes a study on wireless power transfer using induction technique. It describes how electrical power is converted to magnetic energy in a transmitter coil, generating a time-varying magnetic field. When a receiver coil is placed within this field, the magnetic energy is reverted back to electrical energy to power a load without the use of wires. The document outlines the circuit designs for the transmitter and receiver, and analyzes the relationship between current, magnetic flux, and power transfer through mathematical equations and simulation results. Experimental data shows different voltages induced in receiver coils with varying numbers of turns. The summary concludes that induction-based wireless power transfer over short distances is possible by controlling current harmonics to reduce power losses.
The document summarizes key concepts from electromagnetism:
(1) It discusses Faraday's law of induction, which states that a changing magnetic field induces a circulating electric field.
(2) It explains Maxwell's modification of Ampere's law to include displacement current, which resolved inconsistencies between Ampere's law and the continuity equation. This led to the prediction of electromagnetic waves.
(3) It compares conduction and displacement currents, noting that displacement current dominates at high frequencies in good insulators, allowing electromagnetic waves to propagate through space.
This document provides an overview of small scale renewable energy systems focusing on photovoltaic cells. It discusses the cross-section and configuration of solar panels, the ideal and single-diode models for photovoltaic cells, and how to characterize cells using I-V curves to determine key parameters like short circuit current, open circuit voltage, maximum power point, fill factor, and efficiency. It also covers temperature effects and mitigation of shading issues through the use of bypass diodes or microinverters in solar panel configurations.
This document discusses transmission line propagation coefficients including reflection coefficient and transmission coefficient. It defines the reflection coefficient as the ratio of reflected to incident voltage or current. Reflection and transmission coefficients are derived for a transmission line terminated by a load impedance. Standing wave patterns on transmission lines are also analyzed. Key properties of standing waves include maximum and minimum voltages occurring at intervals of half wavelength and voltages/currents being 90 degrees out of phase.
Simulation Model solves exact the Enigma named Generating high Voltages and h...IJERA Editor
Simulation model of Tesla coil has been successfully completed, and has been verified the procedure and functioning. The literature and documentation for the model were taken from the rich sources, especially the copies of Tesla patents. The oscillating system‟s electrical scheme consists of the voltage supply 220/50 Hz, Fe transformer, capacitor and belonging chosen electrical components, the air gap in the primary Tesla coil (air transformer) and spark gap in the exit of the coil. The investigation of the oscillating process Tesla coil‟s system using the simulation model in MATLAB & SIMULINK have given the exact solution the enigma named the generating high voltage and high frequency the Tesla‟s coil. The inductance voltage from the spark current in the primary (coil) with its high voltage impulse excites the oscillating series circuit Ce-L3-R3 on the secondary of the air transformer to its own damped oscillations.
Ch10 - potential difference and electric potential energycpphysics
Electric potential energy and gravitational potential energy both increase as work is done to overcome a field and raise an object or charge. Voltage measures electric potential energy per unit charge and represents how much energy a charge will gain or lose when moving through a potential difference. It is defined as the change in electric potential energy divided by the charge and can be measured in joules per coulomb (volts).
This document discusses the topic of electrostatics and dielectrics in the Electromagnetic Theory course. It defines a dielectric as a material where charge displacement occurs in an external electric field rather than free motion of charges. Dielectrics are classified as polar or nonpolar depending on whether they have a permanent dipole moment. The types of polarization in dielectrics are electronic, orientation, and ionic polarization. Key concepts discussed include polarization density, polarization charge density, the relationship between polarization and electric field through susceptibility and relative permittivity, atomic polarizability, and the relationship between polarization and electric displacement. An example problem calculates the polarization given the electric displacement.
The document provides instructions for navigating a presentation on atomic physics and quantum mechanics. It begins with directions for viewing the presentation as a slideshow and advancing through it. The rest of the document consists of sections from the presentation covering topics like quantization of energy, models of the atom including Bohr's model, quantum mechanics, and the uncertainty principle. Key points, equations, and examples are included for each topic.
This document presents a new method for analyzing transient voltage distributions in transformer windings. It models the electric, magnetic, and current fields as equivalent circuits consisting of interconnected electric and magnetic networks.
The electric network models the voltage distribution within each winding section using resistances and emfs derived from the geometry. The magnetic network models the flux paths using reluctances, which is then converted to an equivalent inductance network. Capacitances model the electric fields within and between sections.
The electric, magnetic, and capacitance networks are coupled through ideal transformers to form a single overall mathematical model without mutual inductances. The model can analyze transient voltages inside the windings resulting from any applied terminal overvoltage waveform. An example application
The document discusses transmission line impedance and input impedance. It defines characteristic impedance as the ratio of voltage to current waves travelling along a transmission line. It provides expressions for characteristic impedance in terms of line parameters R, L, G, C. It then derives expressions for input impedance of open circuit, short circuit, matched and mismatched lossless transmission lines. It shows that input impedance is capacitive for a short open circuit line and inductive for a short circuit line.
This document contains lecture notes from a course on electromagnetic theory taught by Arpan Deyasi. It covers topics on magnetic scalar and vector potentials, including their definitions, properties, and applications to problems involving magnetic fields generated by currents. The notes provide the mathematical relationships between magnetic fields and potentials, and work through examples such as calculating the potentials for an infinite solenoid and current-carrying wire.
Learning Objectives
Define electric charge, and describe how the two types of charge interact.
Desribe three common situations that generate static electricity. State the law of conservation of charge.
Describe three methods for charging an object.
State Coulomb’s law
Describe an electric field diagram of a positive point charge; of a negative point charge with twice the magnitude of positive charge
Draw the electric field lines between two points of the same charge; between two points of opposite charge.
Thank you So much
This document is a report on the applications of derivatives in the career of Electronic and Automation Engineering. It contains an introduction explaining how derivatives are useful for measuring rates of change in electrical situations. It then lists the objectives of introducing derivatives and their graphical interpretation. The theoretical foundations section discusses how derivatives indicate instantaneous rates of change and how slope represents rate of change. The development section contains 4 exercises applying derivatives to calculate current, resistance, voltage and inductance in various electric circuits. It concludes that derivatives are important tools in electrical research and many other fields.
This document summarizes a student project that used the finite difference method and the Laplace equation to model the electrostatic properties of a non-symmetrical surface. The student created an Excel model of an infinitely long magnetic strip surrounded by a conducting box. The model was used to calculate potential, electric field, surface charge density, and capacitance per unit length for different node amounts. The results showed higher potential and charge density near the strip, and flux lines directed towards the box edges rather than another plate. Overall, the model behaved similarly to a parallel-plate capacitor except for non-symmetrical flux lines.
Taller grupal 2_aplicacion de la derivada en la ingeniera electrónica y autom...JHANDRYALCIVARGUAJAL
The document is a report in Spanish for a Calculus course. It discusses applications of the derivative in the career of electronics and automation. It contains 3 problems solved using concepts of maxima, minima, and the first and second derivatives. The problems involve finding the maximum power output of a circuit, determining the maximum net resistance of a parallel circuit, and calculating the maximum error in the equivalent resistance of a parallel circuit based on measurement errors.
This document contains a 30 question physics exam with multiple choice answers. The exam covers topics in mechanics, electromagnetism, optics, modern physics, and waves. For each question, there is a short problem statement followed by 4 possible answer choices. The full solutions to the exam questions are available online at the given website.
1. The document discusses the derivation of the permittivity constant (ε0) and permeability constant (μ0) from first principles, which has never been done before.
2. It shows that ε0 and μ0 can be derived from dimensional analysis and have dimensions of meters per second, corresponding to the speed of light.
3. The product of ε0 and μ0 is inversely proportional to the square of the speed of light, allowing the speed of light to be calculated from Maxwell's equations.
This document provides an overview of key concepts in electrostatics, including:
- Electric charge can be positive, negative, or neutral depending on the balance of protons and electrons in an object. Charging methods include contact and induction.
- Coulomb's law quantitatively describes the electric force between two charges. It is analogous to the gravitational force formula.
- Electric fields are described by field lines and show the influence of electric forces. The strength of an electric field can be determined from its effect on a small test charge.
- Electric potential, or voltage, describes the electric potential energy per unit charge. It is measured in volts. Equipotential lines represent positions of equal electric potential
This document discusses Gauss's law and related electromagnetic theory concepts taught in a course. It provides:
1) An overview of Gauss's law, which states that the total outward electric flux through a closed surface is equal to the total charge enclosed divided by the permittivity of the medium.
2) Derivations and proofs of Gauss's law using calculus theorems like divergence theorem.
3) Applications of Gauss's law to problems involving charge distributions and electric field calculations.
4) Discussions of related concepts like Gauss's law in polarized media, current continuity equation, relaxation time, and Poisson's equation.
5) Example problems demonstrating the application of these electromagnetic theory principles.
This document summarizes a study on wireless power transfer using induction technique. It describes how electrical power is converted to magnetic energy in a transmitter coil, generating a time-varying magnetic field. When a receiver coil is placed within this field, the magnetic energy is reverted back to electrical energy to power a load without the use of wires. The document outlines the circuit designs for the transmitter and receiver, and analyzes the relationship between current, magnetic flux, and power transfer through mathematical equations and simulation results. Experimental data shows different voltages induced in receiver coils with varying numbers of turns. The summary concludes that induction-based wireless power transfer over short distances is possible by controlling current harmonics to reduce power losses.
The document summarizes key concepts from electromagnetism:
(1) It discusses Faraday's law of induction, which states that a changing magnetic field induces a circulating electric field.
(2) It explains Maxwell's modification of Ampere's law to include displacement current, which resolved inconsistencies between Ampere's law and the continuity equation. This led to the prediction of electromagnetic waves.
(3) It compares conduction and displacement currents, noting that displacement current dominates at high frequencies in good insulators, allowing electromagnetic waves to propagate through space.
This document provides an overview of small scale renewable energy systems focusing on photovoltaic cells. It discusses the cross-section and configuration of solar panels, the ideal and single-diode models for photovoltaic cells, and how to characterize cells using I-V curves to determine key parameters like short circuit current, open circuit voltage, maximum power point, fill factor, and efficiency. It also covers temperature effects and mitigation of shading issues through the use of bypass diodes or microinverters in solar panel configurations.
This document discusses transmission line propagation coefficients including reflection coefficient and transmission coefficient. It defines the reflection coefficient as the ratio of reflected to incident voltage or current. Reflection and transmission coefficients are derived for a transmission line terminated by a load impedance. Standing wave patterns on transmission lines are also analyzed. Key properties of standing waves include maximum and minimum voltages occurring at intervals of half wavelength and voltages/currents being 90 degrees out of phase.
Simulation Model solves exact the Enigma named Generating high Voltages and h...IJERA Editor
Simulation model of Tesla coil has been successfully completed, and has been verified the procedure and functioning. The literature and documentation for the model were taken from the rich sources, especially the copies of Tesla patents. The oscillating system‟s electrical scheme consists of the voltage supply 220/50 Hz, Fe transformer, capacitor and belonging chosen electrical components, the air gap in the primary Tesla coil (air transformer) and spark gap in the exit of the coil. The investigation of the oscillating process Tesla coil‟s system using the simulation model in MATLAB & SIMULINK have given the exact solution the enigma named the generating high voltage and high frequency the Tesla‟s coil. The inductance voltage from the spark current in the primary (coil) with its high voltage impulse excites the oscillating series circuit Ce-L3-R3 on the secondary of the air transformer to its own damped oscillations.
Ch10 - potential difference and electric potential energycpphysics
Electric potential energy and gravitational potential energy both increase as work is done to overcome a field and raise an object or charge. Voltage measures electric potential energy per unit charge and represents how much energy a charge will gain or lose when moving through a potential difference. It is defined as the change in electric potential energy divided by the charge and can be measured in joules per coulomb (volts).
This document discusses the topic of electrostatics and dielectrics in the Electromagnetic Theory course. It defines a dielectric as a material where charge displacement occurs in an external electric field rather than free motion of charges. Dielectrics are classified as polar or nonpolar depending on whether they have a permanent dipole moment. The types of polarization in dielectrics are electronic, orientation, and ionic polarization. Key concepts discussed include polarization density, polarization charge density, the relationship between polarization and electric field through susceptibility and relative permittivity, atomic polarizability, and the relationship between polarization and electric displacement. An example problem calculates the polarization given the electric displacement.
The document provides instructions for navigating a presentation on atomic physics and quantum mechanics. It begins with directions for viewing the presentation as a slideshow and advancing through it. The rest of the document consists of sections from the presentation covering topics like quantization of energy, models of the atom including Bohr's model, quantum mechanics, and the uncertainty principle. Key points, equations, and examples are included for each topic.
This document presents a new method for analyzing transient voltage distributions in transformer windings. It models the electric, magnetic, and current fields as equivalent circuits consisting of interconnected electric and magnetic networks.
The electric network models the voltage distribution within each winding section using resistances and emfs derived from the geometry. The magnetic network models the flux paths using reluctances, which is then converted to an equivalent inductance network. Capacitances model the electric fields within and between sections.
The electric, magnetic, and capacitance networks are coupled through ideal transformers to form a single overall mathematical model without mutual inductances. The model can analyze transient voltages inside the windings resulting from any applied terminal overvoltage waveform. An example application
The document discusses transmission line impedance and input impedance. It defines characteristic impedance as the ratio of voltage to current waves travelling along a transmission line. It provides expressions for characteristic impedance in terms of line parameters R, L, G, C. It then derives expressions for input impedance of open circuit, short circuit, matched and mismatched lossless transmission lines. It shows that input impedance is capacitive for a short open circuit line and inductive for a short circuit line.
This document contains lecture notes from a course on electromagnetic theory taught by Arpan Deyasi. It covers topics on magnetic scalar and vector potentials, including their definitions, properties, and applications to problems involving magnetic fields generated by currents. The notes provide the mathematical relationships between magnetic fields and potentials, and work through examples such as calculating the potentials for an infinite solenoid and current-carrying wire.
Learning Objectives
Define electric charge, and describe how the two types of charge interact.
Desribe three common situations that generate static electricity. State the law of conservation of charge.
Describe three methods for charging an object.
State Coulomb’s law
Describe an electric field diagram of a positive point charge; of a negative point charge with twice the magnitude of positive charge
Draw the electric field lines between two points of the same charge; between two points of opposite charge.
Thank you So much
This document is a report on the applications of derivatives in the career of Electronic and Automation Engineering. It contains an introduction explaining how derivatives are useful for measuring rates of change in electrical situations. It then lists the objectives of introducing derivatives and their graphical interpretation. The theoretical foundations section discusses how derivatives indicate instantaneous rates of change and how slope represents rate of change. The development section contains 4 exercises applying derivatives to calculate current, resistance, voltage and inductance in various electric circuits. It concludes that derivatives are important tools in electrical research and many other fields.
This document summarizes a student project that used the finite difference method and the Laplace equation to model the electrostatic properties of a non-symmetrical surface. The student created an Excel model of an infinitely long magnetic strip surrounded by a conducting box. The model was used to calculate potential, electric field, surface charge density, and capacitance per unit length for different node amounts. The results showed higher potential and charge density near the strip, and flux lines directed towards the box edges rather than another plate. Overall, the model behaved similarly to a parallel-plate capacitor except for non-symmetrical flux lines.
Taller grupal 2_aplicacion de la derivada en la ingeniera electrónica y autom...JHANDRYALCIVARGUAJAL
The document is a report in Spanish for a Calculus course. It discusses applications of the derivative in the career of electronics and automation. It contains 3 problems solved using concepts of maxima, minima, and the first and second derivatives. The problems involve finding the maximum power output of a circuit, determining the maximum net resistance of a parallel circuit, and calculating the maximum error in the equivalent resistance of a parallel circuit based on measurement errors.
Aplicación de la derivada en la carrea de telecomunicacionesENelson3
This document discusses applications of derivatives in telecommunications. It presents two examples of optimization problems solved using derivatives that are relevant to telecommunications. The first problem involves finding the dimensions of a copper bar that maximize its area for use as an antenna support. The second problem determines the dimensions of a copper box that maximize its volume for storing internet cabling. Both problems demonstrate how to set up and solve optimization problems using derivatives.
APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...WILIAMMAURICIOCAHUAT1
El cálculo diferencial proporciona información sobre el comportamiento de las funciones
matemáticas. Todos estos problemas están incluidos en el alcance de la optimización de funciones y pueden resolverse aplicando cálculo
Problemas de aplicación ley de ohm y ley de wattKatheryncaicedo1
This document discusses color coding, breadboards, and exercises applying Ohm's law and Watt's law. It begins by explaining color coding for electronic components like resistors and how to use the color bands to determine a resistor's value. It then defines and describes the purpose and structure of a breadboard for prototyping circuits. The document concludes with examples solving for current, voltage, resistance, and power in various circuit scenarios.
The document describes an electronics lab course that focuses on antenna design and implementation of digital communication systems using MATLAB/Simulink. The course covers topics like using a slotted line to determine unknown frequency and reflection coefficients, investigating yagi antennas by examining driven elements, reflectors and directors, and studying the effects of conductor thickness on bandwidth. It also includes implementing digital modulation and coding techniques like PCM, PSK and FSK using MATLAB. Students use antenna hardware and software to study properties of dipole antennas and parasitic elements in yagi antenna systems.
The document describes an electronics lab course that focuses on antenna design and implementation of digital communication systems using MATLAB/Simulink. The course covers topics like using a slotted line to determine unknown frequency and reflection coefficients, investigating yagi antennas by examining driven elements and parasitic directors and reflectors, and implementing modulation/demodulation techniques like PCM, PPM, and various digital coding formats in software simulations. Students will explore how antenna performance is affected by parameters like element thickness and stacking/baying configurations. The course aims to help students understand fundamental antenna properties and simulate key communication systems.
Analysis of Simple Maglev System using SimulinkArslan Guzel
The document is a PowerPoint presentation analyzing a simple maglev system using Simulink. It includes:
- An introduction to maglev systems and their applications.
- Circuit diagrams and explanations of the position sensor, electromagnet actuator, and other system components.
- Derivations of the system's mathematical model and equations in state space form for both nonlinear and linear controller approaches.
- Descriptions of the Simulink models created to simulate the system, including blocks for the generalized system, nonlinear controller, and determining voltage and current.
- Analysis of simulation results for two air gap scenarios, comparing graphs of position, velocity, acceleration, and other signals between the scenarios.
Its a simple project for class 12th science students. This project is collected from various sources including Google, Wikipedia and Slideshare, Youtube and many more.
Leakage Current Paths in PV Transformer-Less Single-Phase Inverter Topology a...IAES-IJPEDS
This document summarizes a research paper on leakage current paths in photovoltaic transformer-less single-phase inverter topologies and mitigation of leakage currents through pulse width modulation switching. It begins with an introduction to renewable energy sources and solar power generation. It then describes the proposed transformer-less inverter topology, identifying potential leakage current paths. Simulation results are presented comparing leakage currents using different duty cycles and decoupling techniques. The paper analyzes AC and DC decoupling topologies at 50% and 75% duty cycles, finding that PWM switching can effectively reduce leakage currents compared to non-PWM conditions.
This lesson reviews key concepts about magnetism, including magnetic fields, forces, and induction. It provides examples of calculating magnetic fields near a straight wire, in the center of a coil, and on the axis of a solenoid. Examples are also given for working problems involving magnetic forces on moving charges and current-carrying wires. The lesson finishes by reviewing Faraday's Law of induction and Lenz's Law, providing examples of applying these concepts. Students are directed to practice problems to reinforce their understanding of these magnetic topics before moving on to a unit on optics.
This document discusses using wavelet transforms to identify and classify faults in underground high voltage cables. It begins by introducing wavelet transforms and their advantages over Fourier transforms for analyzing non-stationary signals like those seen in power cables. The document then provides more details on discrete wavelet transforms and multi-resolution analysis. It describes simulating different fault conditions in underground cables using MATLAB and analyzing the resulting voltage signals using wavelet transforms. Key wavelet coefficients are examined to detect and classify the type of fault (e.g. line-to-ground) and its location along the cable. The results demonstrate the proposed wavelet-based technique can accurately identify and classify faults.
Some Research Notes on developing a Hybrid UAV for space industrialization. Goal is to develop profitable routes, infrastructure and vehicles to harvest power from Venus, Mercury and Sun and transmit power to interests
This document provides an overview of Module 1 of General Physics 2 for Quarter 3. It includes the development team for the module and the key learning competencies, which cover describing charging by rubbing and induction, explaining electron transfer in charging by rubbing, and calculating electric force and field using Coulomb's law. The document then provides introductions to the basic concepts of electrostatics, including how bodies get charged through rubbing or induction. It also explains Coulomb's law and how to calculate electric force and field. Sample problems are provided as examples. Later sections discuss related topics like electric flux and Gauss's law, with more sample problems. A set of activities for students is also included.
This document provides an introduction to three-phase circuits and power. Some key points:
- Real power (P) is the average power supplied to a load over time and is proportional to voltage (V), current (I), and the cosine of the phase angle between them. Reactive power (Q) represents energy that oscillates in inductors and capacitors.
- Apparent power (S) is defined as V×I. Real and reactive power can be expressed as S=P+jQ using phasor notation.
- Power factor is the ratio of real power to apparent power. It indicates how effectively the voltage and current work together to transfer power. A power factor of 1 indicates a purely
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
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TIME DIVISION MULTIPLEXING TECHNIQUE FOR COMMUNICATION SYSTEMHODECEDSIET
Time Division Multiplexing (TDM) is a method of transmitting multiple signals over a single communication channel by dividing the signal into many segments, each having a very short duration of time. These time slots are then allocated to different data streams, allowing multiple signals to share the same transmission medium efficiently. TDM is widely used in telecommunications and data communication systems.
### How TDM Works
1. **Time Slots Allocation**: The core principle of TDM is to assign distinct time slots to each signal. During each time slot, the respective signal is transmitted, and then the process repeats cyclically. For example, if there are four signals to be transmitted, the TDM cycle will divide time into four slots, each assigned to one signal.
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3. **Frame Structure**: TDM data is organized into frames, where each frame consists of a set of time slots. Each frame is repeated at regular intervals, ensuring continuous transmission of data streams. The frame structure helps in managing the data streams and maintaining the synchronization between the transmitter and receiver.
4. **Multiplexer and Demultiplexer**: At the transmitting end, a multiplexer combines multiple input signals into a single composite signal by assigning each signal to a specific time slot. At the receiving end, a demultiplexer separates the composite signal back into individual signals based on their respective time slots.
### Types of TDM
1. **Synchronous TDM**: In synchronous TDM, time slots are pre-assigned to each signal, regardless of whether the signal has data to transmit or not. This can lead to inefficiencies if some time slots remain empty due to the absence of data.
2. **Asynchronous TDM (or Statistical TDM)**: Asynchronous TDM addresses the inefficiencies of synchronous TDM by allocating time slots dynamically based on the presence of data. Time slots are assigned only when there is data to transmit, which optimizes the use of the communication channel.
### Applications of TDM
- **Telecommunications**: TDM is extensively used in telecommunication systems, such as in T1 and E1 lines, where multiple telephone calls are transmitted over a single line by assigning each call to a specific time slot.
- **Digital Audio and Video Broadcasting**: TDM is used in broadcasting systems to transmit multiple audio or video streams over a single channel, ensuring efficient use of bandwidth.
- **Computer Networks**: TDM is used in network protocols and systems to manage the transmission of data from multiple sources over a single network medium.
### Advantages of TDM
- **Efficient Use of Bandwidth**: TDM all
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ML Based Model for NIDS MSc Updated Presentation.v2.pptx
La derivada en las telecomunicaciones
1. DEPARTAMENTO DE CIENCIAS EXACTAS
CÁLCULO DIFERENCIAL E INTEGRAL
Grupo N.º: 5
Nombres:
•ARIAS CAMPOS, Marco Antonio
•CHIU CAYAMBE, Vanessa Katherine
•YÁNEZ LOAIZA, Erick Darío
Nombre del profesor: Dra. Lucía Castro Mgs.
NRC: 4389
Fecha: viernes 12 de febrero 2021
Período: Noviembre 2020 Abril 2021
PARCIAL II
TALLER Nro. 2
TEMA:
APLICACIONES DE LA DERIVADA EN LA
CARRERA DE TELECOMUNICACIONES
2. • Contenido: Aplicaciones de la derivada en las telecomunicaciones
1. INTRODUCCION
1.1. Aplicaciones de la derivada
1.2. Optimización de funciones
1.3. Pasos para resolver problemas con optimización
1.4. Razón de cambio
2. Objetivos
3. Fundamentación teórica
3.1. Diferencial de potencial
3.2. Ley de Faraday
3.3. Ley de Lenz
3.4. Inductancia mutua
3.6. La potencia
3.7. Circuitos eléctricos
3.8. Ondas armónicas
4. Desarrollo
5. Conclusiones
6. Bibliografía
3. 1. Introducción
• En una función la derivada se puede representar geométricamente como la pendiente en una curva, además físicamente se le puede interpretar
como una razón de cambio instantánea. Hablando de otra forma, la derivada de una función nos llega a indicar el ritmo con el que la función puede
llegar a variar.
1.1 Aplicaciones de la derivada
• Según Hernández Juan (2017) dice que: “Mediante el estudio de funciones y, más concretamente, mediante el uso de la derivada podemos
conocer: la variación del espacio en función del tiempo, el crecimiento de una bacteria en función del tiempo, el desgaste de un neumático en función
del tiempo, el beneficio de una empresa en función del tiempo”, se puede afirmar que la derivada llega a ser fundamental en diversas situaciones de la
vida cotidiana.
• Tenemos aplicaciones a la: geometría, física, química, biología, medicina, ingeniería, arquitectura, economía.
1.2 Optimización de funciones
• Es la consecución de máximos y mínimos relativos, sometida a unas restricciones. Dicho de otra forma, tiene como objetivo encontrar áreas
mínimas, la menor resistencia, el mayor alcance y máximo beneficio, todo esto esta dentro de la categoría de optimización de funciones.
• Se puede decir que al momento de realizar problemas de optimización siempre va a ser encontrar un valor mínimo, reducirle o también nos puede
pedir todo lo contrario como es encontrar el valor máximo, maximizar.
• De esta manera se puede calcular medidas con precisión como pueden ser el radio y altura de alguna figura geométrica como latas que tiene forma de
un cilindro o cajas que prácticamente son cubos. Para llegar a obtener extremos relativos se tendrá que hacer uso de la derivada en la función para
luego poder igual esta misma a cero
4. 1.3. Pasos para resolver problemas con optimización
• 1.- Graficar si el problema lo requiere.
• 2.- Se analiza y plantea la función que tendremos que llegar a minimizar o maximizar.
• 3.- Si llega a ver mas de una variable se debe hacer un análisis el cual nos permita relacionar ambas variables.
• 4.- Se tiene que despejar por cualquier método y luego remplazar en la función original para que nos quede en función
de una sola variable
• 5.- Se tiene que derivar la función para luego igualar a cero, esto nos permitirá encontrar los extremos locales
• 6.- En caso de querer comprobar el resultado obtenido, hallar una segunda derivada
1.4. Razón de cambio
• Según Cova Guillermo (2016) dice que “La razón de cambio es la proporción en la que una variable cambia con
respecto a otra, de manera más explícita hablamos de la pendiente de una curva en una gráfica, es decir el cambio en el
eje "y" entre el cambio del eje "x". A esto se le conoce también como la primera derivada”. Se puede definir de una
forma mas sencilla como la medida en la cual una variable se modifica con otra.
• Tenemos también la razón de cambio instantánea que muchos le conocen con su nombre vulgar que viene a ser la
“segunda derivada”
5. 2. Objetivos
Introducir el concepto de la derivada y proporcionar su aplicación en la ingeniería de las
telecomunicaciones.
Conocer en que ramas, las derivadas cumplen funciones importantes y fundamentales que se llegan a
desarrollar, mediante el uso del cálculo.
Aprender conceptos generales sobre la optimización de funciones y la razón de cambio en la cual
esta presente la derivación.
6. 3. Fundamentación teórica
Una vez conocido el concepto de derivada se debe conocer en cómo esta influye o
tiene aplicaciones en la carrera de Electrónica y Telecomunicaciones. Para empezar,
las Telecomunicaciones es una rama de la ingeniería que ayuda a la resolución con
problemas de transmisión y recepción de señales (mayormente electromagnéticas) y
circuitos de menor escala. Si bien las aplicaciones de la derivada e integrales son
extensas en este campo, mayormente se suelen utilizar para análisis de curvas,
máximos y mínimos o formas de onda y sobre todo para el análisis de potenciales
eléctricos y magnéticos en diseños de alto voltaje.
7. Entre algunas importantes aplicaciones de la derivada en la carrera se pueden destacar:
- Cambios instantáneos de corriente eléctrica.
- Variaciones de flujo magnético.
- Variaciones de campos eléctricos.
- Leyes de Maxwell.
- Conversión de energía.
- Leyes de electromagnetismo como la Ley de Ampere, Ley de Gauss, etc.
- Miniaturización de componentes internos.
- Comprensión y digitalización de imágenes, sonidos y videos.
Así, se puede decir que el Cálculo se aplica en casi todas las ramas de ciencias Físico-Matemáticas, con mayor énfasis en Ingenierías.
El uso de las derivadas debe ser correctamente aplicado ya que, si no se lo hace así, no se podrán plantear ecuaciones diferenciales ni resolver
problemas como el Análisis de Fourier el cuál consiste en la transformación de Ecuaciones Diferenciales en Ecuaciones Algebraicas con
coeficientes de fácil resolución.
8. A continuación, se dará ejemplos de la aplicación de las derivadas en carrera:
3.1. Diferencial de Potencial
El trabajo por unidad de carga se lo conoce como potencial eléctrico o voltaje. Para mover una carga desde el infinito hasta
cierto punto de otra carga o campo eléctrico requiere un cierto trabajo (W1). Para mover la misma carga desde el infinito a
otro punto en presencia de otra carga o campo eléctrico se requiere otro trabajo (W2) por tanto se tiene:
𝑊2 − 𝑊1
𝑞
En donde q deberá tender a un valor muy pequeño.
∆𝑉 = 𝑉2 − 𝑉1 = lim
𝑞→0
𝑊2 − 𝑊1
𝑞
Formalmente la diferencia de potencial se define como:
𝑉 =
𝑑𝑊
𝑑𝑞
9. Figura 1: Diferencia de Potencial
Fuente: Anónimo. (s.f.). Diferencia de potencial entre dos puntos. [Imagen]. Recuperado de https://www.calculisto.com/topics/circuitos-electricos/summary/353
3.2. Ley de Faraday
Esta ley relaciona la razón de cambio de flujo magnético que pasa a través de una espira o lazo con la magnitud de la fuerza electromotriz (FEM) ε
inducida en la espira:
𝜀 =
𝑑𝜙
𝑑𝑡
En donde:
ε = Fuerza electromotriz inducida
𝜙 = Flujo magnético.
La FEM es la diferencia de potencial a través de una espira cuando su resistencia es alta.
10. Esta ley va de la mano con la de Faraday, debido a que ésta en cambio, establece la dirección en la que fluye la
corriente y establece que la dirección siempre es tal que se opone al cambio de flujo que la produce.
𝜀 = −
𝑑𝜙
𝑑𝑡
En la práctica se lidia con inducciones magnéticas de espiras múltiples donde cada una contribuye a la FEM. En
donde N representa el número de vueltas.
𝜀 = −𝑁
𝑑𝜙
𝑑𝑡
3.3. Ley de Lenz
Figura 2: Ley de Lenz
Fuente: Anónimo. (s.f.). Aplicaciones de las derivadas en ingeniería Electrónica y de Telecomunicaciones. [Figura]. Recuperado de https://es.scribd.com/document/316309763/240745314-Aplicaciones-de-La-Derivada-en-
Electronica
11. 3.4. Inductancia mutua
Es el efecto de producir una fem en una bobina, esto se debe al cambio de corriente en una bobina acoplada. Su dirección
será siempre opuesta al cambio del campo magnético producido en ella por la bobina acoplada (Ley de Lenz).
La fem en la bobina 1 (izquierda) se debe a su inductancia L, mientras que la fem inducida de la bobina 2 se origina por el
cambio de la corriente I, se puede expresar como:
Figura 3: Bobinas
Fuente: Olmo, M. (s.f.). Inductancia Mutua. [Figura]. Recuperado de http://hyperphysics.phy-astr.gsu.edu/hbasees/magnetic/indmut.html.
𝐹𝑒𝑚2 = −𝑁2𝐴
Δ𝐵
Δ𝑡
= −𝑀
Δ𝐼1
Δ𝑡
12. 3.5. Corriente Eléctrica
La corriente eléctrica es el movimiento de las cargas. Se la define como la razón de flujo de cargas con respecto
al tiempo:
𝑖 𝑡 =
𝑑𝑞
𝑑𝑡
La unidad de la corriente eléctrica es el Ampere, una de las siete unidades básicas. El Ampere es la intensidad de
una corriente constante que manteniéndose en conductores paralelos de sección circular despreciable y situados
a la distancia de un metro uno de otro en el vacío, produce una fuerza igual a 2x10-7 Newton por metro de
longitud.
Figura 4: Generador de Tensión
Fuente: Anónimo. (2017). Corriente eléctrica. [Figura]. Recuperado de https://es.wikipedia.org/wiki/Corriente_el%C3%A9ctrica.
13. 3.6. Potencia
La potencia es la razón de absorber o generar energía por una unidad de tiempo:
𝑃 =
𝑑𝑊
𝑑𝑡
Sin embargo, con la ley de la cadena se obtiene:
𝑃 =
𝑑𝑊 𝑑𝑞
𝑑𝑡 𝑑𝑞
Considerando las ecuaciones 𝑖 𝑡 =
𝑑𝑞
𝑑𝑡
y 𝑉 =
𝑑𝑊
𝑑𝑞
se tiene que:
𝑃 = 𝑣𝑖
Esta será de suma utilidad ya que tanto el voltaje y corriente se miden con sus medidores (usualmente con
multímetro), de tal manera que es sencillo medir o calcular la potencia en cualquier elemento de un circuito.
14. 3.7. Circuitos eléctricos
Los capacitores (condensadores) y los inductores (bobinas) son elementos que almacenan energía, los capacitores
almacenan energía en forma de campo eléctrico (voltaje) y los inductores almacenan energía en forma de campo
magnético (corriente).
Por lo tanto, existen lapsos o tiempos de cargas y descargas, lo cual dependen de una función, con respecto al tiempo.
Figura 5: Ecuaciones de terminal para inductores y capacitores ideales.
Fuente: Anónimo. (s.f.). Práctica 2: Análisis en el tiempo de circuitos RL y RC. [Tabla]. Recuperado de https://fgagor.webs.ull.es/PracticaTC2.pdf.
En el SI la unidad de la inductancia es Henrios (H) y la de capacitancia se mide en Faradios (F).
15. 3.8. Ondas Armónicas
Una onda armónica es aquella que está descrita por una función seno o coseno. Nos
centraremos en aquellas ondas unidimensionales cuyas variables son la posición x y
el tiempo t.
𝑦 = 𝐴. 𝑠𝑒𝑛(𝑘 ±𝑣. 𝑡 )
𝑦 = 𝐴. 𝑐𝑜𝑠(𝑘 ±𝑣. 𝑡 )
16. 4. Desarrollo
1. La corriente que circula a través de un inductor de 0,3 H es i(t)= 𝟐𝟎𝒕𝒆−𝟔𝒕
[𝑨].Halle la tensión y la energía almacenada en él.
Datos Solución
• 𝑣 = 𝐿
𝑑𝑖
𝑑𝑡
• 𝐿 = 0.3 𝐻
• 𝑖 𝑡 = 20𝑡𝑒−6𝑡
[𝐴]
𝑣 𝑡 = 0.3
𝑑
𝑑𝑡
20𝑡𝑒−6𝑡
→ 𝑠𝑎𝑐𝑎𝑟 𝑙𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑒 𝑝𝑎𝑟𝑎 𝑝𝑟𝑜𝑐𝑒𝑑𝑒𝑟 𝑎 𝑑𝑒𝑟𝑖𝑣𝑎𝑟
𝑣(𝑡) = 0,3 . 20
𝑑
𝑑𝑡
(𝑡𝑒−6𝑡
)
𝑣 𝑡 = 6
𝑑
𝑑𝑡
𝑡𝑒−6𝑡 → 𝑑𝑒𝑟𝑖𝑣𝑎𝑟 𝑝𝑜𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑜
𝑣 𝑡 = 6 𝑡 ∗ −6𝑒−6𝑡
+ 𝑒−6𝑡
1
𝑣 𝑡 = 6 −6𝑒−6𝑡𝑡 + 𝑒−6𝑡
𝑣 𝑡 = 6 𝑒−6𝑡 −6𝑒−6𝑡 𝑡
𝑣 𝑡 = 6 𝑒−6𝑡
1 − 6𝑡 [𝑉]
17. 2. La función de onda correspondiente a una onda armónica en una cuerda es Y (x, t) = 0,005
sen(248t+58,5x), escrita en el SI. ¿Cuál es la ecuación de la velocidad y aceleración de una partícula de la
cuerda que se encuentre en el punto x = – 2 cm?
El desplazamiento máximo de un segmento cualquiera de la cuerda viene dado por la amplitud de la función Y
(x, t). Es decir: A = 0,005 m.
La función de onda de una partícula de la cuerda que se encuentra en el punto x = 0,02 m es:
𝛾 0,02 , 𝑡 = 𝛾 𝑡 = 0,005 𝑠𝑒𝑛 248𝑡 − 58,5 0,02 = 0,005 𝑠𝑒𝑛(248𝑡 − 1,17)
La ecuación de su velocidad:
𝑑𝛾
𝑑𝑡
= 0,005 248 cos 248𝑡 − 1,17 = 1,24 cos(248𝑡 − 1,17)
y la de su aceleración:
𝑑2
𝛾
𝑑𝑡2 = −1,24(248 𝑠𝑒𝑛 248𝑡 − 1,17 = −307,52 𝑠𝑒𝑛(248𝑡 − 1,17)
18. 5. Conclusiones
Se llego a relacionar la asignatura de cálculo más específicamente en el tema de derivadas con nuestra carrera, ampliando el
enfoque hacia cualquier tipo de proceso matemático. Suelen usarse para el análisis de curvas, máximos y mínimos o formas
de onda y sobre todo para análisis de potenciales eléctricos y magnéticos en diseños de alto voltaje y antenas.
La aplicación de la derivada en Ingeniería en Electrónica y Telecomunicaciones es fundamental en cálculos tanto básicos
como la Potencia, hasta llegar a más complejos con integrales y derivadas para determinar ondas análogas y transformarlas a
digitales o viceversa. Estas cumplen un papel fundamental en circuitos electrónicos pudiéndose observar gráficamente a
través de un osciloscopio, para futuros cambios en circuitos o imagen.
Podemos asociar las derivadas con las Telecomunicaciones ya que esta es esencial para resolver problemas de corriente
eléctrica, potencia, razón de cambio, entre otras. Particularmente es un elemento utilizado para conocer el cambio de una
variable con respecto a otra, por ello también se ocupa en varios temas relacionados con energía, electricidad, campos, etc.
19. 6. Bibliografía
• Camacho, S. (s.f.). Circuitos eléctricos AC. Recuperado de:
https://www.academia.edu/7188806/LABORATORIO_DE_CIRCUITOS_ELECTRICOS_AC.
• EcuRed. (s.f.). Optimización de funciones. Recuperado de:
https://www.ecured.cu/Optimizaci%C3%B3n_de_funciones#Problemas_de_optimizaci.C3.B3n.
• Fernández, J. (s.f.). Ondas Armónicas. Recuperado de: https://www.fisicalab.com/apartado/ondas-armonicas.
• Galdón, J. (s.f.). Optimización matemática: Una aplicación de la derivada de una función. Recuperado de:
https://www.tusclasesparticulares.com/blog/optimizacion-matematica-aplicacion-derivada-
funcion#:~:text=La%20llamada%20Optimizaci%C3%B3n%20de%20funciones,funci%C3%B3n%2C%20sometida%20a%20unas%20restriccion
es.&text=Una%20vez%20que%20tengamos%20la,funci%C3%B3n%2C%20e%20igual%C3%A1ndola%20a%20cero
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